On a boundary value problem for nonlinear functional differential equations (Q2500872)

The paper deals with the scalar functional boundary value problem \[ u'(t) =H(u)(t)+Q(u)(t),\tag{1} \] \[ u(a)=h(u),\tag{2} \] where \(H,Q:C[a,b]\to L_1[a,b]\) are, in general, nonlinear continuous operators and \(h:C[a,b]\to\mathbb{R}\) is a continuous functional. The author gives sufficient conditions on \(H,Q\) and \(h\) for the solvability and unique solvability of problem (1), (2). The optimality of the conditions is demonstrated on examples. The solvability of problem (1), (2) is proved by a combination of an existence principle by \textit{I. Kiguradze} and \textit{B. Půža} [Mem. Differ. Equ. Math. Phys. 12, 106--113 (1997; Zbl 0909.34054)] with differential and integral inequalities.