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# Abstract fractional integro-differential equations involving nonlocal initial conditions in ╬▒-norm

## Abstract

In the present paper, we deal with the Cauchy problems of abstract fractional integro-differential equations involving nonlocal initial conditions in ╬▒-norm, where the operator A in the linear part is the generator of a compact analytic semigroup. New criterions, ensuring the existence of mild solutions, are established. The results are obtained by using the theory of operator families associated with the function of Wright type and the semigroup generated by A, Krasnoselkii's fixed point theorem and Schauder's fixed point theorem. An application to a fractional partial integro-differential equation with nonlocal initial condition is also considered.

Mathematics subject classification (2000)

26A33, 34G10, 34G20

## 1 Introduction

Let (A, D(A)) be the infinitesimal generator of a compact analytic semigroup of bounded linear operators {T(t)} tŌēź0on a real Banach space (X, ||┬Ę||) and 0 Ōłł Žü(A). Denote by X ╬▒ , the Banach space D(A ╬▒ ) endowed with the graph norm ||u|| ╬▒ = ||A ╬▒ u|| for u Ōłł X ╬▒ . The present paper concerns the study of the Cauchy problem for abstract fractional integro-differential equation involving nonlocal initial condition

$c D t ╬▓ u ( t ) = A u ( t ) + F ( t , u ( t ) , u ( ╬║ 1 ( t ) ) ) + Ōł½ 0 t K ( t - s ) G ( s , u ( s ) , u ( ╬║ 2 ( s ) ) ) d s , t Ōłł [ 0 , T ] , u ( 0 ) + H ( u ) = u 0$
(1.1)

in X ╬▒ , where $c D t ╬▓$, 0 < ╬▓ < 1, stands for the Caputo fractional derivative of order ╬▓, and K : [0, T] ŌåÆ ŌäØ+, ╬║ 1, ╬║ 2 : [0, T] ŌåÆ[0, T], F, G : [0, T] ├Ś X ╬▒ ├Ś X ╬▒ ŌåÆ X, H : C([0, T]; X ╬▒ ) ŌåÆ X ╬▒ are given functions to be specified later. As can be seen, H constitutes a nonlocal condition.

The fractional calculus that allows us to consider integration and differentiation of any order, not necessarily integer, has been the object of extensive study for analyzing not only anomalous diffusion on fractals (physical objects of fractional dimension, like some amorphous semiconductors or strongly porous materials; see [1ŌĆō3] and references therein), but also fractional phenomena in optimal control (see, e.g., [4ŌĆō6]). As indicated in [2, 5, 7] and the related references given there, the advantages of fractional derivatives become apparent in modeling mechanical and electrical properties of real materials, as well as in the description of rheological properties of rocks, and in many other fields. One of the emerging branches of the study is the Cauchy problems of abstract differential equations involving fractional derivatives in time. In recent decades, there has been a lot of interest in this type of problems, its applications and various generalizations (cf. e.g., [8ŌĆō11] and references therein). It is significant to study this class of problems, because, in this way, one is more realistic to describe the memory and hereditary properties of various materials and processes (cf. [4, 5, 12, 13]).

In particular, much interest has developed regarding the abstract fractional Cauchy problems involving nonlocal initial conditions. For example, by using the fractional power of operators and some fixed point theorems, the authors studied the existence of mild solutions in  for fractional differential equations with nonlocal initial conditions and in  for fractional neutral differential equations with nonlocal initial conditions and time delays. The existence of mild solutions for fractional differential equations with nonlocal initial conditions in ╬▒-norm using the contraction mapping principle and the Schauder's fixed point theorem have been investigated in .

We here mention that the abstract problem with nonlocal initial condition was first considered by Byszewski , and the importance of nonlocal initial conditions in different fields has been discussed in [18, 19] and the references therein. Deng , especially, gave the following nonlocal initial values: $H ( u ) = Ōłæ i = 1 p C i u ( t i )$, where C i (i = 1, ..., p) are given constants and 0 < t 1 < ┬Ę┬Ę┬Ę < t p-1< t p < + Ōł× (p Ōłł ŌäĢ), which is used to describe the diffusion phenomenon of a small amount of gas in a transparent tube. In the past several years theorems about existence, uniqueness and stability of Cauchy problem for abstract evolution equations with nonlocal initial conditions have been studied by many authors, see for instance [19ŌĆō28] and references therein.

In this paper, we will study the existence of mild solutions for the fractional Cauchy problem (1.1). New criterions are established. Both Krasnoselkii's fixed point theorem and Schauder's fixed point theorem, and the theory of operator families associated with the function of Wright type and the semigroup generated by A, are employed in our approach. The results obtained are generalizations and continuation of the recent results on this issue.

The paper is organized as follows. In Section 2, some required notations, definitions and lemmas are given. In Section 3, we present our main results and their proofs.

## 2 Preliminaries

In this section, we introduce some notations, definitions and preliminary facts which are used throughout this work.

We first recall some definitions of fractional calculus (see e.g., [6, 13] for more details).

Definition 2.1 The Riemann-Liouville fractional integral operator of order ╬▓ > 0 of function f is defined as

$I ╬▓ f ( t ) = 1 ╬ō ( ╬▓ ) Ōł½ 0 t ( t - s ) ╬▓ - 1 f ( s ) d s ,$

provided the right-hand side is pointwise defined on [0, Ōł×), where ╬ō(┬Ę) is the gamma function.

Definition 2.2 The Caputo fractional derivative of order ╬▓ > 0, m - 1 < ╬▓ < m, m Ōłł ŌäĢ, is defined as

$c D ╬▓ f ( t ) = I m ŌłÆ ╬▓ D t m f ( t ) = 1 ╬ō ( m ŌłÆ ╬▓ ) Ōł½ 0 t ( t ŌłÆ s ) m ŌłÆ ╬▓ ŌłÆ 1 D s m f ( s ) d s ,$

where $D t m := d m d t m$ and f is an abstract function with value in X. If 0 < ╬▓ < 1, then

$c D ╬▓ f ( t ) = 1 ╬ō ( 1 ŌłÆ ╬▓ ) Ōł½ 0 t f ŌĆ▓ ( s ) ( t ŌłÆ s ) ╬▓ d s .$

Throughout this paper, we let A : D(A) ŌåÆ X be the infinitesimal generator of a compact analytic semigroup of bounded linear operators {T(t)} tŌēź0on X and 0 Ōłł Žü(A), which allows us to define the fractional power A ╬▒ for 0 Ōēż ╬▒ < 1, as a closed linear operator on its domain D(A ╬▒ ) with inverse A -╬▒.

Let X ╬▒ denote the Banach space D(A ╬▒ ) endowed with the graph norm ||u|| ╬▒ = ||A ╬▒ u|| for u Ōłł X ╬▒ and let C([0, T];X ╬▒ ) be the Banach space of all continuous functions from [0, T] into X ╬▒ with the uniform norm topology

$| u | ╬▒ = sup { Ōłź u ( t ) Ōłź ╬▒ , t Ōłł [ 0 , T ] } .$

ŌäÆ (X) stands for the Banach space of all linear and bounded operators on X. Let M be a constant such that

$M = sup { Ōłź T ( t ) Ōłź ŌäÆ ( X ) , t Ōłł [ 0 , Ōł× ) } .$

For k > 0, write

$╬ś k = { u Ōłł C ( [ 0 , T ] ; X ╬▒ ) ; | u | ╬▒ Ōēż k } .$

The following are basic properties of A ╬▒ .

