Discrete and Continuous Models and Applied Computational ScienceDiscrete and Continuous Models and Applied Computational Science2658-46702658-7149Peoples' Friendship University of Russia named after Patrice Lumumba (RUDN University)8410Research ArticleUniqueness and Stability of Solutions for Certain Linear Equations of the First Kind with Two VariablesAsanovAvytavyt.asanov@mail.ruKadenovaZ AKadenova71@mail.ruKyrgyz-Turkish University ManasMinistry of Education and Science of the Kyrgyz Republic150320133303508092016Copyright © 2013,2013The article is devoted to the study of uniqueness and stability of solutions of linear integral equations of the first kind with two independent variables. The relevance of the problem is due to the needs in development of new approaches for the regularization and uniqueness of the solution of linear integral equations of the first kind with two independent variables. Integral and operator equations of the first kind with two independent variables arise in theoretical and applied problems.Works of A.N. Tikhonov, M.M. Lavrentyev and B.K. Ivanov, in which a new concept of correctness of setting such targets is given, different from the classical, show tools for research of ill-posed problems, which stimulated the interest to the integral equations that are of great practical importance. At the present time the theory and applications of ill-posed problems have been rapidly developing. One of the classes of such ill-posed problems are integral equations of the first kind with two independent variables. As of approximate solutions of such problems, stable to small variations of the initial data, we use the solutions derived by the method of regularization. In this article we prove the theorem of uniqueness and obtain estimates of stability for such equations in families of sets of correctnesses. For the tasks solution the methods of functional analysis and method of nonnegative quadratic forms are used. The results of the work are new.linearinteqral equationsfirst kindtwo variablessolutionuniqueness and stabilityлинейныйинтегральные уравненияпервого родадвух переменныхрешениеединственность и устойчивость[Aparstyn A.S. Nonclassical Linear Volterra Equations of the First Kind. — Utrecht: VSP, 2003.][Asanov A. Regularization, Uniqueness and Existence of Solutions of Volterra Equations of the First Kind. — Utrecht: VSP, 1998.][Bukhgeim A.L. Volterra Equations and Inverse Problems. — Utrecht: VSP, 1999.][Imanaliev M.I., Asanov A. On Solutions of Systems of Volterra Nonlinear Integral Equations of the First Kind // Doklady Akademii Nauk. — 1989. — Vol. 309, No 5. — Pp. 1052–1055.][Imanaliev M.I., Asanov A. On Solutions of Systems of Nonlinear Two Dimensional Volterra Integral Equations of the First Kind // Doklady Akademii Nauk. — 1991. — Vol. 317, No 2. — Pp. 330–333.][Lavrent’ev M.M., Romanov V.G., Shishatskii S.P. ILL-posed Problems of Mathematical Physics and Analysis. — American Mathematical Society: Providence, R.I, 1986.][Magnitskii N.A. Linear VolterrA Integral Equations of the First and Third Kind // Doklady Akademii Nauk. — 1991. — Vol. 317, No 2. — Pp. 330–333.][Shishatskii S.P., Asanov A., Atamanov E.R. Uniqueness Problems for Degenerating Equations and Nonclassical Problems. — Utrecht: VSP, 2001.]