Theory of Differential Equations by I. M. Gel'fand and G. E. Shilov (Auth.)
By I. M. Gel'fand and G. E. Shilov (Auth.)
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Extra resources for Theory of Differential Equations
Sample text
Indeed, for a given b0 , we can determine b from the condition \/(2p0'b) — b0 . Then we determine a from the condition that the space W^a be nontrivial (Chapter I, Section 1) with M(x) = Ω(χ) = (l/po)\ x \Po\ and finally we determine the interval T from the condition 2Po+1b0T < (l//>0) θν°(θ < a). These quantities allow us to construct the test function spaces Φ and Ψ and to carry out the reasoning which achieves the proof of the theorem. The uniqueness theorem has been proved for p0 > 1. For p0 ^ 1, the proof is carried out according to the same plan, but replacing the functions \ σ \Po> \ τ \Po which define the space W^a , by the functions | σ | r , | T \r with arbitrary positive powers r > 1.
Consider the differential equation du(t)\dt = -At*u(t), (1) where u(t) is an unknown element of the space Φ'. The problem of finding a solution of this equation, satisfying the initial condition u(O) = uoe0' (2) will be called the abstract Cauchy problem. 1 The Cauchy Problem in a Topological Vector Space 33 with y(t) 6 Φ, and satisfying the initial condition Bn in the inequalities (3), written for the functions ψν{ζ), can be chosen independently of v. Replacing the constants a^ by cij — S;· and b^ by bj + pj in the above inequalities keeping the constants a$ and bj fixed, and requiring the validity of the inequalities (l)-(3) for all positive δ;· and pj, we obtain the definitions of the spaces WM a , WQ>b, and W^\ ; these are (countably normed) perfect topological vector spaces and their unions yield the appropriate spaces WM , W°, and WMQ. 2. Operations in Test Function Spaces In all the indicated spaces the operations of differentiation d/dxj and multiplication by Xj (or d/dzj and zi , respectively) are defined and continuous.