Some coincidence theorems in wedges, cones, and convex sets. (English) Zbl 0576.34018

Coincidence degree is used to study the solvability of a semilinear equation in a prescribed convex set. Generalizations of the Schauder fixed point theorem and compression theorem by Krasnosel’skij are given. The author applies some of the abstract results to establish existence of nonnegative solutions to some boundary value problems involving ordinary differential equations.


34B15 Nonlinear boundary value problems for ordinary differential equations
47J05 Equations involving nonlinear operators (general)
47J25 Iterative procedures involving nonlinear operators
Full Text: DOI


[1] Cooke, K; Kaplan, J, A periodicity threshold theorem for epidemics and population growth, Math. biosci., 31, 87-104, (1976) · Zbl 0341.92012
[2] Gaines, R.E; Mawhin, J, Coincidence degree and nonlinear differential equations, () · Zbl 0326.34020
[3] Gaines, R.E; Santanilla, J, A coincidence theorem in convex sets with applications to periodic solutions of ordinary differential equations, Rocky mountain J. math., 12, 669-678, (1982) · Zbl 0508.34030
[4] Gatica, J.A; Smith, H.L, Fixed point techniques in a cone with applications, J. math. anal. appl., 61, 58-71, (1977) · Zbl 0435.47060
[5] {\scG. B. Gustafson and K. Schmitt}, Methods of nonlinear analysis in the theory of differential equations, Lecture Notes, University of Utah. · Zbl 0227.34017
[6] Gustafson, G.B; Schmitt, K, Nonzero solutions of boundary value problems for second order ordinary and delay-differential equations, J. math. anal. appl., 12, 129-147, (1972) · Zbl 0227.34017
[7] Krasnosel’skii, M, Positive solutions of operator equations, (1964), Noordhoff Groningen · Zbl 0121.10604
[8] Leggett, R; Williams, L, A fixed point theorem with applications to an infectious disease model, J. math. anal. appl., 76, 91-97, (1980) · Zbl 0448.47044
[9] Mawhin, J, Topological degree methods in nonlinear boundary value problems, () · Zbl 0414.34025
[10] Mawhin, J; Schmitt, K, Rothe and altman type coincidence theorems and applications to differential equations, Nonlinear anal. theory methods appl., 1, 151-160, (1977) · Zbl 0347.34028
[11] Nussbaum, R, A periodicity threshold theorem for some nonlinear integral equations, SIAM J. math. anal., 9, 356-376, (1978) · Zbl 0385.45007
[12] Schmitt, K, Fixed point and coincidence theorems with applications to nonlinear differential and integral equations, ()
[13] Smith, H.L, On periodic solutions of delay-integral equations, () · Zbl 0354.92035
[14] Smith, H.L, On periodic solutions of a delay-integral equation modeling epidemics, J. math. biol., 4, 69-80, (1977) · Zbl 0354.92035
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.