Existence of mild solutions for fractional evolution equations with nonlocal conditions

Boundary Value Problems20122012:113

DOI: 10.1186/1687-2770-2012-113

Received: 8 August 2012

Accepted: 28 September 2012

Published: 17 October 2012

Abstract

This paper deals with the existence and uniqueness of mild solutions for a class of fractional evolution equations with nonlocal initial conditions. We present some local growth conditions on a nonlinear part and a nonlocal term to guarantee the existence theorems. An example is given to illustrate the applicability of our results.

MSC: 34A12, 35F25.

Keywords

fractional evolution equations nonlocal initial conditions existence uniqueness

1 Introduction

The differential equations involving fractional derivatives in time have recently been proved to be valuable tools in the modeling of many phenomena in various fields of engineering and science. Indeed, we can find numerous applications in electrochemistry, control, porous media, electromagnetic processing etc. (see [15]). Hence, the research on fractional differential equations has become an object of extensive study during recent years; see [611] and references therein.

On the other hand, the nonlocal initial condition, in many cases, has much better effect in applications than the traditional initial condition. As remarked by Byszewski and Lakshmikantham (see [12, 13]), the nonlocal initial value problems can be more useful than the standard initial value problems to describe many physical phenomena.

Let X be a Banach space with norm http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq1_HTML.gif, and let T > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq2_HTML.gif be a constant. Consider the existence and uniqueness of mild solutions of fractional evolution equation with nonlocal condition in the form
{ D q u ( t ) + A u ( t ) = f ( t , u ( t ) ) + 0 t K ( t , s ) h ( s , u ( s ) ) d s , t J = [ 0 , T ] , u ( 0 ) + g ( u ) = x 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_Equ1_HTML.gif
(1)

where D q http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq3_HTML.gif is the Caputo fractional derivative of order q ( 0 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq4_HTML.gif, the linear operator −A is the infinitesimal generator of an analytic semigroup { S ( t ) } t 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq5_HTML.gif in X, the functions f, h and g will be specified later. K C ( Δ , R + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq6_HTML.gif, where Δ = { ( t , s ) | 0 s t T } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq7_HTML.gif, R + = [ 0 , + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq8_HTML.gif. Throughout this paper, we always assume that K = max ( t , s ) Δ K ( t , s ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq9_HTML.gif.

In some existing articles, the fractional differential equations with nonlocal initial conditions were treated under the hypothesis that the nonlocal term is completely continuous or global Lipschitz continuous. It is obvious that these conditions are not easy to verify in many cases. To make the things more applicable, in [6] the authors studied the existence and uniqueness of mild solutions of Eq. (1) under the case K ( t , s ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq10_HTML.gif. In their main results, they did not assume the complete continuity of the nonlocal term, but they needed the following assumptions:

(F1) there exist a constant q 1 [ 0 , q ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq11_HTML.gif and m L 1 q 1 ( J , R + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq12_HTML.gif such that f ( t , x ) m ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq13_HTML.gif for all x X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq14_HTML.gif and almost all t J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq15_HTML.gif;

(F2) there exists a constant L > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq16_HTML.gif such that g ( u ) g ( v ) L u v http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq17_HTML.gif for u , v C ( J , X ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq18_HTML.gif;

and some other conditions.

In this paper, we will improve the conditions (F1) and (F2). We only assume that f and h satisfy local growth conditions (see assumption ( H 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq19_HTML.gif)) and g is local Lipschitz continuous (see assumption ( H 4 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq20_HTML.gif)). We will carry out our investigation in the Banach space X α : = ( D ( A α ) , α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq21_HTML.gif, 0 < α < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq22_HTML.gif, where D ( A α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq23_HTML.gif is the domain of the fractional power of A. Finally, an example is given to illustrate the applicability of our main results. We can see that the main results in [6] cannot be applied to our example.

The rest of this paper is organized as follows. In Section 2, some preliminaries are given on the fractional power of the generator of an analytic semigroup and on the mild solutions of Eq. (1). In Section 3, we study the existence and uniqueness of mild solutions of Eq. (1). In Section 4, we give an example to illustrate the applicability of our results.

2 Preliminaries

Let X be a Banach space with norm http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq1_HTML.gif, and let A : D ( A ) X X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq24_HTML.gif be the infinitesimal generator of an analytic semigroup S ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq25_HTML.gif ( t 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq26_HTML.gif) of a uniformly bounded linear operator in X, that is, there exists M 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq27_HTML.gif such that S ( t ) M http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq28_HTML.gif for all t 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq26_HTML.gif. Without loss of generality, let 0 ρ ( A ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq29_HTML.gif. Then for any α > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq30_HTML.gif, we can define A α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq31_HTML.gif by
A α = 1 Γ ( α ) 0 t α 1 S ( t ) d t . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_Equa_HTML.gif

A α L ( X ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq32_HTML.gif is injective, and A α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq33_HTML.gif can be defined by A α = ( A α ) 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq34_HTML.gif with the domain D ( A α ) = A α ( X ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq35_HTML.gif. For α = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq36_HTML.gif, let A α = I http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq37_HTML.gif.

