A fixed point theorem in partially ordered sets and some applications to matrix equations

Abstract

Abstract. An analogue of Banach's fixed point theorem in partially ordered sets is proved in this paper, and several applications to linear and nonlinear matrix equations are discussed.

Multidimensional coincidence point results for generalized $(\psi ,\theta ,\varphi)$-contraction on ordered metric spaces

“…Furthermore, we show how multidimensional results can be seen as simple consequences of our unidimensional coincidence point theorem. We modify, improve, sharpen, enrich and generalize the results of Alotaibi and Alsulami [1], Alsulami [2], Gnana-Bhaskar and Lakshmikantham [6], Harjani and Sadarangani [13], Harjani et al [14], Lakshmikantham and Ciric [22], Luong and Thuan [23], Nieto and Rodriguez-Lopez [26], Ran and Reurings [27], Razani and Parvaneh [28] and many other famous results in the literature.…”

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

“…On the other hand, Ran and Reurings [6] proved the following Banach-Caccioppoli type principle in ordered metric spaces. Theorem 1.8 (Ran and Reurings [6]) Let (X, ≼) be a partially ordered set such that every pair x, y X has a lower and an upper bound.…”

“…Theorem 1.8 (Ran and Reurings [6]) Let (X, ≼) be a partially ordered set such that every pair x, y X has a lower and an upper bound. Let d be a metric on X such that the metric space (X, d) is complete.…”

“…Fixed point theory is considered as one of the most important tools of nonlinear analysis that widely applied to optimization, computational algorithms, physics, variational inequalities, ordinary differential equations, integral equations, matrix equations and so on (see, for example, [1][2][3][4][5][6]). The Banach contraction principle [7] is a fundamental result in fixed point theory.…”

Coupled fixed point theorems in ordered $\epsilon$-chainable partial metric spaces

“…Subsequently, several authors have proved fixed point theorems in partial metric spaces. (see, e.g., [13]- [14], [17]) Recently, Ran and Reurings [18], Bhaskar and Lakshmikantham [3], Nieto and Lopez [14], Agarwal, El-Gebeily and O'regan [1] and Lakshmikantham and Ciric [11] presented some new results for contraction in partially ordered metric spaces. For a given partially ordered set X, Bhaskar and Lakshmikantham in [3] introduced the concept of coupled fixed point of a mapping F : X × X → X.…”