Oscillatory behavior of integro-dynamic and integral equations on time scales. (English) Zbl 1325.34109
Summary: By making use of asymptotic properties of nonoscillatory solutions, the oscillation behavior of solutions for the integro-dynamic equation
\[
x^{\varDelta}(t)=e(t)-\int_ 0^tk(t,s)f(s,x(s)){\varDelta}s, \quad t\geq 0
\]
and the integral equation
\[
x(t)=e(t)-\int_ 0^tk(t,s)f(s,x(s)){\varDelta}s, \quad t\geq 0
\]
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.
MSC:
| 34N05 | Dynamic equations on time scales or measure chains |
| 34K11 | Oscillation theory of functional-differential equations |
| 45J05 | Integro-ordinary differential equations |
| 26E70 | Real analysis on time scales or measure chains |
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