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Asymptotic conditions for solving non-symmetric problems of third order nonlinear differential systems. (English) Zbl 1176.34022

The author establishes new solvability criteria for the forced third-order nonlinear system
\[ X'''+AX''+BX'+sH(t,X)=P(t),\quad t\in [0,T], \]
subject to the periodic boundary conditions
\[ X(0)-X(T)=X'(0)-X'(T)=X''(0)-X''(T)=0, \]
where \(X=(x_i)_{1\leq i\leq n}:[0,T]\to \mathbb{R}^n\), \(A\) and \(B\) are constant real \(n\times n\) matrices, \(H=(h_i(t,X))_{1\leq i\leq n}:[0,T]\times \mathbb{R}^n\to \mathbb{R}^n\) and \(P=(p_i)_{1\leq i\leq n}:[0,T]\to \mathbb{R}^n\) are \(n\)-vectors, which are \(T\)-periodic in \(t\). Furthermore, \(H\) satisfies the Caratheodory conditions, and \(s\in\{-1,1\}\). The main tool is Mawhin’s continuation theorem.
Reviewer: Minghe Pei (Jilin)

MSC:

34B15 Nonlinear boundary value problems for ordinary differential equations
34C25 Periodic solutions to ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
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