Higher order ordinary differential equations
Let \([a,b]\) be a compact real interval, \(n\) a nonnegative integer, \(f\) a real function defined on \([a,b] \times \mathbb{R}^{n+1}\) and \(\{x_ h\} \subset \mathbb{R}\) be a bounded sequence. The aim of the paper is to prove the existence of a positive integer \(\nu \geq n+1\) such that for every \(k \geq \nu\) and every \(t_ 1,\ldots,t_ k \in[a,b]\) the problem \(x^{(k)}= f(t,x,x', \ldots,x^{(n)})\), \(x^{(i-1)}(t_ i)=x_ i\), \(i=1,2,\ldots,k\), has at least one solution. The function \(f\) is supposed to be an \(L^ 2\)-Carathéodory function, i.e. for every \(z \in \mathbb{R}^{n+1}\) the function \(t \to f(t,z)\) is measurable; for a.e. \(t \in[a,b]\) the function \(z \to f(t,z)\) is continuous; for every \(\rho>0\) the function \(t \to \sup_{\| z \| \leq \rho} | f(t,z) |\) belongs to \(L^ 2([a,b])\). Solutions of the above problem are found in the Sobolev space \(W^{k,2}([a,b])\). The existence results are obtained provided \(b-a< \pi/2\) and the proofs are based on the existence theorem for inclusions by \textit{O. N. Ricceri} and \textit{B. Ricceri} [Appl. Anal. 38, 259-270 (1990; Zbl 0687.47044)].
scientific article; zbMATH DE number 459195