scholarly journals On (Λ,Υ,ℜ)-Contractions and Applications to Nonlinear Matrix Equations

Mathematics ◽  
2019 ◽  
Vol 7 (5) ◽  
pp. 443 ◽  
Author(s):  
Eskandar Ameer ◽  
Muhammad Nazam ◽  
Hassen Aydi ◽  
Muhammad Arshad ◽  
Nabil Mlaiki
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorems ◽  
Positive Definite Matrix ◽  
Positive Definite ◽  
Matrix Equations ◽  
Continuous Map ◽  
Comparison Functions ◽  
Hermitian Positive Definite Matrix ◽  
Nonlinear Matrix ◽  
Common Fixed Point Theorems

In this paper, we study the behavior of Λ , Υ , ℜ -contraction mappings under the effect of comparison functions and an arbitrary binary relation. We establish related common fixed point theorems. We explain the significance of our main theorem through examples and an application to a solution for the following nonlinear matrix equations: X = D + ∑ i = 1 n A i ∗ X A i − ∑ i = 1 n B i ∗ X B i X = D + ∑ i = 1 n A i ∗ γ X A i , where D is an Hermitian positive definite matrix, A i , B i are arbitrary p × p matrices and γ : H ( p ) → P ( p ) is an order preserving continuous map such that γ ( 0 ) = 0 . A numerical example is also presented to illustrate the theoretical findings.

Fixed point theorems for $$F_\mathfrak {R}$$ F R -contractions with applications to solution of nonlinear matrix equations

2016 ◽  
Vol 19 (3) ◽  
pp. 1711-1725 ◽  
Author(s):  
Kanokwan Sawangsup ◽  
Wutiphol Sintunavarat ◽  
Antonio Francisco Roldán López de Hierro
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorems ◽  
Matrix Equations ◽  
Nonlinear Matrix

Binary relation for tripled fixed point theorem in metric spaces

2021 ◽  
Vol 39 (2) ◽  
pp. 9-26
Author(s):  
Animesh Gupta ◽  
Vandana Rai
Keyword(s):  
Fixed Point ◽  
Binary Relation ◽  
Existence And Uniqueness ◽  
Fixed Point Theorems ◽  
Metric Spaces ◽  
Positive Definite ◽  
Nonlinear Matrix Equation ◽  
Tripled Fixed Point ◽  
Contractivity Condition ◽  
Nonlinear Matrix

In this paper we present a new extension of tripled fixed point theorems in metric spaces endowed with a reflexive binary relation that is not necessarily neither transitive nor antisymmetric. The key feature in this tripled fixed point theorems is that the contractivity condition on the nonlinear map is only assumed to hold on elements that are comparable in the binary relation. Next on the basis of the tripled fixed point theorems, we prove the existence and uniqueness of positive definite solutions of a nonlinear matrix equation of type

scholarly journals Common Positive Solution of Two Nonlinear Matrix Equations Using Fixed Point Results

Mathematics ◽  
2021 ◽  
Vol 9 (18) ◽  
pp. 2199
Author(s):  
Hemant Kumar Nashine ◽  
Rajendra Pant ◽  
Reny George
Keyword(s):  
Fixed Point ◽  
Positive Definite ◽  
Matrix Equations ◽  
Trace Norm ◽  
Time Analysis ◽  
Positive Definite Solution ◽  
Cpu Time ◽  
Nonlinear Matrix ◽  
The Given ◽  
Definite Solution

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.

scholarly journals Inequalities for the eigenvalues of the positive definite solutions of the nonlinear matrix equation

Filomat ◽  
2019 ◽  
Vol 33 (9) ◽  
pp. 2667-2671
Author(s):  
Guoxing Wu ◽  
Ting Xing ◽  
Duanmei Zhou
Keyword(s):  
Matrix Equation ◽  
Positive Definite Matrix ◽  
Positive Definite ◽  
Nonlinear Matrix Equation ◽  
The Matrix ◽  
Hermitian Positive Definite Matrix ◽  
Nonlinear Matrix ◽  
Positive Integers ◽  
Equivalent Conditions

