AN ANTI-PERIODIC CAPUTOq FRACTIONAL BOUNDARY VALUE PROBLEM WITH Ap LAPLACIAN OPERATOR

Bahaa El-din Abdallah; T. Abdeljawad

doi:10.1501/Commua1_0000000714

AN ANTI-PERIODIC CAPUTOq FRACTIONAL BOUNDARY VALUE PROBLEM WITH Ap LAPLACIAN OPERATOR

Year 2014, Volume: 63 Issue: 2, 91 - 99, 01.08.2014

Bahaa El-din Abdallah T. Abdeljawad

https://doi.org/10.1501/Commua1_0000000714

Cited By: 2

Abstract

This paper studies the existence of solutions for an anti-periodicboundary value problem for the q-fractional p-Laplacian equation. The existence result depends on Leray-Schaefer’s Fixed Point Theorem

Keywords

p-Laplacian, Caputo q-fractional derivative

References

[1] T. Chen and w. Liu, An anti-periodic boundary value problem for the fractional di§erential equation with a pLaplacian operator, Applied Mathematics Letters, vol. 25, no. 11, pp. 1671-1675, 2012.
[2] H. Aktuglu and M. A. ÷zarslan, On the solvability of Caputo º qfractional boundary value problem involving pLaplacian operator, Hindawi Publishing Corporation, vol. 2013, Article ID 658617, 8 pages.
[3] T. Abdeljawad, D. Baleanu, Caputo q-Fractional Initial Value Problems and a q-Analogue Mittag-Le er Function, Communications in Nonlinear Science and Numerical Simulations, vol. 16 (12), 4682-4688 (2011).
[4] Thabet Abdeljawad, J. Alzabut, The q-fractional analogue for Gronwall-type inequality, Vol. 2013 (2013), Article ID 543839, 7 pages.
[5] T. Abdeljawad, D. Dumitru, Fractional di§erences and integration by parts, Journal of computational Analysis and Applications, vol 13, no. 3, 574-582.
[6] Thabet Abdeljawad, Betul Benli, Dumitru Baleanu, A generalized q-Mittag-Le er function by Caputo fractional linear equations, Abstract and Applied Analysis, vol 2012, 11 pages, Article ID 546062 (2012).
[7] Fahd Jarad, Thabet Abdeljawad, Dumitru Baleanu. Stability of q-farctional nonautonomous systems, Nonlinear Analysis: Real and World Applications, doi: 10.1016/j.nnorwa.2012.08.001, (2012).
[8] T. Abdeljawad, F. Jarad, D. Baleanu, A semigroup-like property for discrete Mittag-Le er functions, Advances in Di§erence Equations, 2012/1/72.
[9] N. Bastos, R. A. C. Ferreira, D. F. M. Torres, Discrete-Time Variational Problems, Signal Processing, Volume 91 Issue 3, March, 2011.
[10] F. M. At¨c¨, P.W. Eloe, A Transform method in discrete fractional calculus , International Journal of Di§erence Equations, vol 2, no 2, (2007), 165ñ176.
[11] F. M. At¨c¨, P.W.Eloe, Initial value problems in discrete fractional calculus, Proceedings of the American Mathematical Society, to appear.
[12] F. M. At¨c¨, P.W. Eloe, Fractional q-calculus on a time scale , Journal of Nonlinear Mathematical Physics 14, 3, (2007), 333ñ344.
[13] K.S. Miller, B. Ross,Fractional di§erence calculus, Proceedings of the International Symposium on Univalent Functions, Fractional Calculus and Their Applications, Nihon University, Koriyama, Japan, (1989), 139-152.
[14] T. Ernst, The history of q-calculus and new method (Licentiate Thesis), U.U.D.M. Report 2000: http://math.uu.se/thomas/Lics.pdf.
[15] R. P. Agrawal, Certain fractional q-integrals and q-derivatives, Proc. Camb. Phil. Soc. (1969), 66,365, 365-370.
