On the existence of mild solutions to the Cauchy problem for a class of fractional evolution equation

  • Jin Liang1Email author,

    Affiliated with

    • Sheng-Hua Yan1,

      Affiliated with

      • Fang Li2 and

        Affiliated with

        • Ting-Wen Huang3

          Affiliated with

          Advances in Difference Equations20122012:40

          DOI: 10.1186/1687-1847-2012-40

          Received: 29 January 2012

          Accepted: 4 April 2012

          Published: 4 April 2012

          The Erratum to this article has been published in Advances in Difference Equations 2015 2015:262

          Abstract

          We are concerned with the existence of mild solutions to the Cauchy problem for fractional evolution equations of neutral type with almost sectorial operators

          d q d t q ( x ( t ) - h ( t , x ( t ) ) ) = - A ( x ( t ) - h ( t , x ( t ) ) ) + f ( t , x ( t ) ) , t > 0 , x (0) =  x 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_Equa_HTML.gif

          where 0 < q < 1, the fractional derivative is understood in the Caputo sense, A is an almost sectorial operator on a complex Banach space, and f, h are given functions. With the help of the theory of measure of noncompactness and a fixed point theorem of Darbo type, we establish a new existence theorem of mild solutions for the Cauchy problem above. By the way, the global attractive property of the solutions is also obtained. Moreover, we give two examples to illustrate our abstract results.

          Keywords

          fractional evolution equations mild solutions almost sectorial operators neutral type measure of noncompactness global attractive

          1 Introduction

          The fractional evolution equations have received increasing attention during recent years and have been studied extensively (see, e.g., [113] and references therein) since they can be used to describe many phenomena arising in engineering, physics, economy, and science.

          We mention that much of the previous research on the evolution equations was done provided that the operator in the linear part is the infinitesimal generator of a strongly continuous operator semigroup, an analytic semigroup, or a compact semigroup, or is a Hille-Yosida operator (see, e.g., [112, 14, 15] and references therein). On the other hand, when the operator in the linear part is an almost sectorial operator, for which the resolvent operators do not satisfy the required estimate to be a sectorial operator (see the example of almost sectorial operators which are not sectorial given by von Wahl in [16]), much less is known about the fractional evolution equations of neutral type with almost sectorial operators.

          In this article, we will pay our attentions to the existence of mild solutions to the following Cauchy problem for fractional evolution equations of neutral type with almost sectorial operators
          d q d t q ( x ( t ) - h ( t , x ( t ) ) ) = - A ( x ( t ) - h ( t , x ( t ) ) ) + f ( t , x ( t ) ) , t > 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_Equ1_HTML.gif
          (1.1)
          x ( 0 ) = x 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_Equ2_HTML.gif
          (1.2)

          where 0 < q < 1, the fractional derivative is understood in the Caputo sense, A is an almost sectorial operator on a complex Banach space, and f, h are given functions. We will use the theory of measure of noncompactness and a fixed point theorem of Darbo type to establish a new existence theorem for the problem (1.1)-(1.2). By the way, the global attractive property of the solutions are also obtained. Moreover, we give two examples to illustrate our abstract results.

          This article is organized as follows: In Section 2, we state some basic concepts, notations and properties about fractional order operator and measure of noncompactness. A new existence result and the global attractive property of the solutions will be given and proved in Section 3. Finally, in Section 4, we present two concrete examples, whose physical background is statistical physics and fractional quantum mechanics (see, e.g., [12, 13]).

          2 Basic concepts, notations and lemmas

          Let X be a complex Banach space with norm ||·|| and B(x, r) denote the closed ball centered at x and with radius r. Suppose M X http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_IEq1_HTML.gif denotes the family of all nonempty and bounded subsets of X and subfamily consisting of all relatively compact sets is denoted by N X http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_IEq2_HTML.gif. As usual, for a linear operator A, we denote by D(A) the domain of A, by the family R(z; A) = (zI - A)-1, zρ(A) of bounded linear operators the resolvent of A. Moreover, we denote by L(X, X) the space of all bounded linear operators from Banach space X to X with the usual operator norm ||·||L(X, X), and we abbreviate this notation to L(X).

          Definition 2.1[12] The fractional integral of order q with the lower limit zero for a function fAC[0, ∞) is defined as
          I q f ( t ) = 1 Γ ( q ) 0 t ( t - s ) q - 1 f ( s ) d s , t > 0 , 0 < q < 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_Equb_HTML.gif

          provided the right side is point-wise defined on [0, ∞), where Γ(·) is the gamma function.

          Definition 2.2[12] Riemann-Liouville derivative of order q with the lower limit zero for a function fAC[0, ∞) can be written as
          L D q f ( t ) = 1 Γ ( 1 - q ) d d t 0 t ( t - s ) - q f ( s ) d s , t > 0 , 0 < q < 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_Equc_HTML.gif
          Definition 2.3[12] The Caputo derivative of order q for a function fAC[0, ∞) can be written as
          c D q f ( t ) = L D q ( f ( t ) - f ( 0 ) ) , t > 0 , 0 < q < 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_Equd_HTML.gif

          where c D q : = d q d t q http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_IEq3_HTML.gif.

