Nonlocal nonlinear integrodifferential equations of fractional orders

  • Amar Debbouche1,

    Affiliated with

    • Dumitru Baleanu2, 3Email author and

      Affiliated with

      • Ravi P Agarwal4

        Affiliated with

        Boundary Value Problems20122012:78

        DOI: 10.1186/1687-2770-2012-78

        Received: 11 May 2012

        Accepted: 9 July 2012

        Published: 24 July 2012

        Abstract

        In this paper, Schauder fixed point theorem, Gelfand-Shilov principles combined with semigroup theory are used to prove the existence of mild and strong solutions for nonlinear fractional integrodifferential equations of Sobolev type with nonlocal conditions in Banach spaces. To illustrate our abstract results, an example is given.

        MSC:35A05, 34G20, 34K05, 26A33.

        Keywords

        fractional evolution equation nonlocal condition Schauder’s fixed point theorem uniformly continuous semigroup

        1 Introduction

        We are concerned with the nonlocal nonlinear fractional problem
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_Equ1_HTML.gif
        (1.1)
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_Equ2_HTML.gif
        (1.2)

        where d α d t α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq1_HTML.gif, 0 < α 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq2_HTML.gif is the Riemann-Liouville fractional derivative, 0 t 1 < < t p a http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq3_HTML.gif, c 1 , , c p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq4_HTML.gif are real numbers, B and A are linear closed operators with domains contained in a Banach space X and ranges contained in a Banach space Y, W ( t ) = ( B 1 ( t ) u ( t ) , , B r ( t ) u ( t ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq5_HTML.gif, { B i ( t ) : i = 1 , , r , t I = [ 0 , a ] } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq6_HTML.gif is a family of linear closed operators defined on dense sets S 1 , , S r D ( A ) D ( B ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq7_HTML.gif respectively in X into X, f : I × X r Y http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq8_HTML.gif and g : Δ × X r Y http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq9_HTML.gif are given abstract functions. Here Δ = { ( s , t ) : 0 s t a } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq10_HTML.gif.

        Fractional differential equations have attracted many authors [1, 68, 21, 24, 25, 30]. This is mostly because it efficiently describes many phenomena arising in engineering, physics, economics and science. In fact, we can find several applications in viscoelasticity, electrochemistry, electromagnetic, etc. For example, Machado [22] gave a novel method for the design of fractional order digital controllers.

        Following Gelfand and Shilov [20], we define the fractional integral of order α > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq11_HTML.gif as
        I a α f ( t ) = 1 Γ ( α ) a t ( t s ) α 1 f ( s ) d s , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_Equa_HTML.gif
        also, the (Riemann-Liouville) fractional derivative of the function f of order 0 < α < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq12_HTML.gif as
        D t α a f ( t ) = 1 Γ ( 1 α ) d d t a t ( t s ) α f ( s ) d s , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_Equb_HTML.gif

        where f is an abstract continuous function on the interval [ a , b ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq13_HTML.gif and Γ ( α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq14_HTML.gif is the Gamma function, see also [14, 24].

        The existence results to evolution equations with nonlocal conditions in a Banach space was studied first by Byszewski [9, 10]; subsequently, many authors were pointed to the same field, see for instance [24, 1113, 19, 28].

        Deng [15] indicated that using the nonlocal condition u ( 0 ) + h ( u ) = u 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq15_HTML.gif to describe, for instance, the diffusion phenomenon of a small amount of gas in a transparent tube can give a better result than using the usual local Cauchy problem u ( 0 ) = u 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq16_HTML.gif. Let us observe also that since Deng’s papers, the function h is considered
        h ( u ) = k = 1 p c k u ( t k ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_Equ3_HTML.gif
        (1.3)

        where c k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq17_HTML.gif k = 1 , 2 , , p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq18_HTML.gif are given constants and 0 t 1 < < t p a http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq3_HTML.gif.

