Abstract
Sufficient conditions are established guaranteeing the existence of a positive ω-periodic solution to the equation u �� + f (u)u � + g(u )= h(t, u),
Highlights
1 Introduction The topic of singular boundary value problems has been of substantial and rapidly growing interest for many scientists and engineers. The importance of such investigation is emphasized by the fact that numerical simulations of solutions to such problems usually break down near singular points
= p(R) – p∞, to model the bubble dynamics in liquid, where R(t) is the ratio of the bubble at the time t, ρ is the liquid density, p∞ is the pressure in the liquid at a large distance from the bubble, and p(R) is the pressure in the liquid at the bubble boundary
The results dealing with the existence of positive ω-periodic solutions of ( . ) were established in [ ] provided h is bounded from above
Summary
The topic of singular boundary value problems has been of substantial and rapidly growing interest for many scientists and engineers. We have proved (see [ ]) that this result cannot be extended to the case when h is a general integrable (and so unbounded) function unless some additional conditions are introduced. After the introduction and basic notation, we recall the definition of lower and upper functions to the problem ) in the case when there exists a couple of well-ordered lower and upper functions. The following definitions of lower and upper functions are suitable for us. Let α and β be lower and upper functions to the problem ). Besides, let us suppose that ρ fulfills at least one of the following conditions: (c) there exists a sequence {zn}+n=∞ of positive numbers such that lim n→+∞. There exists at least one solution to the problem Let g > , g ≥ , ≤ δ < , ν > γ , ν + δ > and either (μ + δ) sgn |c| ≥ or ν + δ ≥
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