Abstract
AbstractIn this paper, we introduce the notion of diagonal operator, we present the historical roots of diagonal operators and we give some fixed point theorems for this class of operators. Our approaches are based on the weakly Picard operator technique, difference equation techniques, and some fixed point theorems for multi-valued operators. Some applications to differential and integral equations are given. We also present some research directions.
Highlights
1 Introduction and preliminary notions and results we will present some useful notions and results concerning diagonal operators, coupled fixed point operators, and iterations of some operators generated by the above concepts
The operator UV : X → X, defined by UV (x) := V (x, x), for all x ∈ X, is called the diagonal operator corresponding to the operator V
The aim of this paper is to present some historical roots of the diagonal operators, to study the fixed points of this class of operators, and to give some applications
Summary
Introduction and preliminary notions and results we will present some useful notions and results concerning diagonal operators, coupled fixed point operators, and iterations of some operators generated by the above concepts.1.1 Diagonal operators Let X be a nonempty set and V : X × X → X be an operator. Let Y be a compact convex subset of a Banach space E and T : Y → P(Y ) be an upper semi-continuous multi-valued operator with acyclic values. (Difference equations [ – ]) Let (X, →) be an L-space and V : X × X → X be an operator.
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