        However, among the previous research on nonlocal Cauchy problems, few authors have been concerned with mild solutions of fractional semilinear differential equations [23].

        Recently, many authors have extended this work to various kinds of nonlinear evolution equations [2, 3, 5, 11, 12, 18]. Balachandran and Uchiyama [3] proved the existence of mild and strong solutions of a nonlinear integrodifferential equation of Sobolev type with nonlocal condition.

        In this paper, motivated by [3, 13, 17, 19], we use Schauder fixed point theorem and the semigroup theory to investigate the existence and uniqueness of mild and strong solutions for nonlinear fractional integrodifferential equations of Sobolev type with nonlocal conditions in Banach spaces, the solutions were obtained by using Gelfand-Shilov approach in fractional calculus and are given in terms of some probability density functions such that their Laplace transforms are indicated [17].

        Our paper is organized as follows. Section 2 is devoted to the review of some essential results. In Section 3, we state and prove our main results; the last section deals with giving an example to illustrate the abstract results.

        2 Preliminary results

        In this section, we mention some results obtained by Balachandran [3], El-Borai [19] and Pazy [26], which will be used to get our new results. Let X and Y be Banach spaces with norm | | http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq19_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq20_HTML.gif respectively. The operator B : D ( B ) X Y http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq21_HTML.gif satisfies the following hypotheses: (H1) = B is bijective,; (H2) = B 1 : Y D ( B ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq22_HTML.gif is compact.. The above fact and the closed graph theorem imply the boundedness of the linear operator A B 1 : Y Y http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq23_HTML.gif. Further E = A B 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq24_HTML.gif generates a uniformly continuous semigroup Q ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq25_HTML.gif t 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq26_HTML.gif such that max t I Q ( t ) K http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq27_HTML.gif Q ( t ) h D ( A ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq28_HTML.gif E Q ( t ) h K t h http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq29_HTML.gif for every h X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq30_HTML.gif and all t ( 0 , a ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq31_HTML.gif, see [29].

        Let λ = B 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq32_HTML.gif, c = k = 1 p | c k | http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq33_HTML.gif and Λ τ = { ( u 1 , , u r ) : u i X , i = 1 r | u i | τ } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq34_HTML.gif.

        It is supposed that (H3) = f and g are continuous in t on I, Δ respectively, and there exist constants M 1 , M 2 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq35_HTML.gif such that f ( t , W ) M 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq36_HTML.gif, g ( t , s , W ) M 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq37_HTML.gif for all t I http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq38_HTML.gif, ( s , t ) Δ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq39_HTML.gif and W Λ τ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq40_HTML.gif..

        Definition 2.1 By a strong solution of the nonlocal Cauchy problem (1.1), (1.2), we mean a function u with values in X such that
        1. (i)

          u is a continuous function in t I http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq38_HTML.gif and u ( t ) D ( A ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq41_HTML.gif,

           
        2. (ii)

          d α u d t α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq42_HTML.gif exists and is continuous on ( 0 , a ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq43_HTML.gif, 0 < α < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq12_HTML.gif, and u satisfies (1.1) on ( 0 , a ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq43_HTML.gif and (1.2).

           
        Remark 2.1 Let us take in the considered problem B is the identity, the inhomogeneous part is equal to an abstract continuous function f ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq44_HTML.gif, and the nonlocal condition is reduced to the initial condition u ( 0 ) = u 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq45_HTML.gifi.e.
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_Equ4_HTML.gif
        (2.1)
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_Equ5_HTML.gif
        (2.2)
        According to El-Borai [1719], we first apply the fractional integral on both sides of (2.1) and then using (2.2), we apply the Laplace transform on the new integral equations by considering a suitable one-sided stable probability density whose Laplace transform is given. Hence we can conclude that a solution of the problem (2.1)-(2.2) can be formally represented by
        u ( t ) = 0 ζ α ( θ ) Q ( t α θ ) u 0 d θ + α 0 t 0 θ ( t s ) α 1 ζ α ( θ ) Q ( ( t s ) α θ ) f ( s ) d θ d s , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_Equ6_HTML.gif
        (2.3)
        where ζ α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq46_HTML.gif is a probability density function defined on ( 0 , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq47_HTML.gif such that its Laplace transform is given by
        0 e θ x ζ α ( θ ) d θ = j = 0 ( x ) j Γ ( 1 + α j ) , 0 < α 1 , x > 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_Equc_HTML.gif

        For more details, we refer to Zhou et al.[27, 31], see also [14, 16].

