Opuscula Math. 46, no. 2 (2026), 219-234
https://doi.org/10.7494/OpMath.202602271
Opuscula Mathematica
On a fixed point theorem for operator systems and eigenvalue criteria for existence of positive solutions
Laura M. Fernández-Pardo
Jorge Rodríguez-López
Abstract. We provide an alternative approach, based on the Leray-Schauder fixed point index in cones, to a fixed point theorem for operator systems due to Precup. Our focus is on the case of operators whose components are entirely of compressive type. The abstract technique is applied to a system of second-order differential equations providing a coexistence positive solution by means of an eigenvalue type criterion.
Keywords: coexistence fixed point, fixed point index, positive solution, nonlinear systems.
Mathematics Subject Classification: 47H10, 47H11, 34B18, 34B16, 34C25.
- H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev. 18 (1976), 620-709. https://doi.org/10.1137/1018114
- L.M. Fernández-Pardo, J. Rodríguez-López, Component-wise Krasnosel'skii type fixed point theorem in product spaces and applications, Nonlinear Anal. Real World Appl. 88 (2026), 104506. https://doi.org/10.1016/j.nonrwa.2025.104506
- A. Granas, The Leray-Schauder index and the fixed point theory for arbitrary ANRs, Bulletin de la S.M. F. 100 (1972), 209-228. https://doi.org/10.24033/bsmf.1737
- D. Guo, V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, Boston, 1988. https://doi.org/10.1016/b978-0-12-293475-9.50004-2
- M.A. Krasnosel'skiĭ, Fixed points of cone-compressing or cone-extending operators, Soviet Math. Dokl. 135 (1960), 527-530. https://doi.org/10.1112/s0024609303002212
- M.A. Krasnosel'skiĭ, Positive Solutions of Operator Equations, Noordhoff, 1964. https://doi.org/10.1142/9789812819642_0002
- R. Precup, A vector version of Krasnosel'skii's fixed point theorem in cones and positive periodic solutions of nonlinear systems, J. Fixed Point Theory Appl. 2 (2007), 141-151. https://doi.org/10.1007/s11784-007-0027-4
- R. Precup, Componentwise compression-expansion conditions for systems of nonlinear operator equations and applications, Math. Models Eng. Biol. Med., AIP Conf. Proc., 1124, Amer. Inst. Phys., Melville, NY, 2009, 284-293. https://doi.org/10.1063/1.3142943
- R. Precup, J. Rodríguez-López, Multiplicity results for operator systems via fixed point index, Results Math. 74 (2019), Article no. 25. https://doi.org/10.1007/s00025-019-0955-5
- J. Rodríguez-López, A fixed point index approach to Krasnosel'skii-Precup fixed point theorem in cones and applications, Nonlinear Anal. 226 (2023), 113138. https://doi.org/10.1016/j.na.2022.113138
- F. Wang, J.Á. Cid, M. Zima, Amann's three solutions theorems for semilinear operator equation and applications, J. Fixed Point Theory Appl. 27 (2025), 103. https://doi.org/10.1007/s11784-025-01255-7
- J.R.L. Webb, Solutions of nonlinear equations in cones and positive linear operators, J. London Math. Soc. 82 (2010), 420-436. https://doi.org/10.1112/jlms/jdq037
- J.R.L. Webb, G. Infante, Non-local boundary value problems of arbitrary order, J. London Math. Soc. 79 (2009), 238-258. https://doi.org/10.1112/jlms/jdn066
- J.R.L. Webb, K.Q. Lan, Eigenvalue criteria for existence of multiple positive solutions of nonlinear boundary value problems of local and nonlocal type, Topol. Methods Nonlinear Anal. 27 (2006), 91-115.
- Laura M. Fernández-Pardo
https://orcid.org/0000-0001-8260-8965- Universidade de Santiago de Compostela, Departamento de Estatística, Análise Matemática e Optimización & CITMAga , 15782, Facultade de Matemáticas, Campus Vida, Santiago, Spain
- Jorge Rodríguez-López (corresponding author)
- https://orcid.org/0000-0002-8453-4397
- Universidade de Santiago de Compostela, Departamento de Estatística, Análise Matemática e Optimización & CITMAga , 15782, Facultade de Matemáticas, Campus Vida, Santiago, Spain
- Communicated by Marek Galewski.
- Received: 2025-09-24.
- Revised: 2026-02-26.
- Accepted: 2026-02-27.
- Published online: 2026-04-10.
