Periodic solution for nonlinear systems with \(p\)-Laplacian-like operators. (English) Zbl 0910.34051

The authors study the existence of solutions to boundary value problems \[ \bigl( \varphi (u')\bigr)'= f(t,u,u'), \quad u(0)= u(T),\;u'(0)= u'(T), \tag{1} \] where the function \(\varphi: \mathbb{R}^N \to \mathbb{R}^N\) satisfies some monotonicity conditions which ensure that \(\varphi\) is an homeomorphism onto \(\mathbb{R}^N\). Let us note, that the class of nonlinear operators \((\varphi (u'))'\) contains some vector versions of \(p\)-Laplacian operators. The function \(f:I\times \mathbb{R}^N \times \mathbb{R}^N \to\mathbb{R}^N\) is assumed to be Carathéodory, \(I-[0,T]\).
The authors extend the continuation theorem by M. Mawhin [J. Differ. Equations 12, 610-636 (1972; Zbl 0244.47049)] onto the problem (1). This general existence theorem generalizes a result proved by Z. Guo [Differ. Integral Equ. 6, No. 3, 705-719 (1993; Zbl 0784.34018)] for nonlinear perturbations of the one-dimensional \(p\)-Laplacian. As a consequence of this generalized continuation theorem the authors present various existence theorems for quasilinear systems (1) with a nonlinearity \(f\) satisfying some one-sided growth conditions.
Further, using degree theory for compact vector fields which are invariant under the action of \(S^1\), see [Th. Bartsch and J. Mawhin, J. Differ. Equations 92, No. 1, 90-99 (1991; Zbl 0729.34064)], the authors prove a second generalization of the continuation theorem. In contrast to the first one, where they use homotopy to the averaged nonlinearity, in the proof of the second one they work with homotopy to an autonomous system. An application of the theorem and a lot of examples illustrating all theorems are given.