        Using Gelfand-Shilov principle [20], it is suitable to rewrite (1.1), (1.2) in the form
        B u ( t ) = B u ( 0 ) + 1 Γ ( α ) 0 t ( t η ) α 1 × [ A u ( η ) + f ( η , W ( η ) ) + 0 η g ( η , s , W ( s ) ) d s ] d η , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_Equ7_HTML.gif
        (2.4)

        where Γ ( α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq48_HTML.gif is the Gamma function.

        According to [1719], the equation (2.4) is equivalent to the integral equation
        B u ( t ) = Ψ ( t ) B u ( 0 ) + 0 t Φ ( t η ) [ f ( η , W ( η ) ) + 0 η g ( η , s , W ( s ) ) d s ] d η , t > 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_Equ8_HTML.gif
        (2.5)
        where
        Ψ ( t ) = 0 ζ α ( θ ) Q ( t α θ ) d θ , Φ ( t ) = α 0 θ t α 1 ζ α ( θ ) Q ( t α θ ) d θ . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_Equd_HTML.gif
        It is assumed that there exists an operator ψ on D ( ψ ) = X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq49_HTML.gif given by the formula
        ψ = [ I + k = 1 p c k B 1 Ψ ( t k ) B ] 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_Eque_HTML.gif
        satisfying ψ u 0 D ( B ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq50_HTML.gif and for k = 1 , , p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq51_HTML.gif
        ψ 0 t k B 1 Φ ( t k η ) [ f ( η , W ( η ) ) + 0 η g ( η , s , W ( s ) ) d s ] d η D ( B ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_Equf_HTML.gif
        also (H4) = K λ B ψ u 0 [ λ 2 K 2 c a α B ψ + λ K a α ] ( M 1 + a M 2 ) τ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq52_HTML.gif.. Further we assume (H5) = There is a number γ ( 0 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq53_HTML.gif such that
        B i ( t 2 ) Q ( t 1 ) h K 1 t 1 γ h , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_Equg_HTML.gif

        where t 1 ( 0 , a ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq54_HTML.gif, t 2 I http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq55_HTML.gif, h X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq30_HTML.gif and K 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq56_HTML.gif is a positive constant, i = 1 , , r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq57_HTML.gif.; (H6) = The functions B 1 ( t ) h , , B r ( t ) h http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq58_HTML.gif are uniformly Hölder continuous in t I http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq38_HTML.gif for every element h in i S i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq59_HTML.gif.. Suppose that { Q ( t ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq60_HTML.gif is a C 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq61_HTML.gif-semigroup of operators on X such that B 1 Q ( t k ) B C e δ t k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq62_HTML.gif, where δ is a positive constant and C 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq63_HTML.gif. Noting that 0 ζ α ( θ ) d θ = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq64_HTML.gif (see [14], p.4]).

        If k = 1 p | c k | e δ t k < 1 C http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq65_HTML.gif, then k = 1 p c k B 1 ψ ( t k ) B < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq66_HTML.gif, which achieves that ψ exists on X.

        3 Main results

        The following is different from [3, 19, 26] and represents the new result.

