On One Class of Nonclassical Linear Volterra Integral Equations of the First Kind
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ON ONE CLASS OF NONCLASSICAL LINEAR VOLTERRA INTEGRAL EQUATIONS OF THE FIRST KIND A. Asanov1 and T. Bekeshov2, 3
UDC 517.968
On the basis of a new approach, we prove the uniqueness theorem and construct Lavrent’ev’s regularizing operators for the solution of nonclassical linear Volterra integral equations of the first kind with nondifferentiable kernels.
Consider a nonclassical linear integral equation of the first kind Zt
K(t, s)u(s) ds = f (t),
(1)
t 2 [t0 , T ],
↵(t)
where ↵(t) 2 C[t0 , T ],
↵(t0 ) = t0 ,
↵(t) t for t 2 [t0 , T ],
K(t, s) and f (t) are known functions in the domains � G = (t, s); t0 t T, ↵(t) s t
and
[t0 , T ],
respectively, f (t0 ) = 0, and u(t) is the required function. The theory and applications of integral equations were studied in numerous works. Thus, in particular, a survey of the results of investigations of the Volterra integral equations of the second kind is presented in [1]. The Volterra integral equations of the first and third kinds with smooth kernels were studied and the existence of a multiparameter family of solutions was proved in [2]. In [3], Lavrent’ev’s regularizing operators were constructed for the solution of linear Fredholm integral equations of the first kind. The theory and numerical methods used for the solution of nonclassical linear Volterra integral equations of the first kind with differentiable and nonzero kernels on a diagonal were analyzed in [4]. In [4–7], one can find the applications of nonclassical Volterra integral equations of the first kind to various practical problems. In [8], by using the Lavrent’ev’s regularization method, approximate solutions were constructed for the Volterra integral equations of the first kind with smooth and nonzero kernels on the diagonal and differentiable solutions. The theorems on uniqueness of solutions were proved and regularizing operators for the solutions of systems of linear and nonlinear Volterra integral equations of the first and third kinds were constructed in [9, 10]. In [11], the uniqueness theorem was proved for a system of linear integral Fredholm equations of the third kind and a regularizing operator was constructed for the solution of this system. On the basis 1
“Manas” Kyrgyz-Turkish University, Bishkek, Kyrgyzstan; e-mail: [email protected]. Osh State University, Osh, Kyrgyzstan; e-mail: [email protected]. 3 Corresponding author. 2
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, No. 2, pp. 161–172, February, 2020. Original article submitted February 21, 2019. 0041-5995/20/7202–0177
© 2020
Springer Science+Business Media, LLC
177
A. A SANOV
178
AND
T. B EKESHOV
of a new approach, the problems of existence and uniqueness of the solutions of scalar Fredholm integral equations of the third kind with multipoint singularities and systems of these equations were investigated in [12, 13]. A survey of the results of investigations of Volterra integral equations of the first kind can be found in [14]. In the present paper, on the basis of a modified method pro
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