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The existence of solutions for a nonlinear mixed problem of singular fractional differential equations

Dumitru Baleanu123*, Hakimeh Mohammadi4 and Shahram Rezapour4

Author Affiliations

1 Department of Chemical and Materials Engineering, Faculty of Engineering, King Abdulaziz University, P.O. Box 80204, Jeddah, 21589, Saudi Arabia

2 Department of Mathematics, Cankaya University, Ogretmenler Cad. 14, Balgat, Ankara, 06530, Turkey

3 Institute of Space Sciences, Magurele, Bucharest, Romania

4 Department of Mathematics, Azarbaijan Shahid Madani University, Azarshahr, Tabriz, Iran

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Advances in Difference Equations 2013, 2013:359  doi:10.1186/1687-1847-2013-359

The electronic version of this article is the complete one and can be found online at: http://www.advancesindifferenceequations.com/content/2013/1/359


Received:27 May 2013
Accepted:28 August 2013
Published:11 December 2013

© 2013 Baleanu et al.; 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

By using fixed point results on cones, we study the existence of solutions for the singular nonlinear fractional boundary value problem

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M1">View MathML</a>

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M2">View MathML</a> is an integer, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M3">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M4">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M5">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M6">View MathML</a>, f is an <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M7">View MathML</a>-Caratheodory function, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M8">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M9">View MathML</a> may be singular at value 0 in one dimension of its space variables x, y, z. Here, cD stands for the Caputo fractional derivative.

Keywords:
boundary value problem; fixed point; fractional differential equation; Green function; regularization; singular

1 Introduction

Fractional differential equations (see, for example, [1-6] and references therein) started to play an important role in several branches of science and engineering. There are some works about existence of solutions for the nonlinear mixed problems of singular fractional boundary value problem (see, for example, [7-11] and [12]). Also, there are different methods for solving distinct fractional differential equations (see, for example, [13-18] and [19]). By using fixed point results on cones, we focus on the existence of positive solutions for a nonlinear mixed problem of singular fractional boundary value problem. For the convenience of the reader, we present some necessary definitions from fractional calculus theory (see, for example, [20]). The Caputo derivative of fractional order α for a function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M10">View MathML</a> is defined by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M11">View MathML</a>

Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M12">View MathML</a>. As you know, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M13">View MathML</a> denotes the space of functions, whose qth powers of modulus are integrable on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M14">View MathML</a>, equipped with the norm <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M15">View MathML</a>. We consider the sup norm

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M16">View MathML</a>

on the space <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M17">View MathML</a>. Also, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M18">View MathML</a> is the set of absolutely continuous functions on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M14">View MathML</a>. Let B be a subset of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M20">View MathML</a>. A function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M21">View MathML</a> is called an <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M7">View MathML</a>-Caratheodory function whenever the real-valued function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M23">View MathML</a> on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M14">View MathML</a> is measurable for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M25">View MathML</a>, the function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M26">View MathML</a> is continuous for almost all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M27">View MathML</a>, and for each compact set <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M28">View MathML</a>, there exists a function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M29">View MathML</a> such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M30">View MathML</a> for almost all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M31">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M32">View MathML</a>. Consider the nonlinear fractional boundary value problem

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M2">View MathML</a> is an integer, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M3">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M4">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M5">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M6">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M8">View MathML</a>. We say that the function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M40">View MathML</a> is a positive solution for the problem whenever <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M41">View MathML</a> on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M14">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M43">View MathML</a> is a function in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M13">View MathML</a>, and u satisfies the boundary conditions almost everywhere on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M14">View MathML</a>. In this paper, we suppose that f is an <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M7">View MathML</a>-Caratheodory function on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M47">View MathML</a>, where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M48">View MathML</a>, there exists a positive constant m such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M49">View MathML</a> for almost all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M31">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M25">View MathML</a>, f satisfies the estimate