34C25 Periodic solutions to ordinary differential equations
Full Text: DOI


[1] Bartsch, T.; Mawhin, J., The Leray-Schauder degree of \(S^1\), J. Differential Equations, 92, 90-99 (1991) · Zbl 0729.34064
[2] Bobisud, L. E., Steady state turbulent flow with reaction, Rocky Mountain J. Math., 21, 993-1007 (1991) · Zbl 0760.34023
[3] Boccardo, L.; Drábek, P.; Giachetti, D.; Kučera, M., Generalization of Fredholm alternative for nonlinear differential operators, Nonlinear Anal., 10, 1083-1103 (1986) · Zbl 0623.34031
[4] Cañada, A.; Martinez-Amores, P., Solvability of some operator equations and periodic solutions of nonlinear functional differential equations, J. Differential Equations, 49, 415-429 (1983) · Zbl 0476.34061
[5] Cañada, A.; Martinez-Amores, P., Periodic solutions of nonlinear vector differential ordinary differential equations at resonance, Nonlinear Anal., 7, 747-761 (1983) · Zbl 0516.34041
[6] Cañada, A.; Ortega, R., Existence theorems for equations in normed spaces and boundary value problems for nonlinear vector ordinary differential equations, Proc. Royal Soc. Edinburgh Ser. A, 98, 1-11 (1984) · Zbl 0558.47046
[7] Capietto, A.; Mawhin, J.; Zanolin, F., Continuation theorems for periodic perturbations of autonomous systems, Trans. Amer. Math. Soc., 329, 41-72 (1992) · Zbl 0748.34025
[8] Dang, H.; Oppenheimer, S. F., Existence and uniqueness results for some nonlinear boundary value problems, J. Math. Anal. Appl., 198, 35-48 (1996) · Zbl 0855.34021
[9] De Coster, C., On pairs of positive solutions for the one-dimensional \(p\), Nonlinear Anal., 23, 669-681 (1994) · Zbl 0813.34021
[10] Deimling, K., Nonlinear Functional Analysis (1985), Springer-Verlag: Springer-Verlag Berlin/New York · Zbl 0559.47040
[11] Del Pino, M.; Elgueta, M.; Manásevich, R., A homotopic deformation along \(pu^puftuuuTp\), J. Differential Equations, 80, 1-13 (1989)
[12] Del Pino, M.; Manásevich, R.; Murua, A., Existence and multiplicity of solutions with prescribed period for a second order quasilinear o.d.e, Nonlinear Anal., 18, 79-92 (1992) · Zbl 0761.34032
[13] Drabek, P., Solvability of boundary value problems with homogeneous ordinary differential operators, Rend. Istit. Math. Univ. Trieste, 18, 105-124 (1996) · Zbl 0633.34015
[14] Fabry, C.; Fayyad, D., Periodic solutions of second order differential equations with a \(p\), Rend. Istit. Mat. Univ. Trieste, 24, 207-227 (1992) · Zbl 0824.34026
[15] Garcı́a-Huidobro, M.; Manásevich, R.; Zanolin, F., Strongly nonlinear second-order ODE’s with unilateral conditions, Differential Integral Equations, 6, 1057-1078 (1993) · Zbl 0785.34023
[16] Garcı́a-Huidobro, M.; Manásevich, R.; Zanolin, F., A Fredholm-like result for strongly nonlinear second order ODE’s, J. Differential Equations, 114, 132-167 (1994) · Zbl 0835.34028
[17] Garcı́a-Huidobro, M.; Manásevich, R.; Zanolin, F., On a pseudo Fučik spectrum for strongly nonlinear second order ODE’s and an existence result, J. Comput. Appl. Math., 52, 219-239 (1994) · Zbl 0811.34014
[18] Garcı́a-Huidobro, M.; Manásevich, R.; Zanolin, F., Strongly nonlinear second-order ODE’s with rapidly growing terms, J. Math. Anal. Appl., 202, 1-26 (1996) · Zbl 0991.34007
[19] Garcı́a-Huidobro, M.; Ubilla, P., Multiplicity of solutions for a class of nonlinear 2nd order equations, Nonlinear Anal., 28, 1509-1520 (1997) · Zbl 0874.34021
[20] Guo, Z., Boundary value problems of a class of quasilinear ordinary differential equations, Differential Integral Equations, 6, 705-719 (1993) · Zbl 0784.34018
[21] Huang, Y. X.; Metzen, G., The existence of solutions to a class of semilinear equations, Differential Integral Equations, 8, 429-452 (1995) · Zbl 0818.34013
[22] Manásevich, R.; Zanolin, F., Time-mappings and multiplicity of solutions for the one-dimensional \(p\), Nonlinear Anal., 21, 269-291 (1993) · Zbl 0792.34021
[23] Mawhin, J., Equivalence theorems for nonlinear operator equations and coincidence degree theory for some mappings in locally convex topological vector spaces, J. Differential Equations, 12, 610-636 (1972) · Zbl 0244.47049
[24] Mawhin, J., Topological Methods in Nonlinear Boundary Value Problems. Topological Methods in Nonlinear Boundary Value Problems, CBMS Regional Conf. Ser. in Math., 40 (1979), Amer. Math. Soc: Amer. Math. Soc Providence · Zbl 0414.34025
[25] Mawhin, J., Compacité, monotonie et convexité dans l’étude de problèmes aux limites semi-linéaires. Compacité, monotonie et convexité dans l’étude de problèmes aux limites semi-linéaires, Sémin. d’analyse moderne, 19 (1981), Universite de Sherbrooke · Zbl 0497.47033
[26] Mawhin, J., Topological degree and boundary value problems for nonlinear differential equations, Topological Methods for Ordinary Differential Equations. Topological Methods for Ordinary Differential Equations, Lecture Notes in Math., 1537 (1993), Springer-Verlag: Springer-Verlag Berlin/New York, p. 74-142 · Zbl 0798.34025
[27] Mawhin, J., Continuation theorems and periodic solutions of ordinary differential equations, (Granas, A.; Frigon, M., Topological Methods in Differential Equations and Inclusions. Topological Methods in Differential Equations and Inclusions, NATO ASI Series C 472 (1995), Kluwer Academic: Kluwer Academic Dordrecht/Norwell), 291-375 · Zbl 0834.34047
[28] O’Regan, D., Some general existence principles and results for \((φy qf tyyt\), SIAM J. of Math. Anal., 24, 648-668 (1993) · Zbl 0778.34013
[29] Rouche, N.; Mawhin, J., Ordinary Differential Equations. Stability and Periodic Solutions (1980), Pitman: Pitman London · Zbl 0433.34001
[30] Villari, G., Soluzioni periodiche di una classe di equazioni differenziali del terz’ordine quasi lineari, Ann. Mat. Pura Appl., 73, 103-110 (1966) · Zbl 0144.11401
[31] Wang, J.; Gao, W.; Lin, Z., Boundary value problems for general second order equations and similarity solutions to the Rayleigh problem, Tóhoku Math. J., 47, 327-344 (1995) · Zbl 0845.34038
[32] Ward, J., Asymptotic conditions for periodic solutions of ordinary differential equations, Proc. Amer. Math. Soc., 81, 415-420 (1981) · Zbl 0461.34029
[33] Zhang, M., Nonuniform nonresonance at the first eigenvalue of the \(p\), Nonlinear Anal., 29, 41-51 (1997) · Zbl 0876.35039
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.