Interpolative Ćirić-Reich-Rus-type best proximity point results with applications

Naeem Saleem; Hüseyin Işık; Sana Khaleeq; Choonkil Park; Naeem Saleem; Hüseyin Işık; Sana Khaleeq; Choonkil Park

doi:10.3934/math.2022542

AIMS Mathematics

2022, Volume 7, Issue 6: 9731-9747. doi: 10.3934/math.2022542

Previous Article Next Article

Research article

Interpolative Ćirić-Reich-Rus-type best proximity point results with applications

1.
Department of Mathematics, University of Management and Technology, Lahore, Pakistan
2.
Department of Engineering Science, Bandırma Onyedi Eylül University, Bandırma 10200, Balıkesir, Turkey
3.
Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Korea

Received: 13 October 2021 Revised: 21 February 2022 Accepted: 07 March 2022 Published: 17 March 2022
MSC : 47H10, 54H25

In this paper, we introduce the notion of $ \omega $-interpolative Ćirić-Reich-Rus-type proximal contraction. We obtain some best proximity point results for these mappings using the concept of $ \omega $-admissibility in complete metric spaces. Some best proximity results are extended to partial ordered metric spaces and graphical metric spaces. Several new definitions are presented by considering the special cases of aforementioned results. The application of these results in fixed point theory is also discussed. The acquired results extend $ \omega $-interpolative Ćirić-Reich-Rus-type contraction for obtaining fixed points.
- complete metric space,
- ordered metric space,
- graph theory,
- interpolative proximal contraction
Citation: Naeem Saleem, Hüseyin Işık, Sana Khaleeq, Choonkil Park. Interpolative Ćirić-Reich-Rus-type best proximity point results with applications[J]. AIMS Mathematics, 2022, 7(6): 9731-9747. doi: 10.3934/math.2022542

Related Papers:

Abstract

In this paper, we introduce the notion of $ \omega $-interpolative Ćirić-Reich-Rus-type proximal contraction. We obtain some best proximity point results for these mappings using the concept of $ \omega $-admissibility in complete metric spaces. Some best proximity results are extended to partial ordered metric spaces and graphical metric spaces. Several new definitions are presented by considering the special cases of aforementioned results. The application of these results in fixed point theory is also discussed. The acquired results extend $ \omega $-interpolative Ćirić-Reich-Rus-type contraction for obtaining fixed points.