          Next, we recall the concept of measure of noncompactness (cf. [17]).

          Definition 2.4 μ M X R + http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_IEq4_HTML.gif is said to be a measure of noncompactness in X if it satisfies the following conditions:
          1. (1)

            the family Ker μ = { Ω M X ; μ Ω = 0 } http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_IEq5_HTML.gif is nonempty and Ker μ N X http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_IEq6_HTML.gif ;

             
          2. (2)

            Ω ⊂ Ω0μ(Ω) ≤ μ0), for Ω and Ω 0 M X http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_IEq7_HTML.gif ;

             
          3. (3)

            μ(Conv(Ω)) = μ(Ω), where Conv(Ω) denotes the convex hull of Ω;

             
          4. (4)

            μ ( Ω ̄ ) = μ ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_IEq8_HTML.gif, where Ω ̄ http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_IEq9_HTML.gif denotes the closure of Ω M X http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_IEq10_HTML.gif ;

             
          5. (5)

            μ(λ Ω + (1 - λ0) ≤ λμ(Ω) + (1 - λ)μ0), for λ ∈ [0, 1] and any Ω , Ω 0 M X http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_IEq11_HTML.gif;

             
          6. (6)

            If n } is a sequence of sets from M X http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_IEq1_HTML.gif such that Ωn+l⊂ Ω n , Ω ̄ n = Ω n ( n = 1 , 2 , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_IEq12_HTML.gif, and if limn→ μ n ) = 0, then the intersection Ω = n = 1 Ω n http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_IEq13_HTML.gif is nonempty.

             

          The following is a fixed point theorem of Darbo type (see [17]).

          Lemma 2.5 Let M http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_IEq14_HTML.gifbe a nonempty, bounded, closed and convex subset of a Banach space X, and let H : M M http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_IEq15_HTML.gifbe a continuous mapping. Assume that there exists a constant k ∈ [0, 1), such that
          μ ( H ( Ω ) ) k μ ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_Eque_HTML.gif

          for any nonempty subset Ω of M http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_IEq14_HTML.gif. Then H has a fixed point in M http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_IEq14_HTML.gif.

          Let -1 < γ < 0, and S μ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_IEq16_HTML.gif with 0 < μ < π be the open sector
          { z C \ { 0 } ; arg  z < μ } http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_Equf_HTML.gif
          and be its closure, that is
          S μ = { z C \ { 0 } ; arg  z μ } { 0 } , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_Equg_HTML.gif

          for more details, we refer to [18, 19].

          As in [18], we state the concept of almost sectorial operators as follows.

          Definition 2.6 Let - 1 < γ < 0 and 0 < ω < π 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_IEq17_HTML.gif. By Θ ω γ ( X ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_IEq18_HTML.gifwe denote the family of all linear closed operators A: D(A) ⊂ X → X which satisfy
          1. (1)

            σ(A) ⊂ S ω = {zC\{0}; | arg z| ≤ ω} ∪ {0} and

             
          2. (2)
            for every ω < μ < π there exists a constant C μ such that
            R ( z ; A ) L ( X ) C μ z γ , f o r a l l z C \ S μ . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_Equh_HTML.gif
             

          A linear operator A will be called an almost sectorial operator on X if A Θ ω γ ( X ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_IEq19_HTML.gif.

          Remark 2.7 Let A Θ ω γ ( X ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_IEq19_HTML.gif. Then the definition implies that 0 ∈ ρ(A).

          We denote the semigroup associated with A by {T (t)}t≥0.For t S π 2 - ω 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_IEq20_HTML.gif,
          T ( t ) = e - t z ( A ) = 1 2 π i Γ θ e - t z R ( z ; A ) d z , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_Equi_HTML.gif

          here ω < θ < μ < π 2 - arg  t http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_IEq21_HTML.gif, forms an analytic semigroup of growth order 1 + γ. We have the following lemma on T (t) [[19], Theorem 3.9].

