Geodesic
On generalized convex contractions of type-2 in b-metric and 2-metric spaces
Khan, M. S.1 ; Singh, Y. M.2 ; Maniu, G.3 ; Postolache, M.4
1 Department of Mathematics and Statistics, Sultan Qaboos University, P. O. Box 36, PCode 123, Al-Khod, Muscat, Sultanate of Oman
2 Department of Humanities and Basic Sciences, Manipur Institute of Technology, Takyelpat–795001, India
3 Department of Computer Science, Information Technology, Mathematics and Physics, , Petroleum-Gas University of Ploieşti, Bucureşti Bvd., No. 39, 100680 Ploieşti, Romania
4 China Medical University, No. 91, Hsueh-Shih Road, Taichung, Taiwan;Department of Mathematics & Informatics, University ”Politehnica” of Bucharest, Splaiul Independentei 313, Bucharest 060042, Romania
Journal of nonlinear sciences and its applications, Tome 10 (2017) no. 6, p. 2902-2913.

Voir la notice de l'article provenant de la source International Scientific Research Publications

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 Miandaragh et al. [M. A. Miandaragh, M. Postolache, S. Rezapour, Fixed Point Theory Appl., 2013 (2013), 8 pages] and the convex contraction mapping of type-2 of Istrăţescu[V. I. Istrăţescu, I, Libertas Math., 1 (1981), 151–163]. Utilizing this class of mappings, we establish approximate fixed point and fixed point theorems in the setting of b-metric and 2-metric spaces.
DOI : 10.22436/jnsa.010.06.05
Classification : 47H10, 54H25
Keywords: Approximate fixed point, fixed point, convex contraction, asymptotic regular, \(\alpha\)-admissible, b-metric and 2-metric spaces.

Khan, M. S. 1 ; Singh, Y. M. 2 ; Maniu, G. 3 ; Postolache, M. 4

1 Department of Mathematics and Statistics, Sultan Qaboos University, P. O. Box 36, PCode 123, Al-Khod, Muscat, Sultanate of Oman
2 Department of Humanities and Basic Sciences, Manipur Institute of Technology, Takyelpat–795001, India
3 Department of Computer Science, Information Technology, Mathematics and Physics, , Petroleum-Gas University of Ploieşti, Bucureşti Bvd., No. 39, 100680 Ploieşti, Romania
4 China Medical University, No. 91, Hsueh-Shih Road, Taichung, Taiwan;Department of Mathematics & Informatics, University ”Politehnica” of Bucharest, Splaiul Independentei 313, Bucharest 060042, Romania 
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Khan, M. S.; Singh, Y. M.; Maniu, G.; Postolache, M. On generalized convex contractions of type-2 in b-metric and 2-metric spaces. Journal of nonlinear sciences and its applications, Tome 10 (2017) no. 6, p. 2902-2913. doi : 10.22436/jnsa.010.06.05. http://geodesic.mathdoc.fr/articles/10.22436/jnsa.010.06.05/

[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, Volume 54 (2011), pp. 2020-2026 | DOI

[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., Volume 2012 (2012), pp. 1-8 | DOI | Zbl

[3] Bakhtin, I. A. The contraction mapping principle in almost metric space, (Russian) Functional analysis, Ulyanovsk. Gos. Ped. Inst., Ulyanovsk (1989), pp. 26-37

[4] Banach, S. Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fund. Math., Volume 3 (1922), pp. 133-181

[5] Berinde, V. Generalized contractions in quasimetric spaces, Seminar on Fixed Point Theory, ”Babeş-Bolyai” Univ., Cluj-Napoca (1993), pp. 3-9

[6] Berinde, M. Approximate fixed point theorems, Stud. Univ. Babeş-Bolyai Math., Volume 51 (2006), pp. 11-25

[7] Boriceanu, M. Fixed point theory for multivalued generalized contraction on a set with two b-metrics, Stud. Univ. Babeş- Bolyai Math., Volume 54 (2009), pp. 3-14 | Zbl

[8] Boriceanu, M.; Bota, M.; Petruşsel, A. Multivalued fractals in b-metric spaces, Cent. Eur. J. Math., Volume 8 (2010), pp. 367-377 | DOI | Zbl

[9] Browder, F. E.; Petryshyn, W. V. The solution by iteration of nonlinear functional equations in Banach spaces, Bull. Amer. Math. Soc., Volume 72 (1966), pp. 571-575 | DOI

