On periodic solutions of sublinear Duffing equations. (English) Zbl 0727.34030

The authors are concerned with the scalar Duffing equation \(x''+g(x)=p(t),\) where g: \({\mathbb{R}}\to {\mathbb{R}}\) is continuous and p: \({\mathbb{R}}\to {\mathbb{R}}\) is continuous and \(2\pi\)-periodic. They prove that the equation has at least one \(2\pi\)-periodic solution if the following assumptions are satisfied: (i) there exist two constants E, \(d>0\), such that \(g(x) sgn(x)\geq E>\sup \{| p(t)| :\;t\in {\mathbb{R}}\}\) for \(| x| \geq d\); \((ii)\quad \liminf_{x\to +\infty}g(x)/x=0;\) (iii) g is continuously differentiable on \([d,+\infty]\) and there is a constant \(M>0\) such that \(xg'(x)/g(x)\leq M\) for \(x>d\). The result is achieved by a continuation lemma based on coincidence degree. An example is presented which shows that this result is not contained in the results of the (about 30) articles quoted in the bibliography.
Reviewer: W.Müller (Berlin)


34C25 Periodic solutions to ordinary differential equations
47H11 Degree theory for nonlinear operators
Full Text: DOI


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