On stability analysis and existence of positive solutions for a general non-linear fractional differential equations

In this article, we deals with the existence and uniqueness of positive solutions of general non-linear fractional differential equations (FDEs) having fractional derivative of different orders involving p-Laplacian operator. Also we investigate the Hyers–Ulam (HU) stability of solutions. For the existence result, we establish the integral form of the FDE by using the Green function and then the existence of a solution is obtained by applying Guo–Krasnoselskii’s fixed point theorem. For our purpose, we also check the properties of the Green function. The uniqueness of the result is established by applying the Banach contraction mapping principle. An example is offered to ensure the validity of our results.


Introduction
Fractional calculus concerns the applications of derivatives and integrals of arbitrary order. During the last few decades, it received great attention because of its various applications in diverse scientific fields. Arbitrary-order models are more flexible than integerorder models. FDEs arise in numerous scientific and engineering fields such as physics, polymer rheology, geophysics, biophysics, aerodynamics, capacitor theory, biology, nonlinear oscillation of earthquake, control theory, blood flow phenomena, viscoelasticity, and electrical circuits. For the exhaustive study of its applications, we refer to extensive work [1][2][3][4][5][6][7]. The fundamental differences between exponential decay, the power law, the Mittag-Leffler law and some possible applications in nature are presented in [8,9].
Nowadays, the existence and uniqueness (EU) of solutions for different type of FDEs is a field of intensive research. Here, we introduce some important and recent work of several researcher about the existence of a positive solution (EPS) of different classes of FDEs. For example, the EU results for Dirichlet and mixed problems of singular FDEs with the Riemann-Liouville sense of fractional derivative were investigated by Agarwal et al. [10,11]. Baleanu et al. in [12] established the existence of a solution on partially ordered Banach spaces for a non-linear FDEs. Vong studied the singular FDEs involving non-local type boundary conditions in [13] by using fixed point techniques. For more details of EU of different types of FDEs with different types of fractional derivatives, see . Numerical solutions for the fractional Fisher's type equations involving the Atangana-Baleanu fractional derivative by methods of spectral collocation are in [26]. As of lately, some authors investigated the FDEs with p-Laplacian operator by diverse types of mathematical techniques. For instance, Khan et al. investigated the existence criterion for solutions for FDEs involving the φ p -Laplacian operator in [36]. The EU of results for FDEs with φ p -Laplacian operator are analyzed by Chuanzhi Bai in [37] via fixed point theorems. Also we present the Green function's properties and two examples to illustrate the results. The EPS for FDEs with the φ p -Laplacian operator is studied by Tian et al. [38] and EPSs are obtained with the help of a monotone iterative method. For more EU results for FDEs with a p-Laplacian operator one may refer to [39][40][41][42] Recently, a great interest has been shown in the study of HU stability of non-linear FDEs with different type of boundary conditions. By HU stability we mean that there exists an exact solution very close to the approximate solution of a FDE and that the error can be calculated. The EU of solutions and HU stability FDEs with p-Laplacian operator and ABCfractional derivative involving a spatial singularity is derived by Khan et al. in [43] using the well-known Guo-Krasnoselskii theorem. Khan et al. [44] discussed the analytical study of existence and stability results of a singular non-linear FDEs with φ p -operator involving fractional integral and differential boundary conditions. The EU and HU stability of solutions for a coupled system of FDEs involving the derivative in Caputo's sense are proved by Khan et al. [45] using a Leray-Schauder-type fixed point theorem and topological degree theory. Li et al. investigated the HU stability of FDEs in [46] and also presented an example to illustrate their result. Stability and EU of solutions for the fractional order HIV model were introduced by Khan et al. in [47]. Existence and stability of solutions for singular delay FDEs with fractional integral initial conditions by using the Green function and the fixed point theorem were established by the Khan et al. in [48]. For more details of stability analysis, see [49][50][51][52][53][54][55][56][57].
Motivated by the above work, we introduce the EU and HU stability results, for nonliner FDEs involving Caputo fractional derivatives of distinct orders with φ * P Laplacian operator: where c D ζ , c D σ denotes the derivative of fractional order ζ and σ in Caputo's sense, respectively, and ψ * , λ i (t) are continuous functions. The orders n -1 < σ ≤ n, 0 < ζ ≤ 1 where n ≥ 4, ψ * ∈ L[0, 1] and φ * p (z) = |z| p-1 z denotes the p-Laplacian operator and satisfies 1 The rest of article is divided in four parts. Basic definitions and desired lemmas are presented in Sect. 1 and properties of the Green functions are discussed in Sect. 2. The EU results are given in Sect. 3. HU stability is discussed in Sect. 4. In Sect. 5, we introduce an example.
Here, we introduce certain definitions, desired lemmas and theorems, which are essential to find the main result.
Applying the q-Laplacian operator further on (2.4) we get the form Again taking the integral operator I σ to both sides of (2.5) and using Lemma 1.1, then (2.5) becomes where a j ∈ R for j = 0, 1, 2, . . . , n -1.

Using the boundary conditions
Putting the values of the constants a i in (2.6), we get where H σ (t, s) is defined in (2.2).
By (3.2) and using (R 2 R 3 ), for any t ∈ [0, 1], we get This proves that G * is uniformly bounded. In order to show that the operator G * is compact, we show the equicontinuity of the operator G * .

Theorem 3.2 Suppose that assumptions
, are satisfied and ∃a, b ∈ R + such that any of the following condition is satisfied: Then Eq. (1.1) has a positive solution z ∈ K * and a ≤ z ≤ b.

Uniqueness result
(3.14) Proof We prove the uniqueness result for p ≥ 2. By (3.10) and, for all t ∈ [0, 1], . (3.15) For each z ∈ (A * (r)) \ A * (r) and using (3.15) we have but in (3.14) we assumed that * < 1. This proves that G * is a contraction map. Hence by the Banach contraction mapping principle there exists a unique fixed point for operator G * . Hence, there exists a unique solution of Eq. (1.1) on [0, 1].

Hyers-Ulam stability
Here, we analyze the HU stability of (1.1). We define the HU stability as follows. If then there exists a function υ(t), which is continuous and satisfies Proof For the HU stability of the problem (1.1), we prove that for the integral equation (4.4) Hence using (4.4) Eq. (3.1) is HU stable. As a result, the FDE (1.1) is HU stable.

Example
Here, we present some examples to illustrate our results.

Conclusion
In this investigation, the existence results for general FDEs (1.1) involving a φ * p -Laplacian operator is established by using Guo-Krasnoselskii's fixed point theorem [58]. The uniqueness results are proved by using the Banach contraction mapping principle and HU stability is also evaluated. The properties of the Green function also proved. The validity of our result is illustrated by examples. Also, one can study the multiple solutions, periodic solutions and controllability for the proposed general non-linear FDEs.