A fixed point theorem for a Meir-Keeler type contraction through rational expression

Samet, Bessem¹ ; Vetro, Calogero² ; Yazidi, Habib³

¹ Department of Mathematics, King Saud University, Riyadh, Saudi Arabia
² Dipartimento di Matematica e Informatica, Universita degli Studi di Palermo, via Archirafi 34, 90123 Palermo, Italy
³ Département de Mathématiques, Ecole Supérieure des Sciences et Techniques de Tunis, 5, Avenue Taha Hussein-Tunis, B. P.: 56, Bab Menara-1008, Tunisie

Journal of nonlinear sciences and its applications, Tome 6 (2013) no. 3, p. 162-169.

Résumé

In this paper, we establish a new fixed point theorem for a Meir-Keeler type contraction through rational expression. The presented theorem is an extension of the result of Dass and Gupta (1975). Some applications to contractions of integral type are given.

Détail

DOI : 10.22436/jnsa.006.03.02

Classification : 54H25, 47H10
Keywords: Fixed point, Meir-Keeler type contraction, Rational expression, Contraction of integral type.

Bibliographie

[1] Alimohammady, M.; Ramzannezhad, M. On \(\Phi\)-fixed point for maps on uniform spaces, J. Nonlinear Sci. Appl., Volume 1 (4) (2008), pp. 241-243

[2] Altun, I.; Türkoğlu, D.; Rhoades, B. E. Fixed points of weakly compatible maps satisfying a general contractive condition of integral type, Fixed Point Theory Appl., Article ID 17301 , Volume 2007 (2007), pp. 1-9

[3] Azam, A.; M. Arshad Kannan fixed point theorem on generalized metric spaces, J. Nonlinear Sci. Appl. , Volume 1 (1) (2008), pp. 45-48

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

[5] Beg, I.; Abbas, M.; Nazir, T. Generalized cone metric spaces, J. Nonlinear Sci. Appl., Volume 3 (1) (2010), pp. 21-31

[6] A. Branciari A fixed point theorem of Banach-Caccippoli type on a class of generalized metric spaces, Publ. Math. Debrecen, Volume 57 (2000), pp. 31-37

[7] Branciari, A. A fixed point theorem for mappings satisfying a general contractive condition of integral type, Int. J. Math. Math. Sci. , Volume 29 (2002), pp. 531-536

[8] Chatterjee, H. On generalization of Banach contraction principle , Indian J. Pure Appl. Math., Volume 10 (4) (1979), pp. 400-403

[9] Ćirić, Lj. B. A generalization of Banach's contraction principle, Proc. Amer. Math. Soc., Volume 45 (1974), pp. 267-273

[10] Ćirić, Lj. B. A new fixed point theorem for contractive mappings, Publ. Inst. Math. (Beograd), Volume 30 (44) (1981), pp. 25-27

[11] Dass, B. K.; S. Gupta An extension of Banach contraction principle through rational expression , Indian J. Pure Appl. Math., Volume 6 (1975), pp. 1455-1458

[12] Djoudi, A.; A. Aliouche Common fixed point theorems of Gregus type for weakly compatible mappings satisfying contractive conditions of integral type, J. Math. Anal. Appl., Volume 329 (2007), pp. 31-45

[13] Edelstein, M. On fixed and periodic points under contractive mappings, J. Lond. math. Soc., Volume 37 (1962), pp. 74-79

[14] Gordji, M. Eshaghi; Baghani, H.; Khodaei, H.; Ramezani, M. A generalization of Nadler's fixed point theorem, J. Nonlinear Sci. Appl., Volume 3 (2) (2010), pp. 148-151

[15] Jachymski, J. Equivalent conditions and the Meir-Keeler type theorems, J. Math. Anal. Appl., Volume 194 (1995), pp. 293-303

[16] Jleli, M.; B. Samet The Kannan's fixed point theorem in a cone rectangular metric space, J. Nonlinear Sci. Appl., Volume 2 (3) (2009), pp. 161-167

[17] Kasahara, S. On some generalizations of the Banach contraction theorem, Publ. Res. Inst. Math. Sci. , Volume 12 (2) (1976), pp. 427-437

[18] Kirk, W. A. Contraction mappings and extensions, in Handbook of Metric Fixed Point Theory, W. A. Kirk and B. Sims, Eds., pp.1-34, Kluwer Academic, Dordrecht, The Netherlands, 2001

[19] Matkowski, J. Fixed point theorems for contractive mappings in metric spaces , Časopis Pro Pěstování Matematiky, Volume 105 (4) (1980), pp. 341-344

[20] Meir, A.; E. Keeler A theorem on contraction mappings , J. Math. Anal. Appl. , Volume 28 (1969), pp. 326-329

[21] Merryfield, J.; Stein, J. D. A generalization of the Banach Contraction Principle, J. Math. Anal. Appl. , Volume 273 (2002), pp. 112-120

[22] Mihet, D. On Kannan fixed point principle in generalized metric spaces, J. Nonlinear Sci. Appl., Volume 2 (2) (2009), pp. 92-96

[23] Rhoades, B. E. Two fixed point theorems for mappings satisfying a general contractive condition of integral type, Int. J. Math. Math. Sci. , Volume 63 (2003), pp. 4007-4013

[24] Samet, B. A fixed point theorem in a generalized metric space for mappings satisfying a contractive condition of integral type, Int. Journal of Math. Analysis, Volume 3 (26) (2009), pp. 1265-1271

[25] Samet, B. Coupled fixed point theorems for a generalized Meir-Keeler contraction in partially ordered metric spaces, Nonlinear Anal., Volume 72 (2010), pp. 4508-4517

[26] B. Samet Ciric's fixed point theorem in a cone metric space, in J. Nonlinear Sci. Appl. (to appear)

[27] Samet, B.; Yazidi, H. An extension of Banach fixed point theorem for mappings satisfying a contractive condition of integral type, J. Pure Appl. Math. ( to appear)

[28] T. Suzuki Meir-Keeler Contractions of Integral Type Are Still Meir-Keeler Contractions, Int. J. Math. Math. Sci., Article ID 39281, Volume 2007 (2007), pp. 1-6

[29] Vetro, C. On Branciari's theorem for weakly compatible mappings, Appl. Math. Lett., Volume 23 (6) (2010), pp. 700-705

[30] Vijayaraju, P.; Rhoades, B. E.; R. Mohanraj A fixed point theorem for a pair of maps satisfying a general contractive condition of integral type, Int. J. Math. Math. Sci. , Volume 15 (2005), pp. 2359-2364