Paper's Title:

On Some Relations Among the Solutions of the Linear Volterra Integral Equations

Author(s):

Ismet Ozdemir and Faruk Temizer

Inönü Üniversitesi Eğitim Fakültesi,
44280-Malatya,
Turkey

ismet.ozdemir@inonu.edu.tr

omer.temizer@inonu.edu.tr
 

Abstract:

The sufficient conditions for y₁(x)≤ y₂(x) were given in [1] such that y_m(x)=f_m(x)+∫_a^x K_m(x, t)y_m(t)dt,(m=1,2) and x∈ [a, b]. Some properties such as positivity, boundedness and monotonicity of the solution of the linear Volterra integral equation of the form f(t)=1-∫₀^tK(t-τ)f(τ)dτ=1-K*f, (0≤ t<∞) were obtained, without solving this equation, in [3,4,5,6]. Also, the boundaries for functions f', f'',..., f⁽ⁿ⁾,(n ∈ N) defined on the infinite interval [0, ∞) were found in [7,8].

In this work, for the given equation f(t)=1-K* f and n≥ 2, it is derived that there exist the functions  L₂, L₃,..., L_n which can be obtained by means of K and some inequalities among the functions f, h₂, h₃,..., h_i for i=2, 3,...., n are satisfied on the infinite interval [0, ∞), where h_i is the solution of the equation h_i(t)=1-L_i* h_i and n is a natural number.

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