New Generalized Contractions by Employing Two Control Functions and Coupled Fixed-Point Theorems with Applications
Abstract
:1. Introduction and Preliminaries
- (i)
- iff
- (ii)
- (iii)
- then the pair is called a —metric space.
- (i)
- iff
- (ii)
- (iii)
- then the pair is called an extended —metric space.
- (i)
- iff
- (ii)
- (iii)
- then is called a -space.
2. Main Results
- (p1)
- and
- (p2)
- and with
- (p3)
- for all and for assume that
- (1)
- and
- (2)
- and with
- (3)
- where for each Further, if for every and are exist and finite. Then, the mapping Z has a unique CFP.
- If or or It is a trivial case.
- If , and we have and it is a trivial case too.
- If or we obtain
- If or we have It is satisfied for any value of ξ and ϱ.
- If or we obtain . It is fulfilled for any value of ξ and ϱ.
3. Fixed-Point Techniques on Graphs
- (a)
- (b)
- for each and for with we have
- (c)
- for any sequences with , we obtain
- (d)
- Z is continuous, or for any sequences with , and we have i.e., there is at least one CFP of
- (e)
- for every we have and exist and are finite,
- (f)
- assume that then, we have and Z has a unique CFP.
- If and then by the condition , we obtain
4. Applications
- (1)
- and
- (2)
- and with
- (3)
- and
- and with
- (1)
- and
- (2)
- clearly, and with
- (3)
5. Conclusions and Future Works
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
- Fredholm, E.I. Sur une classe d’equations fonctionnelles. Acta Math. 1903, 27, 365–390. [Google Scholar] [CrossRef]
- Rus, M.D. A note on the existence of positive solution of Fredholm integral equations. Fixed Point Theory 2004, 5, 369–377. [Google Scholar]
- Berenguer, M.I.; Munoz, M.V.F.; Guillem, A.I.G.; Galan, M.R. Numerical treatment of fixed-point applied to the nonlinear Fredholm integral equation. Fixed Point Theory Appl. 2009, 2009, 735638. [Google Scholar] [CrossRef]
- Hammad, H.A.; De la Sen, M. Tripled fixed point techniques for solving system of tripled-fractional differential equations. AIMS Math. 2021, 6, 2330–2343. [Google Scholar] [CrossRef]
- Hammad, H.A.; Aydi, H.; De la Sen, M. Solutions of fractional differential type equations by fixed point techniques for multivalued contractions. Complexity 2021, 2021, 5730853. [Google Scholar] [CrossRef]
- Czerwik, S. Contraction mappings in b—metric spaces. Acta Math. Inform. Univ. Ostrav. 1993, 1, 5–11. [Google Scholar]
- Kamran, T.; Samreen, M.; Ain, Q.U. A generalization of b —metric space and some fixed point theorems. Mathematics 2017, 5, 19. [Google Scholar] [CrossRef]
- Mlaiki, N.; Aydi, H.; Souayah, N.; Abdeljawad, T. Controlled metric type spaces and the related contraction principle. Mathematics 2018, 6, 194. [Google Scholar] [CrossRef]
- Lateef, D. Kannan fixed point theorem in c—metric spaces. J. Math. Anal. 2019, 10, 34–40. [Google Scholar]
- Kannan, R. Some results on fixed points. Bull. Calcutta Math. Soc. 1968, 60, 71–76. [Google Scholar]
- Hammad, H.A.; De la Sen, M.; Aydi, H. Analytical solution for differential and nonlinear integral equations via Fϖe-Suzuki contractions in modified ϖe-metric-like spaces. J. Funct. Spaces 2021, 2021, 6128586. [Google Scholar]
- Abdeljawad, T.; Mlaiki, N.; Aydi, H.; Souayah, N. Double controlled metric type spaces and some fixed point results. Mathematics 2018, 6, 320. [Google Scholar] [CrossRef]
- Lateef, D. Fisher type fixed point results in controlled metric spaces. J. Math. Comput. Sci. 2020, 20, 234–240. [Google Scholar] [CrossRef]
