Some New Results for (α, β)-Admissible Mappings in 𝔽-Metric Spaces with Applications to Integral Equations
Abstract
:1. Introduction and Preliminaries
- if and only if ;
- ;
- For every , and for every with , we have
- (1)
- A sequence is called -convergent to if
- (2)
- A sequence is called -Cauchy if
- (3)
- The space is said to be -complete if every -Cauchy sequence in Γ is -convergent to some element in
- (i)
- for some implies
- (ii)
- for some implies
2. Main Results
- (i)
- There exists such that and
- (ii)
- Υ is continuous or;
- (iii)
- If is a sequence in Γ such that and for all , then .
- (i)
- There exists such that and
- (ii)
- is continuous or;
- (iii)
- If is a sequence in such that and for all , then .
- (i)
- There exists such that and
- (ii)
- is continuous or;
- (iii)
- If is a sequence in such that and for all , then .
- (i)
- There exists such that and
- (ii)
- is continuous or;
- (iii)
- If is a sequence in such that and for all , then .
- (i)
- There exists such that and ;
- (ii)
- Υ and S are two (-admissible mappings;
- (iii)
- If is a sequence in Γ such that as and for all , then .
- (i)
- There exists such that and ;
- (ii)
- Υ is an -admissible mapping;
- (iii)
- If is a sequence in Γ such that as and for all , then .
3. Application to Integral Equation
- (1)
- For all we have ;
- (2)
- There exist such that if and for some then for every we obtain
- (3)
- There exists such that and ;
- (4)
- for some implies and for some implies ;
- (5)
- If is a sequence in Γ such that as and for all then
4. Conclusions
Author Contributions
Funding
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
- Banach, S. Sur les operations dans les ensembles abstracts et leur application aux equations integrals. Fundam. Math. 1922, 3, 133–181. [Google Scholar] [CrossRef]
- Bakhtin, I.A. The contraction mapping principle in almost metric space. Funct. Ana. Gos. Ped. Inst. Unianowsk 1989, 30, 26–37. [Google Scholar]
- Matthews, S.G. Partial Metric Topology; Research Report; Department of Computer Science, University of Warwick: Coventry, UK, 1992; Volume 212. [Google Scholar]
- Huang, L.G.; Zhang, X. Cone metric spaces and fixed point theorems of contractive mappings. J. Math. Anal. Appl. 2007, 332, 1468–1476. [Google Scholar] [CrossRef] [Green Version]
- Kirk, W.A.; Shahzad, N. Fixed Point Theory in Distances Spaces; Springer: Berlin, Germany, 2014. [Google Scholar]
- Sintunawarat, W. Nonlinear integral equations with new admissibility types in b-metric spaces. J. Fixed Point Theory Appl. 2016, 18, 397–416. [Google Scholar] [CrossRef]
- George, R.; Nabwey, A.H.; Ramaswamy, R.; Radenović, S.; Reshma, K.P. Rectangular cone b-metric spaces over Banach algebra and contraction principle. Fixed Point Theory Appl. 2017, 2017, 14. [Google Scholar] [CrossRef] [Green Version]
- Faraji, H.; Nourouzi, K.; O’Regan, D. A fixed point theorem in uniform spaces generated by a family of b-pseudometrics. Fixed Point Theory 2019, 20, 177–183. [Google Scholar] [CrossRef]
- Faraji, H.; Savić, D.; Radenović, S. Fixed point theorems for Geraghty contraction type mappings in b-metric spaces and applications. Axioms 2019, 8, 34. [Google Scholar] [CrossRef] [Green Version]
- Jleli, M.; Samet, B. On a new generalization of metric spaces. J. Fixed Point Theory Appl. 2018, 20, 1–20. [Google Scholar] [CrossRef] [Green Version]
- Mitrović, Z.D.; Aydi, H.; Hussain, N.; Mukheimer, A. Reich, Jungck, and Berinde common fixed point results on F-metric spaces and an application. Mathematics 2019, 7, 387. [Google Scholar] [CrossRef] [Green Version]
- Gilić, E.; Dolićanin-Djekić, D.; Mitrović, Z.D.; Pučić, D.; Aydi, H. On Some Recent Results Concerning F-Suzuki-Contractions in b-Metric Spaces. Mathematics 2020, 8, 940. [Google Scholar] [CrossRef]
- Vujaković, J.; Mitrović, S.; Mitrović, Z.D.; Radenović, S. On F-Contractions for Weak α-Admissible Mappings in Metric-Like Spaces. Mathematics 2020, 8, 1629. [Google Scholar] [CrossRef]
- Hussain, N.; Latif, A.; Iqbal, I. Fixed point results for generalized F-contractions in modular metric and fuzzy metric spaces. Fixed Point Theory Appl. 2015, 2015, 158. [Google Scholar] [CrossRef] [Green Version]
