## Abstract

An m-point nonlocal boundary value problem is posed for quasi- linear differential equations of first order on the plane. Nonlocal boundary value problems are investigated using the algorithm of reducing nonlocal boundary value problems to a sequence of Riemann-Hilbert problems for a generalized analytic function. The conditions for the existence and uniqueness of a generalized solution in the space are considered.

## References

[1] A. Ashyralyev, O. Gercek: Nonlocal boundary value problem for elliptic-parabolic differential and difference equations. Discrete Dyn. Nat. Soc. (2008) 16p.

[2] R. Beals: Nonlocal Elliptic Boundary Value Problems. Bull. Amer. Math. Soc. 70 (5) (1964) 693-696.

[3] V. Beridze, D. Devadze, H. Meladze: On one nonlocal boundary value problem for quasilinear differential equations. In: Proceedings of A. Razmadze Mathematical Institute. (2014) 31-39.

[4] G. Berikelashvili: Construction and analysisi of difference schemes of the rate convergence. Memoirs of Difi. Euqat. and Math. Physics 38 (2006) 1-36.

[5] G. Berikelashvili: On the solvability of a nonlocal boundary value problem in the weighted Sobolev spaces. Proc. A. Razmadze Math. Inst. 119 (1999) 3-11.

[6] G. Berikelashvili: To a nonlocal generalization of the Dirichlet problem. Journal of Inequalities and Applications 2006 (1) (2006) 6p. Art.ID 93858

[7] A.V. Bitsadze, A.A. Samarskií: On some simple generalizations of linear elliptic boundary problems. Dokl. Akad. Nauk SSSR 185 (1969) 739-740. Translation in Sov. Math. Dokl., 10 (1969), 398-400

[8] F.E. Browder: Non-local elliptic boundary value problems. Amer. J. Math 86 (1964) 735-750.

[9] J.R. Canon: The solution of heat equation subject to the specification of energy. Quart. Appl. Math. 21 (2) (1963) 155-160.

[10] T. Carleman: Sur la théorie des équations intégrales et ses applications. Füssli, Zurich (1932).

[11] R. Courant, D. Hilbert: Methods of mathematical physics. Interscience Publishers, New York (1953).

[12] D. Devadze, V. Beridze: Optimality Conditions and Solution Algorithms of Optimal Control Problems for Nonlocal Boundary-Value Problems. Journal of Mathematical Sciences 218 (6) (2016) 731-736.

[13] D. Sh. Devadze and V. Sh. Beridze: Optimality conditions for quasi-linear differential equations with nonlocal boundary conditions. Uspekhi Mat. Nauk 68 (4) (2013) 179-180. Translation in Russian Math. Surveys, 68 (2013), 773-775

[14] D. Devadze, M. Dumbadze: An Optimal Control Problem for a Nonlocal Boundary Value Problem. Bulletin of The Georgian National Academy Of Sciences 7 (2) (2013) 71-74.

[15] J.I. Diaz, J-M. Rakotoson: On a non-local stationary free-boundary problem arising in the confinement of a plasma in a stellarator geometry. Arch.Rational. Mech. Anal. 134 (1) (1996) 53-95.

[16] D.G. Gordeziani: On methods for solving a class of non-local boundary-value problems. Ed. TSU, Tbilisi (1981) 32p.

[17] D.G. Gordeziani, G.A. Avalishvili: On the constructing of solutions of the nonlocal initial boundary value problems for one dimensional medium oscillation equations. Mathem. Mod. 12 (1) (2000) 93-103.

[18] D.G. Gordeziani, T.Z. Djioev: On solvability of one boundary value problem for a nonlinear elliptic equations. Bull. Georgian Acad. Sci 68 (2) (1972) 189-292.

[19] G. Gordeziani, N. Gordeziani, G. Avalishvili: Non-local boundary value problem for some partial differential equations. Bulletin of the Georgian Academy of Sciences 157 (1) (1998) 365-369.

[20] D. Gordezian, E. Gordeziani, T. Davitashvili, G. Meladze: On the solution of some non-local boundary and initial-boundary value problems. GESJ: Computer Science and Telecommunications 2010 (3) (161-169).

[21] A.V. Gulin, V.A. Marozova: A family of selfjoint difference schemes. Diff. Urav. 44 (9) (2008) 1249-1254.

[22] P.L. Gurevich: Asymptotics of Solution for nonlocal elliptic problems in plane bounded domains. Functional Differential Equations 10 (1-2) (2003) 175-214.

[23] A.K. Gushchin, V.P. Mikhailov: On the stability of nonlocal problems for a second order elliptic equation. Math. Sb. 185 (1) (1994) 121-160.

[24] V.A. Il’in, E.I. Moiseev: A two-dimensional nonlocal boundary value problem for the Poisson operator in the differential and the difference interpretation. Mat. Model. 2 (8) (1990) 139-156. (Russian)

[25] N.I. Ionkin: Solution of boundary-value problem in heat-conduction theory with non-classical boundary conditions. Diff, Urav. 13 (1977) 1177-1182.

[26] D.V. Kapanadze: On a nonlocal Bitsadze-Samarskiĭ boundary value problem. Differentsial’nye Uravneniya 23 (3) (1987) 543-545. (Russian)

[27] G.F. Mandzhavidze, V. Tuchke: Some boundary value problems for first-order nonlinear differential systems on the plane. Boundary value problems of the theory of generalized analytic functions and their applications Tbilis. Gos. Univ., Tbilisi (1983) 79-124. (Russian)

[28] B.P Paneyakh: Some nonlocal boundary value problems for linear differential operators. Mat. Zametki 35 (3) (1984) 425-434. (Russian)

[29] C.V. Pao: Reaction diffusion equations with nonlocal boundary and nonlocal initial conditions. J. Math. Anal. Appl. 195 (3) (1995) 702-718.

[30] M.P. Sapagovas: A difference method of increased order of accuracy for the Poisson equation with nonlocal conditions. Differential Equations 44 (7) (2008) 1018-1028.

[31] M.P. Sapagovas, R.Yu. Chegis: On some Boundary value problems with a non-local condition. Differentsial’nye Uravneniya 23 (7) (1987) 1268-1274. (Russian)

[32] F. Shakeris, M. Dehghan: The method of lines for solution of the one-dimensional wave equation subject to an integral consideration. Computers & Mathematics with Applications 56 (9) (2008) 2175-2188.

[33] V.V. Shelukin: A non-local in time model for radionuclide propagation in Stokes uid. Dinamika Splosh. Sredy 107 (1993) 180-193.

[34] A.L. Skubachevskin: On a spectrum of some nonlocal boundary value problems. Mat. Sb. 117 (7) (1982) 548-562. (Russian)

[35] I.N. Vekua: Generalized analytic functions. Second edition. Nauka, Moscow (1988). (Russian)

[36] V.B. Vasylyk: Exponentially convergent method for the m-point nonlocal problem for an elliptic differential equation in banach space. J. Numer. Appl. Math. 105 (2) (2011) 124-135.