An Extension of Strict Almost Contractions Employing Control Function and Binary Relation with Applications to Boundary Value Problems
Abstract
:1. Introduction
2. Preliminaries
- (i)
- ,
- (ii)
- ,
- (iii)
- .
- (iv)
- (v)
- (vi)
- (vii)
- (I)
- ,
- (II)
- ,
3. Main Results
- (i)
- is ()-complete,
- (ii)
- remains locally -transitive and -closed,
- (iii)
- satisfying ,
- (iv)
- remains ()-continuous, or remains ()-self-closed,
- (v)
- and verifying
4. Consequences
- (i)
- is ()-complete,
- (ii)
- remains locally -transitive and -closed,
- (iii)
- satisfying ,
- (iv)
- remains ()-continuous, or remains ()-self-closed,
- (v)
- verifying
5. Examples
6. Applications to Boundary Value Problems
7. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
- Boyd, D.W.; Wong, J.S.W. On nonlinear contractions. Proc. Am. Math. Soc. 1969, 30, 25. [Google Scholar] [CrossRef]
- Berinde, V. Approximating fixed points of weak contractions using the Picard iteration. Nonlinear Anal. Forum. 2004, 9, 43–53. [Google Scholar]
- Berinde, V.; Păcurar, M. Fixed points and continuity of almost contractions. Fixed Point Theory 2008, 9, 23–34. [Google Scholar]
- Babu, G.V.R.; Sandhy, M.L.; Kameshwari, M.V.R. A note on a fixed point theorem of Berinde on weak contractions. Carpathian J. Math. 2008, 24, 8–12. [Google Scholar]
- Berinde, V.; Takens, F. Iterative Approximation of Fixed Points; Springer: Berlin/Heidelberg, Germany, 2007; Volume 1912. [Google Scholar]
- Berinde, V. General constructive fixed point theorems for Ćirić-type almost contractions in metric spaces. Carpathian J. Math. 2008, 24, 10–19. [Google Scholar]
- Alghamdi, M.A.; Berinde, V.; Shahzad, N. Fixed points of non-self almost contractions. Carpathian J. Math. 2014, 30, 7–14. [Google Scholar] [CrossRef]
- Altun, I.; Acar, Ö. Fixed point theorems for weak contractions in the sense of Berinde on partial metric spaces. Topol. Appl. 2012, 159, 2642–2648. [Google Scholar] [CrossRef]
- Berinde, V. Stability of Picard iteration for contractive mappings satisfying an implicit relation. Carpathian J. Math. 2011, 27, 13–23. [Google Scholar] [CrossRef]
- Alam, A.; Imdad, M. Relation-theoretic contraction principle. J. Fixed Point Theory Appl. 2015, 17, 693–702. [Google Scholar] [CrossRef]
- Alam, A.; Imdad, M. Relation-theoretic metrical coincidence theorems. Filomat 2017, 31, 4421–4439. [Google Scholar] [CrossRef]
- Alam, A.; Imdad, M. Nonlinear contractions in metric spaces under locally T-transitive binary relations. Fixed Point Theory 2018, 19, 13–24. [Google Scholar] [CrossRef]
- Alam, A.; Arif, M.; Imdad, M. Metrical fixed point theorems via locally finitely T-transitive binary relations under certain control functions. Miskolc Math. Notes 2019, 20, 59–73. [Google Scholar] [CrossRef]
- Arif, M.; Imdad, M.; Alam, A. Fixed point theorems under locally T-transitive binary relations employing Matkowski contractions. Miskolc Math. Notes 2022, 23, 71–83. [Google Scholar] [CrossRef]
- Alam, A.; George, R.; Imdad, M. Refinements to relation-theoretic contraction principle. Axioms 2022, 11, 316. [Google Scholar] [CrossRef]
- Hossain, A.; Alam, A.; Sessa, S.; Khan, Q.H. Relation-theoretic weak contractions and applications. Mathematics 2023, 11, 1976. [Google Scholar] [CrossRef]
- Hasanuzzaman, M.; Imdad, M.; Saleh, H.N. On modified L-contraction via binary relation with an application. Fixed Point Theory 2022, 23, 267–278. [Google Scholar] [CrossRef]
- Hasanuzzaman, M.; Sessa, S.; Imdad, M.; Alfaqih, W.M. Fixed point results for a selected class of multi-valued mappings under (θ,R)-contractions with an application. Mathematics 2020, 8, 695. [Google Scholar] [CrossRef]
- Hasanuzzaman, M.; Imdad, M. Relation theoretic metrical fixed point results for Suzuki type ZR-contraction with an application. Aims Math. 2020, 5, 2071–2087. [Google Scholar] [CrossRef]
- Lipschutz, S. Schaum’s Outlines of Theory and Problems of Set Theory and Related Topics; McGraw-Hill: New York, NY, USA, 1964. [Google Scholar]
- Kolman, B.; Busby, R.C.; Ross, S. Discrete Mathematical Structures, 6th ed.; Pearson/Prentice Hall: Hoboken, NJ, USA, 2009. [Google Scholar]
- Samet, B.; Turinici, M. Fixed point theorems on a metric space endowed with an arbitrary binary relation and applications. Commun. Math. Anal. 2012, 13, 82–97. [Google Scholar]
- Jleli, M.; Rajic, V.C.; Samet, B.; Vetro, C. Fixed point theorems on ordered metric spaces and applications to nonlinear elastic beam equations. J. Fixed Point Theory Appl. 2012, 12, 175–192. [Google Scholar] [CrossRef]
- Nieto, J.J.; Rodríguez-López, R. Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. Order 2005, 22, 223–239. [Google Scholar] [CrossRef]
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Filali, D.; Akram, M.; Dilshad, M. An Extension of Strict Almost Contractions Employing Control Function and Binary Relation with Applications to Boundary Value Problems. Mathematics 2023, 11, 4027. https://doi.org/10.3390/math11194027
Filali D, Akram M, Dilshad M. An Extension of Strict Almost Contractions Employing Control Function and Binary Relation with Applications to Boundary Value Problems. Mathematics. 2023; 11(19):4027. https://doi.org/10.3390/math11194027
Chicago/Turabian StyleFilali, Doaa, Mohammad Akram, and Mohammad Dilshad. 2023. "An Extension of Strict Almost Contractions Employing Control Function and Binary Relation with Applications to Boundary Value Problems" Mathematics 11, no. 19: 4027. https://doi.org/10.3390/math11194027
APA StyleFilali, D., Akram, M., & Dilshad, M. (2023). An Extension of Strict Almost Contractions Employing Control Function and Binary Relation with Applications to Boundary Value Problems. Mathematics, 11(19), 4027. https://doi.org/10.3390/math11194027