          Lemma 2.8 Let A Θ ω γ ( X ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_IEq19_HTML.gifwith - 1 < γ < 0 and 0 < ω < π 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_IEq17_HTML.gifThen
          1. (i)
            T(t) is analytic in S π 2 - ω 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_IEq22_HTML.gif and
            d n d t n T ( t ) = ( - A ) n T ( t ) , f o r a l l t S π 2 - ω 0 ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_Equj_HTML.gif
             
          2. (ii)

            T(s + t) = T(s) T(t) for all s , t S π 2 - ω 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_IEq23_HTML.gif ;

             
          3. (ii)
            There exists a constant C 0 = C 0(γ) > 0 such that
            T ( t ) L ( X ) C 0 t - γ - 1 , f o r a l l t > 0 ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_Equk_HTML.gif
             
          4. (iv)
            The range R(T(t)) of T(t) for each t S π 2 - ω 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_IEq24_HTML.gif is contained in D(A ). Particularly, for all αC with Reβ > 0, R(T(t ))D(A β ) and
            A β T ( t ) x = 1 2 π i Γ θ z β e - t z R ( z ; A ) x d z , f o r a l l x X , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_Equl_HTML.gif
             
          and hence there exists a constant C'= C'(γ, β) > 0 such that
          A β T ( t ) L ( X ) C t - γ - R e β - 1 , f o r a l l t > 0 ; http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_Equm_HTML.gif
          1. (v)
            If β > 1 + γ, then D(A β ) ⊂ Σ T , where Σ T is the continuity set of the semigroup {T (t)}t ≥ 0, that is,
            Σ T = x X ; lim t 0 ; t > 0 T ( t ) x = x . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_Equn_HTML.gif
             

          Clearly, we note that the condition (ii) of the Lemma 2.8 does not satisfy for t = 0 or s = 0.

          The relation between the resolvent operators of A and the semigroup T(t) is characterized by

          Lemma 2.9 [[19], Theorem 3.13] Let A Θ ω γ ( X ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_IEq19_HTML.gifwith - 1 < γ < 0 and 0 < ω < π 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_IEq17_HTML.gif. Then for every λC with Reλ > 0, one has
          R ( λ ; - A ) = 0 e - λ t T ( t ) d t . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_Equo_HTML.gif

          Now, we give the definition of mild solution to (1.1)-(1.2).

          Definition 2.10 A continuous function x: (0, T ] → X satisfying the equation
          x ( t ) = S q ( t ) x 0 + h ( t , x ( t ) ) + 0 t ( t - s ) q - 1 P q ( t - s ) f ( s , x ( s ) ) d s http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_Equp_HTML.gif
          for t ∈ (0, T ] is called a mild solution of (1.1)-(1.2), where
          S q ( t ) x = 0 Ψ q ( σ ) T ( σ t q ) x d σ , t S π 2 - ω 0 , x X , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_Equq_HTML.gif
          P q ( t ) x = 0 q σ Ψ q ( σ ) T ( σ t q ) x d σ , t S π 2 - ω 0 , x X , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_Equr_HTML.gif
          and Ψ q (σ) is the function of Wright type such that
          Ψ q ( z ) : = n = 0 ( - z ) n n ! Γ ( - q n + 1 - q ) = 1 π n = 1 ( - z ) n ( n - 1 ) ! Γ ( n q ) sin ( n π q ) , z C , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_Equs_HTML.gif

          with 0 < q < 1.

          Remark 2.11 [[13], Remark 4.1] For every x0D(A β ) (β > 1 + γ), this mild solution (if any) is continuous at t = 0.

          Remark 2.12[13] It is not difficult to verify that for -1 < r < ∞, λ > 0 and -1 < α + γ < 0,
          1. (1)

            Ψ q (t) 0, t > 0;

             
          2. (2)

            0 Ψ q ( t ) t r d t = Γ ( 1 + r ) Γ ( 1 + q r ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_IEq25_HTML.gif.

             
          Then we have
          S q ( t ) x C 0 Γ ( - γ ) Γ ( 1 - q ( 1 + γ ) ) t - q ( 1 + γ ) x , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_Equ3_HTML.gif
          (2.1)
          P q ( t ) x q C 0 Γ ( 1 - γ ) Γ ( 1 - q γ ) t - q ( 1 + γ ) x , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_Equ4_HTML.gif
          (2.2)
          A α P q ( t ) x 0 q σ Ψ q ( σ ) A α T ( σ t q ) d σ x q C 0 Ψ q ( σ ) t - q ( γ + α + 1 ) σ - γ - α d σ x q C Γ ( 1 - γ - α ) Γ ( 1 - q ( γ + α ) ) t - q ( γ + α + 1 ) x . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_Equ5_HTML.gif
          (2.3)

          Lemma 2.13 [[13], Theorem 3.2] For t > 0, S q t http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_IEq26_HTML.gifand P q t http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_IEq27_HTML.gifare continuous in the uniform operator topology.

          Let
          X α = D ( A α ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_Equt_HTML.gif
          and let BC(R+,, X α ) denote the Banach space consisting of all real functions defined bounded and continuous from R+ to X α with the norm
          x = sup t R + x α , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_Equu_HTML.gif

          for xBC(R+, X α ).

          It is clear that D(A β ) ⊂ D(A α ).

          Next, we present a measure of noncompactness introduced in [17].

