Existence results for q-difference inclusions with three-point boundary conditions involving different numbers of q
Discussiones Mathematicae. Differential Inclusions, Control and Optimization, Tome 34 (2014) no. 1, pp. 41-59.

Voir la notice de l'article provenant de la source Library of Science

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: q-difference inclusions, nonlocal boundary conditions, fixed point theorems
@article{DMDICO_2014_34_1_a2,
     author = {Ntouyas, Sotiris and Sitthiwirattham, Thanin and Tariboon, Jessada},
     title = {Existence results for q-difference inclusions with three-point boundary conditions involving different numbers of q},
     journal = {Discussiones Mathematicae. Differential Inclusions, Control and Optimization},
     pages = {41--59},
     publisher = {mathdoc},
     volume = {34},
     number = {1},
     year = {2014},
     zbl = {1327.39006},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DMDICO_2014_34_1_a2/}
}
TY  - JOUR
AU  - Ntouyas, Sotiris
AU  - Sitthiwirattham, Thanin
AU  - Tariboon, Jessada
TI  - Existence results for q-difference inclusions with three-point boundary conditions involving different numbers of q
JO  - Discussiones Mathematicae. Differential Inclusions, Control and Optimization
PY  - 2014
SP  - 41
EP  - 59
VL  - 34
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/DMDICO_2014_34_1_a2/
LA  - en
ID  - DMDICO_2014_34_1_a2
ER  - 
%0 Journal Article
%A Ntouyas, Sotiris
%A Sitthiwirattham, Thanin
%A Tariboon, Jessada
%T Existence results for q-difference inclusions with three-point boundary conditions involving different numbers of q
%J Discussiones Mathematicae. Differential Inclusions, Control and Optimization
%D 2014
%P 41-59
%V 34
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/DMDICO_2014_34_1_a2/
%G en
%F DMDICO_2014_34_1_a2
Ntouyas, Sotiris; Sitthiwirattham, Thanin; Tariboon, Jessada. Existence results for q-difference inclusions with three-point boundary conditions involving different numbers of q. Discussiones Mathematicae. Differential Inclusions, Control and Optimization, Tome 34 (2014) no. 1, pp. 41-59. http://geodesic.mathdoc.fr/item/DMDICO_2014_34_1_a2/

[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.