Monotone Generalized Contractions in Ordered Metric Spaces

Abstract

Abstract. In this paper, we prove some existence and uniqueness results on coincidence points for g-monotone mappings satisfying linear as well as generalized nonlinear contractivity conditions in ordered metric spaces. Our results generalize and extend two classical and well known results due to Ran and Reurings (Proc. Amer. Math. Soc. 132 (2004), no. 5, 1435-1443) and Nieto and Rodríguez-López (Acta Math. Sin. 23 (2007), no. 12, 2205-2212) besides similar other ones. Finally, as an application of one of our newly proved results, we establish the existence and uniqueness of solution of a first order periodic boundary value problem.

Comparable nonlinear contractions in ordered metric spaces

“…Notice that Corollary 4.5 improves the main result of Alam and Imdad [3]. Now, we prove the corresponding result of Theorem 4.1 under linear contractivity condition, as follows.…”

“…Recently Alam and Imdad [3] formulated the variants of bounded sequences and monotone sequences with respect to relation ≺ by introducing the following.…”

“…−→ x, then we denote it symbolically by x n x. Alam and Imdad [3] formulated the following notion by using a suitable property on ordered metric space (in order to avoid the necessity of continuity requirement in Ran-Reurings Theorem) utilized by Nieto and Rodríguez-López [27].…”

Relation-theoretic metrical coincidence theorems

“…Definition 5.19. [46] Given a mapping : X → X, we say that an ordered metric space (X, d, ) has g-TCC (termwise monotone-convergence-c-bound) property if every termwise monotone sequence {x n } in X such that x n d −→ x has a subsequence, whose -image is termwise bounded by -image of limit (of the sequence) as a c-bound, i.e., x n k ≺ x ∀ k ∈ N 0 .…”

Results on Coincidence and Common Fixed Points for (ψ,φ)g-Generalized Weakly Contractive Mappings in Ordered Metric Spaces

“…This noted-paper of Ran and Reurings is well followed by two very useful articles from Nieto and Rodríguez-López [15,16]. Presently, proving an order-theoretic analogue of metric-fixed point results is an area of active research and by now there exists considerable literature on this topic (e.g., [17][18][19][20][21][22][23][24][25][26][27]). Our work in this paper is on similar lines wherein our results are proved using (ψ, ϕ) g -generalized weakly contractive mappings.…”