[16] W.A. Al-Salam, Some fractional q-integrals and q-derivatives, Proc. Edin. Math. Soc., vol 15 (1969), 135
[17] W.A. Al-Salam, A. Verma, A fractional Leibniz q-formula, PaciÖc Journal of Mathematics, vol 60, (1975), 1-9.
[18] W. A. Al-Salam, q-Analogues of Cauchyís formula, Proc. Amer. Math. Soc. 17,182-184,(1952- 1953).
[19] M. R. Predrag,D. M. Sladana, S. S. Miomir, Fractional Integrals and Derivatives in q-calculus, Applicable Analysis and Discrete Mathematics, 1, 311-323, (2007).
[20] I. Podlubny, Fractional Di§erential Equations, Academic Press, 1999.
[21] S. Samko, A. A. Kilbas, Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Yverdon, 1993.
[22] A. A. Kilbas, M. H. Srivastava, J. J. Trujillo, Theory and Application of Fractional Di§erential Equations, North Holland Mathematics Studies 204, 2006.
[23] A. A. Kilbas, M. Saigo, On solution of integral equation of Abel-Volterra type, Di§. Integr. Equat., 8 (5), (1995) 993-1011.
[24] W. Hahn, Beitr‰ge zur Theorie der Heineschen Reihen. Die 24 Integrale der Hypergeometrischen q-Di§erenzengleichung. Das q-Analogon der Laplace-Transformation, Math. Nachr., 2, 340-379 (1949).
[25] Ferhan M. At¨c¨, Paul W. Eloe, Gronwall‚eTM s inequality on discrete fractional calculus, Computers & Mathematics with Applications, Volume 64, Issue 10, November 2012, Pages 3193‚eì3200.
[26] R. P. Agarwal, Di§erence equations and inequalities, 2nd ed., Monographs and Textbooks in Pure and Applied Mathematics, vol. 228, Marcel Dekker Inc., New York, 2000. Theory, methods, and applications.
[27] G. A. Anastassiou, Nabla discrete fractional calculus and nabla inequalities, Math. Comput. Modelling 51 (2010), no. 5-6, 562-571.
[28] F. M. Atici, P. W. Eloe, A transform method in discrete fractional calculus, Int. J. Di§erence Equ. 2 (2007), no. 2, 165-176.
[29] F. M. Atici, Paul W. Eloe, Initial value problems in discrete fractional calculus, Proc. Amer. Math. Soc. 137 (2009), no. 3, 981-989.
[30] F. M. Atici, S. SengÃ 1 4 l, Modeling with fractional di§erence equations, J. Math. Anal. Appl. 369 (2010), no. 1, 1-9.
[31] F. M. Atici, P. W. Eloe, Twoñpoint boundary value problems for Önite fractional di§erence equations, J. Di§erence Equ. Appl. 17 (2011), no. 4, 445-456.
[32] N. R. O. Bastos, R. A. C. Ferreira, D. F. M. Torres, Necessary optimality conditions for fractional di§erence problems of the calculus of variations, Discrete Contin. Dyn. Syst. 29 (2011), no. 2, 417-437.
[33] J. B. Diaz, T. J. Osler, Di§erences of fractional order, Math. Comp. 28 (1974), 185-202.
[34] C. S. Goodrich, Continuity of solutions to discrete fractional initial value problems, Comput. Math. Appl. 59 (2010), no. 11, 3489-3499.
[35] C. S. Goodrich, Solutions to a discrete rightñfocal fractional boundary value problem, Int. J. Di§erence Equ. 5 (2010), no. 2, 195-216.
[36] H. L. Gray, N. F. Zhang, On a new deÖnition of the fractional di§erence, Math. Comp. 50 (1988), no. 182, 513-529.
[37] R. L. Magin, Fractional Calculus in Bioengineering, Begell House, 2006.
[38] Mohamed A. Khamsi, William A. Kirk, An Introduction to Metric Spaces and Fixed Point Theory,A Wiley-Interscience Series of Texts,2001

