On a boundary value problem for integro-differential equations on the halfline

The following boundary value problem equation(0.1)x′(t)=p(t)∫0σ(t)q(s)x(τ(s))ds,equation(0.2)x(t)=φ(t)fort∈[τ0,0),lim inft→+∞|x(t)|<+∞ is considered, where p∈C(R+;(0,+∞))p∈C(R+;(0,+∞)), q∈Lloc(R+;R+)q∈Lloc(R+;R+), 0≤σ(t)≤t0≤σ(t)≤t, τ(t)