Forced oscillation of second-order differential equations with mixed nonlinearities
نویسندگان
چکیده
where t ≥ t > , n ≥ is a natural number, βi ≥ (i = , , . . . ,n) are constants, r ∈ C([t,∞),R), qj, τj, e ∈ C([t,∞),R), r(t) > , r′(t)≥ , qj(t)≥ (j = , , , . . . ,n), e(t)≤ . We also assume that there exists a function τ ∈ C([t,∞),R) such that τ (t) ≤ τj(t) (j = , , , . . . ,n), τ (t)≤ t, limt→∞ τ (t) =∞, and τ ′(t) > . We consider only those solutions x of equation (.) which satisfy condition sup{|x(t)| : t ≥ T} > for allT ≥ t.We assume that (.) possesses such solutions. As usual, a solution of (.) is called oscillatory if it has arbitrarily large zeros on the interval [t,∞); otherwise, it is termed nonoscillatory. Equation (.) is said to be oscillatory if all its solutions are oscillatory. Functional differential equations arise in many applied problems in natural sciences, technology, and automatic control; see, for instance, Hale []. Inmechanical and engineering problems, questions related to the existence of oscillatory and nonoscillatory solutions play an important role. As a result, many theoretical studies have been undertaken during the past few years. We refer the reader to [–] and the references cited therein.
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