Lemma 1 ([14])

A α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq33_HTML.gif has the following properties:
  1. (i)

    D ( A α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq23_HTML.gif is a Banach space with the norm x α : = A α x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq38_HTML.gif for x D ( A α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq39_HTML.gif;

     
  2. (ii)

    S ( t ) : X X α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq40_HTML.gif for each t > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq41_HTML.gif;

     
  3. (iii)

    A α S ( t ) x = S ( t ) A α x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq42_HTML.gif for each x D ( A α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq39_HTML.gif and t 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq26_HTML.gif;

     
  4. (iv)

    A α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq31_HTML.gif is a bounded linear operator on X with D ( A α ) = I m ( A α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq43_HTML.gif;

     
  5. (v)

    If 0 < α β http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq44_HTML.gif, then D ( A β ) D ( A α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq45_HTML.gif.

     
Let X α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq46_HTML.gif be the Banach space of D ( A α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq23_HTML.gif endowed with the norm α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq47_HTML.gif. Denote by C ( J , X α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq48_HTML.gif the Banach space of all continuous functions from J into X α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq46_HTML.gif with the supnorm given by u C = sup t J u ( t ) α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq49_HTML.gif for u C ( J , X α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq50_HTML.gif. From Lemma 1(iv), since A α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq31_HTML.gif is a bounded linear operator for α > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq30_HTML.gif, we denote by C α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq51_HTML.gif the operator norm of A α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq31_HTML.gif in X, that is, C α : = A α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq52_HTML.gif. For any t 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq26_HTML.gif, denote by S α ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq53_HTML.gif the restriction of S ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq25_HTML.gif to X α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq46_HTML.gif. From Lemma 1(ii) and (iii), for any x X α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq54_HTML.gif, we have
S ( t ) x α = A α S ( t ) x = S ( t ) A α x S ( t ) A α x = S ( t ) x α , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_Equb_HTML.gif
and
S ( t ) x x α = A α S ( t ) x A α x = S ( t ) A α x A α x 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_Equc_HTML.gif

as t 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq55_HTML.gif. Therefore, S ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq25_HTML.gif ( t 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq26_HTML.gif) is a strongly continuous semigroup in X α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq46_HTML.gif, and S α ( t ) α S ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq56_HTML.gif for all t 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq26_HTML.gif. To prove our main results, the following lemma is needed.

Lemma 2 ([15])

If S ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq25_HTML.gif ( t 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq26_HTML.gif) is a compact semigroup in X, then S α ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq53_HTML.gif ( t 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq26_HTML.gif) is an immediately compact semigroup in X α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq46_HTML.gif, and hence it is immediately norm-continuous.

For x X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq14_HTML.gif, define two families { U ( t ) } t 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq57_HTML.gif and { V ( t ) } t 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq58_HTML.gif of operators by
U ( t ) x = 0 η q ( θ ) S ( t q θ ) x d θ , V ( t ) x = q 0 θ η q ( θ ) S ( t q θ ) x d θ , 0 < q < 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_Equd_HTML.gif
where
η q ( θ ) = 1 q θ 1 1 q ρ q ( θ 1 q ) , ρ q ( θ ) = 1 π n = 1 ( 1 ) n 1 θ q n 1 Γ ( n q + 1 ) n ! sin ( n π q ) , θ ( 0 , ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_Eque_HTML.gif
where η q http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq59_HTML.gif is the probability density function defined on ( 0 , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq60_HTML.gif, which has properties η q ( θ ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq61_HTML.gif for all θ ( 0 , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq62_HTML.gif and
0 η q ( θ ) d θ = 1 , 0 θ η q ( θ ) d θ = 1 Γ ( 1 + q ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_Equ2_HTML.gif
(2)

The following lemma follows from the results in [68, 10].

Lemma 3 The following properties are valid:
  1. (i)
    For fixed t 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq26_HTML.gif and any x X α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq54_HTML.gif, we have
    U ( t ) x α M x α , V ( t ) x α q M Γ ( 1 + q ) x α = M Γ ( q ) x α . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_Equf_HTML.gif
     
  2. (ii)

    The operators U ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq63_HTML.gif and V ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq64_HTML.gif are strongly continuous for all t 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq26_HTML.gif.

     
  3. (iii)

    If S ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq25_HTML.gif ( t 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq26_HTML.gif) is a compact semigroup in X, then U ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq63_HTML.gif and V ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq64_HTML.gif are norm-continuous in X for t > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq41_HTML.gif.

