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Boundary value problems for higher order ordinary differential equations. (English) Zbl 0809.34034
The authors consider the equation \(x^{(k)}= f(t,x,x',\dots, x^{(n)})\), where \(k\geq n+1\) and \(f: [a,b]\times \mathbb{R}^{n+1}\to\mathbb{R}\) is a Carathéodory function. They consider the question of the existence of solutions of this equation satisfying the boundary conditions \(x^{(i)}(t_ i)= x_ i\), \(t_ i\in [a,b]\) or \(x(t_ i)= x_ i\), \(a\leq t_ 0< t_ 1<\cdots <t_{k-1}\leq b\), where \(x_ i\in \mathbb{R}\), \(i= 0,1,\dots, k-1\). Using the Schauder fixed point theorem and the properties of the Abel-Gontcharoff and Lagrange polynomials, they prove several existence results for each sufficiently large \(k\) and suitable sequences \(\{t_ i\}\), \(\{x_ i\}\). More precisely, they prove the existence of a number \(\nu\geq n+1\) such that for each \(k\geq\nu\) the above problems have solutions, but they do not determine a value of \(\nu\).

34B15 Nonlinear boundary value problems for ordinary differential equations
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
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