Abstract
In this work, several fixed point theorems of set-valued monotone mappings and set-valued Caristi-type mappings are proved in partially ordered Hausdorff topological spaces, which indeed extend and improve many recent results in the setting of metric spaces.
Highlights
Several fixed point theorems of set-valued monotone mappings and set-valued Caristi-type mappings are proved in partially ordered Hausdorff topological spaces, which extend and improve many recent results in the setting of metric spaces
In 1976, Caristi [1] proved Caristi’s fixed point theorem [1, 2], which has been the subject of intensive research in the past decades, and has found many applications in nonlinear analysis
Kirk’s method has been widely used in the generalizations of primitive Caristi’s result and the study of fixed point theorems of monotone mappings with respect to a partial order introduced by a functional and many satisfactory fixed point results have been obtained in metric spaces
Summary
Several fixed point theorems of set-valued monotone mappings and set-valued Caristi-type mappings are proved in partially ordered Hausdorff topological spaces, which extend and improve many recent results in the setting of metric spaces. Recall that this general fixed point theorem states that each mapping T : X → X has a fixed point provided that (X, d) is a complete metric space and there exists a lower semicontinuous and bounded below function φ : X → R such that d(x, Tx) ≤ φ(x) − φ(Tx) for each x ∈ X.
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