Theorem 2.1 (, pp. 69-75)).

1. (a)

T(t) : X ŌåÆ X ╬▒ for each t > 0, and A ╬▒ T(t)x = T(t)A ╬▒ x for each x Ōłł X ╬▒ and t Ōēź 0.

2. (b)

A ╬▒ T(t) is bounded on X for every t > 0 and there exist M ╬▒ > 0 and ╬┤ > 0 such that

$| | A ╬▒ T ( t ) | | ŌäÆ ( X ) Ōēż M ╬▒ t ╬▒ e - ╬┤ t .$
3. (c)

A -╬▒ is a bounded linear operator in X with D(A ╬▒ ) = Im(A -╬▒).

4. (d)

0 < ╬▒ 1 Ōēż ╬▒ 2 , then $X ╬▒ 2$.Ōå¬ $X ╬▒ 1$

Lemma 2.1. The restriction of T(t) to X ╬▒ is exactly the part of T(t) in X ╬▒ and is an immediately compact semigroup in X ╬▒ , and hence it is immediately norm-continuous.

Define two families ${ S ╬▓ ( t ) } t Ōēź 0$ and ${ P ╬▓ ( t ) } t Ōēź 0$ of linear operators by

$S ╬▓ ( t ) x = Ōł½ 0 Ōł× ╬© ╬▓ ( s ) T ( t ╬▓ s ) x d s , P ╬▓ ( t ) x = Ōł½ 0 Ōł× ╬▓ s ╬© ╬▓ ( s ) T ( t ╬▓ s ) x d s$

for x Ōłł X, t Ōēź 0, where

$╬© ╬▓ ( s ) = 1 ŽĆ ╬▓ Ōłæ n = 1 Ōł× ( - s ) n - 1 ╬ō ( 1 + ╬▓ n ) n ! sin ( n ŽĆ ╬▓ ) , s Ōłł ( 0 , Ōł× )$

is the function of Wright type defined on (0, Ōł×) which satisfies

$╬© ╬▓ ( s ) Ōēź 0 , s Ōłł ( 0 , Ōł× ) , Ōł½ 0 Ōł× ╬© ╬▓ ( s ) d s = 1 , a n d Ōł½ 0 Ōł× s ╬Č ╬© ╬▓ ( s ) d s = ╬ō ( 1 + ╬Č ) ╬ō ( 1 + ╬▓ ╬Č ) , ╬Č Ōłł ( - 1 , Ōł× ) .$
(2.1)

The following lemma follows from the results in .

Lemma 2.2. The following properties hold:

1. (1)

For every t Ōēź 0, $S ╬▓ ( t )$ and $P ╬▓ ( t )$ are linear and bounded operators on X, i.e.,

$Ōłź S ╬▓ ( t ) x Ōłź Ōēż M Ōłź x Ōłź , Ōłź P ╬▓ ( t ) x Ōłź Ōēż ╬▓ M ╬ō ( 1 + ╬▓ ) Ōłź x Ōłź$

for all x Ōłł X and 0 Ōēż t < Ōł×.

2. (2)

For every x Ōłł X, $tŌåÆ S ╬▓ ( t ) x$, $tŌåÆ P ╬▓ ( t ) x$ are continuous functions from [0, Ōł×) into X.

3. (3)

$S ╬▓ ( t )$ and $P ╬▓ ( t )$ are compact operators on X for t > 0.

4. (4)

For all x Ōłł X and t Ōłł (0, Ōł×), $Ōłź A ╬▒ P ╬▓ ( t ) xŌłźŌēż C ╬▒ t - ╬▒ ╬▓ ŌłźxŌłź$ , where $C ╬▒ = M ╬▒ ╬▓ ╬ō ( 2 - ╬▒ ) ╬ō ( 1 + ╬▓ ( 1 - ╬▒ ) )$.

We can also prove the following criterion.

Lemma 2.3. The functions $tŌåÆ A ╬▒ P ╬▓ ( t )$ and $tŌåÆ A ╬▒ S ╬▓ ( t )$ are continuous in the uniform operator topology on(0, +Ōł×).

Proof. Let ╬Ą > 0 be given. For every r > 0, from (2.1), we may choose ╬┤ 1 , ╬┤ 2 > 0 such that

$M ╬▒ r ╬▒ ╬▓ Ōł½ 0 ╬┤ 1 ╬© ╬▓ ( s ) s - ╬▒ d s Ōēż ╬Ą 6 , M ╬▒ r ╬▒ ╬▓ Ōł½ ╬┤ 2 Ōł× ╬© ╬▓ ( s ) s - ╬▒ d s Ōēż ╬Ą 6 .$
(2.2)

Then, we deduce, in view of the fact t ŌåÆ A ╬▒ T(t) that is continuous in the uniform operator topology on (0, Ōł×) (see [, Lemma 2.1]), that there exists a constant ╬┤ > such that

$Ōł½ ╬┤ 1 ╬┤ 2 ╬© ╬▓ ( s ) A ╬▒ T t 1 ╬▓ s - A ╬▒ T t 2 ╬▓ s ŌäÆ ( X ) d s Ōēż ╬Ą 3 ,$
(2.3)

for t 1, t 2 Ōēź r and |t 1 - t 2| < ╬┤.

On the other hand, for any x Ōłł X, we write

$S ╬▓ ( t 1 ) x - S ╬▓ ( t 2 ) x = Ōł½ 0 ╬┤ 1 ╬© ╬▓ ( s ) T t 1 ╬▓ s x - T t 2 ╬▓ s x d s + Ōł½ ╬┤ 1 ╬┤ 2 ╬© ╬▓ ( s ) T t 1 ╬▓ s x - T t 2 ╬▓ s x d s + Ōł½ ╬┤ 2 Ōł× ╬© ╬▓ ( s ) ( T ( t 1 ╬▓ s ) x - T ( t 2 ╬▓ s ) x ) d s .$

Therefore, using (2.2, 2.3) and Lemma 2.2, we get

$A ╬▒ S ╬▓ ( t 1 ) x - A ╬▒ S ╬▓ ( t 2 ) x Ōēż Ōł½ 0 ╬┤ 1 ╬© ╬▓ ( s ) A ╬▒ T t 1 ╬▓ s ŌäÆ ( X ) + A ╬▒ T t 2 ╬▓ s ŌäÆ ( X ) Ōłź x Ōłź d s + Ōł½ ╬┤ 1 ╬┤ 2 ╬© ╬▓ ( s ) Ōłź A ╬▒ T t 1 ╬▓ s - A ╬▒ T t 2 ╬▓ s Ōłź ŌäÆ ( X ) Ōłź x Ōłź d s + Ōł½ ╬┤ 2 Ōł× ╬© ╬▓ ( s ) A ╬▒ T t 1 ╬▓ s ŌäÆ ( X ) + A ╬▒ T t 2 ╬▓ s ŌäÆ ( X ) Ōłź x Ōłź d s Ōēż 2 M ╬▒ r ╬▒ ╬▓ Ōł½ 0 ╬┤ 1 ╬© ╬▓ ( s ) s - ╬▒ Ōłź x Ōłź d s + Ōł½ ╬┤ 1 ╬┤ 2 ╬© ╬▓ ( s ) T t 1 ╬▓ s - T t 2 ╬▓ s ŌäÆ ( X ) Ōłź x Ōłź d s + 2 M ╬▒ r ╬▒ ╬▓ Ōł½ ╬┤ 2 Ōł× ╬© ╬▓ ( s ) s - ╬▒ Ōłź x Ōłź d s Ōēż ╬Ą Ōłź x Ōłź ,$

that is,

$Ōłź A ╬▒ S ╬▓ ( t 1 ) - A ╬▒ S ╬▓ ( t 2 ) Ōłź Ōēż ╬Ą , f o r t 1 , t 2 Ōēź r a n d | t 1 - t 2 | < ╬┤$

which together with the arbitrariness of r > 0 implies that $A ╬▒ P ╬▓ ( t )$ is continuous in the uniform operator topology for t > 0. A similar argument enable us to give the characterization of continuity on $A ╬▒ P ╬▓ ( t )$. This completes the proof. Ō¢Ā

Lemma 2.4. For every t > 0, the restriction of $S ╬▓ ( t )$ to X ╬▒ and the restriction of $P ╬▓ ( t )$ to X ╬▒ are compact operators in X ╬▒ .

