Linearized oscillation theory for a nonlinear equation with a distributed delay

We obtain linearized oscillation theorems for the equation with distributed delays (1) x ̇ ( t ) + ∑ k = 1 m r k ( t ) ∫ − ∞ t f k ( x ( s ) ) d s R k ( t , s ) = 0 . The results are applied to logistic, Lasota–Wazewska and Nicholson’s blowflies equations with a distributed delay. In addition, the “Mean Value Theorem” is proved which claims that a solution of (1) also satisfies the linear equation with a variable concentrated delay x ̇ ( t ) + ( ∑ k = 1 m r k ( t ) f k ′ ( ξ k ( t ) ) ) x ( g ( t ) ) = 0 .