One initial boundary-value problem for integro-differential equation of the second order with power nonlinearity

In the Sobolev space $W_\infty^2(\mathbb{R}^+)$ we investigate one initial boundary-value problem for integro-differential equation of the second order with power nonlinearity on a semi-axis. Assuming that summary-difference even kernel serves for the considered kernel as minorant in the sense of M.A. Krasnosel'skii, we prove the existence of nonnegative (nontrivial) solution in the Sobolev space $W_\infty^2(\mathbb{R}^+)$. We also calculate the limits of constructed solution at the infinity.