FIXED POINT THEOREMS FOR A PAIR OF MAPPINGS SATISFYING A GENERALIZED WEAKLY CONTRACTIVE CONDITION IN ORDERED METRIC SPACES

Yıl 2012, Cilt: 41 Sayı: 2, 243 - 253, 01.02.2012

Hemant Kumar Nashine İshak Altun

Öz

Fixed point theorems for a pair of mappings which satisfy the condition of being generalized weakly contractive in complete ordered metric spaces are derived. At the end of the paper, applications of the previous results to new fixed point results of integral type are also shown. 

Anahtar Kelimeler

Fixed point, Complete metric space, Altering distance function, Weakly contractive condition, Partially ordered set

Kaynakça

• Agarwal, R.P., El-Gebeily, M.A. and and O’Regan, D. Generalized contractions in partially ordered metric spaces, Applicable Analysis 87 (1), 109–116, 2008.
• Alber, Ya.E. and Guerre-Delabriere, S. Principles of weakly contractive maps in Hilbert spaces, in: New Results in Operator Theory and its Applications (I.Gohberg and Yu. Lyubich, eds.) (Oper. Theory Adv. Appl. 98, Birkhauser, Basel, 1997), 7–22.
• Abbas, M. and Khan, M.A. Common fixed point theorem of two mappings satisfying a generalized weak contractive condition, Inter. J. Math. Math. Sci., Article ID 131068, 9 pages, 2009.
• Altun, I. and Simsek, H. Some fixed point theorems on ordered metric spaces and application, Fixed Point Theory and Applications, Article ID 621492, 17 pages, 2010.
• Amini-Harandi, A. and Emami, H. A fixed point theorem for contraction type maps in partially ordered metric spaces and application to ordinary differential equations, Nonlinear Analysis 72 (5), 2238–2242, 2010.
• Beg, I. and Butt, A.R. —em Fixed point for set-valued mappings satisfying an implicit relation in partially ordered metric spaces, Nonlinear Analysis 71 (9), 3699–3704, 2009.
• Beg, I. and Butt, A.R. Fixed points for weakly compatible mappings satisfying an implicit relation in partially ordered metric spaces, Carpathian Journal of Mathematics 25 (1), 1–12, 2009.
• Bhaskar, T.G. and Lakshmikantham, V. Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Analysis 65 (7), 1379–1393, 2006.
• Boyd, D.W. and Wong, J.S.W. On nonlinear contractions, Proc. Amer. Math. Soc. 20, 458–464, 1969.
• ´Ciri´c, L., Caki´c, N., Rajovi´c, M. and Ume, J.S. Monotone generalized nonlinear contractions in partially ordered metric spaces, Fixed Point Theory and Applications 2008, Article ID 131294, 11 pages, 2008.
• Dutta, P.N. and Choudhury, B.S. A generalisation of contraction principle in metric spaces, Fixed Point Theory and Applications, Article ID 406368, 8 Pages, 2008.
• Harjani, J. and Sadarangani, K. Fixed point theorems for weakly contractive mappings in partially ordered sets, Nonlinear Analysis 71 (7-8), 3403–3410, 2009.
• Harjani, J. and Sadarangani, K. Generalized contractions in partially ordered metric spaces and applications to ordinary differential equations, Nonlinear Analysis 72 (3-4), 1188–1197, 2010.
• Khan, M.S., Swaleh, M. and Sessa, S. Fixed point theorems by altering distances between the points, Bulletin of the Australian Mathematical Society 30 (1), 1–9, 1984.
• Nieto, J.J. and Rodr´ıguez-Lo´pez, R. Contractive mapping theorems in partially ordered sets and applications to ordianry differential equations, Order 22 (3), 223–239, 2005.
• Nieto, J.J. and Rodr´ıguez-Lo´pez, R. Existence and uniqueness of fixed point in partially ordered sets and applications to ordianry differential equations, Acta Math. Sinica (English Series) 23 (12), 2205–2212, 2007.
• Nieto, J.J., Pouso, R.L. and Rodr´ıguez-Lo´pez, R. Fixed point theorems in ordered abstract spaces, Proceedings of the American Mathematical Society 135 (8), 2505–2517, 2007.
• O’Regan, D. and Petrusel, A. Fixed point theorems for generalized contractions in ordered metric spaces, Journal of Mathematical Analysis and Applications 341 (2), 1241–1252, 2008.
• Ran, A.C.M. and Reurings, M.C.B. A fixed point theorem in partially ordered sets and some applications to matrix equations, Proceedings of the American Mathematical Society 132(5), 1435–1443, 2004.
• Reich, S. Some fixed point problems, Atti. Acad. Naz. Linei 57, 194–198, 1974. [21] Rhoades, B.E. Some theorems on weakly contractive maps, Nonlinear Analysis 47 (4), 2683– 2693, 2001.
• Zhang, Q. and Song, Y. Fixed point theory for generalized ϕ-weakly contraction, Applied Mathematics Letters 22 (1), 75–78, 2009.