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

  • Saïd Abbas1,

    Affiliated with

    • Mouffak Benchohra2 and

      Affiliated with

      • Alberto Cabada3Email author

        Affiliated with

        Boundary Value Problems20122012:128

        DOI: 10.1186/1687-2770-2012-128

        Received: 9 July 2012

        Accepted: 24 October 2012

        Published: 6 November 2012

        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.

        Keywords

        integro-differential equation left-sided mixed Riemann-Liouville integral of fractional order Caputo fractional-order derivative finite delay infinite delay solution fixed point

        1 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. [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 second-order partial functional differential equations of the form
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_Equ1_HTML.gif
        (1)
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_Equ2_HTML.gif
        (2)

        where a , b > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq1_HTML.gif, f : R × B × B × B R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq2_HTML.gif, Φ : E 0 R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq3_HTML.gif, D x : = x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq4_HTML.gif, D y : = y http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq5_HTML.gif, D x y 2 : = 2 x y http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq6_HTML.gif, and ℬ is a vector space of real-valued functions defined in ( , 0 ] × ( , 0 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq7_HTML.gif, equipped with a semi-norm 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 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
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_Equ3_HTML.gif
        (3)
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_Equ4_HTML.gif
        (4)
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_Equ5_HTML.gif
        (5)
        where J : = [ 0 , a ] × [ 0 , b ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq8_HTML.gif; a , b , α , β > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq9_HTML.gif, θ = ( 0 , 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq10_HTML.gif, r = ( r 1 , r 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq11_HTML.gif, r 1 , r 2 ( 0 , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq12_HTML.gif, I θ r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq13_HTML.gif is the left-sided mixed Riemann-Liouville integral of order r (see Section 2 for definition), D θ r c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq14_HTML.gif is the fractional Caputo derivative of order r, f : J × R n × C R n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq15_HTML.gif, g : J × C R n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq16_HTML.gif are given continuous functions, ϕ C ( J ˜ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq17_HTML.gif, φ : [ 0 , a ] R n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq18_HTML.gif, ψ : [ 0 , b ] R n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq19_HTML.gif are given absolutely continuous functions with φ ( x ) = ϕ ( x , 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq20_HTML.gif, ψ ( y ) = ϕ ( 0 , y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq21_HTML.gif for each x [ 0 , a ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq22_HTML.gif, y [ 0 , b ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq23_HTML.gif, and C : = C ( [ α , 0 ] × [ β , 0 ] ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq24_HTML.gif is the Banach space of continuous functions on [ α , 0 ] × [ β , 0 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq25_HTML.gif coupled with the norm
        w C = sup ( x , y ) [ α , 0 ] × [ β , 0 ] w ( x , y ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_Equa_HTML.gif
        If u C ( [ α , a ] × [ β , b ] ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq26_HTML.gif; α , β , a , b > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq27_HTML.gif, then for any ( x , y ) J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq28_HTML.gif, define u ( x , y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq29_HTML.gif by
        u ( x , y ) ( s , t ) = u ( x + s , y + t ) ; ( s , t ) [ α , 0 ] × [ β , 0 ] , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_Equb_HTML.gif

        here u ( x , y ) ( , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq30_HTML.gif represents the history of the state from time ( x α , y β ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq31_HTML.gif up to the present time ( x , y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq32_HTML.gif.

        Next, we consider the system of fractional-order neutral integro-differential equations with infinite delay of the form
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_Equ6_HTML.gif
        (6)
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_Equ7_HTML.gif
        (7)
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_Equ8_HTML.gif
        (8)

        where J, φ, ψ are as in the problem (3)-(5) and ϕ C ( J ˜ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq33_HTML.gif, f : J × R n × B R n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq34_HTML.gif, g : J × B R n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq35_HTML.gif 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 integro-differential 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 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].

        2 Preliminaries

        In this section, we introduce notations, definitions, and preliminary facts which are used throughout this paper. By C ( J ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq36_HTML.gif we denote the Banach space of all continuous functions from J into R n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq37_HTML.gif with the norm
        w J = sup ( x , y ) J w ( x , y ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_Equc_HTML.gif

        where http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq38_HTML.gif denotes the usual supremum norm on R n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq37_HTML.gif.

