Periodic solutions for a Liénard type \(p\)-Laplacian differential equation
The authors investigate existence and uniqueness of a \(T\)-periodic solution of the Liénard type \(p\)-Laplacian differential equation: \[ (\varphi_p(x'(t)))'+f(x(t)) x'(t)+ g(t,x(t))=e(t)\,, \] where \(p>0\), \(\varphi:\mathbb R\rightarrow \mathbb R\) is given by \(\varphi_p(s)=|s|^{p-2}s\) for \(s\neq 0\) and \(\varphi(0)=0\), \(f:\mathbb R\rightarrow \mathbb R\) and \(g:\mathbb R^2\rightarrow \mathbb R\) are continuous functions, \(e:\mathbb R\rightarrow \mathbb R\) is a continuous periodic with period \(T>0\) function, \(g\) is \(T\)-periodic in the first argument. As the authors underline, recently this matter is studied mainly under the assumptions \((H_1)\): \(g(t,u)=\beta(t)g(u)\text{ or } g(t,u)=g(u)\;\text{ for all } t,u\in \mathbb R,\) but there are scarce results these conditions. Using topological degree theory the authors prove: Theorem. Let the following conditions hold: \((A_1)\): The functions \({\partial\over\partial t}g(t,x)\) and \({\partial\over\partial x}g(t,x)\) are continuous, and \({\partial\over\partial x}g(t,x)<0\) for all \(t,\,x\in \mathbb R\). Suppose that there exists a positive constant \(d\) such that \((A_2)\): \(x(g(t,x)-e(t))<0\) for all \(|x|>d\), \(t,\,x\in \mathbb R\). Then the considered equation has a unique \(T\)-periodic solution.
scientific article; zbMATH DE number 5518732