Abstract
In (2008), Zhang proved the existence of fixed points of mixed monotone operators along with certain convexity and concavity conditions. In this paper, mixed monotone single-valued and multi-valued operators of Rhoades type are defined and two fixed point theorems are proved. MSC:47H10, 47H07.
Highlights
1 Introduction and preliminaries In ( ), mixed monotone operators were introduced by Guo and Lakshmikantham [ ]
Many authors studied them in Banach spaces and obtained lots of interesting results
In ( ), Rhoades [ ] introduced a new fixed point theorem as a generalization of Banach fixed point theorem
Summary
Let u , v ∈ S , A : P × P → E be a weak mixed monotone operator of Rhoades type with A(([θ , v ] ∩ S) × ([θ , v ] ∩ S)) ⊂ S satisfying the following conditions:. Let P be a cone of E, let S be a completely ordered closed subset of E with S = S\{θ } ⊂ int P and let λS ⊂ S for all λ ∈ [ , ]. A has at least one fixed point x* ∈ [u , v ] ∩ S
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