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Boundary value problems for systems of implicit differential equations. (English) Zbl 0799.34023

The aim of the authors is to give some sufficient conditions which guarantee the existence of solutions to the nonlinear systems of differential equations of the form \(F(t, y(t), y'(t), \dots, y^{(k)} (t))=0\), \(t\in [0,T]\), \(y\in \mathbb{R}^ n\), under various boundary conditions. Their approach consists in reducing first the given problem to a boundary-value problem for a differential inclusion of the type \(y^{(k)} (t)\in \psi(t, y(t), y'(t),\dots, y^{(k-1)} (t))\), \(t\in [0,T]\), with an appropriate multifunction \(\psi\) and then looking for solutions in the Sobolev space \(W^{k,p} ([0,T], \mathbb{R}^ n)\). To show the necessity of Sobolev space they give an example which shows that even the smoothness of \(F\) does not imply the existence of \(C^ k\)-solutions. In addition to a main theorem of general character the paper includes also four theorems related to a series of special cases such as
i) \(\sum_{j=0}^ m a_ j (z, z(t), z'(t),\dots, z^{(k-1)} (t)) (z^{(k)}(t)- \varepsilon z(t))^ j =0\), \(\varepsilon\geq 0\);
ii) \(F(a(t,y(t), y'(t),\dots, y^{(k-1)} (t)), y^{(k)} (t))=0\), a.e. \(t\) with \(y^{(j)}(t_ j)= r_ j\) \((j=0,1,\dots, k-1)\), where \(t_ j\in [0,\infty)\) and \(r_ j\in \mathbb{R}^ n\) and
iii) \(F(a(t,y(t),y'(t)), y''(t))=0\), a.e. \(t\in [0,1]\) with \(y(0)= r_ 0\), \(y(1)= r_ 1\) or \(y(0)= y(1)\), \(y'(0)= y'(1)\).

MSC:

34B15 Nonlinear boundary value problems for ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
34A09 Implicit ordinary differential equations, differential-algebraic equations
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