Document Type : Research Paper

Authors

1 Department of Applied Mathematics, School of Applied Natural Sciences, Adama Science and Technology University, Post Box No.1888, Adama, Ethiopia.

2 Department of Mathematics, Vignan's Foundation for Science, Technology & Research, Vadlamudi-522213, Andhra Pradesh, India.

Abstract

The aim of this paper is to prove some coupled fixed point  theorems of  a self mapping satisfying a certain rational type contraction along with  strict mixed monotone property in an ordered metric space. Further, a result  is presented for the uniqueness of a coupled fixed point under an order relation in a space. These results generalize and extend known existing results in the literature.

Keywords

###### ##### References
[1] R.P. Agarwal, M.A. El-Gebeily and D. O'Regan, Generalized contractions in partially ordered metric spaces, Appl. Anal., 87 (2008), pp. 1-8.

[2] J. Ahmad, M. Arshad and C. Vetro, On a theorem of Khan in a generalized metric space, Int. J. Anal., 2013, Article ID 852727, (2013).

[3] I. Altun, B. Damjanovic and D. Djoric, Fixed point and common fixed point theorems on ordered cone metric spaces, Appl. Math. Lett., 23 (2010), pp. 310-316.

[4] A. Amini-Harandi and H. Emami, A fixed point theorem for contraction type maps in partially ordered metric spaces and application to ordinary differential equations, Nonlinear Anal., Theory Methods Appl., 72 (2010), pp. 2238-2242.

[5] M. Arshad, A. Azam and P. Vetro, Some common fixed results in cone metric spaces, Fixed Point Theory Appl., 2009, Article ID 493965 (2009).

[6] M. Arshad, J. Ahmad and E. Karapinar, Some common fixed point results in rectangular metric spaces, Int. J. Anal., 2013, Article ID 307234 (2013).

[7] M. Arshad, E. Karapinar and J. Ahmad, Some unique fixed point theorems for rational contractions in partially ordered metric spaces, Journal of Inequalities and Applications, 2013:248, 2013.

[8] H. Aydi, E. Karapinar and W. Shatanawi, Coupled fixed point results for ($psi, varphi$)-weakly contractive condition in ordered partial metric spaces, Comput. Math. Appl., 62(12) (2011), pp. 4449-4460.

[9] A. Azam, B. Fisher and M. Khan, Common fixed point theorems in complex valued metric spaces, Numer. Funct. Anal. Optim., 32(3) (2011), pp. 243--253.

[10] I. Beg and A.R. Butt, Fixed point for set-valued mappings satisfying an implicit relation in partially ordered metric spaces, Nonlinear Anal., 71 (2009), pp. 3699-3704.

[11] T.G. Bhaskar and V. Lakshmikantham, Fixed point theory in partially ordered metric spaces and applications, Nonlinear Anal., Theory Methods Appl., 65 (2006), pp. 1379-1393.

[12] S. Chandok, T.D. Narang and M.A. Taoudi, Some coupled fixed point theorems for mappings satisfying a generalized contractive condition of rational type, Palestine Journal of Mathematics, 4(2) (2015), pp. 360-366.

[13] B.S. Choudhury and A. Kundu, A coupled coincidence point result in partially ordered metric spaces for compatible mappings, Nonlinear Anal., Theory Methods Appl., 73 (2010), pp. 2524--2531.

[14] L. Ciric, M.O. Olatinwo, D. Gopal and G. Akinbo, Coupled fixed point theorems for mappings satisfying a contractive condition of rational type on a partially ordered metric space, Advances in Fixed Point Theory, 2(1) (2012), pp. 1-8.

[15] Z. Dricia, F.A. McRaeb and J.V. Devi, Fixed-point theorems in partially ordered metric spaces for operators with PPF dependence, Nonlinear Anal., Theory Methods Appl., 67 (2007), pp. 641-647.

[16] M. Edelstein, On fixed points and periodic points under contraction mappings, J. Lond. Math. Soc., 37 (1962), pp. 74-79.

[17] D. Gopal, M. Abbas, D. K. Patel and C. Vetro, Fixed points of $alpha$-type $F$-contractive mappings with an application to nonlinear fractional differential equation, Acta Math. Sci., 36(3) (2016), pp. 957-970.

