Triple positive solutions for two classes of delayed nonlinear fractional FDEs with nonlinear integral boundary value conditions

Kaihong Zhao

doi:10.1186/s13661-015-0445-y

Abstract

This paper is concerned with two classes of delayed nonlinear fractional functional differential equations (FDEs) with nonlinear Riemann-Stieltjes integral boundary value conditions. By employing the well-known Leggett-Williams fixed point theorem and a generalization of Leggett-Williams fixed point theorem, some new sufficient criteria are established to guarantee the existence of at least triple positive solutions. As applications, some interesting examples are presented to illustrate our main results.

Highlights

1 Introduction 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, Bode’s analysis of feedback amplifiers, capacitor theory, electrical circuits, electron-analytical chemistry, biology, control theory, fitting of experimental data, and so forth
Fractional differential equations serve as an excellent tool for the description of hereditary properties of various materials and processes
In physics, the traditional way to deal with the behavior of certain materials under the influence of external forces in mechanics is to use the laws of Hooke and Newton

Summary

Introduction

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, Bode’s analysis of feedback amplifiers, capacitor theory, electrical circuits, electron-analytical chemistry, biology, control theory, fitting of experimental data, and so forth. In Section , one sufficient condition is given by the well-known Leggett-Williams fixed point theorem to guarantee the existence multiple positive solutions for BVP (Leggett-Williams fixed point theorem, see [ ]) Let P be a cone in a real Banach space E, ψ be a nonnegative continuous concave functional on P such that ψ(x) ≤ x for x ∈ Pc. Suppose that A : Pc → Pc is completely continuous and there exist < a < b < d ≤ c such that (i) {x ∈ P(ψ, b, d) : ψ(x) > b} = ∅ and ψ(Ax) > b for all x ∈ P(ψ, b, d); (ii) Ax < a for all x ∈ Pa; (iii) ψ(Ax) > b for all x ∈ P(ψ, b, c) with Ax > d.

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