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On stability, ultimate boundedness, and existence of periodic solutions of certain third order differential equations with delay. (English) Zbl 1331.34145

The authors consider a non-autonomous delay differential equation of third order of the form \[ (g(x){x}')'' + (h(x) x')' + \phi (x)x' + f(x(t - \tau )) = e(t), \tag{1} \] where the primes in equation (1) denote differentiation with respect to \(t\), \(t \in \mathbb R ^+\), \(\mathbb R^+ =[0,\infty );\tau \) is a positive constant, \(g\), \(h,\phi\), \(f:\mathbb R \to \mathbb R \) and \(e:\mathbb R ^+\to \mathbb R\) are continuous functions. The continuity of the functions \(g\), \(h\), \(\phi\), \(f\) and \(e\) guarantees the existence of a solution of equation (1). The functions \(g\), \(h\), \(\phi\) and \(f\) satisfy a Lipschitz condition. Then the solution is unique. The authors prove three new theorems which include sufficient conditions and guarantee the uniform asymptotical stability of the zero solution for the case \(e(t) \equiv 0\) in equation (1) and the boundedness of all solutions and the existence of at the least one periodic solution of equation (1) when \(e(t) \neq 0\).

MSC:

34K20 Stability theory of functional-differential equations
34K12 Growth, boundedness, comparison of solutions to functional-differential equations
34K13 Periodic solutions to functional-differential equations
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