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This article is part of the series Proceedings of the International Congress in Honour of Professor Hari M. Srivastava.

Open Access Research

An estimate of Sumudu transforms for Boehmians

Shrideh Khalaf Qasem Al-Omari1 and Adem Kılıçman2*

Author Affiliations

1 Department of Applied Sciences, Faculty of Engineering Technology, Al-Balqa Applied University, Amman, 11134, Jordan

2 Department of Mathematics and Institute of Mathematical Research, Universiti Putra Malaysia (UPM), 43400 UPM, Serdang, Selangor, Malaysia

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Advances in Difference Equations 2013, 2013:77  doi:10.1186/1687-1847-2013-77

The electronic version of this article is the complete one and can be found online at: http://www.advancesindifferenceequations.com/content/2013/1/77


Received:14 January 2013
Accepted:5 March 2013
Published:26 March 2013

© 2013 Al-Omari and Kılıçman; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The space of Boehmians is constructed using an algebraic approach that utilizes convolution and approximate identities or delta sequences. A proper subspace can be identified with the space of distributions. In this paper, we first construct a suitable Boehmian space on which the Sumudu transform can be defined and the function space S can be embedded. In addition to this, our definition extends the Sumudu transform to more general spaces and the definition remains consistent for S elements. We also discuss the operational properties of the Sumudu transform on Boehmians and finally end with certain theorems for continuity conditions of the extended Sumudu transform and its inverse with respect to δ- and Δ-convergence.

MSC: 54C40, 14E20, 46E25, 20C20.

Keywords:
Sumudu transforms; Boehmian spaces; the space <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M1">View MathML</a>; the space <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M2">View MathML</a>; distributions

1 Introduction

The Sumudu transform of one variable function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M3">View MathML</a> is introduced as a new integral transform by Watugala in [1] and is given by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M4">View MathML</a>

over the set of functions

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M5">View MathML</a>

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M6">View MathML</a> is a function that can be expressed as a convergent series [2,3]. The Sumudu transform was applied to solve the ordinary differential equations in control engineering problems; see [3].

The Sumudu transform of the convolution product of f and u is given by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M7">View MathML</a>

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M8">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M9">View MathML</a> are the Sumudu transforms of f and u, respectively.

Some of the properties were established by Weerakoon in [4,5]. In [6], further fundamental properties of this transform were also established by Asiru. Similarly, this transform was applied to a one-dimensional neutron transport equation in [7] by Kadem.

In [8], the Sumudu transform was extended to the distributions and some of their properties were also studied. Recently, this transform has been applied to solve the system of differential equations; see Kılıçman et al. in [9].

Note that a very interesting fact about Sumudu transform is that the original function and its Sumudu transform have the same Taylor coefficients except the factor n; see Zhang [10]. Similarly, the Sumudu transform sends combinations <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M10">View MathML</a> into permutations <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M11">View MathML</a>, and hence it will be useful in the discrete systems.

The following are the general properties of the Sumudu transform which are auxiliary from the substitution method and the properties of integral operators.

(i) If<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M12">View MathML</a>and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M13">View MathML</a>are non-negative integers and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M14">View MathML</a>and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M15">View MathML</a>are the corresponding Sumudu transforms of<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M16">View MathML</a>and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M17">View MathML</a>, respectively, then

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M18">View MathML</a>

(ii) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M19">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M20">View MathML</a>.

(iii) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M21">View MathML</a>, where<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M22">View MathML</a>is the Sumudu transform off.

More properties of the Sumudu transforms a long with a some of applications were given in [11] and [12].

2 Boehmian space

Boehmians were first constructed as a generalization of regular Mikusinski operators [13]. The minimal structure necessary for the construction of Boehmians consists of the following elements:

(i) a nonempty set <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M23">View MathML</a>;

(ii) a commutative semigroup <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M24">View MathML</a>;

(iii) an operation <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M25">View MathML</a> such that for each <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M26">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M27">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M28">View MathML</a>;

(iv) a collection <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M29">View MathML</a> such that

(a) If <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M30">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M31">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M32">View MathML</a> for all n, then <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M33">View MathML</a>;

(b) If <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M34">View MathML</a>, then <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M35">View MathML</a>.

