Existence for Nonautonomous Fractional Integrodifferential Equations with Nonlocal Conditions

and Applied Analysis 3 By using the family {ψ(t, s)}, we denote φ 1 (t, s) = [A (t) − A (s)] ψ (t − s, s) ,


Introduction
In this paper, we denote that  is a positive constant and assume that a family of closed linear operators {()} ∈[0,] satisfies the following.
Fractional calculus is a generalization of ordinary differentiation and integration to arbitrary noninteger order.The field of the application of fractional calculus is very broad.
We can see it in the study of the memorial materials, earthquake analysis, robots, electric fractal network, fractional sine oscillator, electrolysis chemical, fractional capacitance theory, electrode electrolyte interface description, fractal theory, especially in the dynamic process description of porous structure, fractional controller design, vibration control of viscoelastic system and pliable structure objects, fractional biological neurons, and probability theory.For details, see the monographs of Kilbas et al. [2], Kiryakova [3], Lakshmikantham and Vatsala [4], Miller and Ross [5], Samko et al. [6] and Podlubny [7], and the references therein.Some recent contributions to the theory of fractional differential equations can be seen in [8][9][10][11][12][13][14][15][16][17][18][19][20] and the references therein.Among the previous researches, most of researchers focus on the case that the differential operators (possibly unbounded) in the main parts are independent of time .However, when treating some parabolic evolution problems, it is usually assumed that the partial differential operators depend on time  (i.e., it is the case of the problems under considerations being nonautonomous), since this class of operators appears frequently in the applications (see [21] and the references therein).
Moreover, since the work of Byszewski [22], the nonlocal Cauchy problems have been investigated in many papers (cf., e.g., [13-15, 20, 23-25] and the references therein).The nonlocal conditions give a better description in applications than standard ones, and the Cauchy problem with nonlocal initial condition can be applied in physics with better effect than the classical Cauchy problem with traditional initial conditions.The existence of mild solutions of nonautonomous fractional evolution equations with nonlocal conditions of the form (4) is an untreated original topic, which in fact is the main motivation of the present paper.
In this paper, using a pair of evolution families {(, )} and {(, )} associated with the semigroup {exp(−())} ( > 0), we give a reasonable concept of solution to problem (4) in Section 2.Moreover, in general, the semigroup {exp(−())} ( > 0) generated by −() ( ∈ [0, ]) is not compact, so we obtain the main result based on the theory of measures of noncompactness and the condensing maps.These techniques are often used to deal with abstract integer order differential equations but rarely used in abstract fractional order differential equations(e.g., [8][9][10][11][12][13][14][15][16][17][18][19][20] and the references therein).We will study (4) under suitable hypotheses based on a special noncompactness measure and the properties of fixed points set of condensing operators [26,27] and establish a new existence result for (4) without the assumptions that the nonlinearity  satisfies a Lipschitz type condition and the semigroup {exp(−())} generated by −() is compact (see Theorem 14).As one can see, our result is obtained under assumptions weaker than those required previously in the similar literature.The result is new even for the case of () ≡  (autonomous).Moreover, an example is given to show an application of the abstract result.

Preliminaries
Throughout this paper, we set  = [0, ], a compact interval in R. We denote by  a separable Banach space with norm ‖ ⋅ ‖, by () the Banach space of all linear and bounded operators on , and by ([, ], ) the space of all -valued continuous functions on [, ] with the supremum norm as follows: for any  ∈  ([, ] , ) .
We set Next, we recall the definition of the Riemann-Liouville integral.
Definition 3 (see [7]).The Riemann-Liouville derivative of order  with the lower limit zero for a function  ∈ [0, ∞) can be written as Based on the work in [12], we give the following definition of the operator family {(, )}.Definition 4. Let   be a probability density function defined on (0, ∞) such that its Laplace transform is given by We define operator families {(, )} by the semigroup exp(−()) associated with () as follows: By using the family {(, )}, we denote and construct the family {(, )} by Lemma 5 (see [12]).The operator-valued functions ( − , ) and ()( − , ) are continuous in uniform topology in the variables , , where 0 ≤  ≤  − , 0 ≤  ≤ , for any  > 0. Clearly, Moreover, we have A mild solution of ( 4) can be defined as follows.
We will need the following facts from the theory of measures of noncompactness and condensing maps (see, e.g., [26,27]) which are used later in this paper.Definition 7. Let  be a Banach space, 2  the family of all nonempty subsets of , (A, ≥) a partially ordered set, and then we say that ] is a measure of noncompactness (MNC) in .
As an example of the MNC, we may consider the Hausdorff MNC: We know that  is monotone, nonsingular, invariant with respect to union with compact sets, algebraically semiadditive, and regular.This means that In Section 3, we will establish an existence result to the problem (4) using the following assertion about -estimates for a multivalued integral (Theorem 4.2.3 of [27]).
Let G : [0, h] → 2  be a multifunction.It is called where Let  be a Banach space and ] a monotone nonsingular MNC in .Definition 9. A continuous map B :  ⊆  →  is called condensing with respect to a MNC ] (or ]-condensing) if, for every bounded set Ω ⊆  which is not relatively compact, we have The application of the topological degree theory for condensing maps (see, e.g., [26,27]) yields the following fixed point principle which will be used later.

Main Result
We need the hypotheses as follows.
for almost all  ∈ .
(H3) The function  is completely continuous and there exists a positive constant  such that      (V)     ≤ , V ∈  (, ) .
Define the operator F : (, ) → (, ) as follows: It is clear that the operator F is well defined.For some MNC ], we will show that the operator F is ]-condensing on every bounded subset of (, ).To this end, we divided the proof into three propositions.
Proposition 11.The operator F is continuous.

Conclusion
This paper deals with the existence of mild solution of a class of nonlinear nonautonomous fractional integrodifferential equations with nonlocal conditions in an abstract space.Sufficient conditions for the existence of mild solution are derived with the help of the fixed point theorem for condensing maps.An example is provided to illustrate the obtained result.

Theorem 10 .
Let M be a bounded convex closed subset of  and B : M → M a ]-condensing map.Then fix B = { :  = B()} is nonempty.