        Lemma 3.1 If u is a continuous solution of (2.5), then u satisfies the integral equation
        u ( t ) = B 1 Ψ ( t ) B ψ u 0 k = 1 p c k B 1 Ψ ( t ) B ψ × 0 t k B 1 Φ ( t k s ) [ f ( s , W ( s ) ) + 0 s g ( s , η , W ( η ) ) d η ] d s + 0 t B 1 Φ ( t s ) [ f ( s , W ( s ) ) + 0 s g ( s , η , W ( η ) ) d η ] d s . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_Equ9_HTML.gif
        (3.1)
        Proof Using (2.5) and (1.2), we get
        k = 1 p c k B u ( t k ) = k = 1 p c k Ψ ( t k ) B u 0 k = 1 p c k Ψ ( t k ) B k = 1 p c k u ( t k ) + k = 1 p c k 0 t k Φ ( t k s ) [ f ( s , W ( s ) ) + 0 s g ( s , η , W ( η ) ) d η ] d s . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_Equh_HTML.gif
        Then
        k = 1 p c k u ( t k ) [ I + k = 1 p c k B 1 Ψ ( t k ) B ] = k = 1 p c k B 1 Ψ ( t k ) B u 0 + k = 1 p c k 0 t k B 1 Φ ( t k s ) × [ f ( s , W ( s ) ) + 0 s g ( s , η , W ( η ) ) d η ] d s . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_Equi_HTML.gif
        Thus
        Ψ ( t ) B u ( 0 ) = Ψ ( t ) [ B u 0 k = 1 p c k B u ( t k ) ] = Ψ ( t ) B u 0 Ψ ( t ) B ψ k = 1 p c k B 1 Ψ ( t k ) B u 0 Ψ ( t ) B ψ k = 1 p c k 0 t k B 1 Φ ( t k s ) [ f ( s , W ( s ) ) + 0 s g ( s , η , W ( η ) ) d η ] d s = Ψ ( t ) B ψ u 0 [ ψ 1 k = 1 p c k B 1 Ψ ( t k ) B ] Ψ ( t ) B ψ k = 1 p c k 0 t k B 1 Φ ( t k s ) [ f ( s , W ( s ) ) + 0 s g ( s , η , W ( η ) ) d η ] d s . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_Equj_HTML.gif

        Hence the required result. □

        Definition 3.1 A continuous solution of the integral equation (3.1) is called a mild solution of the nonlocal problem (1.1), (1.2) on I.

        Theorem 3.2 If the assumptions (H1)∼(H4) hold and W ( t ) = u ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq67_HTML.gif, then the problem (1.1), (1.2) has a mild solution on I.

        Proof Let Z = C ( I , X ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq68_HTML.gif and Z 0 = { u Z : u ( t ) Λ τ , t I } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq69_HTML.gif. It is easy to see that Z 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq70_HTML.gif is a bounded closed convex subset of Z. We define a mapping φ : Z 0 Z 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq71_HTML.gif by
        ( φ u ) ( t ) = B 1 Ψ ( t ) B ψ u 0 k = 1 p c k B 1 Ψ ( t ) B ψ 0 t k B 1 Φ ( t k s ) [ f ( s , W ( s ) ) + 0 s g ( s , η , W ( η ) ) d η ] d s + 0 t B 1 Φ ( t s ) [ f ( s , W ( s ) ) + 0 s g ( s , η , W ( η ) ) d η ] d s , t I . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_Equk_HTML.gif
        Noting also that 0 θ ζ α ( θ ) d θ = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq72_HTML.gif (see [14], p.4]), we have
        ( φ u ) ( t ) K λ B ψ u 0 + ( M 1 + a M 2 ) λ 2 K 2 c a α B ψ + ( M 1 + a M 2 ) λ K a α τ . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_Equl_HTML.gif
        We deduce that φ is continuous and maps Z 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq70_HTML.gif into itself. Moreover, φ maps Z 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq70_HTML.gif into a precompact subset of Z 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq70_HTML.gif. Note that the set Z 0 ( t ) = { ( φ u ) ( t ) : u Z 0 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq73_HTML.gif is precompact in X, for every fixed t I http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq38_HTML.gif. We shall show that φ ( Z 0 ) = S = { φ u : u Z 0 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq74_HTML.gif is an equicontinuous family of functions. For 0 < s < t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq75_HTML.gif, we have
        ( φ u ) ( t ) ( φ u ) ( s ) [ λ B ψ u 0 + c λ 2 K a α ( M 1 + a M 2 ) B ψ ] Ψ ( t ) Ψ ( s ) + λ ( M 1 + a M 2 ) s t Φ ( t η ) d η + λ ( M 1 + a M 2 ) 0 s Φ ( t η ) Φ ( s η ) d η . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_Equm_HTML.gif

