On mild solutions to fractional differential equations with nonlocal conditions ∗

We prove new existence results of mild solutions to fractional differential equa- tions with nonlocal conditions in Banach spaces. The nonlocal item is only assumed to be continuous. This generalizes some recent results in this area.


Introduction
In this paper, we are concerned with the existence of mild solutions for a fractional differential equation with nonlocal conditions of the form: where D q is the Caputo fractional derivatives of order q with 0 < q ≤ 1, A : D(A) ⊂ X → X is the infinitesimal generator of a strongly continuous semigroup T (t), t ≥ 0, X a real Banach * The work was supported by the NSF of China (11001034), the Research Fund for China Postdoctoral Scientific Program (20100480036), the Research Fund for Shanghai Postdoctoral Scientific Program (10R21413700). † Corresponding author: fzbmath@yahoo.com.cn (Z. Fan) space endowed with the norm · , f and g are appropriate continuous functions to be specified later.
Recently, the fractional differential equations are appropriate models for describing real world problems, which cannot be described using classical integer order differential equations.
So, they have been studied by many researchers. And, some recent contributions to the theory of fractional differential equations can be seen in [1-5, 13, 16-18, 20, 21].
On the other hand, the following differential equations with nonlocal conditions have been studied extensively in the literature, since it is demonstrated that the nonlocal problems have better effects in applications than the classical ones. . (1.2) Many authors developed different techniques and methods to solve the above nonlocal problem.
For more details on this topic we refer the reader to [7, 9-12, 14, 15, 19] and references therein.
Naturally, some researchers combined the above two directions and studied the fractional differential equation (1.1) with nonlocal conditions. In [8,23], the authors studied the existence of mild solutions to equation (1.1) when the nonlocal item g was assumed to be Lipschitz or compact function in different frameworks. In this paper, we study further the existence of mild solutions to nonlocal problem (1.1). By using the ideas in [10,22], we prove the existence of mild solutions to equation (1.1) without the Lipschitz or compact assumption on the nonlocal item g. Actually, the continuity of g is only assumed and g is completely determined on [δ, T ] for some small δ > 0 or g is continuous in C([0, T ], X) with L 1 ([0, T ], X) topology (see Corollaries 3.5-3.7). Our results extend some existing ones in this area.
This paper has three sections. In the next section, we recall some definitions on Caputo fractional derivatives and mild solutions to equation (1.1). In the last section, we establish the existence of mild solutions to equation (1.1) via the techniques developed in [10,22].

Preliminaries
Throughout this paper, let N, R and R + be the set of positive integers, real numbers and positive real numbers, respectively. We denote by X a Banach spaces with norm · , where Γ is the Gamma function.
If f takes values in Banach space X, the integrals which appear in above three definitions are taken in Bochner's sense.
In this paper, we always suppose that the linear operator A : D(A) ⊂ X → X generates a compact strongly continuous semigroup {T (t) : t ≥ 0}, i.e., T (t) is compact for any t > 0.

Moreover, we denote
Now, using the probability density function and its Laplace transform developed in [6] (also see [8,21]), we can give the following definition of mild solutions to equation (1.1).
EJQTDE, 2011 No. 53, p. 3 Definition 2.4. A continuous function u is said to be a mild solution of (1.1) if u satisfies