     
  4. (iv)

    If S ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq25_HTML.gif ( t 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq26_HTML.gif) is a compact semigroup in X, then U ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq63_HTML.gif and V ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq64_HTML.gif are compact operators in X for t > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq41_HTML.gif.

     

In this paper, we adopt the following definition of a mild solution of Eq. (1).

Definition 1 By a mild solution of Eq. (1), we mean a function u C ( J , X α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq50_HTML.gif satisfying
u ( t ) = U ( t ) ( x 0 g ( u ) ) + 0 t ( t s ) q 1 V ( t s ) [ f ( s , u ( s ) ) + 0 s K ( s , τ ) h ( τ , u ( τ ) ) d τ ] d s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_Equg_HTML.gif

for all t J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq15_HTML.gif.

To prove our main results, we also need the following two lemmas.

Lemma 4 A measurable function H : [ 0 , T ] X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq65_HTML.gif is Bochner integrable if | H | http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq66_HTML.gif is Lebesgue integrable.

Lemma 5 (Krasnoselskii’s fixed point theorem)

Let X be a Banach space, let B be a bounded closed and convex subset of X and let Q 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq67_HTML.gif and Q 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq68_HTML.gif be mappings from B into X such that Q 1 x + Q 2 y B http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq69_HTML.gif for every pair x , y B http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq70_HTML.gif. If Q 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq67_HTML.gif is a contraction and Q 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq68_HTML.gif is completely continuous, then the operator equation Q 1 x + Q 2 x = x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq71_HTML.gif has a solution on B.

Lemmas 4 and 5, which can be found in many books, are classical.

The following are the basic assumptions of this paper.

( H 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq72_HTML.gif) S ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq25_HTML.gif ( t 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq26_HTML.gif) is a compact operator semigroup in X.

( H 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq73_HTML.gif) There exists a constant β [ α , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq74_HTML.gif such that the functions f , h : J × X α X β http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq75_HTML.gif satisfy the following conditions:
  1. (i)

    For each x X α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq54_HTML.gif, the functions f ( , x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq76_HTML.gif, h ( , x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq77_HTML.gif are measurable.

     
  2. (ii)

    For each t J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq15_HTML.gif, the functions f ( t , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq78_HTML.gif, h ( t , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq79_HTML.gif are continuous.

     
( H 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq19_HTML.gif) For t J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq15_HTML.gif and r > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq80_HTML.gif, there exist positive functions φ r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq81_HTML.gif satisfying φ r ( ) ( t ) 1 q L 1 ( [ 0 , t ] , R + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq82_HTML.gif and ϕ r L 1 ( J , R + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq83_HTML.gif such that
sup x α r f ( t , x ) β φ r ( t ) , sup x α r h ( t , x ) β ϕ r ( t ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_Equh_HTML.gif
and there are positive constants σ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq84_HTML.gif and σ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq85_HTML.gif such that
lim inf r + 1 r 0 t φ r ( s ) ( t s ) 1 q d s σ 1 < + , lim inf r + 1 r 0 t 0 s ϕ r ( τ ) d τ ( t s ) 1 q d s σ 2 < + , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_Equi_HTML.gif
( H 4 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq20_HTML.gif) g : C ( J , X α ) X α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq86_HTML.gif and for r > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq80_HTML.gif, there exists a positive constant L such that
g ( u ) g ( v ) α L u v C http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_Equj_HTML.gif

for all u , v B r : = { u C ( J , X α ) : u ( t ) α r , t J } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq87_HTML.gif.

3 Main results

In this section, we introduce the existence and uniqueness theorems of mild solutions of Eq. (1). The discussions are based on fractional calculus and Krasnoselskii’s fixed point theorem. Our main results are as follows.

Theorem 1 Suppose that the assumptions ( H 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq72_HTML.gif)-( H 4 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq20_HTML.gif) hold. If x 0 X α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq88_HTML.gif and the following inequality holds:
M L + M C β α Γ ( q ) ( σ 1 + K σ 2 ) < 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_Equ3_HTML.gif
(3)

then Eq. (1) has at least one mild solution on J.

