Sturm theorems and distance between adjacent zeros for second order integro-differential equations

Between two adjacent zeros of any nontrivial solution of the second order ordinary differential equation x⁰⁰(t) + a(t)x⁰(t) + b(t)x(t) = 0 there is one and only one zero of every nonproportional solution. This principle of zeros’ distribution is known as the Sturm separation theorem which is a basis of many classical results on oscillation and asymptotic properties and on boundary value problems for ordinary differential equations. For delay and integro-differential equations this principle of zeros’ distribution is not true. In this paper, the assertion on validity of the Sturm separation theorem are proposed. Distance between two zeros of nontrivial solutions to integro-differential equations is estimated.