Convex contractions on extended $ b $-metric spaces

Dan Ricinschi; Dan Ricinschi

doi:10.3934/math.2024887

AIMS Mathematics

2024, Volume 9, Issue 7: 18163-18185. doi: 10.3934/math.2024887

Previous Article Next Article

Research article

Convex contractions on extended $ b $-metric spaces

Dan Ricinschi ^,

Faculty of Applied Sciences, National University of Sciences and Technology Politehnica Bucharest, Splaiul Independenţei 311, Bucharest 060042, Romania

Received: 08 February 2024 Revised: 02 April 2024 Accepted: 09 April 2024 Published: 29 May 2024
MSC : 47H09, 47H10, 54H25

This research investigated different types of convex contractions in the setting of extended $ b $-metric spaces from the point of view of the existence and uniqueness of their fixed points. The assumptions imposed on involved mappings refer to convexity of order $ 2 $, two-sided convexity or Ćirić-type convexity, which also fulfill a continuity type condition. An example was provided to emphasize the usability of the results.
- extended $ b $-metric space,
- generalized metric space,
- convex contraction,
- generalized contraction
Citation: Dan Ricinschi. Convex contractions on extended $ b $-metric spaces[J]. AIMS Mathematics, 2024, 9(7): 18163-18185. doi: 10.3934/math.2024887

Related Papers:

Abstract

This research investigated different types of convex contractions in the setting of extended $ b $-metric spaces from the point of view of the existence and uniqueness of their fixed points. The assumptions imposed on involved mappings refer to convexity of order $ 2 $, two-sided convexity or Ćirić-type convexity, which also fulfill a continuity type condition. An example was provided to emphasize the usability of the results.

