Abstract
The principal goal of this work is to investigate new sufficient conditions for the existence and convergence of positive definite solutions to certain classes of matrix equations. Under specific assumptions, the basic tool in our study is a monotone mapping, which admits a unique fixed point in the setting of a partially ordered Banach space. To estimate solutions to these matrix equations, we use the Krasnosel’skiĭ iterative technique. We also discuss some useful examples to illustrate our results.
Highlights
Matrix equations are often used in the study of ladder networks, control theory, stochastic filtering, dynamic programming, statistics, and other fields, according to Anderson [1]
Let B = R be the Banach space equipped with the usual norm and ξ : B → B be a mapping defined by ξ (θ ) = θ for all θ ∈ B
We extend Definition 3 in the setting of partially ordered Banach spaces as follows: Definition 5
Summary
Matrix equations are often used in the study of ladder networks, control theory, stochastic filtering, dynamic programming, statistics, and other fields, according to Anderson [1]. Nieto and Rodríguez-López [3] used partially ordered spaces and fixed point theorems to find solutions of some differential equations [4] The advantage of this strategy is that the mapping requirements only need to be satisfied for comparable elements, and the relevance of this viewpoint is to govern the essence of the solutions, whether they are negative or positive, which leads to a variety of interesting applications. We extend the concept of (b, θ )-enriched contraction mapping in the setting of partially ordered Banach spaces and establish some existence and convergence results. Thereafter, we use these findings to solve the matrix Equations (1) and (2).
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