Abstract |
This dissertation consists of two chapters. Chapter I concerns the equation (1) du/dt = Au + f(t,u), u(0) = u(,0), where, for X a Banach space, u(,0) (ELEM) X, f : {0,(tau)} x X (--->) X is jointly continuous and uniformly approximable by functions which are Lipschitz continuous in X, and A generates a C(,0)-semigroup, {S(t) : t (GREATERTHEQ) 0}, on X such that for t > 0, S(t) is compact In the case where f is bounded, it is shown that mild solutions of (1) exist on {0,(tau)} and that the set of all mild solutions of (1), i.e. {u (ELEM) C({0,(tau)},X) : u is a mild solution of (1)} is a compact set and is homeomorphic to the intersection of a decreasing sequence of absolute retracts In the general case, it is shown that mild solutions of (1) exist on a fixed interval {0,(eta)} (L-HOOK EQ) {0,(tau)}, and the set of all mild solutions on {0,(eta)} is compact and is homeomorphic to the intersection of a decreasing sequence of absolute retracts Chapter II concerns an ergodic theorem due to N. Wiener. He proved that if u is a finite Borel measure, u = u(,c) + u(,d) the Lebesgue decomposition of u into continuous and discrete parts and if u(t) = 1/(2(pi))(' 1/2) (INT)(,(//R)) e('itx) u(dx) is the Fourier transform of u, then (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) as T (--->) (INFIN) There is a natural reformulation of this result. Let iA generate a C(,0)-unitary group, {U(t) : t (ELEM) (//R)}, on a Hilbert space H, where A is a self-adjoint operator with the spectral family {E(,A)((gamma)) : (gamma) (ELEM) (//R)}. Let (sigma)(,p)(A) be the point spectrum of A and f, g (ELEM) H. Then (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) as T (--->) (INFIN) The main generalization of this is as follows. Let A generate a C(,0)-contraction semigroup, {S(t) : t (GREATERTHEQ) 0}, on a Hilbert space H. Let P(,(gamma)) be the projection onto ker(A-(gamma)I), and f, g (ELEM) H. Then (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) as T (--->) (INFIN) Formula (2) was known to N. Wiener {21}. Chapter II contains simple proofs of (2) and (3). Moreover, necessary and sufficient conditions are given for the convergence in (2) and (3) to occur at a rate 0((alpha)) where (alpha) : (//R)('+) (--->) (//R)('+) and (alpha)(t) (--->) 0 as t (--->) (INFIN) Results are obtained with the concept of almost convergence replacing the slightly weaker Cesaro convergence of (2) and (3). An extension of these results to the case of a semigroup which is similar to a contraction semigroup is also given |