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A generalized upper and lower solutions method and multiplicity results for nonlinear first-order ordinary differential equations. (English) Zbl 0674.34009
Summary: We introduce a generalized upper and lower solutions method for the solvability of first-order ordinary differential equations \(u'(t)=f(t,u(t))\), \(u(0)=u(1)\) in order to cover the case when the function f satisfies Carathéodory conditions. This method is then applied to obtain multiplicity results when the nonlinearity f interacts with the real eigenvalue of the linearized problem. Our proofs are based on differential inequalities and classical Leray-Schauder degree.

MSC:
34A34 Nonlinear ordinary differential equations and systems, general theory
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[1] Ambrosetti, A.; Prodi, G., On the inversion of some differentiable mappings with singularities between Banach spaces, Ann. mat. pura appl., 93, 231-246, (1972) · Zbl 0288.35020
[2] Brezis, H., Analyse fonctionnelle, théorie et applications, (1983), Masson Paris · Zbl 0511.46001
[3] Cafagna, V.; Donati, F., Un résultat global de multiplicité pour un problème différentiel non linéaire du premier ordre, C. R. acad. sci. Paris Sér. I math., 300, No. 15, 523-526, (1985) · Zbl 0582.34009
[4] Fabry, C.; Habets, P., Upper and lower solutions for second-order boundary value problems with non-linear boundary conditions, Nonlinear anal. TMA, 10, 985-1007, (1986) · Zbl 0612.34015
[5] Fabry, C.; Mawhin, J.; Nkashama, M.N., A multiplicity result for periodic solutions of forced nonlinear second order ordinary differential equations, Bull. London math. soc., 18, 173-180, (1986) · Zbl 0586.34038
[6] Gaines, R.; Mawhin, J., Coincidence degree and nonlinear differential equations, () · Zbl 0339.47031
[7] Gossez, J.P., Some nonlinear differential equations with resonance at the first eigenvalue, (), 355-389 · Zbl 0438.35058
[8] Lakshmikantham, V.; Leela, S., Existence and monotone method for periodic solutions of first-order differential equations, J. math. anal. appl., 91, 237-243, (1983) · Zbl 0525.34031
[9] Leela, S., Monotone technique for periodic solutions of differential equations, J. math. phys. sci., 18, 73-82, (1984) · Zbl 0557.34037
[10] Lepin, L.A., Concept of lower and upper functions, Differential equations, 16, 1133-1139, (1980) · Zbl 0512.34003
[11] Mawhin, J., Topological degree methods in nonlinear boundary value problems, () · Zbl 0414.34025
[12] Mawhin, J., First order ordinary differential equations with several periodic solutions, J. appl. math. physics, 38, 257-265, (1987) · Zbl 0644.34035
[13] Rouche, N.; Mawhin, J., Ordinary differential equations: stability and periodic solutions, (1980), Pitman Boston · Zbl 0433.34001
[14] Sadyrbaev, F.Zh., Lyapunov functions and the solvability of the first boundary-value problem for ordinary second-order differential equations, Differential equations, 16, 387-391, (1980) · Zbl 0475.34012
[15] Vatsala, A.S., On the existence of periodic quasi-solutions for first order systems, Nonlinear anal. TMA, 7, 1283-1289, (1983) · Zbl 0526.34029
[16] Walter, W., Differential and integral inequalities, (1970), Springer-Verlag Berlin
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