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The Lazer-Solimini equation with state-dependent delay. (English) Zbl 1252.34077

Consider the differential-delay equations \[ \begin{aligned} x''+ g[x](t) &= p(t),\\ x''- g[x](t) &= -p(t),\end{aligned}\tag{\(*\)} \] where \(g[x](t):= g(x(t-\tau(t, x(t))))\), \(p: \mathbb{R}\to\mathbb{R}\) is continuous and \(T\)-periodic, \(\tau: \mathbb{R}\times \mathbb{R}_+\to \mathbb{R}_+\) is continuous and \(T\)-periodic in the first variable, \(g: \mathbb{R}_+\to \mathbb{R}_+\) is continuous and satisfies \[ \lim_{x\to+\infty} g(x)= 0,\quad\lim_{x\to 0^+} g(x)=+\infty. \] The authors establish the existence of \(T\)-periodic solutions of \((*)\). The proof relies on a combination of Leray-Schauder degree and a priori bounds.

MSC:

34K13 Periodic solutions to functional-differential equations
47N20 Applications of operator theory to differential and integral equations
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References:

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