^{}Young Researchers and Elite club, Central Tehran Branch, Islamic Azad University, Tehran, Iran. Department of Mathematics, Faculty of Science, Islamic Azad University, Central Tehran Branch, PO. Code 13185-768, Tehran, Iran.

Abstract

Recently, Rahimi et al. [Comp. Appl. Math. 2013, In press] dened the concept of quadrupled xed point in K-metric spaces and proved several quadrupled xed point theorems for solid cones on K-metric spaces. In this paper some quadrupled xed point results for T-contraction on K-metric spaces without normality condition are proved. Obtained results extend and generalize well-known comparable results in the literature.

[1] M. Abbas, B.E. Rhoades, Fixed and periodic point results in cone metric spaces, Appl. Math. Lett. 22 (2009), pp. 511-515. [2] S. Banach, Sur les oprations dans les ensembles abstraits et leur application aux quations intgrales, Fund. Math. J. 3 (1922), pp. 133-181. [3] V. Berinde, M. Borcut, Tripled xed point theorems for contractive type mappings in partially ordered metric spaces, Nonlinear Anal. 74 (2011), pp. 4889-4897. [4] T. Bhaskar, V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal. 65 (2006), pp. 1379-1393. [5] M. Filipovic, L. Paunovic, S. Radenovic, M. Rajovic, Remarks on Cone metric spaces and xed point theorems of T-Kannan and T-Chatterjea contractive mappings", Math. Comput. Modelling. 54 (2011), pp. 1467-1472. [6] K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, 1985. [7] G.E. Hardy, T.D. Rogers, A generalization of a xed point theorem of Reich, Canad. Math. Bull. 1 (6) (1973), pp. 201{206. [8] L.G. Huang, X. Zhang, Cone metric spaces and xed point theorems of contractive mappings, J. Math. Anal. Appl. 332 (2007), pp. 1467{1475. [9] E. Karapinar, N.V. Luong, Quadruple xed point theorems for nonlinear contractions, Comput. Math. Appl. 64 (6) (2012) pp. 1839{1848. [10] J.R. Morales, E. Rojas, Cone metric spaces and xed point theorems of T-Kannan contractive mappings, Int. J. Math. Anal. 4 (4) (2010), pp. 175{184. [11] E.M. Mukhamadiev, V.J. Stetsenko, Fixed point principle in generalized metric space, Izvestija AN Tadzh. SSR, z.-mat.igeol.-chem.nauki. 10 (4) (1969), pp. 8{19 (in Russian). [12] A.I. Perov, The Cauchy problem for systems of ordinary dierential equations, Approximate Methods of Solving Dierential Equations. Kiev. Naukova Dumka. (1964), pp. 115{134 (in Russian). [13] H. Rahimi, S. Radenovic, G. Soleimani Rad, P. Kumam, Quadrupled xed point results in abstract metric spaces, Comp. Appl. Math. (2013), DOI 10.1007/s40314-013-0088-5. [14] H. Rahimi, B.E. Rhoades, S. Radenovic, G. Soleimani Rad, Fixed and periodic point theorems for T- contractions on cone metric spaces, Filomat. 27 (5) (2013), pp. 881{888 (DOI 10.2298/FIL1305881R). [15] H. Rahimi, G. Soleimani Rad, Common xed point theorems and c-distance in ordered cone metric spaces, Ukrainian Mathematical Journal, (2013), to appear. [16] H. Rahimi, G. Soleimani Rad, Fixed point theory in various spaces, Lambert Academic Publishing, Germany, 2013. [17] H. Rahimi, G. Soleimani Rad, New xed and periodic point results on cone metric spaces, Journal of Linear and Topological Algebra 1 (1) (2012), pp. 33{40. [18] H. Rahimi, G. Soleimani Rad, Note on Common xed point results for noncommuting mappings without continuity in cone metric spaces", Thai. J. Math. 11 (3) (2013), pp. 589{599. [19] H. Rahimi, G. Soleimani Rad, Some xed point results in metric type space, J. Basic Appl. Sci. Res. 2 (9) (2012), pp. 9301{9308. [20] H. Rahimi, G. Soleimani Rad, P. Kumam, Coupled common xed point theorems under weak contractions in cone metric type spaces , Thai. J. Math, In press.

[21] H. Rahimi, P. Vetro, G. Soleimani Rad, Some common xed point results for weakly compatible mappings in cone metric type space, Miskolc Math. Notes. 14 (1) (2013), pp. 233{243. [22] S. Rezapour, R. Hamlbarani, Some note on the paper cone metric spaces and xed point theorems of con- tractive mappings, J. Math. Anal. Appl. 345 (2008), pp. 719-724. [23] B.E. Rhoades, A comparison of various denition of contractive mappings, Trans. Amer. Math. Soc. 266 (1977), pp. 257-290. [24] B. Samet, C. Vetro, Coupled xed point, f-invariant set and xed point of N-order, Ann. Funct. Anal. 1 (2) (2010), pp. 46-56. [25] P. P. Zabrejko, K-metric and K-normed linear spaces: survey, Collect. Math. 48 (4-06)(1997), pp. 825-859.