\(R_{\delta}\)-set of solutions to a boundary value problem (Q5929988)

The author considers the generalized boundary value problem NEWLINE\[NEWLINEx^{(n)}+ p_1(t) x^{(n-1)}+\cdots+ p_n(t) x+ f(t, x,\dots, x^{(m)})= q(t),\;a\leq t\leq b,\tag{1}NEWLINE\]NEWLINE NEWLINE\[NEWLINE\ell_i(x)= 0,\qquad i= 1,\dots, n;NEWLINE\]NEWLINE with \(n\geq 1\); \(0\leq m\leq n-1\); \(-\infty< a< b<\infty\); \(p_k, q\in C([a, b])\), \(k= 1,2,\dots, n\); \(f: [a,b]\times \mathbb{R}^{m+1}\to \mathbb{R}\) is continuous; \(\ell_i: C^{n-1}([a, b])\to \mathbb{R}\), \(i= 1,\dots, n\) are linearly independent linear continuous functionals.NEWLINENEWLINENEWLINEHere, a sufficient condition for the existence of an \(R_\delta\)-set of solutions to problem (1) on a compact interval is established.