×
Khan, M. S.; Singh, Y. Mahendra; Maniu, Georgeta; Postolache, Mihai

On generalized convex contractions of type-2 in \(b\)-metric and 2-metric spaces. (English) Zbl 1412.47141

J. Nonlinear Sci. Appl. 10, No. 6, 2902-2913 (2017).
Summary: In this paper, we present the notion of generalized convex contraction mapping of type-2, which includes the generalized convex contraction (resp. generalized convex contraction of order-2) of M. Miandaragh et al. [Fixed Point Theory Appl. 2013, Paper No. 255, 8 p. (2013; Zbl 1321.54089)] and the convex contraction mapping of type-2 of V. I. Istratescu [Libertas Math. 1, 151–163 (1981; Zbl 0477.54032)]. Utilizing this class of mappings, we establish approximate fixed point and fixed point theorems in the setting of \(b\)-metric and 2-metric spaces.

Cited in 5 Documents

MSC:

47H10 Fixed-point theorems
54H25 Fixed-point and coincidence theorems (topological aspects)
54E40 Special maps on metric spaces

Keywords:

approximate fixed point; fixed point; convex contraction; asymptotic regular; \(\alpha\)-admissible; \(b\)-metric and 2-metric spaces

Citations:

Zbl 1321.54089; Zbl 0477.54032
Cite Review PDF
Full Text: DOI

References:

