On the oscillation of Volterra integral equations with positive and negative nonlinearities (Q2804388)

The author considers the question when the solutions to the Volterra integral equation NEWLINE\[NEWLINE x(t)=f(t)-\int_0^t k(t,s)(h_1(s,x(s))-h_2(s,x(s)))\, ds NEWLINE\]NEWLINE are oscillatory, i.e., have arbitrarily large zeros. A number of different results concerning this question are stated under the common assumptions that \(0 \leq k(t,s)\leq k_1(t)k_2(s)\), \(h_1(t,u)\geq p(t)\mathcal F(u)\), \(h_2(t,u)\leq q(t)\mathcal G(u)\), where \(x\mathcal F(x)>0\), \(x\mathcal G(x)>0\), when \(x\neq 0\) and \(\lim_{|x|\to \infty}x^{-1}\mathcal F(x) >1\), \(\lim_{|x|\to \infty} x^{-1}\mathcal G(x) <1\), \(\lim_{x\to 0} x^{-1}\mathcal F(x) <1\), \(\lim_{x\to 0} x^{-1}\mathcal G(x) >1\).