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Existence of periodic solutions for \(p\)-Laplacian equation under the frame of Fučik spectrum. (English) Zbl 1219.34053

The paper deals with the second order ordinary differential equation with \(p\)-Laplacian \(\phi_p\)
\[ (\phi_p(x'))'+g(x)=p(t) \tag{1} \]
and establishes conditions for the existence of periodic solutions of (1) within the theory of Fučik spectrum \(S\) (\(T\)-periodic) for the equation
\[ (\phi_p(x'))'+\beta\phi_p(x^+)-\alpha \phi_p(x^-)=0. \]
In particular, the assumptions on the asymptotic behavior of the potential function \(G(x):=\int_0^xg(s)\,ds\)
\[ \lim_{x\to\infty}\frac{pG(x)}{|x|^p}=a,\quad \lim_{x\to-\infty}\frac{pG(x)}{|x|^p}=b, \]
for \((a,b)\not\in S\) yield \(T\)-periodic solutions of (1). The proofs are based on the topological degree arguments and some limit processes.

MSC:

34C25 Periodic solutions to ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
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References:

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