          For any nonempty and bounded subset Y of the space BC(R+, X) and a positive number T, we denote ωT (x, ε) as the modulus of continuity of function x on the interval [0, T ], where xY and ε ≥ 0. Namely,
          ω T ( x , ε ) = sup { x ( t ) - x ( s ) ; t , s [ 0 , T ] , t - s ε } . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_Equv_HTML.gif
          We then assume additionally
          ω T ( Y , ε ) = sup { ω T ( x , ε ) ; x Y } , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_Equw_HTML.gif
          ω 0 T ( Y ) = lim ε 0 ω T ( Y , ε ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_Equx_HTML.gif
          ω 0 ( Y ) = lim T ω 0 T ( Y ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_Equy_HTML.gif
          and
          diam ( Y ) = sup { x ( t ) - y ( t ) ; x , y Y } . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_Equz_HTML.gif
          Finally, consider the function μ defined on the family M B C ( R + , X ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_IEq28_HTML.gif by the formula:
          μ ( Y ) = ω 0 ( Y ) + lim sup t diam(Y) . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_Equ6_HTML.gif
          (2.4)

          It is known that μ is a measure of noncompactness.

          Definition 2.14 The solution x(t) of (1.1)-(1.2) is said to be globally attractive, if
          lim t ( x ( t ) - y ( t ) ) = 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_Equaa_HTML.gif

          for any solution y(t) of equation (1.1)-(1.2).

          3 Main result

          In this section, we assume -1 < α + γ < 0 and 0 < α < β < 1.

          Theorem 3.1 Let A Θ ω γ ( X ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_IEq19_HTML.gifand 0 < ω < π 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_IEq17_HTML.gif. Assume that

          (H1) f: R+× X α → X is continuous, and there exists a positive function ν(·): R+ R+such that
          f ( t , x ) ν ( t ) , t h e f u n c t i o n s ν ( s ) ( t - s ) 1 +  q ( γ  +  α ) b e l o n g s t o L 1 ( [ 0 , t ] , R  +  ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_Equ7_HTML.gif
          (3.1)
          lim t η ( t ) : = lim t 0 t ν ( s ) ( t - s ) 1 + q ( γ + α ) = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_Equ8_HTML.gif
          (3.2)
          (H2) The function hBC(R+, X α ) and there exists a constant L ∈ (0, 1) such that
          h ( t 1 , x ( t 1 ) ) - h ( t 2 , x ( t 2 ) ) α L ( t 1 - t 2 + x ( t 1 ) - x ( t 2 ) α ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_Equab_HTML.gif
          (H3) For each nonempty, bounded set DBC(R+, X α ), the family of functions
          { t h ( t , φ ) ; φ D } http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_Equac_HTML.gif

          is equicontinuous.

          Then
          1. (1)

            for every x 0D(A β ) with β > 1 + γ, the problem (1.1)-(1.2) has at least a mild solution on BC(R +, X α );

             
          2. (2)

            all solutions are globally attractive.

             
          Proof. Consider the operator as follows:
          ( H x ) ( t ) = S q ( t ) x 0 + h ( t , x ( t ) ) + 0 t ( t - s ) q - 1 P q ( t - s ) f ( s , x ( s ) ) d s , t 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_Equad_HTML.gif
          Step 1: We prove that there exists a ball
          B r = { x B C ( R + , X α ) ; x r } http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_Equae_HTML.gif

          with radius r and centered at 0, such that H(B r ) ⊂ B r .

          For any r > 0 and xB r , in view of (H2),
          h ( t , x ( t ) ) α h ( t , x ( t ) ) - h ( t , 0 ) α + h ( t , 0 ) α L r + M 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_Equ9_HTML.gif
          (3.3)
          where
          M 1 = sup t R + h ( t , 0 ) α . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_Equaf_HTML.gif
          By (3.2), we get
          sup { η ( t ) } K http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_Equag_HTML.gif

          for a positive constant K.

          Moreover, for arbitrary xB r , by (2.3) and (3.1) we have
          ( H x ) ( t ) α S q ( t ) x 0 α + h ( t , x ( t ) ) α + 0 t ( t - s ) q - 1 P q ( t - s ) f ( s , x ( s ) ) α d s S q ( t ) x 0 α + L r + M 1 + q C Γ ( 1 - γ - α ) Γ ( 1 - q ( γ + α ) ) 0 t ( t - s ) - 1 - q ( γ + α ) ν ( s ) d s sup t R + S q ( t ) A α x 0 + L r + M 1 + q C Γ ( 1 - γ - α ) Γ ( 1 - q ( γ + α ) ) K http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_Equah_HTML.gif
          Choose r such that
          r sup t R + S q ( t ) A α x 0 + M 1 + q C K Γ ( 1 - γ - α ) Γ ( 1 - q ( γ + α ) ) 1 - L . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_Equai_HTML.gif
          Then
          ( H x ) ( t ) α r , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_Equaj_HTML.gif

          that is H(B r ) ⊂ B r .

          Step 2: We prove that the operator H is continuous on B r .

