Nonlinear Riemann-Liouville Fractional Differential Equations Research Articles

Existence, stability and global attractivity results for nonlinear Riemann-Liouville fractional differential equations

Existence, attractivity, and stability of solutions of a non-linear fractional differential equation of Riemann-Liouville type are proved using the classical Schauder fixed point theorem and a fixed point result due to Dhage. The results are illustrated with examples.

Existence and uniqueness of positive solutions for nonlinear fractional mixed problems

Abstract This paper is devoted to study the existence and uniqueness of solutions of a one parameter family of nonlinear Riemann–Liouville fractional differential equations with mixed boundary value conditions. An exhaustive study of the sign of the related Green’s function is carried out. Under suitable assumptions on the asymptotic behavior of the nonlinear part of the equation at zero and at infinity, and by application of the fixed point theory of compact operators defined in suitable cones, it is proved that there exists at least one solution of the considered problem. Moreover, the method of lower and upper solutions is developed and the existence of solutions is deduced by a combination of both techniques. In particular cases, the Banach contraction principle is used to ensure the uniqueness of solutions.

Practical stability for Riemann\u2013Liouville delay fractional differential equations

In this paper, we study a system of nonlinear Riemann–Liouville fractional differential equations with delays. First, we define in an appropriate way initial conditions which are deeply connected with the fractional derivative used. We introduce an appropriate generalization of practical stability which we call practical stability in time. Several sufficient conditions for practical stability in time are obtained using Lyapunov functions and the modified Razumikhin technique. Two types of derivatives of Lyapunov functions are used. Some examples are given to illustrate the introduced definitions and results.

Lyapunov Functions and Lipschitz Stability for Riemann–Liouville Non-Instantaneous Impulsive Fractional Differential Equations

In this paper a system of nonlinear Riemann–Liouville fractional differential equations with non-instantaneous impulses is studied. We consider a Riemann–Liouville fractional derivative with a changeable lower limit at each stop point of the action of the impulses. In this case the solution has a singularity at the initial time and any stop time point of the impulses. This leads to an appropriate definition of both the initial condition and the non-instantaneous impulsive conditions. A generalization of the classical Lipschitz stability is defined and studied for the given system. Two types of derivatives of the applied Lyapunov functions among the Riemann–Liouville fractional differential equations with non-instantaneous impulses are applied. Several sufficient conditions for the defined stability are obtained. Some comparison results are obtained. Several examples illustrate the theoretical results.

Spectral collocation method for nonlinear Riemann–Liouville fractional differential system

The spectral collocation method is investigated for the system of nonlinear Riemann–Liouville fractional differential equations (FDEs). The main idea of the presented method is to solve the corresponding system of nonlinear weakly singular Volterra integral equations obtained from the system of FDEs. In order to carry out convergence analysis for the presented method, we investigate the regularity of the solution to the system of FDEs. The provided convergence analysis result shows that the presented method has spectral convergence. Theoretical results are confirmed by numerical experiments. The presented method is applied to solve multi-term nonlinear Riemann–Liouville fractional differential equations and multi-term nonlinear Riemann–Liouville fractional integro-differential equations.

Spectral collocation method for nonlinear Riemann-Liouville fractional differential equations

We investigate the spectral collocation method for Riemann-Liouville fractional differential equation (FDE). The main step in the presented method is to transform the FDE to the corresponding nonlinear weakly singular Volterra integral equation (VIE). We provide convergence analysis to show that the presented method has spectral convergency. We carry out numerical experiments to confirm theoretical results.

Computer Simulation and Iterative Algorithm for Approximate Solving of Initial Value Problem for Riemann-Liouville Fractional Delay Differential Equations

The main aim of this paper is to suggest an algorithm for constructing two monotone sequences of mild lower and upper solutions which are convergent to the mild solution of the initial value problem for Riemann-Liouville fractional delay differential equation. The iterative scheme is based on a monotone iterative technique. The suggested scheme is computerized and applied to solve approximately the initial value problem for scalar nonlinear Riemann-Liouville fractional differential equations with a constant delay on a finite interval. The suggested and well-grounded algorithm is applied to a particular problem and the practical usefulness is illustrated.

