Solvability and construction of solutions of integral equations
Кілттік сөздер:
integral equation, general solution, existence of a solution, necessary and sufficient condition, solvability criterion, extremal problem, minimizing sequenceАннотация
A class of integral equations with respect to one variable function as well as to multivariable
function that are solvable for any right hand side of an equation has been singled out. A necessary
and sufficient condition for existence of a solution has been obtained for the class of integral
equations and the general form of their exact solutions has been found. Necessary and sufficient
conditions for existence of solutions to the mentioned equations with a given right hand side
are obtained by reducing them to solving an extremal problem. An algorithm for solving the
extremal problem by constructing a minimizing sequence has been developed and a convergence
rate estimation has been obtained. A solvability criterion as a requirement on infimum of functional
has been formulated. A necessary and sufficient condition for solvability of an integral equation
with parameter has been obtained and its general solution has been found.
Библиографиялық сілтемелер
[2] Aisagaliev S.A., Belogurov A.P. Controllability and speed of the process described by a parabolic equation with bounded control // Siberian Mathematical Journal. - 2012. - Vol. 53, No. 1. - P. 13-28.
[3] Aisagaliev S.A. Controllability theory for dynamical systems. – Almaty: Kazakh universiteti, 2014. – 158 p. (in Russian)
[4] Aisagaliev S.A. Controllability and Optimal Control in Nonlinear Systems// Journal of Computer and Systems Sciences International. - 1994. - No 32(5). - P. 73-80.
[5] Aisagaliev S.A., Kabidoldanova A.A. Optimal control of dynamical systems. - Saarbrucken: Palmarium Academic Publishing, 2012. – 288 p. (in Russian)
[6] Aisagaliev S.A., Kabidoldanova A.A. On the Optimal Control of Linear Systems with Linear Performance Criterion and Constraints // Differential Equations. - 2012. - Vol. 48, No 6. - P. 832-844).
[7] Aisagaliev S.A., Aisagaliev T.S. Boundary value problems solving methods. – Almaty: Kazakh universiteti, 2002. – 348 p. (in Russian)
[8] Aisagaliev S.A., Kalimoldayev M.N. Constructive method for solving a boundary value problem for ordinary differential equations //Differential Equations / MAIK NAUKA. INTERPERIODICA. SPRINGER, 233 SPRING ST, NEW YORK, NY 10013-1578 USA. - 2015. - Vol. 51, Issue 2. - P. 149-162.
[9] Aisagaliev S.A., Zhunussova Zh.Kh. To the boundary value problem of ordinary differential equations [Electronic Journal of Qualitative Theory of Differential Equations (EJQTDE)]. - 2015. - No. 57. - P.1-17. - URL: http://www.math.u-szeged.hu/ejqtde
[10] Aisagaliev S.A. General solution of one class of integral equations // Mathematical journal. - 2005. - V. 5, No. 4 (18). - P. 17-34. (in Russian)
[11] Aisagaliev S.A. Constructive theory of boundary value optimal control problem. – Almaty: Kazakh universiteti, 2007. – 328 p. (in Russian)
[12] Aisagaliev S.A., Belogurov A.P., Sevryugin I.V. On solving the first kind Fredholm integral equation for multivariable function // Vestnik KazNU, ser. math., mech., inf. – 2011. - No 1(68). - P. 3-16. (in Russian)
[13] Vasiliyev F.P. Methods for solving extremal problems. – M: Nauka, 1981. – 400 p. (in Russian)
