Document Type : Research Paper
Authors
- Vahid Parvaneh ^{} ^{1}
- Nawab Hussain ^{2}
- Hasan Hosseinzadeh ^{} ^{3}
- Peyman Salimi ^{4}
^{1} Department of Mathematics, Gilan-E-Gharb Branch, Islamic Azad University, Gilan-E-Gharb, Iran.
^{2} Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia.
^{3} Department of Mathematics, Ardebil Branch, Islamic Azad University, Ardebil, Iran.
^{4} Peyman Salimi: Young Researchers and Elite Club, Rasht Branch,Islamic Azad University, Rasht, Iran.
Abstract
The aim of this paper is to establish some fixed point theorems for $\alpha$-admissible Mizoguchi-Takahashi contractive mappings defined on a ${b}$-metric space which generalize the results of Gordji and Ramezani \cite{Roshan6}. As a result, we obtain some coupled fixed point theorems which generalize the results of '{C}iri'{c} {et al.} \cite{Ciric3}. We also present an application in order to illustrate the effectiveness of our results.
Keywords
Main Subjects
[1] A. Aghajani, M. Abbas, and J.R. Roshan, Common fixed point of generalized weak contractive mappings in partially ordered b-metric spaces, to appear in Math. Slovaca.
[2] M.U. Ali, T. Kamran, W. Sintunavarat, and Ph. Katchang, Mizoguchi-Takahashi's Fixed Point Theorem with α,η Functions, Abstract and Applied Analysis, vol. 2013, Article ID 418798, 4 pages, 2013. doi:10.1155/2013/418798.
[3] A. Amini-Harandi and H. Emami, A fixed point theorem for contraction type maps in partially ordered metric spaces and application to ordinary diferential equations, Nonlinear Anal., 72 (2010) 2238-2242.
[4] T.G. Bhaskar and V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal., 65 (2006) 1379-1393.
[5] M. Boriceanu, Strict fixed point theorems for multivalued operators in b-metric spaces, Int. J. Modern Math., 4 (3) (2009), 285-301.
[6] Lj. Ciric, B. Damjanovic, M. Jleli, and B. Samet, Coupled fixed point theorems for generalized Mizoguchi-Takahashi contraction and applications to ordinary differential equations, Fixed Point Theory Appl., 2012, 2012:51.
[7] S. Czerwik, Contraction mappings in b-metric spaces, Acta Math. Inf. Univ. Ostraviensis, 1 (1993) 5-11.
[8] W.S. Du, Coupled fixed point theorems for nonlinear contractions satisfied Mizoguchi-Takahashi's condition in quasi-ordered metric spaces, Fixed Point Theory Appl., 2010 (2010) Article ID 876372, 9 pages.
[9] M.E. Gordji and M. Ramezani, A generalization of Mizoguchi and Takahashi's theorem for single-valued mappings in partially ordered metric spaces, Nonlinear Anal., (2011), doi: 10.1016/j.na. 2011.04.020.
[10] J. Harjani, B. Lopez, and K. Sadarangani, Fixed point theorems for mixed monotone operators and applications to integral equations, Nonlinear Anal. 74 (2011) 1749-1760.
[11] N. Hussain, D. Doric, Z. Kadelburg, and S. Radenovic, Suzuki-type fixed point results in metric type spaces, Fixed Point Theory Appl., (2012), 2012:126.
[12] N. Hussain, E. Karapinar, P. Salimi, and P. Vetro, Fixed point results for G^{m}-Meir-Keeler contractive and G-(α,ψ)-Meir-Keeler contractive mappings, Fixed Point Theory and Applications 2013, 2013:34.
[13] N. Hussain, V. Parvaneh, J.R. Roshan, and Z Kadelburg, Fixed points of cyclic weakly (ψ, α, L, A, B)-contractive mappings in ordered b-metric spaces with applications, Fixed Point Theory Appl,. 2013 :256, 2013.
[14] N. Hussain, J.R. Roshan, V. Parvaneh, and M. Abbas, Common fixed point results for weak contractive mappings in ordered b-dislocated metric spaces with applications, Journal of Inequalities and Applications, 2013, 486, 2013.
[15] E. Karapinar and R.P. Agarwal, A note on Coupled fixed point theorems for α-ψ-contractive-type mappings in partially ordered metric spaces, Fixed Point Theory and Applications. 2013, 2013:216.
[16] E. Karapinar, P. Kumam, and P. Salimi, On α-ψ-Meir-Keeler contractive mappings, Fixed Point Theory Appl., 2013, 2013:94.
[17] E. Karapinar and B. Samet, Generalized (α-ψ)-contractive type mappings and related fixed point theorems with applications, Abstr. Appl. Anal., 2012 (2012) Article ID: 793486.
[18] N.V. Luong and N.X. Thuan, Coupled fixed points in partially ordered metric spaces and application, Nonlinear Anal. 74 (2011) 983-992.
[19] G. Minak and I. Altun, Some new generalizations of Mizoguchi-Takahashi type fixed point theorem, Journal of Inequalities and Applications, 2013, 2013:493.
[20] N. Mizoguchi and W. Takahashi, Fixed point theorems for multi-valued mappings on complete metric space, J. Math. Anal. Appl., 141 (1989) 177-188.
[21] M. Mursaleen, S.A. Mohiuddine, and R.P. Agarwal, Coupled fixed point theorems for α-ψ-contractive type mappings in partially ordered metric spaces, Fixed Point Theory Appl. 2012, 124 (2012).
[22] Z. Mustafa, J.R. Roshan, V. Parvaneh, and Z. Kadelburg, Fixed point theorems for weakly T-Chatterjea and weakly T-Kannan contractions in b-metric spaces, Journal of Inequalities and Applications, 2014:46, 2014.
[23] Z. Mustafa, J.R. Roshan, V. Parvaneh, and Z. Kadelburg, Some common fixed point results in ordered partial b-metric spaces, Journal of Inequalities and Applications, 2013, 562, 2013.
[24] S.B. Nadler Jr., Multivalued contraction mappings, Pacific J. Math. 30 (1969) 475-88.
[25] J.J. Nieto and R. Rodriguez-Lopez, Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations, Acta Mathematica Sinica, 23 (12) (2007) 2205-2212.
[26] J.J. Nieto and R. Rodriguez-Lopez, Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order, 22 (3) (2005) 223-239.
[27] V. Parvaneh, J.R. Roshan, and S. Radenovic, Existence of tripled coincidence points in ordered b-metric spaces and an application to a system of integral equations, Fixed Point Theory and Applications, 2013, 2013:130.
[28] A.C.M. Ran and M.C.B. Reurings, A Fixed point theorem in partially ordered metric sets and some applications to matrix equetions, Proc. Amer. Math. Soc., 132 (5)(2003) 1435-1443.
[29] S. Reich, Fixed points of contractive functions, Boll. Unione Mat. Ital. 4 (5) (1972) 26-42.
[30] J.R. Roshan, V. Parvaneh, and I. Altun, Some coincidence point results in ordered b-metric spaces and applications in a system of integral equations, Applied Mathematics and Computation, 2014, 226, 725-737.
[31] J.R. Roshan, V. Parvaneh, S. Sedghi, N. Shobkolaei, and W. Shatanawi, Common fixed points of almost generalized (α-ψ)_s-contractive mappings in ordered b-metric spaces, Fixed Point Theory Appl., 2013, 2013:159.
[32] B. Samet, C. Vetro, and P. Vetro, Fixed point theorem for α-ψ-contractive type mappings, Nonlinear Anal., 75 (2012) 2154-2165.