Abstract singular equations and applications
The author considers the degenerate operational equation in the unknown u: (1) \(BMu+Lu=h,\) where h belongs to a complex Banach space E, B is a closed linear operator from E into itself, L,M are closed linear operators from another Banach space F into E, and M in general, has neither an inverse nor bounded inverse. An equation like (1) was considered by the author [Math. Ann. 273, 17-44 (1985; Zbl 0556.34067)] under spectral hypotheses of parabolic type, that is, the modified resolvent \((zM+L)^{-1}\) exists \(\forall z\) in the sector \(Rez>-a_ 0-b_ 0| Imz|,\) \(a_ 0,b_ 0>0\), and the norm of \(L(zM+L)^{-1}\) as a bounded operator from E into itself has a polynomial growth. The present paper is devoted to handle equation (1) in a particular case, which occurs in various applications, that is, when 0 in an isolated singularity of \((zL+M)^{-1}\). To this end, both an extension of the technique developed by the author [J. Differ. Equations 39, 212-225 (1981; Zbl 0422.34025] for \(B=d/dt\), and the approach in the paper previously quoted, are used. As a particular case, the singular differential equations in a Hilbert space considered by \textit{J. Lagnese} [SIAM J. Math. Anal. 4, 623-637 (1973; Zbl 0235.34128)] can be easily treated. In fact, various examples of applications to differential problems are given, of abstract or concrete type: they concern both initial value problems (for first and second order differential equations) and boundary problems.
scientific article; zbMATH DE number 3973370