Oscillatory behavior of integro-dynamic and integral equations on time scales

By making use of asymptotic properties of nonoscillatory solutions, the oscillation behavior of solutions for the integro-dynamic equation xΔ(t)=e(t)−∫0tk(t,s)f(s,x(s))Δs,t≥0 and the integral equation x(t)=e(t)−∫0tk(t,s)f(s,x(s))Δs,t≥0 on time scales is investigated. Easily verifiable sufficient conditions are established for the oscillation of all solutions. The results are new for both continuous and discrete cases. The paper is concluded by an open problem.