Operator Differential Equations in Hilbert Space

Citation

Lopes, Louis Aloysius (1964) Operator Differential Equations in Hilbert Space. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/1N8N-Y957. https://resolver.caltech.edu/CaltechETD:etd-10182002-082821

Abstract

NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document. In this paper the theory of dissipative linear operators in Hilbert space developed by R. S. Phillips has been applied in the study of the Cauchy problem [...](t) + A(t)x(t) = f(t), x(o) = x[subscript o] where A(t), t [epsilon] [o,[tau]], is a family of unbounded linear operators with a common dense domain D in a Hilbert space H, f [epsilon] [...], the Hilbert space of measurable functions on [o, [tau]] with values in H which have square integrable norm, and x[subscript o] [epsilon] H. It is assumed that for each t [epsilon] [o,[tau]] A(t) is maximal dissipative, satisfying for each x [epsilon] D, Re (A(t)x,x) [greater than or equal to] [alpha] [...], [alpha] > o, and A(t)x is strongly continuous and has a bounded measurable strong derivative on J. Let A[subscript o] be any maximal dissipative linear operator with domain D satisfying Re (A[subscript o]x,x) [greater than or equal to] [alpha] [...] for all x [epsilon] D. Then B(t) = A(t)A[subscript o][superscript -1] is a one-to-one continuous linear transformation of H onto itself. It is assumed that B[superscript -1](t) is bounded on [o, [tau]]. Under these conditions it is shown that, first, there exists a weak solution to the Cauchy problem, and, second, that the weak solution is a unique strong solution which is the limit of a sequence of classical solutions. The theory is applied to a time-dependent hyperbolic system of partial differential equations.

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