        The right-hand side of the above inequality is independent of u Z 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq76_HTML.gif and tends to zero as s t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq77_HTML.gif as a consequence of the continuity of Ψ ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq78_HTML.gif and Φ ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq79_HTML.gif in the uniform operator topology for t > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq80_HTML.gif. It is clear that S is bounded in Z. Thus by Arzela-Ascoli’s theorem, S is precompact. Hence by the Schauder fixed point theorem, φ has a fixed point in Z 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq70_HTML.gif and any fixed point of φ is a mild solution of (1.1), (1.2) on I such that u ( t ) X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq81_HTML.gif for all t I http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq38_HTML.gif. □

        Theorem 3.3 Assume that
        1. (i)

          Conditions (H 1)∼(H 6) hold,

           
        2. (ii)

          Y is a reflexive Banach space with norm http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq82_HTML.gif,

           
        3. (iii)
          there are numbers L 1 , L 2 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq83_HTML.gif and p , q ( 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq84_HTML.gif such that
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_Equn_HTML.gif
           

        for all t 1 , t 2 I http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq85_HTML.gif, ( s 1 , η ) , ( s 2 , η ) Δ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq86_HTML.gifand all W , W Λ τ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq87_HTML.gif, where w i = B i u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq88_HTML.gifand w i = B i u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq89_HTML.gif.

        Then the problem (1.1), (1.2) has a unique strong solution on I.

        Proof Applying Theorem 3.2, the problem (1.1), (1.2) has a mild solution u C ( I , Λ τ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq90_HTML.gif. Now, we shall show that u is a unique strong solution of the considered problem on I.

        According to (H6), i = 1 r | w i w i | http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq91_HTML.gif is uniformly Hölder continuous in t I http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq38_HTML.gif for every element u in i S i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq59_HTML.gif combined with (iii), which implies that t f ( t , W ( t ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq92_HTML.gif and t 0 t g ( t , s , W ( s ) ) d s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq93_HTML.gif are uniformly Hölder continuous on I.

        Set
        V ( t ) = f ( t , W ( t ) ) + 0 t g ( t , s , W ( s ) ) d s . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_Equo_HTML.gif
        From (3.1), the solution u of the considered problem can be written in the form
        u ( t ) = B 1 Ψ ( t ) B ψ u 0 B 1 Ψ ( t ) B ψ k = 1 p c k 0 t k B 1 Φ ( t k s ) V ( s ) d s + 0 t B 1 Φ ( t s ) V ( s ) d s . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_Equp_HTML.gif
        Noting that Ψ and ψ are bounded, using our assumptions, we observe that there exists a unique function V C ( I , X ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq94_HTML.gif which satisfies the equation
        d α ( B u ( t ) ) d t α + A u ( t ) = V ( t ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_Equq_HTML.gif
        Also as in [19], p.409], we deduce that
        0 t B 1 Φ ( t s ) V ( s ) d s D ( E ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_Equr_HTML.gif

        for all t I http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq38_HTML.gif and ψ u 0 D ( E ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq95_HTML.gif. It follows that u ( t ) D ( E ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq96_HTML.gif for all t I http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq38_HTML.gif. □

