Common Fixed Point Theorem for Three Mappings in Banach Valued Norm Spaces

Abstract

In this paper, we give a generalized theorem on point of coincidence and common fixed point for three weakly compatiblemappings  in Banach valued norm spaces. We give a new method  for
construction of the sequence, which is convergence to the common fixed point of these three mappings.

Keywords

• Banach valued normed spaces
• common fixed point
• weakly compatible mappings

References

1. Banach, S., “Sur les operations dans les ensembles abstraits et leur application aux equations itegrales”, Fund. Math., 3(1922)133–181.
2. Huang, L. G., Zhang, X., “Cone metric spaces and fixed point theorems of contractive mappings”, J. Math. Anal. Appl. 332 (2007), 1468–1476.
3. Fréchet, M., “Sur quelques points du calcul fonctionnel”, Rend. Circ. Mat. Palermo 22 (1906) 1–-74.
4. Zabrejko, P. P., “K-metric and K-normed linear spaces: survey”, Collect. Math. 48 (1997), 825– 859.
5. Kurepa, D. R., “Tableaux ramifiés d’ensembles. Espace pseudo-distanciés”, Comptes Rendus l’Académie des Sciences, 198,(1934) 1563–1565.
6. Krein, M. G., Rutman, M. A., “Linear operators leaving invariant a cone in a Banach space”, Uspekhi Matematicheskikh Nauk,3(1),(1948) 3– 95.
7. Kantorovic, L. V., “The principle of the majorant and Newton’s method”, Doklady Akademii Nauk SSSR, 76(1951) 17–20.
8. Rzepecki, B., “On fixed point theorems of Maia type”, Publications De L’institut Mathématique 28 (1980), 179–-186.

9. Lin, S. D., “A common fixed point theorem in abstract spaces”, Indian J. Pure Appl. Math. 18 (8) (1987) 685–690.
10. Abdeljawad, T., Karapnar, E., “Quasi-Cone Metric Spaces and Generalizations of Caristi Kirk’s Theorem”, doi:10.1155/2009/574387 (2009). Theory Appl.
11. Abdeljawad, T., “Completion of cone metric spaces”, Hacet. J. Math. Stat., (to appear).
12. Abdeljawad, T., Karapınar, E., “A Common Fixed Point Theorem of a Gregus Type on Convex Cone Metric Spaces”, J. Comput. Anal. Appl. ,13(4), 609-621(2011).
13. Abdeljawad, T., Turkoglu, D., Abuloha, M., “Some theorems and examples of cone Banach spaces”, J. Comput. Anal. Appl., 12(4), 739-753 (2010).
14. Abdeljawad, T., Karapınar, E., Tas, K., “Common fixed point theorems in cone banach spaces”, Hacet. J. Math. Stat. 40 (2) (2011), 211–217.
15. Cvetkovic, A. S., Stanic, M. P., Dimitrijevic, S., Simic, S., “Common Fixed Point Theorems for Four Mappings on Cone Metric Type Space”, Fixed point Theory Appl. 15 pages (2011) Article ID 589725.
16. Ding, H.-S., Kadelburg, Z., Karapınar, E., Radenovic, S., “Common Fixed Points of Weak Contractions in Cone Metric Spaces”, Abstr. Appl. Anal., 2012 2012 Article id: 793862
17. Karapinar, E., “Fixed point theorems in cone Banach spaces”, Fixed Point Theory and Appl. 9 pages (2009) Article ID 60928.
18. Karapinar, E., Yuksel, U., “On Common Fixed Point Theorems Without Commuting Conditions in tvs-Cone Metric Spaces”, J. Comput. Anal. Appl ,13(6), 1115-1122 (2011)
19. E. Karapınar, Some Nonunique Fixed Point Theorems of Ciric type on Cone Metric Spaces, Abstr. Appl. Anal., vol. 2010, Article ID 123094, 14 pages (2010).
20. Karapınar, E., “Couple Fixed Point Theorems For Nonlinear Contractions in Cone Metric Spaces”, Comput. Math. Appl. , 59( 12), , Pages 3656–3668 (2010).
21. Karapınar, E., Türkoğlu, Approximations Theorem For a Couple in Cone Banach Space”, Fixed Point Theory Appl., Volume 2010 (2010), Article ID 784578, 9 pages.
22. Rezapour, Sh., Hamlbarani, R., “Some notes on the paper "Cone metric spaces and fixed point theorems of contractive mappings"”, J. Math. Anal. Appl. 347 (2008), 719–724.
23. Sonmez, A., “On paracompactness in cone metric spaces”, Applied Mathematics Letters 23 (2010) 494, 497.
24. Türkoğlu, D., Abuloha, “Cone metric spaces and fixed point theorems in diametrically contractive mappings”, Acta. Math. Sin. 26 (3) (2010), 489– 496.
25. Türkoğlu, D., Abuloha, M., “Cone Metric Spaces and Fixed Point Theorems in Diametrically Contractive Mappings”, Acta Mathematica Sinica, English Series, 26(3), 489-496 (2010).
26. Turkoglu, D., Abuloha, M., Abdeljawad, T., “KKM mappings in cone metric spaces and some fixed point theorems”, Nonlinear Analysis: Theory, Methods & Applications, 72(1), 348-353 (2010).