[10] Czerwik, S. Contraction mappings in b-metric spaces, Acta Math. Inform. Univ. Ostraviensis, Volume 1 (1993), pp. 5-11

[11] Czerwik, S. Nonlinear set-valued contraction mappings in b-metric spaces, Atti Sem. Mat. Fis. Univ. Modena, Volume 46 (1998), pp. 263-276

[12] Gähler, S. 2-metrische Räume und ihre topologische Struktur, (German) Math. Nachr., Volume 26 (1963), pp. 115-148 | DOI | Zbl

[13] Gähler, S. Lineare 2-normierte Räume, (German) Math. Nachr., Volume 28 (1964), pp. 1-43 | Zbl | DOI

[14] Gähler, S. Über 2-Banach-Räume, (German) Math. Nachr., Volume 42 (1969), pp. 335-347 | Zbl | DOI

[15] Ghorbanian, V.; Rezapour, S.; Shahzad, N. Some ordered fixed point results and the property (P), Comput. Math. Appl., Volume 63 (2012), pp. 1361-1368 | DOI

[16] Iséki, K. Fixed point theorem in 2-metric spaces, Math. Sem. Notes Kobe Univ., Volume 3 (1975), pp. 133-136

[17] Istrăţescu, V. I. Some fixed point theorems for convex contraction mappings and convex nonexpansive mappings, I, Libertas Math., Volume 1 (1981), pp. 151-163 | Zbl

[18] Karapınar, E. \(\alpha-\psi\)-Geraghty contraction type mappings and some related fixed point results, Filomat, Volume 28 (2014), pp. 37-48 | Zbl | DOI

[19] Khan, M. S. On fixed point theorems in 2-metric space, Publ. Inst. Math. (Beograd) (N.S.), Volume 41 (1980), pp. 107-113

[20] Kohlenbach, U.; Leuştean, L. The approximate fixed point property in product spaces, Nonlinear Anal., Volume 66 (2007), pp. 806-818 | DOI

[21] Kumar, P.; Poonam Fixed point for \(\alpha-\psi\) contractive mapping in 2-metric spaces, Int. J. Math. Trends Tech., Volume 10 (2014), pp. 34-37

[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., Volume 19 (2015), pp. 315-333 | Zbl | DOI

[23] Liu, Z.-Q.; Zhang, F.-R.; Mao, J.-F. Common fixed points for compatible mappings of type (A), Bull. Malaysian Math. Soc., Volume 22 (1999), pp. 67-86

[24] Miandaragh, M. A.; Postolache, M.; Rezapour, S. Approximate fixed points of generalized convex contractions, Fixed Point Theory Appl., Volume 2013 (2013), pp. 1-8 | DOI | Zbl

[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., Volume 75 (2013), pp. 3-10 | Zbl

[26] Miculescu, R.; Mihail, A. A generalization of Istrăţescu’s fixed point theorem for convex contractions, ArXiv, Volume 2015 (2015), pp. 1-17

[27] Ramezani, M. Orthogonal metric space and convex contractions, Int. J. Nonlinear Anal. Appl., Volume 6 (2015), pp. 127-132 | Zbl

[28] Reich, S.; Zaslavski, A. J. Genericity and porosity in fixed point theory: a survey of recent results, Fixed Point Theory Appl., Volume 2015 (2015), pp. 1-21 | DOI | Zbl

[29] Samet, B.; Vetro, C.; Vetro, P. Fixed point theorems for \(\alpha-\psi\)-contractive type mappings, Nonlinear Anal., Volume 75 (2012), pp. 2154-2165 | DOI

[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., Volume 6 (2012), pp. 263-271

[31] Shatanawi, W.; Pitea, A.; Lazović, R. Contraction conditions using comparison functions on b-metric spaces, Fixed Point Theory Appl., Volume 2014 (2014), pp. 1-11 | DOI | Zbl

[32] Singh, S. L.; Prasad, B. Some coincidence theorems and stability of iterative procedures, Comput. Math. Appl., Volume 55 (2008), pp. 2512-2520 | DOI

[33] Tijs, S.; Torre, A.; Brânzei, R. Approximate fixed point theorems, Miron Nicolescu (1903–1975) and Nicolae Ciorânescu(1903–1957), Libertas Math., Volume 23 (2003), pp. 35-39

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