Discontinuity, Nonlinearity, and Complexity
On the Solvability of Nonlocal Boundary Value Problem for the Systems of Impulsive Hyperbolic Equations with Mixed Derivatives
Discontinuity, Nonlinearity, and Complexity 5(2) (2016) 153165  DOI:10.5890/DNC.2016.06.005
A.T. Assanova
Department of Differential equations, Institute of Mathematics and Mathematical Modelling, Almaty, 050010, Pushkin str., 125, Kazakhstan
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Abstract
A nonlocal boundary value problem for a system of impulsive hyperbolic equations at the fixed times is considered. The questions of existence, uniqueness, and construction of algorithms for finding the solutions to this problem are studied. By introducing the additional parameters as values of solutions on specific lines the considered problem is reduced to the problem consisting of the Goursat problem for a system of hyperbolic equations and the Cauchy problem for ordinary differential equations. The algorithms for finding the approximate solutions of latter problem are obtained and their convergence to the solution of original problem is proved. Conditions for existence of a unique solution to the nonlocal boundary value problem with impulse effects are set in the terms of initial data.
Acknowledgments
The author expresses her sincere appreciation to the reviewers for their helpful comments and suggestions that allowed improve the content of article.
References

[1]  Ptashnyck, B.I. (1984), Illposed boundary value problems for partial differential equations, Naukova Dumka: Kiev, Ukraine. (in Russian). 

[2]  Kiguradze, T. (1994), Some boundary value problems for systems of linear partial differential equations of hyperbolic type, Memoires Differential Equations and Mathematical Physics, 1, 1144. 

[3]  Cesari, L. (1963), Periodic solutions of hyperbolic partial differential equations, Proc. Internat. Sympos. Nonlinear Vibrations (Kiev 1961), Izd. Akad. Nauk Ukrain. SSR: Kiev, 2, 440457. 

[4]  Vejvoda, O. Herrmann, L., Lovicar, V. et al. (1982), Partial differential equations: time  periodic solutions, Martinus Nijhoff Publ: Prague  Hague  Boston  London. 

[5]  Samoilenko, A.M. and Tkach, B.P. (1992), Numericalanalytical methods in the theory periodical solutions of equations with partial derivatives, Naukova Dumka: Kiev, Ukraine. (in Russian). 

[6]  Kolesov, A. Ju., Mishchenko, E.F., Rozov, N.Kh. (1998), Asymptotic methods of investigation of periodic solutions of nonlinear hyperbolic equations, Proc. Steklov Inst. of Math., 222, 3188. 

[7]  Samoilenko, A.M. and Perestyuk, N.A. (1982), Periodic and almost periodic solutions of a differential equations with impulse effect, Ukranian mathematical journal, 34 (1), 6673. (in Russian). 

[8]  Samoilenko, A.M., Perestyuk, N.A. and Akhmetov,M.U. (1983), Almost periodic solutions of a differential equations with impulse effect, Preprint/ Ukranian Academy of Sciences. Institute of Mathematics; No. 83.26, Kiev. (in Russian). 

[9]  Akhmetov, M.U. and Perestyuk, N.A. (1984), On almost periodic solutions of a class of systems with impulse effect, Ukranian mathematical journal, 36 (4), 486490. (in Russian). 

[10]  Perestyuk, N.A. and Akhmetov, M.U. (1987), On almost periodic solutions of a impulsive systems, Ukranian mathematical journal, 39 (1), 7480. (in Russian). 

[11]  Akhmetov, M.U. and Perestyuk, N.A. (1989), Stability of periodic solutions of differential equations with impulse effect on surfaces, Ukranian mathematical journal, 41 (12), 15961601. (in Russian). 

[12]  Bainov, D.D. and Simeonov, P.S. (1989), Systems with Impulse Effect: Stability, Theory and Applications, Halsted Press: New York  Chichester  Brisbane  Toronto. 

[13]  Hu, S. and Lakshmikantham, V. (1989), Periodic boundary value problems for second order impulsive differential systems, Nonlinear Analysis, 13 (1), 7585. 

[14]  Lakshmikantham, V., Bainov, D.D. and Simeonov, P.S. (1989), Theory of Impulsive Differential Equations, World Scientific: Singapore. 

[15]  Samoilenko, A.M. and Perestyuk, N.A. (1995), Impulsive Differential Equations,World Scientific: Singapore. 

[16]  Rogovchenko, S.P. (1988), Periodic Solutions for Hyperbolic Impulsive Systems, Preprint/ Ukranian Academy of Sciences. Institute of Mathematics; No 88.3, Kiev. (in Russian). 

[17]  Perestyuk, N.A. and Tkach, A.B. (1997), Periodic solutions for weakly nonlinear partial system with pulse influense, Ukranian Mathematical Journal, 49 (4), 601605. 

[18]  Bainov, D.D., Minchev, E. and Myshkis, A. (1997), Periodic boundary value problems for impulsive hyperbolic systems, Commun. Appl. Anal., 1 (4), 114. 

[19]  Tkach, A.B. (2001), Numericalanalytic method of finding periodic solutions for systems of partial differential equations with pulse influence, Nonlinear oscillations, 4 (2), 278288. 

[20]  Asanova, A.T. (2013), On a nonlocal boundaryvalue problem for systems of impulsive hyperbolic equations, Ukranian Mathematical Journal, 65 (2), 349365. 

[21]  Asanova, A.T., Dzhumabaev, D.S. (2002), Unique Solvability of the Boundary Value Problem for Systems of Hyper bolic Equations with Data on the Characteristics, Computational Mathematics and Mathematical Physics, 42 (11), 16091621. 

[22]  Asanova, A.T., Dzhumabaev, D.S. (2003), Correct Solvability of a Nonlocal Boundary Value Problem for Systems of Hyperbolic Equations, Doklady Mathematics, 68 (1), 4649. 

[23]  Asanova, A.T., Dzhumabaev, D.S. (2003), Unique Solvability of Nonlocal Boundary Value Problems for Systems of Hyperbolic Equations, Differential Equations, 39 (10), 14141427. 

[24]  Asanova, A.T., Dzhumabaev, D.S., (2004), Periodic solutions of systems of hyperbolic equations bounded on a plane, Ukrainian Mathematical Journal, 56 (4), 682694. 

[25]  Asanova, A.T., Dzhumabaev, D.S. (2005),WellPosed Solvability of Nonlocal Boundary Value Problems for Systems of Hyperbolic Equations, Differential Equations, 41 (3), 352363. 

[26]  Asanova, A.T. and Dzhumabaev, D.S. (2013), Wellposedness of nonlocal boundary value problems with integral condition for the system of hyperbolic equations, Journal of Mathematical Analysis and Applications, 402 (1), 167 178. 