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Periodic boundary value problem for a matrix differential equation. (English) Zbl 0810.34018

Farkas, M. (ed.) et al., Differential equations and its applications. Proceedings of the colloquium on differential equations and applications, held in Budapest, August 21-24, 1991. Amsterdam: North-Holland Publishing Company. Colloq. Math. Soc. János Bolyai. 62, 87-99 (1991).
The author considers the system of nonlinear differential equations of the form \((Lx)(t)= \phi(t)\), \(t\in [0,w]\) written for \(x(t)\in \mathbb{R}^ n\) \((n\geq 1)\) with the boundary conditions \(x^{(i)}(0)= x^{(i)}(w)\), \(i= 0,1,\dots, n-1\), where \((Lx)(t)= \sum^ n_{i=0} A_ i(t) x^{(n- i)}(t)\), \(A_ 0=\) (unit matrix), and \[ \phi(t)= \sum^{n-1}_{i=0} P_ i(t,x(t),\dots, x^{(n-1)}(t)) x^{(i)}(t)+ Q(t,x(t),\dots, x^{(n-1)}(t)). \] He assumes that \(A_ i\) and \(P_ i\) are \(n\times n\) matrices of continuous functions while \(Q\) is an \(n\times 1\) vector of such functions. His aim is to give sufficient conditions which guarantee the existence of a solution. He gives also the conditions under which \(x(t)\) is \(w\)-periodic when \(A_ i\), \(P_ i\) and \(Q\) are \(w\)-periodic in \(t\). The results are stated in two theorems and are applied to an example which consists of a generalization of the scalar Liénard problem.
For the entire collection see [Zbl 0792.00006].

MSC:

34B15 Nonlinear boundary value problems for ordinary differential equations
34C25 Periodic solutions to ordinary differential equations
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