Existence of periodic solutions for quasilinear differential equations
The paper deals with the nonlinear differential equation with \(p\)-Laplacian \[ ( \phi_{p}( u^{\prime}) ) ^{\prime}+f(t,u,u^{\prime })=0, \] where \(\phi_{p}(u)=| u| ^{p-2}u,\) \(p>1\) and \(f:[0,T]\times \mathbb{R}^{2}\to\mathbb{R}\) is a continuous, \(T\)-periodic function in \(t\) for some \(T>0.\) Using a fixed-point theorem due to \textit{T. Ding} [Chin. Ann. Math. 2, 281--300 (1981; Zbl 0484.34025)], the author obtains new sufficient conditions for the existence of \(T\)-periodic solutions. The paper concludes with an illustrative example. It should be noted that during the last decade there has been an increasing interest in the study of differential equations with \(p\)-Laplacian, and many papers on the subject have been published. Unfortunately, the bibliography contains only three items with the most recent paper dated 1992.
scientific article; zbMATH DE number 2081731
scientific article