Interval criteria for forced oscillation with nonlinearities given by Riemann–Stieltjes integrals

We study the oscillation of second-order forced differential equations with nonlinearity given by a Riemann–Stieltjes integral of the form (p(t)x′)′+q(t)x+∫0br(t,s)|x(t)|α(s)sgnx(t)dξ(s)=e(t), where b∈(0,∞]b∈(0,∞], α∈C[0,b)α∈C[0,b) is strictly increasing such that 0≤α(0)<1<α(b−)0≤α(0)<1<α(b−), p,q,e∈C[0,∞)p,q,e∈C[0,∞) with p(t)>0p(t)>0, r∈C([0,∞)×[0,b))r∈C([0,∞)×[0,b)), and ξ:[0,b)→Rξ:[0,b)→R is nondecreasing. Interval oscillation criteria of the El-Sayed type and the Kong type are obtained. As a special case, the work in this paper unifies and improves the existing results in the literature for equations with a finite number of nonlinear terms. We also extend our results to equations with delays.