Abstract
In this paper we obtain sufficient conditions for the existence of solutions of some classes of partial neutral integrodifferential equations of fractional order by using suitable fixed point theorems.
MSC: 26A33.
Keywords:
integrodifferential equation; leftsided mixed RiemannLiouville integral of fractional order; Caputo fractionalorder derivative; finite delay; infinite delay; solution; fixed point1 Introduction
Fractional differential and integral equations have recently been applied in various areas of engineering, science, finance, applied mathematics, bioengineering and others. There has been a significant development in ordinary and partial fractional differential equations in recent years; see the monographs of Abbas et al.[1], Baleanu et al.[2], Kilbas et al.[3], Lakshmikantham et al.[4], Podlubny [5], and the references therein.
In [6], Czlapinski proved some results for the following system of the Darboux problem for the secondorder partial functional differential equations of the form
where , , , , , , and ℬ is a vector space of realvalued functions defined in , equipped with a seminorm and satisfying some suitable axioms, which was introduced by Hale and Kato [7]; see also [810] with rich bibliography concerning functional differential equations with infinite delay. Recently, Abbas et al. studied some existence results for the Darboux problem for several classes of fractionalorder partial differential equations with finite delay [11,12] and others with infinite delay [13,14].
Motivated by the above papers, in this article we deal with the existence of solutions for two systems of neutral integrodifferential equations of fractional order with delay. First, we consider the system of fractionalorder neutral integrodifferential equations with finite delay of the form
where ; , , , , is the leftsided mixed RiemannLiouville integral of order r (see Section 2 for definition), is the fractional Caputo derivative of order r, , are given continuous functions, , , are given absolutely continuous functions with , for each , , and is the Banach space of continuous functions on coupled with the norm
If ; , then for any , define by
here represents the history of the state from time up to the present time .
Next, we consider the system of fractionalorder neutral integrodifferential equations with infinite delay of the form
where J, φ, ψ are as in the problem (3)(5) and , , are given continuous functions, and ℬ is called a phase space that will be specified in Section 4.
During the last two decades, many authors have considered the questions of existence, uniqueness, estimates of solutions, and dependence with respect to initial conditions of the solutions of differential and integral equations of two and three variables (see [1519] and the references therein).
It is clear that more complicated partial differential systems with deviated variables and partial differential integral systems can be obtained from (3) and (6) by a suitable definition of f and g. Barbashin [20] considered a class of partial integrodifferential equations which appear in mathematical modeling of many applied problems (see [21], Section 19). Recently Pachpatte [22,23] considered some classes of partial functional differential equations which occur in a natural way in the description of many physical phenomena.
We present the existence results for our problems based on the nonlinear alternative of the LeraySchauder theorem. The present results extend those considered with integer order derivative [6,9,16,24,25] and those with fractional derivative [11,12,26].
2 Preliminaries
In this section, we introduce notations, definitions, and preliminary facts which are used throughout this paper. By we denote the Banach space of all continuous functions from J into with the norm
where denotes the usual supremum norm on .
Also, is a Banach space with the norm
As usual, by we denote the space of absolutely continuous functions from J into and is the space of Lebesgueintegrable functions with the norm
Definition 2.1 ([27])
Let , , and . The leftsided mixed RiemannLiouville integral of order r of u is defined by
where is the (Euler’s) gamma function defined by ; .
In particular,
Note that if , then exists for all . Moreover, provided , and
Definition 2.3 ([27])
Let and . The Caputo fractionalorder derivative of order r of u is defined by the expression
The case is included, and we have
In the sequel, we need the following lemma.
Lemma 2.5 ([26])
Letand. Then the unique solutionof the problem
is given by the following expression:
where
As a consequence of Lemma 2.5, it is not difficult to verify the following result.
Corollary 2.6Letand. A functionis a solution of the problem (3)(5) if and only ifusatisfies
Also, we need the following theorem.
Theorem 2.7 (Nonlinear alternative of LeraySchauder type [28])
Byand∂Uwe denote the closure ofUand the boundary ofUrespectively. LetXbe a Banach space andCa nonempty convex subset ofX. LetUbe a nonempty open subset ofCwithandbe a completely continuous operator.
Then either
(a) Thas fixed points or
3 Existence results with finite delay
Let us start by defining what we mean by a solution of the problem (3)(5).
Definition 3.1 A function is said to be a solution of the problem (3)(5) if u satisfies equations (3), (5) on J and the condition (4) on .
Further, we present conditions for the existence of a solution of the problem (3)(5).
(H_{1}) There exist nonnegative functions such that
(H_{2}) For any bounded set B in E, the set is equicontinuous in E, and there exist constants such that
Set
Theorem 3.2Assume that the hypotheses (H_{1}) and (H_{2}) hold. Then if
the problem (3)(5) has at least one solution.
