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

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.

Hybrid Ćirić Type Graphic Υ,Λ-Contraction Mappings with Applications to Electric Circuit and Fractional Differential Equations

“…(0; 1) : Remark 3.1. Theorems 2.1 and 2.2 generalize and extend results of Liu et al [6], Jleli and Samet [3] and Wardowski [8] for partial b-metric spaces and partial metric spaces along with a power graphic contraction pair, respectively. 4.…”

“…Ameer et al [4,5] introduced common fixed point results for generalized multivalued (Υ, Λ)contractions in complete metric, b-metric spaces. Ameer et al [6] initiated the notion of rational (Υ, Λ, )-contractive pair of mappings (where is a binary relation) and established new common fixed point results for these mappings in complete metric spaces. On the other hand, Bakhtin [7] investigated the concept of b-metric spaces.…”

Some Singular Value Inequalities for Sector Matrices Involving Operator Concave Functions

“…Matrix inequalities have many important applications in solving matrix equations; for more information, we refer the readers to see [3,4]. In this paper, we mainly deal with matrix singular value inequalities.…”

Fixed Point Theorems for Nonexpansive Mappings under Binary Relations

“…In 2015, Alam and Imdad [17] established a novel version of the Banach contraction principle, employing an amorphous binary relation. In recent years, various metrical fixed theorems were proved under different types of contractivity conditions, employing certain binary relations (e.g., [18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36]). In such results, the involved contraction conditions remain relatively weaker than the usual contraction conditions, as these are required to hold merely for those elements which are related in the underlying binary relation.…”