Relation-theoretic contraction principle in cone metric spaces with Banach algebra
Authors: Malhotra S.K., Sharma J.B., Shukla S.
Keywords: Cone metric space; Relation-theoretic contraction principle; Solid cone; Banach algebra; Fixed point
Abstract:
In this work, we introduce the notion of relation-theoretic contractions in cone metric spaces with Banach algebra and prove some fixed point results for such contractions. Our results generalize and unify several known results in the setting of cone metric spaces with Banach algebra. An example is provided which illustrate the results proved herein and shows that how the new results are different from existing ones.
References:
[1] Alam, A., Imdad, M. (2015) Relation-theoretic contraction principle. Journal of Fixed Point Theory and Applications, 17(4): 693-702
[2] Ben-El-Mechaiekh, H. (2014) The Ran-Reurings fixed point theorem without partial order: A simple proof. Journal of Fixed Point Theory and Applications, 16(1-2): 373-383
[3] Çakallı, H., Sönmez, A., Genç, Ç. (2012) On an equivalence of topological vector space valued cone metric spaces and metric spaces. Applied Mathematics Letters, 25(3): 429-433
[4] Du, W. (2010) A note on cone metric fixed point theory and its equivalence. Nonlinear Analysis: Theory, Methods & Applications, 72(5): 2259-2261
[5] Huang, L., Zhang, X. (2007) Cone metric spaces and fixed point theorems of contractive mappings. Journal of Mathematical Analysis and Applications, 332(2): 1468-1476
[6] Kadelburg, Z., Pavlović, M., Radenović, S. (2010) Common fixed point theorems for ordered contractions and quasicontractions in ordered cone metric spaces. Comput. Math. Appl., vol. 59, br. 9, str. 3148-3159
[7] Kadelburg, Z., Radenović, S., Rakočević, V. (2011) A note on the equivalence of some metric and cone metric fixed point results. Applied Mathematics Letters, 24(3): 370-374
[8] Kirk, W.A., Srinivasan, P.S., Veeramani, P. (2003) Fixed points for mappings satisfying cyclical contractive conditions. Fixed Point Theory, Volume 4, br. 1, 79-89
[9] Kolman, B., Busby, R.C., Ross, S. (2000) Discrete mathematical structures. New Delhi: PHI Pvt. Ltd, 3rd ed
[10] Latif, A., Shaddad, F. (2010) Fixed Point Results for Multivalued Maps in Cone Metric Spaces. Fixed Point Theory and Applications, 2010(1): 941371
[11] Lipschutz, S. (1964) Schaum’s outlines of theory and problems of set theory and related topics. New York: McGraw-Hill
[12] Liu, H., Xu, S. (2013) Cone metric spaces with Banach algebras and fixed point theorems of generalized Lipschitz mappings. Fixed Point Theory and Applications, 2013(1): 320
[13] Liu, H., Xu, S. (2013) Fixed Point Theorems of Quasicontractions on Cone Metric Spaces with Banach Algebras. Abstract and Applied Analysis, 2013: 1-5
[14] Nieto, J.J., Rodríguez-López, R. (2005) Contractive Mapping Theorems in Partially Ordered Sets and Applications to Ordinary Differential Equations. Order, 22(3): 223-239
[15] Nieto, J.J., Rodríguez-López, R. (2006) Existence and Uniqueness of Fixed Point in Partially Ordered Sets and Applications to Ordinary Differential Equations. Acta Mathematica Sinica, English Series, 23(12): 2205-2212
[16] Radenovic, S., Rhoades, B.E. (2009) Fixed point theorem for two non-self mappings in cone metric spaces. Computers & Mathematics with Applications, vol. 57, br. 10, str. 1701-1707
[17] Ran, A.C.M., Reurings, M.C.B. (2004) A fixed point theorem in partially ordered sets and some application to matrix equations. Proceedings of the American Mathematical Society, 132(05): 1435-1444
[18] Rezapour, S., Hamlbarani, R. (2008) Some notes on the paper ‘Cone metric spaces and fixed point theorems of contractive mappings’. Journal of Mathematical Analysis and Applications, 345(2): 719
[19] Rudin, W. (1991) Functional Analysis. u: International Series in Pure and Applied Mathematics, New York: McGraw-Hill
[20] Samet, B., Turinici, M. (2012) Fixed point theorems on a metric space endowed with an arbitrary binary relation and applications. Commun. Math. Anal, 13, 82-97
[21] Turinici, M. (2011) Ran-Reurings fixed point results in ordered metric space. Libertas Math, 31, 49-55
[22] Turinici, M. (2012) Nieto-Lopez theorems in ordered metric space. Math. Student, 81, 219-229
[23] Turinici, M. (2012) Linear contractions in product ordered metric spaces. Annali dell’universita’ di ferrara, 59(1): 187-198
[24] Xu, S., Radenović, S. (2014) Fixed point theorems of generalized Lipschitz mappings on cone metric spaces over Banach algebras without assumption of normality. Fixed Point Theory and Applications, 2014(1): 102