The existence and uniqueness of solutions and asymptotic estimate of solution formulas are studied for the following initial value problem: g ( t ) y ′ ( t ) = a y ( t ) [ 1 + f ( t , y ( t ) , ∫ 0 + t K ( t , s , y ( t ) , y ( s ) ) d s ) ] , y ( 0 + ) = 0 , t ∈ ( 0 , t 0 ] , where a > 0 is a constant and t 0 > 0 . An approach which combines topological method of T. Ważewski and Schauder's fixed point theorem is used.