Existence of Positive Solutions for Boundary Value Problem of Nonlinear Fractional *q*-Difference Equation

Author(s)
Liu Yang

Affiliation(s)

Department of Mathematics and Computing Sciences, Hengyang Normal University, Hengyang, China.

Department of Mathematics and Computing Sciences, Hengyang Normal University, Hengyang, China.

ABSTRACT

In this paper, we investigate the existence of positive
solutions for a class of nonlinear *q*-fractional
boundary value problem. By using some fixed point theorems on cone, some
existence results of positive solutions are obtained.

Cite this paper

Yang, L. (2013) Existence of Positive Solutions for Boundary Value Problem of Nonlinear Fractional*q*-Difference Equation. *Applied Mathematics*, **4**, 1450-1454. doi: 10.4236/am.2013.410195.

Yang, L. (2013) Existence of Positive Solutions for Boundary Value Problem of Nonlinear Fractional

References

[1] I. Podlubny, “Fractional Differential Equations,” Academic Press, San Diego, 1999.

[2] M. El-Shahed, “Existence of Solution for a Boundary Value Problem of Fractional Order,” Advances in Applied Mathematical Analysis, Vol. 2, No. 1, 2007, pp. 1-8.

[3] S. Zhang, “Existence of Solution for a Boundary Value Problem of Fractional Order,” Acta Mathematica Scientia, Vol. 26, No. 2, 2006, pp. 220-228. http://dx.doi.org/10.1016/S0252-9602(06)60044-1

[4] M. El-Shahed and F. M. Al-Askar, “On the Existence of Positive Solutions for a Boundary Value Problem of Fracntional Order,” International Journal of Mathematical Analysis, Vol. 4, No. 13-16, 2010, pp. 671-678.

[5] Z. B. Bai and H. S. Lü, “Positive Solutions for Boundary Value Problem of Nonlinear Fractional Differential Equation,” Journal of Mathematical Analysis and Applications, Vol. 311, No. 2, 2005, pp. 495-505. http://dx.doi.org/10.1016/j.jmaa.2005.02.052

[6] F. H. Jackson, “On q-Functions and a Certain Difference Operator,” Transactions of the Royal Society Edinburgh, Vol. 46, No. 2, 1909, pp. 253-281. http://dx.doi.org/10.1017/S0080456800002751

[7] R. Jackson, “On q-Definite Integrals,” Quarterly Journal of Pure and Applied Mathematics, Vol. 41, 1910, pp. 193-203.

[8] V. Kac and P. Cheung, “Quantum Calculus,” Springer, New York, 2002.

http://dx.doi.org/10.1007/978-1-4613-0071-7

[9] W. A. Al-Salam, “Some Fractional q-Integrals and qDerivatives,” Proceedings of the Edinburgh Mathematical Society, Vol. 15, No. 2, 1967, pp. 135-140.

http://dx.doi.org/10.1017/S0013091500011469

[10] R. P. Agarwal, “Certain Fractional q-Integrals and qDerivatives,” Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 66, No. 2, 1969, pp. 365-370.

http://dx.doi.org/10.1017/S0305004100045060

[11] P. M. Rajkovi, S. D. Marinkovi and M. S. Stankovi, “On q-Analogues of Caputo Derivative and Mittag-Leffler Function,” Fractional Calculus Applied Analysis, Vol. 10, No. 4, 2007, pp. 359-373.

[12] F. M. Atici and P. W. Eloe, “Fractional q-Calculus on a Time Scale,” Journal of Nonlinear Mathematical Physics, Vol. 14, No. 3, 2007, pp. 333-370.

[13] M. El-Shahed and H. A. Hassan, “Positive Solutions of q-Difference Equation,” Proceedings of the American Mathematical Society, Vol. 138, No. 5, 2010, pp. 17331738.

http://dx.doi.org/10.1090/S0002-9939-09-10185-5

[14] R. A. C. Ferreira, “Nontrivial Solutions for Fractional qDifference Boundary Value Problems,” Electronic Journal of Qualitative Theory of Differential Equations, No. 70, 2010, pp. 1-10.

[15] R. A. C. Ferreira, “Positive Solutions for a Class of Boundary Value Problems with Fractional q-Differences,” Computers and Mathematics with Applications, Vol. 61, No. 2, 2011, pp. 367-373.

http://dx.doi.org/10.1016/j.camwa.2010.11.012

[16] M. El-Shahed and F. M. Al-Askar, “Positive Solutions for Boundary Value Problem of Nonlinear Fractional q-Differences,” ISRN Mathematical Analysis, Vol. 2011, 2011, pp. 1-12.

http://dx.doi.org/10.5402/2011/385459

[17] H. Gauchman, “Integral Inequality in q-Calculus,” Computers & Mathematics with Applications, Vol. 47, No. 2-3, 2004, pp. 281-300. http://dx.doi.org/10.1016/S0898-1221(04)90025-9

[18] M. D. Rus, “A Note on the Existence of Positive Solutions of Fredholm Integral Equations,” Fixed Point Theory, Vol. 5, No. 2, 2004, pp. 369-377.

