Abstract
We study the periodic Dirichlet problem for a semilinear wave equation with discontinuous nonlinearity. First we establish a continuation theorem for a semilinear operator equation in a Hilbert space, where a key tool is the Berkovits-Tienari degree theory for a class of perturbations of monotone type of a densely defined closed linear operator. Applying the continuation theorem, we prove the main results on the solvability of the given semilinear wave equation, with the aid of spectral theory for densely defined closed linear operators.
Highlights
The study of nonlinear wave equations has been developed in various ways of approach by many researchers; for instance, by Brézis and Nirenberg [, ], Rabinowitz [ ], Berkovits and Mustonen [ – ]
To find periodic solutions of a nonlinear wave equation, Mawhin and Willem [ ] established a Leray-Schauder type continuation theorem for abstract equations involving some perturbations of monotone type of a linear operator in a Hilbert space, where the Galerkin approximation method was used; see [ ]
We study a semilinear wave equation of the form
Summary
The study of nonlinear wave equations has been developed in various ways of approach by many researchers; for instance, by Brézis and Nirenberg [ , ], Rabinowitz [ ], Berkovits and Mustonen [ – ]. ), by applying the continuation theorem to the reference maps L + P and L + cI, where P denotes the orthogonal projection onto the kernel of L, I denotes the identity operator, and –c is a positive regular value of L.
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