Elements of Δ are called delta sequences. Consider

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M36">View MathML</a>

Now if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M37">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M38">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M39">View MathML</a>, then we say <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M40">View MathML</a>. The relation ∼ is an equivalence relation in ℚ. The space of equivalence classes in ℚ is denoted by β. Elements of β are called Boehmians.

We note that between <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M23">View MathML</a> and β there is a canonical embedding expressed as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M42">View MathML</a>. The operation ⊙ can also be extended to <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M43">View MathML</a> by <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M44">View MathML</a>. The relationship between the notion of convergence and the product ⊙ is given by:

(i) If <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M45">View MathML</a> as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M46">View MathML</a> in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M23">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M48">View MathML</a> is any fixed element, then <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M49">View MathML</a> in A (as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M50">View MathML</a>);

(ii) If <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M45">View MathML</a> as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M46">View MathML</a> in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M23">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M54">View MathML</a>, then <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M55">View MathML</a> in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M23">View MathML</a> (as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M46">View MathML</a>).

The operation ⊙ can be extended to <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M58">View MathML</a> as follows: If<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M59">View MathML</a>and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M60">View MathML</a>, then<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M61">View MathML</a>. In β, there are two types of convergence as follows.

(1) A sequence<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M62">View MathML</a>inβis said to beδ-convergent tohinβ, denoted by<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M63">View MathML</a>, if there exists<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M64">View MathML</a>such that<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M65">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M66">View MathML</a>, and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M67">View MathML</a>as<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M68">View MathML</a>in<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M23">View MathML</a>for every<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M70">View MathML</a>.

(2) A sequence<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M62">View MathML</a>inβis said to be Δ-convergent tohinβ, denoted by<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M72">View MathML</a>, if there exists a<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M73">View MathML</a>such that<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M74">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M75">View MathML</a>, and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M76">View MathML</a>as<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M46">View MathML</a>in<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M23">View MathML</a>.

For further discussion, see [14-16].

3 The Boehmian space <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M79">View MathML</a>

Denote by <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M80">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M81">View MathML</a> the space of all rapidly decreasing functions over <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M82">View MathML</a> (<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M83">View MathML</a>) and the space of all test functions of compact support, respectively. In what follows, we obtain preliminary results required to construct the Boehmian space <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M84">View MathML</a>, where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M85">View MathML</a>.

Lemma 3.1

(1) If<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M86">View MathML</a>, then<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M87">View MathML</a>.

(2) If<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M88">View MathML</a>and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M89">View MathML</a>, then<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M90">View MathML</a>.

(3) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M91">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M92">View MathML</a>.

(4) If<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M93">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M86">View MathML</a>, then<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M95">View MathML</a>.

Proofs are analogous to those of classical cases and details are omitted.

Definition 3.2 A sequence <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M96">View MathML</a> of functions from <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M97">View MathML</a> is said to be in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M98">View MathML</a> if and only if

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M99">View MathML</a>

This means that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M100">View MathML</a> shrinks to zero as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M46">View MathML</a>. Each member of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M102">View MathML</a> is called a delta sequence or an approximate identity or, sometimes, a summability kernel. Delta sequences, in general, appear in many branches of mathematics, but probably the most important applications are those in the theory of generalized functions. The basic use of delta sequences is the regularization of generalized functions, and further, they can be used to define the convolution product and the product of generalized functions.

Lemma 3.3If<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M103">View MathML</a>, then<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M104">View MathML</a>.

Lemma 3.4If<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M105">View MathML</a>, then so is<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M106">View MathML</a>and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M107">View MathML</a>

Theorem 3.5Let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M108">View MathML</a>and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M109">View MathML</a>such that<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M110">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M111">View MathML</a> , then<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M112">View MathML</a>in<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M113">View MathML</a>.

