Nonlinear Caputo Fractional Differential Equations Research Articles

Nonlinear Sequential Riemann-Liouville and Caputo Fractional Differential Equations with Nonlocal and Integral Boundary Conditions

In this paper, we discuss the existence and uniqueness of solutions for a new class of sequential fractional differential equations of Riemann-Liouville and Caputo types with nonlocal integral boundary conditions, by using standard fixed point theorems. We also demonstrate the application of the obtained results with the aid of examples.

Applications of Lyapunov Functions to Caputo Fractional Differential Equations

One approach to study various stability properties of solutions of nonlinear Caputo fractional differential equations is based on using Lyapunov like functions. A basic question which arises is the definition of the derivative of the Lyapunov like function along the given fractional equation. In this paper, several definitions known in the literature for the derivative of Lyapunov functions among Caputo fractional differential equations are given. Applications and properties are discussed. Several sufficient conditions for stability, uniform stability and asymptotic stability with respect to part of the variables are established. Several examples are given to illustrate the theory.

Lyapunov Functions and Stability of Caputo Fractional Differential Equations with Delays

The direct Lyapunov method is extended to nonlinear Caputo fractional differential equations with variable bounded delays. A brief overview of the literature on derivatives of Lyapunov functions is given and applications to fractional equations are discussed. Advantages and disadvantages are illustrated with examples. Sufficient conditions using three derivatives of Lyapunov functions are given and our results are compared with results in the literature. Also fractional order extensions of comparison principle are established.

Iterative techniques for the initial value problem for Caputo fractional differential equations with non-instantaneous impulses

Two types of algorithms for constructing monotone successive approximations for solutions to initial value problems for a scalar nonlinear Caputo fractional differential equation with non-instantaneous impulses are given. The impulses start abruptly at some points and their action continue on given finite intervals. Both algorithms are based on the application of lower and upper solutions to the problem. The first one is a generalization of the monotone iterative technique and it requires an application of the Mittag-Leffler function with one and two parameters. The second one is easier from a practical point of view and is applicable when the right hand sides of the equation are monotone. We prove that the functional sequences are convergent and their limits are minimal and maximal solutions of the problem. An example is given to illustrate the results.

Existence of Solutions for a System of Fractional Differential Equations with Coupled Nonlocal Boundary Conditions

AbstractWe study the existence of solutions for a system of nonlinear Caputo fractional differential equations with coupled boundary conditions involving Riemann-Liouville fractional integrals, by using the Schauder fixed point theorem and the nonlinear alternative of Leray-Schauder type. Two examples are given to support our main results.

A Linearized Stability Theorem for Nonlinear Delay Fractional Differential Equations

In this paper, we prove a theorem of linearized asymptotic stability for fractional differential equations with a time delay. More precisely, using the method of linearization of a nonlinear equation along an orbit (Lyapunov's first method), we show that an equilibrium of a nonlinear Caputo fractional differential equation with a time delay is asymptotically stable if its linearization at the equilibrium is asymptotically stable. Our approach based on a technique which converts the linear part of the equation into a diagonal one. Then using properties of generalized Mittag-Leffler functions, the construction of an associated Lyapunov--Perron operator and the Banach contraction mapping theorem, we obtain the desired result.

Iterative techniques with computer realization for the initial value problem for Caputo fractional differential equations

The main aim of the paper is to suggest some algorithms and to use them in an appropriate computer environment to solve approximately the initial value problem for scalar nonlinear Caputo fractional differential equations on a finite interval. The iterative schemes are based on appropriately defined lower and upper solutions to the given problem. A number of different cases depending on the type of lower and upper solutions are studied and various schemes for constructing successive approximations are provided. In some of the algorithms we do not use Mittag-Leffler functions and as a result the practical application of the algorithms are easier. Also we suggest two algorithms generalizing the monotone-iterative techniques and using Mittag-Leffler functions and in a practical problem computer software is build and used.

A STUDY OF NONLINEAR FRACTIONAL-ORDER BOUNDARY VALUE PROBLEM WITH NONLOCAL ERDELYI-KOBER AND GENERALIZED RIEMANN-LIOUVILLE TYPE INTEGRAL BOUNDARY CONDITIONS

We investigate a new kind of nonlocal boundary value problems of nonlinear Caputo fractional differential equations supplemented with integral boundary conditions involving Erdelyi-Kober and generalized Riemann-Liouville fractional integrals. Existence and uniqueness results for the given problem are obtained by means of standard fixed point theorems. Examples illustrating the main results are also discussed.

Noninstantaneous impulses in Caputo fractional differential equations and practical stability via Lyapunov functions

Practical stability of a nonlinear Caputo fractional differential equation with noninstantaneous impulses is studied using Lyapunov like functions. We present a new definition of the derivative of a Lyapunov like function along the given fractional differential equation with noninstantaneous impulses. Sufficient conditions for practical stability, practical quasi stability and strongly practical stability are established and several examples are given to illustrate the results.

