Abstract
PurposeThe authors prove the existence and uniqueness of fixed point of mappings satisfying Geraghty-type contractions in the setting of preordered modularG-metric spaces. The authors apply the results in solving nonlinear Volterra-Fredholm-type integral equations. The results extend generalize compliment and include several known results as special cases.Design/methodology/approachThe results of this paper are theoretical and analytical in nature.FindingsThe authors prove the existence and uniqueness of fixed point of mappings satisfying Geraghty-type contractions in the setting of preordered modularG-metric spaces. apply the results in solving nonlinear Volterra-Fredholm-type integral equations. The results extend, generalize, compliment and include several known results as special cases.Research limitations/implicationsThe results are theoretical and analytical.Practical implicationsThe results were applied to solving nonlinear integral equations.Social implicationsThe results has several social applications.Originality/valueThe results of this paper are new.
Highlights
In 1973, Geraghty [1] introduced an interesting generalization of Banach contraction mapping principle using the concept of class S of functions, that is α : Rþ → 1⁄20; 1Þ with the condition that αðtnÞ → 10tn → 0 where Rþ is the set of all nonnegative real numbers and t ∈ Rþ for all n ∈ N
In 2012, Gordji et al [2] proved some fixed point theorems for generalized Geraghty contraction in partially ordered complete metric spaces
We prove some fixed point theorems for Geraghty-type contraction mappings in the setting of preordered modular G-metric spaces
Summary
In 1973, Geraghty [1] introduced an interesting generalization of Banach contraction mapping principle using the concept of class S of functions, that is α : Rþ → 1⁄20; 1Þ with the condition that αðtnÞ → 10tn → 0 where Rþ is the set of all nonnegative real numbers and t ∈ Rþ for all n ∈ N. In 2012, Gordji et al [2] proved some fixed point theorems for generalized Geraghty contraction in partially ordered complete metric spaces. Yolacan [4] established some new fixed point theorems in 0-complete ordered partial metric spaces He remarked on coupled generalized Banach contraction mapping. We prove some fixed point theorems for Geraghty-type contraction mappings in the setting of preordered modular G-metric spaces. Let ðX ; ωGÞ be a complete modular G-metric space with a preorder, 6 and a nondecreasing self-mapping T : XωG → XωG on XωG such that for each λ > 0, there is νðλÞ ∈ 1⁄20; λÞ such that the following conditions hold:.
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