Schaefer type theorem and periodic solutions of evolution equations (Q819703)

The authors introduce the concept of a separate contraction mapping which is a particular case of the notion of weak contraction introduced by \textit{J. Dugundji} and \textit{A. Granas} [Bull. Greek Math. Soc. 19, 141--151 (1978; Zbl 0417.54010)], and hence their fixed point result for separate contraction is a corollary of Dugundji-Granas' result. Then, they obtain a fixed point result and a generalization of the Leray-Schauder alternative for compact and completely continuous perturbations of separate contraction, respectively. Finally, they apply the Leray-Schauder alternative to obtain \(T\)-periodic solutions to the following problem: \[ \frac{\partial}{\partial t}\left[u-k\int_{t-r}^t e^{-c(t-s)}u(s,x)\,ds\right] =\frac{\partial^2u}{\partial x} + au- bu^3 + f(t,x),\quad 0\leq x\leq 1, \] \[ u(t,0)=u(t,1)=0 \] for \(k=0\) or \(1\). The problem is converted in a fixed point problem of the form \(u=Lu\) for an appropriate completely continuous operator \(L\).