[1] Alghamdi, M. A.; Alnafei, S. H.; Radenović, S.; Shahzad, N., Fixed point theorems for convex contraction mappings on cone metric spaces, Math. Comput. Modelling, 54, 2020-2026 (2011) · Zbl 1235.54021 · doi:10.1016/j.mcm.2011.05.010
[2] Aydi, H.; Bota, M. F.; Karapınar, E.; Mitrović, S., A fixed point theorem for set-valued quasi-contractions in b-metric spaces, Fixed Point Theory Appl., 2012, 1-8 (2012) · Zbl 1457.54032 · doi:10.1186/1687-1812-2012-88
[3] Bakhtin, I. A., The contraction mapping principle in almost metric space, (Russian) Functional analysis, Ulyanovsk. Gos. Ped. Inst., Ulyanovsk, 26-37 (1989) · Zbl 0748.47048
[4] Banach, S., Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fund. Math., 3, 133-181 (1922) · JFM 48.0201.01
[5] Berinde, V., Generalized contractions in quasimetric spaces, Seminar on Fixed Point Theory, “Babeş-Bolyai” Univ., Cluj-Napoca, 3-9 (1993) · Zbl 0878.54035
[6] Berinde, M., Approximate fixed point theorems, Stud. Univ. Babeş-Bolyai Math., 51, 11-25 (2006) · Zbl 1164.54028
[7] Boriceanu, M., Fixed point theory for multivalued generalized contraction on a set with two b-metrics, Stud. Univ. Babeş- Bolyai Math., 54, 3-14 (2009) · Zbl 1240.54118
[8] Boriceanu, M.; Bota, M.; Petruşsel, A., Multivalued fractals in b-metric spaces, Cent. Eur. J. Math., 8, 367-377 (2010) · Zbl 1235.54011 · doi:10.2478/s11533-010-0009-4
[9] Browder, F. E.; Petryshyn, W. V., The solution by iteration of nonlinear functional equations in Banach spaces, Bull. Amer. Math. Soc., 72, 571-575 (1966) · Zbl 0138.08202 · doi:10.1090/S0002-9904-1966-11544-6
[10] Czerwik, S., Contraction mappings in b-metric spaces, Acta Math. Inform. Univ. Ostraviensis, 1, 5-11 (1993) · Zbl 0849.54036
[11] Czerwik, S., Nonlinear set-valued contraction mappings in b-metric spaces, Atti Sem. Mat. Fis. Univ. Modena, 46, 263-276 (1998) · Zbl 0920.47050
[12] Gähler, S., 2-metrische Räume und ihre topologische Struktur, (German) Math. Nachr., 26, 115-148 (1963) · Zbl 0117.16003 · doi:10.1002/mana.19630260109
[13] Gähler, S., Lineare 2-normierte Räume, (German) Math. Nachr., 28, 1-43 (1964) · Zbl 0142.39803 · doi:10.1002/mana.19640280102
[14] Gähler, S., Über 2-Banach-Räume, (German) Math. Nachr., 42, 335-347 (1969) · Zbl 0191.41202 · doi:10.1002/mana.19690420414
[15] Ghorbanian, V.; Rezapour, S.; Shahzad, N., Some ordered fixed point results and the property (P), Comput. Math. Appl., 63, 1361-1368 (2012) · Zbl 1248.54025 · doi:10.1016/j.camwa.2011.12.071
[16] Iséki, K., Fixed point theorem in 2-metric spaces, Math. Sem. Notes Kobe Univ., 3, 133-136 (1975) · Zbl 0314.54050
[17] Istrăţescu, V. I., Some fixed point theorems for convex contraction mappings and convex nonexpansive mappings, I, Libertas Math., 1, 151-163 (1981) · Zbl 0477.54032
[18] Karapınar, E., \( \alpha-\psi \)-Geraghty contraction type mappings and some related fixed point results, Filomat, 28, 37-48 (2014) · Zbl 1450.54017 · doi:10.2298/FIL1401037K
[19] Khan, M. S., On fixed point theorems in 2-metric space, Publ. Inst. Math. (Beograd) (N.S.), 41, 107-113 (1980) · Zbl 0462.54031
[20] Kohlenbach, U.; Leuştean, L., The approximate fixed point property in product spaces, Nonlinear Anal., 66, 806-818 (2007) · Zbl 1118.47047 · doi:10.1016/j.na.2005.12.020
[21] Kumar, P.; Poonam, Fixed point for \(\alpha-\psi\) contractive mapping in 2-metric spaces, Int. J. Math. Trends Tech., 10, 34-37 (2014)
[22] Latif, A.; Sintunavarat, W.; Ninsri, A., Approximate fixed point theorems for partial generalized convex contraction mappings in \(\alpha \)-complete metric spaces, Taiwanese J. Math., 19, 315-333 (2015) · Zbl 1357.54034 · doi:10.11650/tjm.19.2015.4746
[23] Liu, Z.-Q.; Zhang, F.-R.; Mao, J.-F., Common fixed points for compatible mappings of type (A), Bull. Malaysian Math. Soc., 22, 67-86 (1999) · Zbl 1001.54028
[24] Miandaragh, M. A.; Postolache, M.; Rezapour, S., Approximate fixed points of generalized convex contractions, Fixed Point Theory Appl., 2013, 1-8 (2013) · Zbl 1321.54089 · doi:10.1186/1687-1812-2013-255
[25] Miandaragh, M. A.; Postolache, M.; Rezapour, S., Some approximate fixed point results for generalized \(\alpha \)-contractive mappings, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys., 75, 3-10 (2013) · Zbl 1289.54135
[26] Miculescu, R.; Mihail, A., A generalization of Istrăţescu’s fixed point theorem for convex contractions, ArXiv, 2015, 1-17 (2015)
[27] Ramezani, M., Orthogonal metric space and convex contractions, Int. J. Nonlinear Anal. Appl., 6, 127-132 (2015) · Zbl 1321.54096 · doi:10.22075/ijnaa.2015.261
[28] Reich, S.; Zaslavski, A. J., Genericity and porosity in fixed point theory: a survey of recent results, Fixed Point Theory Appl., 2015, 1-21 (2015) · Zbl 1347.54002 · doi:10.1186/s13663-015-0444-9
[29] Samet, B.; Vetro, C.; Vetro, P., Fixed point theorems for \(\alpha-\psi \)-contractive type mappings, Nonlinear Anal., 75, 2154-2165 (2012) · Zbl 1242.54027 · doi:10.1016/j.na.2011.10.014
[30] Sastry, K. P. R.; Rao, C. S.; Sekhar, A. Chandra; Balaiah, M., A fixed point theorem for cone convex contractions of order \(m \geq 2\), Int. J. Math. Sci. Eng. Appl., 6, 263-271 (2012)
[31] Shatanawi, W.; Pitea, A.; Lazović, R., Contraction conditions using comparison functions on b-metric spaces, Fixed Point Theory Appl., 2014, 1-11 (2014) · Zbl 1310.54071 · doi:10.1186/1687-1812-2014-135
[32] Singh, S. L.; Prasad, B., Some coincidence theorems and stability of iterative procedures, Comput. Math. Appl., 55, 2512-2520 (2008) · Zbl 1142.65360 · doi:10.1016/j.camwa.2007.10.026
[33] Tijs, S.; Torre, A.; Brânzei, R., Approximate fixed point theorems, Miron Nicolescu (1903-1975) and Nicolae Ciorânescu(1903-1957), Libertas Math., 23, 35-39 (2003) · Zbl 1056.47046
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.