Some stability and boundedness conditions for non-autonomous differential equations with deviating arguments

In this article, the author studies the stability and boundedness of solutions for the non-autonomous third order differential equation with a deviating argument, r: x ′′′ (t) + a(t)x ′′ (t) + b(t)g1(x ′ (t − r)) + g2(x ′ (t)) + h(x(t − r)) = p(t,x(t),x(t − r),x ′ (t),x ′ (t − r),x ′′ (t)), where r > 0 is a constant. Sufficient conditions are obtained; a stability result in the literature is improved and extended to the preceding equation for the case p(t,x(t),x(t − r),x ′ (t),x ′ (t −r),x ′′ (t)) = 0, and a new boundedness result is also established for the case p(t,x(t),x(t − r),x ′ (t),x ′ (t − r),x ′′ (t)) 6 0.