[18] G.C. Hardy and T. Rogers, A generalization of fixed point theorem of S. Reich, Can. Math. Bull., 16 (1973), pp. 201-206.

[19] S. Hong, Fixed points of multivalued operators in ordered metric spaces with applications, Nonlinear Anal., Theory Methods Appl., 72 (2010), pp. 3929-3942.

[20] R. Kannan, Some results on fixed points-II, Am. Math. Mon., 76 (1969), pp. 71-76.

[21] E. Karapinar and N.V. Luong, Quadruple fixed point theorems for nonlinear contractions, Comput. Math. Appl., 64(6) (2012), pp. 1839-1848.

[22] E. Karapinar, Coupled fixed point on cone metric spaces, Gazi Univ. J. Sci., 24(1) (2011), pp. 51-58.

[23] P. Kumam, F. Rouzkard, M. Imdad and D. Gopal, Fixed Point Theorems on Ordered Metric Spaces through a Rational Contraction, Abstract and Applied Analysis, 2013, Article ID 206515, 9 pages.

[24] V. Lakshmikantham and L.B. Ciric, Couple fixed point theorems for nonlinear contractions in partially ordered metric spaces, Nonlinear Anal., Theory Methods Appl., 70 (2009), pp. 4341-4349.

[25] H. Lakzian, D. Gopal and W. Sintunavarat, New fixed point results for mappings of contractive type with an application to nonlinear fractional differential equations, J. Fixed Point Theory Appl., 18(2) (2016), pp. 251-266.

[26] N.V. Luong and N.X. Thuan, Coupled fixed points in partially ordered metric spaces and application, Nonlinear Anal., Theory Methods Appl., 74 (2011), pp. 983-992.

[27] B. Monjardet, Metrics on partially ordered sets-a survey, Discrete Math., 35 (1981), pp. 173-184.

[28] J.J. Nieto and R.R. Lopez, Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order, 22 (2005), pp. 223-239.

[29] J.J. Nieto and R.R. Lopez, Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equation, Acta Math. Sin. Engl. Ser., 23(12) (2007), pp. 2205-2212.

[30] J.J. Nieto, L. Pouso and R. Rodríguez-López, Fixed point theorems in ordered abstract spaces, Proc. Am. Math. Soc., 135 (2007), pp. 2505-2517.

[31] M. Ozturk and M. Basarir, On some common fixed point theorems with rational expressions on cone metric spaces over a Banach algebra, Hacet. J. Math. Stat., 41(2) (2012), pp. 211-222.

[32] A.C.M. Ran and M.C.B. Reurings, A fixed point theorem in partially ordered sets and some application to matrix equations, Proc. Am. Math. Soc., 132 (2004), pp. 1435-1443.

[33] S. Reich, Some remarks concerning contraction mappings, Can. Math. Bull., 14 (1971), pp. 121-124.

[34] F. Rouzkard and M. Imdad, Some common fixed point theorems on complex valued metric spaces, Comput. Math. Appl., (2012). doi:10.1016/j.camwa.2012.02.063.

[35] B. Samet, Coupled fixed point theorems for a generalized Meir-Keeler contraction in partially ordered metric spaces, Nonlinear Anal., 74(12) (2010), pp. 4508-4517.

[36] P.L. Sharma and A.K. Yuel, A unique fixed point theorem in metric space, Bull. Cal. Math. Soc., 76 (1984), pp. 153-156.

[37] S. Shukla, D. Gopal and J. Martinez-Moreno, Fixed Points of set valued $alpha-F$-contractions and its application to nonlinear integral equations, Filomat, 31 (11) (2017), pp. 3377-3390.

[38] D.R. Smart, Fixed Point Theorems, Cambridge University Press, Cambridge, 1974.

[39] E.S. Wolk, Continuous convergence in partially ordered sets, Gen. Topol. Appl., 5 (1975), pp. 221-234.

[40] C.S. Wong, Common fixed points of two mappings, Pac. J. Math., 48 (1973), pp. 299-312.

[41] X. Zhang, Fixed point theorems of multivalued monotone mappings in ordered metric spaces, Appl. Math. Lett., 23 (2010), pp. 235-240.