Proof Define two operators Q 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq67_HTML.gif and Q 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq68_HTML.gif as follows:
( Q 1 u ) ( t ) = U ( t ) ( x 0 g ( u ) ) , t J , ( Q 2 u ) ( t ) = 0 t ( t s ) q 1 V ( t s ) [ f ( s , u ( s ) ) + 0 s K ( s , τ ) h ( τ , u ( τ ) ) d τ ] d s , t J . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_Equk_HTML.gif
Obviously, u is a mild solution of Eq. (1) if and only if u is a solution of the operator equation u = Q 1 u + Q 2 u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq89_HTML.gif on J. To prove the operator equation u = Q 1 u + Q 2 u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq89_HTML.gif has solutions, we first show that there is a positive number r such that Q 1 u + Q 2 v B r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq90_HTML.gif for every pair u , v B r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq91_HTML.gif. If this were not the case, then for each r > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq80_HTML.gif, there would exist u r , v r B r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq92_HTML.gif and t r J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq93_HTML.gif such that ( Q 1 u r ) ( t r ) + ( Q 2 v r ) ( t r ) α > r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq94_HTML.gif. Thus, from Lemma 3, ( H 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq19_HTML.gif) and ( H 4 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq20_HTML.gif), we have
r < ( Q 1 u r ) ( t r ) + ( Q 2 v r ) ( t r ) α U ( t r ) ( x 0 g ( u r ) ) α + 0 t r ( t r s ) q 1 V ( t r s ) [ f ( s , v r ( s ) ) + 0 s K ( s , τ ) h ( τ , v r ( τ ) ) d τ ] α d s M x 0 [ g ( u r ) g ( 0 ) ] g ( 0 ) α + M Γ ( q ) 0 t r ( t r s ) q 1 A α β A β [ f ( s , v r ( s ) ) + 0 s K ( s , τ ) h ( τ , v r ( τ ) ) d τ ] d s M x 0 α + M L r + M g ( 0 ) α + M C β α Γ ( q ) 0 t r ( t r s ) q 1 [ φ r ( s ) + K 0 s ϕ r ( τ ) d τ ] d s . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_Equl_HTML.gif
Dividing on both sides by r and taking the lower limit as r + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq95_HTML.gif, we have
M L + M C β α Γ ( q ) ( σ 1 + K σ 2 ) 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_Equm_HTML.gif

which contradicts (3). Hence, for some r > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq80_HTML.gif, Q 1 u + Q 2 v B r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq90_HTML.gif for every pair u , v B r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq91_HTML.gif.

The next proof will be given in two steps.

Step 1. Q 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq67_HTML.gif is a contraction on B r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq96_HTML.gif.

For any u , v B r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq91_HTML.gif and t J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq15_HTML.gif, according to Lemma 3 and assumption ( H 4 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq20_HTML.gif), we have
( Q 1 u ) ( t ) ( Q 1 v ) ( t ) α = U ( t ) [ g ( v ) g ( u ) ] α M g ( u ) g ( v ) α M L u v C , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_Equn_HTML.gif

which implies that Q 1 u Q 1 v C M L u v C http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq97_HTML.gif. It follows from (3) that M L < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq98_HTML.gif, hence Q 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq67_HTML.gif is a contraction on B r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq96_HTML.gif.

Step 2. Q 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq68_HTML.gif is a completely continuous operator on B r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq96_HTML.gif.

We first prove that Q 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq68_HTML.gif is continuous on B r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq96_HTML.gif. Let { u n } B r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq99_HTML.gif with u n u B r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq100_HTML.gif as n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq101_HTML.gif. Then for any t J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq15_HTML.gif, s [ 0 , t ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq102_HTML.gif, by assumption ( H 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq73_HTML.gif), we have
f ( s , u n ( s ) ) f ( s , u ( s ) ) , h ( s , u n ( s ) ) h ( s , u ( s ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_Equo_HTML.gif
as n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq101_HTML.gif, and from assumption ( H 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq19_HTML.gif), we have
f ( s , u n ( s ) ) f ( s , u ( s ) ) β 2 φ r ( s ) , h ( s , u n ( s ) ) h ( s , u ( s ) ) β 2 ϕ r ( s ) , 0 s K ( s , τ ) h ( τ , u n ( τ ) ) h ( τ , u ( τ ) ) β d τ 2 K ϕ r L 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_Equp_HTML.gif
This together with the Lebesgue dominated convergence theorem gives that
( Q 2 u n ) ( t ) ( Q 2 u ) ( t ) α 0 t ( t s ) q 1 V ( t s ) [ f ( s , u n ( s ) ) f ( s , u ( s ) ) ] α d s + 0 t ( t s ) q 1 V ( t s ) 0 s K ( s , τ ) [ h ( τ , u n ( τ ) ) h ( τ , u ( τ ) ) ] d τ α d s M Γ ( q ) 0 t ( t s ) q 1 A α β A β [ f ( s , u n ( s ) ) f ( s , u ( s ) ) ] d s + M Γ ( q ) 0 t ( t s ) q 1 A α β 0 s K ( s , τ ) A β [ h ( τ , u n ( τ ) ) h ( τ , u ( τ ) ) ] d τ d s M C β α Γ ( q ) 0 t ( t s ) q 1 f ( s , u n ( s ) ) f ( s , u ( s ) ) β d s + M C β α Γ ( q ) 0 t ( t s ) q 1 0 s K ( s , τ ) h ( τ , u n ( τ ) ) h ( τ , u ( τ ) ) β d τ d s 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_Equq_HTML.gif

as n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq101_HTML.gif. Hence, lim n Q 2 u n Q 2 u C = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq103_HTML.gif. This means that Q 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq68_HTML.gif is continuous on B r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq96_HTML.gif.

