# Solvability and construction of solutions of integral equations

### Abstract

A class of integral equations with respect to one variable function as well as to multivariablefunction that are solvable for any right hand side of an equation has been singled out. A necessaryand sufficient condition for existence of a solution has been obtained for the class of integralequations and the general form of their exact solutions has been found. Necessary and sufficientconditions for existence of solutions to the mentioned equations with a given right hand sideare obtained by reducing them to solving an extremal problem. An algorithm for solving theextremal problem by constructing a minimizing sequence has been developed and a convergencerate estimation has been obtained. A solvability criterion as a requirement on infimum of functionalhas been formulated. A necessary and sufficient condition for solvability of an integral equationwith parameter has been obtained and its general solution has been found.### References

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[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.

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[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).

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[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)

[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)

Published

2017-11-24

How to Cite

АЙСАГАЛИЕВ, С. А.; АЙСАГАЛИЕВА, С. С.; КАБИДОЛДАНОВА, А. А..
Solvability and construction of solutions of integral equations.

**KazNU Bulletin. Mathematics, Mechanics, Computer Science Series**, [S.l.], v. 89, n. 2, p. 3-18, nov. 2017. ISSN 1563-0277. Available at: <http://bm.kaznu.kz/index.php/kaznu/article/view/347>. Date accessed: 17 aug. 2018.
Section

Mathematics

**Keywords:**integral equation, general solution, existence of a solution, necessary and sufficient condition, solvability criterion, extremal problem, minimizing sequence