On (?,?,?)-Contractions and Applications to Nonlinear Matrix Equations

Eskandar Ameer, Muhammad Nazam, Hassen Aydi, Muhammad Arshad and Nabil Mlaiki
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Eskandar Ameer: Department of Mathematics and Statistics, International Islamic University Islamabad, Islamabad 44000, Pakistan
Muhammad Nazam: Department of Mathematics, Allama Iqbal Open University, Islamabad 44000, Pakistan
Hassen Aydi: Université de Sousse, Institut Supérieur d’Informatique et des Techniques de Communication, H. Sousse 4000, Tunisia
Muhammad Arshad: Department of Mathematics and Statistics, International Islamic University, H-10, Islamabad 44000, Pakistan
Nabil Mlaiki: Department of Mathematics and General Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia

Mathematics, 2019, vol. 7, issue 5, 1-18

Abstract: 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.

Keywords: fixed point; binary relation; ?-contraction; comparison function; nonlinear matrix equation (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2019
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