Fractional Differential Equation Boundary Value Research Articles

Unique positive solutions for fractional differential equation boundary value problems

In this work, we consider the uniqueness of positive solutions for fractional differential equation boundary value problems. Our results can not only guarantee the existence of a unique positive solution, but also be applied to construct an iterative scheme for approximating it.

On solutions of some fractional m-point boundary value problems at resonance

In this paper, the following fractional order ordinary differential equation boundary value problem: D � ( �)

Multiple positive solutions for the boundary value problem of a nonlinear fractional differential equation

In this paper, we consider the properties of Green’s function for the nonlinear fractional differential equation boundary value problem D 0 + α u ( t ) = f ( t , u ( t ) ) , 0 < t < 1 , u ( 0 ) = u ( 1 ) = u ′ ( 0 ) = u ′ ( 1 ) = 0 , where 3 < α ≤ 4 is a real number, and D 0 + α is the standard Riemann–Liouville differentiation. As an application of Green’s function, we give some multiple positive solutions for singular and nonsingular boundary value problems, and we also give the uniqueness of solution for a singular problem by means of the Leray–Schauder nonlinear alternative, a fixed-point theorem on cones and a mixed monotone method.

Upper and lower solutions method and a fractional differential equation boundary value problem

The method of lower and upper solutions for fractional differential equation Du(t)+ g(t,u(t)) = 0,t 2 (0,1),1 < � � 2, with Dirichlet boundary condition u(0) = a,u(1) = b is used to give sufficient conditions for the existence of at least onesolution.

Positive solutions for boundary value problem of nonlinear fractional differential equation

In this paper, we investigate the existence and multiplicity of positive solutions for nonlinear fractional differential equation boundary value problem: D 0 + α u ( t ) + f ( t , u ( t ) ) = 0 , 0 < t < 1 , u ( 0 ) = u ( 1 ) = 0 , where 1 < α ⩽ 2 is a real number, D 0 + α is the standard Riemann–Liouville differentiation, and f : [ 0 , 1 ] × [ 0 , ∞ ) → [ 0 , ∞ ) is continuous. By means of some fixed-point theorems on cone, some existence and multiplicity results of positive solutions are obtained. The proofs are based upon the reduction of problem considered to the equivalent Fredholm integral equation of second kind.