In this paper, the Hermitian positive definite solutions of the matrix equation Xs + A*X-tA = Q are considered, where Q is a Hermitian positive definite matrix, s and t are positive integers. Bounds for the sum of eigenvalues of the solutions to the equation are given. The equivalent conditions for solutions of the equation are obtained. The eigenvalues of the solutions of the equation with the case AQ = QA are investigated.

scholarly journals Extended Branciari quasi-b-distance spaces, implicit relations and application to nonlinear matrix equations

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Reena Jain ◽  
Hemant Kumar Nashine ◽  
Vahid Parvaneh
Keyword(s):  
Positive Definite Matrix ◽  
Weak Limit ◽  
Positive Definite ◽  
Contractive Condition ◽  
Sufficient Condition ◽  
Nonlinear Matrix Equation ◽  
Property A ◽  
Underlying Space ◽  
Hermitian Positive Definite Matrix ◽  
Nonlinear Matrix

AbstractThis study introduces extended Branciari quasi-b-distance spaces, a novel implicit contractive condition in the underlying space, and basic fixed-point results, a weak well-posed property, a weak limit shadowing property and generalized Ulam–Hyers stability. The given notions and results are exemplified by suitable models. We apply these results to obtain a sufficient condition ensuring the existence of a unique positive-definite solution of a nonlinear matrix equation (NME) $\mathcal{X}=\mathcal{Q} + \sum_{i=1}^{k}\mathcal{A}_{i}^{*} \mathcal{G(X)}\mathcal{A}_{i}$ X = Q + ∑ i = 1 k A i ∗ G ( X ) A i , where $\mathcal{Q}$ Q is an $n\times n$ n × n Hermitian positive-definite matrix, $\mathcal{A}_{1}$ A 1 , $\mathcal{A}_{2}$ A 2 , …, $\mathcal{A}_{m}$ A m are $n \times n$ n × n matrices, and $\mathcal{G}$ G is a nonlinear self-mapping of the set of all Hermitian matrices that are continuous in the trace norm. We demonstrate this sufficient condition for the NME $\mathcal{X}= \mathcal{Q} +\mathcal{A}_{1}^{*}\mathcal{X}^{1/3} \mathcal{A}_{1}+\mathcal{A}_{2}^{*}\mathcal{X}^{1/3} \mathcal{A}_{2}+ \mathcal{A}_{3}^{*}\mathcal{X}^{1/3}\mathcal{A}_{3}$ X = Q + A 1 ∗ X 1 / 3 A 1 + A 2 ∗ X 1 / 3 A 2 + A 3 ∗ X 1 / 3 A 3 , and visualize this through convergence analysis and a solution graph.

scholarly journals Hybrid Coupled Fixed Point Theorems in Metric Spaces with Applications

2019 ◽  
Vol 2019 ◽  
pp. 1-15
Author(s):  
Muhammad Shoaib ◽  
Muhammad Sarwar ◽  
Thabet Abdeljawad
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorems ◽  
Metric Spaces ◽  
Coupled Fixed Point ◽  
Coupled System ◽  
Matrix Equations ◽  
Common Coupled Fixed Point ◽  
Nonlinear Matrix ◽  
Two Hybrid ◽  
Coupled Coincidence

In this manuscript, using CLR property, coupled coincidence and common coupled fixed point results for two-hybrid pairs satisfying (F,φ)- contraction are demonstrated. Using the established results existence of solution to the coupled system of functional and nonlinear matrix equations is also discussed. We provide examples where the main theorem is applicable but most current relevant results in literature fail to have a common coupled fixed point.