Main Results
Let r be a fixed positive real number. Write Clearly, B r , W r are bounded closed and convex sets. We make the following assumptions. ( (H) The set g(convQW r ) is pre-compact, where convB denotes the convex closed hull of set Remark 3.1. It is easy to see that condition (H) is weaker than the compactness and convexity of g. The same hypothesis can be seen from [10,22], where the authors considered the existence of mild solutions for semilinear nonlocal problems of integer order when A is a linear, densely defined operator on X which generates a C 0 -semigroup. After the proof of our main results, we will give some special types of nonlocal item g which is neither Lipschitz nor compact, but satisfies the condition (H) in the next Corollaries.
Under these assumptions, we can prove the main results in this paper. Proof. For u ∈ C([0, T ], X), from the properties of probability density function ξ q and condition (H1), it follows that It is easy to see that the fixed point of Q is a mild solution of nonlocal problem (1.1). Subsequently, we will prove that Q has a fixed point by using Schauder's fixed point theorem.
Firstly, we prove that the mapping Q is continuous on C([0, T ], X). For this purpose, let By the continuity of f , we deduce that f (s, u n (s)) converges to f (s, u(s)) in X uniformly for s ∈ [0, T ], it follows Then by the continuity of g and f , we get lim n→∞ Qu n = Qu in C([0, T ], X), which implies that the mapping Q is continuous on C([0, T ], X).
Secondly, we claim that QW r ⊆ W r . In fact, for any u ∈ W r , by (3.1), we have For t ∈ (0, T ] and δ > 0, set By (3.1) and the compactness of T (t), t > 0, we deduce that Q 1 δ W r (t) is relatively compact in X for any δ > 0. Moreover, we have as δ → 0, which implies that Q 1 W r (t) is relatively compact in X for every t ∈ (0, T ] since there are a family of relatively compact sets arbitrarily close to it. Next, we prove that Q 1 W r is equicontinuous on [η, T ] for any small positive number η. For u ∈ W r and η ≤ t 1 < t 2 ≤ T , there exist positive numbers δ and N such that Now, as T (·) is compact, T (t) is operator norm continuous for t > 0. Thus T (t) is operator norm continuous uniformly for t ∈ [η q δ, T q N]. Combining this with the absolute continuity of ξ q (·) on [0, ∞), it follows that Q 1 W r is equicontinuous on [η, T ].
For Q 2 : W r → C([0, T ], X), we claim that it is a compact mapping. In fact, Q 2 W r (0) is relatively compact. For t ∈ (0, T ], let δ ∈ (0, t) and define a mapping on W r by for u ∈ W r . We get that Q 2 δ W r (t) is relatively compact for any δ ∈ (0, t) since T (δ q δ) is compact. Moreover, for u ∈ W r , we obtain as δ → 0, which implies that Q 2 W r (t) is relatively compact in X for every t ∈ (0, T ] since there are a family of relatively compact sets arbitrarily close to it. Next, we prove that Q 2 W r is equicontinuous on [0, T ]. For u ∈ W r and 0 ≤ t 1 < t 2 ≤ T , we have For the last expression of the right side of the above inequality, if t 1 = 0, then it equals to zero; if t 1 > 0, then there exist positive numbers δ and N such that f (s, u(s)) . In summary, we have proven that QW r (t) is relatively compact for every t ∈ (0, T ] and QW r is equicontinuous on [η, T ] for any small positive number η.
EJQTDE, 2011 No. 53, p. 8 Now, let W = convQW r , we get that W is a bounded closed and convex subset of C([0, T ], X) and QW ⊆ W . It is easy to see that QW (t) is relatively compact in X for every t ∈ (0, T ] and QW is equicontinuous on [η, T ] for any small positive number η. Moreover, we know that g(W ) = g(convQW r ) is pre-compact due to the condition (H).
Thus, we claim that Q : W → W is a compact mapping. In fact, it is easy to see that ) dσ is relatively compact since g(W ) = g(convQW r ) is pre-compact. It remains to prove that Q 1 W is equicontinuous on [0, T ]. For that, let u ∈ W and 0 ≤ t 1 < t 2 ≤ T , there exists positive number N such that In The following theorem is a direct consequence of Theorem 3.2.
for all t ∈ [0, T ], then the nonlocal problem (1.1) has at least one mild solution.
Remark 3.4. It is easy to see that if there exist constants L 1 , L 2 > 0 and α, β ∈ [0, 1) such Next, we will give special types of nonlocal item g which is neither Lipschitz nor compact, but satisfies the condition (H).
We give the following assumptions.
Proof. It is easy to see that the mapping g with g(u) = p j=1 c j u(t j ) satisfies condition (H3). And all the conditions in Corollary 3.5 are satisfied. So the conclusion holds. Proof. According to Theorem 3.2, we should only to prove that the hypothesis (H) is satisfied.
Remark 3.8. Our results extend some recent ones about the fractional differential equations with nonlocal conditions, since neither the Lipschitz continuity nor the compactness assumption on the nonlocal item is required.