Strict stability with respect to initial time difference for Caputo fractional differential equations by Lyapunov functions

Abstract The strict stability properties are generalized to nonlinear Caputo fractional differential equations in the case when both initial points and initial times are changeable. Using Lyapunov functions, some criteria for strict stability, eventually strict stability and strict practical stability are obtained. A brief overview of different types of derivatives in the literature related to the application of Lyapunov functions to Caputo fractional equations are given, and their advantages and disadvantages are discussed with several examples. The Caputo fractional Dini derivative with respect to to initial time difference is used to obtain some sufficient conditions.

Existence of positive solutions for multi-point boundary value problems of Caputo fractional differential equation

In this paper, we investigate the existence of positive solutions for the non-linear Caputo fractional differential equation. Using the five-functional fixed-point theorem, the existence of multiple positive solutions is obtained for the boundary value problems. An example is also given to illustrate our main result.

Practical stability with respect to initial time difference for Caputo fractional differential equations

Practical stability with initial data difference for nonlinear Caputo fractional differential equations is studied. This type of stability generalizes known concepts of stability in the literature. It enables us to compare the behavior of two solutions when both initial values and initial intervals are different. In this paper the concept of practical stability with initial time difference is generalized to Caputo fractional differential equations. A definition of the derivative of Lyapunov like function along the given nonlinear Caputo fractional differential equation is given. Comparison results using this definition and scalar fractional differential equations are proved. Sufficient conditions for several types of practical stability with initial time difference for nonlinear Caputo fractional differential equations are obtained via Lyapunov functions. Some examples are given to illustrate the results.

Linearized asymptotic stability for fractional differential equations

We prove the theorem of linearized asymptotic stability for fractional differential equations. More precisely, we show that an equilibrium of a nonlinear Caputo fractional differential equation is asymptotically stable if its linearization at the equilibrium is asymptotically stable. As a consequence we extend Lyapunov's first method to fractional differential equations by proving that if the spectrum of the linearization is contained in the sector $\{\lambda \in \C : |\arg \lambda| > \frac{\alpha \pi}{2}\}$ where $\alpha > 0$ denotes the order of the fractional differential equation, then the equilibrium of the nonlinear fractional differential equation is asymptotically stable.

Stability of nonlinear Caputo fractional differential equations

In this paper, by using the fractional differential inequality, we obtain a Caputo type fractional comparison result. Further, We investigate the stability and instability of nonlinear Caputo fractional differential equations by using the Caputo type fractional comparison principle.

Lyapunov functions and strict stability of Caputo fractional differential equations

One of the main properties studied in the qualitative theory of differential equations is the stability of solutions. The stability of fractional order systems is quite recent. There are several approaches in the literature to study stability, one of which is the Lyapunov approach. However, the Lyapunov approach to fractional differential equations causes many difficulties. In this paper a new definition (based on the Caputo fractional Dini derivative) for the derivative of Lyapunov functions to study a nonlinear Caputo fractional differential equation is introduced. Comparison results using this definition and scalar fractional differential equations are presented, and sufficient conditions for strict stability and uniform strict stability are given. Examples are presented to illustrate the theory.

Positive Solutions for a System of Fractional Differential Equations with Nonlocal Integral Boundary Conditions

In this paper, we discuss by means of a fixed point theorem, the existence of positive solutions of a system of nonlinear Caputo fractional differential equations with integral boundary conditions. An example is given to illustrate the main results.

Positive solutions for multi-point boundary value problems of fractional differential equations with p-Laplacian

This paper investigates the existence of solutions for multi-point boundary value problems of higher-order nonlinear Caputo fractional differential equations with p-Laplacian. Using the five functionals fixed-point theorem, the existence of multiple positive solutions is proved. An example is also given to illustrate the effectiveness of ourmain result. Copyright © 2015 John Wiley & Sons, Ltd.

Positive solutions for nonlinear Caputo fractional differential equations with integral boundary conditions

In this paper, we consider the properties of Green’s function for a class of nonlinear Caputo fractional differential equations with integral boundary conditions by constructing an available integral operator. By means of well-known fixed point theorems and lower and upper solutions method, some new existence and nonexistence criteria of single or multiple positive solutions for fractional differential equation boundary value problems are established. As applications, some interesting examples are presented to illustrate the main results.

Initial time difference quasilinearization for Caputo Fractional Differential Equations

This paper deals with an application of the method of quasilinearization by not demanding the Hölder continuity assumption of functions involved and by choosing upper and lower solutions with initial time difference for nonlinear Caputo fractional differential equations. Thus, we construct monotone flows that are generated by solutions of linear fractional differential equations which converge uniformly and quadratically to the unique solution of the problem. Also, necessary comparison result concerning lower and upper solutions are proved without using Hölder continuity.Mathematics subject classification: 34A12, 34A45, 34C11.