Abstract
Some questions about abstract methods for initial value problems lead us to a study of the equation (*) u′(t)=(A+B)u(t), where A is m-dissipative and B is compact. Does a solution to (*) necessarily exist? Earlier studies of this question, reviewed and then continued here, depend on an analysis of the related quasiautonomous equation (**) u′(t)=Au(t)+f(t). We say A has Gihman's property if the mapping f ↦ u is continuous from L 1 w ([0,T],K) into C([0,T];X) for every compact K⊂X; this condition is closely related to the Lie-Trotter-Kato product formula. If A has this property, then (*) is known to have a solution. In this paper, we consider linear, m-dissipative operators A which lack Gihman's property. We obtain partial results regarding the existence of solutions of (*); but in general, the existence question remains open. Our method applies the variation of parameters formula to (**), but this requires a weakened topology when Range (f) \(\nsubseteq \overline {D(A)}\). Two examples are studied: one in ℓ∞, the other in the space of bounded continuous functions.
Keywords
- Banach Space
- Weak Topology
- Norm Topology
- Semi Group
- Bounded Continuous Function
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Schechter, E. (1987). Compact perturbations of linear m-dissipative operators which lack Gihman's property. In: Gill, T.L., Zachary, W.W. (eds) Nonlinear Semigroups, Partial Differential Equations and Attractors. Lecture Notes in Mathematics, vol 1248. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0077423
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DOI: https://doi.org/10.1007/BFb0077423
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