Stability of Generalized Proportional Caputo Fractional Differential Equations by Lyapunov Functions

In this paper, nonlinear nonautonomous equations with the generalized proportional Caputo fractional derivative (GPFD) are considered. Some stability properties are studied by the help of the Lyapunov functions and their GPFDs. A scalar nonlinear fractional differential equation with the GPFD is considered as a comparison equation, and some comparison results are proven. Sufficient conditions for stability and asymptotic stability were obtained. Examples illustrating the results and ideas in this paper are also provided.

Convergence and stability of an iteration process and solution of a fractional differential equation

In this paper, we prove that a three-step iteration process is stable for contractive-like mappings. It is also proved analytically and numerically that the considered process converges faster than some remarkable iterative processes for contractive-like mappings. Furthermore, some convergence results are proved for the mappings satisfying Suzuki’s condition (C) in uniformly convex Banach spaces. A couple of nontrivial numerical examples are presented to support the main results and the visualization is showed by Matlab. Finally, by utilizing our main result the solution of a nonlinear fractional differential equation is approximated.

Existence and U-H-R Stability of Solutions to the Implicit Nonlinear FBVP in the Variable Order Settings

In this paper, the existence of the solution and its stability to the fractional boundary value problem (FBVP) were investigated for an implicit nonlinear fractional differential equation (VOFDE) of variable order. All existence criteria of the solutions in our establishments were derived via Krasnoselskii’s fixed point theorem and in the sequel, and its Ulam–Hyers–Rassias (U-H-R) stability is checked. An illustrative example is presented at the end of this paper to validate our findings.

On a fractional problem of Lane–Emden type: Ulam type stabilities and numerical behaviors

In this work, we study some types of Ulam stability for a nonlinear fractional differential equation of Lane–Emden type with anti periodic conditions. Then, by using a numerical approach for the Caputo derivative, we investigate behaviors of the considered problem.

A Study on the Solutions of a Multiterm FBVP of Variable Order

In the present research study, for a given multiterm boundary value problem (BVP) involving the nonlinear fractional differential equation (NnLFDEq) of variable order, the uniqueness-existence properties are analyzed. To arrive at such an aim, we first investigate some specifications of this kind of variable order operator and then derive required criteria confirming the existence of solution. All results in this study are established with the help of two fixed-point theorems and examined by a practical example.

On Extended Branciari b -Distance Spaces and Applications to Fractional Differential Equations

In this work, we define new α − λ -rational contractive conditions and establish fixed-points results based on aforesaid contractive conditions for a mapping in extended Branciari b -distance spaces. We furnish two examples to justify the work. Further, we discuss results on weak well-posed property, weak limit shadowing property, and generalized w -Ulam-Hyers stability in the underlying space. Finally, as an application of our main result, we obtain sufficient conditions for the existence of solutions of a nonlinear fractional differential equation with integral boundary conditions.

STABILITY OF NONLINEAR FRACTIONAL DIFFERENTIAL EQUATIONS IN THE FRAME OF ATANGANA-BALEANU OPERATOR

Theory of fractional calculus with singular and non-singular kernels is pioneering and has garnered significant interest recently. Fair amount of literature on the qualitative properties of fractional differential and integral equations involving different types of operators is available. This manuscript aims to analyze the stability of a class of nonlinear fractional differential equation in terms of Atangana-Baleanu-Caputo operator. Sufficient conditions for the existence and uniqueness of solutions are obtained by employing classical fixed point theorems and Banach contraction principle. Also adequate conditions for Hyers-Ulam stability are established. To substantiate our analytic results, an example is provided with numerical simulation.

A Common Fixed Point Theorem for Generalised F -Kannan Mapping in Metric Space with Applications

This paper is aimed at proving a common fixed point theorem for F -Kannan mappings in metric spaces with an application to integral equations. The main result of the paper will extend and generalise the recent existing fixed point results in the literature. We also provided illustrative examples and some applications to integral equation, nonlinear fractional differential equation, and ordinary differential equation for damped forced oscillations to support the results.