Year 2014, Volume: 63 Issue: 2, 91 - 99, 01.08.2014

Bahaa El-din Abdallah T. Abdeljawad

https://doi.org/10.1501/Commua1_0000000714

Cited By: 2

Abstract

References

[1] T. Chen and w. Liu, An anti-periodic boundary value problem for the fractional di§erential equation with a pLaplacian operator, Applied Mathematics Letters, vol. 25, no. 11, pp. 1671-1675, 2012.
[2] H. Aktuglu and M. A. ÷zarslan, On the solvability of Caputo º qfractional boundary value problem involving pLaplacian operator, Hindawi Publishing Corporation, vol. 2013, Article ID 658617, 8 pages.
[3] T. Abdeljawad, D. Baleanu, Caputo q-Fractional Initial Value Problems and a q-Analogue Mittag-Le er Function, Communications in Nonlinear Science and Numerical Simulations, vol. 16 (12), 4682-4688 (2011).
[4] Thabet Abdeljawad, J. Alzabut, The q-fractional analogue for Gronwall-type inequality, Vol. 2013 (2013), Article ID 543839, 7 pages.
[5] T. Abdeljawad, D. Dumitru, Fractional di§erences and integration by parts, Journal of computational Analysis and Applications, vol 13, no. 3, 574-582.
[6] Thabet Abdeljawad, Betul Benli, Dumitru Baleanu, A generalized q-Mittag-Le er function by Caputo fractional linear equations, Abstract and Applied Analysis, vol 2012, 11 pages, Article ID 546062 (2012).
[7] Fahd Jarad, Thabet Abdeljawad, Dumitru Baleanu. Stability of q-farctional nonautonomous systems, Nonlinear Analysis: Real and World Applications, doi: 10.1016/j.nnorwa.2012.08.001, (2012).
[8] T. Abdeljawad, F. Jarad, D. Baleanu, A semigroup-like property for discrete Mittag-Le er functions, Advances in Di§erence Equations, 2012/1/72.
[9] N. Bastos, R. A. C. Ferreira, D. F. M. Torres, Discrete-Time Variational Problems, Signal Processing, Volume 91 Issue 3, March, 2011.
[10] F. M. At¨c¨, P.W. Eloe, A Transform method in discrete fractional calculus , International Journal of Di§erence Equations, vol 2, no 2, (2007), 165ñ176.
[11] F. M. At¨c¨, P.W.Eloe, Initial value problems in discrete fractional calculus, Proceedings of the American Mathematical Society, to appear.
[12] F. M. At¨c¨, P.W. Eloe, Fractional q-calculus on a time scale , Journal of Nonlinear Mathematical Physics 14, 3, (2007), 333ñ344.
[13] K.S. Miller, B. Ross,Fractional di§erence calculus, Proceedings of the International Symposium on Univalent Functions, Fractional Calculus and Their Applications, Nihon University, Koriyama, Japan, (1989), 139-152.
[14] T. Ernst, The history of q-calculus and new method (Licentiate Thesis), U.U.D.M. Report 2000: http://math.uu.se/thomas/Lics.pdf.
[15] R. P. Agrawal, Certain fractional q-integrals and q-derivatives, Proc. Camb. Phil. Soc. (1969), 66,365, 365-370.
[16] W.A. Al-Salam, Some fractional q-integrals and q-derivatives, Proc. Edin. Math. Soc., vol 15 (1969), 135
[17] W.A. Al-Salam, A. Verma, A fractional Leibniz q-formula, PaciÖc Journal of Mathematics, vol 60, (1975), 1-9.
[18] W. A. Al-Salam, q-Analogues of Cauchyís formula, Proc. Amer. Math. Soc. 17,182-184,(1952- 1953).
[19] M. R. Predrag,D. M. Sladana, S. S. Miomir, Fractional Integrals and Derivatives in q-calculus, Applicable Analysis and Discrete Mathematics, 1, 311-323, (2007).
[20] I. Podlubny, Fractional Di§erential Equations, Academic Press, 1999.
[21] S. Samko, A. A. Kilbas, Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Yverdon, 1993.
[22] A. A. Kilbas, M. H. Srivastava, J. J. Trujillo, Theory and Application of Fractional Di§erential Equations, North Holland Mathematics Studies 204, 2006.
[23] A. A. Kilbas, M. Saigo, On solution of integral equation of Abel-Volterra type, Di§. Integr. Equat., 8 (5), (1995) 993-1011.
[24] W. Hahn, Beitr‰ge zur Theorie der Heineschen Reihen. Die 24 Integrale der Hypergeometrischen q-Di§erenzengleichung. Das q-Analogon der Laplace-Transformation, Math. Nachr., 2, 340-379 (1949).
[25] Ferhan M. At¨c¨, Paul W. Eloe, Gronwall‚eTM s inequality on discrete fractional calculus, Computers & Mathematics with Applications, Volume 64, Issue 10, November 2012, Pages 3193‚eì3200.
[26] R. P. Agarwal, Di§erence equations and inequalities, 2nd ed., Monographs and Textbooks in Pure and Applied Mathematics, vol. 228, Marcel Dekker Inc., New York, 2000. Theory, methods, and applications.
[27] G. A. Anastassiou, Nabla discrete fractional calculus and nabla inequalities, Math. Comput. Modelling 51 (2010), no. 5-6, 562-571.
[28] F. M. Atici, P. W. Eloe, A transform method in discrete fractional calculus, Int. J. Di§erence Equ. 2 (2007), no. 2, 165-176.
[29] F. M. Atici, Paul W. Eloe, Initial value problems in discrete fractional calculus, Proc. Amer. Math. Soc. 137 (2009), no. 3, 981-989.
[30] F. M. Atici, S. SengÃ 1 4 l, Modeling with fractional di§erence equations, J. Math. Anal. Appl. 369 (2010), no. 1, 1-9.
[31] F. M. Atici, P. W. Eloe, Twoñpoint boundary value problems for Önite fractional di§erence equations, J. Di§erence Equ. Appl. 17 (2011), no. 4, 445-456.
[32] N. R. O. Bastos, R. A. C. Ferreira, D. F. M. Torres, Necessary optimality conditions for fractional di§erence problems of the calculus of variations, Discrete Contin. Dyn. Syst. 29 (2011), no. 2, 417-437.
[33] J. B. Diaz, T. J. Osler, Di§erences of fractional order, Math. Comp. 28 (1974), 185-202.
[34] C. S. Goodrich, Continuity of solutions to discrete fractional initial value problems, Comput. Math. Appl. 59 (2010), no. 11, 3489-3499.
[35] C. S. Goodrich, Solutions to a discrete rightñfocal fractional boundary value problem, Int. J. Di§erence Equ. 5 (2010), no. 2, 195-216.
[36] H. L. Gray, N. F. Zhang, On a new deÖnition of the fractional di§erence, Math. Comp. 50 (1988), no. 182, 513-529.
[37] R. L. Magin, Fractional Calculus in Bioengineering, Begell House, 2006.
[38] Mohamed A. Khamsi, William A. Kirk, An Introduction to Metric Spaces and Fixed Point Theory,A Wiley-Interscience Series of Texts,2001

