A Nonlocal Cauchy Problem for Fractional Integrodifferential Equations

This paper is concerned with a nonlocal Cauchy problem for fractional integrodifferential equations in a separable Banach space X. We establish an existence theorem for mild solutions to the nonlocal Cauchy problem, by virtue of measure of noncompactness and the fixed point theorem for condensing maps. As an application, the existence of the mild solution to a nonlocal Cauchy problem for a concrete integrodifferential equation is obtained.


Introduction
Nonlocal Cauchy problem for equations is an initial problem for the corresponding equations with nonlocal initial data.Such a Cauchy problem has better effects than the normal Cauchy problem with the classical initial data when we deal with many concrete problem coming from engineering and physics cf., e.g., 1-10 and references therein .Therefore, the study of this type of Cauchy problem is important and significant.Actually, as we have seen from the just mentioned literature, there have been many significant developments in this field.
On the other hand, fractional differential and integrodifferential equations arise from various real processes and phenomena appeared in physics, chemical technology, materials, earthquake analysis, robots, electric fractal network, statistical mechanics biotechnology, medicine, and economics.They have in recent years been an object of investigations with much increasing interest.For more information on this subject see for instance 9, 11-18 and references therein.c D q x t Ax t f t, x t t 0 k t, s h t, s, x s ds, t ∈ 0, T , where k and g are given functions to be specified later and the fractional derivative is understood in the Caputo sense, this means that, the fractional derivative is understood in the following sense: c D q x t : L D q x t − x 0 , t > 0, 0 < q < 1, 1.3 and where L D q x t : 1 Γ 1 − q d dt t 0 t − s −q x s ds, t > 0, 0 < q < 1 1.4 is the Riemann-Liouville derivative of order q of x t , where Γ • is the Gamma function.
Our main purpose is to establish an existence theorem for the mild solutions to the nonlocal Cauchy problem based on a special measure of noncompactness under weak assumptions on the nonlinearity f and the semigroup {T t } t≥0 generated by A.

Existence Result and Proof
As usual, we abbreviate u L p 0,T , R with u L p , for any u ∈ L p 0, T , R .
As in 16, 17 , we define the fractional integral of order q with the lower limit zero for a function f ∈ AC 0, ∞ as provided the right side is point-wise defined on 0, ∞ .Now we recall some very basic concepts in the theory of measures of noncompactness and condensing maps see, e.g., 19, 20 .
Definition 2.1.Let E be a Banach space, 2 E the family of all nonempty subsets of E, A, ≥ a partially ordered set, and α : 2 then we say that α is a measure of noncompactness in E.
Definition 2.2.Let E be a Banach space, and If for every bounded set Ω ⊆ Y which is not relatively compact, then we say that F is condensing with respect to the measure of noncompactness α or α-condensing .
be a one-sided stable probability density, and For any z ∈ X, we define operators {Y t } t≥0 and {Z t } t≥0 by Y t z ∞ 0 ξ q σ T t q σ zdσ, Z t z q ∞ 0 σt q−1 ξ q σ T t q σ zdσ.

2.8
If 2.16 Proof.First of all, let us prove our definition of the mild solution to problem 1.2 is well defined and reasonable.Actually, the proof is basic.We present it here for the completeness of the proof as well as the convenience of reading.Write

2.17
Clearly, the nonlocal Cauchy problem 1.2 can be written as the following equivalent integral equation: provided that the integral in 2.18 exists.Formally taking the Laplace transform to 2.18 , we have Therefore, if the related integrals exist, then we obtain

2.20
Now using the uniqueness of the Laplace transform cf., e.g., 21, Theorem 1.1.6, we deduce that ξ q σ T t − s q σ a x s dσ ds.

2.21
Consequently, we see that the mild solution to problem 1.2 given by Definition 2.3 is well defined.
Next, we define the operator F : C 0, T , X → C 0, T , X as follows: It is clear that the operator F is well defined.
The operator F can be written in the form F F 1 F 2 , where the operators F i , i 1, 2 are defined as follows:

2.23
The following facts will be used in the proof.
x n − x 0,T 0, 2.28 for an x ∈ C 0, T , X .Then by the assumptions, we know that for almost every t ∈ 0, T and t, s ∈ Δ:

2.34
By 2.33 and our assumptions, we see that F is continuous.
Since χ is the Hausdorff measure of noncompactness in X, we know that χ is monotone, nonsingular, invariant with respect to union with compact sets, algebraically semiadditive, and regular.This means that i for any Noting that for any ψ ∈ L 1 0, T , X , we have lim So, there exists a positive constant L such that

10 Journal of Applied Mathematics
For every bounded subset Ω ⊂ C 0, T , X , we define mod c Ω : lim

2.41
Then mod c Ω is the module of equicontinuity of Ω, and α is a measure of noncompactness in the space C 0, T , X with values in the cone R 2 .

