Krasnosel’skii Type Hybrid Fixed Point Theorems and Their Applications to Fractional Integral Equations

H M Srivastava,B S Desale,Sachin V Bedre,S M Khairnar

doi:10.1155/2014/710746

Abstract

Some hybrid fixed point theorems of Krasnosel’skii type, which involve product of two operators, are proved in partially ordered normed linear spaces. These hybrid fixed point theorems are then applied to fractional integral equations for proving the existence of solutions under certain monotonicity conditions blending with the existence of the upper or lower solution.

Highlights

Nieto and Rodrıguez-Lopez [1] proved the following hybrid fixed point theorem for the monotone mappings in partially ordered metric spaces using the mixed arguments from algebra and geometry.Theorem 1 (Nieto and Rodrıguez-Lopez [1])
We apply the hybrid fixed point theorems proved in the preceding sections to some nonlinear fractional integral equations of mixed type
Given a closed and bounded interval J = [t0, t0 + a] in R, R being the set of real numbers or some real numbers t0 ∈ R and a ∈ R with a > 0 and given a real number 0 < q < 1, consider the following nonlinear hybrid fractional integral equation: x (t) = [f (t, x (t))] (

Summary

Introduction

Nieto and Rodrıguez-Lopez [1] proved the following hybrid fixed point theorem for the monotone mappings in partially ordered metric spaces using the mixed arguments from algebra and geometry. Suppose that either T is continuous or X is such that if xn → x is a sequence in X whose consecutive terms are comparable, there every term ecxoimstspaarasubblesetqoutehnecelim{xintkx}k.∈INf of {xn}n∈N such that there exists x0 ∈ X with x0 ≦ T(x0) or x0 ≧ T(x0), T has a fixed point which is unique if “every pair of elements in X has a lower and an upper bound.”. The main object of this paper is first to establish some hybrid fixed point theorems of Krasnosel’skii type in partially ordered normed linear spaces, which involve product of two operators We apply these hybrid fixed point theorems to fractional integral equations for proving the existence of solutions under certain monotonicity conditions blending with the existence of the upper or lower solution

Hybrid Fixed Point Theorems

Krasnosel’skii Type Fixed Point Theorems

Fractional Integral Equations of Mixed Type

An Illustrative Example