          Let {x n } be a sequence of B r such that x n → × in B r as n → ∞. Then
          f ( s , x n ( s ) ) f ( s , x ( s ) ) , as n http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_Equ10_HTML.gif
          (3.4)

          since the function f is continuous on R+× X α .

          For every t ∈ [0, T], using (H2) and (2.3), we obtain
          ( H x n ) ( t ) - ( H x ) ( t ) α h ( t , x n ( t ) ) - h ( t , x ( t ) ) α + 0 t ( t - s ) q - 1 P q ( t - s ) [ f ( s , x n ( s ) ) - f ( s , x ( s ) ) ] d s α L x n - x + 0 t ( t - s ) q - 1 P q ( t - s ) [ f ( s , x n ( s ) ) - f ( s , x ( s ) ) ] α d s L x n - x + M 2 0 t ( t - s ) - 1 - q ( γ + α ) f ( s , x n ( s ) ) - f ( s , x ( s ) ) d s , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_Equ11_HTML.gif
          (3.5)
          where
          M 2 = q C Γ ( 1 - γ - α ) Γ ( 1 - q ( γ + α ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_Equak_HTML.gif
          Clearly, the first term of (3.5) tends to zero as n → ∞. From the fact that
          f ( s , x n ( s ) ) - f ( s , x ( s ) ) 2 ν ( s ) , s R + , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_Equal_HTML.gif

          (3.4), and the Lebesgue Dominated Convergence Theorem, it follows that the second term of (3.5) tends to zero too as n → ∞.

          Therefore, H is continuous on B r .

          Step 3: Let Ω be arbitrary nonempty subset of B r , we prove that
          μ ( H ( Ω ) ) μ ( Ω ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_Equam_HTML.gif

          Let us choose x ∈ Ω and tl, t2 with |t2 - tl| < ε. Without loss of generality we may assume that tl< t2.

          For any T > 0, when 0 = tl< t2≤ T, we have
          0 t 2 ( t 2 - s ) q - 1 P q ( t 2 - s ) f ( s , x ( s ) ) α d s M 2 0 t 2 ( t 2 - s ) - 1 - q ( γ + α ) ν ( s ) d s . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_Equan_HTML.gif

          Hence ||(Hx)(t2)|| is small as t2 is small independently of x ∈ Ω.