2.42
By the assumptions and the continuity of T t in the uniform operator topology for t > 0, we get mod c F 1 Ω 0.

2.43
Clearly, f s, x s a x s ≤ μ s m * k * x 0,T .

2.45
It is not hard to see that the right-hand side of 2.45 tend to 0 as t 2 → t 1 .Thus, the set Combining with 2.43 , we have mod c FΩ 0, which implies mod c Ω 0 from 2.42 .Next, we show that Ψ Ω 0. It is easy to see that

2.46
For any t ∈ 0, T , we define

2.47
We consider the multifunction s ∈ 0, t G s : Obviously, G is integrable, that is, G admits a Bochner integrable selection g : 0, h → E, and

2.51
Therefore, since X is a separable Banach space, we know by 20, Theorem 4.2.3 that

2.53
Similarly, if we set is integrable and integrably bounded.Thus, we obtain the following estimate for a.e.s ∈ 0, t :

2.56
Journal of Applied Mathematics 13 Now, from 2.53 and 2.56 , it follows that where 0 < L < 1.Then by 2.42 , we get Ψ Ω 0. Hence α Ω 0, 0 .Thus, Ω is relatively compact due to the regularity property of α.This means that F is α-condensing.
Let us introduce in the space C 0, T , X the equivalent norm defined as x * sup t∈ 0,T e −Lt x t .

2.59
Next, we show that there exists some r > 0 such that FB r ⊂ B r .Suppose on the contrary that for each r > 0 there exist x r • ∈ B r , and some t ∈ 0, T such that Fx r t * > r.
From the assumptions, we have Moreover,

2.62
Dividing both sides of 2.62 by r, and taking r → ∞, we have qM Γ 1 q sup t∈ 0,T t 0 t − s q−1 μ s m * k * e −L t−s ds ≥ 1.

2.63
This is a contradiction.Hence for some positive number r, FB r ⊂ B r .According to the following known fact.
Let M be a bounded convex closed subset of E and F : M → M a α-condensing map.Then Fix F {x : x F x } is nonempty.we see that problem 1.2 has at least one mild solution.
Next, for c ∈ 0, 1 , we consider the following one-parameter family of maps:

2.64
We will demonstrate that the fixed point set of the family H, is a priori bounded.Indeed, let x ∈ Fix H, for t ∈ 0, T , we have x τ ds .

2.66
Noting that the H ölder inequality, we have

Journal of Applied Mathematics 15
We denote y t : sup s∈ 0,t x s .

2.69
Let t ∈ 0, t such that y t x t .Then, by 2.68 , we can see y s ds.

2.70
By a generalization of Gronwall's lemma for singular kernels 22, Lemma 7.1.1, we deduce that there exists a constant κ κ q such that

2.71
Hence, sup t∈ 0,T x t ≤ w.Now we consider a closed ball:

2.72
We take the radius R > 0 large enough to contain the set Fix H inside itself.Moreover, from the proof above, F : B R → C 0, T , X is α-condensing.Consequently, the following known fact implies our conclusion: Let V ⊂ E be a bounded open neighborhood of zero and F : V → E a α-condensing map satisfying the boundary condition: x / λF x , 2.73 for all x ∈ ∂V and 0 < λ ≤ 1.Then, Fix F is nonempty compact.

Example
In this section, let X L 2 0, π , we consider the following nonlocal Cauchy problem for an integrodifferential problem: where ∂ q t is the Caputo fractional partial derivative of order 0 < q < 1; ξ ∈ 0, π ; k > 0 is a constant to be specified later; 1, . . ., j are continuous functions and there exists a positive constant b such that generates an analytic semigroup and uniformly bounded semigroup {T t } t≥0 on X with T t L X ≤ 1.Therefore, 3.1 is a special case of 1.2 .