## Abstract and Applied Analysis

### A Note on Nonlocal Boundary Value Problems for Hyperbolic Schrödinger Equations

#### Abstract

The nonlocal boundary value problem ${d}^{2}u(t)/d{t}^{2}+Au(t)=f(t) (0\le t\le 1)$, $i(du(t)/dt)+Au(t)=g(t) (-1\le t\le 0)$, $u({0}^{+})=u({0}^{-})$, ${u}_{t}({0}^{+})={u}_{t}({0}^{-})$, $Au(-1)=\alpha u(\mu )+\phi$, $0<\mu \le 1$, for hyperbolic Schrödinger equations in a Hilbert space $H$ with the self-adjoint positive definite operator $A$ is considered. The stability estimates for the solution of this problem are established. In applications, the stability estimates for solutions of the mixed-type boundary value problems for hyperbolic Schrödinger equations are obtained.

#### Article information

Source
Abstr. Appl. Anal., Volume 2012, Special Issue (2012), Article ID 687321, 12 pages.

Dates
First available in Project Euclid: 5 April 2013

https://projecteuclid.org/euclid.aaa/1365174045

Digital Object Identifier
doi:10.1155/2012/687321

Mathematical Reviews number (MathSciNet)
MR2935138

Zentralblatt MATH identifier
1246.35035

#### Citation

Ozdemir, Yildirim; Kucukunal, Mehmet. A Note on Nonlocal Boundary Value Problems for Hyperbolic Schrödinger Equations. Abstr. Appl. Anal. 2012, Special Issue (2012), Article ID 687321, 12 pages. doi:10.1155/2012/687321. https://projecteuclid.org/euclid.aaa/1365174045

#### References

• M. S. Salakhitdinov, Equations of Mixed-Composite Type, FAN, Tashkent, Uzbekistan, 1974.
• T. D. Djuraev, Boundary Value Problems for Equations of Mixed and Mixed-Composite Types, FAN, Tashkent, Uzbekistan, 1979.
• M. G. Karatopraklieva, “A nonlocal boundary value problem for an equation of mixed type,” Differensial'nye Uravneniya, vol. 27, no. 1, p. 68, 1991 (Russian).
• D. Bazarov and H. Soltanov, Some Local and Nonlocal Boundary Value Problems for Equations of Mixed and Mixed-Composite Types, Ylym, Ashgabat, Turkmenistan, 1995.
• S. N. Glazatov, “Nonlocal boundary value problems for linear and nonlinear equations of variable type,” Sobolev Institute of Mathematics SB RAS, no. 46, p. 26, 1998.
• A. Ashyralyev and N. Aggez, “A note on the difference schemes of the nonlocal boundary value problems for hyperbolic equations,” Numerical Functional, vol. 25, no. 5-6, pp. 439–462, 2004.
• A. Ashyralyev and Y. Ozdemir, “On nonlocal boundary value problems for hyperbolic-parabolic equations,” Taiwanese Journal of Mathematics, vol. 11, no. 4, pp. 1075–1089, 2007.
• A. Ashyralyev and O. Gercek, “Nonlocal boundary value problems for elliptic-parabolic differential and difference equations,” Discrete Dynamics in Nature and Society, vol. 2008, Article ID 904824, 16 pages, 2008.
• A. Ashyralyev and A. Sirma, “Nonlocal boundary value problems for the Schrödinger equation,” Computers & Mathematics with Applications, vol. 55, no. 3, pp. 392–407, 2008.
• A. Ashyralyev and O. Yildirim, “On multipoint nonlocal boundary value problems for hyperbolic differential and difference equations,” Taiwanese Journal of Mathematics, vol. 14, no. 1, pp. 165–194, 2010.
• A. Ashyralyev and B. Hicdurmaz, “A note on the fractional Schrödinger differential equations,” Kybernetes, vol. 40, no. 5-6, pp. 736–750, 2011.
• A. Ashyralyev and F. Ozger, “The hyperbolic-elliptic equation with the nonlocal condition,” AIP Conference Proceedings, vol. 1389, pp. 581–584, 2011.
• Z. Zhao and X. Yu, “Hyperbolic Schrödinger equation,” Advances in Applied Clifford Algebras, vol. 14, no. 2, pp. 207–213, 2004.
• A. A. Oblomkov and A. V. Penskoi, “Laplace transformations and spectral theory of two-dimensional semidiscrete and discrete hyperbolic Schrödinger operators,” International Mathematics Research Notices, no. 18, pp. 1089–1126, 2005.
• A. Avila and R. Krikorian, “Reducibility or nonuniform hyperbolicity for quasiperiodic Schrödinger cocycles,” Annals of Mathematics, vol. 164, no. 3, pp. 911–940, 2006.
• M. Kozlowski and J. M. Kozlowska, “Development on the Schrodinger equation for attosecond laser pulse interaction with planck gas,” Laser in Engineering, vol. 20, no. 3-4, pp. 157–166, 2010.
• K. Tselios and T. E. Simos, “Runge-Kutta methods with minimal dispersion and dissipation for problems arising from computational acoustics,” Journal of Computational and Applied Mathematics, vol. 175, no. 1, pp. 173–181, 2005.
• D. P. Sakas and T. E. Simos, “Multiderivative methods of eighth algebraic order with minimal phase-lag for the numerical solution of the radial Schrödinger equation,” Journal of Computational and Applied Mathematics, vol. 175, no. 1, pp. 161–172, 2005.
• G. Psihoyios and T. E. Simos, “A fourth algebraic order trigonometrically fitted predictor-corrector scheme for IVPs with oscillating solutions,” Journal of Computational and Applied Mathematics, vol. 175, no. 1, pp. 137–147, 2005.
• Z. A. Anastassi and T. E. Simos, “An optimized Runge-Kutta method for the solution of orbital problems,” Journal of Computational and Applied Mathematics, vol. 175, no. 1, pp. 1–9, 2005.
• T. E. Simos, “Closed Newton-Cotes trigonometrically-fitted formulae of high order for long-time integration of orbital problems,” Applied Mathematics Letters, vol. 22, no. 10, pp. 1616–1621, 2009.
• S. Stavroyiannis and T. E. Simos, “Optimization as a function of the phase-lag order of nonlinear explicit two-step $P$-stable method for linear periodic IVPs,” Applied Numerical Mathematics, vol. 59, no. 10, pp. 2467–2474, 2009.
• T. E. Simos, “Exponentially and trigonometrically fitted methods for the solution of the Schrödinger equation,” Acta Applicandae Mathematicae, vol. 110, no. 3, pp. 1331–1352, 2010.
• M. E. Koksal, “Recent developments on operator-difference schemes for solving nonlocal BVPs for the wave equation,” Discrete Dynamics in Nature and Society, vol. 2011, Article ID 210261, 14 pages, 2011.
• A. Ashyralyev and M. E. Koksal, “On the numerical solution of hyperbolic PDEs with variable space operator,” Numerical Methods for Partial Differential Equations, vol. 25, no. 5, pp. 1086–1099, 2009.
• H. O. Fattorini, Second Order Linear Differential Equations in Banach Spaces, vol. 108 of North-Holland Mathematics Studies, North-Holland, Amsterdam, The Netherlands, 1985.
• S. Piskarev and S.-Y. Shaw, “On certain operator families related to cosine operator functions,” Taiwanese Journal of Mathematics, vol. 1, no. 4, pp. 527–546, 1997.
• P. E. Sobolevskii, Difference Methods for the Approximate Solution of Differential Equations, Izdat, Voronezh Gosud University, Voronezh, Russia, 1975.