- Shoaib, A.; Kumam, P.; Alshoraify, S.S.; Arshad, M. Fixed point results in double controlled quasi metric type spaces. AIMS Math. 2021, 6, 1851–1864. [Google Scholar] [CrossRef]
- Sezen, M.S. Controlled fuzzy metric spaces and some related fixed point results. Numer. Methods Partial. Differ. Equ. 2021, 37, 583–593. [Google Scholar] [CrossRef]
- Tasneem, S.; Gopalani, K.; Abdeljawad, T. A different approach to fixed point theorems on triple controlled metric type spaces with a numerical experiment. Dyn. Syst. Appl. 2021, 30, 111–130. [Google Scholar]
- Hammad, H.A.; Agarwal, P.; Guirao, L.G.J. Applications to boundary value problems and homotopy theory via tripled fixed point techniques in partially metric spaces. Mathematics 2021, 9, 2012. [Google Scholar] [CrossRef]
- Bhaskar, T.G.; Lakshmikantham, V. Fixed point theorems in partially ordered metric spaces and applications. Nonlinear Anal. Theory Methods Appl. 2006, 65, 1379–1393. [Google Scholar] [CrossRef]
- Abbas, M.; Khan, M.A.; Radenović, S. Common coupled fixed point theorems in cone metric spaces for w-compatible mapping. Appl. Math. Comput. 2010, 217, 195–202. [Google Scholar] [CrossRef]
- Lakshmikantham, V.; Cirić, L. Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces. Nonlinear Anal. 2009, 70, 4341–4349. [Google Scholar] [CrossRef]
- Radenović, S. Some coupled coincidence points results of monotone mappings in partially ordered metric spaces. Int. J. Anal. Appl. 2014, 5, 174–184. [Google Scholar]
- Sgroi, M.; Vetro, C. Multivalued F-contractions and the solution of certain functional and integral equations. Filomat 2013, 27, 1259–1268. [Google Scholar] [CrossRef]
- Chifu, C.; Petruşel, G. Coupled fixed point results for (φ,G)-contractions of type (b) in b—metric spaces endowed with a graph. J. Nonlinear Sci. Appl. 2017, 10, 671–683. [Google Scholar] [CrossRef]
- Shatanawi, W.; Samet, B.; Abbas, M. Coupled fixed point theorems for mixed monotone mappings in ordered partial metric spaces. Math. Comput. Model. 2012, 55, 680–687. [Google Scholar] [CrossRef]
- Rashwan R., A.; Hammad H., A.; Mahmoud, M.G. Common fixed point results for weakly compatible mappings under implicit relations in complex valued g—metric spaces. Inf. Sci. Lett. 2019, 8, 111–119. [Google Scholar]
- Jachymski, J. The contraction principle for mappings on a metric space with a graph. Proc. Am. Math. Soc. 2008, 136, 1359–1373. [Google Scholar] [CrossRef]
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Hammad, H.A.; Zayed, M. New Generalized Contractions by Employing Two Control Functions and Coupled Fixed-Point Theorems with Applications. Mathematics 2022, 10, 3208. https://doi.org/10.3390/math10173208
Hammad HA, Zayed M. New Generalized Contractions by Employing Two Control Functions and Coupled Fixed-Point Theorems with Applications. Mathematics. 2022; 10(17):3208. https://doi.org/10.3390/math10173208
Chicago/Turabian StyleHammad, Hasanen A., and Mohra Zayed. 2022. "New Generalized Contractions by Employing Two Control Functions and Coupled Fixed-Point Theorems with Applications" Mathematics 10, no. 17: 3208. https://doi.org/10.3390/math10173208
APA StyleHammad, H. A., & Zayed, M. (2022). New Generalized Contractions by Employing Two Control Functions and Coupled Fixed-Point Theorems with Applications. Mathematics, 10(17), 3208. https://doi.org/10.3390/math10173208