Iterative techniques with computer realization for initial value problems for Riemann–Liouville fractional differential equations

Abstract The main aim of this 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 Riemann–Liouville 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. The suggested schemes are applied to some problems and their practical usefulness is illustrated.

Existence and Ulam type stability for nonlinear Riemann–Liouville fractional differential equations with constant delay

A nonlinear Riemann–Liouville fractional differential equation with constant delay is studied. Initially, some existence results are proved. Three Ulam type stability concepts are defined and studied. Several sufficient conditions are obtained. Some of the obtained results are illustrated on fractional biological models.

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.

Positive solutions for a system of fractional differential equations with p-Laplacian operator and multi-point boundary conditions

We investigate the existence and nonexistence of positive solutions for a system of nonlinear Riemann–Liouville fractional differential equations with parameters and p-Laplacian operator subject to multi-point boundary conditions, which contain fractional derivatives. The proof of our main existence results is based on the Guo–Krasnosel'skii fixed-point theorem.

Positive solutions for a system of Riemann-Liouville fractional differential equations with multi-point fractional boundary conditions

We study the existence and nonexistence of positive solutions for a system of nonlinear Riemann-Liouville fractional differential equations subject to multi-point boundary conditions which contain fractional derivatives.

Systems of Riemann–Liouville fractional equations with multi-point boundary conditions

We study the existence and multiplicity of positive solutions for a system of nonlinear Riemann–Liouville fractional differential equations, subject to multi-point boundary conditions which contain fractional derivatives. The nonsingular and singular cases are investigated.

Lyapunov type inequalities for mixed nonlinear Riemann–Liouville fractional differential equations with a forcing term

In this paper, we present some new Lyapunov and Hartman type inequalities for Riemann–Liouville fractional differential equations of the form (aDαx)(t)+p(t)∣x(t)∣μ−1x(t)+q(t)∣x(t)∣γ−1x(t)=f(t), where p, q, f are real-valued functions and 0<γ<1<μ<2. No sign restrictions are imposed on the potential functions p, q and the forcing term f. The inequalities obtained generalize and compliment the existing results for the special cases of this equation in the literature.

Positive solutions for a system of semipositone coupled fractional boundary value problems

We study the existence of positive solutions for a system of nonlinear Riemann-Liouville fractional differential equations with sign-changing nonlinearities, subject to coupled integral boundary conditions.

Existence of positive solutions for a system of semipositone fractional boundary value problems

We investigate the existence of positive solutions for a system of nonlinear Riemann-Liouville fractional differential equations with sign-changing nonlinearities, subject to coupled integral boundary conditions.

Existence and Nonexistence of Positive Solutions for Coupled Riemann-Liouville Fractional Boundary Value Problems

We investigate the existence and nonexistence of positive solutions for a system of nonlinear Riemann-Liouville fractional differential equations with two parameters, subject to coupled integral boundary conditions.

Nonexistence of positive solutions for a system of coupled fractional boundary value problems

We investigate the nonexistence of positive solutions for a system of nonlinear Riemann-Liouville fractional differential equations with coupled integral boundary conditions.

On a System of Fractional Differential Equations with Coupled Integral Boundary Conditions

Abstract We investigate the existence and multiplicity of positive solutions for a system of nonlinear Riemann-Liouville fractional differential equations, subject to coupled integral boundary conditions. The nonsingular and singular cases are studied.

Positive Solutions for Systems of Coupled Fractional Boundary Value Problems

We investigate the existence and nonexistence of positive solutions for a system of nonlinear Riemann-Liouville fractional differential equations with coupled integral boundary conditions which contain some positive constants.