        Also, E : = C ( [ α , a ] × [ β , b ] ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq39_HTML.gif is a Banach space with the norm
        w E = sup ( x , y ) [ α , a ] × [ β , b ] w ( x , y ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_Equd_HTML.gif
        As usual, by A C ( J ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq40_HTML.gif we denote the space of absolutely continuous functions from J into R n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq37_HTML.gif and L 1 ( J ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq41_HTML.gif is the space of Lebesgue-integrable functions w : J R n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq42_HTML.gif with the norm
        w L 1 = 0 a 0 b w ( x , y ) d y d x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_Eque_HTML.gif

        Definition 2.1 ([27])

        Let r = ( r 1 , r 2 ) ( 0 , ) × ( 0 , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq43_HTML.gif, θ = ( 0 , 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq10_HTML.gif, and u L 1 ( J ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq44_HTML.gif. The left-sided mixed Riemann-Liouville integral of order r of u is defined by
        ( I θ r u ) ( x , y ) = 1 Γ ( r 1 ) Γ ( r 2 ) 0 x 0 y ( x s ) r 1 1 ( y t ) r 2 1 u ( s , t ) d t d s , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_Equf_HTML.gif

        where Γ ( ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq45_HTML.gif is the (Euler’s) gamma function defined by Γ ( ξ ) = 0 t ξ 1 e t d t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq46_HTML.gif; ξ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq47_HTML.gif.

        In particular,
        ( I θ θ u ) ( x , y ) = u ( x , y ) , ( I θ σ u ) ( x , y ) = 0 x 0 y u ( s , t ) d t d s ; for almost all  ( x , y ) J , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_Equg_HTML.gif

        where σ = ( 1 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq48_HTML.gif.

        Note that if u L 1 ( J ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq44_HTML.gif, then I θ r u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq49_HTML.gif exists for all r 1 , r 2 ( 0 , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq50_HTML.gif. Moreover, I θ r u C ( J ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq51_HTML.gif provided u C ( J ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq52_HTML.gif, and
        ( I θ r u ) ( x , 0 ) = ( I θ r u ) ( 0 , y ) = 0 ; x [ 0 , a ] , y [ 0 , b ] . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_Equh_HTML.gif
        Example 2.2 Let λ , ω ( 1 , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq53_HTML.gif and r = ( r 1 , r 2 ) ( 0 , ) × ( 0 , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq54_HTML.gif. Then
        I θ r x λ y ω = Γ ( 1 + λ ) Γ ( 1 + ω ) Γ ( 1 + λ + r 1 ) Γ ( 1 + ω + r 2 ) x λ + r 1 y ω + r 2 for almost all  ( x , y ) J . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_Equi_HTML.gif

        By 1 r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq55_HTML.gif we mean ( 1 r 1 , 1 r 2 ) [ 0 , 1 ) × [ 0 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq56_HTML.gif.

        Definition 2.3 ([27])

        Let r ( 0 , 1 ] × ( 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq57_HTML.gif and u L 1 ( J ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq44_HTML.gif. The Caputo fractional-order derivative of order r of u is defined by the expression
        D θ r c u ( x , y ) = ( I θ 1 r D x y 2 u ) ( x , y ) = 1 Γ ( 1 r 1 ) Γ ( 1 r 2 ) 0 x 0 y D s t 2 u ( s , t ) ( x s ) r 1 ( y t ) r 2 d t d s . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_Equj_HTML.gif
        The case σ = ( 1 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq48_HTML.gif is included, and we have
        ( c D θ σ u ) ( x , y ) = ( D x y 2 u ) ( x , y ) for almost all  ( x , y ) J . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_Equk_HTML.gif
        Example 2.4 Let λ , ω ( 1 , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq53_HTML.gif and r = ( r 1 , r 2 ) ( 0 , 1 ] × ( 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq58_HTML.gif. Then
        D θ r c x λ y ω = Γ ( 1 + λ ) Γ ( 1 + ω ) Γ ( 1 + λ r 1 ) Γ ( 1 + ω r 2 ) x λ r 1 y ω r 2 for almost all  ( x , y ) J . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_Equl_HTML.gif

        In the sequel, we need the following lemma.

