Differential Equations, Vol. 39, No. 3, 2003, pp. 344–352. Translated from Differentsial’nye Uravneniya, Vol. 39, No. 3, 2003, pp. 320–327. Original Russian Text Copyright c  2003 by Lomtatidze, Hakl, P ˚ uˇ za. ORDINARY DIFFERENTIAL EQUATIONS On the Periodic Boundary Value Problem for First-Order Functional-Differential Equations A. G. Lomtatidze, R. Hakl, and B. P˚ uˇ za Masaryk University, Brno, Czech Republic Received August 6, 2002 1. STATEMENT OF THE PROBLEM AND BASIC NOTATION In the present paper, we consider the boundary value problem u ′ (t)= ℓ(u)(t)+ F (u)(t), (1.1) u(a)= u(b), (1.2) where ℓ : C ([a, b]; R) → L([a, b]; R) is a linear bounded operator and F : C ([a, b]; R) → L([a, b]; R) is a continuous operator, not necessarily linear. This problem, which is the subject of numerous studies, has long been attracting mathematicians’ attention. Interesting results about its solvability can be found, e.g., in [1–11]. Nevertheless, problem (1.1), (1.2) has not been completely analyzed yet even for the linear case, in which Eq. (1.1) has the form u ′ (t)= ℓ(u)(t)+ g(t). (1.3) In a sense, we fill the gap. More precisely, new effective criteria for the solvability and unique solvability of problems (1.3), (1.2) and (1.1), (1.2) are given in Sections 2 and 3. In Section 4, we construct examples justifying the optimality of these criteria. The results are further specialized for the equations u ′ (t)= p(t)u(τ (t)) + g(t), (1.4) u ′ (t)= p(t)u(τ (t)) + f (t, u(µ(t)),u(t)) (1.5) with deviating arguments. We use the following notation: R is the set of real numbers; R + = [0, +∞[; C ([a, b]; R) is the space of continuous functions u :[a, b] → R with the norm ‖u‖ C = max{|u(t)| : a ≤ t ≤ b}; C ([a, b]; R + )= {u ∈ C ([a, b]; R): u(t) ≥ 0 for t ∈ [a, b]}; C 0 ([a, b]; R)= {u ∈ C ([a, b]; R): u(a)= u(b)}; ˜ C ([a, b]; R) is the set of absolutely continuous functions u :[a, b] → R; L([a, b]; R) is the space of Lebesgue integrable functions p :[a, b] → R with the norm ‖p‖ L = b  a |p(s)|ds; L ([a, b]; R + )= {p ∈ L([a, b]; R): p(t) ≥ 0 for t ∈ ]a, b[ }; M ab is the set of measurable functions τ :[a, b] → [a, b]; 0012-2661/03/3903-0344$25.00 c  2003 MAIK “Nauka/Interperiodica”