Monotone Generalized Nonlinear Contractions in Partially Ordered Metric Spaces

Abstract

A concept of g-monotone mapping is introduced, and some fixed and common fixed point theorems for g-non-decreasing generalized nonlinear contractions in partially ordered complete metric spaces are proved. Presented theorems are generalizations of very recent fixed point theorems due to Agarwal et al. 2008 .

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]). …”

“…Definition 2.8 (Ćirić et al [29]) Let (X, ≼) be a partially ordered set and f, g : X X are two giving mappings. The mapping f is said to be g-nondecreasing if for all x, y X, we have gx gy ⇒ fx fy.…”

Common fixed point theorems under implicit relations on ordered metric spaces and application to integral equations

“…Thereafter, several authors worked in this direction and proved fixed point theorems in ordered metric spaces. For more details see [1][2][3][4][5][6][7][8]10,11,14,18,21,23,24,27,29,[31][32][33][34] and the references cited therein.…”

Fixed point and multidimensional fixed point theorems with applications to nonlinear matrix equations in terms of weak altering distance functions

“…In the recent years, many authors extended fixed point results for weak contractions and generalized contractions, which are generalizations of Banach contraction mapping principle to partially ordered metric spaces (see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]). Some of the above results involve altering distance functions presented by Khan et al in [16].…”