Proof. First consider the restriction of $S ╬▓ ( t )$ to X ╬▒ . For any r > 0 and t > 0, it is sufficient to show that the set ${ S ╬▓ ( t ) u ; u Ōłł B r }$ is relatively compact in X ╬▒ , where B r := {u Ōłł X ╬▒ ; ||u|| ╬▒ Ōēż r}.

Since by Lemma 2.1, the restriction of T(t) to X ╬▒ is compact for t > 0 in X ╬▒ , for each t > 0 and ╬Ą Ōłł (0, t),

$Ōł½ ╬Ą Ōł× ╬© ╬▓ ( s ) T t ╬▓ s u d s ; u Ōłł B r = T t ╬▓ ╬Ą Ōł½ ╬Ą Ōł× ╬© ╬▓ ( s ) T t ╬▓ s - t ╬▓ ╬Ą u d s ; u Ōłł B r$

is relatively compact in X ╬▒ . Also, for every u Ōłł B r , as

$Ōł½ ╬Ą Ōł× ╬© ╬▓ ( s ) T t ╬▓ s u d s ŌåÆ S ╬▓ ( t ) u , ╬Ą ŌåÆ 0$

in X ╬▒ , we conclude, using the total boundedness, that the set ${ S ╬▓ ( t ) u ; u Ōłł B r }$ is relatively compact, which implies that the restriction of $S ╬▓ ( t )$ to X ╬▒ is compact. The same idea can be used to prove that the restriction of $P ╬▓ ( t )$ to X ╬▒ is also compact. Ō¢Ā

The following fixed point theorems play a key role in the proofs of our main results, which can be found in many books.

Lemma 2.5 (Krasnoselskii's Fixed Point Theorem). Let E be a Banach space and B be a bounded closed and convex subset of E, and let F 1, F 2 be maps of B into E such that F 1 x + F 2 y Ōłł B for every pair x, y Ōłł B. If F 1 is a contraction and F 2 is completely continuous, then the equation F 1 x + F 2 x = x has a solution on B.

Lemma 2.6 (Schauder Fixed Point Theorem). If B is a closed bounded and convex subset of a Banach space E and F : B ŌåÆ B is completely continuous, then F has a fixed point in B.

## 3 Main results

Based on the work in [, Lemma 3.1 and Definition 3.1], in this paper, we adopt the following definition of mild solution of Cauchy problem (1.1).

Definition 3.1. By a mild solution of Cauchy problem (1.1), we mean a function u Ōłł C([0, T]; X ╬▒ ) satisfying

$u ( t ) = S ╬▓ ( t ) ( u 0 ŌłÆ H ( u ) ) + Ōł½ 0 t ( t ŌłÆ s ) ╬▓ ŌłÆ 1 P ╬▓ ( t ŌłÆ s ) ( F ( s , u ( s ) , u ( ╬║ 1 ( s ) ) ) + Ōł½ 0 s K ( s ŌłÆ Žä ) G ( Žä , u ( Žä ) , u ( ╬║ 2 ( Žä ) ) ) d Žä ) d s$

for t Ōłł [0, T].

Let us first introduce our basic assumptions.

(H 0) ╬║ 1, ╬║ 2 Ōłł C([0, T]; [0, T]) and K Ōłł C([0, T]; ŌäØ+).

(H 1) F, G : [0, T] ├Ś X ╬▒ ├Ś X ╬▒ ŌåÆ X are continuous and for each positive number k Ōłł ŌäĢ, there exist a constant ╬│ Ōłł [0, ╬▓(1 - ╬▒)) and functions Žå k (┬Ę) Ōłł L 1/╬│(0, T; ŌäØ+), ŽĢ k (┬Ę) Ōłł L Ōł×(0, T; ŌäØ+) such that

$sup Ōłź u Ōłź ╬▒ , Ōłź v Ōłź ╬▒ Ōēż k Ōłź F ( t , u , v ) Ōłź Ōēż Žå k ( t ) a n d liminf k ŌåÆ + Ōł× Ōłź Žå k Ōłź L 1 ŌłĢ ╬│ ( 0 , T ) k = Žā 1 < Ōł× , sup Ōłź u Ōłź ╬▒ , Ōłź v Ōłź ╬▒ Ōēż k Ōłź G ( t , u , v ) Ōłź Ōēż ŽĢ k ( t ) a n d liminf k ŌåÆ + Ōł× Ōłź ŽĢ k Ōłź L Ōł× ( 0 , T ) k = Žā 2 < Ōł× .$

(H 2) F, G : [0, T] ├Ś X ╬▒ ├Ś X ╬▒ ŌåÆ X are continuous and there exist constants L F , L K such that

$Ōłź F ( t , u 1 , v 1 ) - F ( t , u 2 , v 2 ) Ōłź Ōēż L F ( Ōłź u 1 - u 2 Ōłź ╬▒ + Ōłź v 1 - v 2 Ōłź ╬▒ ) , Ōłź G ( t , u 1 , v 1 ) - G ( t , u 2 , v 2 ) Ōłź Ōēż L G ( Ōłź u 1 - u 2 Ōłź ╬▒ + Ōłź v 1 - v 2 Ōłź ╬▒ )$

for all (t, u 1, v 1), (t, u 2, v 2) Ōłł [0, T] ├Ś X ╬▒ ├Ś X ╬▒ .

(H 3) H : C([0, T]; X ╬▒ ) ŌåÆ X ╬▒ is Lipschitz continuous with Lipschitz constant L H .

(H 4) H : C([0, T]; X ╬▒ ) ŌåÆ X ╬▒ is continuous and there is a ╬Ę Ōłł (0, T) such that for any u, w Ōłł C([0, T]; X ╬▒ ) satisfying u(t) = w(t)(t Ōłł[╬Ę, T]), H(u) = H(w).

(H 5) There exists a nondecreasing continuous function ╬” : ŌäØ+ ŌåÆ ŌäØ+ such that for all u Ōłł ╬ś k ,

$Ōłź H ( u ) Ōłź ╬▒ Ōēż ╬” ( k ) , a n d liminf k ŌåÆ + Ōł× ╬” ( k ) k = ╬╝ < Ōł× .$

Remark 3.1. Let us note that (H 4) is the case when the values of the solution u(t) for t near zero do not affect H(u). We refer tofor a case in point.

In the sequel, we set $k ╠ā : = Ōł½ 0 T K ( t ) d t$. We are now ready to state our main results in this section.

