Existence results for forced nonlinear periodic BVPs at resonance. (English) Zbl 0589.34005
Results concerning the existence of p-periodic solutions for systems of nonlinear differential equations at resonance, of the kind \(x''+Dx+Ag(t,x)=h(t),\) are shown in the paper. In the equation D and A are \(m\times m\) constant matrices, D being diagonal, h is a p-periodic forcing term and g is a vector field, not necessarily bounded. In particular, a classical theorem by Lazer and Leach is extended to the aforementioned systems, under the most general hypotheses. The proofs are based on the method of the topological degree (Mawhin’s continuation theorem).
Reviewer: V.C.Boffi
MSC:
| 34A12 | Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations |
Keywords:
second order differential equation; p-periodic solutions; resonance; method of the topological degree; Mawhin’s continuation theoremReferences:
| [1] | Amann, H., Existence theorems for equations of Hammerstein type, Appl. Anal., 2, 385-397 (1973) · Zbl 0244.47047 |
| [2] | Amann, H.; Mancini, G., Some applications of monotone operator theory to resonance problems, J. Nonlinear Anal., T.M.A., 3, 815-830 (1979) · Zbl 0418.47028 |
| [3] | Amaral, L.; Pera, M. P., A note on periodic solutions at resonance, Boll. Un. Mat. Ital. (Analisi Funzionale e Applicazioni), 18-C, 107-117 (1981) · Zbl 0472.34028 |
| [4] | Ambrosetti, A.; Mancini, G., Existence and multiplicity results for nonlinear elliptic problems with linear part at resonance. The case of the simple eigenvalue, J. Differential Equations, 28, 220-245 (1978) · Zbl 0393.35032 |
| [5] | Ambrosetti, A.; Mancini, G., Theorems of existence and multiplicity for nonlinear elliptic problems with noninvertible linear part, Ann. Scuola Norm. Sup. Pisa (Rend. Cl. Sci.), 5, 15-28 (1978) · Zbl 0375.35024 |
| [6] | Brézis, H.; Nirenberg, L., Characterizations of the ranges of some nonlinear operators and applications to boundary value problems, Ann. Scuola Norm. Sup. Pisa (Rend. Cl. Sci.), 5, 225-326 (1978) · Zbl 0386.47035 |
| [7] | Cesari, L., Functional analysis and periodic solutions of nonlinear differential equations, Contribution to differential Equations, 149-187 (1963), New York: J. Wiley & Sons, New York · Zbl 0132.07101 |
| [8] | Cesari, L., Existence theorems across a point of resonance, Bull. Amer. Mat. Soc., 82, 903-906 (1976) · Zbl 0341.47040 |
| [9] | Cesari, L.; Cesari, L.; Kannan, R.; Schuur, J. D., Functional analysis, nonlinear differential equations, and the alternative method, Nonlinear Functional Analysis and Differential Equations, 1-197 (1976), New York: Dekker, New York · Zbl 0343.47038 |
| [10] | Cesari, L.; Kannan, R., An abstract existence theorem at resonance, Proc. Amer. Math. Soc., 63, 221-225 (1977) · Zbl 0361.47021 |
| [11] | Cesari, L., Nonlinear oscillations across a point of resonance for nonselfadjoint systems, J. Differential Equations, 28, 43-59 (1978) · Zbl 0395.34034 |
| [12] | Cesari, L.; Kannan, R., Existence of solutions of a nonlinear differential equation, Proc. Amer. Math. Soc., 88, 605-613 (1983) · Zbl 0529.34005 |
| [13] | Ding, T. R., Unbounded perturbations of forced harmonic oscillations at resonance, Proc. Amer. Mat. Soc., 88, 59-66 (1983) · Zbl 0538.34028 |
| [14] | Fabry, C.; Franchetti, C., Nonlinear equations with growth restrictions on the nonlinear term, J. Differential Equations, 20, 283-291 (1976) · Zbl 0326.47060 |
| [15] | Fučik, S., Remarks on some nonlinear boundary value problems, Comm. Math. Univ. Carolin., 17, 721-730 (1976) · Zbl 0353.34023 |
| [16] | Fučik, S.; Hess, P., Nonlinear perturbations of linear operators having nullspaee with strong unique continuation property, J. Nonlinear Anal., TMA, 3, 271-277 (1979) · Zbl 0426.47042 |
| [17] | Fučik, S., Solvability of nonlinear equations and boundary value problems (1980), Boston: Reidel, Boston · Zbl 0453.47035 |
| [18] | Gaines, R. E.; Mawhin, J., Coincidence degree and nonlinear differential equations, Lecture Notes in Math. (1977), Berlin and New York: Springer-Verlag, Berlin and New York · Zbl 0326.34021 |
| [19] | Gupta, C. P., On functional equations of Fredholm and Hammerstein type with applications to existence of periodic solutions of certain ordinary differential equations, J. Integral Equations, 3, 21-41 (1981) · Zbl 0457.34040 |