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M52">View MathML</a>

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M53">View MathML</a> are positive and non-increasing, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M54">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M55">View MathML</a> are positive, w is non-decreasing in all its variables, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M56">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M57">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M58">View MathML</a>, and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M59">View MathML</a>. Since we suppose that problem (∗) is singular, that is, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M9">View MathML</a> may be singular at the value 0 of its space variables x, y, z, we use regularization and sequential techniques for the existence of positive solutions of the problem. In this way, for each natural number n define the function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M61">View MathML</a> by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M62">View MathML</a>

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M31">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M64">View MathML</a>, where

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M65">View MathML</a>

It is easy to see that each <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M61">View MathML</a> is an <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M7">View MathML</a>-Caratheodory function on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M68">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M69">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M70">View MathML</a>

and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M71">View MathML</a>

for almost all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M31">View MathML</a> and all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M25">View MathML</a>. In 2012, Agarwal et al. proved the following result.

Lemma 1.1[7]

Let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M74">View MathML</a>and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M75">View MathML</a>. Then we have<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M76">View MathML</a>for all<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M31">View MathML</a>and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M78">View MathML</a>

where<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M79">View MathML</a>.

2 Main results

Now, we are ready to investigate the problem in regular and singular cases. First, we give the following result.

Lemma 2.1Let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M80">View MathML</a>. Then the boundary value problem

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M81">View MathML</a>

is equivalent to the fractional integral equation<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M82">View MathML</a>, where

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M83">View MathML</a>

whenever<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M84">View MathML</a>and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M85">View MathML</a>

whenever<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M86">View MathML</a>.

Proof From <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M87">View MathML</a> and the boundary conditions, we obtain

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M88">View MathML</a>

By properties of the Caputo derivative, we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M89">View MathML</a>

Thus, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M90">View MathML</a> and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M91">View MathML</a>

By using the boundary conditions <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M92">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M93">View MathML</a>, we get <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M94">View MathML</a> and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M95">View MathML</a>

Hence, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M96">View MathML</a> and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M97">View MathML</a>

Thus,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M98">View MathML</a>

This completes the proof. □

Put <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M99">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M100">View MathML</a>. It is easy to check that the Green function G in the last result belongs to <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M101">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M102">View MathML</a> for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M103">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M104">View MathML</a>

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M105">View MathML</a>. Consider the Banach space <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M106">View MathML</a> with the norm <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M107">View MathML</a> and the cone

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M108">View MathML</a>

For each natural number n, define the operator <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M109">View MathML</a> on P by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M110">View MathML</a>

Now, we prove that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M109">View MathML</a> is a completely continuous operator (see [2]).

Lemma 2.2The operator<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M109">View MathML</a>is a completely continuous operator.

Proof Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M113">View MathML</a>. Then, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M114">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M115">View MathML</a>. Now, define <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M116">View MathML</a> for almost all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M31">View MathML</a>. Then <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M74">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M119">View MathML</a> for almost all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M31">View MathML</a>. By using the properties of fractional integral <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M121">View MathML</a>, it is easy to see that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M122">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M123">View MathML</a> and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M124">View MathML</a>

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M31">View MathML</a>. This implies that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M126">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M127">View MathML</a> on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M14">View MathML</a>. Consequently, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M109">View MathML</a> maps P into P. In order to prove that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M109">View MathML</a> is a continuous operator, let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M131">View MathML</a> be a convergent sequence in P and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M132">View MathML</a>. Thus, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M133">View MathML</a> uniformly on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M14">View MathML</a> for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M135">View MathML</a>. Since

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M136">View MathML</a>

we get <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M137">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M138">View MathML</a> uniformly on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M14">View MathML</a>. Also, we have <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M140">View MathML</a> on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M14">View MathML</a>, and so <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M142">View MathML</a>. Now, put

