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EN
We consider a bitopological vector space (X, v, II.II), where (X, v) is a topological vector space, and II.II is a norm defined on X. This paper deals with the existence and uniqueness of solution for initial value problem of first differential equation: (P)( ˙ x(t) = f(t), t is an element of]alpha, beta[ x(alpha) = x1, where the vector valued function f:]alpha,beta[-› X is assumed to be not necessarily in the classical Lebesgue-Bochner space L1(]alpha,beta[, (X, II.II). Here, by the solution of problem (P), we mean a vector valued function x acting from ]alpha,beta[ into X satisfying the conditions: 1) x is absolutely continuous with respect to the norm II.II; 2) x is almost everywhere differentiable on ]alpha,beta[ with respect to the topology v; 3) ˙ x = f(t) almost everywhere on ]alpha,beta[; 4) x(alpha) = x1. For this, we introduce a special class of integrable functions called generalized Lebesgue- Bochner space denoted L1(]alpha,beta[, (Xv, II.II)) containing (in general, strictly containing, [see the example given at the end of the paper]) the classical Lebesgue-Bochner space L1(]alpha,beta[, (X, II.II). Thus, under some conditions on the pair of topologies (v, II.II) , we prove that if f is an element of L1(]alpha,beta[, (Xv,II.II)), then the initial value problem (P) has an unique solution in the above mentioned sense. Finally, we give an example to illustrate the result given in this paper.
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