Stability and existence the solution for a coupled system of hybrid fractional differential equation with uniqueness

Abstract In this literature, we are investigating the existence and uniqueness of solutions for a coupled system of hybrid differential equations with p-Laplacian operator involving the fractional derivatives Caputo and Riemann–Liouville derivatives of various orders. For this purpose, the proposed problem will be converted into integral equations by using two Green functions for ], where . The existence of a solution is proved by using topological degree and Leray–Schauder fixed point theorems. Uniqueness of solution is obtained with the help of Banach principle. Some adequate conditions for Hyers–Ulam- stability of the solution also are investigated. To verify the results. An expressive example will be presented to clarify the results.

Since an unstable solution is not useful and does not provide the necessary information within the specified domain, its stability is just as important as its nature. Conversely, Stable solutions often include significant details within the prescribed domain. In this regard, researchers have studied several stability theories to FDEs such as Lyapunov, exponential and Mittag-Laffer in addition to other important kinds of stability analysis are HU-stability and UH-Rassias for various aspects of couple system of hybrid FDEs for EUS. Here, are some studies of a couple of systems in FDE. For example, see these studies like Pesticides in Soil and Trees, Chemical Kinetics, Irregular Heartbeats and Chemostats (Ahmad & Nieto, 2009;Bai & Fang, 2004;Hu, Liu, & Liu, 2013;Iqbal, Li, Shah, & Khan, 2017;Khan, Li, Sun, & Khan, 2017;Samina, Shah, & Khan, 2017;Shah & Tunc, 2017;Su, 2009). At first, we introduce some modern and important contributions of researchers for the investigation of EUS of FDEs of different types of FDEs. For example. Abdellaoui and Dahmani (2015) used fractional calculus and fixed point theorems to investigate EUS for fractional coupled systems where D k and D are caputo derivatives with mÀ1 < k, <m, m 2 N Ã , j, d 2 R Ã , a, b, P, q are real unmbers, Q 1 , Q 2 , Y 1 and Y 2 are functions. Hu et al. (2013) studied BVP for a coupled system of FDEs by using the degree theorem given by c D k fðsÞÀ/ðs, lðsÞ, l 0 ðsÞÞ ¼ 0, s 2 ð0, 1Þ, c D a lðsÞÀgðs, fðsÞ, f 0 ðsÞÞ ¼ 0, s 2 ð0, 1Þ, where 1<k, a<2, c D k and c D a are Caputo derivatives and g, / : ½0, 1 Â R 2 ! R:  studied the EUS for a couple system of FDEs by degree theory as follows where c D r and c D k are the Caputo derivatives, 0<r, k 1, Q 1 , Q 2 2 C½0, 1 Â R 2 , R, and -, . : ðI, RÞ ! R are continuous function, 0<, l<1, d i , a i , i ði ¼ 1, 2Þ are real number. Lei Hu (2018) studied the EUS for ðnÀ1, 1Þ-type coupled systems of FDEs by degree theorem, given by c D r XðÞ ¼ Pð, xðÞ, x ðÞ, :::, x kÀ1 ðÞÞ, 2 ð0, 1Þ, c D d xðÞ ¼ Yð, XðÞ, X ðÞ, :::, X kÀ1 ðÞÞ, 2 ð0, 1Þ, where c D d and c D r are Caputo derivatives, kÀ1<r, d<k, P, Y : ½0, 1 Â R k ! R are continuous functions, g, n 2 ð0, 1 are given constants satisfying kg k ¼ ln k ¼ k: Hu et al. (2013) investigated existence of the solution of BVP for a coupled system of FDEs with p-Laplacian at resonance given by where c D r , c D d , c D and c D k are the Caputo derivative of orders 0<r, d, , k 1, 1<d þ r<2, 1< þ k<2 and n, g : ½0, 1 Â R n ! R is continuous. In recent years, the investigation of HU-stability analysis for non-linear FDEs is a topic of hot research. Therefore, Hyers-Ulam type stability plays important roles for FDEs. The HU stability can be defined as an exact solution exist very near to the approximate solution for FDEs with very small error. Recently, nonlinear differential equations with quadratic perturbations have received much attention. This type of differential equations is called hybrid differential equations. The hybrid FDEs have been considered more significant in different scientific fields and taking as special cases of dynamic systems. The first-order hybrid differential equation is investigated by Dhage and Lakshmikantham (2010). For more detail about hybrid FDEs, see the articles (B aleanu, Agarwal, B aleanu, Agarwal, Mohammadi, & Rezapor, 2013;B aleanu & Mustafa, 2010;B aleanu, Mustafa, & Agarwal, 2010a, 2010bIsaia, 2006;Royden & Fitzpatrick, 1988;Stamova, 2015;Zada, Faisal, & Li, 2017). Inspired from the aboveaforementioned contributions, in this article, we investigate a coupled system of HFDEs with p-Laplacian by Leray-Schauder and topological degree theories. In addition to studying some important conditions for the HUstability of the solution to our suggested problem c D k ð/ p ð c D r ðlðfÞÀQ 2 ðf, ðfÞÞÞÞÞ ¼ ÀQ 1 ðf, ðfÞÞ, (1.1) where kÀ1<k, r<k, c D k , c D r , are Caputo derivatives, Q 1 , Q 2 2 L½0, 1 and / p ðgÞ ¼ gjgj ðpÀ2Þ is the p-Laplacian and / À1 p ¼ / q , such that 1=p þ 1=q ¼ 1: To the best of our knowledge, the topological degree and Leray-Schauder theories have not been widely used for the study of EUS for a coupled system of HFDEs with IBCs having orders in ðkÀ1, k for k ! 3 involving the p-Laplacian operator. Therefore, we prove EUS and HU-stability for the coupled system with the help of degree theorem as proposed by Deimling (1985). We study three main aspects of the HFDEs with p-Laplacian (1.1), including existence of solution, uniqueness and HU-stability for our proposed problem. To achieve these aims, we'll use Green functions to convert the problem (1.1) into an integral equation. After that, we'll use the topological degree approach to prove existence and uniqueness. Our proposed problem will become complex and more generic compared to the problems previously studied before and aforementioned. This work could draw researchers' attention to the analysis of HU-stability and various other kinds of stability for more complicated problems. We also advise readers that the issue (1.1) has the potential to be investigated further for other aims, such as multiplicity results.

Preliminaries
Here, we will offer some notations, proposition, theorems, definitions and Lammas, which has an important role to study and prove this the article.
Lemma 3. Let / p be p-Laplacian. Then, we have

Hyers-Ulam stability of system
Here, we will study HU-stability for nonlinear coupled system of HFDEs with p-Laplacian operator and integral IBCs for the solution.

Illustrative example
In the part we will offer an example to prove our results of the proposed problem in sections 3 and 4. To verify our work, we have drawn an example in the following two cases.

Conclusion the article
In this literature, we have investigated three aspects to a coupled system HFDEs (1.1) involving Caputos derivative with / p Laplacian through using degree theorem and analysis on Banach space functional. For these purposes, we first transformed our problem (1.1) into integral equations by Green functions. Then, we proved existence of the solution with the help of the Leray-Schauder and topological degree theorems. We proved the uniqueness of the solution by using the Banach principle. As well as, we used the Hyers-Ulam technique to prove stability. To verify the validity of existing results, we included an example as an application of our findings using Mathematica. additional objectives In the future, we intend to look into the outcomes of its multiplicity. Also, we plan to study the existence and uniqueness of solution to this problem by different derivatives like Atangana-Baleanu-Caputo and Riemann-Liouville derivatives of various orders.