Operator Differential Equations in Hilbert Space

Author: Lopes, Louis Aloysius

Year: 1964

Degree: Dissertation (Ph.D.)

Advisors: De Prima, Charles R.; Erdélyi, Arthur

Committee Member: Unknown, Unknown

Option: Mathematics

DOI: 10.7907/1N8N-Y957

Abstract

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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 Bsuperscript -1 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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