Abstract
We discuss a pair of nonlinear matrix equations (NMEs) of the form X=R1+∑i=1kAi*F(X)Ai, X=R2+∑i=1kBi*G(X)Bi, where R1,R2∈P(n), Ai,Bi∈M(n), i=1,⋯,k, and the operators F,G:P(n)→P(n) are continuous in the trace norm. We go through the necessary criteria for a common positive definite solution of the given NME to exist. We develop the concept of a joint Suzuki-implicit type pair of mappings to meet the requirement and achieve certain existence findings under weaker assumptions. Some concrete instances are provided to show the validity of our findings. An example is provided that contains a randomly generated matrix as well as convergence and error analysis. Furthermore, we offer graphical representations of average CPU time analysis for various initializations.
Highlights
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Nonlinear matrix equations have long been a popular topic in nonlinear analysis
Motivated by the preceding work, we present the concept of joint Suzuki-implicit type pair of mappings on arbitrary binary relation and establish certain existence findings under weaker assumptions
Summary
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. T is said to be R-continuous at ν if for every R-preserving sequence (νn ) converging to ν, we get T (νn ) → T (ν) as n → ∞. We fix following notations for (Ξ, R) a relational space, self-mappings T , S on Ξ and a R-directed subset D of Ξ, Fix (T ) := the set of all fixed points of T ; X(T , R) := {ν ∈ Ξ : (ν, T ν) ∈ R}; P(ν, θ, R) := the class of all paths in R from ν to θ in R, where ν, θ ∈ Ξ; PS (ν, θ, R) := the class of all S -paths in R from ν to θ in R, where ν, θ ∈ Ξ; PS (ν, θ, T , R) := the class of all S -paths {ω0 , ω1 , ω2 , .
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