Journal of Applied Mathematics

Sequential Derivatives of Nonlinear q-Difference Equations with Three-Point q-Integral Boundary Conditions

Nittaya Pongarm, Suphawat Asawasamrit, and Jessada Tariboon

Full-text: Open access

Abstract

This paper studies sufficient conditions for the existence of solutions to the problem of sequential derivatives of nonlinear q-difference equations with three-point q-integral boundary conditions. Our results are concerned with several quantum numbers of derivatives and integrals. By using Banach's contraction mapping, Krasnoselskii's fixed-point theorem, and Leray-Schauder degree theory, some new existence results are obtained. Two examples illustrate our results.

Article information

Source
J. Appl. Math., Volume 2013 (2013), Article ID 605169, 9 pages.

Dates
First available in Project Euclid: 14 March 2014

Permanent link to this document
https://projecteuclid.org/euclid.jam/1394807921

Digital Object Identifier
doi:10.1155/2013/605169

Mathematical Reviews number (MathSciNet)
MR3039730

Zentralblatt MATH identifier
1266.39009

Citation

Pongarm, Nittaya; Asawasamrit, Suphawat; Tariboon, Jessada. Sequential Derivatives of Nonlinear $q$ -Difference Equations with Three-Point $q$ -Integral Boundary Conditions. J. Appl. Math. 2013 (2013), Article ID 605169, 9 pages. doi:10.1155/2013/605169. https://projecteuclid.org/euclid.jam/1394807921


Export citation

References

  • F. H. Jackson, “On q-difference equations,” American Journal of Mathematics, vol. 32, no. 4, pp. 305–314, 1910.
  • R. D. Carmichael, “The general theory of linear q-difference equations,” American Journal of Mathematics, vol. 34, no. 2, pp. 147–168, 1912.
  • T. E. Mason, “On properties of the solutions of linear q-difference equations with entire function coefficients,” American Journal of Mathematics, vol. 37, no. 4, pp. 439–444, 1915.
  • C. R. Adams, “On the linear ordinary q-difference equation,” American Mathematical Series II, vol. 30, pp. 195–205, 1929.
  • W. J. Trjitzinsky, “Analytic theory of linear q-differece equations,” Acta Mathematica, vol. 61, no. 1, pp. 1–38, 1933.
  • T. Ernst, “A new notation for q-calculus and a new q-Taylor formula,” U.U.D.M. Report, Department of Mathematics, Uppsala University, 1999.
  • R. J. Finkelstein, “q-field theory,” Letters in Mathematical Physics, vol. 34, no. 2, pp. 169–176, 1995.
  • R. J. Finkelstein, “q-deformation of the Lorentz group,” Journal of Mathematical Physics, vol. 37, no. 2, pp. 953–964, 1996.
  • R. Floreanini and L. Vinet, “Automorphisms of the q-oscillator algebra and basic orthogonal polynomials,” Physics Letters A, vol. 180, no. 6, pp. 393–401, 1993.
  • R. Floreanini and L. Vinet, “Symmetries of the q-difference heat equation,” Letters in Mathematical Physics, vol. 32, no. 1, pp. 37–44, 1994.
  • R. Floreanini and L. Vinet, “q-gamma and q-beta functions in quantum algebra representation theory,” Journal of Computational and Applied Mathematics, vol. 68, no. 1-2, pp. 57–68, 1996.
  • P. G. O. Freund and A. V. Zabrodin, “The spectral problem for the q-Knizhnik-Zamolodchikov equation and continuous q-Jacobi polynomials,” Communications in Mathematical Physics, vol. 173, no. 1, pp. 17–42, 1995.
  • G. Gasper and M. Rahman, Basic Hypergeometric Series, vol. 35 of Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, UK, 1990.
  • G.-N. Han and J. Zeng, “On a q-sequence that generalizes the median Genocchi numbers,” Annales des Sciences Mathématiques du Québec, vol. 23, no. 1, pp. 63–72, 1999.
  • V. Kac and P. Cheung, Quantum Calculus, Universitext, Springer, New York, NY, USA, 2002.
  • G. Bangerezako, “Variational q-calculus,” Journal of Mathematical Analysis and Applications, vol. 289, no. 2, pp. 650–665, 2004.
  • A. Dobrogowska and A. Odzijewicz, “Second order q-difference equations solvable by factorization method,” Journal of Computational and Applied Mathematics, vol. 193, no. 1, pp. 319–346, 2006.
  • G. Gasper and M. Rahman, “Some systems of multivariable orthogonal q-Racah polynomials,” The Ramanujan Journal, vol. 13, no. 1–3, pp. 389–405, 2007.
  • M. E. H. Ismail and P. Simeonov, “q-difference operators for orthogonal polynomials,” Journal of Computational and Applied Mathematics, vol. 233, no. 3, pp. 749–761, 2009.
  • M. Bohner and G. Sh. Guseinov, “The h-Laplace and q-Laplace transforms,” Journal of Mathematical Analysis and Applications, vol. 365, no. 1, pp. 75–92, 2010.
  • M. El-Shahed and H. A. Hassan, “Positive solutions of q-difference equation,” Proceedings of the American Mathematical Society, vol. 138, no. 5, pp. 1733–1738, 2010.
  • B. Ahmad, “Boundary-value problems for nonlinear third-order q-difference equations,” Electronic Journal of Differential Equations, vol. 94, pp. 1–7, 2011.
  • B. Ahmad, A. Alsaedi, and S. K. Ntouyas, “A study of second-order q-difference equations with boundary conditions,” Advances in Difference Equations, vol. 2012, article 35, 2012.
  • B. Ahmad, S. K. Ntouyas, and I. K. Purnaras, “Existence results for nonlinear q-difference equations with nonlocal boundary conditions,” Communications on Applied Nonlinear Analysis, vol. 19, no. 3, pp. 59–72, 2012.
  • B. Ahmad and J. J. Nieto, “On nonlocal boundary value problems of nonlinear q-difference equations,” Advances in Difference Equations, vol. 2012, article 81, 2012.
  • B. Ahmad and S. K. Ntouyas, “Boundary value problems for q-difference inclusions,” Abstract and Applied Analysis, vol. 2011, Article ID 292860, 15 pages, 2011.
  • M. A. Krasnoselskii, “Two remarks on the method of successive approximations,” Uspekhi Matematicheskikh Nauk, vol. 10, no. 1, pp. 123–127, 1955.