Existence and uniqueness of solution for initial value problem of first order differential equation involving generalized Lebesgue-Bochner spaces Lp(I,(Xv, II.II))

• Autorzy: Lahrech S. , Ouahab A. , Benbrik A. , Mbarki A.
• Języki publikacji: EN

Abstrakty

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.

Słowa kluczowe

EN

bitopological vector spaces; generalized Lebesgue-Bochner spaces; initial value problem; differential equations

PL

bitopologiczna przestrzeń wektora; przestrzenie Lebesgue-Bochnera; zagadnienie początkowe; równania różniczkowe

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2007
• Tom: Vol. 40, nr 2
• Strony: 303--315
• Opis fizyczny: Bibliogr. 7 poz.

Twórcy

autor

Lahrech S.

autor

Ouahab A.

autor

Benbrik A.

autor

Mbarki A.

• Dynamical System and Optimization Group, Gafo Laboratory Department of Mathematics , Faculty of Science Mohamed First University Oujda, Morocco,

Bibliografia

• [1] D. H. Hyers, Pseudo-normed linear spaces and abelian groups, Duke Math. J. 5 (1939), 628-634.
• [2] S. Lahrech, A. Ouahab, A. Benbrik, A. Mbarki, Some Properties of Sequentially Continuous Linear Mappings Acting in Topological Vector Spaces, accepted for publication in IJPAM. Sofia, Bulgaria, 2005.
• [3] A. Taylor, D. Lay, Introduction to Functional Analysis, Second Edition. Krieger Publishing Company, Malabar, Florida, 1980.
• [4] A. N. Kolmogorov, C. V. Fomin, Functional Analysis, Nayka, (1976) (in Russian).
• [5] J. L. Kelly, T. P. Srinivasan, Measure and Integral, Volume 1. Springer-Verlag, New York, 1988.
• [6] A. J. Weir, Lebesgue Integration and Measure, Cambridge University Press, 1973.
• [7] J. Mikusinski, The Bochner Integral, Birkhauser, 1978.