Theorem 3.1. Let the assumptions (H 0), (H 1) and (H 3) be satisfied. Then, for u 0 Ōłł X ╬▒ , the fractional Cauchy problem (1.1) has at least one mild solution provided that

$M L H + C ╬▒ Žā 1 T ( 1 - ╬▒ ) ╬▓ - ╬│ 1 - ╬│ ( 1 - ╬▒ ) ╬▓ - ╬│ 1 - ╬│ + C ╬▒ Žā 2 k ╠ā T ( 1 - ╬▒ ) ╬▓ ( 1 - ╬▒ ) ╬▓ < 1 .$
(3.1)

Proof. Let v Ōłł C([0, T]; X ╬▒ ) be fixed with |v| ╬▒ ŌēĪ 0. From (3.1) and (H 1), it is easy to see that there exists a k 0 > 0 such that

$M ( Ōłź u 0 Ōłź ╬▒ + L H k 0 + Ōłź H ( ╬Į ) Ōłź ╬▒ ) + C ╬▒ 1 - ╬│ ( 1 - ╬▒ ) ╬▓ - ╬│ 1 - ╬│ T ( 1 - ╬▒ ) ╬▓ - ╬│ Ōłź Žå k 0 Ōłź L 1 ŌłĢ ╬│ ( 0 , T ) + C ╬▒ k ╠ā T ( 1 - ╬▒ ) ╬▓ ( 1 - ╬▒ ) ╬▓ Ōłź ŽĢ k 0 Ōłź L Ōł× ( 0 , T ) Ōēż k 0 .$

Consider a mapping ╬ō defined on $╬ś k 0$ by

It is easy to verify that (╬ōu)(┬Ę) Ōłł C([0, T]; X ╬▒ ) for every $uŌłł ╬ś k 0$. Moreover, for every pair $v,uŌłł ╬ś k 0$ and t Ōłł [0, T], by (H 1) a direct calculation yields

That is, $╬ō 1 v+ ╬ō 2 uŌłł ╬ś k 0$ for every pair $v,uŌłł ╬ś k 0$. Therefore, the fractional Cauchy problem (1.1) has a mild solution if and only if the operator equation ╬ō1 u + ╬ō2 u = u has a solution in $╬ś k 0$.

In what follows, we will show that ╬ō1 and ╬ō2 satisfy the conditions of Lemma 2.5. From (H 3) and (3.1), we infer that ╬ō1 is a contraction. Next, we show that ╬ō2 is completely continuous on $╬® k 0$.

We first prove that ╬ō2 is continuous on $╬ś k 0$. Let ${ u n } n = 1 Ōł× ŌŖé ╬ś k 0$ be a sequence such that u n ŌåÆ u as n ŌåÆ Ōł× in C([0, T]; X ╬▒ ). Therefore, it follows from the continuity of F, G, ╬║ 1 and ╬║ 2 that for each t Ōłł [0, T],

$F ( t , u n ( t ) , u n ( ╬║ 1 ( t ) ) ) ŌåÆ F ( t , u ( t ) , u ( ╬║ 1 ( t ) ) ) a s n ŌåÆ Ōł× , G ( t , u n ( t ) , u n ( ╬║ 1 ( t ) ) ) ŌåÆ G ( t , u ( t ) , u ( ╬║ 2 ( t ) ) ) a s n ŌåÆ Ōł× .$

Also, by (H 1), we see

$Ōł½ 0 t ( t - s ) ╬▓ - 1 - ╬▒ ╬▓ Ōłź F ( s , u n ( s ) , u n ( ╬║ 1 ( s ) ) ) - F ( s , u ( s ) , u ( ╬║ 1 ( s ) ) ) Ōłź d s Ōēż 2 Ōł½ 0 t ( t - s ) ╬▓ - 1 - ╬▒ ╬▓ Žå k 0 ( s ) d s Ōēż 2 1 - ╬│ ( 1 - ╬▒ ) ╬▓ - ╬│ 1 - ╬│ T ( 1 - ╬▒ ) ╬▓ - ╬│ Ōłź Žå k 0 Ōłź L 1 ŌłĢ ╬│ ( 0 , T ) ,$

and

$Ōł½ 0 t ( t - s ) ╬▓ - 1 - ╬▒ ╬▓ Ōł½ 0 s K ( s - Žä ) Ōłź G ( Žä , u n ( Žä ) , u n ( ╬║ 2 ( Žä ) ) ) - G ( Žä , u ( Žä ) , u ( ╬║ 2 ( Žä ) ) ) Ōłź d Žä d s Ōēż 2 k ╠ā Ōłź ŽĢ k 0 Ōłź L Ōł× ( 0 , T ) Ōł½ 0 t ( t - s ) ╬▓ - 1 - ╬▒ ╬▓ d s Ōēż 2 k ╠ā T ( 1 - ╬▒ ) ╬▓ ( 1 - ╬▒ ) ╬▓ Ōłź ŽĢ k 0 Ōłź L Ōł× ( 0 , T ) .$

Hence, as

$Ōłź ( ╬ō 2 u n ) ( t ) - ( ╬ō 2 u ) ( t ) Ōłź ╬▒ Ōēż C ╬▒ Ōł½ 0 t ( t - s ) ╬▓ - 1 - ╬▒ ╬▓ Ōłź F ( s , u n ( s ) , u n ( ╬║ 1 ( s ) ) ) - F ( s , u ( s ) , u ( ╬║ 1 ( s ) ) ) Ōłź d s + C ╬▒ Ōł½ 0 t ( t - s ) ╬▓ - 1 - ╬▒ ╬▓ Ōł½ 0 s K ( s - Žä ) Ōłź G ( Žä , u n ( Žä ) , u n ( ╬║ 2 ( Žä ) ) ) - G ( Žä , u ( Žä ) , u ( ╬║ 2 ( Žä ) ) ) Ōłź d Žä d s ,$

we conclude, using the Lebesgue dominated convergence theorem, that for all t Ōłł [0, T],

$Ōłź ( ╬ō 2 u n ) ( t ) - ( ╬ō 2 u ) ( t ) Ōłź ╬▒ ŌåÆ 0 , a s n ŌåÆ Ōł× ,$

which implies that

$| ╬ō 2 u n - ╬ō 2 u | ╬▒ ŌåÆ 0 , a s n ŌåÆ Ōł× .$

This proves that ╬ō2 is continuous on $╬ś k 0$.