Next, we will show that the set { Q 2 u : u B r } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq104_HTML.gif is relatively compact. It suffices to show that the family of functions { Q 2 u : u B r } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq104_HTML.gif is uniformly bounded and equicontinuous, and for any t J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq15_HTML.gif, the set { ( Q 2 u ) ( t ) : u B r } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq105_HTML.gif is relatively compact.

For any u B r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq106_HTML.gif, we have Q 2 u C r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq107_HTML.gif for some r > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq80_HTML.gif, which means that { Q 2 u : u B r } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq104_HTML.gif is uniformly bounded. In what follows, we show that { Q 2 u : u B r } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq104_HTML.gif is a family of equicontinuous functions. For t [ 0 , T ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq108_HTML.gif, we have
( Q 2 u ) ( t ) ( Q 2 u ) ( 0 ) α M C β α Γ ( q ) 0 t ( t s ) q 1 [ φ r ( s ) + K 0 s ϕ r ( τ ) d τ ] d s . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_Equr_HTML.gif
Hence, it is only necessary to consider t > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq41_HTML.gif. For 0 < t 1 < t 2 T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq109_HTML.gif, from Lemma 3 and assumption ( H 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq19_HTML.gif), we have
( Q 2 u ) ( t 2 ) ( Q 2 u ) ( t 1 ) α 0 t 1 ( t 2 s ) q 1 [ V ( t 2 s ) V ( t 1 s ) ] [ f ( s , u ( s ) ) + 0 s K ( s , τ ) h ( τ , u ( τ ) ) d τ ] d s α + 0 t 1 [ ( t 2 s ) q 1 ( t 1 s ) q 1 ] V ( t 1 s ) × [ f ( s , u ( s ) ) + 0 s K ( s , τ ) h ( τ , u ( τ ) ) d τ ] d s α + t 1 t 2 ( t 2 s ) q 1 V ( t 2 s ) [ f ( s , u ( s ) ) + 0 s K ( s , τ ) h ( τ , u ( τ ) ) d τ ] d s α 0 t 1 ( t 2 s ) q 1 A α β [ V ( t 2 s ) V ( t 1 s ) ] × A β [ f ( s , u ( s ) ) + 0 s K ( s , τ ) h ( τ , u ( τ ) ) d τ ] d s + M Γ ( q ) 0 t 1 | ( t 2 s ) q 1 ( t 1 s ) q 1 | × A α β A β [ f ( s , u ( s ) ) + 0 s K ( s , τ ) h ( τ , u ( τ ) ) d τ ] d s + M Γ ( q ) t 1 t 2 ( t 2 s ) q 1 A α β A β [ f ( s , u ( s ) ) + 0 s K ( s , τ ) h ( τ , u ( τ ) ) d τ ] d s C β α 0 t 1 ( t 2 s ) q 1 V ( t 2 s ) V ( t 1 s ) [ φ r ( s ) + K 0 s ϕ r ( τ ) d τ ] d s + M C β α Γ ( q ) 0 t 1 | ( t 2 s ) q 1 ( t 1 s ) q 1 | [ φ r ( s ) + K 0 s ϕ r ( τ ) d τ ] d s + M C β α Γ ( q ) t 1 t 2 ( t 2 s ) q 1 [ φ r ( s ) + K 0 s ϕ r ( τ ) d τ ] d s I 1 + I 2 + I 3 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_Equs_HTML.gif
where
I 1 = C β α 0 t 1 ( t 2 s ) q 1 V ( t 2 s ) V ( t 1 s ) [ φ r ( s ) + K 0 s ϕ r ( τ ) d τ ] d s , I 2 = M C β α Γ ( q ) 0 t 1 | ( t 2 s ) q 1 ( t 1 s ) q 1 | [ φ r ( s ) + K 0 s ϕ r ( τ ) d τ ] d s , I 3 = M C β α Γ ( q ) t 1 t 2 ( t 2 s ) q 1 [ φ r ( s ) + K 0 s ϕ r ( τ ) d τ ] d s . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_Equt_HTML.gif
For any ϵ ( 0 , t 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq110_HTML.gif, we have
I 1 C β α 0 t 1 ϵ ( t 2 s ) q 1 V ( t 2 s ) V ( t 1 s ) [ φ r ( s ) + K 0 s ϕ r ( τ ) d τ ] d s + C β α t 1 ϵ t 1 ( t 2 s ) q 1 V ( t 2 s ) V ( t 1 s ) [ φ r ( s ) + K 0 s ϕ r ( τ ) d τ ] d s C β α 0 t 1 ϵ ( t 1 s ) q 1 [ φ r ( s ) + K 0 s ϕ r ( τ ) d τ ] d s sup s [ 0 , t 1 ϵ ] V ( t 2 s ) V ( t 1 s ) + 2 M C β α Γ ( q ) t 1 ϵ t 1 ( t 1 s ) q 1 [ φ r ( s ) + K 0 s ϕ r ( τ ) d τ ] d s . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_Equu_HTML.gif

