### Tripled Fixed Point Results in Generalized Metric Spaces under Nonlinear Type Contractions Depended on another Function

#### Abstract

In 2006, Mustafa and Sims[18-19] introduced an improved version of the generalized metric space structure which they called G-metric spaces and in 2011; Berinde and Borcut [11] introduced the concept of triple fixed point. The intent of this paper is to establish some tripled fixed point theorems for mappings having mixed monotone property under nonlinear type contractions depended on another function in the framework of a G-metric space X enclosed with partial order. The presented results generalize, improve and extend corresponding results of Hassen et al. [13] ( Tripled Fixed Point Results in Generalized Metric Spaces” Journal of Applied Mathematics Volume 2012, Article ID 314279, 10 pages, doi:10.1155/2012/ 314279). Moreover, some examples are provided to illustrate the usability of the obtained results.

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E. S. Wolk, Continuous convergence in partially ordered sets, General Topology and its Applications, Vol. 5, No. 3, pp. 221-234, 1975.

B. Monjardet, Metrics on partially ordered sets –a survey, Discrete mathematics, Vol. 35, pp. 173-184, 1981.

A. C. M. Ran and M. C. B. Reurings, A fixed point theorem in partially ordered sets and some applications to matrix equations, proceedings of the American Mathematical society, Vol. 132, No. 5, pp. 1435-1443, 2004.

T. GnanaBhaskar and V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Analysis, Vol. 65, No. 7, pp. 1379-1393, 2006.

V. Lakshmikantham and L. Ciric, Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces, Nonlinear Analysis: Theory, Method & Applications, Vol. 70, No. 12, pp. 4341-4349, 2009.

B. Samet and C. Vetro, Coupled fixed point, f- invariant set and fixed point of N-order, Annals of Functional Analysis, Vol. 1, No. 2, pp. 46-56, 2010.

W. Shatanawi, Coupled fixed point theorems in generalized metric spaces, Hacettepe Journal of Mathematics and Statistics, Vol. 40, No. 3, pp. 441–447, 2011.

H. Aydi, B. Damjanovi´c, B. Samet, and W. Shatanawi, Coupled fixed point theorems for nonlinear contractions in partially ordered G-metric spaces, Mathematical and Computer Modelling, Vol. 54, No. 9- 10, pp. 2443–2450, 2011.

H. Aydi, W. Shatanawi, and M. Postolache, Coupled fixed point results for (ψ, φ)-weakly contractive mappings in ordered G-metric spaces, Computers & Mathematics with Applications, Vol. 63, pp. 298–309, 2012.

B. S. Choudhury and P. Maity, Coupled fixed point results in generalized metric spaces, Mathematical and Computer Modelling, Vol. 54, No. 1-2, pp. 73–79, 2011. [11]. V. Berinde and M. Borcut, Tripled fixed point theorems for contractive type mappings in partially ordered metric spaces, Nonlinear Analysis: Theory, Method & Applications, Vol. 74, No. 15, pp. 4889-4897, 2011.

V. Berinde and M. Borcut, Tripled coincidence theorems for contractive type mappings in partially ordered metric spaces, Applied Mathematics & Computation, Vol. 218, No. 10, pp. 5929–5936, 2012.

H. Aydi, E. Karapınar, and W. Shatanawi, Tripled Fixed Point Results in Generalized Metric Spaces, Journal of Applied Mathematics, Volume 2012, Article ID 314279, 10 pages, doi:10.1155/2012/314279.

B. C. Dhage, Generalized metric space and mapping with fixed point, Bulletin of Calcutta Mathematical Society, Vol. 84, pp. 329–336, 1992.

B. C. Dhage, On generalized metric spaces and topological structure-II, Pure and Applied Mathematika Sciences, Vol. 40, No. 1-2, pp. 37–41, 1994.

B. C. Dhage, A common fixed point principle in D- metric spaces, Bulletin of the Calcutta Mathematical Society, Vol. 91, No. 6, pp. 475–480, 1999.

