Some Fixed Point Theorems of Leggett–Williams Type for Multivalued Mappings and Applications to Second-Order Differential Inclusions
DOI:
https://doi.org/10.61383/ejam.202532104Keywords:
fixed point index, fixed point theorem of Leggett--Williams type, multivalued mapping, second-order differential inclusionsAbstract
This paper examines the application of fixed point index theory to set-valued mappings in ordered Banach spaces, with a specific focus on Leggett-Williams type fixed point theorems. The study provides novel conditions for the existence of one and multiple fixed points under a range of assumptions. Additionally, it applies these results to prove the existence of positive solutions for the second-order differential inclusion problem \( - x''(t) \in f\left( {t,x(t)} \right),t \in [0,1]\) with boundary conditions \(x(0)=0=x'(1)\). The obtained results contribute to expanding the scope of application of fixed point theory, providing new techniques for solving boundary value problems in different cases.References
[1] A. Lasota, Z. Opial, An application of the Kakutani-Ky Fan theorem in the theory of ordinary differential equations, Bull. Pol. Acad. Sci. Math. 13 (1965), 781–786.
[2] A. Myshkis, B. Gelman, V. Obukhovskii and Y. Borisovich, Introduction to the Theory of Multivalued Maps and Differential Inclusions, Moscow: LIBROKOM, 2011.
[3] D. Anderson, J. Henderson, R. Avery, Functional expansion-compression fixed point theorem of Leggett-Williams type, Electron. J. Differential Equations 2010 (2010), no. 63, 1–9.
[4] D. Anderson, J. Henderson, R. Avery, Some Fixed Point Theorems of Leggett-Williams Type, Rocky Mountain J. Math. 41 (2011), no. 2, 371–386. DOI: https://doi.org/10.1216/RMJ-2011-41-2-371
[5] D. Guo, V. Lakshmikantham, Nonlinear Problems in Abstract Cone, Academic Press, San Diego, 1988.
[6] D. O'Regan, M. Zima, Leggett-Williams norm-type fixed point theorems for multivalued mappings, Appl. Math. Comput. 187 (2017), 1238–1249. DOI: https://doi.org/10.1016/j.amc.2006.09.035
[7] J. Jahn, Truong X. D. H., New order relations in set optimization, J. Optim. Theory Appl. 148 (2011), no. 2, 209–236. DOI: https://doi.org/10.1007/s10957-010-9752-8
[8] K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, 1985. DOI: https://doi.org/10.1007/978-3-662-00547-7
[9] K. Deimling, Multivalued Differential Equations, Walter de Gruyter, Berlin–New York, 1992. DOI: https://doi.org/10.1515/9783110874228
[10] K. Mebarki, S. Benslimane, S. Djebali, On the fixed point for sums of operators, Fixed Point Theory 23 (2022), no. 1, 143–162. DOI: https://doi.org/10.24193/fpt-ro.2022.1.09
[11] K. Mebarki, S. Benslimane, S. Georgiev, Expansion-compression fixed point theorem of Leggett-Williams type for the sum of two operators and applications for some classes of BVPs, Stud. Univ. Babes-Bolyai Math. 68 (2023), no. 3, 631–647. DOI: https://doi.org/10.24193/subbmath.2023.3.12
[12] K. Mebarki, S. Georgiev, Leggett-Williams fixed point theorem type for sums of two operators and application in PDEs, J. Difference Equ. Appl. 13 (2021), no. 3, 321–344. DOI: https://doi.org/10.7153/dea-2021-13-18
[13] N. B. Huy, T. T. Binh, V. V. Tri, The monotone minoraut method and eigenvalue problem for multivalued operators in cones, Fixed Point Theory 19 (2018), no. 1, 275–286. DOI: https://doi.org/10.24193/fpt-ro.2018.1.22
[14] P. Fitzpatrick, W. Petryshyn, Fixed point theorems and the fixed point index for multivalued mappings in cones, J. London Math. Soc. 12 (1975), no. 2, 75–85. DOI: https://doi.org/10.1112/jlms/s2-12.1.75
Downloads
Published
Issue
Section
License
Copyright (c) 2025 Nguyen Dang Quang
This work is licensed under a Creative Commons Attribution 4.0 International License.
