Basic Theory of Nonlinear Third-Order q-Difference Equations and Inclusions

Mathematical Modelling and Analysis

Volumn 18, Issue 1, 2013, Pages 122-135

  Ahmad, Bashir ^a   Nieto, Juan Jose ^a,b  

^a KING ABDULAZIZ UNIVERSITY   (Saudi Arabia)

^b UNIVERSITY OF SANTIAGO DE COMPOSTELA   (Spain)

Author keywords

extremal solution; fixed point; inclusions; q difference equations

Indexed keywords

BOUNDARY VALUE PROBLEMS; DIFFERENCE EQUATIONS; FIXED POINT ARITHMETIC; INCLUSIONS;

BASIC THEORY; EXISTENCE OF SOLUTIONS; EXISTENCE RESULTS; EXTREMAL SOLUTIONS; FIXED POINT THEOREMS; FIXED POINTS; INCLUSION PROBLEM; Q-DIFFERENCE EQUATION;

NONLINEAR EQUATIONS;

EID: 84873345335     PISSN: 13926292     EISSN: 16483510     Source Type: Journal    
DOI: 10.3846/13926292.2013.760012     Document Type: Article

References (42)

1. T. Abdeljawad and D. Baleanu. Caputo q-fractional initial value problems and a q-analogue Mittag-Leffler function. Commun. Nonlinear Sci. Numer. Simul., 16: 4682-4688, 2011. http://dx.doi.org/10.1016/j.cnsns.2011.01.026.
2. T. Abdeljawad, B. Benli and D. Baleanu. A generalized q-Mittag-Leffler function by q-Captuo fractional linear equations. Abstr. Appl. Anal., p. 11 pages, 2012. http://dx.doi.org/10.1155/2012/546062. Art. No. 546062
3. N. Abrashina-Zhadaeva and N. Romanova. Algorithms for numerical solving of 2D anomalous diffusion problems. Math. Model. Anal., 17: 447-455, 2012. http://dx.doi.org/10.3846/13926292.2012.686123
4. C.R. Adams. On the linear ordinary q-difference equation. Ann. Math., 30: 195-205, 1928. http://dx.doi.org/10.2307/1968274
5. Agarwal, R., Dhage, B. and O'Regan, D. 2003. The upper and lower solution method for differential inclusions via a lattice fixed point theorem. Dynam. Systems Appl., 12: 1-7.
6. Ahmad, B. 2011. Boundary value problems for nonlinear third-order q-difference equations. Electron. J. Differential Equations, 94: 1-7.
7. B. Ahmad and J.J. Nieto. On nonlocal boundary value problems of nonlinear q-difference equations, Adv. Difference Equ., 81: 10 pages, 2012
8. B. Ahmad, J.J. Nieto and J. Pimentel. Some boundary value problems of fractional differential equations and inclusions. Comput. Math. Appl., 62: 1238-1250, 2011. http://dx.doi.org/10.1016/j.camwa.2011.02.035
9. B. Ahmad and S.K. Ntouyas Boundary value problems for q-difference inclusions. Abstr. Appl. Anal., p. 15 pages, 2011. http://dx.doi.org/10.1155/2011/292860. Article ID 292860
10. Alzabut, J.O. and Abdeljawad, T. 2011. Perron's theorem for q-delay difference equations. Appl. Math. Information Sci., 5: 74-84.
11. M.H. Annaby and Z.S. Mansour. q-Taylor and interpolation series for Jackson q-difference operators. J. Math. Anal. Appl., 344: 472-483, 2008. http://dx.doi.org/10.1016/j.jmaa.2008.02.033.
12. Aubin, J.P. and Cellina, A. 1984. Differential Inclusions, Berlin-Heidelberg, New York: Springer-Verlag.
13. Aubin, J.P. and Frankowska, H. 1990. Set-Valued Analysis, Boston: Birkhäuser.
14. Avramescu, C. 2004. A fixed point theorem for multivalued mappings. Electron. J. Qual. Theory Differ. Equ., 17: 1-10.
15. G. Bangerezako. Variational q-calculus. J. Math. Anal. Appl., 289: 650-665, 2004. http://dx.doi.org/10.1016/j.jmaa.2003.09.004.
16. R.D. Carmichael. The general theory of linear q-difference equations. Amer. J. Math., 34: 147-168, 1912. http://dx.doi.org/10.2307/2369887.
17. Castaing, C. and Valadier, M. 1977. Convex Analysis and Measurable Multifunctions, volume 580 of Lecture Notes in Math, Berlin, Heidelberg, New York: Springer-Verlag.
18. Deimling, K. 1992. Multivalued Differential Equations, Berlin, New York: Walter De Gruyter.