There are 38 citations in total.

Details

Primary Language	English
Journal Section	Research Articles
Authors	Bahaa El-din Abdallah This is me T. Abdeljawad This is me
Publication Date	August 1, 2014
Published in Issue	Year 2014 Volume: 63 Issue: 2

Cite

APA	Abdallah, B. E.-d., & Abdeljawad, T. (2014). AN ANTI-PERIODIC CAPUTOq FRACTIONAL BOUNDARY VALUE PROBLEM WITH Ap LAPLACIAN OPERATOR. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 63(2), 91-99. https://doi.org/10.1501/Commua1_0000000714
AMA	Abdallah BEd, Abdeljawad T. AN ANTI-PERIODIC CAPUTOq FRACTIONAL BOUNDARY VALUE PROBLEM WITH Ap LAPLACIAN OPERATOR. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. August 2014;63(2):91-99. doi:10.1501/Commua1_0000000714
Chicago	Abdallah, Bahaa El-din, and T. Abdeljawad. “AN ANTI-PERIODIC CAPUTOq FRACTIONAL BOUNDARY VALUE PROBLEM WITH Ap LAPLACIAN OPERATOR”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 63, no. 2 (August 2014): 91-99. https://doi.org/10.1501/Commua1_0000000714.
EndNote	Abdallah BE-d, Abdeljawad T (August 1, 2014) AN ANTI-PERIODIC CAPUTOq FRACTIONAL BOUNDARY VALUE PROBLEM WITH Ap LAPLACIAN OPERATOR. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 63 2 91–99.
IEEE	B. E.-d. Abdallah and T. Abdeljawad, “AN ANTI-PERIODIC CAPUTOq FRACTIONAL BOUNDARY VALUE PROBLEM WITH Ap LAPLACIAN OPERATOR”, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., vol. 63, no. 2, pp. 91–99, 2014, doi: 10.1501/Commua1_0000000714.
ISNAD	Abdallah, Bahaa El-din - Abdeljawad, T. “AN ANTI-PERIODIC CAPUTOq FRACTIONAL BOUNDARY VALUE PROBLEM WITH Ap LAPLACIAN OPERATOR”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 63/2 (August 2014), 91-99. https://doi.org/10.1501/Commua1_0000000714.
JAMA	Abdallah BE-d, Abdeljawad T. AN ANTI-PERIODIC CAPUTOq FRACTIONAL BOUNDARY VALUE PROBLEM WITH Ap LAPLACIAN OPERATOR. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2014;63:91–99.
MLA	Abdallah, Bahaa El-din and T. Abdeljawad. “AN ANTI-PERIODIC CAPUTOq FRACTIONAL BOUNDARY VALUE PROBLEM WITH Ap LAPLACIAN OPERATOR”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, vol. 63, no. 2, 2014, pp. 91-99, doi:10.1501/Commua1_0000000714.
Vancouver	Abdallah BE-d, Abdeljawad T. AN ANTI-PERIODIC CAPUTOq FRACTIONAL BOUNDARY VALUE PROBLEM WITH Ap LAPLACIAN OPERATOR. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2014;63(2):91-9.

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