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M143">View MathML</a>

Then, it is easy to see that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M144">View MathML</a> for almost all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M31">View MathML</a>, and there exists <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M146">View MathML</a> such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M147">View MathML</a> for almost all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M31">View MathML</a> and all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M149">View MathML</a>. Since <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M61">View MathML</a> is an <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M7">View MathML</a>-Caratheodory function, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M152">View MathML</a> is bounded in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M153">View MathML</a>, and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M154">View MathML</a> is bounded in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M17">View MathML</a>. Therefore, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M156">View MathML</a> uniformly on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M14">View MathML</a>. Since <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M158">View MathML</a> is <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M7">View MathML</a>-convergent on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M14">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M161">View MathML</a>

uniformly on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M14">View MathML</a>. Hence, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M109">View MathML</a> is a continuous operator. Now, we have to show that for each bounded sequence <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M152">View MathML</a> in P, the sequence <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M165">View MathML</a> is relatively compact in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M17">View MathML</a>. Choose a positive constant k such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M167">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M168">View MathML</a> for all m. Note that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M169">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M170">View MathML</a> for all m, where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M79">View MathML</a>. But we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M172">View MathML</a>

and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M173">View MathML</a>

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M31">View MathML</a> and m. This implies that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M165">View MathML</a> is bounded in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M153">View MathML</a>. Also, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M177">View MathML</a>

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M178">View MathML</a>, where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M79">View MathML</a>. Hence, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M180">View MathML</a> is equicontinuous on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M14">View MathML</a>. Thus, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M165">View MathML</a> is relatively compact in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M153">View MathML</a> by the Arzela-Ascoli theorem. Hence, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M109">View MathML</a> is a completely continuous operator. □

We need the following result (see [2] and [21]).

Lemma 2.3[21]

LetYbe a Banach space, Pa cone inYand<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M185">View MathML</a>and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M186">View MathML</a>bounded open balls inYcentered at the origin with<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M187">View MathML</a>. Suppose that<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M188">View MathML</a>is a completely continuous operator such that<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M189">View MathML</a>for all<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M190">View MathML</a>and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M191">View MathML</a>for all<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M192">View MathML</a>. ThenThas a fixed point in<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M193">View MathML</a>.

Theorem 2.4For each natural numbern, problem (∗) has a solution<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M194">View MathML</a>such that<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M195">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M196">View MathML</a>and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M197">View MathML</a>for all<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M31">View MathML</a>.

Proof Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M199">View MathML</a>. It is sufficient to show that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M109">View MathML</a> has a fixed point <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M201">View MathML</a> in P with the desired conditions. In this way, note that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M202">View MathML</a>

and so <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M203">View MathML</a>. Put <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M204">View MathML</a>. Then <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M205">View MathML</a> for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M206">View MathML</a>. If <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M207">View MathML</a>, then

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M208">View MathML</a>

and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M209">View MathML</a>

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M113">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M31">View MathML</a>, because w is non-decreasing in all its variables. Since <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M212">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M213">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M214">View MathML</a> and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M215">View MathML</a>

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M79">View MathML</a>, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M217">View MathML</a>

and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M218">View MathML</a>

Hence, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M219">View MathML</a>, where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M220">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M221">View MathML</a>. Since

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M222">View MathML</a>

there exists a positive constant L such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M223">View MathML</a>

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M224">View MathML</a>. Thus, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M225">View MathML</a> for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M113">View MathML</a> with <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M227">View MathML</a>. Put <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M228">View MathML</a>. Then <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M229">View MathML</a> for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M192">View MathML</a>. By using last result, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M109">View MathML</a> has a fixed point <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M201">View MathML</a> in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M233">View MathML</a>. But <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M234">View MathML</a> and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M235">View MathML</a>

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M31">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M113">View MathML</a>. Since <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M238">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M239">View MathML</a>

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M31">View MathML</a>. This completes the proof. □

Now, we give our last result.

Theorem 2.5Problem (∗) has a solutionusuch that<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M241">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M242">View MathML</a>and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M243">View MathML</a>for all<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M31">View MathML</a>.

