By Carl Faith

ISBN-10: 3540057056

ISBN-13: 9783540057055

E-book by way of religion, Carl

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**Additional resources for Algebra II. Ring Theory: Ring Theory**

**Sample text**

101) is bounded on [0,oo). ■ 4. 103) we also consider x'(t) = A(t)x{t). 105) 52 1. 104) has a unique solution. 105) at the moment t is denoted as x(t) = U(t)c0. 106) for the differential equation in the space £ ( E , E) of bounded operators acting in a Banach space E. For each t there exists a bounded inverse operator U~l(t). 104) is represented in the form x(t) = U(t)U-\s)c0 = U(t,s)c0, where U(t, s) is the evolution operator. 45. 107) where c[t] = II U(k) (u~\k k=[t] V - 1) + / * U-\s)B{s) ds) co.

30), and the assumption c„ G T>(A) enables us to maintain 24 1. 30) in the given interval. 30) for n — 1 < t < n. The recursion relation c„ = (u(l) + £_x U(n - s)B ds) c„_! 38). 30) on the interval [0,1], then c\ = £o(l) £ 1)(A). Now we consider the solution x\(t) on [1,2] and see that ci = ii(2) G D(^l)- Continuing the solution to the right we conclude that cn G U(-A) for any n. ■ If the correctness of the Cauchy problem is not known beforehand, we may resort to certain restrictions on the resolvent of the operator A in order to prove the existence and uniqueness of the solution.

59) on t > 0 satisfying the conditions x(n + j) = cn+j, -N

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