Weak ψ-Contractions on Directed Graphs with Applications to Integral Equations
Abstract
:1. Introduction
- —the set of natural numbers;
- —the set of real numbers;
- ;
- ;
- BCP—Banach contraction principle;
- MS—metric space;
- CMS—complete metric space;
- fpt—fixed-point theorem;
- NIE—nonlinear integral equation;
- —the set of fixed points of a self-map .
2. Preliminaries
- ;
- contains all loops;
- G admits no parallel edge.
- .
- (i)
- ;
- (ii)
- ;
- ;
- .
- ;
- ;
- ;
3. Main Results
4. Special Cases
- Case-2. Let be a poset. Define a digraph on by ; our main results reduce to Theorems 2 and 3 of Harjani and Sadarangani [17]. Condition (i) of Definition 11 means that is ⪯-nondecreasing, and Condition (ii) of Definition 11 means that the weak -contraction inequality holds for merely ⪯-preserving elements of .
- Case-3. Let be a poset. Define a digraph on by . Theorem 2 reduces to Theorem 6 of Harjani and Sadarangani [17]. Condition (i) of Definition 11 means that map two ⪯-comparable elements to two ⪯-comparable elements, and Condition (ii) of Definition 11 means that the weak -contraction inequality holds for merely ⪯-comparable elements of .
- Case-4. Let be fixed. Define a digraph on by . Elements of are termed e-closed (c.f. [14]). We see that contains all loops. Under this setting, our results deduce certain new results wherein the map takes e-closed elements of to e-closed elements, and the weak -contraction inequality must be satisfied for merely e-closed elements of .
- Case-5. Particularly for , (after removing the transitivity requirement on G), Theorems 1 and 2 reduce to the corresponding results of Jachymski [15].
- Case-6. If is continuous and increasing such that for , and , then . Under this substitution, Theorems 1 and 2 deduce the corresponding results of Samreen and Kamran [9].
- Case-7. If , where is a right upper semi-continuous function such that for , then . Under this substitution, Theorems 1 and 2 deduce the recent results because of Filali et al. [10].
5. Examples
6. An Application to NIEs
- (a)
- F, Θ and are continuous;
- (b)
- ;
- (c)
- There exists that satisfies for any with that
- (d)
- .
7. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
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Filali, D.; Dilshad, M.; Akram, M. Weak ψ-Contractions on Directed Graphs with Applications to Integral Equations. Mathematics 2024, 12, 2675. https://doi.org/10.3390/math12172675
Filali D, Dilshad M, Akram M. Weak ψ-Contractions on Directed Graphs with Applications to Integral Equations. Mathematics. 2024; 12(17):2675. https://doi.org/10.3390/math12172675
Chicago/Turabian StyleFilali, Doaa, Mohammad Dilshad, and Mohammad Akram. 2024. "Weak ψ-Contractions on Directed Graphs with Applications to Integral Equations" Mathematics 12, no. 17: 2675. https://doi.org/10.3390/math12172675