        Lemma 2.5 ([26])

        Let f L 1 ( J ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq59_HTML.gif and g A C ( J ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq60_HTML.gif. Then the unique solution u A C ( J ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq61_HTML.gif of the problem
        { D θ r c ( u g ) ( x , y ) = f ( x , y ) ; ( x , y ) J , u ( x , 0 ) = φ ( x ) ; x [ 0 , a ] , u ( 0 , y ) = ψ ( y ) ; y [ 0 , b ] , φ ( 0 ) = ψ ( 0 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_Equm_HTML.gif
        is given by the following expression:
        u ( x , y ) = μ ( x , y ) + g ( x , y ) g ( x , 0 ) g ( 0 , y ) + g ( 0 , 0 ) + ( I θ r f ) ( x , y ) ; ( x , y ) J , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_Equn_HTML.gif
        where
        μ ( x , y ) = φ ( x ) + ψ ( y ) φ ( 0 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_Equo_HTML.gif

        As a consequence of Lemma 2.5, it is not difficult to verify the following result.

        Corollary 2.6 Let f L 1 ( J × R n × C ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq62_HTML.gif and g A C ( J × C ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq63_HTML.gif. A function u A C ( [ α , a ] × [ β , b ] ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq64_HTML.gif is a solution of the problem (3)-(5) if and only if u satisfies
        { u ( x , y ) = Φ ( x , y ) ; ( x , y ) J ˜ , u ( x , y ) = μ ( x , y ) + g ( x , y , u ( x , y ) ) + g ( 0 , 0 , u ( 0 , 0 ) ) u ( x , y ) = g ( x , 0 , u ( x , 0 ) ) g ( 0 , y , u ( 0 , y ) ) u ( x , y ) = + I θ r f ( x , y , I θ r u ( x , y ) , u ( x , y ) ) ; ( x , y ) J . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_Equp_HTML.gif

        Also, we need the following theorem.

        Theorem 2.7 (Nonlinear alternative of Leray-Schauder type [28])

        By U ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq65_HTML.gif 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 0 U http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq66_HTML.gif and T : U ¯ C http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq67_HTML.gif be a completely continuous operator.

        Then either
        1. (a)

          T has fixed points or

           
        2. (b)

          there exist u U http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq68_HTML.gif and λ ( 0 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq69_HTML.gif with u = λ T ( u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq70_HTML.gif.

           

        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 u E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq71_HTML.gif is said to be a solution of the problem (3)-(5) if u satisfies equations (3), (5) on J and the condition (4) on J ˜ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq72_HTML.gif.

        Further, we present conditions for the existence of a solution of the problem (3)-(5).

        (H1) There exist nonnegative functions p , q , d C ( J ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq73_HTML.gif such that
        f ( x , y , u , v ) p ( x , y ) + q ( x , y ) u + d ( x , y ) v C http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_Equq_HTML.gif

        for all ( x , y ) J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq28_HTML.gif, u R n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq74_HTML.gif, and v C http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq75_HTML.gif.

        (H2) For any bounded set B in E, the set { ( x , y ) g ( x , y , u ( x , y ) ) : u B } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq76_HTML.gif is equicontinuous in E, and there exist constants L 1 , L 2 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq77_HTML.gif such that
        g ( x , y , u ) L 1 + L 2 u C ; ( x , y ) J  and  u C . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_Equr_HTML.gif
        Set
        L = a r 1 b r 2 Γ ( 1 + r 1 ) Γ ( 1 + r 2 ) , p = p J , q = q J and d = d J . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_Equs_HTML.gif
        Theorem 3.2 Assume that the hypotheses (H1) and (H2) hold. Then if
        4 L 2 + q L 2 + d L < 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_Equ9_HTML.gif
        (9)

        the problem (3)-(5) has at least one solution u A C ( [ α , a ] × [ β , b ] ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq78_HTML.gif.

        Proof Transform the problem (3)-(5) into a fixed point problem. Define the operator N : E E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq79_HTML.gif by
        ( N u ) ( x , y ) = { Φ ( x , y ) ; ( x , y ) J ˜ , μ ( x , y ) + g ( x , y , u ( x , y ) ) + g ( 0 , 0 , u ( 0 , 0 ) ) g ( x , 0 , u ( x , 0 ) ) g ( 0 , y , u ( 0 , y ) ) + I θ r f ( x , y , I θ r u ( x , y ) , u ( x , y ) ) ; ( x , y ) J . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_Equ10_HTML.gif
        (10)

        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 N ( u ) = u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq80_HTML.gif. 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.

