EXISTENCE THEOREMS OF SOLUTION OF FRACTIONAL DIFFERENTIAL EQUATIONS WITH NONLOCAL CONDITIONS

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. This paper establishes a study on some important latest innovations in the existence of mild solution of semilinear for differential and fractional differential equations subject to nonlocal initial conditions. To apply this, the study uses Hausdorff measure of non-compactness and fixed point theorems. A wider applicability of these techniques are based on their reliability and reduction in the size of the mathematical work.


INTRODUCTION
In recent years there has been a growing interest in the differential equation. The differential equations be an important branch of modern mathematics. It arises frequently in many applied areas which include engineering, electrostatics, mechanics, the theory of elasticity, potential, and mathematical physics [11,12,14,15,16,33].
During the last decades, mathematical modeling has been supported by the field of fractional calculus, with several successful results and fractional operators showing to be an excellent tool to describe the hereditary properties of various materials and processes. Recently, this combination has gained a lot of importance, mainly because fractional differential equations have become powerful tools in modeling several complex phenomena in numerous seemingly diverse and widespread fields of science and engineering [4,5,6,9,20,27].
The concept of nonlocal initial condition has been introduced to extend the study of classical initial value problems. The earliest works related with problems submitted to nonlocal initial conditions were made by Byszewski [1,2,3].
Motivated by above works, in this paper, we discuss new existence results for nonlocal differential equations of the form: and where D q is the standard Riemann-Liouville fractional derivative of order q, 0 < q ≤ 1 and f : J ×X −→ X, g : C[0, 1] −→ X are functions and A is a semi-group of bounded linear operators strongly continuous that generated by A in the Banach space X, with norm . .
The main objective of the present paper is to study the new existence results of the solution for nonlocal differential equations and nonlocal fractional differential equations.
The rest of the paper is organized as follows: In Section 2, some preliminaries, basic definitions and Lemma related to fractional calculus are recalled. In Section 3, the new existence results of the solution for nonlocal differential equations and nonlocal fractional differential equations have been proved. Finally, we will give a report on our work and a brief conclusion is given in Section 4.

PRELIMINARIES
Let (X, . ) be a real Banach space. We denote by C(J, X) the space of X-valued continuous functions on J with the norm y = sup{ y(t) , t ∈ J}, and by L 1 (J, X) the space of X-valued Bochner functions on J with the norm y = 1 0 y(s) ds. A C 0 -semigroup T (t) is said to be compact if T (t) is compact for any t > 0. If the semigroup T (t) is compact then t −→ T (t)y are equicontinuous at all t > 0 with respect to y in all bounded subsets of X, i.e. the semigroup T (t) is equicontinuous. In this paper, we suppose that A generates a C 0 semigroup T (t) on X. Since no confusion may occur, we denote by α the Hausdorff measure of noncompactness on both X and C(J, X).
By a mild solution of the nonlocal initial value problems (1) and (2), we mean the function y ∈ C(J, X) which satisfies
where R + is the set of positive real numbers.
where the parameter α is the order of the derivative and is allowed to be real or even complex.
In this paper, only real and positive α will be considered.
Then F has a fixed point inŪ.

MAIN RESULTS
In this section, we shall give an existence results of Eq.(1), with the initial condition (2) and prove it.
Before starting and proving the main results, we introduce the following hypotheses: (H1): The C 0 semigroup T (t) generated by A is equicontinuous. We denote N = sup{ T (t) ,t ∈ J}.
(H2): The function g : C(J, X) → X is continuous and compact, there exists positive constants c and d such that g(y) ≤ c y + d, for all y ∈ C(J, X).
(H3): The function f (., y) is measurable for all y ∈ X, and f (t, ) is continuous for a.e.
Theorem 2. Assume that (H1)-(H5) hold. If there exist a constant R with Then there is at least one mild solution of the problem (1).
Proof. Firstly, we transform (3) for all y ∈ C(J, X). We can show that F is continuous by the usual techniques (see, e.g. [6,7]).
We denote W = {y ∈ C(J, X), y(t) ≤ R, for all t ∈ J}, then W ⊆ C(J, X) is bounded and convex. For any y ∈ W , we have (Ty)(t) ≤ T (t)g(y) + Let B 0 =co(TW ). For any B ⊂ B 0 , we know from Lemma 4, for any ε > 0, there is a sequence We know there is a continuous function Φ : J −→ R + (M = max |Φ(t)| : t ∈ J) such that for any So, Since ε > 0 is arbitrary, it follows from the above inequality that From Lemma 4, for any ε > 0, there is a sequence {z n } ∞ n=1 ⊂co(T 1 B), such that α(T 2 B(t)) = α(T ((co (T 1 B(t))))) Hence, by the method of mathematical induction, for any positive integer n and t ∈ J, we obtain α(T n B(t)) ≤ a n +C 1 n a n−1 bt +C 2 n a n−2 (bt) 2 2! + · · · + (bt) n n! α(B).
It follows from Lemma 6, that F has at least one fixed point in B 0 , i.e. the nonlocal initial value problem (1) has at least one mild solution in B 0 . Thus, the proof is completed.
We shall next discuss the existence result for the nonlocal initial value problem (2). Here we list the following hypotheses. (A5): g(y) ≤ a y + b, ∀y ∈ C(I, X) and for some positive constants a, b where g : C(I, X) −→ X is a continuous compact map.
where L * z 1 , L * z 2 and L * g are positive constants. Proof. Let the operator ϒ : C(I, X) → C(I, X) be defined by To prove the operator T is continuous on C(I, X), we suppose y n −→ y in C(I, X) then by (A2), we have that.
Firstly, we prove that ϒ is satisfied Monch's conditions , let B r = {y ∈ C(J, X) : y ∞ ≤ r}, and suppose D ⊆ B r is a countable such that D ⊆ (CO)({0} ∪ ϒ(D)). Let Q is the Hausdorff MNC.
Then by (A6) there exist ι > 0 such that ι = y , let the set Λ = {y ∈ C(I, X) : |y < ι}. So for all y ∈ δ Λ, we get y = δ ϒ(y) to some β ∈ (0, 1). Thus by theorem 1, we get a fixed point of ϒ inΛ and this fixed point is a mild solution to the problem (2), and the proof is completed.

CONCLUSIONS
The main purpose of this work was to present new existence of mild solution of semilinear for differential and fractional differential equations subject to nonlocal initial conditions. To apply this, the study uses Hausdorff measure of non-compactness and fixed point theorems.
Moreover, the results of references [34,10] appear as a special case of our results.