In this paper we obtain sufficient conditions for the existence of solutions of some classes of partial neutral integro-differential equations of fractional order by using suitable fixed point theorems.
Keywords:integro-differential equation; left-sided mixed Riemann-Liouville integral of fractional order; Caputo fractional-order derivative; finite delay; infinite delay; solution; fixed point
Fractional differential and integral equations have recently been applied in various areas of engineering, science, finance, applied mathematics, bio-engineering 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., Baleanu et al., Kilbas et al., Lakshmikantham et al., Podlubny , and the references therein.
In , Czlapinski proved some results for the following system of the Darboux problem for the second-order partial functional differential equations of the form
where , , , , , , and ℬ is a vector space of real-valued functions defined in , equipped with a semi-norm and satisfying some suitable axioms, which was introduced by Hale and Kato ; see also [8-10] 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 fractional-order 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 integro-differential equations of fractional order with delay. First, we consider the system of fractional-order neutral integro-differential equations with finite delay of the form
where ; , , , , is the left-sided mixed Riemann-Liouville 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 fractional-order neutral integro-differential 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 [15-19] 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  considered a class of partial integro-differential equations which appear in mathematical modeling of many applied problems (see , 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 Leray-Schauder theorem. The present results extend those considered with integer order derivative [6,9,16,24,25] and those with fractional derivative [11,12,26].
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 Lebesgue-integrable functions with the norm
Definition 2.1 ()
Let , , and . The left-sided mixed Riemann-Liouville integral of order r of u is defined by
where is the (Euler’s) gamma function defined by ; .
Note that if , then exists for all . Moreover, provided , and
Example 2.2 Let and . Then
By we mean .
Definition 2.3 ()
Let and . The Caputo fractional-order derivative of order r of u is defined by the expression
The case is included, and we have
Example 2.4 Let and . Then
In the sequel, we need the following lemma.
Lemma 2.5 ()
Let and . Then the unique solution of the problem
is given by the following expression:
As a consequence of Lemma 2.5, it is not difficult to verify the following result.
Corollary 2.6Let and . A function is a solution of the problem (3)-(5) if and only ifusatisfies
Also, we need the following theorem.
Theorem 2.7 (Nonlinear alternative of Leray-Schauder type )
By and∂Uwe denote the closure ofUand the boundary ofUrespectively. LetXbe a Banach space andCa nonempty convex subset ofX. LetUbe a nonempty open subset ofCwith and be a completely continuous operator.
(a) Thas fixed points or
(b) there exist and with .
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).
(H1) There exist nonnegative functions such that
for all , , and .
(H2) For any bounded set B in E, the set is equicontinuous in E, and there exist constants such that
Theorem 3.2Assume that the hypotheses (H1) and (H2) 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 (H2) 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.
Claim 1: is continuous.
Let be a sequence such that in E. Then for each , we have
Hence, for each , we get
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 (H2) and (H3), we have that for each and ,
Claim 3: maps bounded sets inEinto equicontinuous sets inE.
Let , , , , and let be such that . Then
As , , the right-hand 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 Arzelá-Ascoli 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
Then for , we have
It is obvious that
As consequence, if , then .
On the contrary, when , we have that . So, from the previous inequalities and the condition (9), we arrive at
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 semi-normed space satisfying suitable axioms, which was introduced by Hale and Kato (see ). 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 semi-normed 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:
(A1) If is a continuous function on J and for all , then there are constants such that for any , the following conditions hold:
(i) is in ℬ;
(A2) For the function in (A1), is a ℬ-valued continuous function on J.
(A3) The space ℬ is complete.
Example 4.1 Let ℬ be the set of all functions which are continuous on , , with the semi-norm
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
Then we have and .
Example 4.3 Let , and let
be the semi-norm for the space of all functions which are continuous on measurable on , and such that . Then
5 Existence results with infinite delay
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 all , , and .
( ) For any bounded set B in Ω, the set is equicontinuous in Ω, and there exist constants such that
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.
Let be a function defined by
Then for all .
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
and let be the norm in defined by
is a Banach space with the norm .
Note that if and only if .
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 integro-differential equations of fractional order:
We have ; . For each and , we have
Hence, the condition (H1) is satisfied with , , . Also, the condition (H2) 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 .
The authors declare that they have no competing interests.
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.
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 MTM2010-15314.
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