        Using (H2) we deduce that g is a complete continuous operator from E to R n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq37_HTML.gif, so it suffices to show that the operator N 1 : E E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq81_HTML.gif defined by
        ( N 1 u ) ( x , y ) = { Φ ( x , y ) ; ( x , y ) J ˜ , μ ( x , y ) + I θ r f ( x , y , I θ r u ( x , y ) , u ( x , y ) ) ; ( x , y ) J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_Equ11_HTML.gif
        (11)

        is continuous and completely continuous. The proof will be given in several claims.

        Claim 1: N 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq82_HTML.gif is continuous.

        Let { u n } n N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq83_HTML.gif be a sequence such that u n u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq84_HTML.gif in E. Then for each ( x , y ) [ α , a ] × [ β , b ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq85_HTML.gif, we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_Equt_HTML.gif
        Hence, for each ( x , y ) J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq28_HTML.gif, we get
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_Equu_HTML.gif
        Since u n u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq84_HTML.gif as n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq86_HTML.gif and f, I θ r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq87_HTML.gif are continuous, then
        N 1 ( u n ) N 1 ( u ) E 0 as  n . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_Equv_HTML.gif

        Claim 2: N 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq82_HTML.gif maps bounded sets into bounded sets in E.

        Indeed, it is enough to show that for any η > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq88_HTML.gif, there exists a positive constant > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq89_HTML.gif such that if u E η http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq90_HTML.gif, we have that N 1 ( u ) E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq91_HTML.gif.

        By (H2) and (H3), we have that for each ( x , y ) J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq28_HTML.gif and u E η http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq92_HTML.gif,
        ( N 1 u ) ( x , y ) μ ( x , y ) + p a r 1 b r 2 Γ ( 1 + r 1 ) Γ ( 1 + r 2 ) + q a r 1 b r 2 Γ ( r 1 ) Γ ( r 2 ) Γ ( 1 + r 1 ) Γ ( 1 + r 2 ) × 0 x 0 y ( x s ) r 1 1 ( y t ) r 2 1 u ( s , t ) d t d s + d Γ ( r 1 ) Γ ( r 2 ) 0 x 0 y ( x s ) r 1 1 ( y t ) r 2 1 u ( s , t ) C d t d s μ ( x , y ) + p a r 1 b r 2 Γ ( 1 + r 1 ) Γ ( 1 + r 2 ) + q a r 1 b r 2 Γ ( r 1 ) Γ ( r 2 ) Γ ( 1 + r 1 ) Γ ( 1 + r 2 ) × 0 x 0 y ( x s ) r 1 1 ( y t ) r 2 1 u J d t d s + d Γ ( r 1 ) Γ ( r 2 ) 0 x 0 y ( x s ) r 1 1 ( y t ) r 2 1 u E d t d s μ J + p L + ( q L 2 + d L ) u E μ J + p L + ( q L + d ) L η : = 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_Equw_HTML.gif
        Thus,
        N 1 ( u ) E max { 1 , Φ C } : = . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_Equx_HTML.gif

        Claim 3: N 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq82_HTML.gif maps bounded sets in E into equicontinuous sets in E.

        Let ( x 1 , y 1 ) , ( x 2 , y 2 ) J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq93_HTML.gif, x 1 < x 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq94_HTML.gif, y 1 < y 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq95_HTML.gif, η > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq88_HTML.gif, and let u E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq71_HTML.gif be such that u E η http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq96_HTML.gif. Then
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_Equy_HTML.gif

        As x 1 x 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq97_HTML.gif, y 1 y 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq98_HTML.gif, the right-hand side of the above inequality tends to zero with the same rate of convergence for all u E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq99_HTML.gif with u E η http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq96_HTML.gif.

        The equicontinuity for the cases x 1 < x 2 < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq100_HTML.gif, y 1 < y 2 < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq101_HTML.gif and x 1 0 x 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq102_HTML.gif, y 1 0 y 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq103_HTML.gif is obvious. As a consequence of Claims 1 to 3 together with the Arzelá-Ascoli theorem, we can conclude that N 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq82_HTML.gif is continuous and completely continuous.

        Step 2: A priori bounds.