[19] D. J. Guo and V. Lakshmikantham, “Nonlinear Problems in abstrAct Cones,” Academic Press, San Diego, 1988.

[1] I. Podlubny, “Fractional Differential Equations,” Academic Press, San Diego, 1999.

[2] M. El-Shahed, “Existence of Solution for a Boundary Value Problem of Fractional Order,” Advances in Applied Mathematical Analysis, Vol. 2, No. 1, 2007, pp. 1-8.

[3] S. Zhang, “Existence of Solution for a Boundary Value Problem of Fractional Order,” Acta Mathematica Scientia, Vol. 26, No. 2, 2006, pp. 220-228. http://dx.doi.org/10.1016/S0252-9602(06)60044-1

[4] M. El-Shahed and F. M. Al-Askar, “On the Existence of Positive Solutions for a Boundary Value Problem of Fracntional Order,” International Journal of Mathematical Analysis, Vol. 4, No. 13-16, 2010, pp. 671-678.

[5] Z. B. Bai and H. S. Lü, “Positive Solutions for Boundary Value Problem of Nonlinear Fractional Differential Equation,” Journal of Mathematical Analysis and Applications, Vol. 311, No. 2, 2005, pp. 495-505. http://dx.doi.org/10.1016/j.jmaa.2005.02.052

[6] F. H. Jackson, “On q-Functions and a Certain Difference Operator,” Transactions of the Royal Society Edinburgh, Vol. 46, No. 2, 1909, pp. 253-281. http://dx.doi.org/10.1017/S0080456800002751

[7] R. Jackson, “On q-Definite Integrals,” Quarterly Journal of Pure and Applied Mathematics, Vol. 41, 1910, pp. 193-203.

[8] V. Kac and P. Cheung, “Quantum Calculus,” Springer, New York, 2002.

http://dx.doi.org/10.1007/978-1-4613-0071-7

[9] W. A. Al-Salam, “Some Fractional q-Integrals and qDerivatives,” Proceedings of the Edinburgh Mathematical Society, Vol. 15, No. 2, 1967, pp. 135-140.

http://dx.doi.org/10.1017/S0013091500011469

[10] R. P. Agarwal, “Certain Fractional q-Integrals and qDerivatives,” Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 66, No. 2, 1969, pp. 365-370.

http://dx.doi.org/10.1017/S0305004100045060

[11] P. M. Rajkovi, S. D. Marinkovi and M. S. Stankovi, “On q-Analogues of Caputo Derivative and Mittag-Leffler Function,” Fractional Calculus Applied Analysis, Vol. 10, No. 4, 2007, pp. 359-373.

[12] F. M. Atici and P. W. Eloe, “Fractional q-Calculus on a Time Scale,” Journal of Nonlinear Mathematical Physics, Vol. 14, No. 3, 2007, pp. 333-370.

[13] M. El-Shahed and H. A. Hassan, “Positive Solutions of q-Difference Equation,” Proceedings of the American Mathematical Society, Vol. 138, No. 5, 2010, pp. 17331738.

http://dx.doi.org/10.1090/S0002-9939-09-10185-5

[14] R. A. C. Ferreira, “Nontrivial Solutions for Fractional qDifference Boundary Value Problems,” Electronic Journal of Qualitative Theory of Differential Equations, No. 70, 2010, pp. 1-10.

[15] R. A. C. Ferreira, “Positive Solutions for a Class of Boundary Value Problems with Fractional q-Differences,” Computers and Mathematics with Applications, Vol. 61, No. 2, 2011, pp. 367-373.

http://dx.doi.org/10.1016/j.camwa.2010.11.012

[16] M. El-Shahed and F. M. Al-Askar, “Positive Solutions for Boundary Value Problem of Nonlinear Fractional q-Differences,” ISRN Mathematical Analysis, Vol. 2011, 2011, pp. 1-12.

http://dx.doi.org/10.5402/2011/385459

[17] H. Gauchman, “Integral Inequality in q-Calculus,” Computers & Mathematics with Applications, Vol. 47, No. 2-3, 2004, pp. 281-300. http://dx.doi.org/10.1016/S0898-1221(04)90025-9

[18] M. D. Rus, “A Note on the Existence of Positive Solutions of Fredholm Integral Equations,” Fixed Point Theory, Vol. 5, No. 2, 2004, pp. 369-377.

[19] D. J. Guo and V. Lakshmikantham, “Nonlinear Problems in abstrAct Cones,” Academic Press, San Diego, 1988.