It follows from Lemma 3 that I 1 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq111_HTML.gif as t 2 t 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq112_HTML.gif and ϵ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq113_HTML.gif independently of u B r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq106_HTML.gif. From the expressions of I 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq114_HTML.gif and I 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq115_HTML.gif, it is clear that I 2 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq116_HTML.gif and I 3 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq117_HTML.gif as t 2 t 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq112_HTML.gif independently of u B r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq106_HTML.gif. Therefore, we prove that { Q 2 u : u B r } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq104_HTML.gif is a family of equicontinuous functions.

It remains to prove that for any t J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq15_HTML.gif, the set W ( t ) : = { ( Q 2 u ) ( t ) : u B r } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq118_HTML.gif is relatively compact.

Obviously, W ( 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq119_HTML.gif is relatively compact in X α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq46_HTML.gif. Let 0 < t T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq120_HTML.gif be fixed. For each δ ( 0 , t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq121_HTML.gif, ϵ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq122_HTML.gif and u B r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq106_HTML.gif, we define an operator Q 2 δ , ϵ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq123_HTML.gif by
( Q 2 δ , ϵ u ) ( t ) = 0 t δ ( t s ) q 1 ϵ q θ η q ( θ ) S ( ( t s ) q θ ) f ( s , u ( s ) ) d θ d s + 0 t δ ( t s ) q 1 ϵ q θ η q ( θ ) S ( ( t s ) q θ ) 0 s K ( s , τ ) h ( τ , u ( τ ) ) d τ d θ d s = S ( δ q ϵ ) [ 0 t δ ( t s ) q 1 ϵ q θ η q ( θ ) S ( ( t s ) q θ δ q ϵ ) f ( s , u ( s ) ) d θ d s + 0 t δ ( t s ) q 1 ϵ q θ η q ( θ ) S ( ( t s ) q θ δ q ϵ ) × 0 s K ( s , τ ) h ( τ , u ( τ ) ) d τ d θ d s ] . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_Equv_HTML.gif
Then the sets W δ , ϵ ( t ) : = { ( Q 2 δ , ϵ u ) ( t ) : u B r } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq124_HTML.gif are relatively compact in X α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq46_HTML.gif since by Lemma 2, the operator S α ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq53_HTML.gif is compact for t > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq41_HTML.gif in X α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq46_HTML.gif. Moreover, for every u B r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq106_HTML.gif, from Lemma 3 and assumption ( H 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq19_HTML.gif), we have
( Q 2 u ) ( t ) ( Q 2 δ , ϵ u ) ( t ) α 0 t ( t s ) q 1 0 ϵ q θ η q ( θ ) S ( ( t s ) q θ ) × [ f ( s , u ( s ) ) + 0 s K ( s , τ ) h ( τ , u ( τ ) ) d τ ] d θ d s α + t δ t ( t s ) q 1 ϵ q θ η q ( θ ) S ( ( t s ) q θ ) × [ f ( s , u ( s ) ) + 0 s K ( s , τ ) h ( τ , u ( τ ) ) d τ ] d θ d s α 0 t ( t s ) q 1 A α β 0 ϵ q θ η q ( θ ) S ( ( t s ) q θ ) d θ [ φ r ( s ) + K 0 s ϕ r ( τ ) d τ ] d s + t δ t ( t s ) q 1 A α β ϵ q θ η q ( θ ) S ( ( t s ) q θ ) d θ [ φ r ( s ) + K 0 s ϕ r ( τ ) d τ ] d s q M C β α 0 t ( t s ) q 1 [ φ r ( s ) + K 0 s ϕ r ( τ ) d τ ] d s 0 ϵ θ η q ( θ ) d θ + M C β α Γ ( q ) t δ t ( t s ) q 1 [ φ r ( s ) + K 0 s ϕ r ( τ ) d τ ] d s . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_Equw_HTML.gif

Therefore, there are relatively compact sets arbitrarily close to the set W ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq125_HTML.gif for t ( 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq126_HTML.gif and since it is compact at t = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq127_HTML.gif, we have the relative compactness of W ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq125_HTML.gif in X α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq46_HTML.gif for all t J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq15_HTML.gif.

