Existence of solutions to \((k,n-k-2)\) boundary value problems
The boundary value problem considered has the form \[ u^{(n)}(t)+ f\bigl(t,u(t), u'(t), \dots,u^{(n-2)} (t)\bigr)=0, \quad t\in(0,1), \] \[ u^{(r_i-1)}(0)=0,\;1\leq i\leq k,\quad u^{(s_j-1)}(1)=0,\;1\leq j\leq n-k-2, \] \[ \alpha u^{(n-2)}(0)-\beta u^{(n-1)}(0)=0,\quad\gamma u^{(n-2)}(1)+\delta u^{(n-1)} (1)= 0. \] Here, \(\{r_1,\dots,r_k\}\) and \(\{s_1,\dots,s_{n-k-2}\}\) are disjoint sets whose union is \(\{1,\dots,n-2\}\). Sufficient conditions for existence and nonuniqueness of solutions are given. Results of J. Mawhin and M. A. Krasnosel'skij are employed in the proofs.
higher-order ordinary differential equations
scientific article; zbMATH DE number 1311051
scientific article