References

[1]	S. Banach, Sur les opérations dans les ensembles abstraits et leur applications aux équations intégrales, Fund. Math., 3 (1922), 133–181. https://doi.org/10.4064/FM-3-1-133-181 doi: 10.4064/FM-3-1-133-181
[2]	R. Caccioppoli, Un teorema generale sull' esistenza di elementi uniti in una transformazione funzionale, Rend. Accad. Lincei, 11 (1930), 794–799.
[3]	E. Rakotch, A note on contractive mappings, Proc. Amer. Math. Soc., 13 (1962), 459–465. https://doi.org/10.1090/S0002-9939-1962-0148046-1 doi: 10.1090/S0002-9939-1962-0148046-1
[4]	D. Boyd, J. Wong, On nonlinear contractions, Proc. Amer. Math. Soc., 20 (1969), 458–464. https://doi.org/10.1090/S0002-9939-1969-0239559-9 doi: 10.1090/S0002-9939-1969-0239559-9
[5]	A. Meir, E. Keeler, A theorem on contraction mappings, J. Math. Anal. Appl., 28 (1969), 326–329. https://doi.org/10.1016/0022-247X(69)90031-6 doi: 10.1016/0022-247X(69)90031-6
[6]	R. Kannan, Some results on fixed points, Bull. Calcutta Math. Soc., 60 (1968), 71–76.
[7]	V. Subrahmanyam, Completeness and fixed points, Monatsh. Math., 80 (1975), 325–330. https://doi.org/10.1007/BF01472580 doi: 10.1007/BF01472580
[8]	S. K. Chatterjea, Fixed-point theorems, C. R. Acad. Bulg. Sci., 25 (1972), 727–730.
[9]	S. Reich, Some remarks concerning contraction mappings, Canad. Math. Bull., 14 (1971), 121–124. https://doi.org/10.4153/CMB-1971-024-9 doi: 10.4153/CMB-1971-024-9
[10]	G. E. Hardy, T. D. Rogers, A generalization of a fixed point theorem of Reich, Canad. Math. Bull., 16 (1973), 201–206. https://doi.org/10.4153/CMB-1973-036-0 doi: 10.4153/CMB-1973-036-0
[11]	L. B. Ćirić, A generalization of Banach's contraction principle, Proc. Amer. Math. Soc., 45 (1974), 267–273.
[12]	H. Aydi, W. Shatanawi, M. Postolache, Z. Mustafa, N. Tahat, Theorems for Boyd-Wong-type contractions in ordered metric spaces, Abstr. Appl. Anal., 2012 (2012), 359054. https://doi.org/10.1155/2012/359054 doi: 10.1155/2012/359054
[13]	N. V. Luong, N. X. Thuan, Fixed point theorem for generalized weak contractions satisfying rational expressions in ordered metric spaces, Fixed Point Theory Appl., 2011 (2011), 46. https://doi.org/10.1186/1687-1812-2011-46 doi: 10.1186/1687-1812-2011-46
[14]	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
[15]	V. I. Istrăţescu, Some fixed point theorems for convex contraction mappings and mappings with convex diminishing diameters. I, Ann. Mat. Pura Appl., 130 (1982), 89–104. https://doi.org/10.1007/BF01761490 doi: 10.1007/BF01761490
[16]	M. A. Miandaragh, M. Postolache, S. Rezapour, Approximate fixed points of generalized convex contractions, Fixed Point Theory Appl., 2013 (2013), 255. https://doi.org/10.1186/1687-1812-2013-255 doi: 10.1186/1687-1812-2013-255
[17]	I. A. Bakhtin, The contraction mapping principle in almost metric spaces, Funct. Anal., 30 (1989), 26–37.
[18]	S. Czerwik, Contraction mappings in $b$-metric spaces, Acta Math. Inf. Univ. Ostraviensis, 1 (1993), 5–11.
[19]	M. U. Ali, T. Kamran, M. Postolache, Solution of Volterra integral inclusion in $b$-metric spaces via new fixed point theorem, Nonlinear Anal. Modell. Control, 22 (2017), 17–30. https://doi.org/10.15388/NA.2017.1.2 doi: 10.15388/NA.2017.1.2
[20]	T. Kamran, M. Postolache, M. U. Ali, Q. Kiran, Feng and Liu type $F$-contraction in $b$-metric spaces with application to integral equations, J. Math. Anal., 7 (2016), 18–27.
[21]	W. Shatanawi, A. Pitea, R. Lazović, Contraction conditions using comparison functions on $b$-metric spaces, Fixed Point Theory Appl., 2014 (2014), 135. https://doi.org/10.1186/1687-1812-2014-135 doi: 10.1186/1687-1812-2014-135
[22]	W. Kirk, N. Shahzad, Fixed point theory in distance spaces, Springer, 2014. https://doi.org/10.1007/978-3-319-10927-5
[23]	A. Branciari, A fixed point theorem of Banach-Caccioppoli type on a class of generalized metric spaces, Publ. Math. Debrecen, 57 (2000), 31–37. https://doi.org/10.5486/pmd.2000.2133 doi: 10.5486/pmd.2000.2133
[24]	P. Hitzler, A. K. Seda, Dislocated topologies, J. Electr. Eng., 51 (2000), 3–7.
[25]	F. Khojasteh, E. Karapınar, S. Radenovic, $\theta$-metric space: a generalization, Math. Probl. Eng., 2013 (2013), 504609. https://doi.org/10.1155/2013/504609 doi: 10.1155/2013/504609