        We shall show that there exists an open set U E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq104_HTML.gif with u λ N ( u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq105_HTML.gif for all λ ( 0 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq69_HTML.gif and all u U http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq68_HTML.gif.

        Let u E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq71_HTML.gif be such that u = λ N ( u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq106_HTML.gif for some 0 < λ < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq107_HTML.gif. Thus, for each ( x , y ) J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq28_HTML.gif, we have
        u ( x , y ) = λ ( μ ( x , y ) + g ( x , y , u ( x , y ) ) + g ( 0 , 0 , u ( 0 , 0 ) ) g ( x , 0 , u ( x , 0 ) ) g ( 0 , y , u ( 0 , y ) ) ) + λ I θ r f ( x , y , I θ r u ( x , y ) , u ( x , y ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_Equz_HTML.gif
        Then for ( x , y ) J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq108_HTML.gif, we have
        u ( x , y ) μ ( x , y ) + 4 L 1 + L 2 ( u ( x , y ) C + u ( x , 0 ) C + u ( 0 , y ) C + u ( 0 , 0 ) C ) + p L + q L Γ ( r 1 ) Γ ( r 2 ) 0 x 0 y ( x s ) r 1 1 ( y t ) r 2 1 u ( s , t ) d t d s + d Γ ( r 1 ) Γ ( r 2 ) 0 x 0 y ( x s ) r 1 1 ( y t ) r 2 1 u ( s , t ) C d t d s μ ( x , y ) + 4 L 1 + 4 L 2 u E + p L + q L Γ ( r 1 ) Γ ( r 2 ) 0 x 0 y ( x s ) r 1 1 ( y t ) r 2 1 u J d t d s + d Γ ( r 1 ) Γ ( r 2 ) 0 x 0 y ( x s ) r 1 1 ( y t ) r 2 1 u E d t d s μ J + 4 L 1 + p L + ( 4 L 2 + q L 2 + d L ) u E . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_Equaa_HTML.gif
        It is obvious that
        u E = max { u J ˜ , u J } max { ϕ J ˜ , u J } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_Equab_HTML.gif

        As consequence, if u J ϕ J ˜ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq109_HTML.gif, then u E ϕ J ˜ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq110_HTML.gif.

        On the contrary, when u J > ϕ J ˜ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq111_HTML.gif, we have that u E = u J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq112_HTML.gif. So, from the previous inequalities and the condition (9), we arrive at
        u E μ J + 4 L 1 + p L 1 4 L 2 q L 2 d L : = M . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_Equac_HTML.gif
        Thus,
        u E max { ϕ J ˜ , M } : = M . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_Equad_HTML.gif
        Set
        U = { u E : u E < M + 1 } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_Equae_HTML.gif

        By our choice of U, there is no u U http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq68_HTML.gif such that u = λ N ( u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq106_HTML.gif for λ ( 0 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq69_HTML.gif.

        As a consequence of Steps 1 and 2 together with Theorem 2.7, we deduce that N has a fixed point u in U ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq65_HTML.gif 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 [7]). For further applications, see, for instance, the books [10, 29, 30] and their references. For any ( x , y ) J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq28_HTML.gif, denote E ( x , y ) : = ( [ 0 , x ] × { 0 } ) ( { 0 } × [ 0 , y ] ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq113_HTML.gif. Furthermore, in case x = a http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq114_HTML.gif, y = b http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq115_HTML.gif, we write simply ℰ. Consider the space ( B , ( , ) B ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq116_HTML.gif a semi-normed linear space of functions mapping ( , 0 ] × ( , 0 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq7_HTML.gif into R n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq37_HTML.gif and satisfying the following fundamental axioms which were adapted from those introduced by Hale and Kato for ordinary differential functional equations:

        (A1) If z : ( , a ] × ( , b ] R n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq117_HTML.gif is a continuous function on J and z ( x , y ) B http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq118_HTML.gif for all ( x , y ) E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq119_HTML.gif, then there are constants H , K , M > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq120_HTML.gif such that for any ( x , y ) J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq28_HTML.gif, the following conditions hold:
        1. (i)

          z ( x , y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq121_HTML.gif is in ℬ;

           
        2. (ii)

          z ( x , y ) H z ( x , y ) B http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq122_HTML.gif;

           
        3. (iii)

          z ( x , y ) B K sup ( s , t ) [ 0 , x ] × [ 0 , y ] z ( s , t ) + M sup ( s , t ) E ( x , y ) z ( s , t ) B http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq123_HTML.gif.

