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

• Main Authors: Eskandar Ameer, Muhammad Nazam, Hassen Aydi, Muhammad Arshad, Nabil Mlaiki
• Format: Article
• Language: English
• Published: MDPI AG 2019-05-01
• Series: Mathematics
• Subjects: fixed point; binary relation; Λ-contraction; comparison function; nonlinear matrix equation
• Online Access: https://www.mdpi.com/2227-7390/7/5/443

In this paper, we study the behavior of <inline-formula> <math display="inline"> <semantics> <mfenced separators="" open="(" close=")"> <mo>&#923;</mo> <mo>,</mo> <mo>&#933;</mo> <mo>,</mo> <mo>&real;</mo> </mfenced> </semantics> </math> </inline-formula>-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: <inline-formula> <math display="inline"> <semantics> <mrow> <mi>X</mi> <mo>=</mo> <mi>D</mi> <mo>+</mo> <msubsup> <mo>&#8721;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </msubsup> <msubsup> <mi>A</mi> <mrow> <mi>i</mi> </mrow> <mo>&#8727;</mo> </msubsup> <mi>X</mi> <msub> <mi>A</mi> <mi>i</mi> </msub> <mo>&#8722;</mo> <msubsup> <mo>&#8721;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </msubsup> <msubsup> <mi>B</mi> <mrow> <mi>i</mi> </mrow> <mo>&#8727;</mo> </msubsup> <mi>X</mi> <msub> <mi>B</mi> <mi>i</mi> </msub> </mrow> </semantics> </math> </inline-formula> <inline-formula> <math display="inline"> <semantics> <mrow> <mi>X</mi> <mo>=</mo> <mi>D</mi> <mo>+</mo> <msubsup> <mo>&#8721;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </msubsup> <msubsup> <mi>A</mi> <mrow> <mi>i</mi> </mrow> <mo>&#8727;</mo> </msubsup> <mi>&#947;</mi> <mfenced open="(" close=")"> <mi>X</mi> </mfenced> <msub> <mi>A</mi> <mi>i</mi> </msub> <mo>,</mo> </mrow> </semantics> </math> </inline-formula> where <i>D</i> is an Hermitian positive definite matrix, <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>A</mi> <mi>i</mi> </msub> <mo>,</mo> <msub> <mi>B</mi> <mi>i</mi> </msub> </mrow> </semantics> </math> </inline-formula> are arbitrary <inline-formula> <math display="inline"> <semantics> <mrow> <mi>p</mi> <mo>&#215;</mo> <mi>p</mi> </mrow> </semantics> </math> </inline-formula> matrices and <inline-formula> <math display="inline"> <semantics> <mrow> <mi>&#947;</mi> <mo>:</mo> <mi>H</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>&#8594;</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </semantics> </math> </inline-formula> is an order preserving continuous map such that <inline-formula> <math display="inline"> <semantics> <mrow> <mi>&#947;</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mrow> </semantics> </math> </inline-formula>. A numerical example is also presented to illustrate the theoretical findings.