          For 0 < tl< t2≤ T, taking into account our assumptions, we get
          ( H x ) ( t 2 ) - ( H x ) ( t 1 ) α ( S q ( t 2 ) - S q ( t 1 ) ) x 0 α + h ( t 2 , x ( t 2 ) ) - h ( t 1 , x ( t 1 ) ) α + 0 t 1 [ ( t 2 - s ) q - 1 - ( t 1 - s ) q - 1 ] P q ( t 2 - s ) f ( s , x ( s ) ) d s α + t 1 t 2 ( t 2 - s ) q - 1 P q ( t 2 - s ) f ( s , x ( s ) ) d s α + 0 t 1 ( t 1 - s ) q - 1 [ P q ( t 2 - s ) - P q ( t 1 - s ) ] f ( s , x ( s ) ) d s α  =  I 1  +  I 2  +  I 3  +  I 4  +  I 5 . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_Equ12_HTML.gif
          (3.6)
          As a consequence of the continuity of { S q t } http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_IEq26_HTML.gif in the uniform operator topology for t > 0, we know that
          I 1 0 ,  as  t 2 t 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_Equao_HTML.gif
          By (H3), we see that
          I 2 0 ,  as  t 2 t 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_Equap_HTML.gif
          Using (2.3) and (H1), we have
          I 3 = 0 t 1 [ ( t 2 - s ) q - 1 - ( t 1 - s ) q - 1 ] P q ( t 2 - s ) f ( s , x ( s ) ) d s α q C Γ ( 1 - γ - α ) Γ ( 1 - q ( γ + α ) ) 0 t 1 ( t 2 - s ) q - 1 - ( t 1 - s ) q - 1 ( t 2 - s ) q - 1 ν ( s ) ( t 2 - s ) 1 + q ( γ + α ) d s . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_Equaq_HTML.gif
          Therefore, by (3.2), we get
          I 3 0 ,  as  t 2 t 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_Equar_HTML.gif
          Moreover, we have
          I 4 = t 1 t 2 ( t 2 - s ) q - 1 P q ( t 2 - s ) f ( s , x ( s ) ) d s α q C Γ ( 1 - γ - α ) Γ ( 1 - q ( γ + α ) ) t 1 t 2 ( t 2 - s ) - 1 - q ( γ + α ) ν ( s ) d s 0 , as t 2 t 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_Equas_HTML.gif
          Finally, for ε > 0 small enough, we obtain
          I 5 = 0 t 1 ( t 1 - s ) q - 1 [ P q ( t 2 - s ) - P q ( t 1 - s ) ] f ( s , x ( s ) ) d s α q 0 t 1 0 σ Ψ q ( σ ) ( t 1 - s ) q - 1 T ( ( t 2 - s ) q σ ) - T ( ( t 1 - s ) q σ ) α ν ( s ) d σ d s q 0 t 1 - 2 ε 0 σ Ψ q ( σ ) ( t 1 - s ) q - 1 T ( ( t 2 - s ) q σ - ε q σ ) - T ( ( t 1 - s ) q σ - ε q σ ) A α T ( ε q σ ) ν ( s ) d σ d s + M 2 t 1 - 2 ε t 1 ( t 1 - s ) q - 1 ( t 1 - s ) q ( α + γ + 1 ) + ( t 1 - s ) q - 1 ( t 2 - s ) q ( α + γ + 1 ) ν ( s ) d s q C ε q ( γ + α + 1 ) 0 t 1 - 2 ε 0 σ - γ - α Ψ q ( σ ) T ( ( t 2 - s ) q σ - ε q σ ) - T ( ( t 1 - s ) q σ - ε q σ ) ν ( s ) ( t 1 - s ) 1 - q d σ d s + M 2 t 1 - 2 ε t 1 ( t 1 - s ) q - 1 ( t 1 - s ) q ( α + γ + 1 ) + ( t 1 - s ) q - 1 ( t 2 - s ) q ( α + γ + 1 ) ν ( s ) d s  =  I 5  +  I 5 . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_Equat_HTML.gif
          The continuity of the function t → ||T (t) ||k for t ∈ (0, T) implies that
          I 5 0 ,  as  t 2 t 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_Equau_HTML.gif
          Furthermore, it is easy to see that
          I 5 ′′ 0 ,  as  ε 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_Equav_HTML.gif
          Thus, we obtain
          ω 0 T ( H Ω ) = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_Equaw_HTML.gif
          Consequently, we have
          ω 0 ( H Ω ) = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_Equ13_HTML.gif
          (3.7)
          Now, by our assumptions, for arbitrarily fixed tR+ and x, y ∈ Ω we deduce that
          ( H x ) ( t ) - ( H y ) ( t ) α h ( t , x ( t ) ) - h ( t , y ( t ) ) α + 0 t ( t - s ) q - 1 P q ( t - s ) [ f ( s , x ( s ) ) - f ( s , y ( s ) ) ] α d s L x ( t ) - y ( t ) α + 2 M 2 η ( t ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_Equax_HTML.gif
          By (3.2), we have
          lim sup t diam ( H Ω ) ( t ) L lim sup t diam Ω ( t ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_Equ14_HTML.gif
          (3.8)
          Therefore, using the measure of noncompactness μ defined by the formula (2.4) and keeping in mind (3.7) and (3.8), we obtain
          μ ( H Ω ) L μ ( Ω ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_Equ15_HTML.gif
          (3.9)

          Step 4: We prove that the conclusion (1) is true.

          Since 0 < L < 1, in view of (3.9) and Lemma 2.5, we deduce that the operator H has a fixed point x in the ball B r . Hence equation (1.1)-(1.2) has at least one mild solution x(t).

          Step 5: We prove that the conclusion (2) is true.

          Clearly, for any other mild solution y(t) of Equation (1.1)-(1.2), we have
          x ( t ) - y ( t ) α = ( H x ) ( t ) - ( H y ) ( t ) α L x ( t ) - y ( t ) α + 2 M 2 η ( t ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_Equay_HTML.gif
          Then by (3.2) we have
          lim t x ( t ) - y ( t ) α 2 M 2 1 - L lim t η ( t ) = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_Equaz_HTML.gif

          That is, all mild solutions of (1.1)-(1.2) are globally attractive. □

          From the proof of Theorem 3.1, we can also see that the following theorem holds.

          Theorem 3.2 Let A Θ ω γ ( X ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_IEq19_HTML.gifand 0 < ω < π 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_IEq17_HTML.gif. If the maps f and h satisfy

          (H1) The function f: R+× X → X is continuous, and there exists a positive function v(·): R+ R+such that
          f ( t , x ) ν ( t ) , t h e f u n c t i o n s ν ( s ) ( t  -  s ) 1 +  q γ b e l o n g s t o L 1 ( [ 0 , t ] , R  +  ) , lim t η ( t ) :  =  lim t 0 t ν ( s ) ( t  -  s ) 1 +  q γ d s  =  0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_Equba_HTML.gif
          (H2) The function hBC(R+, X) and there exists a constant L ∈ (0, 1) such that
          h ( t 1 , x ( t 1 ) ) - h ( t 2 , x ( t 2 ) ) L ( t 1 - t 2 + x ( t 1 ) - x ( t 2 ) ) , t 1 , t 2 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_Equbb_HTML.gif
          (H3) For each nonempty, bounded set DBC(R+, X), the family of functions
          { t h ( t , φ ) ; φ D } http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_Equbc_HTML.gif

          is equicontinuous.