The Hermitian Positive Definite Solution of the Nonlinear Matrix Equation

2017 ◽  
Vol 18 (5) ◽  
pp. 293-301
Author(s):  
Xindong Zhang ◽  
Xinlong Feng
Keyword(s):  
Matrix Equation ◽  
Iterative Algorithms ◽  
Positive Definite Matrix ◽  
Positive Definite ◽  
Nonlinear Matrix Equation ◽  
Positive Definite Solution ◽  
Hermitian Positive Definite Solution ◽  
Hermitian Positive Definite Matrix ◽  
Nonlinear Matrix ◽  
Definite Solution

Abstract:In this paper, we study the nonlinear matrix equation $X^{s}\pm\sum^{m}_{i=1}A^{T}_{i}X^{\delta_{i}}A_{i}=Q$, where $A_{i}\;(i=1,2,\ldots,m)$ is $n\times n$ nonsingular real matrix and $Q$ is $n\times n$ Hermitian positive definite matrix. It is shown that the equation has an unique Hermitian positive definite solution under some conditions. Iterative algorithms for obtaining the Hermitian positive definite solution of the equation are proposed. Finally, numerical examples are reported to illustrate the effectiveness of algorithms.

scholarly journals An implicit relation, relational theoretic approach under w-distance and application to nonlinear matrix equations

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Reena Jain ◽  
Hemant Kumar Nashine ◽  
Manuel De la Sen
Keyword(s):  
Metric Spaces ◽  
Sufficient Conditions ◽  
Positive Definite Matrix ◽  
Positive Definite ◽  
Contractive Condition ◽  
Theoretic Approach ◽  
Nonlinear Matrix Equation ◽  
Surface Plot ◽  
Hermitian Positive Definite Matrix ◽  
Nonlinear Matrix

AbstractWe propose a new class of implicit relations and an implicit type contractive condition based on it in the relational metric spaces under w-distance functional. Further we derive fixed points results based on them. Useful examples illustrate the applicability and effectiveness of the presented results. We apply these results to discuss sufficient conditions ensuring the existence of a unique positive definite solution of the nonlinear matrix equation (NME) of the form $\mathcal{U}=\mathcal{Q} + \sum_{i=1}^{k}\mathcal{A}_{i}^{*} \mathcal{G}\mathcal{(U)}\mathcal{A}_{i}$ U = Q + ∑ i = 1 k A i ∗ G ( U ) A i , where $\mathcal{Q}$ Q is an $n\times n$ n × n Hermitian positive definite matrix, $\mathcal{A}_{1}$ A 1 , $\mathcal{A}_{2}$ A 2 , …, $\mathcal{A}_{m}$ A m are $n \times n$ n × n matrices and $\mathcal{G}$ G is a nonlinear self-mapping of the set of all Hermitian matrices which are continuous in the trace norm. In order to demonstrate the obtained conditions, we consider an example together with convergence and error analysis and visualisation of solutions in a surface plot.

scholarly journals Positive solution to a generalized Lyapunov equation via a coupled fixed point theorem in a metric space endowedwith a partial order

Filomat ◽  
2015 ◽  
Vol 29 (8) ◽  
pp. 1831-1837 ◽  
Author(s):  
Maher Berzig ◽  
Bessem Samet
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Coupled Fixed Point ◽  
Positive Definite Matrix ◽  
Lyapunov Equation ◽  
Positive Definite ◽  
Uniqueness Result ◽  
Coupled Fixed Point Theorem ◽  
Hermitian Positive Definite Matrix

We consider the generalized continuous-time Lyapunov equation: A*XB + B*XA = -Q, where Q is an N x N Hermitian positive definite matrix and A,B are arbitrary N x N matrices. Under certain conditions, using a coupled fixed point theorem du to Bhaskar and Lakshmikantham combined with the Schauder fixed point theorem, we establish an existence and uniqueness result of Hermitian positive definite solution to such equation. Moreover, we provide an iteration method to find convergent sequences which converge to the solution if one exists. Numerical experiments are presented to illustrate our theoretical results.

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