Abstract
AbstractThe purpose of this paper is to provide sufficient conditions for the existence and uniqueness of fixed points for non-decreasing and mixed monotone mappings with auxiliary functions in the framework of metric space endowed with a partial order. As applications of our results we obtain several interesting corollaries and fixed point theorems in the underlying spaces. In order to illustrate our results, we provide two examples in which other theorems from the literature cannot be applied. In addition, the equivalence property between unidimensional and multidimensional fixed point theorems is investigated. We also present some applications to the existence of solutions of integral and differential equations.
Highlights
One of the newest branches of fixed point theory is devoted to the study of coupled fixed points, introduced by Guo and Lakshmikantham [ ] in
Harjani and Sadarangani [ ] investigated some unidimensional fixed point theorems for generalized contractions in complete partially ordered metric spaces and applications to ordinary differential equations
In [ ] and [ ] the authors obtained some multidimensional fixed point theorems for mixed monotone mappings, which extended the corresponding coupled, tripled, and quadrupled fixed point results appearing in the literature
Summary
One of the newest branches of fixed point theory is devoted to the study of coupled fixed points, introduced by Guo and Lakshmikantham [ ] in. Thereafter, Gnana Bhaskar and Lakshmikantham [ ] introduced the concept of the mixed monotone property for contractive operators in two variables in the setting of a partially ordered metric space, and they established some coupled fixed point theorems. Harjani and Sadarangani [ ] investigated some unidimensional fixed point theorems for generalized contractions in complete partially ordered metric spaces and applications to ordinary differential equations.
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