Existence results and the monotone iterative technique for systems of nonlinear fractional differential equations

AbstractBy establishing a comparison result and using the monotone iterative technique combined with the method of upper and lower solutions, we investigate the existence of solutions for systems of nonlinear fractional differential equations

Solitary Wave Solutions for a Time-Fraction Generalized Hirota-Satsuma Coupled KdV Equation by a New Analytical Technique

A new iterative technique is employed to solve a system of nonlinear fractional partial differential equations. This new approach requires neither Lagrange multiplier like variational iteration method (VIM) nor polynomials like Adomian's decomposition method (ADM) so that can be more easily and effectively established for solving nonlinear fractional differential equations, and will overcome the limitations of these methods. The obtained numerical results show good agreement with those of analytical solutions. The fractional derivatives are described in Caputo sense

New unique existence criteria for higher-order nonlinear singular fractional differential equations

In this paper, a nonlinear three-point boundary value problem of higher-order singular fractional differential equations is discussed. By applying the properties of Green function and some fixed point theorems for sum-type operator on cone, some new criteria on the existence and uniqueness of solutions are obtained. Moreover, two iterative sequences are given for uniformly approximating the positive solution, which are important for practical application. At last, we give two examples to illustrate the main results

Iterative positive solutions for singular nonlinear fractional differential equation with integral boundary conditions

In this article, we study the existence of iterative positive solutions for a class of singular nonlinear fractional differential equations with Riemann-Stieltjes integral boundary conditions, where the nonlinear term may be singular both for time and space variables. By using the properties of the Green function and the fixed point theorem of mixed monotone operators in cones we obtain some results on the existence and uniqueness of positive solutions. We also construct successively some sequences for approximating the unique solution. Our results include the multipoint boundary problems and integral boundary problems as special cases, and we also extend and improve many known results including singular and non-singular cases

Existence and Uniqueness of the Solutions for Fractional Differential Equations with Nonlinear Boundary Conditions

We study the existence and uniqueness of the solutions for the boundary value problem of fractional differential equations with nonlinear boundary conditions. By using the upper and lower solutions method in reverse order and monotone iterative techniques, we obtain the sufficient conditions of both the existence of the maximal and minimal solutions between an upper solution and a lower solution and the uniqueness of the solutions for the boundary value problem and present the iterative sequence for calculating the approximate analytical solutions of the boundary value problem and the error estimate. An example is also given to illustrate the main results