Singular Lidstone boundary value problem with given maximal values for solutions.
The singular Lidstone boundary value problem \[ \begin{gathered} (-1)^n x^{(2n)}(t)= \mu f(t,x(t),\dots, x^{(2n-2)}(t)),\quad t\in J= [0,T],\\ x^{(2j)}(0)= x^{(2j)}(T)= 0,\quad 0\leq j\leq n-1,\end{gathered} \] together with the condition \(\max\{x(t): t\in J\}= A\), \(A> 0\), is considered. The function \(f\in \text{Car}(J\times D)\) with \(D\) depending on the parameter \(\mu\in \mathbb{R}\) may be singular at the zero value of all its phase variables. It is assumed also that \(f\) is such that for suitable functions \(\psi\), \(\phi\), \(h_j\), \(q_j\), \(\omega_j\) and \(\alpha_j\in (0,1)\) \[ 0< \psi(t)\leq f(t,x_0,\dots, x_{2n-2}), \] \[ f(t, x_0, x_1,\dots, x_{2n-2})\leq \phi(t)+ \sum^{2n-2}_{j=0} q_j(t)\omega_j(| x_j|)+ \sum^{2n-2}_{j=0} h_j(t)| x_j|^{\alpha_j}, \] for a.e. \(t\in J\) and each \((t,x_0,x_1,\dots, x_{2n-2})\in D\). Using Borsuk's theorem and the Leray-Schauder degree, the authors prove that a sequence of auxiliary regular boundary value problems (BVPs) has a sequence of solutions. Next, by the Arzela-Ascoli theorem, they select a convergent subsequence of solutions and applying Vitali's convergence theorem, show that its limit is positive on \((0,T)\) and is a solution of the above BVP.
Singular Lidstone boundary value problem
scientific article; zbMATH DE number 2032561