On the Existence and Uniqueness of Solutions for Local Fractional Differential Equations

Hossein Jafari 1,2,*, Hassan Kamil Jassim 3, Maysaa Al Qurashi 4 and Dumitru Baleanu 5,6 1 Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar 47416, Iran 2 Department of Mathematical Sciences, University of South Africa (UNISA), Pretoria 0003, South Africa 3 Department of Mathematics, Faculty of Education for Pure Sciences, University of Thi-Qar, Nasiriyah 64001, Iraq; hassan.kamil28@yahoo.com 4 Department of Mathematics, King Saud University, P.O. Box 22452, Riyadh 11495, Saudi Arabia; Maysaa@ksu.edu.sa 5 Department of Mathematics, Çankaya University, Ankara 06530, Turkey; dumitru@cankaya.edu.tr 6 Institute of Space Sciences, P.O. Box MG-23, Magurele-Bucharest RO-76911, Romania * Correspondence: jafari@umz.ac.ir; Tel: +98-11-3530-2466

The existence and uniqueness of solutions of differential equations with the Riemann-Liouville fractional derivative and the Caputo fractional derivative using the Schauder fixed point theorem, the lower and upper solution method, the contracting mapping principle and the Leray-Schauder theory have been investigated in some papers [12][13][14][15].
Very recently in [16], the author studied the existence and uniqueness of solutions of some classes of differential equations with local fractional derivative operators.In this paper, we are interested in the existence and uniqueness of DEs with LFDOs of the form: and a system of DEs with LFDOs of the form where and D α , D 2α , and D 3α are the LFDOs of order α , 2α, and 3α respectively and υ, µ ∈ [σ, ω].By using a variety of tools including the (CMT) and (IDT), existence and uniqueness results are obtained.The rest of this paper is organized as follows.In Section 2, we give some necessary notations, definitions, and theorems.In Section 3, we study the existence and uniqueness of solutions of local fractional differential Equations ( 1)-( 3) by using the contracting mapping theorem.Examples are given to illustrate our results in Section 4. Finally, in Section 5 we outline the main conclusions.

Basic Definitions and Preliminaries
In this section, we present some basic definitions and theorems that are used to prove our new results (see [16]).

Definition 1. Let us consider that
Let (X, || • || α ) be a generalized Banach space (GBS), and let T : X → X be a map.If a number

Theorem 3. (Increasing and Decreasing Theorem)
1. If then Ω(χ) is an increasing on that interval.

Main Results
Here, we investigate the existence and uniqueness of solutions of the LFDEs (1), ( 2) and ( 3).First, we prove the existence and uniqueness of solutions of the LFDEs by applying (the CMT).
Theorem 4. Let us consider that T : X → X is a map on the complete GBS (X, || • || α ) such that for some m ≥ 1 , T m is contracting.Then T has a unique FP.
Proof.Since T m is CM on X, then we have T m has a unique FP ξ α .
Then Ω is LC.
Proof.Since ∂ α Ω ∂γ α is LF continuous, then it attains a maximum value, denoted by Now, let us consider (χ, γ 1 ) , (χ, γ 2 ) ∈ Dom Ω.Using Theorem 2, there is a point (χ, τ) ∈ Dom Ω such that: an LF continuous function and satisfies a LC, then the LFDE: subject to the initial conditions and We first prove by induction that In fact, for n = 1, we obtain We suppose the desired inequality holds for n = k.
Hence, the estimates (7) hold.Now, we have Therefore T has a unique fixed point in C α [σ, ω] , which gives a unique solution to the local fractional differential.
is a local fractional continuous function and satisfies a Lipschitz continuous, then the local fractional differential equation: and We claim that for all n, The case is n = 1 has already shown.The induction step is as follows: Hence, the estimates (11) hold.

Now, we have
. This gives an unique solution to the local fractional differential.
Next, we apply the increasing and decreasing test to prove the uniqueness of the solution of the system of LFDEs.
n is LF continuous and satisfies a Lipschitz continuous, then there is at most one solution X(τ) of the local fractional differential system that satisfies a given initial condition X Proof.Suppose that Φ(τ, X) satisfies the LC.

Applications
To illustrate the application of our results, let us consider the following examples.
Example 1.The local fractional IVP has a unique solution.
For this initial value problem, the integral operator T is defined as In other words, T ζ(χ) = ζ(χ), so ζ(χ) is a unique fixed point of T, which gives a unique solution to the local fractional IVP (13).
Example 2. The LFDE has a unique solution.
For this initial value problem, the integral operator T is defined as

Conclusions
We have presented some existence and uniqueness results for an initial value problem of local fractional differential equations (LFDEs) and a system of LFDEs with local fractional derivative operators.The proof of the existence and uniqueness of the solutions is proved by applying the contracting mapping theorem while the uniqueness of solutions for system of LFDEs is proved by applying the increasing and decreasing theorem.The present work can be extended to nonlinear differential equations with local fractional derivative operators.