          Then for every x0D(A β ) with β > 1 + γ, the problem (1.1)-(1.2) has at least a mild solution on BC (R+, X) and all solutions are globally attractive.

          4 Applications

          Example 4.1: Let Ω be a bounded domain in R N (N ≥ 1) with boundary Ω of class C4. Let X = C l ( Ω ̄ ) ( 0 < l < 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_IEq29_HTML.gif. Set
          A ̃ = - Δ , D ( A ̃ ) = { v C 2 + l ( Ω ̄ ) ; v = 0  on  Ω } . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_Equbd_HTML.gif
          It follows from [[13], Example 1.2] that there exist ν, ε > 0 such that
          A ̃ + ν Θ π 2 - ε γ ( C l ( Ω ̄ ) ) , γ = l 2 - 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_Eqube_HTML.gif
          We consider the fractional initial boundary value problem
          q t q u ( t , x ) - h ( t , u ( t , x ) ) = Δ u ( t , x ) - h ( t , u ( t , x ) ) + f ( t , u ( t , x ) ) , x Ω , ( u - h ) | Ω = 0 , u ( 0 , x ) = u 0 ( x ) , x Ω , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_Equ16_HTML.gif
          (4.1)
          where
          h ( t , u ( t , x ) ) = arctan  t A ̃ - α sin ( 1 + A ̃ α u ( t , x ) ) 0 ζ ( t ) 1 + u ( t , x ) d t , f ( t , u ( t , x ) ) = ( t + r 0 ) a u ( t , x ) 1 + u ( t , x ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_Equbf_HTML.gif
          here t > 0, r0 is a positive constant,
          l 2 < α < 1 , 0 < α + l 2 < 1 , - 1 < a < q α + l 2 - 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_Equbg_HTML.gif

          ζ(·) ∈ L1(R + , R) and π 2 0 ζ ( t ) d t L < 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_IEq30_HTML.gif

          The problem (4.1) can be written abstractly as (1.1)-(1.2).

          Moreover, for t ≥ 0, we can see
          f ( t , u ( t ) ) v ( t ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_Equbh_HTML.gif

          where v(t): = (t + r0) a .

          It is clear that the function s ν ( s ) ( t - s ) 1 + q ( γ + α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_IEq31_HTML.gif belongs to L1([0, t], R+) and
          0 t ν ( s ) ( t - s ) 1 + q ( γ + α ) d s 0 t s a ( t - s ) 1 + q ( γ + α ) d s = t a - q ( γ + α ) 0 1 s a ( 1 - s ) - 1 - q ( γ + α ) d s = t a - q ( γ + α ) B ( a + 1 , - q ( γ + α ) ) 0 , t . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_Equbi_HTML.gif

          where B(·, ·) is the Beta function.

          Moreover, for tl, t2 ≥ 0 we have
          h ( t 1 , u ( t 1 ) ) - h ( t 2 , u ( t 2 ) ) α = arctan t 1 sin ( 1 + A ̃ α u ( t 1 , x ) ) 0 ζ ( t ) 1 + u ( t , x ) d t - arctan t 2 sin ( 1 + A ̃ α u ( t 2 , x ) ) 0 ζ ( t ) 1 + u ( t , x ) d t arctan t 1 - arctan t 2 sin ( 1 + A ̃ α u ( t 1 , x ) ) + arctan t 2 sin ( 1 + A ̃ α u ( t 1 , x ) ) - sin ( 1 + A ̃ α u ( t 2 , x ) ) 0 ζ ( t ) d t t 1 - t 2 + π 2 u ( t 1 ) - u ( t 2 ) α 0 ζ ( t ) d t L ( t 1 - t 2 + u ( t 1 ) - u ( t 2 ) α ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_Equbj_HTML.gif

          Consequently, it follows from Theorem 3.1 that, for every u 0 D ( A ̃ α + β ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_IEq32_HTML.gif with 1 > β > α > l 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_IEq33_HTML.gif, the Equation (4.1) has at least a mild solution on BC(R+, X α ) and all solutions are globally attractive.

          For example, if we put
          l = 1 12 , α = 1 8 , a = - 8 9 , q = 1 2 , ζ ( t ) = e - π t , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_Equbk_HTML.gif

          then the assumptions can be satisfied.

          Example 4.2: Let
          A ^ = ( - i Δ + σ ) 1 2 , D ( A ^ ) = W 1 , 3 ( R 2 ) ( a Sobolev space ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_Equbl_HTML.gif

          where i Δ is the Schro" dinger operator, σ > 0 is a suitable constant.

