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Positive solutions of nonlocal boundary value problem for high-order nonlinear fractional \(q\)-difference equations. (English) Zbl 1470.39021

Summary: We study the nonlinear \(q\)-difference equations of fractional order \((D_q^\alpha u)(t) + f(t, u(t)) = 0\), \(0 < t < 1\), \((D_q^i u)(0) = 0\), \((D_q^\beta u)(1) = a(D_q^\beta u)(\eta)\), \(0 \leq i \leq n - 2\), where \(D_q^\alpha\) is the fractional \(q\)-derivative of the Riemann-Liouville type of order \(\alpha\), \(n - 1 < \alpha \leq n\), \(\alpha > 2\), \(1 \leq \beta \leq n - 2\), and \(0 \leq a \leq 1\). We obtain the existence and multiplicity results of positive solutions by using some fixed point theorems. Finally, we give examples to illustrate the results.

MSC:

39A13 Difference equations, scaling (\(q\)-differences)
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
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[1] Jackson, F. H., On \(q\)-functions and a certain difference operator, Transactions of the Royal Society of Edinburgh, 46, 253-281 (1908)
[2] Jackson, F. H., On \(q\)-definite integrals, The Quarterly Journal of Pure and Applied Mathematics, 41, 193-203 (1910) · JFM 41.0317.04
[3] Ernst, T., The history of \(q\)-calculus and a new method, UUDM Report 2000, 16 (2000), Department of Mathematics, Uppsala University
[4] Ahmad, B., Boundary-value problems for nonlinear third-order \(q\)-difference equations, Electronic Journal of Differential Equations, 94 (2011) · Zbl 1226.39003
[5] Ahmad, B.; Ntouyas, S. K., Boundary value problems for \(q\)-difference inclusions, Abstract and Applied Analysis, 2011 (2011) · Zbl 1216.39012 · doi:10.1155/2011/292860
[6] Ahmad, B.; Alsaedi, A.; Ntouyas, S. K., A study of sencond-order \(q\)-difference equations with bounday conditions, Advances in Difference Equations, 2012 (2012) · Zbl 1293.34061 · doi:10.1186/1687-1847-2012-94
[7] Yu, C. L.; Wang, J. F., Existence of solutions for nonlinear second-order \(q\)-difference equations with first-order \(q\)-derivatives, Advances in Difference Equations, 2013 (2013) · Zbl 1380.92026 · doi:10.1186/1687-1847-2013-94
[8] Al-Salam, W. A., Some fractional \(q\)-integrals and \(q\)-derivatives, Proceedings of the Edinburgh Mathematical Society, 15, 135-140 (1966-1967) · Zbl 0171.10301 · doi:10.1017/S0013091500011469
[9] Agarwal, R. P., Certain fractional \(q\)-integrals and \(q\)-derivatives, Proceedings of the Cambridge Philosophical Society, 66, 365-370 (1969) · Zbl 0179.16901
[10] Rajković, P. M.; Marinković, S. D.; Stanković, M. S., Fractional integrals and derivatives in \(q\)-calculus, Applicable Analysis and Discrete Mathematics, 1, 1, 311-323 (2007) · Zbl 1199.33013 · doi:10.2298/AADM0701311R
[11] Atici, F. M.; Eloe, P. W., Fractional \(q\)-calculus on a time scale, Journal of Nonlinear Mathematical Physics, 14, 3, 333-344 (2007) · Zbl 1157.81315 · doi:10.2991/jnmp.2007.14.3.4
[12] Bai, Z.; Lü, H., Positive solutions for boundary value problem of nonlinear fractional differential equation, Journal of Mathematical Analysis and Applications, 311, 2, 495-505 (2005) · Zbl 1079.34048 · doi:10.1016/j.jmaa.2005.02.052
[13] Campos, L. M. B. C., On the solution of some simple fractional differential equations, International Journal of Mathematics and Mathematical Sciences, 13, 3, 481-496 (1990) · Zbl 0711.34019 · doi:10.1155/S0161171290000709
[14] Delbosco, D.; Rodino, L., Existence and uniqueness for a nonlinear fractional differential equation, Journal of Mathematical Analysis and Applications, 204, 2, 609-625 (1996) · Zbl 0881.34005 · doi:10.1006/jmaa.1996.0456
