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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.

MSC:

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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