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Some remarks on semilinear problems at resonance where the nonlinearity depends only on the derivatives. (English) Zbl 0853.34021
The author considers the Neumann and the periodic boundary value problems for the second order nonlinear system $$u'' + g(t,u') = f(t)$$, $$t \in [a,b]$$, where $$f \in L^1 ([a,b], \mathbb{R}^n)$$ and $$g : [a,b] \times \mathbb{R}^n \to \mathbb{R}^n$$ is a Carathéodory function such that (1) $$g(t,v)/ |v |\to 0$$ uniformly a.e. in $$t \in [a,b]$$ whenever $$|v |\to \infty$$. In particular, he shows that for each $$\widetilde f \in L^1 ([a,b], \mathbb{R}^n)$$ such that $$(\int^b_a \widetilde fds)/(b - a) = 0$$ there exist $$k \in \mathbb{R}^n$$ such that the given problem with $$f = \widetilde f + k$$ possesses a family of solutions of the form $$u(t) + c$$, $$c \in \mathbb{R}^n$$. In the scalar case $$n = 1$$ analogous existence results are obtained if $$g$$ is continuous ((1) need not be satisfied) and $$f \in L^2 ([a,b], \mathbb{R})$$. Some uniqueness results are given, as well.
Reviewer: M.Tvrdý (Praha)

##### MSC:
 34B15 Nonlinear boundary value problems for ordinary differential equations 34C25 Periodic solutions to ordinary differential equations
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##### References:
 [1] Canada A., Drabek P.: On semilinear problems with nonlinearities depending only on derivatives. Západočeská Univerzita Plzeň, Preprint vědeckých prací 50 (), 1994. [2] Fučík S.: Solvability of Nonlinear Equations and Boundary Value Problems. Reidel, Dordrecht, 1980. [3] Rouche N., Mawhin J.: Ordinary Differential Equations. Stability and Periodic Solutions. Pitman, Boston, 1980. · Zbl 0433.34001 [4] Zeidler E.: Nonlinear functional Analysis and its Applications, vol. I. Springer, New York, 1986. · Zbl 0583.47050
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