        4 Example

        Consider the nonlinear integro-partial differential equation of fractional order
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_Equ10_HTML.gif
        (4.1)
        with nonlocal condition
        u ( x , 0 ) + k = 1 p c k u ( x , t k ) = g ( x ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_Equ11_HTML.gif
        (4.2)
        where 0 < α 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq2_HTML.gif, 0 t 1 < < t p a http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq3_HTML.gif, x R n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq97_HTML.gif, D x q = D x 1 q 1 D x n q n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq98_HTML.gif, D x i = x i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq99_HTML.gif, q = ( q 1 , , q n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq100_HTML.gif is an n-dimensional multi-index, | q | = q 1 + + q n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq101_HTML.gif, W = ( w 1 , , w r ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq102_HTML.gif,
        w i ( x , t ) = | q | 2 m 1 b q i ( x , t ) D x q u ( x , t ) + Ω | q | 2 m 1 c q i ( x , t ) D y q u ( y , t ) d y , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_Equs_HTML.gif
        and Ω is an open subset of R n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq103_HTML.gif. Let L 2 ( R n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq104_HTML.gif be the set of all square integrable functions on R n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq103_HTML.gif. We denote by C m ( R n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq105_HTML.gif the set of all continuous real-valued functions defined on R n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq103_HTML.gif which have continuous partial derivatives of order less than or equal to m. By C 0 m ( R n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq106_HTML.gif we denote the set of all functions f C m ( R n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq107_HTML.gif with compact supports. Let H m ( R n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq108_HTML.gif be the completion of C 0 m ( R n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq106_HTML.gif with respect to the norm
        f m 2 = | q | m R n | D x q f ( x ) | 2 d x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_Equt_HTML.gif
        It is supposed that
        1. (i)
          The operator E = | q | = 2 m e q ( x ) D x q http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq109_HTML.gif is uniformly elliptic on R n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq103_HTML.gif. In other words, all the coefficients e q http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq110_HTML.gif, | q | = 2 m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq111_HTML.gif are continuous and bounded on R n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq103_HTML.gif, and there is a positive number c such that
          ( 1 ) m + 1 | q | = 2 m e q ( x ) ξ q c | ξ | 2 m , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_Equu_HTML.gif
           

        for all x R n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq97_HTML.gif and all ξ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq112_HTML.gif, ξ R n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq113_HTML.gif, where e q = a q b q 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq114_HTML.gif, ξ q = ξ 1 q 1 ξ n q n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq115_HTML.gif and | ξ | 2 = ξ 1 2 + + ξ n 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq116_HTML.gif.

        (ii) All the coefficients e q http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq110_HTML.gif | q | = 2 m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq111_HTML.gif, satisfy a uniform Hölder condition on R n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq103_HTML.gif. Under these conditions, the operator E with the domain of definition D ( E ) = H 2 m ( R n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq117_HTML.gif generates an analytic semigroup Q ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq25_HTML.gif defined on L 2 ( R n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq104_HTML.gif, and it is well known that H 2 m ( R n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq118_HTML.gif is dense in Y = L 2 ( R n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq119_HTML.gif, see [17], p.438].

        Lemma 4.1 The solution representation of (4.1), (4.2) can be written explicitly.

        Proof Let { E q ( x ) : | q | 2 m } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq120_HTML.gif be a family of deterministic square matrices of order k and let L ( x , D ) = { E q ( x ) : | q | = 2 m } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq121_HTML.gif. We assume that
        det { ( 1 ) m L ( x , σ ) λ I } = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_Equv_HTML.gif

        has roots which satisfy the inequality Re λ < δ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq122_HTML.gif, δ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq123_HTML.gif for all x R n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq97_HTML.gif and for any real vector σ, σ 1 2 + + σ n 2 = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq124_HTML.gif. If Θ is a matrix of order m × n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq125_HTML.gif, then we introduce | Θ | = i , j | b i j | http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq126_HTML.gif.

