Search

An initial value problem for a scalar nonlinear differential equation with a variable order for the generalized proportional Caputo fractional derivative is studied. We consider the case of a piecewise constant variable order of the fractional derivative. Since the order of the fractional integrals and derivatives depends on time, we will consider several different cases. The argument of the variable order could be equal to the current time or it could be equal to the variable of the integral determining the fractional derivative. We provide three different definitions of generalized proportional fractional integrals and Caputo-type derivatives, and the properties of the defined differentials/integrals are discussed and compared with what is known in the literature. Appropriate auxiliary systems with constant-order fractional derivatives are defined and used to construct solutions of the studied problem in the three cases of fractional derivatives. Existence and uniqueness are studied. Also, the Ulam-type stability is defined in the three cases, and sufficient conditions are obtained. The suggested approach is more broadly based, and the same methodology can be used in a number of additional issues. [ABSTRACT FROM AUTHOR]

In the current paper, we intend to check the existence aspects of solutions for a category of the multi‐point boundary value problem (BVP) involving a p$$ p $$‐Laplacian differential operator within the generalized fractional derivatives depending on another function. Based on two fixed point results attributed to Schaefer and Banach, the desired results are verified. Further, the continuity of solutions in terms of inputs (fractional orders, associated parameters, and appropriate function) is extensively discussed. A simulative example is prepared to demonstrate the applications of the obtained results. [ABSTRACT FROM AUTHOR]

In the existing article, the existence of solutions to nonlinear fractional differential inclusions in the sense of the Atangana-Baleanu-Caputo (ABC) fractional derivatives in Banach space is studied. The investigation of the main results relies on the set-valued issue of Monch fixed point theorem incorporated with the Kuratowski measure of non-compactness. A simulated example is proposed to explain the obtained results. [ABSTRACT FROM AUTHOR]

Anew class of implicit Hadamard fractional differential equations with Riemann-Stieltjes integral boundary conditions is studied in this research paper. The existence and uniqueness results of the aforesaid problem are investigated using Schauder's fixed point theorem and Banach's contraction mapping principle. A simulative example is given to highlight the acquired outcomes. [ABSTRACT FROM AUTHOR]

The existence, uniqueness and the Hyers-Ulam stability of solutions for nonlinear p–Laplacian implicit Caputo-Katugampola fractional differential equations (FDEs) with multi-point generalized fractional integral boundary conditions are established. With the aid of the Leray-Schauder nonlinear alternative and the Banach contraction principle, the desired results are obtained. An example is set forth to make efficient the main results. [ABSTRACT FROM AUTHOR]

This paper is concerned with the Ulam-Hyers and the generalized Ulam-Hyers-Rassias stability results for linear fractional differential equations with hybrid proportional-Caputo derivatives in framwork of the Laplace transform method. The existence and uniqueness of solutions for nonlinear fractional differential equations with hybrid proportional-Caputo derivatives are established by means of Schaefer's fixed point theorem, Banach's fixed point theorem, and generalized Gronwall's inequality. Two examples are proposed to illustrate the main results. [ABSTRACT FROM AUTHOR]

This paper concerns the investigation of controllability and Hyers-Ulam stability of initial value problems for fractional differential equations via generalized proportional-Caputo fractional derivatives. The main results are obtained by means of the Krasnoselskii's fixed point theorem. [ABSTRACT FROM AUTHOR]

This paper is devoted to initial value problems for impulsive fractional differential equations of Caputo–Fabrizio type fractional derivative. By means of Banach's fixed point theorem and Schaefer's fixed point theorem, the existence and uniqueness results are obtained. Finally, an example is given to illustrate one of the main results. [ABSTRACT FROM AUTHOR]

We discuss the existence, uniqueness and continuous dependence of solution for a non-autonomous semilinear Hilfer fractional differential equation with nonlocal conditions in the space of weighted continuous functions. By means of the Krasnoselskii's fixed point theorem and the generalized Gronwall's inequality, we establish the desired results. [ABSTRACT FROM AUTHOR]

We prove the existence and uniqueness of solution for fractional differential equations with Riemann-Liouville fractional integral boundary conditions. The first existence and uniqueness result is based on Banach’s contraction principle. Moreover, other existence results are also obtained by using the Krasnoselskii fixed point theorem. An example is given to illustrate the main results. [ABSTRACT FROM AUTHOR]

A generalized proportional vector-order fractional derivative in the Caputo sense is defined and studied. Two types of existence results for the mild solutions of the initial value problem for nonlinear Caputo-type generalized proportional vector-order fractional differential equations are obtained. With the aid of the Leray–Schauder nonlinear alternative and the Banach contraction principle, the main results are established. In the case of a local Lipschitz right hand side part function, the existence of a bounded mild solution is proved. Some examples illustrating the main results are provided. [ABSTRACT FROM AUTHOR]

The object of investigation in this paper is a scalar linear fractional differential equation with generalized proportional derivative of Riemann–Liouville type (LFDEGD). The main goal is the obtaining an explicit solution of the initial value problem of the studied equation. Note that the locally solvability, being the same as the existence of solutions to the initial value problem, is connected with the symmetry of a transformation of a system of differential equations. At the same time, several criteria for existence of the initial value problem for nonlinear fractional differential equations with generalized proportional derivative are connected with the linear ones. It leads to the necessity of obtaining an explicit solution of LFDEGD. In this paper two cases are studied: the case of no impulses in the differential equation are presented and the case when instantaneous impulses at initially given points are involved. All obtained formulas are based on the application of Mittag–Leffler function with two parameters. In the case of impulses, initially the appropriate impulsive conditions are set up and later the explicit solutions are obtained. [ABSTRACT FROM AUTHOR]