On the solutions of the linear integral equations of Volterra type

Some boundaries about the solution of the linear Volterra integral equations of the form f(t)=1-K*f were obtained as |f(t)|=<1, |f(t)|=<2 and |f(t)|=<4 in (J. Math. Anal. Appl. 1978; 64:381-397; Int. J. Math. Math. Sci. 1982; 5(1):123-131). The boundary of the solution function of an equation in this type was found as |f(t)|=<2n in (Integr. Equ. Oper. Theory 2002; 43:466-479), where t(element of)[0, (infinity)) and n is a natural number such that n>=2. In (Math. Comp. 2006; 75:1175-1199), it is shown that the boundary of the solution function of an equation in the same form can also be derived as that of (Integr. Equ. Oper. Theory 2002; 43:466-479) under different conditions than those of (Integr. Equ. Oper. Theory 2002; 43:466-479). In the present paper, the sufficient conditions for the boundedness of functions f, fʹ, fʹ, ..., f(n+3), (n(element of)N) defined on the infinite interval [0, (infinity)) are given by our method, where f is the solution of the equation f(t)=1-K*f.