           

        (A2) For the function z ( , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq124_HTML.gif in (A1), z ( x , y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq121_HTML.gif is a ℬ-valued continuous function on J.

        (A3) 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 ϕ : ( , 0 ] × ( , 0 ] R n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq125_HTML.gif which are continuous on [ α , 0 ] × [ β , 0 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq25_HTML.gif, α , β 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq126_HTML.gif, with the semi-norm
        ϕ B = sup ( s , t ) [ α , 0 ] × [ β , 0 ] ϕ ( s , t ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_Equaf_HTML.gif

        Then we have H = K = M = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq127_HTML.gif. The quotient space B ˆ = B / B http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq128_HTML.gif is isometric to the space C http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq129_HTML.gif of all continuous functions from [ α , 0 ] × [ β , 0 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq25_HTML.gif into R n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq37_HTML.gif with the supremum norm. This means that partial differential functional equations with finite delay are included in our axiomatic model.

        Example 4.2 Let γ R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq130_HTML.gif, and let C γ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq131_HTML.gif be the set of all continuous functions ϕ : ( , 0 ] × ( , 0 ] R n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq125_HTML.gif, for which a limit lim ( s , t ) e γ ( s + t ) ϕ ( s , t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq132_HTML.gif exists, with the norm
        ϕ C γ = sup ( s , t ) ( , 0 ] × ( , 0 ] e γ ( s + t ) ϕ ( s , t ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_Equag_HTML.gif

        Then we have H = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq133_HTML.gif and K = M = max { e γ ( a + b ) , 1 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq134_HTML.gif.

        Example 4.3 Let α , β , γ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq135_HTML.gif, and let
        ϕ C L γ = sup ( s , t ) [ α , 0 ] × [ β , 0 ] ϕ ( s , t ) + 0 0 e γ ( s + t ) ϕ ( s , t ) d t d s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_Equah_HTML.gif
        be the semi-norm for the space C L γ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq136_HTML.gif of all functions ϕ : ( , 0 ] × ( , 0 ] R n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq125_HTML.gif which are continuous on [ α , 0 ] × [ β , 0 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq25_HTML.gif measurable on ( ( , α ] × ( , 0 ] ) ( ( , 0 ] × ( , β ] ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq137_HTML.gif, and such that ϕ C L γ < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq138_HTML.gif. Then
        H = 1 , K = α 0 β 0 e γ ( s + t ) d t d s , M = 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_Equai_HTML.gif

        5 Existence results with infinite delay

        Set
        Ω : = { u : ( , a ] × ( , b ] R n : u ( x , y ) B  for  ( x , y ) E  and  u | J C ( J ) } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_Equaj_HTML.gif

        Let us start by defining what we mean by a solution of the problem (6)-(8).

        Definition 5.1 A function u Ω http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq139_HTML.gif is said to be a solution of (6)-(8) if u satisfies equations (6) and (8) on J and the condition (7) on J ˜ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq140_HTML.gif.

        Now, we present conditions for the existence of a solution of the problem (6)-(8).

        ( H 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq141_HTML.gif) There exist nonnegative functions p ¯ , q ¯ , d ¯ C ( J ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq142_HTML.gif such that
        f ( x , y , u , v ) p ¯ ( x , y ) + q ¯ ( x , y ) u + d ¯ ( x , y ) v B http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_Equak_HTML.gif

        for all ( x , y ) J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq28_HTML.gif, u R n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq74_HTML.gif, and v B http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq143_HTML.gif.

        ( H 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq144_HTML.gif) For any bounded set B in Ω, the set { ( x , y ) g ( x , y , u ( x , y ) ) : u B } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq145_HTML.gif is equicontinuous in Ω, and there exist constants L ¯ 1 , L ¯ 2 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq146_HTML.gif such that
        g ( x , y , u ) L ¯ 1 + L ¯ 2 u B ; ( x , y ) J  and  u B . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_Equal_HTML.gif
        Set
        p ¯ = p ¯ J , q ¯ = q ¯ J and d ¯ = d ¯ J . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_Equam_HTML.gif
        Theorem 5.2 Assume that the hypotheses ( H 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq141_HTML.gif) and ( H 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq144_HTML.gif) hold. If
        4 L ¯ 2 + q ¯ L 2 + d ¯ L < 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_Equ12_HTML.gif
        (12)

        then the problem (6)-(8) has at least one solution on ( , a ] × ( , b ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq147_HTML.gif.