| [20] | Gupta, C. P., Perturbations of second order linear elliptic problems by unbounded nonlinearities, J. Nonlinear Anal., TMA, 6, 919-933 (1982) · Zbl 0509.35035 |
| [21] | Hale, J. K., Ordinary differential equations (1969), New York: Wiley-Interscience, New York · Zbl 0186.40901 |
| [22] | Hess, P., On nonlinear equations of Hammerstein type in Banach spaces, Proc. Amer. Math. Soc., 31, 308-312 (1971) · Zbl 0229.47041 |
| [23] | Invernizzi, S.; Zanolin, F., On the existence and uniqueness of periodic solutions of differential delay equations, Math. Z., 163, 25-37 (1978) · Zbl 0369.34035 |
| [24] | Kannan, R., Existence of periodic solutions of nonlinear differential equations, Trans. Amer. Math. Soc., 217, 225-236 (1976) · Zbl 0338.34035 |
| [25] | Kannan, R.; Locker, J., Nonlinear boundary value problems and operators TT^*, J. Differential Equations, 28, 60-103 (1978) · Zbl 0375.34016 |
| [26] | Lazer, A. C.; Sánchez, D. A., On periodically perturbed conservative systems, Michigan Math. J., 16, 193-200 (1969) · Zbl 0187.34501 |
| [27] | Lazer, A. C.; Leach, D. E., Bounded perturbations of forced harmonic oscillations at resonance, Ann. Mat. Pura App., 82, 49-68 (1969) · Zbl 0194.12003 |
| [28] | Lin, S. S., Some results for semilinear differential equations at resonance, J. Math. Anal. Appl., 93, 574-592 (1983) · Zbl 0531.47039 |
| [29] | Mawhin, J., Degré topologique et solutions périodiques des sistèmes différentiels non linéaires, Bull. Soc. Roy. Sci. Liège, 38, 308-398 (1969) · Zbl 0186.41704 |
| [30] | Mawhin, J., Periodic solutions of strongly nonlinear differential systems, Proc. Fifth Intern. Conf. Nonlinear Oscill., 380-400 (1970), Kiev: Izd. Inst. Mat. Akad. Nauk. Ukr. S.S.R., Kiev · Zbl 0243.34075 |
| [31] | Mawhin, J., Periodic solutions of nonlinear functional differential equations, J. Differential Equations, 10, 240-261 (1971) · Zbl 0223.34055 |
| [32] | Mawhin, J., The solvability of some operator equations with a quasi-bounded nonlinearity in normed spaces, J. Math. Anal. Appl., 45, 455-467 (1974) · Zbl 0307.47060 |
| [33] | Mawhin, J., Contractive mappings and periodically perturbed conservative systems, Arch. Math. (Brno), 12, 67-74 (1976) · Zbl 0353.47034 |
| [34] | J.Mawhin,Landesman-Lazer’s type problems for nonlinear equations, Conf. Sem. Mat. Univ. Bari,147 (1977). · Zbl 0436.47050 |
| [35] | Mawhin, J.; Cesari, L.; Hale, J. K.; Lasalle, J. P., Topology and nonlinear boundary value problems, Dynamical Systems: An International Symposium, 51-82 (1976), New York: Academic Press, New York · Zbl 0334.47040 |
| [36] | Mawhin, J., Topological degree methods in nonlinear boundary value problems, Reg. Conf. Ser. in Math., CBMS, n.40 (1979), Providence, R.I.: Amer. Math. Soc., Providence, R.I. · Zbl 0414.34025 |
| [37] | J.Mawhin,Compacité, monotonie et convexité dans étude de problèmes aux limites semilinéaires, Séminaire d’Analyse Moderne,19, Univ. de Sherbrooke, 1981. · Zbl 0497.47033 |
| [38] | Mawhin, J.; Ward, J. R. Jr., Nonuniform nonresonance conditions at the two first eigenvalues for periodic solutions of forced Liénard and Duffing equations, Rocky Mountain J. Math., 12, 643-654 (1982) · Zbl 0536.34022 |
| [39] | Omari, P.; Zanolin, P., On forced nonlinear oscillations in n-th order differential systems with geometric conditions, J. Nonlinear Anal., TMA, 8, 723-748 (1984) · Zbl 0542.34037 |
| [40] | Omari, P.; Zanolin, F., Periodic solutions of Liénard equations, Rend. Sem. Mat. Univ. Padova, 72, 203-230 (1984) · Zbl 0584.34045 |
| [41] | Reissig, R., Schwingungssatze für die verallgemeinerte Liénardsche Differentialgleichung, Abh. Math. Sem. Univ. Hamburg, 44, 45-51 (1975) · Zbl 0323.34033 |
| [42] | Reissig, R., Contraction mappings and periodically perturbed nonconservative systems, Atti Accad. Naz. Lincei (Rend. Cl. Sci.), 58, 696-702 (1975) · Zbl 0344.34033 |
| [43] | R. T.Rockafellar,Convex Analysis, Princeton Univ. Press, 1970. · Zbl 0193.18401 |
| [44] | Taylor, A. E., Introduction of Functional Analysis (1963), New York: Wiley, New York |
| [45] | Ward, J. R. Jr., Asymptotic conditions for periodic solutions of ordinary differential equations, Proc. Amer. Math. Soc., 81, 415-420 (1981) · Zbl 0461.34029 |
| [46] | Willem, M., Periodic solutions of wave equations with jumping nonlinearities, J. Differential Equations, 36, 20-27 (1980) · Zbl 0429.35049 |
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