Proof We show that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M114">View MathML</a> in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M115">View MathML</a>. Let K be a compact set containing the <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M116">View MathML</a> for every <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M117">View MathML</a>. Using <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M118">View MathML</a>, we write

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M119">View MathML</a>

(3.1)

The mapping <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M120">View MathML</a>, where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M121">View MathML</a>, is uniformly continuous from <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M122">View MathML</a>. From the hypothesis that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M123">View MathML</a> as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M46">View MathML</a> (by <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M125">View MathML</a>), we choose <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M126">View MathML</a> such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M127">View MathML</a> for large n and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M128">View MathML</a>. This implies

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M129">View MathML</a>

(3.2)

Hence using <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M130">View MathML</a> and (3.2), (3.1) becomes

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M131">View MathML</a>

Thus <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M132">View MathML</a> in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M133">View MathML</a>. Similarly, we show that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M134">View MathML</a>. This completes the proof of the theorem. □

Theorem 3.6If<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M135">View MathML</a>in<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M136">View MathML</a>and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M137">View MathML</a>, then

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M138">View MathML</a>

Proof

In view of the hypothesis of the theorem, we write

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M139">View MathML</a>

(3.3)

The last equation follows from the fact that [17]

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M140">View MathML</a>

Hence, for each <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M141">View MathML</a>, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M142">View MathML</a>

The proof of the theorem is completed. □

Theorem 3.7If<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M135">View MathML</a>in<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M136">View MathML</a>and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M145">View MathML</a>, then<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M146">View MathML</a>.

Proof

In view of the analysis employed for Theorem 3.5, we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M147">View MathML</a>

Hence

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M148">View MathML</a>

This completes the proof. The Boehmian space <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M149">View MathML</a> is therefore constructed. □

The canonical embedding between <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M150">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M151">View MathML</a> is expressed as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M152">View MathML</a>. The extension of ⋆ to <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M153">View MathML</a> is given by <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M154">View MathML</a>. Convergence in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M151">View MathML</a> is defined in a natural way:

δ-convergence: A sequence<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M156">View MathML</a>in<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M151">View MathML</a>is said to beδ-convergent tohin<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M158">View MathML</a>, denoted by<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M159">View MathML</a>, if there exists a delta sequence<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M160">View MathML</a>such that<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M161">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M162">View MathML</a>, and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M163">View MathML</a>as<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M46">View MathML</a>in<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M150">View MathML</a>for every<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M70">View MathML</a>.

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M102">View MathML</a>-convergence: A sequence<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M168">View MathML</a>in<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M169">View MathML</a>is said to be<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M102">View MathML</a>-convergent tohin<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M151">View MathML</a>, denoted by<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M172">View MathML</a>, if there exists a<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M173">View MathML</a>such that<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M174">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M175">View MathML</a>, and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M176">View MathML</a>as<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M50">View MathML</a>in<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M80">View MathML</a>.

Theorem 3.8The mapping<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M179">View MathML</a>is a continuous embedding of<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M80">View MathML</a>into<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M158">View MathML</a>.

Proof The mapping is one-to-one. For detailed proof, let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M182">View MathML</a>, then <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M183">View MathML</a>. Then since <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M184">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M185">View MathML</a>. Using Theorem 3.5, we get <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M112">View MathML</a>. To show the mapping is continuous, let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M187">View MathML</a> as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M46">View MathML</a> in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M150">View MathML</a>. Then we show that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M190">View MathML</a>

From Theorem 3.5, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M191">View MathML</a> as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M68">View MathML</a>. This completes the proof of the theorem. □

Theorem 3.9Let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M193">View MathML</a>and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M194">View MathML</a>, then

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M195">View MathML</a>

4 The Boehmian space <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M196">View MathML</a>

We describe another Boehmian space as follows. Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M80">View MathML</a> be the space of rapidly decreasing functions [17]. Define

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M198">View MathML</a>

(4.1)

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M9">View MathML</a> denotes the Sumudu transform of u. We also define <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M200">View MathML</a> by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M201">View MathML</a>

(4.2)

Lemma 4.1Let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M202">View MathML</a>and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M203">View MathML</a>, then<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M204">View MathML</a>.