        Proof Transform the problem (6)-(8) into a fixed point problem. Let u Ω http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq139_HTML.gif and define the operator N ¯ : Ω Ω http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq148_HTML.gif by
        ( N ¯ u ) ( x , y ) = { Φ ( x , y ) ; ( x , y ) J ˜ , μ ( x , y ) + g ( x , y , u ( x , y ) ) + g ( 0 , 0 , u ( 0 , 0 ) ) g ( x , 0 , u ( x , 0 ) ) g ( 0 , y , u ( 0 , y ) ) + I θ r f ( x , y , I θ r u ( x , y ) , u ( x , y ) ) ; ( x , y ) J . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_Equ13_HTML.gif
        (13)

        As in Theorem 3.2, we can easily see that N ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq149_HTML.gif maps Ω into itself.

        Let v ( , ) : ( , a ] × ( , b ] R n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq150_HTML.gif be a function defined by
        v ( x , y ) = { ϕ ( x , y ) , ( x , y ) J ˜ , μ ( x , y ) , ( x , y ) J . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_Equan_HTML.gif

        Then v ( x , y ) = ϕ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq151_HTML.gif for all ( x , y ) E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq152_HTML.gif.

        For each w C ( J ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq153_HTML.gif with w ( x , y ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq154_HTML.gif for each ( x , y ) E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq152_HTML.gif, we denote by w ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq155_HTML.gif the function defined by
        w ¯ ( x , y ) = { 0 , ( x , y ) J ˜ , w ( x , y ) , ( x , y ) J . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_Equao_HTML.gif
        If u ( , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq156_HTML.gif satisfies the integral equation, u ( x , y ) = ( N ¯ u ) ( x , y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq157_HTML.gif; ( x , y ) [ α , a ] × [ β , b ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq85_HTML.gif, we can decompose u ( , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq156_HTML.gif as u ( x , y ) = w ¯ ( x , y ) + v ( x , y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq158_HTML.gif; ( x , y ) [ α , a ] × [ β , b ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq85_HTML.gif, which implies u ( x , y ) = w ¯ ( x , y ) + v ( x , y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq159_HTML.gif for every ( x , y ) J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq28_HTML.gif, and the function w ( , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq160_HTML.gif satisfies
        w ( x , y ) = g ( x , y , u ( x , y ) ) + g ( 0 , 0 , w ¯ ( 0 , 0 ) + v ( 0 , 0 ) ) g ( x , 0 , w ¯ ( x , 0 ) + v ( x , 0 ) ) g ( 0 , y , w ¯ ( 0 , y ) + v ( 0 , y ) ) + I θ r f ( x , y , I θ r ( w ¯ ( x , y ) + v ( x , y ) ) , w ¯ ( x , y ) + v ( x , y ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_Equap_HTML.gif
        Set
        C 0 = { w C ( J ) : w ( x , y ) = 0  for  ( x , y ) E } , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_Equaq_HTML.gif
        and let ( a , b ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq161_HTML.gif be the norm in C 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq162_HTML.gif defined by
        w ( a , b ) = sup ( x , y ) E w ( x , y ) B + sup ( x , y ) J w ( x , y ) = sup ( x , y ) J w ( x , y ) , w C 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_Equar_HTML.gif

        C 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq162_HTML.gif is a Banach space with the norm ( a , b ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq161_HTML.gif.

        Note that u Ω http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq163_HTML.gif if and only if w C 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq164_HTML.gif.