It suffice to prove that ╬ō2 is compact on $╬® k 0$. For the sake of brevity, we write

$N ( t , u ( t ) ) = F ( t , u ( t ) , u ( ╬║ 1 ( t ) ) ) + Ōł½ 0 t K ( t - Žä ) G ( Žä , u ( Žä ) , u ( ╬║ 2 ( Žä ) ) ) d Žä .$

Let t Ōłł [0, T] be fixed and ╬Ą, ╬Ą 1 > 0 be small enough. For $uŌłł ╬® k 0$, we define the map $╬ō ╬Ą , ╬Ą 1$ by

$( ╬ō ╬Ą , ╬Ą 1 u ) ( t ) = Ōł½ 0 t ŌłÆ ╬Ą Ōł½ ╬Ą 1 Ōł× ╬▓ Žä ╬© ╬▓ ( Žä ) T ( ( t ŌłÆ s ) ╬▓ Žä ) N ( s , u ( s ) ) d Žä d s = T ( ╬Ą ╬▓ ╬Ą 1 ) Ōł½ 0 t ŌłÆ ╬Ą Ōł½ ╬Ą 1 Ōł× ╬▓ Žä ╬© ╬▓ ( Žä ) T ( ( t ŌłÆ s ) ╬▓ Žä ŌłÆ ╬Ą ╬▓ ╬Ą 1 ) N ( s , u ( s ) ) d Žä d s .$

Therefore, from Lemma 2.1 we see that for each t Ōłł (0, T], the set ${ ╬ō ╬Ą , ╬Ą 1 u ) ( t ) ; u Ōłł ╬® k 0 }$ is relatively compact in X ╬▒ . Then, as

$Ōłź ( ╬ō 2 u ) ( t ) ŌłÆ ( ╬ō ╬Ą , ╬Ą 1 u ) ( t ) Ōłź ╬▒ Ōēż ŌĆ¢ Ōł½ 0 t Ōł½ 0 ╬Ą 1 ╬▓ Žä ( t ŌłÆ s ) ╬▓ ŌłÆ 1 ╬© ╬▓ ( Žä ) T ( ( t ŌłÆ s ) ╬▓ Žä ) N ( s , u ( s ) ) d Žä d s ŌĆ¢ ╬▒ + ŌĆ¢ Ōł½ t ŌłÆ ╬Ą t Ōł½ ╬Ą 1 Ōł× ╬▓ Žä ( t ŌłÆ s ) ╬▓ ŌłÆ 1 ╬© ╬▓ ( Žä ) T ( ( t ŌłÆ s ) ╬▓ Žä ) N ( s , u ( s ) ) d Žä d s ŌĆ¢ ╬▒ Ōēż ╬▓ M ╬▒ [ Ōł½ 0 t ( t ŌłÆ s ) ╬▓ ( 1 ŌłÆ ╬▒ ) ŌłÆ 1 ( Žå k 0 ( s ) + k ╦£ Ōłź ŽĢ k 0 Ōłź L Ōł× ( 0, T ) ) d s Ōł½ 0 ╬Ą 1 Žä 1 ŌłÆ ╬▒ ╬© ╬▓ ( Žä ) d Žä + Ōł½ t ŌłÆ ╬Ą t ( t ŌłÆ s ) ╬▓ ( 1 ŌłÆ ╬▒ ) ŌłÆ 1 ( Žå k 0 ( s ) + k ╦£ Ōłź ŽĢ k 0 Ōłź L Ōł× ( 0, T ) ) d s Ōł½ ╬Ą 1 Ōł× Žä 1 ŌłÆ ╬▒ ╬© ╬▓ ( Žä ) d Žä ] Ōēż ╬▓ M ╬▒ [ ( 1 ŌłÆ ╬│ ( 1 ŌłÆ ╬▒ ) ╬▓ ŌłÆ ╬│ ) 1 ŌłÆ ╬│ T ( 1 ŌłÆ ╬▒ ) ╬▓ ŌłÆ ╬│ Ōłź Žå k 0 Ōłź L 1 / ╬│ ( 0, T ) + k ╦£ T ( 1 ŌłÆ ╬▒ ) ╬▓ ( 1 ŌłÆ ╬▒ ) ╬▓ Ōłź ŽĢ k 0 Ōłź L Ōł× ( 0, T ) ] Ōł½ 0 ╬Ą 1 Žä 1 ŌłÆ ╬▒ ╬© ╬▓ ( Žä ) d Žä + ╬▓ M ╬▒ ╬ō ( 2 ŌłÆ ╬▒ ) ╬ō ( 1 + ╬▓ ( 1 ŌłÆ ╬▒ ) ) [ ( 1 ŌłÆ ╬│ ( 1 ŌłÆ ╬▒ ) ╬▓ ŌłÆ ╬│ ) 1 ŌłÆ ╬│ Ōłź Žå k 0 Ōłź L 1 / ╬│ ( 0, T ) ╬Ą ( 1 ŌłÆ ╬▒ ) ╬▓ ŌłÆ ╬│ + k ╦£ ( 1 ŌłÆ ╬▒ ) ╬▓ Ōłź ŽĢ k 0 Ōłź L Ōł× ( 0, T ) ╬Ą ( 1 ŌłÆ ╬▒ ) ╬▓ ] ŌåÆ 0 a s ╬Ą , ╬Ą 1 ŌåÆ 0 +$

in view of (2.1), we conclude, using the total boundedness, that for each t Ōłł [0, T], the set ${ ╬ō 2 u ) ( t ) ;uŌłł ╬® k 0 }$ is relatively compact in X ╬▒ .

