Open Access

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

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 , https://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 , https://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 , https://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 https://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 https://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, z ρ(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 f AC[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 , https://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 f AC[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 . https://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 f AC[0, ∞) can be written as
c D q f ( t ) = L D q ( f ( t ) - f ( 0 ) ) , t > 0 , 0 < q < 1 . https://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 https://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 + https://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 } https://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 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_IEq6_HTML.gif ;

     
  2. (2)

    Ω Ω0 μ(Ω) ≤ μ0), for Ω and Ω 0 M X https://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 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_IEq8_HTML.gif, where Ω ̄ https://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 https://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 https://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 https://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 , ) https://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 https://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 https://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 https://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 ) https://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 https://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 https://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 https://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 } https://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 } , https://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 https://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 ) https://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 = {z C\{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 . https://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 ) https://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 ) https://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 https://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 , https://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 https://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 ) https://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 https://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 https://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 ; https://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 https://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 ; https://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 https://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 α C 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 , https://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 ; https://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 . https://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 ) https://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 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_IEq17_HTML.gif. Then for every λ C with Reλ > 0, one has
R ( λ ; - A ) = 0 e - λ t T ( t ) d t . https://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 https://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 , https://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 , https://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 , https://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 x0 D(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 ) https://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 , https://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 , https://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 . https://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 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_IEq26_HTML.gifand P q t https://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 ) , https://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 , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_Equu_HTML.gif

for x BC(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 x Y and ε ≥ 0. Namely,
ω T ( x , ε ) = sup { x ( t ) - x ( s ) ; t , s [ 0 , T ] , t - s ε } . https://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 } , https://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 ) , https://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 ) , https://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 } . https://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 ) https://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) . https://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 , https://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 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_IEq19_HTML.gifand 0 < ω < π 2 https://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  +  ) , https://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 . https://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 h BC(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 ) . https://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 D BC(R+, X α ), the family of functions
{ t h ( t , φ ) ; φ D } https://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 0 D(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 . https://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 } https://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 x B 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 , https://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 . https://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 https://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 x B 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 https://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 . https://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 , https://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 https://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 , https://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 ) ) . https://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 + , https://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 ) . https://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 . https://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 . https://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 } https://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 . https://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 . https://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 . https://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 . https://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 . https://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 . https://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 . https://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 . https://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 . https://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 . https://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 t R+ 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 ) . https://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 ) . https://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 ) . https://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 ) . https://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 . https://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 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_IEq19_HTML.gifand 0 < ω < π 2 https://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 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_Equba_HTML.gif
(H2) The function h BC(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 . https://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 D BC(R+, X), the family of functions
{ t h ( t , φ ) ; φ D } https://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 x0 D(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 ) https://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 } . https://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 . https://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 , https://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 ) , https://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 , https://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 . https://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 ) , https://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 ) https://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 . https://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 ) . https://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 ) https://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 https://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 , https://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 ) , https://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 ̃ https://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 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_IEq35_HTML.gif with β ̃ = 5 12 https://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 ̃ https://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 https://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 ) ) https://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 } , https://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 https://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 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-40/MediaObjects/13662_2012_Article_184_Equbo_HTML.gif
for all z C\S μ . Thus, it follows from [[19], Proposition 3.6] that A ^ Θ ω γ ( L 3 ( R 2 ) ) https://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 https://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 . https://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 , https://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 https://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 ) ) . https://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 ) , https://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 https://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 . https://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 ) ) . https://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 ) https://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 https://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 , https://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

