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)


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
Full Text: EuDML EMIS