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Manásevich, Raúl; Mawhin, Jean

Periodic solution for nonlinear systems with \(p\)-Laplacian-like operators. (English) Zbl 0910.34051

J. Differ. Equations 145, No. 2, 367-393 (1998).
The authors study the existence of solutions to boundary value problems \[ \bigl( \varphi (u')\bigr)'= f(t,u,u'), \quad u(0)= u(T),\;u'(0)= u'(T), \tag{1} \] where the function \(\varphi: \mathbb{R}^N \to \mathbb{R}^N\) satisfies some monotonicity conditions which ensure that \(\varphi\) is an homeomorphism onto \(\mathbb{R}^N\). Let us note, that the class of nonlinear operators \((\varphi (u'))'\) contains some vector versions of \(p\)-Laplacian operators. The function \(f:I\times \mathbb{R}^N \times \mathbb{R}^N \to\mathbb{R}^N\) is assumed to be Carathéodory, \(I-[0,T]\).
The authors extend the continuation theorem by M. Mawhin [J. Differ. Equations 12, 610-636 (1972; Zbl 0244.47049)] onto the problem (1). This general existence theorem generalizes a result proved by Z. Guo [Differ. Integral Equ. 6, No. 3, 705-719 (1993; Zbl 0784.34018)] for nonlinear perturbations of the one-dimensional \(p\)-Laplacian. As a consequence of this generalized continuation theorem the authors present various existence theorems for quasilinear systems (1) with a nonlinearity \(f\) satisfying some one-sided growth conditions.
Further, using degree theory for compact vector fields which are invariant under the action of \(S^1\), see [Th. Bartsch and J. Mawhin, J. Differ. Equations 92, No. 1, 90-99 (1991; Zbl 0729.34064)], the authors prove a second generalization of the continuation theorem. In contrast to the first one, where they use homotopy to the averaged nonlinearity, in the proof of the second one they work with homotopy to an autonomous system. An application of the theorem and a lot of examples illustrating all theorems are given.
Reviewer: I.Rachůnková (Olomouc)

Cited in 11 Reviews
Cited in 223 Documents

MSC:

34C25 Periodic solutions to ordinary differential equations

Keywords:

continuation theorem; \(p\)-Laplacian-like operators; periodic solutions; existence

Citations:

Zbl 0244.47049; Zbl 0784.34018; Zbl 0729.34064
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\textit{R. Manásevich} and \textit{J. Mawhin}, J. Differ. Equations 145, No. 2, 367--393 (1998; Zbl 0910.34051)
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References:

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