19. A. Dobrogowska and A. Odzijewicz. Second order q-difference equations solvable by factorization method. J. Comput. Appl. Math., 193: 319-346, 2006. http://dx.doi.org/10.1016/j.cam.2005.06.009.
20. M. El-Shahed and H.A. Hassan. Positive solutions of q-difference equation Proc. Amer. Math. Soc., 138: 1733-1738, 2010. http://dx.doi.org/10.1090/S0002-9939-09-10185-5.
21. R. Finkelstein. The q-Coulomb problem J. Math. Phys., 37: 2628-2636, 1996. http://dx.doi.org/10.1063/1.531532.
22. R. Finkelstein and E. Marcus. Transformation theory of the q-oscillator. J. Math. Phys., 36: 2652-2672, 1995. http://dx.doi.org/10.1063/1.531057.
23. R. Floreanini and L. Vinet. Automorphisms of the q-oscillator algebra and basic orthogonal polynomials. Phys. Lett. A, 180: 393-401, 1993. http://dx.doi.org/10.1016/0375-9601(93)90289-C.
24. R. Floreanini and L. Vinet. Symmetries of the q-difference heat equation. Lett. Math. Phys., 32: 37-44, 1994. http://dx.doi.org/10.1007/BF00761122.
25. R. Floreanini and L. Vinet. Quantum symmetries of q-difference equations. J. Math. Phys., 36: 3134-3156, 1995. http://dx.doi.org/10.1063/1.531017.
26. P.G.O. Freund and A.V. Zabrodin. The spectral problem for the q-Knizhnik- Zamolodchikov equation and continuous q-Jacobi polynomials. Comm. Math. Phys., 173: 17-42, 1995. http://dx.doi.org/10.1007/BF02100180.
27. Gasper, G. and Rahman, M. 1990. Basic Hypergeometric Series, Cambridge: Cambridge University Press.
28. Heikkila, S. and Lakshmikantham, V. 1994. Monotone Iterative Technique for Nonlinear Discontinuous Differential Equations, New York: Marcel Dekker Inc.
29. Hu, S. and Papageorgiou, N.S. 1997. Handbook of Multivalued Analysis, Vol. I, Dordrecht: Theory Kluwer.
30. Hu, S. and Papageorgiou, N.S. 2000. Handbook of Multivalued Analysis, Vol. II, Dordrecht: Applications. Kluwer.
31. M.E.H. Ismail and P. Simeonov. q-Difference operators for orthogonal polynomials. J. Comput. Appl. Math., 233: 749-761, 2009. http://dx.doi.org/10.1016/j.cam.2009.02.044.
32. F.H. Jackson. On q-difference equations. Amer. J. Math., 32: 305-314, 1910. http://dx.doi.org/10.2307/2370183.
33. F. Jarad, T. Abdeljawad and D. Baleanu. Stability of q-fractional non-autonomous systems. Nonlinear Anal. Real World Appl., 14: 780-784, 2013. http://dx.doi.org/10.1016/j.nonrwa.2012.08.001.
34. Kac, V. and Cheung, P. 2002. Quantum Calculus, New York: Springer.
35. Kisielewicz, M. 1991. Differential Inclusions and Optimal Control, Dordrecht,The Netherlands: Kluwer.
36. W.-S. Li, J.J. Nieto and Y.-K. Chang. On impulsive hyperbolic differential inclusions with nonlocal initial conditions. J. Optim. Theory Appl., 140: 431-442, 2009. http://dx.doi.org/10.1007/s10957-008-9468-1.
37. W.-S. Li, J.J. Nieto and Y.-K. Chang. Solvability of impulsive neutral evolution differential inclusions with state-dependent delay. Math. Comput. Modelling, 49: 1920-1927, 2009. http://dx.doi.org/10.1016/j.mcm.2008.12.010.
38. Martelli, M. 1975. Rothe's type theorem for non compact acyclic-valued maps. Boll. Math. Ital., 4: 70-76.
39. T.E. Mason. On properties of the solutions of linear q-difference equations with entire function coefficients. Amer. J. Math., 37: 439-444, 1915. http://dx.doi.org/10.2307/2370216.
40. J.J. Nieto. An abstract monotone iterative technique. Nonlinear Anal., 28: 1923-1933, 1997. http://dx.doi.org/10.1016/S0362-546X(97)89710-6.
41. Smirnov, G.V. 2002. Introduction to the Theory of Differential Inclusions, Providence, RI: American Mathematical Society.
42. Tarski, A. 1955. A lattice-theoretical fixpoint theorem and its applications. Pacific J. Math., 5: 285-309.