On the other hand, for 0 < t 1 < t 2 Ōēż T and ╬Ą' > 0 small enough, we have

$Ōłź ( ╬ō 2 u ) ( t 1 ) - ( ╬ō 2 u ) ( t 2 ) Ōłź ╬▒ Ōēż A 1 + A 2 + A 3 + A 4 ,$

where

Therefore, it follows from (H 1), Lemma 2.2, and Lemma 2.3 that

$A 1 Ōēż C ╬▒ Ōł½ t 1 t 2 ( t 2 ŌłÆ s ) ╬▓ ŌłÆ 1 ŌłÆ ╬▒ ╬▓ ( Žå k 0 ( s ) + k ╦£ Ōłź ŽĢ k 0 Ōłź L Ōł× ( 0, T ) ) d s Ōēż C ╬▒ ( 1 ŌłÆ ╬│ ( 1 ŌłÆ ╬▒ ) ╬▓ ŌłÆ ╬│ ) 1 ŌłÆ ╬│ Ōłź Žå k 0 Ōłź L 1 / ╬│ ( 0, T ) ( t 2 ŌłÆ t 1 ) ( 1 ŌłÆ ╬▒ ) ╬▓ ŌłÆ ╬│ + C ╬▒ k ╦£ Ōłź ŽĢ k 0 Ōłź L Ōł× ( 0, T ) ( 1 ŌłÆ ╬▒ ) ╬▓ ( t 2 ŌłÆ t 1 ) ( 1 ŌłÆ ╬▒ ) ╬▓ , A 2 Ōēż sup s Ōłł [ 0, t 1 ŌłÆ ╬Ą ŌĆ▓ ] Ōłź A ╬▒ ╬▒ ( t 2 ŌłÆ s ) ŌłÆ A ╬▒ ╬▒ ( t 1 ŌłÆ s ) Ōłź ŌäÆ ( X ) ├Ś Ōł½ 0 t 1 ŌłÆ ╬Ą ŌĆ▓ ( t 1 ŌłÆ s ) ╬▓ ŌłÆ 1 ( Žå k 0 ( s ) + k ╦£ Ōłź ŽĢ k 0 Ōłź L Ōł× ( 0, T ) ) d s Ōēż [ ( 1 ŌłÆ ╬│ ╬▓ ŌłÆ ╬│ ) 1 ŌłÆ ╬│ Ōłź Žå k 0 Ōłź L 1 / ╬│ ( 0, T ) ( t 1 ╬▓ ŌłÆ ╬│ 1 ŌłÆ ╬│ ŌłÆ ╬Ą ŌĆ▓ ╬▓ ŌłÆ ╬│ 1 ŌłÆ ╬│ ) 1 ŌłÆ ╬│ + k ╦£ Ōłź ŽĢ k 0 Ōłź L Ōł× ( 0, T ) ╬▓ ( t 1 ╬▓ ŌłÆ ╬Ą ŌĆ▓ ╬▓ ) ] ├Ś sup s Ōłł [ 0, t 1 ŌłÆ ╬Ą ŌĆ▓ ] Ōłź A ╬▒ ╬▒ ( t 2 ŌłÆ s ) ŌłÆ A ╬▒ ╬▒ ( t 1 ŌłÆ s ) Ōłź ŌäÆ ( X ) , A 3 Ōēż C ╬▒ Ōł½ t 1 ŌłÆ ╬Ą ŌĆ▓ t 1 ( t 1 ŌłÆ s ) ╬▓ ŌłÆ 1 ( ( t 2 ŌłÆ s ) ŌłÆ ╬▒ ╬▓ + ( t 1 ŌłÆ s ) ŌłÆ ╬▒ ╬▓ ) ├Ś ( Žå k 0 ( s ) + k ╦£ Ōłź ŽĢ k 0 Ōłź L Ōł× ( 0, T ) ) d s Ōēż 2 C ╬▒ Ōł½ t 1 ŌłÆ ╬Ą ŌĆ▓ t 1 ( t 1 ŌłÆ s ) ╬▓ ŌłÆ 1 ŌłÆ ╬▒ ╬▓ ( Žå k 0 ( s ) + k ╦£ Ōłź ŽĢ k 0 Ōłź L Ōł× ( 0, T ) ) d s Ōēż C ╬▒ ( 1 ŌłÆ ╬│ ( 1 ŌłÆ ╬▒ ) ╬▓ ŌłÆ ╬│ ) 1 ŌłÆ ╬│ Ōłź Žå k 0 Ōłź L 1 / ╬│ ( 0, T ) ╬Ą ŌĆ▓ ( 1 ŌłÆ ╬▒ ) ╬▓ ŌłÆ ╬│ + C ╬▒ k ╦£ Ōłź ŽĢ k 0 Ōłź L Ōł× ( 0, T ) ( 1 ŌłÆ ╬▒ ) ╬▓ ╬Ą ŌĆ▓ ( 1 ŌłÆ ╬▒ ) ╬▓ , A 4 Ōēż C ╬▒ Ōł½ 0 t 1 ( ( t 1 ŌłÆ s ) ╬▓ ŌłÆ 1 ŌłÆ ( t 2 ŌłÆ s ) ╬▓ ŌłÆ 1 ) ( t 2 ŌłÆ s ) ŌłÆ ╬▒ ╬▓ ├Ś ( Žå k 0 ( s ) + k ╦£ Ōłź ŽĢ k 0 Ōłź L Ōł× ( 0, T ) ) d s Ōēż C ╬▒ Ōł½ 0 t 1 ( ( t 1 ŌłÆ s ) ( 1 ŌłÆ ╬▒ ) ╬▓ ŌłÆ 1 ŌłÆ ( t 2 ŌłÆ s ) ( 1 ŌłÆ ╬▒ ) ╬▓ ŌłÆ 1 ) ( Žå k 0 ( s ) + k ╦£ Ōłź ŽĢ k 0 Ōłź L Ōł× ( 0, T ) ) d s Ōēż C ╬▒ ( 1 ŌłÆ ╬│ ( 1 ŌłÆ ╬▒ ) ╬▓ ŌłÆ ╬│ ) 1 ŌłÆ ╬│ Ōłź Žå k 0 Ōłź L 1 / ╬│ ( 0, T ) ├Ś [ t 1 ( 1 ŌłÆ ╬▒ ) ╬▓ ŌłÆ ╬│ ŌłÆ ( t 2 ( 1 ŌłÆ ╬▒ ) ╬▓ ŌłÆ ╬│ 1 ŌłÆ ╬│ ŌłÆ ( t 2 ŌłÆ t 1 ) ( 1 ŌłÆ ╬▒ ) ╬▓ ŌłÆ ╬│ 1 ŌłÆ ╬│ ) 1 ŌłÆ ╬│ ] + 2 k ╦£ ( 1 ŌłÆ ╬▒ ) ╬▓ Ōłź ŽĢ k 0 Ōłź L Ōł× ( 0, T ) ( t 1 ( 1 ŌłÆ ╬▒ ) ╬▓ ŌłÆ t 2 ( 1 ŌłÆ ╬▒ ) ╬▓ + ( t 2 ŌłÆ t 1 ) ( 1 ŌłÆ ╬▒ ) ╬▓ ) ,$

from which it is easy to see that A i (i = 1, 2, 3, 4) tends to zero independently of $uŌłł ╬® k 0$ as t 2 - t 1 ŌåÆ 0 and ╬Ą' ŌåÆ 0. Hence, we can conclude that

$Ōłź ( ╬ō 2 u ) ( t 1 ) - ( ╬ō 2 u ) ( t 2 ) Ōłź ╬▒ ŌåÆ 0 , a s t 2 - t 1 ŌåÆ 0 ,$

and the limit is independently of $uŌłł ╬® k 0$.

For the case when 0 = t 1 < t 2 Ōēż T, since

$Ōłź ( ╬ō 2 u ) ( t 1 ) - ( ╬ō 2 u ) ( t 2 ) Ōłź ╬▒ = Ōł½ 0 t 2 ( t 2 - s ) ╬▓ - 1 P ╬▓ ( t 2 - s ) N ( s , u ( s ) ) d s ╬▒ Ōēż C ╬▒ Ōł½ 0 t 2 ( t 2 - s ) ╬▓ - 1 - ╬▒ ╬▓ ( Žå k 0 ( s ) + k ╠ā Ōłź ŽĢ k 0 Ōłź L Ōł× ( 0 , T ) ) d s Ōēż C ╬▒ 1 - ╬│ ( 1 - ╬▒ ) ╬▓ - ╬│ 1 - ╬│ Ōłź Žå k Ōłź L 1 ŌłĢ ╬│ ( 0 , T ) t 2 ( 1 - ╬▒ ) ╬▓ - ╬│ + C ╬▒ k ╠ā Ōłź ŽĢ k 0 Ōłź L Ōł× ( 0 , T ) ( 1 - ╬▒ ) ╬▓ t 2 ( 1 - ╬▒ ) ╬▓ .$

||(╬ō2 u)(t 1) - (╬ō2 u)(t 2)|| ╬▒ can be made small when t 2 is small independently of $uŌłł ╬® k 0$. Consequently, the set ${ ( ╬ō 2 ) ( Ōŗģ ) ; Ōŗģ Ōłł [ 0 , T ] , u Ōłł ╬® k 0 }$ is equicontinuous. Now applying the Arzela-Ascoli theorem, it follows that ╬ō2 is compact on $╬® k 0$.

Therefore, applying Lemma 2.5, we conclude that ╬ō has a fixed point, which gives rise to a mild solution of Cauchy problem (1.1). This completes the proof. Ō¢Ā

The second result of this paper is the following theorem.

Theorem 3.2. Let the assumptions (H 0), (H 2), (H 4) and (H 5) be satisfied. Then, for u 0 Ōłł X ╬▒ , the fractional Cauchy problem (1.1) has at least one mild solution provided that

$M ╬╝ + 2 C ╬▒ T ( 1 - ╬▒ ) ╬▓ ( L F + k ╠ā L G ) ( 1 - ╬▒ ) ╬▓ < 1 .$
(3.2)

Proof. The proof is divided into the following two steps.