Therefore, the set { Q 2 u : u B r } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq104_HTML.gif is relatively compact by the Ascoli-Arzela theorem. Thus, the continuity of Q 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq68_HTML.gif and relative compactness of the set { Q 2 u : u B r } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq104_HTML.gif imply that Q 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq68_HTML.gif is a completely continuous operator. Hence, Krasnoselskii’s fixed point theorem shows that the operator equation u = Q 1 u + Q 2 u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq89_HTML.gif has a solution on B r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq96_HTML.gif. Therefore, Eq. (1) has at least one mild solution. The proof is completed. □

The following existence and uniqueness theorem for Eq. (1) is based on the Banach contraction principle. We will also need the following assumptions.

( H 5 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq128_HTML.gif) There exists a constant β [ α , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq129_HTML.gif such that the functions f , h : J × X α X β http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq130_HTML.gif are strongly measurable.

( H 6 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq131_HTML.gif) For r > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq80_HTML.gif, there exist functions ρ 1 , ρ 2 L 1 ( J , R + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq132_HTML.gif such that
f ( t , x ) f ( t , y ) β ρ 1 ( t ) x y α , h ( t , x ) h ( t , y ) β ρ 2 ( t ) x y α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_Equx_HTML.gif

for any x , y B r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq133_HTML.gif and t J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq15_HTML.gif.

Theorem 2 Let the assumptions ( H 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq19_HTML.gif)-( H 6 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq131_HTML.gif) be satisfied. If x 0 X α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq88_HTML.gif and the inequalities (3) and
μ M L + M T q C β α Γ ( q + 1 ) ( ρ 1 L 1 + K ρ 2 L 1 ) < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_Equ4_HTML.gif
(4)

hold, then Eq. (1) has a unique mild solution.

Proof From Lemma 4 and assumption ( H 5 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq128_HTML.gif), it is easy to see that ( t s ) q 1 V ( t s ) [ f ( s , u ( s ) ) + 0 s K ( s , τ ) h ( τ , u ( τ ) ) d τ ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq134_HTML.gif is Bochner integrable with respect to s [ 0 , t ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq102_HTML.gif for all t J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq15_HTML.gif. For any u B r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq106_HTML.gif, we define an operator Q by
( Q u ) ( t ) = U ( t ) ( x 0 g ( u ) ) + 0 t ( t s ) q 1 V ( t s ) × [ f ( s , u ( s ) ) + 0 s K ( s , τ ) h ( τ , u ( τ ) ) d τ ] d s , t J . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_Equy_HTML.gif
According to the proof of Theorem 1, we know that Q ( B r ) B r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq135_HTML.gif for some r > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq80_HTML.gif. For any u , v B r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq91_HTML.gif and t J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq15_HTML.gif, from Lemma 3, assumptions ( H 4 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq20_HTML.gif) and ( H 6 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq131_HTML.gif), we have
( Q u ) ( t ) ( Q v ) ( t ) α U ( t ) [ g ( v ) g ( u ) ] α + 0 t ( t s ) q 1 V ( t s ) [ f ( s , u ( s ) ) f ( s , v ( s ) ) ] α d s + 0 t ( t s ) q 1 V ( t s ) 0 s K ( s , τ ) [ h ( τ , u ( τ ) ) h ( τ , v ( τ ) ) ] d τ α d s M L u v C + q M C β α Γ ( q + 1 ) 0 t ( t s ) q 1 ρ 1 ( s ) u ( s ) v ( s ) α d s + q M C β α K Γ ( q + 1 ) 0 t ( t s ) q 1 0 s ρ 2 ( τ ) u ( τ ) v ( τ ) α d τ d s M L u v C + q M C β α Γ ( q + 1 ) 0 t ( t s ) q 1 d s ρ 1 L 1 u v C + q M C β α K Γ ( q + 1 ) 0 t ( t s ) q 1 d s ρ 2 L 1 u v C = [ M L + M T q C β α Γ ( q + 1 ) ( ρ 1 L 1 + K ρ 2 L 1 ) ] u v C . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_Equz_HTML.gif
Thus,
Q u Q v C μ u v C , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_Equaa_HTML.gif

which means that Q is a contraction according to (4). By applying the Banach contraction principle, we know that Q has a unique fixed point on B r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq96_HTML.gif, which is the unique mild solution of Eq. (1). This completes the proof. □

4 An example

Let X = L 2 [ 0 , π ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq136_HTML.gif equip with its natural norm 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq137_HTML.gif and inner product , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq138_HTML.gif. Consider the following system:
{ 1 2 t 1 2 u ( t , x ) 2 x 2 u ( t , x ) = u ( t , x ) sin t 1 2 + 0 t K ( t , s ) u ( s , x ) cos s d s , t [ 0 , T ] , x [ 0 , π ] , u ( t , 0 ) = u ( t , π ) = 0 , t [ 0 , T ] , u ( 0 , x ) + i = 1 m 0 π K 0 ( x , y ) u ( t i , y ) d y = u 0 ( x ) , x [ 0 , π ] , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_Equ5_HTML.gif
(5)

where T > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq2_HTML.gif is a constant, 0 < t 1 < < t m < T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq139_HTML.gif, u 0 X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq140_HTML.gif and K 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq141_HTML.gif will be specified later.

