Existence and uniqueness theorems for integro-differential equations on the half-axis with non-difference kernel of a certain type. Upper and lower bounds for solutions (Q1914814)

The author considers the integro-differential equation \[ -{{d^2y } \over {dx^2}}+ y= \int^\infty_0 R(x-t) y(t) dt+ \int^\infty_0 R_1 (x+t) y(t) dt, \qquad x>0 \tag{E} \] and proves some existence and uniqueness theorems for both the initial value and the boundary value problem associated with (E). Some upper and lower bounds for solutions are also evaluated.