An extension of Krasnoselskii’s cone fixed point theorem for a sum of two operators and applications to nonlinear boundary value problems
DOI:
https://doi.org/10.24193/subbmath.2023.2.16Keywords:
Fixed point, Banach space, cone, expansive mapping, sum of operators, nonlinear boundary value problem, coincidence problems.Abstract
The purpose of this work is to establish a new generalized form of the Krasnoselskii type compression-expansion fixed point theorem for a sum of an expansive operator and a completely continuous one. Applications to three non- linear boundary value problems associated to second order differential equations of coincidence type are included to illustrate the main results.
Mathematics Subject Classification (2010): 47H10, 54H25, 34B18.
Received 21 March 2020; Accepted 10 April 2020.
References
Anderson, D.R., Avery, R.I., Fixed point theorem of cone expansion and compression of functional type, J. Difference Equ. Appl., 8(2002), no. 11, 1073-1083.
Avery, R.I., Anderson, D.R., Krueger, R.J., An extension of the fixed point theorem of cone expansion and compression of functional type, Comm. Appl. Nonlinear Anal., 13(2006), no. 1, 15-26.
Benzenati, L., Mebarki, K., Multiple positive fixed points for the sum of expansive mappings and k-set contractions, Math. Methods Appl. Sci., 42(2019), no. 13, 4412-4426.
Benzenati, L., Mebarki, K., Precup, R., A vector version of the fixed-point theorem of cone compression and expansion for a sum of two operators, Nonlinear Studies, 27(2020), no. 3, 563-575.
Deimling, K., Nonlinear Functional Analysis, Springer-Verlag, Berlin, Heidelberg, 1985.
Djebali, S., Mebarki, K., Fixed point index theory for perturbation of expansive mappings by k-set contractions, Topol. Methods Nonlinear Anal., 54(2019), no. 2, 613-640.
Djebali, S., Mebarki, K., Multiple positive solutions for singular BVPs on the positive half-line, Comput. Math. Appl., 55(2008), 2940-2952.
Djebali, S., Moussaoui, T., A class of second order bvps on infinite intervals, Electron. J. Qual. Theory Differ. Equ., 4(2006), 1-19.
Feng, M., Zhang, X., Ge, W., Positive fixed point of strict set contraction operators on ordered Banach spaces and applications, Abstr. Appl. Anal., (2010), vol. Article ID 439137, 13 pages, doi:10.1155/2010/439137.
Guo, D., Lakshmikantham, V., Nonlinear Problems in Abstract Cones, Notes and Re- ports in Mathematics in Science and Engineering, vol. 5, Academic Press, Boston, Mass, USA, 1988.
Krasnoselskii, M.A., Positive Solutions of Operator Equations, Noordhoff, Groningen, 1964.
Kwong, M.K., On Krasnoselskii’s cone fixed point theorem, Fixed Point Theory Appl., 2008(2008), Article ID 164537, 18 pp.
Kwong, M.K., The topological nature of Krasnoselskii’s cone fixed point theorem, Nonlinear Anal., 69(2008), 891-897.
O’Regan, D., Precup, R., Compression-expansion fixed point theorem in two norms and applications, J. Math. Anal. Appl., 309(2005), no. 2, 383-391.
Precup, R., A vector version of Krasnoselskii’s fixed point theorem in cones and positive periodic solutions of nonlinear systems, J. Fixed Point Theory Appl., 2(2007), 141-151.
Precup, R., Componentwise compression-expansion conditions for systems of nonlinear operator equations and applications, in Mathematical Models in Engineering, Biology, and Medicine, AIP Conference Proceedings 1124, Melville–New York, 2009, 284-293.
Zima, M., On a Certain Boundary Value Problem. Annales Societas Mathematicae Polonae, Series I: Commentationes Mathematicae XXIX, 1990, 331-340.
Zima, M., On positive solutions of boundary value problems on the half-line, J. Math. Anal. Appl., 259(2001), no. 1, 127-136.
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2023 Studia Universitatis Babeș-Bolyai Mathematica
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
