On a nonlocal implicit problem under Atangana\u2013Baleanu\u2013Caputo fractional derivative

In this paper, we derive the equivalent fractional integral equation to the nonlinear implicit fractional differential equations involving Ψ‐Hilfer fractional derivative subject to nonlocal fractional integral boundary conditions. The existence of a solution, Ulam–Hyers, and Ulam–Hyers–Rassias stability have been acquired by means of an equivalent fractional integral equation. Our investigations depend on the fixed‐point theorem due to Krasnoselskii and the Gronwall inequality involving Ψ‐Riemann–Liouville fractional integral. Finally, examples are provided to show the utilization of primary outcomes.

On the existence for a class of periodic boundary value problems of nonlinear fractional hybrid differential equations

By means of the Green function, the boundary value problem of a fractional differential equation can be reduced to the equivalent integral equation. Recently, this method has been used successfully to discuss the existence of the solution to the boundary value problem of a nonlinear fractional differential equation. By applying the nonlinear alternative of the Leray-Schauder type and the Krasnoselskii fixed point theorem, we investigate the boundary value problem of a nonlinear impulsive fractional differential equation, and we obtain two existence results for the solution. MSC:26A33, 34B15.

This manuscript is devoted to proving some results concerning the existence of solutions to a class of boundary value problems for nonlinear implicit fractional differential equations with non-instantaneous impulses and generalized Hilfer fractional derivatives. The results are based on Banach’s contraction principle and Krasnosel’skii’s fixed point theorem. To illustrate the results, an example is provided.

This paper studies the existence and uniqueness of solutions for a class of boundary value problems of nonlinear fractional order differential equations involving the Caputo fractional derivative by employing the Banach’s contraction principle and the Schauder’s fixed point theorem. In addition, an example is given to demonstrate the application of our main results.

In this paper, we study the existence and uniqueness of solutions for Riemann–Stieltjes integral boundary value problems of nonlinear implicit Hadamard fractional differential equations. The investigation of the main results depends on Schauder’s fixed point theorem and Banach’s contraction principle. An illustrative example is given to show the applicability of theoretical results.

We establish a coincidence point theorem in complete metric spaces. As a consequence, we show the existence of solutions to a system of initial value problem of fractional differential equations involving Riemann–Liouville fractional derivatives. Next, we derive several coincidence point theorems for new classes of sublinear and superlinear operators, in the context of ordered Banach spaces. Finally, we apply these results to discuss the existence of positive solutions to a class of initial value problem of fractional differential equations.

In this paper, we establish sufficient conditions for the existence of solutions for a class of initial value problems for impulsive fractional differential equations involving the deformable fractional derivative. We achieve our results using classical fixed point theorems such as the Banach contraction principle, the Shaefer and the alternative Leray–Schauder theorems. We provide an example to illustrate our abstract results.

This paper considers a class of nonlinear hybrid fractional differential equations with the Caputo derivative. The hybrid feature means that the fractional derivative is taken for the corrected state z(t)-f(t,z(t)), which can model memory effects together with a feedback-type term. We first convert the initial value problem into an equivalent Volterra integral equation. Using this form, we define an operator on C([0,T],R) and prove existence and uniqueness of solutions by the Banach fixed point theorem under simple Lipschitz conditions. We also establish well-posedness by showing continuous dependence on the initial data and deriving an explicit a priori bound. Finally, we develop a three-step Adams--Bashforth product-integration scheme for numerical approximation and provide numerical experiments to support the theoretical results.

Study of evolution problem under Mittag–Leffler type fractional order derivative

This paper focuses on a singular boundary value (SBV) problem of nonlinear fractional differential (NFD) equation defined as follows: D 0 + β υ τ + f τ , υ τ = 0 , τ ∈ 0,1 , υ 0 = υ ′ 0 = υ ″ 0 = υ ″ 1 = 0 , where 3 < β ≤ 4 , D 0 + β is the standard Riemann–Liouville fractional (RLF) derivative. The nonlinear function f τ , υ τ might be singular on the spatial and temporal variables. This paper proves that a positive solution to the SBV problem exists and is unique, taking advantage of Green’s function through a fixed-point (FP) theory on cones and mixed monotone operators.

Piecewise continuous solutions of initial value problems of singular fractional differential equations with impulse effects

The purpose of this paper is to investigate a class of initial value problems of fuzzy fractional coupled partial differential equations with Caputo gH-type derivatives. Firstly, using Banach fixed point theorem and the mathematical inductive method, we prove the existence and uniqueness of two kinds of gH-weak solutions of the coupled system for fuzzy fractional partial differential equations under Lipschitz conditions. Then we give an example to illustrate the correctness of the existence and uniqueness results. Furthermore, because of the coupling in the initial value problems, we develop Gronwall inequality of the vector form, and creatively discuss continuous dependence of the solutions of the coupled system for fuzzy fractional partial differential equations on the initial values and ε-approximate solution of the coupled system. Finally, we propose some work for future research.

In this study, we focus on nonlinear forward fractional difference equations with order ν∈(0,1] and construct stability analysis regarding h-stability and Mittag-Leffler stability notions. The main results of the paper are obtained by equiparating the equation in the spotlight with an auxiliary fractional difference equation. The outcomes of the manuscript provide an alternative approach to the ongoing theory of discrete fractional equations since the method used in the main results deviates from the fundamental tools of stability theory, namely fixed point theory and Liapunov's direct method.

The idea proposed in this work is to extend the ZZ transform method to resolve the nonlinear fractional partial differential equations by combining them with the so-called homotopy perturbation method (HPM). We apply this technique to solve some nonlinear fractional equations as: nonlinear time-fractional Fokker-Planck equation, the cubic nonlinear time-fractional Schrodinger equation and the nonlinear timefractional KdV equation. The fractional derivative is described in the Caputo sense. The results show that this is the appropriate method to solve somme models of nonlinear partial differential equations with time-fractional derivative.