- Zoto, K.; Vardhami, I.; Bajović, D.; Mitrović, Z.D.; Radenović, S. On Some Novel Fixed Point Results for Generalized F-Contractions in b-Metric-Like Spaces with Application. CMES Comput. Model. Eng. Sci. 2023, 135, 673–686. [Google Scholar] [CrossRef]
- Saleh, H.N.; Sessa, S.; Alfaqih, W.M.; Imdad, M.; Mlaiki, N. Fixed Circle and Fixed Disc Results for New Types of Θc-Contractive Mappings in Metric Spaces. Symmetry 2020, 12, 1825. [Google Scholar] [CrossRef]
- Mlaiki, N.; Taş, N.; Özgür, N.Y. On the Fixed-Circle Problem and Khan Type Contractions. Axioms 2018, 7, 80. [Google Scholar] [CrossRef] [Green Version]
- Taş, N.; Özgür, N.Y.; Mlaiki, N. New Types of Fc-Contractions and the Fixed-Circle Problem. Mathematics 2018, 6, 188. [Google Scholar] [CrossRef] [Green Version]
- Samet, B.; Vetro, C.; Vetro, P. Fixed point theorems for α-ψ-contractive type mappings. Nonlinear Anal. 2012, 75, 2154–2165. [Google Scholar] [CrossRef] [Green Version]
- Alizadeh, S.; Moradlou, F.; Salimi, P. Some fixed point results for (α, β)-(ψ, ϕ)-contractive mappings. Filomat 2014, 28, 3635–3647. [Google Scholar]
- An, T.V.; Dung, N.V.; Kadelburg, Z.; Radenović, S. Various generalizations of metric spaces and fixed point theorems. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. 2015, 109, 175–198. [Google Scholar] [CrossRef]
- Radenović, S.; Došenović, T.; Öztürk, V.; Dolićanin, Ć. A note on the paper: “Integral equations with new admissability types in b-metric spaces”. J. Fixed Point Theory Appl. 2017, 19, 2287–2295. [Google Scholar] [CrossRef]
- Agarwal, R.P.; Karapınar, E.; O’Regan, D.; de Hierro, A.F.R.L. Fixed Point Theory in Metric Type Spaces; Springer International Publishing: Cham, Switzerland, 2015. [Google Scholar]
- Asif, A.; Nazam, M.; Arshad, M.; Kim, S.O. F-Metric, F-Contraction and Common Fixed-Point Theorems with Applications. Mathematics 2019, 7, 586. [Google Scholar] [CrossRef] [Green Version]
- Bera, A.; Garai, H.; Damjanović, B.; Chanda, A. Some Interesting Results on F-metric Spaces. Filomat 2019, 33, 3257–3268. [Google Scholar] [CrossRef]
- Jahangir, F.; Haghmaram, P.; Nourouzi, K. A note on F-metric spaces. J. Fixed Point Theory Appl. 2021, 23, 1–14. [Google Scholar] [CrossRef]
- Kanwal, T.; Hussain, A.; Baghani, H.; de la Sen, M. New Fixed Point Theorems in Orthogonal F-Metric Spaces with Application to Fractional Differential Equation. Symmetry 2020, 12, 832. [Google Scholar] [CrossRef]
- Lateefa, D.; Ahmadb, J. Dass and Guptas fixed point theorem in F-metric spaces. J. Nonlinear Sci. Appl. 2019, 12, 405–411. [Google Scholar] [CrossRef]
- Ma, Z.; Asif, A.; Aydi, H.; Khan, S.U.; Arshad, M. Analysis of F-contractions in function weighted metric spaces with an application. Open Math. 2020, 18, 582–594. [Google Scholar] [CrossRef]
- Hussain, A.; Kanwal, T. Existence and uniqueness for a neutral differential problem with unbounded delay via fixed point results. Trans. A Razmadze Math. Inst. 2018, 172, 481–490. [Google Scholar] [CrossRef]
i | Exact Solution | Approximation Solution | Approximation Solution |
---|---|---|---|
(m = 64) | (m = 128) | ||
0.0 | 0 | 0.010417 | 0.005208 |
0.1 | 0.199667 | 0.197570 | 0.192399 |
0.2 | 0.397339 | 0.382942 | 0.398412 |
0.3 | 0.591040 | 0.605205 | 0.589930 |
0.4 | 0.778837 | 0.781174 | 0.785758 |
0.5 | 0.958851 | 0.967335 | 0.963098 |
0.6 | 1.129285 | 1.126666 | 1.122812 |
0.7 | 1.288435 | 1.276056 | 1.289847 |
0.8 | 1.434712 | 1.446451 | 1.433200 |
0.9 | 1.566654 | 1.569934 | 1.572171 |
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Faraji, H.; Mirkov, N.; Mitrović, Z.D.; Ramaswamy, R.; Abdelnaby, O.A.A.; Radenović, S. Some New Results for (α, β)-Admissible Mappings in 𝔽-Metric Spaces with Applications to Integral Equations. Symmetry 2022, 14, 2429. https://doi.org/10.3390/sym14112429
Faraji H, Mirkov N, Mitrović ZD, Ramaswamy R, Abdelnaby OAA, Radenović S. Some New Results for (α, β)-Admissible Mappings in 𝔽-Metric Spaces with Applications to Integral Equations. Symmetry. 2022; 14(11):2429. https://doi.org/10.3390/sym14112429
Chicago/Turabian StyleFaraji, Hamid, Nikola Mirkov, Zoran D. Mitrović, Rajagopalan Ramaswamy, Ola A. Ashour Abdelnaby, and Stojan Radenović. 2022. "Some New Results for (α, β)-Admissible Mappings in 𝔽-Metric Spaces with Applications to Integral Equations" Symmetry 14, no. 11: 2429. https://doi.org/10.3390/sym14112429