References

  1. Băleanu D, Mustafa OG, Agarwal RP: Asymptotically linear solutions for some linear fractional differential equations. Abstr Appl Anal 2010, 2010: 8. (Article ID 865139)
  2. Cuevas C, de Souza JC: S -asymptotically -periodic solutions of semilinear fractional integro-differential equations. Appl Math Lett 2009, 22: 865–870. 10.1016/j.aml.2008.07.013MATHMathSciNetView Article
  3. Cuevas C, Lizama C: Almost automorphic solutions to a class of semilinear fractional differential equations. Appl Math Lett 2008, 21: 1315–1319. 10.1016/j.aml.2008.02.001MATHMathSciNetView Article
  4. Diagana T, Mophou GM, NGuérékata GM: On the existence of mild solutions to some semilinear fractional integro-differential equations. Electron J Qual Theory Diff Equ 2010, 58: 1–17.
  5. El-Borai MM, Amar D: On some fractional integro-differential equations with analytic semigroups. Int J Contemp Math Sci 2009, 4: 1361–1371.MATHMathSciNet
  6. El-Borai MM, El-Said El-Nadi K, El-Akabawy EG: On some fractional evolution equations. Comput Math Appl 2010, 59(3):1352–1355. 10.1016/j.camwa.2009.05.005MATHMathSciNetView Article
  7. Henderson J, Ouahab A: Impulsive differential inclusions with fractional order. Comput Math Appl 2010, 59: 1191–1226. 10.1016/j.camwa.2009.05.011MATHMathSciNetView Article
  8. Li F: Mild solutions for fractional differential equations with nonlocal conditions. Adv Diff Equ 2010, 2010: 9. (Article ID 287861)MATHView Article
  9. Lv ZW, Liang J, Xiao TJ: Solutions to fractional differential equations with nonlocal initial condition in Banach spaces. Adv Diff Equ 2010, 2010: 10.MathSciNetView Article
  10. Mophou GM: Optimal control of fractional diffusion equation. Comput Math Appl 2011, 61: 68–78. 10.1016/j.camwa.2010.10.030MATHMathSciNetView Article
  11. Mophou GM, N'Guérékata GM: On some classes of almost automorphic functions and applications to fractional differential equations. Comput Math Appl 2010, 59: 1310–1317. 10.1016/j.camwa.2009.05.008MATHMathSciNetView Article
  12. Podlubny I: Fractional Differential Equations. Academic Press, San Diego; 1999.
  13. Wang RN, Chen DH, Xiao TJ: Abstract fravtional Cauchy problems with almost sectorial operators. J Diff Equ 2012, 252: 202–235. 10.1016/j.jde.2011.08.048MATHMathSciNetView Article
  14. Liang J, Xiao TJ: Semilinear integrodifferential equations with nonlocal initial conditions. Comp Math Appl 2004, 47(6–7):863–875. 10.1016/S0898-1221(04)90071-5MATHMathSciNetView Article
  15. Pazy A: Semigroups of Linear Operators and Applications to Partial Differential Equations, vol. 44 of Applied Mathematical Sciences. Springer, New York, NY, USA; 1983.View Article
  16. von Wahl W: Gebrochene potenzen eines elliptischen operators und parabolische diffren-tialgleichungen in Räuumen hölderstetiger Funktionen. Nachr Akad Wiss Göttingen, Math Phys Klasse 1972, 11: 231–258.
  17. Banaś J, Goebel K: Measures of Noncompactness in Banach space, Lecture Notes in Pure and Applied Mathematics. Volume 60. Dekker, New York; 1980.
  18. Carvalho AN, Dlotko T, Nescimento MJD: Nonautonomous semilinear evolution equations with almost sectorial operators. J Evol Equ 2008, 8: 631–659. 10.1007/s00028-008-0394-3MATHMathSciNetView Article
  19. Periago F, Stadub B: A functional calculus for almost sectorial operators and applications to abstract evolution equations. J Evol Equ 2002, 2: 41–68. 10.1007/s00028-002-8079-9MATHMathSciNetView Article
  20. van Neerven JMAM, Straub B: On the existence and growth of mild solutions of the abstract Cauchy problem for operators with polynomially bounded resolvent. Houston J Math 1998, 24: 137–171.MATHMathSciNet
  21. Xiao TJ, Liang J: The Cauchy Problem for Higher Order Abstract Differential Equations. In Lecture Notes in Math. Volume 1701. Springer, Berlin, New York; 1998.

Copyright

© 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.