Positive Solutions for Coupled Nonlinear Fractional Differential Equations

Wenning Liu,Xingjie Yan,Wei Qi

doi:10.1155/2014/790862

Abstract

We consider the existence of positive solutions for a coupled system of nonlinear fractional differential equations with integral boundary values. Assume the nonlinear term is superlinear in one equation and sublinear in the other equation. By constructing two conesK1,K2and computing the fixed point index in product coneK1×K2, we obtain that the system has a pair of positive solutions. It is remarkable that it is established on the Cartesian product of two cones, in which the feature of two equations can be opposite.

Highlights

Fractional calculus is a very old concept dating back to 17th century; it involves fractional integration and fractional differentiation
Fractional differential equations have attracted increasing interests for their extensive applications, which leads to intensive development of the theory of fractional calculus
By using the cone extension method Feng et al [15] studied the existence of solutions for higher-order nonlinear fractional differential equation with integral boundary conditions: D0α+u (t) + g (t) f (t, u (t)) = 0, 0 < t < 1, u (0) = u󸀠 (0) = ⋅ ⋅ ⋅ = u(n−2) (0) = 0, (1)

Summary

Introduction

Fractional calculus is a very old concept dating back to 17th century; it involves fractional integration and fractional differentiation. The existence of positive solutions for boundary value problem of nonlinear fractional differential equation has attracted attentions from many researches; see [9,10,11,12,13,14].

Results

Conclusion