Proof If <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M205">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M206">View MathML</a>, then using the topology of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M80">View MathML</a> and Leibnitz’ theorem, we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M208">View MathML</a>

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M209">View MathML</a> and K is a compact subset containing the <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M210">View MathML</a>. Hence

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M211">View MathML</a>

for some positive constant M. This completes the proof of the lemma. □

Lemma 4.2The mapping

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M212">View MathML</a>

satisfies the following properties:

(1) If<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M213">View MathML</a>, then<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M214">View MathML</a>.

(2) If<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M215">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M216">View MathML</a>, then<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M217">View MathML</a>.

(3) For<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M218">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M219">View MathML</a>.

(4) For<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M220">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M221">View MathML</a>, then<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M222">View MathML</a>.

Proof The proof of the above lemma is straightforward. Detailed proof is as follows.

Proof of (1). Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M223">View MathML</a>, then <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M224">View MathML</a>. Hence <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M225">View MathML</a> by (4.1). Theorem 3.9 implies <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M214">View MathML</a>.

Proof of (2) is obvious.

Proof of (3). We have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M227">View MathML</a>

Hence <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M219">View MathML</a>.

Proof of (4). Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M229">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M230">View MathML</a>, then

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M231">View MathML</a>

that is,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M232">View MathML</a>

This completes the proof of the theorem. □

Denote by <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M233">View MathML</a> the set of all Sumudu transforms of delta sequences from <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M102">View MathML</a>. That is,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M235">View MathML</a>

(4.3)

Lemma 4.3Let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M236">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M237">View MathML</a>be such that<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M238">View MathML</a>, ∀n, then<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M112">View MathML</a>in<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M113">View MathML</a>.

Proof Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M241">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M242">View MathML</a>. Since <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M238">View MathML</a>, (4.2) implies <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M244">View MathML</a>. Hence <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M245">View MathML</a> for all x. The proof is completed. □

Lemma 4.4For each<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M103">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M247">View MathML</a>.

Proof Since <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M184">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M249">View MathML</a> for all n. Hence, from Theorem 3.9, we get <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M250">View MathML</a> for every n. This completes the proof of the lemma. □

By aid of Lemma 4.3. and Lemma 4.4, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M233">View MathML</a> can be regarded as a delta sequence.

Lemma 4.5Let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M45">View MathML</a>in<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M80">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M254">View MathML</a>, then<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M255">View MathML</a>in<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M256">View MathML</a>.

Proof It is clear that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M9">View MathML</a> is bounded in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M258">View MathML</a>. Further,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M259">View MathML</a>

Hence <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M260">View MathML</a>. □

Lemma 4.6Let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M45">View MathML</a>in<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M80">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M242">View MathML</a>, then<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M264">View MathML</a>in<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M80">View MathML</a>.

Proof Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M145">View MathML</a>, then <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M267">View MathML</a> uniformly on compact subsets of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M82">View MathML</a>. Hence

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M269">View MathML</a>

as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M46">View MathML</a>. Thus <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M271">View MathML</a> as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M46">View MathML</a>. This yields <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M264">View MathML</a> in the topology of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M150">View MathML</a>. The proof is therefore completed. The space <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M275">View MathML</a> can be regarded as a Boehmian space, where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M276">View MathML</a>. □

Lemma 4.7The mapping

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M277">View MathML</a>

(4.4)

is a continuous embedding of<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M278">View MathML</a>into<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M279">View MathML</a>.

