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

Abstract

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.

Presence and diversity of positive solutions for a Caputo-type fractional order nonlinear differential equation with an advanced argument

“…Furthermore, if we define a cone K on B in the following way 17) and the operator T : K → B in the following way Proof. By Lemma 2.9, it is obvious that T (K) ⊆ K. According to the definition of G 1 (x, z), G 2 (ξ, z) and b(x)g(y(ϕ(x))), it is clear that T is continuous.…”

“…During the last few decades diversity of positive solutions of different BVPs for fractional order nonlinear differential equation (FONLDE for short) has extensively considered by using various techniques, for instance see the articles of Agarwal et al [2,3], Afshari et al [1], Asaduzzaman and Ali [5], Bai [8], Chen et al [12], Cu et al [13], Devi et al [17], Sun et al [34], and Torres [36] as well as for lower and upper solutions to the integro-differential and iterative hybrid type fractional differential equations see, Damag et al [14] and Damag et al [15] and for positive solutions of nonlinear dissipative type equations, see Asaduzzaman et al [6].…”

On the existence and stability of two positive solutions of a hybrid differential system of arbitrary fractional order via Avery–Anderson–Henderson criterion on cones

“…The in-depth qualitative behavior of the solution for fractional BVPs is the positivity of such solutions. The study of existence and stability of positive solution in boundary value problems is characterized by more investigation in all components of the fractional models along with the involved boundary conditions [44,45]. Most researchers avoid the multi nonzero components in initial or boundary conditions such as constants, functions, integrals, or even derivatives of functions.…”

“…Sun et al [46] investigated the required conditions for confirming the existence and uniqueness of the solution to a nonlinear fractional differential equation (FDE) whose nonlinearity involves an explicit fractional derivative using Avery-Anderson-Henderson fixed point theorem. Devi et al [44] studied the existence and uniqueness along with the Ulam-Hyers (UH) stability of positive solution of general nonlinear FDEs containing p-Laplacian operator. The authors of [47] turned to the existence and multiplicity of positive solutions for a system consisting of Riemann-Liouville FDEs equipped with the p-Laplacian operators and singular nonnegative nonlinearities, and also furnished with nonlocal boundary conditions which possess the integrals of Riemann-Stieltjes type.…”

Nonlinear Differential Problem with p-Laplacian and via Phi-Hilfer Approach: Solvability and Stability Analysis

“…A. Devi, A. Kumar, D. Baleanu and A. Khan [7]. worked on the EU and HU stability results, for nonliner FDEs involving Caputo fractional derivatives of distinct orders with ψ p Laplacian operator:…”