On a periodic boundary value problem for cyclic feedback type linear functional differential systems

Abstract

Nonimprovable effective sufficient conditions are established for the unique solvability of the periodic problemwhere ω > 0, i : C ([0, ω]) → L ([0, ω]) are linear bounded operators, and q i ∈ L([0, ω]). Statement of problem and formulation of main results. Consider on [0, ω] the systemwith the periodic boundary conditions2) where ω > 0, i : C([0, ω]) → L([0, ω]) are linear bounded operators and q i ∈ L([0, ω]). By a solution of the problem ( 1.1), ( 1.2) we understand a function u ∈ C([0, ω]) which satisfies the system ( 1.1) almost everywhere on [0, ω] and satisfies the conditions ( 1.2).Much work has been carried out on the existence and uniqueness of the solution for a periodic boundary value problem for systems of ordinary differential equations, and many interesting results have been obtained (see, for instance, [1], [2], [3], [4], [5], [6], [7], [8] and the references therein). However, the analogous problem for functional differential equations, even in the case of linear equations, remains little investigated. Mathematics Subject Classification (2000): 34K06, 34K13.

An optimal condition for the uniqueness of a periodic solutionfor systems of higher order linear functional differential equations

“…In the present paper, we study problem (1.1) (1.2) under the assumption that l n;1 ; l i;i C1 .i D 1; n 1/ are monotone linear operators. We establish new unimprovable integral conditions sufficient for unique solvability of the problem (1.1), (1.2) which generalize the well-known results of A. Lasota and Z. Opial (see Remark 1.2) obtained for ordinary differential equations in [7] and, on the other hand, extend our results obtained for linear functional differential equations in [8,9,11]. These results are also new if (1.1) is the system of ordinary differential equations of the form…”

“…where q i ; l i;j 2 L.OE0; !/: The method used for the investigation of the problem considered is based on that developed in our previous papers [8][9][10][11] for functional differential equations. The following notation is used throughout the paper: N (resp., R) is the set of all the natural (resp., real) numbers; R n is the space of n-dimensional column vectors…”

A combined variational-topological approach for dispersion-managed solitons in optical fibers

“…The results dealing with general nonlinear two-dimensional system one can find, e.g., in [4]. In [8,9] one can find conditions guaranteeing the existence of a periodic solution to the n-dimensional linear system of both ordinary and functional differential equations. …”

On the solvability of the periodic boundary value problem for a linear functional differential equation

“…Условиям разрешимости периодической краевой задачи для различных видов функци-онально-дифференциальных уравнений за последние годы было посвящено значительное чис-ло работ [1][2][3][4][5][6][7][8][9][10][11][12]. В частности, для функционально-дифференциальных уравнений первого [1,3], второго [5], третьего [11] порядков получены неулучшаемые условия однозначной разрешимо-сти.…”

“…Для обыкновенных дифференциальных уравнений различные неулучшаемые условия од-нозначной разрешимости периодической задачи были получены, например, в работах [16,17,18], причем результаты работы А. Лясоты и З. Опяля [16] послужили основой для многих дальнейших обобщений, в частности, [1][2][3][4][5][6][7][8][9][10][11][12]17]. § 2 .…”