Partial neutral functional integro-differential equations of fractional order with delay
© Abbas et al.; licensee Springer. 2012
Received: 9 July 2012
Accepted: 24 October 2012
Published: 6 November 2012
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.
Keywordsintegro-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.
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].
here represents the history of the state from time up to the present time .
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].
where denotes the usual supremum norm on .
Definition 2.1 ()
where is the (Euler’s) gamma function defined by ; .
By we mean .
Definition 2.3 ()
In the sequel, we need the following lemma.
Lemma 2.5 ()
As a consequence of Lemma 2.5, it is not difficult to verify the following result.
Also, we need the following theorem.
Theorem 2.7 (Nonlinear alternative of Leray-Schauder type )
By and ∂U we denote the closure of U and the boundary of U respectively. Let X be a Banach space and C a nonempty convex subset of X. Let U be a nonempty open subset of C with and be a completely continuous operator.
T has fixed points or
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).
for all , , and .
the problem (3)-(5) has at least one solution .
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: N is continuous and completely continuous.
is continuous and completely continuous. The proof will be given in several claims.
Claim 1: is continuous.
Claim 2: maps bounded sets into bounded sets in E.
Indeed, it is enough to show that for any , there exists a positive constant such that if , we have that .
Claim 3: maps bounded sets in E into equicontinuous sets in E.
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 .
As consequence, if , then .
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:
is in ℬ;
(A2) For the function in (A1), is a ℬ-valued continuous function on J.
(A3) The space ℬ is complete.
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.
Then we have and .
5 Existence results with infinite delay
Let us start by defining what we mean by a solution of the problem (6)-(8).
Now, we present conditions for the existence of a solution of the problem (6)-(8).
for all , , and .
then the problem (6)-(8) has at least one solution on .
As in Theorem 3.2, we can easily see that maps Ω into itself.
Then for all .
is a Banach space with the norm .
Note that if and only if .
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
Hence, the condition (H1) is satisfied with , , . Also, the condition (H2) is satisfied with and .
Consequently, Theorem 3.2 implies that the problem (16)-(18) has at least one solution defined on .
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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