Step 1. Assume that w Ōłł C([╬Ę, T]; X ╬▒ ) is fixed and set

$w ╠ā ( t ) = w ( t ) , t Ōłł [ ╬Ę , T ] , w ( ╬Ę ) , t Ōłł [ 0 , ╬Ę ] .$

It is clear that w Ōłł C([0, T]; X ╬▒ ). We define a mapping ╬ō w on C([0, T]; X ╬▒ ) by

$( ╬ō w u ) ( t ) = S ╬▓ ( t ) ( u 0 ŌłÆ H ( w ╦£ ) ) + Ōł½ 0 t ( t ŌłÆ s ) ╬▓ ŌłÆ 1 P ╬▓ ( t ŌłÆ s ) ( F ( s , u ( s ) , u ( ╬║ 1 ( s ) ) ) + Ōł½ 0 s K ( s ŌłÆ Žä ) G ( Žä , u ( Žä ) , u ( ╬║ 2 ( Žä ) ) ) d Žä ) d s , t Ōłł [ 0, T ] .$

Clearly, (╬ō w u)(┬Ę) Ōłł C([0, T]; X ╬▒ ) for every u Ōłł C([0, T]; X ╬▒ ). Moreover, for u Ōłł ╬ś k , from (H 2), it follows that

$Ōłź ( ╬ō w u ) ( t ) Ōłź ╬▒ Ōēż Ōłź S ╬▓ ( t ) ( u 0 ŌłÆ H ( w ╦£ ) ) Ōłź ╬▒ + Ōł½ 0 t ( t ŌłÆ s ) ╬▓ ŌłÆ 1 Ōłź P ╬▓ ( t ŌłÆ s ) ( F ( s , u ( s ) , u ( ╬║ 1 ( s ) ) ) + Ōł½ 0 s K ( s ŌłÆ Žä ) G ( Žä , u ( Žä ) , u ( ╬║ 2 ( Žä ) ) ) d Žä ) Ōłź ╬▒ d s Ōēż M ( Ōłź u 0 Ōłź ╬▒ + Ōłź H ( w ╦£ ) Ōłź ╬▒ ) + C ╬▒ Ōł½ 0 t ( t ŌłÆ s ) ╬▓ ( 1 ŌłÆ ╬▒ ) ŌłÆ 1 [ L F ( Ōłź u ( s ) Ōłź ╬▒ + Ōłź u ( ╬║ 1 ( s ) ) Ōłź ╬▒ ) + Ōłź F ( s , ╬Į , ╬Į ) Ōłź + L G Ōł½ 0 s K ( s ŌłÆ Žä ) ( Ōłź u ( Žä ) Ōłź ╬▒ + Ōłź u ( ╬║ 2 ( Žä ) ) Ōłź ╬▒ + Ōłź G ( s , ╬Į , ╬Į ) Ōłź ) d Žä ] d s Ōēż M ( Ōłź u 0 Ōłź ╬▒ + Ōłź H ( w ╦£ ) Ōłź ╬▒ ) + 2 k C ╬▒ ( L F + k ╦£ L G ) t ( 1 ŌłÆ ╬▒ ) ╬▓ ( 1 ŌłÆ ╬▒ ) ╬▓ + C ╬▒ ( max 0 Ōēż s Ōēż T Ōłź F ( s , ╬Į , ╬Į ) Ōłź + k ╦£ max 0 Ōēż s Ōēż T Ōłź G ( s , ╬Į , ╬Į ) Ōłź ) T ( 1 ŌłÆ ╬▒ ) ╬▓ ( 1 ŌłÆ ╬▒ ) ╬▓ ,$

where v Ōłł C([0, T]; X ╬▒ ) is fixed with |v| ╬▒ ŌēĪ 0, which implies that there exists a integer k 0 > 0 such that ╬ō w maps $╬ś k 0$ into itself. In fact, if this is not the case, then for each k > 0, there would exist u k Ōłł ╬ś k and t k Ōłł [0, T] such that ||(╬ō w u k )(t k )|| ╬▒ > k. Thus, we have

$k < Ōłź ( ╬ō w u k ) ( t k ) Ōłź ╬▒ Ōēż M ( Ōłź u 0 Ōłź ╬▒ + Ōłź H ( u ╠ā ) Ōłź ╬▒ ) + 2 k C ╬▒ ( L F + k ╠ā L G ) T ( 1 - ╬▒ ) ╬▓ ( 1 - ╬▒ ) ╬▓ + C ╬▒ max 0 Ōēż s Ōēż T Ōłź F ( s , ╬Į , ╬Į ) Ōłź + k ╠ā max 0 Ōēż s Ōēż T Ōłź G ( s , ╬Į , ╬Į ) Ōłź T ( 1 - ╬▒ ) ╬▓ ( 1 - ╬▒ ) ╬▓$

Dividing on both sides by k and taking the lower limit as k ŌåÆ +Ōł×, we get

$1 Ōēż 2 C ╬▒ ( L F + k ╠ā L G ) T ( 1 - ╬▒ ) ╬▓ ( 1 - ╬▒ ) ╬▓ ,$

this contradicts (3.2). Also, for $u,vŌłł ╬ś k 0$, a direct calculation yields

which together with (3.2) implies that ╬ō w is a contraction mapping on $╬ś k 0$. Thus, by the Banach contraction mapping principle, ╬ō w has a unique fixed point $u w Ōłł ╬ś k 0$, i.e.,

$u w = S ╬▓ ( t ) ( u 0 ŌłÆ H ( w ╦£ ) ) + Ōł½ 0 t ( t ŌłÆ s ) ╬▓ ŌłÆ 1 P ╬▓ ( t ŌłÆ s ) ( F ( s , u w ( s ) , u w ( ╬║ 1 ( s ) ) ) + Ōł½ 0 s K ( s ŌłÆ Žä ) G ( Žä , u w ( Žä ) , u w ( ╬║ 2 ( Žä ) ) ) d Žä ) d s$

for t Ōłł [0, T].

Step 2. Write

$╬ś k 0 ╬Ę = { u Ōłł C ( [ ╬Ę , T ] ; X ╬▒ ) ; Ōłź u ( t ) Ōłź ╬▒ Ōēż k 0 f o r a l l t Ōłł [ ╬Ę , T ] } .$

It is clear that $╬ś k 0 ╬Ę$ is a bounded closed convex subset of C([╬Ę, T]; X ╬▒ ).

Based on the argument in Step 1, we consider a mapping $Ōä▒$ on $╬ś k 0 ╬Ę$ defined by

$( Ōä▒ w ) ( t ) = u w , t Ōłł [ ╬Ę , T ] .$

It follows from (H 5) and (3.2) that $Ōä▒$ maps $╬ś k 0 ╬Ę$ into itself. Moreover, for $w 1 , w 2 Ōłł ╬ś k 0 ╬Ę$, from Step 1, we have

$1 - 2 C ╬▒ T ( 1 - ╬▒ ) ╬▓ ( L F + k ╠ā L G ) ( 1 - ╬▒ ) ╬▓ | u w 1 - u w 2 | ╬▒ Ōēż M Ōłź H ( w ┬» 1 ) - H ( w ┬» 2 ) Ōłź ╬▒ ,$

that is,

$sup t Ōłł [ ╬Ę , T ] Ōłź Ōä▒ w 1 ( t ) - Ōä▒ w 2 ( t ) Ōłź ╬▒ ŌåÆ 0 a s w 1 ŌåÆ w 2 i n ╬ś k 0 ╬Ę ,$

which yields that $Ōä▒$ is continuous. Next, we prove that $Ōä▒$ has a fixed point in $╬ś k 0 ╬Ę$. It will suffice to prove that $Ōä▒$ is a compact operator. Then, the result follows from Lemma 2.6.