[26]	N. V. Dung, Further results on $\theta$-metric spaces, Fixed Point Theory, 25 (2024), 99–110.
[27]	M. Jleli, B. Samet, A generalized metric space and related fixed point theorems, Fixed Point Theory Appl., 2015 (2015), 61. https://doi.org/10.1186/s13663-015-0312-7 doi: 10.1186/s13663-015-0312-7
[28]	M. Jleli, B. Samet, On a new generalization of metric spaces, Fixed Point Theory Appl., 20 (2018), 128. https://doi.org/10.1007/s11784-018-0606-6 doi: 10.1007/s11784-018-0606-6
[29]	T. Kamran, M. Samreen, Q. U. Ain, A generalization of $b$-metric space and some fixed point theorems, Mathematics, 5 (2017), 19. https://doi.org/10.3390/math5020019 doi: 10.3390/math5020019
[30]	M. Samreen, T. Kamran, M. Postolache, Extened $b$-metric space, extended $b$-comparison function and nonlinear contractions, Sci. Bull., Ser. A, Appl. Math. Phys., Politeh. Univ. Buchar., 80 (2018), 21–28.
[31]	M. Samreen, Q. Kiran, T. Kamran, Fixed point theorems for $\varphi$-contractions, J. Inequal. Appl., 2014 (2014), 266. https://doi.org/10.1186/1029-242X-2014-266 doi: 10.1186/1029-242X-2014-266
[32]	P. D. Proinov, A generalization of the Banach contraction principle with high order of convergence of succesive approximations, Nonlinear Anal., 67 (2007), 2361–2369. https://doi.org/10.1016/j.na.2006.09.008 doi: 10.1016/j.na.2006.09.008
[33]	T. L. Hicks, B. E. Rhoades, A Banach type fixed point theorem, Math. Japonica, 24 (1979), 327–330.
[34]	B. Alqahtani, A. Fulga, E. Karapınar, Common fixed point results on an extended $b$-metric space, J. Inequal. Appl., 2018 (2018), 158. https://doi.org/10.1186/s13660-018-1745-4 doi: 10.1186/s13660-018-1745-4
[35]	T. Abdeljawad, R. P. Agarwal, E. Karapınar, S. K. Panda, Solution of the nonlinear integral equation and fractional differential equation using the technique of a fixed point with a numerical experiment in extended $b$-metric space, Symmetry, 11 (2019), 686. https://doi.org/10.3390/sym11050686 doi: 10.3390/sym11050686
[36]	W. Shatanawi, K. Abodayeh, A. Mukheimer, Some fixed point theorems in extended $b$-metric spaces, UPB Sci. Bull., 80 (2018), 71–78.
[37]	B. Alqahtani, A. Fulga, E. Karapınar, Non-unique fixed point results in extended $b$-metric space, Mathematics, 6 (2018), 68. https://doi.org/10.3390/math6050068 doi: 10.3390/math6050068
[38]	Z. D. Mitrović, H. Işik, S. Radenović, The new results in extended $b$-metric spaces and applications, Int. J. Nonlinear Anal. Appl., 11 (2020), 473–482. https://doi.org/10.22075/IJNAA.2019.18239.1998 doi: 10.22075/IJNAA.2019.18239.1998
[39]	S. B. Nadler, Multivalued contraction mappings, Pacific J. Math., 30 (1969), 475–488. https://doi.org/10.2140/PJM.1969.30.475 doi: 10.2140/PJM.1969.30.475
[40]	H. Huang, Y. M. Singh, M. S. Khan, S. Radenović, Rational type contractions in extended $b$-metric spaces, Symmetry, 13 (2021), 614. https://doi.org/10.3390/sym13040614 doi: 10.3390/sym13040614
[41]	B. Alqahtani, A. Fulga, E. Karapınar, V. Rakočević, Contractions with rational inequalities in the extended $b$-metric space, J. Inequal. Appl., 2019 (2019), 220. https://doi.org/10.1186/s13660-019-2176-6 doi: 10.1186/s13660-019-2176-6
[42]	Q. Kiran, N. Alamgir, N. Mlaiki, H. Aydi, On some new fixed-point results in complete extended $b$-metric spaces, Mathematics, 7 (2019), 476. https://doi.org/10.3390/math7050476 doi: 10.3390/math7050476
[43]	N. Haokip, N. Goswami, B. C. Tripathy, Common fixed point results for ($\alpha$-$\beta$)-orbital-cyclic admissible triplet in extended $b$-metric spaces, Thai J. Math., 20 (2022), 563–576.
[44]	L. B. Ćirić, On contraction type mappings, Math. Balkanica, 1 (1971), 52–57.
[45]	H. Aydi, A. Felhi, T. Kamran, E. Karapınar, M. U. Ali, On nonlinear contractions in new extended $b$-metric spaces, Appl. Appl. Math., 14 (2019), 537–547.
[46]	N. Mlaiki, H. Aydi, N. Souayah, T. Abdeljawad, Controlled metric type spaces and the related contraction principle, Mathematics, 6 (2018), 194. https://doi.org/10.3390/math6100194 doi: 10.3390/math6100194
[47]	T. Abdeljawad, N. Mlaiki, H. Aydi, N. Souayah, Double controlled metric type spaces and some fixed point results, Mathematics, 6 (2018), 320. https://doi.org//10.3390/math6120320 doi: 10.3390/math6120320

Reader Comments

Your name:*

Email:*
© 2024 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)