        Let the operator P : C 0 C 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq165_HTML.gif be defined by
        ( P w ) ( x , y ) = g ( x , y , u ( x , y ) ) + g ( 0 , 0 , w ¯ ( 0 , 0 ) + v ( 0 , 0 ) ) g ( x , 0 , w ¯ ( x , 0 ) + v ( x , 0 ) ) g ( 0 , y , w ¯ ( 0 , y ) + v ( 0 , y ) ) + 1 Γ ( r 1 ) Γ ( r 2 ) 0 x 0 y ( x s ) r 1 1 ( y t ) r 2 1 × f ( s , t , I θ r ( w ¯ ( s , t ) + v ( s , t ) ) , w ¯ ( s , t ) + v ( s , t ) , g ( s , t ) ) d t d s . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_Equ14_HTML.gif
        (14)
        Then the operator N ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq149_HTML.gif has a fixed point in Ω if and only if P has a fixed point in C 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq162_HTML.gif. 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 ( H 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq144_HTML.gif), it suffices to show that the operator P ¯ : F F http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq166_HTML.gif defined by
        ( P ¯ w ) ( x , y ) = 1 Γ ( r 1 ) Γ ( r 2 ) 0 x 0 y ( x s ) r 1 1 ( y t ) r 2 1 × f ( s , t , I θ r ( w ¯ ( s , t ) + v ( s , t ) ) , w ¯ ( s , t ) + v ( s , t ) , g ( s , t ) ) d t d s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_Equ15_HTML.gif
        (15)

        is continuous and completely continuous. Also, we can show that there exists an open set U F http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq167_HTML.gif with u λ P ( u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq168_HTML.gif for λ ( 0 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq69_HTML.gif and u U http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq169_HTML.gif. Consequently, by Theorem 2.7, we deduce that N ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq149_HTML.gif has a fixed point u in U ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq170_HTML.gif which is a solution to the problem (6)-(8). □

        6 An example

        Consider the following neutral integro-differential equations of fractional order:
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_Equ16_HTML.gif
        (16)
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_Equ17_HTML.gif
        (17)
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_Equ18_HTML.gif
        (18)
        Set
        f ( x , y , u , v ) = 3 + ( x + y ) | u | + x y | v | e 10 ( 1 + | u | + | v | ) ; ( x , y ) [ 0 , 1 ] × [ 0 , 1 ] , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_Equas_HTML.gif
        and
        g ( x , y , u ) = 1 + 2 | u | 4 e x + y + 10 ; ( x , y ) [ 0 , 1 ] × [ 0 , 1 ] . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_Equat_HTML.gif
        We have μ ( x , y ) = x + y 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq171_HTML.gif; ( x , y ) [ 0 , 1 ] × [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq172_HTML.gif. For each u , v R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq173_HTML.gif and ( x , y ) [ 0 , 1 ] × [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq172_HTML.gif, we have
        | f ( x , y , u , v ) | e 10 ( 3 + ( x + y ) | u | + x y | v | ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_Equau_HTML.gif
        and
        | g ( x , y , u ) | 1 4 e 10 ( 1 + 2 | u | ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_Equav_HTML.gif

        Hence, the condition (H1) is satisfied with p = 3 e 10 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq174_HTML.gif, q = 2 e 10 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq175_HTML.gif, d = e 10 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq176_HTML.gif. Also, the condition (H2) is satisfied with L 1 = 1 4 e 10 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq177_HTML.gif and L 2 = 1 2 e 10 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq178_HTML.gif.

        We shall show that the condition (9) holds for each ( r 1 , r 2 ) ( 0 , 1 ] × ( 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq179_HTML.gif with a = b = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq180_HTML.gif. Indeed, Γ ( r i ) > 0.7 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq181_HTML.gif, Γ ( 1 + r i ) > 0.7 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq182_HTML.gif; i = 1 , 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq183_HTML.gif, and L = 1 Γ ( 1 + r 1 ) Γ ( 1 + r 2 ) < 2.1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq184_HTML.gif. Then
        4 L 2 + q L 2 + d L = 2 e 10 + e 10 ( 2 L 2 + L ) < 12.92 e 10 < 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_Equaw_HTML.gif

        Consequently, Theorem 3.2 implies that the problem (16)-(18) has at least one solution defined on [ 1 , 1 ] × [ 2 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-128/MediaObjects/13661_2012_Article_245_IEq185_HTML.gif.

        Declarations

        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 MTM2010-15314.

        Authors’ Affiliations

        (1)
        Laboratoire de Mathématiques, Université de Saïda
        (2)
        Laboratoire de Mathématiques, Université de Sidi Bel-Abbès
        (3)
        Departamento de Análise Matemática, Facultade de Matemáticas, Universidade de Santiago de Compostela

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