          Then i Δ generates a β ̃ http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_IEq34_HTML.gif-times integrated semigroup S β ̃ ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_IEq35_HTML.gif with β ̃ = 5 12 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_IEq36_HTML.gif on L3(R2) such that
          S β ̃ ( t ) L ( L 3 ( R 2 ) ) M ^ t β ̃ http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_Equbm_HTML.gif
          for all t ≥ 0 and some constant M ^ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_IEq37_HTML.gif (see [20]). Therefore, by virtue of [[21], Theorem 1.3.5 (P. 15)], [[21], Definition 1.3.1 (P. 12)] for C = I, we deduce that the operator i Δ + σ belongs to Θ π 2 β ̃ - 1 ( L 3 ( R 2 ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_IEq38_HTML.gif, which denotes the family of all linear closed operators A: D(A) ⊂ L3(R2) →L3(R2) satisfying
          σ ( A ) S π 2 = z C \ { 0 } ; arg  z π 2 { 0 } , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_Equbn_HTML.gif
          and for every π 2 < μ < π http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_IEq39_HTML.gif there exists a constant such that
          R ( z ; A ) C μ z β ̃ - 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_Equbo_HTML.gif
          for all zC\S μ . Thus, it follows from [[19], Proposition 3.6] that A ^ Θ ω γ ( L 3 ( R 2 ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_IEq40_HTML.gif for some 0 < ω < π 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_IEq17_HTML.gif, where
          γ = - 1 + 2 β ̃ = - 1 6 . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_Equbp_HTML.gif
          Let X = L3(R2), we consider the following equation:
          q t q u ( t , x ) - sin  t e - ( 1 + | u ( t , x ) | ) 0 k ( t ) u ( t , x ) 1 + u ( t , x ) d t = - A ^ u ( t , x ) - sin  t e - ( 1 + u ( t , x ) ) 0 k ( t ) u ( t , x ) 1 + u ( t , x ) d t + ( t + 1 ) b cos ( 1 + u ( t , x ) ) , x R 2 , u ( 0 , x ) = u 0 ( x ) , x R 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_Equ17_HTML.gif
          (4.2)

          where t > 0, -1 < b < qγ and k(·) ∈ L1(R+, R) and 0 k ( t ) d t L < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_IEq41_HTML.gif.

          Set
          u ( t ) ( x ) = u ( t , x ) , h ( t , u ( t ) ) ( x ) = sin t e - ( 1 + u ( t , x ) ) 0 k ( t ) u ( t , x ) 1 + u ( t , x ) d t , f ( t , u ( t ) ) ( x ) = ( t + 1 ) b cos ( 1 + u ( t , x ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_Equbq_HTML.gif

          Then the above Equation (4.2) can be reformulated as the abstract (1.1)-(1.2).

          Moreover, for t ≥ 0, we can see
          f ( t , u ( t ) ) ν ( t ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_Equbr_HTML.gif

          where v(t): = (t + 1) b .

          It is clear that the function s ν ( s ) ( t - s ) 1 + q γ http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_IEq42_HTML.gifbelongs to L1([0, t], R+) and
          0 t ν ( s ) ( t - s ) 1 + q γ d s 0 t s b ( t - s ) 1 + q γ d s = t b - q γ 0 1 s b ( 1 - s ) - 1 - q γ d s = t b - q γ B ( b + 1 , - q γ ) 0 , t . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_Equbs_HTML.gif
          Moreover, for tl, t2 ≥ 0 we have
          h ( t 1 , u ( t 1 ) ) - h ( t 2 , u ( t 2 ) ) = sin t 1 e - ( 1 + u ( t 1 , x ) ) 0 k ( t ) u ( t , x ) 1 + u ( t , x ) d t - sin t 2 e - ( 1 + u ( t 2 , x ) ) 0 k ( t ) u ( t , x ) 1 + u ( t , x ) d t sin t 1 - sin t 2 e - ( 1 + u ( t 1 , x ) ) + sin t 2 e - ( 1 + u ( t 1 , x ) ) - e - ( 1 + u ( t 2 , x ) ) 0 k ( t ) d t t 1 - t 2 + u ( t 1 ) - u ( t 2 ) 0 k ( t ) d t L ( t 1 - t 2 + u ( t 1 ) - u ( t 2 ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_Equbt_HTML.gif

          Consequently, it follows from Theorem 3.2 that, for every u 0 D ( A ^ β ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_IEq43_HTML.gif with 1 > β > 5 6 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_IEq44_HTML.gif, the Equation (4.2) has at least a mild solution on BC(R+, X) and all solutions are globally attractive.

          For example, if we put
          q = 1 2 , b = - 1 2 , k ( t ) = e - 2 t , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_Equbu_HTML.gif

          then the assumptions can be satisfied.

          Notes

          Declarations

          Acknowledgements

          The authors would like to thank the referees for helpful suggestions. The work was supported by the NSF of China (11171210).

          Authors’ Affiliations

          (1)
          Department of Mathematics, Shanghai Jiao Tong University
          (2)
          School of Mathematics, Yunnan Normal University
          (3)
          Texas A & M University at Qatar, c/o Qatar Foundation

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          © Liang et al; licensee Springer. 2012

          This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://​creativecommons.​org/​licenses/​by/​2.​0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.