Let the operator A : D ( A ) X X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq142_HTML.gif be defined by
D ( A ) : = { v X : v , v X , v ( 0 ) = v ( π ) = 0 } , A u = 2 u x 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_Equab_HTML.gif
Then −A generates a compact analytic semigroup S ( ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq143_HTML.gif of uniformly bounded linear operators and S ( t ) L ( X ) e t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq144_HTML.gif for all t 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq26_HTML.gif. Hence, we take M = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq145_HTML.gif. Moreover, the eigenvalues of A are n 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq146_HTML.gif, n N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq147_HTML.gif and the corresponding normalized eigenvectors are z n ( x ) = 2 π sin ( n x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq148_HTML.gif, n = 1 , 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq149_HTML.gif . The operator A 1 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq150_HTML.gif is given by
A 1 2 ξ = n = 1 n ξ , z n z n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_Equac_HTML.gif

for each ξ D ( A 1 2 ) : = { v X : n = 1 n v , z n z n X } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq151_HTML.gif and A 1 2 L ( X ) = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq152_HTML.gif.

Lemma 6 ([16])

If ξ D ( A 1 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq153_HTML.gif, then ξ is absolutely continuous, ξ X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq154_HTML.gif and ξ 2 = A 1 2 ξ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq155_HTML.gif.

Let X 1 2 = ( D ( A 1 2 ) , 1 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq156_HTML.gif, where ξ 1 2 : = A 1 2 ξ X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq157_HTML.gif for all ξ D ( A 1 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq153_HTML.gif. Assume that

(P1) The function K 0 L 2 ( [ 0 , π ] × [ 0 , π ] , R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq158_HTML.gif, K 0 ( 0 , y ) = K 0 ( π , y ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq159_HTML.gif, and the partial derivative ( x , y ) x K 0 ( x , y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq160_HTML.gif belongs to L 2 ( [ 0 , π ] × [ 0 , π ] , R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq161_HTML.gif.

Define
f ( t , u ( t ) ) ( x ) = u ( t , x ) sin t 1 2 , h ( t , u ( t ) ) ( x ) = u ( t , x ) cos t , g ( u ) ( x ) = i = 1 m 0 π K 0 ( x , y ) u ( t i , y ) d y . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_Equad_HTML.gif
Let ξ X 1 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq162_HTML.gif, it follows from
f ( t , ξ ) , z n = 0 π u ( t , x ) sin t 1 2 2 π sin ( n x ) d x = 1 n 0 π [ x u ( t , x ) ] sin t 1 2 2 π cos ( n x ) d x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_Equae_HTML.gif
and
h ( t , ξ ) , z n = 0 π u ( t , x ) cos t 2 π sin ( n x ) d x = 1 n 0 π [ x u ( t , x ) ] cos t 2 π cos ( n x ) d x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_Equaf_HTML.gif

that f and h are functions from [ 0 , T ] × X 1 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq163_HTML.gif into X 1 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq164_HTML.gif and they are continuous. Moreover, a similar computation of [17] together with Lemma 6 and assumption (P1) shows that g ( u ) X 1 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq165_HTML.gif whenever u C ( [ 0 , T ] , X 1 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq166_HTML.gif.

Then for any r > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq80_HTML.gif, we see that the assumptions ( H 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq19_HTML.gif)-( H 6 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq131_HTML.gif) are satisfied with
φ r ( t ) = r sin t 1 2 , ϕ r ( t ) = r cos t , σ 1 = σ 2 = 2 T 1 2 , L = ( n + 1 ) ( 0 π 0 π [ x K 0 ( x , y ) ] 2 d x d y ) 1 2 , ρ 1 ( t ) = ρ 2 ( t ) = 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_Equag_HTML.gif

Thus, the system (5) has at least one mild solution due to Theorem 1 provided that L + 2 T 1 2 π ( 1 + K ) < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq167_HTML.gif. And by Theorem 2, this mild solution of the system (5) is unique on [ 0 , T ] × [ 0 , π ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-113/MediaObjects/13661_2012_Article_208_IEq168_HTML.gif.

Declarations

Acknowledgements

Research was supported by the Fundamental Research Funds for the Gansu Universities.

Authors’ Affiliations

(1)
Department of Mathematics, Northwest Normal University

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