[15] Kilbas, A. A.; Trujillo, J. J., Differential equations of fractional order: methods, results and problems. I, Applicable Analysis, 78, 1-2, 153-192 (2001) · Zbl 1031.34002 · doi:10.1080/00036810108840931
[16] Li, C. F.; Luo, X. N.; Zhou, Y., Existence of positive solutions of the boundary value problem for nonlinear fractional differential equations, Computers & Mathematics with Applications, 59, 3, 1363-1375 (2010) · Zbl 1189.34014 · doi:10.1016/j.camwa.2009.06.029
[17] Yi, L.; Shusen, D., A class of analytic functions defined by fractional derivation, Journal of Mathematical Analysis and Applications, 186, 2, 504-513 (1994) · Zbl 0813.30016 · doi:10.1006/jmaa.1994.1313
[18] Miller, K. S.; Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equations (1993), New York, NY, USA: John Wiley & Sons, New York, NY, USA · Zbl 0789.26002
[19] Qiu, T.; Bai, Z., Existence of positive solutions for singular fractional differential equations, Electronic Journal of Differential Equations, 146, 1-9 (2008) · Zbl 1172.34313
[20] Samko, S. G.; Kilbas, A. A.; Marichev, O. I., Fractional Integrals and Derivatives. Fractional Integrals and Derivatives, Theory and Applications (1993), Yverdon, Switzerland: Gordon and Breach Science, Yverdon, Switzerland · Zbl 0818.26003
[21] Zhang, S., The existence of a positive solution for a nonlinear fractional differential equation, Journal of Mathematical Analysis and Applications, 252, 2, 804-812 (2000) · Zbl 0972.34004 · doi:10.1006/jmaa.2000.7123
[22] Ferreira, R. A. C., Nontrivial solutions for fractional \(q\)-difference boundary value problems, Electronic Journal of Qualitative Theory of Differential Equations, 70, 1-10 (2010) · Zbl 1207.39010
[23] Ferreira, R. A. C., Positive solutions for a class of boundary value problems with fractional \(q\)-differences, Computers & Mathematics with Applications, 61, 2, 367-373 (2011) · Zbl 1216.39013 · doi:10.1016/j.camwa.2010.11.012
[24] El-Shahed, M.; Al-Askar, F. M., Positive solutions for boundary value problem of nonlinear fractional \(q\)-difference equation, ISRN Mathematical Analysis, 2011 (2011) · Zbl 1213.39008
[25] Ma, J.; Yang, J., Existence of solutions for multi-point boundary value problem of fractional \(q\)-difference equation, Electronic Journal of Qualitative Theory of Differential Equations, 92, 1-10 (2011) · Zbl 1340.39010
[26] Graef, J. R.; Kong, L., Positive solutions for a class of higher order boundary value problems with fractional \(q\)-derivatives, Applied Mathematics and Computation, 218, 19, 9682-9689 (2012) · Zbl 1254.34010 · doi:10.1016/j.amc.2012.03.006
[27] Ahmad, B.; Ntouyas, S. K.; Purnaras, L. K., Existence results for nonlocal boundary value problem of nonlinear fractional q-difference equations, Advances in Difference Equations, 2012, article 140 (2012) · Zbl 1388.39003 · doi:10.1186/1687-1847-2012-140
[28] Kac, V.; Cheung, P., Quantum Calculus (2002), New York, NY, USA: Springer, New York, NY, USA · Zbl 0986.05001 · doi:10.1007/978-1-4613-0071-7
[29] Agarwal, R. P.; Meehan, M.; O’Regan, D., Fixed Point Theory and Applications (2001), Cambridge, UK: Cambridge University Press, Cambridge, UK · Zbl 0960.54027 · doi:10.1017/CBO9780511543005
[30] Guo, D. J.; Lakshmikantham, V., Nonlinear Problems in Abstract Cones (1988), Boston, Mass, USA: Academic Press, Boston, Mass, USA · Zbl 0661.47045
[31] Leggett, R. W.; Williams, L. R., Multiple positive fixed points of nonlinear operators on ordered Banach spaces, Indiana University Mathematics Journal, 28, 4, 673-688 (1979) · Zbl 0421.47033 · doi:10.1512/iumj.1979.28.28046
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