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Higher order ordinary differential equations. (English) Zbl 0785.34021
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 O. N. Ricceri and B. Ricceri [Appl. Anal. 38, 259-270 (1990; Zbl 0687.47044)].

##### MSC:
 34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations 34B15 Nonlinear boundary value problems for ordinary differential equations