Let's decompose the mapping $Ōä▒= Ōä▒ 1 + Ōä▒ 2$ as

$( Ōä▒ 1 w ) ( t ) = S ╬▓ ( t ) ( u 0 ŌłÆ H ( w ╦£ ) ) , ( Ōä▒ 2 w ) ( t ) = Ōł½ 0 t ( t ŌłÆ s ) ╬▓ ŌłÆ 1 P ╬▓ ( t ŌłÆ s ) ( F ( s , u w ( s ) , u w ( ╬║ 1 ( s ) ) ) + Ōł½ 0 s K ( s ŌłÆ Žä ) G ( Žä , u w ( Žä ) , u w ( ╬║ 2 ( Žä ) ) ) d Žä ) d s .$

Since assumption (H 5) implies that the set $H ( w ╠ā ) ; w Ōłł ╬ś k 0 ╬Ę$ is bounded in X ╬▒ , it follows from Lemma 2.4 that for each t Ōłł [╬Ę, T], $( Ōä▒ 1 w ) ( t ) ; w Ōłł ╬ś k 0 ╬Ę$ is relatively compact in X ╬▒ . Also, for ╬Ę Ōēż t 1 Ōēż t 2 Ōēż T,

$( S ╬▓ ( t 2 ) - S ╬▓ ( t 1 ) ) ( u 0 - H ( w ╠ā ) ) ╬▒ ŌåÆ 0 a s t 2 - t 1 ŌåÆ 0$

independently of $wŌłł ╬ś k 0 ╬Ę$. This proves that the set $( Ōä▒ 1 w ) ( Ōŗģ ) ; w Ōłł ╬ś k 0 ╬Ę$ is equicontinuous. Thus, an application of Arzela-Ascoli's theorem yields that $Ōä▒ 1$ is compact.

Observe that the set

$F ( t , u ( t ) , u ( ╬║ 1 ( t ) ) ) + Ōł½ 0 t K ( t - Žä ) G ( Žä , u ( Žä ) , u ( ╬║ 2 ( Žä ) ) ) d Žä ; t Ōłł [ 0 , T ] , w Ōłł ╬ś k 0 ╬Ę$

is bounded in X. Therefore, using Lemma 2.1, Lemma 2.2 and Lemma 2.3, it is not difficult to prove, similar to the argument with ╬ō2 in Theorem 3.1, that $Ōä▒ 2$ is compact. Hence, making use of Lemma 2.6 we conclude that $Ōä▒$ has a fixed point $w * Ōłł ╬ś k 0 ╬Ę$. Put $q= u w *$. Then,

$q ( t ) = S ╬▓ ( t ) ( u 0 ŌłÆ H ( w ŌłŚ ╦£ ) ) + Ōł½ 0 t ( t ŌłÆ s ) ╬▓ ŌłÆ 1 P ╬▓ ( t ŌłÆ s ) ( F ( s , q ( s ) , q ( ╬║ 1 ( s ) ) ) + Ōł½ 0 s K ( s ŌłÆ Žä ) G ( Žä , q ( Žä ) , q ( ╬║ 2 ( Žä ) ) ) d Žä ) ) d s , t Ōłł [ 0, T ] .$

Since $u w * =Ōä▒ w * = w * ( t Ōłł [ ╬Ę , T ] ) ,H ( w * ) =H ( q )$ and hence q is a mild solution of the fractional Cauchy problem (1.1). This completes the proof. Ō¢Ā

## 4 Example

In this section, we present an example to our abstract results, which do not aim at generality but indicate how our theorem can be applied to concrete problem.

We consider the partial differential equation with Dirichlet boundary condition and nonlocal initial condition in the form

$c Ōłé t 1 2 u ( t , x ) = Ōłé 2 u ( t , x ) Ōłé x 2 + q 1 ( t ) | u ( sin ( t ) , x ) | 1 + | u ( sin ( t ) , x ) | + Ōł½ 0 t q 2 ( s ) 1 + ( t - s ) 2 | u ( s , x ) | 1 + | u ( s , x ) | d s , 0 Ōēż t Ōēż 1 , 0 Ōēż x Ōēż ŽĆ , u ( t , 0 ) = u ( t , ŽĆ ) = 0 , 0 Ōēż t Ōēż 1 , u ( 0 , x ) = Ōł½ ╬Ę 1 ln [ e u ( s , x ) ( | u ( s , x ) | + 1 ) ] d s + u 0 ( x ) , 0 Ōēż x Ōēż ŽĆ ,$
(4.1)

where the functions q 1, q 2 are continuous on [0, 1] and 0 < ╬Ę < 1.

Let X = L 2[0, ŽĆ] and the operators $A= Ōłé 2 Ōłé x 2 :D ( A ) ŌŖéXŌå”X$ be defined by

$D ( A ) = { u Ōłł X ; u , u ŌĆ▓ are absolutely continuous , u ŌĆ▓ ŌĆ▓ Ōłł X , and u ( 0 ) = u ( ŽĆ ) = 0 } .$

Then, A has a discrete spectrum and the eigenvalues are -n 2, n Ōłł ŌäĢ, with the corresponding normalized eigenvectors $y n ( x ) = 2 ŽĆ sin ( n x )$. Moreover, A generates a compact, analytic semigroup {T(t)} tŌēź0. The following results are well also known (see  for more details):

1. (1)

$T ( t ) u= Ōłæ n = 1 Ōł× e - n 2 t ( u , y n ) y n ,ŌłźT ( t ) Ōłź ŌäÆ ( X ) Ōēż e - t foralltŌēź0$.

2. (2)

$A - 1 2 u= Ōłæ n = 1 Ōł× 1 n ( u , y n ) y n$ for each u Ōłł X. In particular, $A - 1 2 ŌäÆ ( X ) =1$.

3. (3)

$A 1 2 u= Ōłæ n = 1 Ōł× n ( u , y n ) y n$ with the domain

$D A 1 2 = u Ōłł X ; Ōłæ n = 1 Ōł× n ( u , y n ) y n Ōłł X .$

Denote by E ╬Č, ╬▓ , the generalized Mittag-Leffler special function (cf., e.g., ) defined by

$E ╬Č , ╬▓ ( t ) = Ōłæ k = 0 Ōł× t k ╬ō ( ╬Č k + ╬▓ ) ╬Č , ╬▓ > 0 , t Ōłł ŌäØ .$

Therefore, we have

$S ╬▓ ( t ) u = Ōłæ n = 1 Ōł× E ╬▓ ( - n 2 t ╬▓ ) ( u , y n ) y n , u Ōłł X ; Ōłź S ╬▓ ( t ) Ōłź ŌäÆ ( X ) Ōēż 1 , P ╬▓ ( t ) u = Ōłæ n = 1 Ōł× e ╬▓ ( - n 2 t ╬▓ ) ( u , y n ) y n , u Ōłł X ; Ōłź P ╬▓ ( t ) Ōłź ŌäÆ ( X ) Ōēż ╬▓ ╬ō ( 1 + ╬▓ )$

for all t Ōēź 0, where E ╬▓ (t) := E ╬▓,1(t) and e