Proof For <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M280">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M281">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M282">View MathML</a> is a quotient of sequences in the sense that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M283">View MathML</a>. We show that the map (4.4) is one-to-one. Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M284">View MathML</a>, then <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M285">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M286">View MathML</a>. Using Lemma 4.2 and Lemma 4.3, we conclude <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M112">View MathML</a>. □

To establish the continuity of (4.4), let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M187">View MathML</a> as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M46">View MathML</a> in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M136">View MathML</a>. Then <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M291">View MathML</a> as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M46">View MathML</a> by Lemma 4.6, and hence

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M293">View MathML</a>

as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M46">View MathML</a> in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M279">View MathML</a>. This completes the proof of the lemma.

5 The Sumudu transform of Boehmians

Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M296">View MathML</a>, then we define the Sumudu transform of β by the relation

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M297">View MathML</a>

(5.1)

Theorem 5.1<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M298">View MathML</a>is well defined.

Proof Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M299">View MathML</a>, where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M300">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M301">View MathML</a>. Then the concept of quotients yields <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M302">View MathML</a>. Employing Theorem 3.9, we get <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M303">View MathML</a>, i.e., <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M304">View MathML</a>. Equivalently, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M305">View MathML</a>. Thus <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M306">View MathML</a>. This completes the proof of the theorem. □

Theorem 5.2<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M307">View MathML</a>is continuous with respect toδ-convergence.

Proof Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M308">View MathML</a> in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M149">View MathML</a>, then by [14], <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M310">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M311">View MathML</a> as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M46">View MathML</a> in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M80">View MathML</a>. Applying the Sumudu transform to both sides yields <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M314">View MathML</a> as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M46">View MathML</a>. Hence

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M316">View MathML</a>

as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M46">View MathML</a> in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M279">View MathML</a>. This proves the theorem. □

Theorem 5.3<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M319">View MathML</a>is a one-to-one mapping.

Proof Assume <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M320">View MathML</a>, then <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M304">View MathML</a>. Hence

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M322">View MathML</a>

Since the Sumudu transform is one-to-one, we get <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M302">View MathML</a>. Thus

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M324">View MathML</a>

Hence

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M325">View MathML</a>

This completes the proof of the theorem. □

Theorem 5.4Let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M326">View MathML</a>, then

(1) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M327">View MathML</a>;

(2) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M328">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M329">View MathML</a>.

Proof is immediate from the definitions.

Theorem 5.5<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M319">View MathML</a>is continuous with respect to<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M102">View MathML</a>-convergence.

Proof Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M332">View MathML</a> in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M151">View MathML</a> as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M46">View MathML</a>. Then there exist <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M335">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M145">View MathML</a> such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M337">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M187">View MathML</a> as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M46">View MathML</a>. Employing Eq. (5.1), we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M340">View MathML</a>

Hence, we have <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M341">View MathML</a> as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M46">View MathML</a> in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M275">View MathML</a>. Therefore

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M344">View MathML</a>

Hence, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M345">View MathML</a> as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M46">View MathML</a>. □

Theorem 5.6<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M347">View MathML</a>is onto.

Proof Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M348">View MathML</a> be arbitrary, then <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M349">View MathML</a> for every <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M286">View MathML</a>. Then <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M351">View MathML</a>. That is, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M352">View MathML</a> is the corresponding quotient of sequences of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M353">View MathML</a>. Thus <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M354">View MathML</a> is such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M355">View MathML</a> in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M196">View MathML</a>. This completes the proof of the lemma.

Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M357">View MathML</a>, then we define the inverse Sumudu transform of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M358">View MathML</a> by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M359">View MathML</a>

in the space <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M79">View MathML</a>. □

Theorem 5.7Let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M361">View MathML</a>and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M137">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M363">View MathML</a>

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/77/mathml/M364">View MathML</a>

Proof is immediate from the definitions.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Both of the authors contributed equally to the manuscript and read and approved the final draft.

Acknowledgements

Dedicated to Professor Hari M Srivastava.

The authors express their sincere thanks to the referee(s) for the careful and detailed reading of the manuscript and very helpful suggestions that improved the manuscript substantially.

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