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 Δconvergence.
MSC: 54C40, 14E20, 46E25, 20C20.
Keywords:
Sumudu transforms; Boehmian spaces; the space1 Introduction
The Sumudu transform of one variable function
over the set of functions
where
The Sumudu transform of the convolution product of f and u is given by
where
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 onedimensional 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
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
(ii)
(iii)
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
(ii) a commutative semigroup
(iii) an operation
(iv) a collection
(a) If
(b) If
Elements of Δ are called delta sequences. Consider
Now if
We note that between
(i) If
(ii) If
The operation ⊙ can be extended to
(1) A sequence
(2) A sequence
3 The Boehmian space
H
(
Y
)
Denote by
Lemma 3.1
(1) If
(2) If
(3)
(4) If
Proofs are analogous to those of classical cases and details are omitted.
Definition 3.2 A sequence
This means that
Lemma 3.3If
Lemma 3.4If
Theorem 3.5Let
Proof We show that
The mapping
Hence using
Thus
Theorem 3.6If
Proof
In view of the hypothesis of the theorem, we write
The last equation follows from the fact that [17]
Hence, for each
The proof of the theorem is completed. □
Theorem 3.7If
Proof
In view of the analysis employed for Theorem 3.5, we get
Hence
This completes the proof. The Boehmian space
The canonical embedding between
δconvergence: A sequence
Theorem 3.8The mapping
Proof The mapping is onetoone. For detailed proof, let
From Theorem 3.5,
Theorem 3.9Let
4 The Boehmian space
H
(
Y
s
)
We describe another Boehmian space as follows. Let
where
Lemma 4.1Let
Proof If
where
for some positive constant M. This completes the proof of the lemma. □
Lemma 4.2The mapping
satisfies the following properties:
(1) If
(2) If
(3) For
(4) For
Proof The proof of the above lemma is straightforward. Detailed proof is as follows.
Proof of (1). Let
Proof of (2) is obvious.
Proof of (3). We have
Hence
Proof of (4). Let
that is,
This completes the proof of the theorem. □
Denote by
Lemma 4.3Let
Proof Let
Lemma 4.4For each
Proof Since
By aid of Lemma 4.3. and Lemma 4.4,
Lemma 4.5Let
Proof It is clear that
Hence
Lemma 4.6Let
Proof Let
as
Lemma 4.7The mapping
is a continuous embedding of
Proof For
To establish the continuity of (4.4), let
as
5 The Sumudu transform of Boehmians
Let
Theorem 5.1
Proof Let
Theorem 5.2
Proof Let
as
Theorem 5.3
Proof Assume
Since the Sumudu transform is onetoone, we get
Hence
This completes the proof of the theorem. □
Theorem 5.4Let
(1)
(2)
Proof is immediate from the definitions.
Theorem 5.5
Proof Let
Hence, we have
Hence,
Theorem 5.6
Proof Let
Let
in the space
Theorem 5.7Let
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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