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Periodic solutions for some nonlinear differential equations of neutral type. (English) Zbl 0735.34066
The main results of the paper deal with the existence of periodic solutions of the problem $x'(t)+ax'(t-\tau)=f(t,x(t))$, $x$ $2\pi$- periodic, where $f:\bbfR\times\bbfR\to\bbfR$ is Carathéodory, $f(\cdot,y)$ is $2\pi$-periodic for $x$, $a\in\bbfR$ and $\tau\in(0,2\pi)$. All existence theorems are considered corresponding to one from the following two cases $\vert a\vert<1$ and $\vert a\vert=1$.

34K40Neutral functional-differential equations
34C25Periodic solutions of ODE
34K99Functional-differential equations
Full Text: DOI
[1] Brezis, H.; Nirenberg, L.: Forced vibrations for a nonlinear wave equation. Communs pure appl. Math. 31, 1-30 (1978) · Zbl 0378.35040
[2] Coron, J. M.: Periodic solutions of a nonlinear wave equation without assumption of monotonicity. Math. annln 262, 273-285 (1983) · Zbl 0489.35061
[3] Fonda, A.; Habets, P.: Periodic solutions of asymptotically positive homogeneous differential equations. J. diff. Eqns 81, 68-97 (1989) · Zbl 0692.34041
[4] Gaines, R. E.; Mawhin, J.: Coincidence degree and nonlinear differential equations. Lecture notes in mathematics 568 (1977) · Zbl 0339.47031
[5] Hale, J.: Theory of functional differential equations. (1977) · Zbl 0352.34001
[6] Hale, J. K.; Mawhin, J.: Coincidence degree and periodic solutions of neutral equations. J. diff. Eqns 15, 295-307 (1974) · Zbl 0274.34070
[7] Hetzer, G.: A degree continuation theorem for a class of compactly perturbed differentiable Fredholm maps of index zero. Lecture notes in mathematics 730 (1979) · Zbl 0426.34057
[8] Kannan, R.: Nonlinear perturbations of linear problems with infinite dimensional kernel. Recent advances in differential equations (1981) · Zbl 0599.47098
[9] Mawhin, J.: Une généralisation du théoreme de J.A. Marlin. Int. J. Non-linear mech. 5, 335-339 (1970) · Zbl 0202.09501
[10] Mawhin, J.: Periodic solutions of nonlinear functional differential equations. J. diff. Eqns 10, 240-261 (1972) · Zbl 0223.34055
[11] Mawhin, J.: Perturbations non-linéaires d’opérateurs linéaires à noyeau de dimension infinie. Recent contr. Nonlinear partial differential equations 50 (1981)
[12] Mawhin, J.; Ward, J. R.: Nonresonance and existence for nonlinear elliptic boundary value problems. Nonlinear analysis 6, 677-684 (1981) · Zbl 0475.35047
[13] Mawhin, J.; Willem, M.: Operators of monotone type and alternative problems with infinite dimensional kernels. Recent advances in differential equations (1981) · Zbl 0602.47043
[14] Metzen, G.: Existence of periodic solutions of second order differential equations with delay. Proc. am. Math. soc. 3, 765-772 (1988) · Zbl 0671.34059
[15] Sadovskii, B. N.: Application of topological methods in the theory of periodic solutions of nonlinear differential-operator equations of neutral type. Soviet math. Dokl. 12, 1543-1547 (1971) · Zbl 0238.47044
[16] SCHMITT K., Fixed point and coincidence theorems with applications to nonlinear differential and integral equations. Sém. Math. Appl. et Mécan. Univ. Cathol. Louvain 97.
[17] Serra, E.: Periodic solutions for some nonlinear retarded functional differential equations. Ph.d. thesis (1989)