Abstract
In this paper, some fixed point theorems for monotone operators in partially ordered complete metric spaces are proved. Especially, a sufficient and necessary condition for the existence of a fixed point for a class of monotone operators is presented. The main results of this paper are generalizations of the recent results in the literature. Also, the main results can be applied to solve the nonlinear elliptic problems and the delayed hematopoiesis models. MSC:47H10, 54H25.
Highlights
In the last decades, the fixed point theorems for the contraction mappings have been improved and generalized in different directions
During the extensive applications to the nonlinear integral equations, there were many researchers to investigate the existence of a fixed point for contraction-type mappings in partially ordered metric spaces
In, Bhaskar and Lakshmikantham [ ] introduced the notion of coupled fixed point and proved some coupled fixed point theorems for mixed monotone mappings
Summary
The fixed point theorems for the contraction mappings have been improved and generalized in different directions. In , Bhaskar and Lakshmikantham [ ] introduced the notion of coupled fixed point and proved some coupled fixed point theorems for mixed monotone mappings. More details on the direction of the coupled fixed point theory and its applications can be found in the literature (see, e.g., [ – ]) In this manuscript, we give a common method to deal with the existence of a coupled fixed point and the coincidence point for a class of mixed monotone mappings in a partially ordered complete metric space. We establish some fixed point theorems for the monotone operators in the partially ordered complete metric space. (Bhaskar and Lakshmikantham [ ]) An element (x, y) ∈ X is said to be a coupled fixed point of the mapping F : X → X if F(x, y) = x and F(y, x) = y.
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