On the asymptotic behavior of positive solutions of certain integral equations

We present the conditions under which every positive solution xx of the integral equation x(t)=a(t)+∫ct(t−s)α−1k(t,s)f(s,x(s))ds,c>1,α>0 satisfies x(t)=O(a(t))as  t→∞,i.e.,lim supt→∞x(t)a(t)<∞. From the obtained results, we derive a technique which can be applied to some related integral equations that are equivalent to certain fractional differential equations of Caputo derivative of any order.