Some two-point boundary value problems for second-order functional-differential equations (Q2748481)

The differential equation NEWLINE\[NEWLINEu''(t)= F(u)(t)\tag{1}NEWLINE\]NEWLINE with the boundary conditions NEWLINE\[NEWLINEu(a)= 0,\quad u(b)= 0,\tag{2}NEWLINE\]NEWLINE or NEWLINE\[NEWLINEu(a)= 0,\quad u'(b)= 0,\tag{3}NEWLINE\]NEWLINE is considered, where \(F: C'([a, b];\mathbb{R})\to L(]a,b[;\mathbb{R})\) is a continuous operator satisfying the condition NEWLINE\[NEWLINE\sup\{|F(v)(\cdot):\|v\|_{c'}\leq r\}\in L(]a,b[; \mathbb{R}_+)\quad\text{for }r> 0.NEWLINE\]NEWLINE Sufficient conditions for the solvability of problems (1), (2) and (1), (3) are given.