A continuation theorem for periodic boundary value problems with oscillatory nonlinearities
type: Item
instance of: scholarly article
title: A continuation theorem for periodic boundary value problems with oscillatory nonlinearities
zbMATH Open document ID: 0824.34025
DOI: 10.1007/BF01295308
author: Anna Capietto
author: Fabio Zanolin
author: Jean Mawhin
published in: NoDEA. Nonlinear Differential Equations and Applications
publication date: 1995-11-09T00:00:00Z
review text: The authors investigate whether the differential equation \[ x''+ F(t, x, x')= 0,\quad x(0)- x(T)= x'(0)- x'(T)= 0\tag{1} \] has a solution, where \(F: \mathbb{R}^ 3\to \mathbb{R}\) is continuous and \(T\)-periodic in the first variable. There is an extensive literature about this problem, and in this paper, a generalization of previous results is obtained. First, the authors consider a linear (unbounded) Fredholm operator \(L\) of index 0 from a Banach space \(X\) into a Banach space \(Z\) and an \(L\)-completely continuous operator \(N: X\times [0, 1]\to Z\). Then \(\Sigma_ \lambda:= \{x\in \text{dom } L\times [0, 1]: Lx= N(x,\lambda)\}\), \(\lambda\in [0, 1]\}\), is the set of solutions of \(Lx= N(x,\lambda)\), and it is assumed that \(\Sigma_ 0\neq \emptyset\). It is shown that also \(\Sigma_ 1\neq \emptyset\) if certain conditions are satisfied. In the application to (1) one replaces \(F(t,x,x')\) by \(f(t,x,x',\lambda)\), where \(f(t,x,x',0)= f_ 0(x,x')\) and \(f(t,x,x',1)= F(t,x,x')\). For the autonomous problem at \(\lambda=0\), a topological degree argument gives \(\Sigma_ 1\neq \emptyset\). As an application, the Duffing equation is considered, i.e., (2) \(F(t,x,x')= g(x)- p(t,x,x')\), where \(g\) and \(p\) are continuous functions, and \(p\) is additionally bounded. Setting \(G(x):= \int^ x_ 0 g(s) ds\), \(h(c)< 0\) for \(c> 0\) such that \(G(h(c))= G(c)\), and \(\tau_ g(c)= \sqrt 2 \int^ c_{h(c)} {du\over \sqrt{G(c)-G(u)}}\), the following theorem is proved: Assume \(\lim_{| x|\to +\infty} g(x)\text{sign}(x)= +\infty\) and \[ \forall c_ 1> 0\;\exists c_ 2> 0: AB> 0 \& | \sqrt{G(B)}- \sqrt{G(A)}|< c_ 1\Rightarrow | B- A|< c_ 2, \] and suppose that there is a sequence \(M_ n\to +\infty\) such that \(\lim_{n\to+\infty} {T\over \tau_ g(M_ n)}= \rho\in [0,+\infty]\), with \(\rho\) not a positive integer. Then (1), (2) has at least one solution. Finally, some special cases are considered, where the above conditions are replaced by stronger conditions which mig…
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 34C10
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 47H11
zbMATH DE Number: 762090
zbMATH Keywords: periodic solutions
zbMATH Keywords: Duffing equation
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01295308
OpenAlex ID: W1982104381
description: scientific article; zbMATH DE number 762090