References

[1]	H. Aydi, C. M. Chen, E. Karapınar, Interpolative Ćirić-Reich-Rus type contractions via the Branciari distance, Mathematics, 7 (2019), 84. https://doi.org/10.3390/math7010084 doi: 10.3390/math7010084
[2]	A. Amini-Harandi, H. Emami, A fixed point theorem for contraction type maps in partially ordered metric spaces and applications to ordinary differential equations, Nonlinear Anal., 72 (2010), 2238–2242. https://doi.org/10.1016/j.na.2009.10.023 doi: 10.1016/j.na.2009.10.023
[3]	H. Aydi, E. Karapinar, A. F. Roldán López de Hierro, $\omega $-interpolative Ćirić-Reich-Rus-type contractions, Mathematics, 7 (2019), 57. https://doi.org/10.3390/math7010057 doi: 10.3390/math7010057
[4]	H. Aydi, H. Lakzian, Z. D. Mitrović, S. Radenović, Best proximity points of MT-cyclic contractions with property UC, Numer. Funct. Anal. Optim., 41 (2020), 871–882. https://doi.org/10.1080/01630563.2019.1708390 doi: 10.1080/01630563.2019.1708390
[5]	S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fund. Math., 3 (1922), 133–181.
[6]	S. S. Basha, Best proximity point theorems on partially ordered sets, Optim. Lett., 7 (2013), 1035–1043. https://doi.org/10.1007/s11590-012-0489-1 doi: 10.1007/s11590-012-0489-1
[7]	S. S. Basha, Common best proximity points: Global minimization of multi-objective functions, J. Glob. Optim., 54 (2012), 367–373. https://doi.org/10.1007/s10898-011-9760-8 doi: 10.1007/s10898-011-9760-8
[8]	V. Berinde, Iterative approximation of fixed points, Berlin, Heidelberg: Springer, 2007. https://doi.org/10.1007/978-3-540-72234-2
[9]	T. G. Bhaskar, V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal., 65 (2006), 1379–1393. https://doi.org/10.1016/j.na.2005.10.017 doi: 10.1016/j.na.2005.10.017
[10]	S. K. Chatterjea, Fixed point theorems, C. R. Acad. Bulgare Sci., 25 (1972), 727–730.
[11]	L. B. Ćirić, Generalized contractions and fixed-point theorems, Publ. Inst. Math. 12 (1971), 19–26.
[12]	M. Edelstein, An extension of Banach's contraction principle, Proc. Amer. Math. Soc., 12 (1961), 7–10. https://doi.org/10.2307/2034113 doi: 10.2307/2034113
[13]	A. A. Eldred, P. Veeramani, Existence and convergence of best proximity points, J. Math. Anal. Appl., 323 (2006), 1001–1006. https://doi.org/10.1016/j.jmaa.2005.10.081 doi: 10.1016/j.jmaa.2005.10.081
[14]	M. Gabeleh, Global optimal solutions of non-self mappings, U. P. B. Sci. Bull. Ser. A, 75 (2013), 67–74.
[15]	M. Gabeleh, J. Markin, Some notes on the paper "On best proximity points of interpolative proximal contractions", Quaest. Math., 2021, 1–6. https://doi.org/10.2989/16073606.2021.1951872
[16]	M. Gabeleh, C. Vetro, A note on best proximity point theory using proximal contractions, J. Fixed Point Theory Appl., 20 (2018), 1–11. https://doi.org/10.1007/s11784-018-0624-4 doi: 10.1007/s11784-018-0624-4
[17]	G. E. Hardy, T. D. Rogers, A generalization of a fixed point theorem of Reich, Can. Math. Bull., 16 (1973), 201–206. https://doi.org/10.4153/CMB-1973-036-0 doi: 10.4153/CMB-1973-036-0
[18]	H. Işık, H. Aydi, N. Mlaiki, S. Radenović, Best proximity point results for Geraghty type $Z$-proximal contractions with an application, Axioms, 8 (2019), 81. https://doi.org/10.3390/axioms8030081 doi: 10.3390/axioms8030081
[19]	H. Işık, D. Turkoglu, Generalized weakly $\alpha $-contractive mappings and applications to ordinary differential equations, Miskolc Math. Notes, 17 (2016), 365–379. https://doi.org/10.18514/MMN.2016.1384 doi: 10.18514/MMN.2016.1384
[20]	M. Jleli, B. Samet, Best proximity points for $\alpha -\psi $-proximal contractive type mappings and application, Bull. Sci. Math., 137 (2013), 977–995. https://doi.org/10.1016/j.bulsci.2013.02.003 doi: 10.1016/j.bulsci.2013.02.003
[21]	G. Jungck, Commuting mappings and fixed points, Amer. Math. Mon., 83 (1976), 261–263. https://doi.org/10.1080/00029890.1976.11994093 doi: 10.1080/00029890.1976.11994093
[22]	R. Kannan, Some results on fixed points, Bull. Cal. Math. Soc., 60 (1968), 71–76.
[23]	E. Karapinar, Revisiting the Kannan type contractions via interpolation, Adv. Theory Nonlinear Anal. Appl., 2 (2018), 85–87. https://doi.org/10.31197/atnaa.431135 doi: 10.31197/atnaa.431135
[24]	E. Karapinar, R. Agarwal, H. Aydi, Interpolative Reich-Rus-Ćirić type contractions on partial metric spaces, Mathematics, 6 (2018), 1–7. https://doi.org/10.3390/math6110256 doi: 10.3390/math6110256
[25]	H. Kaddouri, H. Işık, S. Beloul, On new extensions of $ F$-contraction with application to integral inclusions, U. P. B. Sci. Bull. Ser. A, 81 (2019), 31–42.
[26]	K. Khammahawong, P. Kumam, A best proximity point theorem for Roger-Hardy type generalized $F$-contractive mappings in complete metric spaces with some examples, RACSAM, 112 (2018), 1503–1519. https://doi.org/10.1007/s13398-017-0440-5 doi: 10.1007/s13398-017-0440-5
[27]	S. Komal, P. Kumam, K. Khammahawong, K. Sitthithakerngkiet, Best proximity coincidence point theorems for generalized non-linear contraction mappings, Filomat, 32 (2018), 6753–6768. https://doi.org/10.2298/FIL1819753K doi: 10.2298/FIL1819753K
[28]	K. Khammahawong, P. Kumam, D. M. Lee, Y. J. Cho, Best proximity points for multi-valued Suzuki $\alpha$-$F$-proximal contractions, J. Fixed Point Theory Appl., 19 (2017), 2847–2871. https://doi.org/10.1007/s11784-017-0457-6 doi: 10.1007/s11784-017-0457-6
[29]	W. A. Kirk, P. S. Srinavasan, P. Veeramani, Fixed points for mapping satisfying cyclical contractive conditions, Fixed Point Theory, 4 (2003), 79–89.
[30]	B. E. Rhoades, A comparison of various definitions of contractive mappings, Trans. Amer. Math. Soc., 226 (1977), 257–290. https://doi.org/10.1090/S0002-9947-1977-0433430-4 doi: 10.1090/S0002-9947-1977-0433430-4
[31]	I. A. Rus, Generalized contractions and applications, Cluj-Napoca: Cluj University Press, 2001.
[32]	S. Reich, Some remarks concerning contraction mappings, Can. Math. Bull., 14 (1971), 121–124. https://doi.org/10.4153/CMB-1971-024-9 doi: 10.4153/CMB-1971-024-9
[33]	S. Reich, Kannan's fixed point theorem, Boll. Un. Mat. Ital., 4 (1971), 1–11.
[34]	N. Saleem, B. Ali, M. Abbas, Z. Raza, Fixed points of Suzuki type generalized multivalued mappings in fuzzy metric spaces with applications, Fixed Point Theory Appl., 36 (2015), 2015. https://doi.org/10.1186/s13663-015-0284-7 doi: 10.1186/s13663-015-0284-7
[35]	N. Saleem, M. Abbas, Z. Raza, Fixed fuzzy point results of generalized Suzuki type $F$-contraction mappings in ordered metric spaces, Georgian Math. J., 27 (2017), 307–320. https://doi.org/10.1515/gmj-2017-0048 doi: 10.1515/gmj-2017-0048
[36]	N. Saleem, I. Iqbal, B. Iqbal, S. Radenović, Coincidence and fixed points of multivalued $F$-contractions in generalized metric space with application, J. Fixed Point Theory Appl., 22 (2020), 1–24. https://doi.org/10.1007/s11784-020-00815-3 doi: 10.1007/s11784-020-00815-3
[37]	B. Samet, C. Vetro, P. Vetro, Fixed point theorems for $ \alpha-\psi $-contractive type mappings, Nonlinear Anal., 75 (2012), 2154–2165. https://doi.org/10.1016/j.na.2011.10.014 doi: 10.1016/j.na.2011.10.014

Reader Comments

Your name:*

Email:*
© 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)