Topological degree methods for perturbations of operators generating compact \(C_0\) semigroups. (English) Zbl 1086.47030

Let \(E\) be a Banach space, \(A:D(A)\subseteq E\to E\) a closed linear operator such that \(-A\) generates a compact \(C_0\)-semigroup, \(M\subset E\) a neighborhood retract and \(F:M\to E\) a locally Lipschitz map. The present paper is devoted to the construction of a topological degree theory for maps of the form \(-A+F:M\cap D(A)\to E\). By using the introduced topological degree and an abstract result concerning the branching of fixed points, the author studies the bifurcation of periodic points of a parameterized boundary value problem of the form \[ \begin{gathered} \dot{u}=- {\mathcal A}(\lambda)u+{\mathcal F}(t,u,\lambda), \\ u(t)\in M,\; u(0)=u(T).\end{gathered} \] As applications, the author considers periodic problems for some classes of partial differential equations.


47H11 Degree theory for nonlinear operators
47J35 Nonlinear evolution equations
47J15 Abstract bifurcation theory involving nonlinear operators
37L05 General theory of infinite-dimensional dissipative dynamical systems, nonlinear semigroups, evolution equations
47N20 Applications of operator theory to differential and integral equations
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