On the bounded derivatives of the solutions of the linear Volterra integral equations

摘要

The boundaries for the solution of the linear Volterra integral equations of the second type of the form f(t) = 1 - ∫_0~t K(t - τ)f(τ)dι = 1 - K * f with unit source term and positive monotonically increasing convolution kernel were obtained as ∣f(t)∣ ≤ 1, ∣f(t)∣ ≤ 2 and ∣f(t)∣ ≤ 4 in [R. Ling, Integral equations of Volterra type, J. Math. Anal. Appl. 64 (1978), pp. 381-397, R. Ling, Solutions of singular integral equations, Internat. J. Math. & Math. Sci. 5 (1982), pp. 123-131.]. The sufficient conditions which are useful for finding the boundary such as ∣f(t)∣ ≤ 2~n of the solution of this equation were given, where 0 ≤ t < ∞ and n is a natural number, [I. OEzdemir and OE. F. Temizer, The boundaries of the solutions of the linear Volterra integral equations with convolution kernel, Math. Comp. 75 (2006), pp. 1175-1199.]. In this paper, a method which ensures finding the boundaries of the derivative functions f', f",..., f~(n+2) for n € N of the solution of the same equation has been developed.