On boundary value problems for functional-differential equations. (English) Zbl 0909.34054

Kiguradze, I.; Půža, B.

Mem. Differ. Equ. Math. Phys. 12, 106-113 (1997).

A general theorem (the principle of a priori boundedness) on solvability of the boundary value problem \[ \frac{dx(t)}{dt}=f(x)(t),\qquad h(t)=0, \] is established, where \[ f: C([a,b]);\mathbb{R}^n)\to L([a,b]);\mathbb{R}^n),\qquad h: C([a,b]);\mathbb{R}^n)\to \mathbb{R}^n \] are continuous operators. As an application, effective criteria for the solvability of the boundary value problem \[ \frac{dx(t)}{dt}=f_0(t,x(t)), \]
\[ x(t_1(x))=A(x)x(t_2(x))+c_0, \] are obtained, where \( f_0: I=[a,b] \times \mathbb{R}^n\to \mathbb{R}^n \) is a vector function satisfying local Carathéodory conditions, \(c_0\in \mathbb{R}^n\) and \(t_i: C(I,\mathbb{R}^n)\to I\), \(i=1,2\), \(A: C(I,\mathbb{R}^n)\to \mathbb{R}^n\) are continuous operators.

Reviewer: J.Diblík (Brno)

MSC:

34K10

Boundary value problems for functional-differential equations

Keywords:

functional-differential equation; boundary value problem; existence of a solution; principle of a priori boundedness