Partial neutral functional integro-differential equations of fractional order with delay

*Correspondence: alberto.cabada@usc.es 3Departamento de Análise Matemática, Facultade de Matemáticas, Universidade de Santiago de Compostela, Santiago de Compostela, Spain Full list of author information is available at the end of the article Abstract 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. MSC: 26A33


Introduction
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. 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 c D r θ u(x, y)g(x, y, u (x,y) ) = f x, y, I r θ u(x, y), u (x,y) ; (x, y) ∈ J, (  ) http://www.boundaryvalueproblems.com/content/2012/1/128 w(x, y) .
here u (x,y) (·, ·) represents the history of the state from time (xα, yβ) up to the present time (x, y). Next, we consider the system of fractional-order neutral integro-differential equations with infinite delay of the form c D r θ u(x, y)g(x, y, u (x,y) ) = f x, y, I r θ u(x, y), u (x,y) ; (x, y) ∈ J, where J, ϕ, ψ are as in the problem ()-() and φ ∈ C(J ), f : J × R n × B → R n , g : J × B → R n are given continuous functions, and B is called a phase space that will be specified in Section .
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 [-] 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 () and () 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 ). Recently Pachpatte [, ] considered some classes of partial functional differential equations which occur in a natural way in the description of many physical phenomena. http://www.boundaryvalueproblems.com/content/2012/1/128 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 [, , , , ] and those with fractional derivative [, , ].

Preliminaries
In this section, we introduce notations, definitions, and preliminary facts which are used throughout this paper. By C(J) we denote the Banach space of all continuous functions from J into R n with the norm where · denotes the usual supremum norm on R n .
As usual, by AC(J) we denote the space of absolutely continuous functions from J into R n and L  (J) is the space of Lebesgue-integrable functions w : J → R n with the norm In particular, Example . Let λ, ω ∈ (-, ∞) and r = (r  , r  ) ∈ (, ∞) × (, ∞). Then x λ+r  y ω+r  for almost all (x, y) ∈ J.
The case σ = (, ) is included, and we have In the sequel, we need the following lemma.

Lemma . ([]) Let f ∈ L  (J) and g ∈ AC(J). Then the unique solution u ∈ AC(J) of the problem
is given by the following expression: As a consequence of Lemma ., it is not difficult to verify the following result.

is a solution of the problem ()-() if and only if u satisfies
Also, we need the following theorem. .

Theorem . Assume that the hypotheses (H  ) and (H  ) hold. Then if
Proof Transform the problem ()-() into a fixed point problem. Define the operator N : μ(x, y) + g(x, y, u (x,y) ) + g(, , u (,) ) g(x, , u (x,) )g(, y, u (,y) ) + I r θ f (x, y, I r θ u(x, y), u (x,y) ); (x, y) ∈ J. () It is clear that N maps E into itself. By Corollary ., the problem of finding the solutions of the problem ()-() is reduced to finding the solutions of the operator equation N(u) = u. We shall show that the operator N satisfies all the conditions of Theorem .. The proof will be given in two steps. http://www.boundaryvalueproblems.com/content/2012/1/128 Step : N is continuous and completely continuous.
Using (H  ) we deduce that g is a complete continuous operator from E to R n , so it suffices to show that the operator N  : E → E defined by is continuous and completely continuous. The proof will be given in several claims. Claim : N  is continuous.
Since u n → u as n → ∞ and f , I r θ are continuous, then

Claim : N  maps bounded sets into bounded sets in E.
Indeed, it is enough to show that for any η > , there exists a positive constant >  such that if u E ≤ η, we have that N  (u) E ≤ .
By (H  ) and (H  ), we have that for each (x, y) ∈ J and u E ≤ η, Thus,

Claim : N  maps bounded sets in E into equicontinuous sets in E.
Let (x  , y  ), (x  , y  ) ∈ J, x  < x  , y  < y  , η > , and let u ∈ E be such that u E ≤ η. Then As x  → x  , y  → y  , the right-hand side of the above inequality tends to zero with the same rate of convergence for all u ∈ E with u E ≤ η.
The equicontinuity for the cases x  < x  < , y  < y  <  and x  ≤  ≤ x  , y  ≤  ≤ y  is obvious. As a consequence of Claims  to  together with the Arzelá-Ascoli theorem, we can conclude that N  is continuous and completely continuous.
Step : A priori bounds.
We shall show that there exists an open set U ⊆ E with u = λN(u) for all λ ∈ (, ) and all u ∈ ∂U.
Let u ∈ E be such that u = λN(u) for some  < λ < . Thus, for each (x, y) ∈ J, we have u(x, y) = λ μ(x, y) + g(x, y, u (x,y) ) + g(, , u (,) ) g(x, , u (x,) )g(, y, u (,y) ) + λI r θ f x, y, I r θ u(x, y), u (x,y) . http://www.boundaryvalueproblems.com/content/2012/1/128 Then for (x, y) ∈ J, we have It is obvious that On the contrary, when u J > φ J , we have that u E = u J . So, from the previous inequalities and the condition (), we arrive at Thus, By our choice of U, there is no u ∈ ∂U such that u = λN(u) for λ ∈ (, ). As a consequence of Steps  and  together with Theorem ., we deduce that N has a fixed point u in U which is a solution to the problem ()-().

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  J and z (x,y) ∈ B for all (x, y) ∈ E, then there are constants H, K, M >  such that for any (x, y) ∈ J, the following conditions hold: μ(x, y) + g(x, y, u (x,y) ) + g(, , u (,) ) g(x, , u (x,) )g(, y, u (,y) ) + I r θ f (x, y, I r θ u(x, y), u (x,y) ); (x, y) ∈ J. () As in Theorem ., we can easily see thatN maps into itself.
Then v (x,y) = φ for all (x, y) ∈ E. For each w ∈ C(J) with w(x, y) =  for each (x, y) ∈ E, we denote by w the function defined by w(x, y), (x, y) ∈ J.
If u(·, ·) satisfies the integral equation, u(x, y) = (Nu)(x, y); (x, y) ∈ [-α, a] × [-β, b], we can http://www.boundaryvalueproblems.com/content/2012/1/128 decompose u(·, ·) as u(x, y) = w(x, y) + v(x, y); (x, y) ∈ [-α, a] × [-β, b], which implies u (x,y) = w (x,y) + v (x,y) for every (x, y) ∈ J, and the function w(·, ·) satisfies w(x, y) = g(x, y, u (x,y) ) + g(, , w (,) + v (,) ) g(x, , w (x,) + v (x,) )g(, y, w (,y) + v (,y) ) and let · (a,b) be the norm in C  defined by C  is a Banach space with the norm · (a,b) . Note that u ∈ if and only if w ∈ C  . Let the operator P : C  → C  be defined by Then the operatorN has a fixed point in if and only if P has a fixed point in C  . As in the proof of Theorem ., we can show that the operator P satisfies all the conditions of Theorem .. Indeed, to prove that P is continuous and completely continuous and by using (H  ), it suffices to show that the operatorP : F → F defined by (Pw)(x, y) =  (r  ) (r  ) x  y  (xs) r  - (yt) r  - × f s, t, I r θ w(s, t) + v(s, t) , w (s,t) + v (s,t) , g(s, t) dt ds () is continuous and completely continuous. Also, we can show that there exists an open set U ⊆ F with u = λP(u) for λ ∈ (, ) and u ∈ ∂U . Consequently, by Theorem ., we deduce thatN has a fixed point u in U which is a solution to the problem ()-().

An example
Consider the following neutral integro-differential equations of fractional order: