Almost monotone contractions on weighted graphs


Monther R. Alfuraidan - Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals Dhahran 31261, Saudi Arabia.
Mostafa Bachar - Department of Mathematics, College of Sciences, King Saud University, Riyadh, Saudi Arabia.
Mohamed A. Khamsi - Department of Mathematical Sciences, University of Texas at El Paso, El Paso, TX 79968, USA.


Almost contraction mappings were introduced as an extension to the contraction mappings for which the conclusion of the Banach contraction principle (BCP in short) holds. In this paper, the concept of monotone almost contractions defined on a weighted graph is introduced. Then a fixed point theorem for such mappings is given.



[1] M. R. Alfuraidan, Remarks on monotone multivalued mappings on a metric space with a graph, J. Inequal. Appl., 2015 (2015), 7 pages.
[2] S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fund. Math., 3 (1922), 133-181.
[3] V. Berinde, On the approximation of fixed points of weak contractive mappings, Carpathian J. Math., 19 (2003), 7-22.
[4] V. Berinde, M. Păcurar, Fixed points and continuity of almost contractions, Fixed Point Theory, 9 (2008), 23-34.
[5] S. K. Chatterjea, Fixed-point theorems, C. R. Acad. Bulgare Sci., 25 (1972), 727-730.
[6] S. M. El-Sayed, A. C. M. Ran, On an iteration method for solving a class of nonlinear matrix equations, SIAM J. Matrix Anal. Appl., 23 (2002), 632-645.
[7] Y. Feng, S. Liu, Fixed point theorems for multi-valued contractive mappings and multi-valued Caristi type mappings, J. Math. Anal. Appl., 317 (2006), 103-112.
[8] K. Goebel, W. A. Kirk, Topics in metric fixed point theory, Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, (1990).
[9] M. T. Goodrich, R. Tamassia, Algorithm design: foundations, analysis and internet examples, John Wiley & Sons, Inc., New York, (2001).
[10] J. Jachymski, The contraction principle for mappings on a metric space with a graph, Proc. Amer. Math. Soc., 136 (2008), 1359-1373.
[11] R. Kannan, Some results on fixed points, Bull. Calcutta Math. Soc., 60 (1968), 71-76.
[12] M. A. Khamsi, W. A. Kirk, An introduction to metric spaces and fixed point theory, Pure and Applied Mathematics, Wiley-Interscience, New York, (2001).
[13] D. Klim, D. Wardowski, Fixed point theorems for set-valued contractions in complete metric spaces, J. Math. Anal. Appl., 334 (2007), 132-139.
[14] S. B. Nadler, Multi-valued contraction mappings, Pacific J. Math., 30 (1969), 475-488.
[15] J. J. Nieto, R. Rodríguez-López, Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order, 22 (2005), 223-239.
[16] A. C. M. Ran, M. C. B. Reurings, A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. Amer. Math. Soc., 132 (2004), 1435-1443.
[17] I. A. Rus, Generalized contractions, Seminar on Fixed Point Theory, 3 (1983), 1-130, Preprint, 83-3, Univ. ''Babe-Bolyai'', Cluj-Napoca, (1983).
[18] I. A. Rus, Generalized contractions and applications, Cluj University Press, Cluj-Napoca, (2001).
[19] M. R. Tasković, Osnove teorije fiksne take, (Serbo-Croatian) [Foundations of fixed-point theory] With an English summary, Matematicka Biblioteka [Mathematical Library], Zavod za Udzbenike i Nastavna Sredstva, Belgrade, (1986), 272 pages.
[20] M. Turinici, Fixed points for monotone iteratively local contractions, Demonstratio Math., 19 (1986), 171-180.
[21] X. A. Udo-utun, Z. U. Siddiqui, M. Y. Balla, An extension of the contraction mapping principle to Lipschitzian mappings, Fixed Point Theory Appl., 2015 (2015), 7 pages.
[22] D. W, Wallis, A beginner's guide to graph theory, Second edition, Birkhäuser Boston, Inc., Boston, MA, (2007).
[23] T. Zamfirescu, Fix point theorems in metric spaces, Arch. Math. (Basel), 23 (1972), 292-298.