Results for Mild solution of fractional coupled hybrid boundary value problems

Abstract The study of coupled system of hybrid fractional differential equations (HFDEs) needs the attention of scientists for the exploration of its different important aspects. Our aim in this paper is to study the existence and uniqueness of mild solution (EUMS) of a coupled system of HFDEs. The novelty of this work is the study of a coupled system of fractional order hybrid boundary value problems (HBVP) with n initial and boundary hybrid conditions. For this purpose, we are utilizing some classical results, Leray–Schauder Alternative (LSA) and Banach Contraction Principle (BCP). Some examples are given for the illustration of applications of our results.


Introduction
There are many scientific contributions to the applications of fractional differential equations (FDEs) in different scientific fields including economics, polymer rheology, chemistry, mechanics, control theory, aerodynamics, biophysics, regular variation in thermodynamics, etc [2,3,6,10]. These applications play a vital role in popularity of FDEs in particular and fractional calculus in general. Scientist are interested in getting highly precised and accurate results in their research models. For this reason they utilize different mathematical tools by the help of which they receive exact solutions of their models as well as numerical approximations. For the exact solutions of different classes of FDEs, we refer to valuable efforts of Y. J. Yang et al. regarding Laplace equation [11]. In [12], C. G. Zhao et al. produced exact solutions of many initial value problems of local FDEs. Besides the exact solutions, we also have valuable efforts of scientist for numerical approximations of FDEs. For example, in [8] H. Khan et al. produced the approximate solution of factional order Logistic equations by the help of operational matrices of Bernstein polynomials.
Various aspects of FDEs have been considered. Recently, a valuable interest has been observed on the existence, uniqueness, multiplicity of results, infinite solutions and no solutions of different classes of FDEs [7,9]. In this paper, our interest is in the EUMS of coupled systems of HFDEs with n hybrid initial and boundary conditions. This area of FDEs is one of important problems drawing the attention of many scientists. For some important studies in this area we refer to B. Ahmad et al. in [1] and M. A. E. Herzallah and D. Baleanu in [5]. We were influenced by the cited work for the study of EUMS of the coupled system of HFDEs of the type D T 1 .z; ‰; ƒ/; ! 2 .n 1; n; i; j D 2; 3; : : : ; n 1; (1) where D ! 1 , D ! 2 are Caputo's fractional derivatives of orders ! 1 , ! 2 respectively, T 1 ; T 2 2 C.OE0; 1 R 2 ; R/, H; G 2 C.OE0; 1 R 2 ; R f0g/ and ‰; ƒ 2 C.OE0; 1; R/. We utilize LSA and BCP.

Organization of the paper
This paper is divided into three sections. The first section is the introduction, which is based on the literature review.
In the second section we present our main results, namely the existence of mild solution (EMS) of the problem (1) and the uniqueness of mild solution (UMS) of the problem (1). These results are based on Leray-Schauder Alternative (LSA) and Banach contraction principle. The third section is dedicated to the applications of our results. These applications are shown on two illustrative examples. We give the following definitions and results from the available literature for the readers and the detailed study on this work can be found in [2,10].
provided the integral converges. Definition 1.2. For J 2 C k OE0; 1, the Caputo fractional derivative for ! order is defined by provided that the integral on .0; 1/ is defined.

Lemma 1.3 ([10]). For
! > 0 and J .t / 2 L 1 OEa; b, we have the following: ::: C k n z n 1 ; k i 2 R; i D 1; 2; 3; :::; n: , LSA). Let F W V ! V be a completely continuous operator (i.e., a map that is restricted to any bounded set in V). Let Y D fx 2 V W x D Fx for some 2 .0; 1/g. Then either Y is unbounded or F has at least one fixed point. Lemma 1.6. Let J .z/ be a continuous function then the mild solution of for i D 2; : : : ; n 1, is given by Proof. Applying the operator I ! 1 0 on (3) and using Lemma 1.4, we obtain Applying the boundary condition (7), we have c 2 D I ! 1 J .1/ and (7) gets the form Applying the condition Putting the values of c 0 ; c 1 ; : : : ; c n in (6), we get and the integral form of the mild solution ‰.z/, is given by (10) thus, the proof is completed.
2 Results for EUMS of the system (1) In this section we focus on two lemmas for the EMS and UMS of the coupled system (1). For this purpose, we utilize LSA and BCP. With respect to these two lemmas we divided this section into two subsections. From [4] we consider the Banach space Y D f‰.z/ W ‰.z/ 2 C 1 OE0; 1g with norm k‰.z/k D maxfj‰.z/j for z 2 OE0; 1g and .Y Y; k.:; :/k/ with norm k.‰; ƒ/.z/k D k‰.z/k C kƒ.z/k. We define an operator F W Y Y ! Y Y by F.‰; ƒ/.z/ D .F 1 .‰; ƒ/.z/; F 2 .‰; ƒ/.z//; We define the following terms to simplify our calculations: (1) In this subsection we focus on the EMS of HFDEs (1). For this, we assume the following two conditions:
Proof. We give the proof in the following three steps.
Step 1: In this step, we prove that the operator F maps bounded subset of Y Y into a bounded. For this we consider .‰; ƒ/.z/, then for z 2 OE0; 1, we have similarly, from (14) and (15), we have thus, F is uniformly bounded.
Step .! 1 1/ T 1 .y; ‰; ƒ/dyj; and jƒ.z/j Ä jF 2 .‰; ƒ/.z/j Ä jG.z; ‰; ƒ/jOEj From (20) and (21) we have Consequently, this implies that S is bounded. Thus, by the help of Steps 1-3 we conclude that F is completely continuous and by Lemma 1.5, F has fixed the point .‰; ƒ/.z/ and is the mild solution of the coupled system of HFDEs (1). (1) In this subsection, we are giving a proof for the UMS of the coupled system of HFDEs (1). For this we use the BCP.
Thus the function F is a contraction mapping and, consequently, F fixes a point which is the UMS of the coupled system of HFDEs (1).