scholarly journals Best Proximity Point Results for Generalized Θ-Contractions and Application to Matrix Equations

Symmetry ◽  
2019 ◽  
Vol 11 (1) ◽  
pp. 93
Author(s):  
Zhenhua Ma ◽  
Azhar Hussain ◽  
Muhammad Adeel ◽  
Nawab Hussain ◽  
Ekrem Savas
Keyword(s):  
Metric Spaces ◽  
Best Proximity Point ◽  
Positive Definite ◽  
Matrix Equations ◽  
Nonlinear Matrix Equation ◽  
Complete Metric Spaces ◽  
Partially Ordered ◽  
Positive Definite Solution ◽  
Nonlinear Matrix ◽  
Definite Solution

In this paper, we introduce the notion of C ´ iri c ´ type α - ψ - Θ -contraction and prove best proximity point results in the context of complete metric spaces. Moreover, we prove some best proximity point results in partially ordered complete metric spaces through our main results. As a consequence, we obtain some fixed point results for such contraction in complete metric and partially ordered complete metric spaces. Examples are given to illustrate the results obtained. Moreover, we present the existence of a positive definite solution of nonlinear matrix equation X = Q + ∑ i = 1 m A i * γ ( X ) A i and give a numerical example.

scholarly journals Optimal solutions and applications to nonlinear matrix and integral equations via simulation function

Filomat ◽  
2018 ◽  
Vol 32 (17) ◽  
pp. 6087-6106 ◽  
Author(s):  
Azhar Hussain ◽  
Tanzeela Kanwal ◽  
Zoran Mitrovic ◽  
Stojan Radenovic
Keyword(s):  
Metric Spaces ◽  
Best Proximity Point ◽  
Optimal Solutions ◽  
Simulation Function ◽  
Nonlinear Matrix Equation ◽  
Positive Definite Solution ◽  
Nonlinear Matrix ◽  
The Given ◽  
Coupled Best Proximity Point ◽  
Definite Solution

Based on the concepts of ?-proximal admissible mappings and simulation function, we establish some best proximity point and coupled best proximity point results in the context of b-complete b-metric spaces. We also provide some concrete examples to illustrate the obtained results. Moreover, we prove the existence of the solution of nonlinear integral equation and positive definite solution of nonlinear matrix equation X = Q + ?m,i=1 A*i?(X)Ai-?m,i=1 B*i(X)Bi. The given results not only unify but also generalize a number of existing results on the topic in the corresponding literature.

On the Nonlinear Matrix Equation Based on Engineering Calculation

2012 ◽  
Vol 450-451 ◽  
pp. 158-161
Author(s):  
Dong Jie Gao
Keyword(s):  
Matrix Equation ◽  
Iteration Method ◽  
Engineering Calculation ◽  
Positive Definite ◽  
Nonlinear Matrix Equation ◽  
Positive Definite Solution ◽  
Nonlinear Matrix ◽  
Definite Solution

We consider the positive definite solution of the nonlinear matrix equation . We prove that the equation always has a unique positive definite solution. The iteration method for the equation is given.

scholarly journals Latest Inversion-Free Iterative Scheme for Solving a Pair of Nonlinear Matrix Equations

2021 ◽  
Vol 2021 ◽  
pp. 1-22
Author(s):  
Sourav Shil ◽  
Hemant Kumar Nashine
Keyword(s):  
Convergence Rate ◽  
Convergence Analysis ◽  
Iterative Scheme ◽  
Positive Definite ◽  
Matrix Equations ◽  
Identity Matrix ◽  
Positive Definite Solution ◽  
Nonlinear Matrix ◽  
Definite Solution

In this work, the following system of nonlinear matrix equations is considered, X 1 + A ∗ X 1 − 1 A + B ∗ X 2 − 1 B = I  and  X 2 + C ∗ X 2 − 1 C + D ∗ X 1 − 1 D = I , where A , B , C ,  and  D are arbitrary n × n matrices and I is the identity matrix of order n . Some conditions for the existence of a positive-definite solution as well as the convergence analysis of the newly developed algorithm for finding the maximal positive-definite solution and its convergence rate are discussed. Four examples are also provided herein to support our results.

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