Research on Asymptotic Behavior of a Class of Second-Order Impulsive Differential Equations Based on Principal and Nonprincipal Solutions

Source abstract

We study the asymptotic integration of second-order impulsive differential equations whose impulse conditions are of the general linear type Δx(θi)+aix(θi)=0,Δp(t)x′(θi)+bix(θi)+cix′(θi)=0, so that the state and the quasi-derivative are coupled simultaneously at the impulse moments. Our main contributions are as follows. First, we show that a first-order derivative term h(t)x′ in the differential operator can be absorbed into a renormalized quasi-derivative P(t)x′=p(t)exp∫h/px′; this transformation leaves the dimensionless pair ci/p(θi),bi/p(θi) and hence the monodromy factor (1−ai)1−ci/p(θi) unchanged, so the monodromy data alone carry no information on bi: within the present framework bi is a free parameter, and it is the simultaneous nonvanishing of bi and ci that places the equation outside the two impulse classes treated in the literature. Second, for the associated homogeneous linear impulsive equation we establish a generalized Wronskian (Abel) identity and prove the existence of positive principal and nonprincipal solutions, characterized by the divergence and convergence of the weighted integral ∫∞γ(t,t0)/(p(t)u2(t))dt, where γ(t,t0) is the product of the monodromy factors, together with the ordering of the quasi-logarithmic derivatives pu′/u<pv′/v. Third, using these solutions and the Schauder fixed point theorem, we prove that the associated nonlinear impulsive equations admit solutions with prescribed asymptotic behavior x(t)=av(t)+bu(t)+o(u(t)) and x(t)=av(t)+bu(t)+o(v(t)); in the second representation the coefficient a is determined by the leading term, while b is not identifiable since bu=o(v), and this is stated precisely in Theorem 4. The separated and mixed impulse cases are recovered as particular limits of our framework. Examples with explicit principal and nonprincipal solutions and numerical simulations, for which the integration scheme and its tolerances are specified, illustrate the results.