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ArticleOriginal scientific text

Title

Existence results for q-difference inclusions with three-point boundary conditions involving different numbers of q

Authors 1, 2, 2

Affiliations

  1. Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece
  2. Department of Mathematics, Faculty of Applied Science, King Mongkut's University of Technology, North Bangkok, Bangkok, Thailand

Abstract

In this paper, we study a new class of three-point boundary value problems of nonlinear second-order q-difference inclusions. Our problems contain different numbers of q in derivatives and integrals. By using fixed point theorems, some new existence results are obtained in the cases when the right-hand side has convex as well as noncovex values.

Keywords

ENG
q-difference inclusions, nonlocal boundary conditions, fixed point theorems

Bibliography

  1. C.R. Adams, On the linear ordinary q-difference equation, Annals Math. 30 (1928) 195-205. doi: 10.2307/1968274
  2. B. Ahmad, Boundary value problems for nonlinear third-order q-difference equations, Electron. J. Diff. Equ. 2011 (94) (2011) 1-7. doi: 10.1155/2011/107384
  3. B. Ahmad and S.K. Ntouyas, Boundary value problems for q-difference inclusions, Abstr. Appl. Anal. 2011 Article ID 292860, 15 pages.
  4. B. Ahmad, A. Alsaedi and S.K. Ntouyas, A study of second-order q-difference equations with boundary conditions, Adv. Difference Equ. 2012 (2012) 35. doi: 10.1186/1687-1847-2012-35
  5. B. Ahmad and J.J. Nieto, Basic theory of nonlinear third-order q-difference equations and inclusions, Math. Model. Anal. 18 (1) (2013) 122-135. doi: 10.3846/13926292.2013.760012
  6. M.H. Annaby and Z.S. Mansour, q-Taylor and interpolation series for Jackson q-difference operators, J. Math. Anal. Appl. 344 (2008) 472-483. doi: 10.1016/j.jmaa.2008.02.033
  7. G. Bangerezako, Variational q-calculus, J. Math. Anal. Appl. 289 (2004) 650-665. doi: 10.1016/j.jmaa.2003.09.004
  8. H.F. Bohnenblust and S. Karlin, On a theorem of Ville, in: Contributions to the Theory of Games. Vol. I, pp. 155-160 (Princeton Univ. Press, 1950).
  9. R.D. Carmichael, The general theory of linear q-difference equations, American J. Math. 34 (1912) 147-168. doi: 10.2307/2369887
  10. C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Mathematics 580 (Springer-Verlag, Berlin-Heidelberg-New York, 1977). doi: 10.1007/BFb0087685
  11. H. Covitz and S.B. Nadler Jr., Multivalued contraction mappings in generalized metric spaces, Israel J. Math. 8 (1970) 5-11. doi: 10.1007/BF02771543
  12. K. Deimling, Multivalued Differential Equations (Walter De Gruyter, Berlin-New York, 1992). doi: 10.1515/9783110874228
  13. A. Dobrogowska and A. Odzijewicz, Second order q-difference equations solvable by factorization method, J. Comput. Appl. Math. 193 (2006) 319-346. doi: 10.1016/j.cam.2005.06.009
  14. T. Ernst, The history of q-calculus and a new method, UUDM Report 2000:16, Department of Mathematics, Uppsala University, 2000, ISSN:1101-3591.
  15. M. El-Shahed and H.A. Hassan, Positive solutions of q-difference equation, Proc. Amer. Math. Soc. 138 (2010) 1733-1738. doi: 10.1090/S0002-9939-09-10185-5
  16. R. Ferreira, Nontrivial solutions for fractional q-difference boundary value problems, E.J. Qualitative Theory Diff. Equ. 70 (2010) 1-10.
  17. G. Gasper and M. Rahman, Basic Hypergeometric Series (Cambridge University Press, Cambridge, 1990).
  18. G. Gasper and M. Rahman, Some systems of multivariable orthogonal q-Racah polynomials, Ramanujan J. 13 (2007) 389-405. doi: 10.1007/s11139-006-0259-8
  19. A. Granas and J. Dugundji, Fixed Point Theory (Springer-Verlag, New York, 2005).
  20. Sh. Hu and N. Papageorgiou, Handbook of Multivalued Analysis, Theory I (Kluwer, Dordrecht, 1997). doi: 10.1007/978-1-4615-6359-4
  21. M.E.H. Ismail and P. Simeonov, q-difference operators for orthogonal polynomials, J. Computat. Appl. Math. 233 (2009) 749-761. doi: 10.1016/j.cam.2009.02.044
  22. F.H. Jackson, On q-functions and a certain difference operator, Trans. Roy. Soc. Edinburgh 46 (1908) 253-281. doi: 10.1017/S0080456800002751
  23. F.H. Jackson, On q-difference equations, American J. Math. 32 (1910) 305-314. doi: 10.2307/2370183
  24. V. Kac and P. Cheung, Quantum Calculus (Springer, New York, 2002). doi: 10.1007/978-1-4613-0071-7
  25. M. Kisielewicz, Differential Inclusions and Optimal Control (Kluwer, Dordrecht, The Netherlands, 1991).
  26. A. Lasota and Z. Opial, An application of the Kakutani-Ky Fan theorem in the theory of ordinary differential equations, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 13 (1965) 781-786.
  27. J. Ma and J. Yang, Existence of solutions for multi-point boundary value problem of fractional q-difference equation, E.J. Qualitative Theory Diff. Equ. 92 (2011) 1-10.
  28. T.E. Mason, On properties of the solutions of linear q-difference equations with entire function coefficients, American J. Math. 37 (1915) 439-444. doi: 10.2307/2370216
  29. T. Sitthiwirattham, J. Tariboon and S.K. Ntouyas, Three-point boundary value problems of nonlinear second-order q-difference equations involving different numbers of q, J. Appl. Math. 2013, Article ID 763786, 12 pages.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
2014/34/1
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Pages:
41-59
Main language of publication
English
Received
2013-10-30
Published
2014

DOI: 10.7151/dmdico1157

Exact and natural sciences