## Upper and lower solutions and multiplicity results.(English)Zbl 0961.34004

Using the topological degree theory, the author obtains some multiplicity results for the second-order differential equation $$x''=f(t,x,x')$$, with both linear (periodic and Neumann) and nonlinear boundary conditions. Here, $$f:[a,b]\times \mathbb{R}^2 \to \mathbb{R}$$ is continuous and satisfies certain growth conditions. The results extend those from a previous paper by the author, established for $$f$$ bounded [J. Math. Anal. Appl. 234, No. 1, 311-327 (1999)].
Reviewer: Eduardo Liz (Vigo)

### MSC:

 34B15 Nonlinear boundary value problems for ordinary differential equations
Full Text:

### References:

 [1] Avery, R.I, Existence of multiple positive solutions to a conjugate boundary problem, MSR hot-line, 2, 1-6, (1998) · Zbl 0960.34503 [2] De Coster, C; Habets, P, Lower and upper solutions in the theory of ODE boundary value problems: classical and recent results, Nonlinear analysis and boundary value problems for ordinary differential equations, CISM courses and lectures, 371, (1996), Springer-Verlag Vienna, p. 1-78 · Zbl 0889.34018 [3] Fabry, C; Mawhin, J; Nkashama, M.N, A multiplicity result for periodic solutions of forced nonlinear boundary value problem, Bull. London math. soc., 18, 173-186, (1986) · Zbl 0586.34038 [4] Gaines, R.E; Mawhin, J.L, Coincidence degree and nonlinear differential equations, Lecture notes in mathematics, 568, (1977), Springer-Verlag Berlin [5] Gossez, J.P; Omari, P, Periodic solutions of a second order ordinary differential equation: A necessary and sufficient condition for nonresonance, J. differential equations, 94, 67-82, (1991) · Zbl 0743.34045 [6] Habets, P; Omari, P, Existence and localization of solutions of second order elliptic problems using lower and upper solutions in the reversed order, Topol. methods nonlinear anal., 8, 25-56, (1996) · Zbl 0897.35030 [7] Kiguradze, I, Some singular boundary value problems for ordinary differential equations, (1975), Izdatel’stvo Tbilisskogo Universiteta, (University of Tbilisi Press) Tbilisi [8] Mawhin, J, Points fixes, points critiques et problèmes aux limités, Séminaire de mathematiques superieures, 92, (1985), Presses Univ. Montréal Montréal · Zbl 0561.34001 [9] Omari, P, Non-ordered lower and upper solutions and solvability of the periodic problem for the Liénard and the Rayleigh equations, Rend. instit. mat. univ. treiste, 20, 54-64, (1988) [10] Rachůnková, I, On a transmission problem, Acta univ. palack. olomuc. fac. rerum natur. math., 105, 45-59, (1992) · Zbl 0807.34018 [11] Rachůnková, I, Upper and lower solution and topological degree, J. math. anal. appl., 234, 311-327, (1998) · Zbl 1086.34017 [12] Rachůnková, I, On the existence of two solutions of the periodic problem for the ordinary second-order differential equation, Nonlinear anal. theory methods appl., 22, 1315-1322, (1994) · Zbl 0808.34023 [13] Rudolf, B, A multiplicity result for a periodic boundary value problem, Nonlinear anal. theory methods appl., 28, 137-144, (1997) · Zbl 0859.34016 [14] Thompson, H.B, Second order ordinary differential equations with fully nonlinear two point boundary conditions, Pacific J. math., 172, 255-277, (1996) · Zbl 0855.34024 [15] Thompson, H.B, Second order ordinary differential equations with fully nonlinear two point boundary conditions, II, Pacific J. math., 172, 279-297, (1996) · Zbl 0862.34015
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.