Almost Nonlinear Contractions under Locally Finitely Transitive Relations with Applications to Integral Equations
Abstract
:1. Introduction
2. Preliminaries
- (i)
- There exists satisfying ;
- (ii)
- £ is locally finitely ϕ-transitive and ϕ-closed;
- (iii)
- remains £-complete;
- (iv)
- Y remains £-continuous, or £ remains ς-self-closed;
- (v)
- There exists verifying.
- (i)
- ,
- (ii)
- ,
- (iii)
- for all .
- (A)
- (B)
3. Main Results
- (i)
- There exists satisfying ;
- (ii)
- £ is locally finitely ϕ-transitive and ϕ-closed;
- (iii)
- remains £-complete;
- (iv)
- Y remains £-continuous, or £ remains ς-self-closed;
- (v)
- There exist and verifying.
4. Examples
5. An Application to Nonlinear Integral Equations
- (a)
- H, ℏ and ¥ are continuous;
- (b)
- ;
- (c)
- There exist and an increasing function such that
- (d)
- .
- (i)
- Let be a lower solution of (17). Then
- (ii)
- Take such that . Employing item (c), we obtain
- (iii)
- is £-complete as it remains complete.
- (iv)
- Assuming that is a £-preserving sequence, which converges to . Then for every , is an increasing sequence (in ) which converges to . This implies that and . Using (19), we conclude ; thereby, £ will be is -self-closed.
- (v)
- As , using increasing property of , we get
6. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Khan, F.A.; Aldhabani, M.S.; Alamer, A.; Alshaban, E.; Alamrani, F.M.; Mohammed, H.I.A. Almost Nonlinear Contractions under Locally Finitely Transitive Relations with Applications to Integral Equations. Mathematics 2023, 11, 4749. https://doi.org/10.3390/math11234749
Khan FA, Aldhabani MS, Alamer A, Alshaban E, Alamrani FM, Mohammed HIA. Almost Nonlinear Contractions under Locally Finitely Transitive Relations with Applications to Integral Equations. Mathematics. 2023; 11(23):4749. https://doi.org/10.3390/math11234749
Chicago/Turabian StyleKhan, Faizan Ahmad, Musaad S. Aldhabani, Ahmed Alamer, Esmail Alshaban, Fahad M. Alamrani, and Hamid I. A. Mohammed. 2023. "Almost Nonlinear Contractions under Locally Finitely Transitive Relations with Applications to Integral Equations" Mathematics 11, no. 23: 4749. https://doi.org/10.3390/math11234749