Abstract
Unimprovable effective efficient conditions are estab- lished for the unique solvability of the periodic problem u '(t) = i+1 X j=2 li,j(uj)(t) + qi(t) for 1 ≤ i ≤ n − 1, u ' (t) = n X j=1 ln,j(uj)(t) + qn(t),
Highlights
Much work had been carried out on the investigation of 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
Opial obtained for ordinary differential equations in [13], and on the other hand, extend results obtained for linear functional differential equations in [5,14,15,16]
All the conditions of Theorem 1.1 are fulfilled for the system (1.16)
Summary
Opial (see Remark 1.1) obtained for ordinary differential equations in [13], and on the other hand, extend results obtained for linear functional differential equations in [5,14,15,16]. Let ln,, li,i+1 (i = 1, n − 1) be linear monotone operators, the conditions (1.8) hold and ω. Let measurable functions τi : [0, 1] → [0, 1] and the linear nonnegative operators li : C([0, 1]) → L([0, 1])(i = 1, n) be given by the equalities τi(t) =. From the relations u′0(t) = |u′0(t)|u0(τi(t)) = li(u0)(t) (i = 1, n), it follows that the vector function (ui(t))ni=1 if ui(t) ≡ u0(t) (i = 1, n) is a nontrivial solution of problem (1.1), (1.2) with ω = 1, q(t) ≡ 0, which contradicts the conclusion of Corollary 1.3
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