B. C. Dhage, Generalized metric spaces and topological structure-I,Annalele Stintificeale Universitatii Al. I. Cuza, Vol. 46, No. 1, pp. 3–24, 2000.

Z. Mustafa and B. Sims, Some remarks concerning D-metric spaces, in Proceedings of the International Conference on Fixed Point Theory and Applications, pp. 189–198, Yokohama, Japan, 2004.

Z. Mustafa and B. Sims, A new approach to generalized metric spaces, Journal of Nonlinear and Convex Analysis, Vol. 7, No. 2, pp. 289–297, 2006.

M. Abbas and B. E. Rhoades, Common fixed point results for non-commuting mappings without continuity in generalized metric spaces, Applied Mathematics and Computation, Vol. 215, No. 1, pp.262–269, 2009.

H. Aydi, W. Shatanawi, and C. Vetro, On generalized weakly G-contraction mapping in G-metric spaces, Computers & Mathematics with Applications, Vol. 62, pp. 4222–4229, 2011.

H. Aydi, A fixed point result involving a generalized weakly contractive condition in G-metric spaces, Bulletin of Mathematical Analysis and Applications, Vol. 3, No. 4, pp. 180–188, 2011.

H. Aydi, A common fixed point of integral type contraction in generalized metric spaces, Journal of Advanced Mathematical Studies, Vol. 5, No. 1, pp. 111– 117, 2012.

Z. Mustafa, A new structure for generalized metric spaces with applications to fixed point theory, Ph.D. thesis, The University of Newcastle, Callaghan, Australia, 2005.

Z. Mustafa, H. Obiedat, and F. Awawdeh, Some fixed point theorem for mapping on complete G-metric spaces, Fixed Point Theory and Applications, Vol. 2008, Article ID 189870, 12 pages, 2008

Z. Mustafa and B. Sims, Fixed point theorems for contractive mappings in complete G-metric spaces, Fixed Point Theory and Applications, Vol. 2009, Article ID 917175, 10 pages, 2009.

Z. Mustafa, W. Shatanawi, and M. Bataineh, Existence of fixed point results in G-metric spaces, International Journal of Mathematics and Mathematical Sciences, Vol. 2009, Article ID 283028, 10 pages,2009.

R. Saadati, S. M. Vaezpour, P. Vetro, and B. E. Rhoades, Fixed point theorems in generalized partially ordered G-metric spaces, Mathematical and Computer Modeling, Vol. 52, No. 5-6, pp. 797–801, 2010.

W. Shatanawi, Fixed point theory for contractive mappings satisfying Φ-maps in G-metric spaces, Fixed Point Theory and Applications, Vol. 2010, Article ID 181650, 9 pages, 2010.

W. Shatanawi, Some fixed point theorems in ordered G-metric spaces and applications, Abstract and Applied Analysis, Vol. 2011, Article ID 126205, 11 pages, 2011.

K. P. Chi, On a fixed point theorem for certain class of maps satisfying a contractive condition depended on another function,” Labachevskii journal of mathematics, Vol.30, No. 4, pp. 289-291, 2009.

K. P. Chi and H. T. Thuy, A fixed point theorem in 2-metric spaces for certain class of maps that satisfy a contractive condition dependent on an another function, Labachevskii journal of mathematics, Vol.31, No. 4, pp. 338-346, 2010.

H. Aydi, N. Bilgili and E. Karapinar, Common fixed point results from quasi –metric spaces to G-metric spaces, Journal of Egyptian Mathematical society (2014), (in press) http://dx.doi.org/10.1016/j.joems.2014.06.009. [34] N. Bilgili and E. Karapinar, Cyclic contractions via auxiliary functions on G-metric spaces, Fixed Point Theory 10.1186/1687-1812-2013-49. 2013, 2013:49

doi: [35] N. Bilgili, I.M. Erhan, E. Karapinar and D. Turkoglu, Cyclic contractions and related fixed point theorems on G-metric spaces, Appl. Math. Inf. Sci.8, No.4, 1541-1551 (2014).