        It is well known that there exists a fundamental matrix solution Z ( x , y , t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq127_HTML.gif which satisfies the system
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_Equw_HTML.gif
        This fundamental matrix also satisfies the inequality
        | D x q Z ( x , y , t ) | K 1 t ρ 1 exp ( K 2 ρ 2 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_Equx_HTML.gif
        where | q | 2 m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq128_HTML.gif ρ 1 = n + | q | 2 m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq129_HTML.gif ρ 2 = i = 1 n | x i y i | λ t 1 2 m 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq130_HTML.gif λ = 2 m 2 m 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq131_HTML.gif and K 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq56_HTML.gif K 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq132_HTML.gif are positive constants. From [13], p.58], if the nonlocal function g ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq133_HTML.gif is an element in Hilbert space H 2 m ( R n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq118_HTML.gif, then we can write
        Q ( t ) g ( x ) = R n Z ( x y , t ) g ( y ) d y . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_Equy_HTML.gif
        It can be proved that
        D x q Q ( t ) g M t β g , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_Equz_HTML.gif

        where 0 < β < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq134_HTML.gifM is a positive constant, | q | 2 m 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq135_HTML.gif t > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq80_HTML.gif and g 2 = R n g 2 ( x ) d x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq136_HTML.gif.

        ([17]) The nonlocal Cauchy problems (4.1), (4.2) are equivalent to the integral equation
        u ( x , t ) = 0 R n ζ α ( θ ) Q ( x ξ , t α θ ) u ( ξ , 0 ) d ξ d θ + α 0 t 0 R n θ ( t η ) α 1 ζ α ( θ ) Q ( x ξ , ( t η ) α θ ) × [ F ( ξ , η , W ) + 0 η G ( ξ , η , s , W ( s ) ) d s ] d ξ d θ d η , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_Equaa_HTML.gif
        where the explicit form of Q is given by
        Q ( x , t ) = e | x | 2 / 4 t ( 4 π t ) n , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_Equab_HTML.gif
        | x | 2 = x 1 2 + x 2 2 + + x n 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq137_HTML.gif. Applying Theorem 3.2, we achieve the proof of the existence of mild solutions of the problems (4.1), (4.2). In addition, if the operators F and G satisfy the following:
        1. (iii)
          There are numbers L 1 , L 2 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq138_HTML.gif and 0 < p , q 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq139_HTML.gif such that
          | q | 2 m 1 R n | F ( x , t , D x q W ) F ( x , s , D x q W ) | 2 d x L 1 ( | t s | p + i = 1 r | w i w i | 2 d x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_Equac_HTML.gif
           
        and
        | q | 2 m 1 R n | G ( x , t , η , D x q W ) G ( x , s , η , D x q W ) | 2 d x L 2 | t s | q http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_Equad_HTML.gif

        for all t , s I http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq140_HTML.gif, ( t , η ) , ( s , η ) Δ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq141_HTML.gif, W , W Λ τ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq87_HTML.gif and all x R n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-78/MediaObjects/13661_2012_Article_176_IEq97_HTML.gif. Then applying Theorem 3.3, we deduce that (4.1), (4.2) has a unique strong solution. □

        5 Conclusion

        In this article, a new solution representation for Sobolev type fractional evolution equation has been proved using Deng’s nonlocal condition, a suitable explicit form of the semigroup has been discussed. Moreover, the existence result of mild solutions for nonlinear fractional integrodifferential equations of Sobolev type with nonlocal conditions in Banach spaces has been established by using Arzela-Ascoli’s theorem and Schauder fixed point theorem. Further, the uniformly Hölder continuous condition has been applied for the existence of strong solution.

        Declarations

        Acknowledgements

        The authors would like to thank the referees for their valuable comments and remarks.

        Authors’ Affiliations

        (1)
        Department of Mathematics, Faculty of Science, Guelma University
        (2)
        Department of Mathematics and Computer Science, Cankaya University
        (3)
        Institute of Space Sciences
        (4)
        Department of Mathematics, Texas A&M University

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