Proof Transform the problem (3)(5) into a fixed point problem. Define the operator by
It is clear that N maps E into itself. By Corollary 2.6, the problem of finding the solutions of the problem (3)(5) is reduced to finding the solutions of the operator equation . We shall show that the operator N satisfies all the conditions of Theorem 2.7. The proof will be given in two steps.
Step 1: Nis continuous and completely continuous.
Using (H_{2}) we deduce that g is a complete continuous operator from E to , so it suffices to show that the operator defined by
is continuous and completely continuous. The proof will be given in several claims.
Let be a sequence such that in E. Then for each , we have
Since as and f, are continuous, then
Claim 2: maps bounded sets into bounded sets inE.
Indeed, it is enough to show that for any , there exists a positive constant such that if , we have that .
By (H_{2}) and (H_{3}), we have that for each and ,
Thus,
Claim 3: maps bounded sets inEinto equicontinuous sets inE.
Let , , , , and let be such that . Then
As , , the righthand side of the above inequality tends to zero with the same rate of convergence for all with .
The equicontinuity for the cases , and , is obvious. As a consequence of Claims 1 to 3 together with the ArzeláAscoli theorem, we can conclude that is continuous and completely continuous.
Step 2: A priori bounds.
We shall show that there exists an open set with for all and all .
Let be such that for some . Thus, for each , we have
It is obvious that
On the contrary, when , we have that . So, from the previous inequalities and the condition (9), we arrive at
Thus,
Set
By our choice of U, there is no such that for .
As a consequence of Steps 1 and 2 together with Theorem 2.7, we deduce that N has a fixed point u in which is a solution to the problem (3)(5). □
4 The phase space ℬ
The notation of the phase space ℬ plays an important role in the study of both qualitative and quantitative theory for functional differential equations. A usual choice is a seminormed space satisfying suitable axioms, which was introduced by Hale and Kato (see [7]). For further applications, see, for instance, the books [10,29,30] and their references. For any , denote . Furthermore, in case , , we write simply ℰ. Consider the space a seminormed linear space of functions mapping into and satisfying the following fundamental axioms which were adapted from those introduced by Hale and Kato for ordinary differential functional equations:
(A_{1}) If is a continuous function on J and for all , then there are constants such that for any , the following conditions hold:
(A_{2}) For the function in (A_{1}), is a ℬvalued continuous function on J.
(A_{3}) The space ℬ is complete.
Now, we present some examples of phase spaces [6,9].
Example 4.1 Let ℬ be the set of all functions which are continuous on , , with the seminorm
Then we have . The quotient space is isometric to the space of all continuous functions from into with the supremum norm. This means that partial differential functional equations with finite delay are included in our axiomatic model.
Example 4.2 Let , and let be the set of all continuous functions , for which a limit exists, with the norm
be the seminorm for the space of all functions which are continuous on measurable on , and such that . Then
5 Existence results with infinite delay
Set
Let us start by defining what we mean by a solution of the problem (6)(8).
Definition 5.1 A function is said to be a solution of (6)(8) if u satisfies equations (6) and (8) on J and the condition (7) on .
Now, we present conditions for the existence of a solution of the problem (6)(8).
() There exist nonnegative functions such that
() For any bounded set B in Ω, the set is equicontinuous in Ω, and there exist constants such that
Set
Theorem 5.2Assume that the hypotheses () and () hold. If
then the problem (6)(8) has at least one solution on.
Proof Transform the problem (6)(8) into a fixed point problem. Let and define the operator by
As in Theorem 3.2, we can easily see that maps Ω into itself.
For each with for each , we denote by the function defined by
If satisfies the integral equation, ; , we can decompose as ; , which implies for every , and the function satisfies
Set
and let be the norm in defined by
is a Banach space with the norm .
Let the operator be defined by
Then the operator has a fixed point in Ω if and only if P has a fixed point in . As in the proof of Theorem 3.2, we can show that the operator P satisfies all the conditions of Theorem 2.7. Indeed, to prove that P is continuous and completely continuous and by using (), it suffices to show that the operator defined by
is continuous and completely continuous. Also, we can show that there exists an open set with for and . Consequently, by Theorem 2.7, we deduce that has a fixed point u in which is a solution to the problem (6)(8). □
6 An example
Consider the following neutral integrodifferential equations of fractional order:
Set
and
We have ; . For each and , we have
and
Hence, the condition (H_{1}) is satisfied with , , . Also, the condition (H_{2}) is satisfied with and .
We shall show that the condition (9) holds for each with . Indeed, , ; , and . Then
Consequently, Theorem 3.2 implies that the problem (16)(18) has at least one solution defined on .
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The three authors have participated into the obtained results. The collaboration of each one cannot be separated in different parts of the paper. All of them have made substantial contributions to the theoretical results. The three authors have been involved in drafting the manuscript and revising it critically for important intellectual content. All authors have given final approval of the version to be published.
Acknowledgements
The authors are grateful to the referees for their helpful remarks. Third author is partially supported by FEDER and Ministerio de Educación y Ciencia, Spain, project MTM201015314.
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