Contractive-Like Mapping Principles in Ordered Metric Spaces and Application to Ordinary Differential Equations

Abstract

The purpose of this paper is to present a fixed point theorem for generalized contractions in partially ordered complete metric spaces. We also present an application to first-order ordinary differential equations.

On coupled generalised Banach and Kannan type contractions

“…The method applied by Geraghty was utilized to obtain further new fixed point results works like [2] and [10]. Definition 1.1.…”

Fixed points for non-self operators in gauge spaces

“…Remark 3.8 It is worth to mention that could be of interest to extend the above technique for other metrical fixed point theorems, see [3], [8], etc. It is also an open problem to present a fixed point theory (in the sense of [20]) for contractions and generalized contractions in ordered complete gauge spaces.…”

Fixed point theorems on ordered gauge spaces with applications to nonlinear integral equations

“…there exists k [0, 1) such that d(Tx, Ty) ≤ kd(x, y) for all x, y X with x ≼ y; 2. there exists x 0 X such that x 0 ≼ Tx 0 ; 3. if {x n } is a nondecreasing sequence in X such that x n x X as n ∞, then x n ≼ x for all n. Then T has a fixed point. Since then, several authors considered the problem of existence (and uniqueness) of a fixed point for contraction type operators on partially ordered metric spaces (see, for example, [2,3,5,[15][16][17]19,[21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38]). …”