Proof By using Theorem 2.4, one gets that for each natural number n, problem (∗) has a solution <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M194">View MathML</a> with the desired conditions. Thus, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M246">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M247">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M248">View MathML</a> for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M31">View MathML</a> and n. Also, we have <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M250">View MathML</a>. Suppose that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M251">View MathML</a>

Then

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M252">View MathML</a>

for almost all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M31">View MathML</a> and n. Since <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M254">View MathML</a>, we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M255">View MathML</a>

and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M256">View MathML</a>

We show that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M257">View MathML</a> is bounded on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M14">View MathML</a>. Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M79">View MathML</a>. Note that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M260">View MathML</a>

and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M261">View MathML</a>

Thus, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M262">View MathML</a> for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M31">View MathML</a>, where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M264">View MathML</a>. Also, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M265">View MathML</a>

Since

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M266">View MathML</a>

and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M267">View MathML</a>

we get <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M268">View MathML</a> for all n, where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M269">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M270">View MathML</a> and also <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M271">View MathML</a>. On the other hand, there exists a positive constant L such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M272">View MathML</a> for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M224">View MathML</a>, and so <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M274">View MathML</a> for all n. Thus, for almost all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M31">View MathML</a> and all n, we have <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M276">View MathML</a>, where

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M277">View MathML</a>

Note that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M278">View MathML</a>. We show that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M279">View MathML</a> is equicontinuous on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M14">View MathML</a>. Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M281">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M282">View MathML</a>. Then

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M283">View MathML</a>

and so

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M284">View MathML</a>

Hence, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M279">View MathML</a> is equicontinuous on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M14">View MathML</a>. Since <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M287">View MathML</a> is a bounded sequence in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M17">View MathML</a>, by using the Arzela-Ascoli theorem, without loss of generality, we can assume that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M287">View MathML</a> is convergent in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M17">View MathML</a>. Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M291">View MathML</a>. Then, it is easy to see that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M292">View MathML</a>, and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M293">View MathML</a> uniformly converges to <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M294">View MathML</a> on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M14">View MathML</a>. Thus, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M296">View MathML</a> converges to <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M297">View MathML</a> in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M17">View MathML</a>. Hence,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M299">View MathML</a>

for almost all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M31">View MathML</a>. Since <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M278">View MathML</a>, by using the dominated convergence theorem on the relation

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M302">View MathML</a>

we get <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M303">View MathML</a> for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M31">View MathML</a>. This completes the proof. □

2.1 Examples for the problem

Example 2.1 Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M305">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M306">View MathML</a> for almost all t in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M14">View MathML</a>. Suppose that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M308">View MathML</a>

on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M47">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M310">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M311">View MathML</a> whenever <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M312">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M313">View MathML</a> whenever <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M314">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M315">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M316">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M317">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M318">View MathML</a>. Then Theorem 2.5 guarantees that problem (∗) has a positive solution.

Example 2.2 Consider the nonlinear mixed problem of singular fractional boundary value problem

via boundary value conditions <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M320">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M321">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M322">View MathML</a>, where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M323">View MathML</a>. Let

Then the map f is singular at <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M325">View MathML</a>, and f satisfies the desired conditions, where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M326">View MathML</a> whenever <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M327">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M313">View MathML</a> whenever <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M329">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M330">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M316">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M332">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M333">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M334">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/359/mathml/M318">View MathML</a>. Then Theorem 2.5 guarantees that this problem has a positive solution.

3 Conclusions

One of the most interesting branches is obtaining solutions of singular fractional differential via boundary value problems. Having these things in mind, we study the existence of solutions for a singular nonlinear fractional boundary value problem. Two illustrative examples illustrate the applicability of the proposed method. It seems that the obtained results could be extended to more general functional spaces. Finally, note that all calculations in proofs of the results depend on the definition of the fractional derivative.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors have equal contributions. All authors read and approved the final manuscript.

Acknowledgements

Research of the second and third authors was supported by Azarbaijan Shahid Madani University. Also, the authors express their gratitude to the referees for their helpful suggestions, which improved the final version of this paper.

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