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On a periodic-type boundary value problem for first-order nonlinear functional differential equations. (English) Zbl 1022.34058
Here, the solvability and the unique solvability of the boundary value problem $u'(t)= F(u)(t),\quad u(a)-\lambda u(b)= h(u),\tag{$$*$$}$ are discussed, where $$F\in K_{ab}$$, $$K_{ab}$$ is the set of continuous operators $$F: C([a, b],\mathbb{R})\to L([a,b]),\mathbb{R})$$ satisfying the Carathéodory condition, that is, for every $$r> 0$$ there exists $$q_r\in L([a, b],\mathbb{R}_+)$$ such that $|F(v)(t)|\leq q_r(t)\text{ for }t\in [a,b],\quad\|v\|_C\leq r,$ $$\lambda\in \mathbb{R}_+$$ and $$h: C([a, b],\mathbb{R})\to \mathbb{R}$$ is continuous. Different types of sufficient conditions are obtained for the solvability of $$(*)$$. Examples are given to illustrate the results obtained.

##### MSC:
 34K10 Boundary value problems for functional-differential equations
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