On a periodic-type boundary value problem for first-order nonlinear functional differential equations. (English) Zbl 1022.34058

Hakl, R.; Lomtatidze, A.; Šremr, J.

Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 51, No. 3, 425-447 (2002).

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.

Reviewer: N.Parhi (Berhampur)

MSC:

34K10

Boundary value problems for functional-differential equations

Keywords:

periodic-type boundary value problem; first-order nonlinear functional-differential equations; solvability

References:

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