## Journal of Applied Mathematics

### Three-Point Boundary Value Problems of Nonlinear Second-Order $q$-Difference Equations Involving Different Numbers of $q$

#### Abstract

We study a new class of three-point boundary value problems of nonlinear second-order q-difference equations. Our problems contain different numbers of q in derivatives and integrals. By using a variety of fixed point theorems (such as Banach’s contraction principle, Boyd and Wong fixed point theorem for nonlinear contractions, Krasnoselskii’s fixed point theorem, and Leray-Schauder nonlinear alternative) and Leray-Schauder degree theory, some new existence and uniqueness results are obtained. Illustrative examples are also presented.

#### Article information

Source
J. Appl. Math., Volume 2013 (2013), Article ID 763786, 12 pages.

Dates
First available in Project Euclid: 14 March 2014

https://projecteuclid.org/euclid.jam/1394808159

Digital Object Identifier
doi:10.1155/2013/763786

Mathematical Reviews number (MathSciNet)
MR3115287

Zentralblatt MATH identifier
06950860

#### Citation

Sitthiwirattham, Thanin; Tariboon, Jessada; Ntouyas, Sotiris K. Three-Point Boundary Value Problems of Nonlinear Second-Order $q$ -Difference Equations Involving Different Numbers of $q$. J. Appl. Math. 2013 (2013), Article ID 763786, 12 pages. doi:10.1155/2013/763786. https://projecteuclid.org/euclid.jam/1394808159

#### 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 Mathematics II, vol. 30, pp. 195–205, 1929.
• W. J. Trjitzinsky, “Analytic theory of linear q-difference 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 1999:25, Department of Mathematics, Uppsala University, Uppsala, Sweden, 1999.
• R. J. Finkelstein, “q-Field theory,” Letters in Mathematical Phy-sics, 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, 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, pp. 63–72, 1999.
• V. Kac and P. Cheung, Quantum Calculus, 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,” 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. S. 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 sec-ond-order q-difference equations with boundary conditions,” Advances in Difference Equations, vol. 2012, p. 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, 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, p. 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.
• W. Liu and H. Zhou, “Existence solutions for boundary value problem of nonlinear fractional q-difference equations,” Adva-nces in Difference Equations, vol. 2013, 113 pages, 2013.
• C. Yu and J. Wang, “Existence of solutions for nonlinear second-order q-difference equations with first-order q-derivatives,” Advances in Difference Equations, vol. 2013, p. 124, 2013.
• N. Pongarm, S. Asawasamrit, and J. Tariboon, “Sequential deri-vatives of nonlinear q-difference equations with three-point q-integral boundary conditions,” Journal of Applied Mathematics, vol. 2013, Article ID 605169, 9 pages, 2013.
• D. W. Boyd and J. S. W. Wong, “On nonlinear contractions,” Proceedings of the American Mathematical Society, vol. 20, pp. 458–464, 1969.
• M. A. Krasnoselskii, “Two remarks on the method of successive approximations,” Uspekhi Matematicheskikh Nauk, vol. 10, pp. 123–127, 1955.
• A. Granas and J. Dugundji, Fixed Point Theory, Springer, New York, NY, USA, 2003.