Nonlinear eigenvalue problems involving two classes of functional differential equations (Q2716457)

Consider the eigenvalue problem NEWLINE\[NEWLINE(-1)^{n- k}u^{(n)}(t)=\lambda a(t) f(u_t),\quad 0\leq t\leq 1,\tag{1}NEWLINE\]NEWLINE NEWLINE\[NEWLINEu(s)= \varphi(s),\quad -r\leq s\leq 0,\tag{2}NEWLINE\]NEWLINE NEWLINE\[NEWLINEu^{(i)}(0)= 0,\quad i= 0,\dots, k-1,\quad u^{(i)}(1)= 0,\quad i= 0,\dots, n-k-1,\tag{3}NEWLINE\]NEWLINE where the functional \(f: C[-r,0]\to [0,+\infty)\) is continuous. Sufficient conditions are given for problem (1)--(3) to have a positive solution.