Existence of multiple positive solutions of a nonlinear arbitrary order boundary value problem with advanced arguments

Guotao Wang,Sotiris Ntouyas,Lihong Zhang

doi:10.14232/ejqtde.2012.1.15

Abstract

In this paper, we investigate nonlinear fractional differential equations of arbitrary order with advanced arguments    D 0+u(t) + a(t)f(u(θ(t))) = 0, 0 3 (n ∈ N), D 0+ is the standard Riemann-Liouville fractional derivative of order α, f : [0,∞) → [0,∞), a : [0, 1] → (0,∞) and θ : (0, 1) → (0, 1] are continuous functions. By applying fixed point index theory and Leggett-Williams fixed point theorem, sufficient conditions for the existence of multiple positive solutions to the above boundary value problem are established.

Highlights

Fractional order differential equations have proved to be better for the description of hereditary properties of various materials and processes than integer order differential equations
We investigate nonlinear fractional differential equations of arbitrary order with advanced arguments
As a matter of fact, fractional differential equations arise in many engineering and scientific disciplines as the mathematical modeling of systems and processes in the fields of physics, chemistry, aerodynamics, electrodynamics of complex medium, polymer rheology, etc.[22, 23, 31, 32]

Summary

Introduction

Fractional order differential equations have proved to be better for the description of hereditary properties of various materials and processes than integer order differential equations. 1 existence and multiplicity of solutions (or positive solutions) of boundary value problems for nonlinear fractional differential equations [3, 11, 12, 15, 16, 24]. As a matter of fact, the theory of integer order differential equations with deviated arguments has found its extensive applications in realistic mathematical modelling of a wide variety of practical situations and has emerged as an important area of investigation. Motivated by some recent work on advanced arguments and boundary value problems of fractional order, in this paper, we investigate the following nonlinear fractionalorder differential equation with advanced arguments. By applying the well-known Banach contraction principle and Guo-Krasnoselskii fixed point theorem, Ntouyas, Wang and Zhang [30] have successfully investigated the existence of at least one positive solutions to the nonlinear fractional boundary value problem (1.1). The main tools employed are the fixed point index theory (Theorem 2.8) and the well-known Leggett-Williams fixed point theorem (Theorem 2.9)

Preliminaries

Main results

Example