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Ahmad, Bashir; Ntouyas, Sotiris K.

Boundary value problems for \(q\)-difference inclusions. (English) Zbl 1216.39012

Abstr. Appl. Anal. 2011, Article ID 292860, 15 p. (2011).
Summary: We investigate the existence of solutions for a class of second-order \(q\)-difference inclusions with nonseparated boundary conditions. By using suitable fixed-point theorems, we study the cases when the right-hand side of the inclusions has convex as well as nonconvex values.

Cited in 51 Documents

MSC:

39A13 Difference equations, scaling (\(q\)-differences)
39A12 Discrete version of topics in analysis
34A60 Ordinary differential inclusions

Keywords:

boundary value problems; second-order \(q\)-difference inclusions; fixed-point theorems
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\textit{B. Ahmad} and \textit{S. K. Ntouyas}, Abstr. Appl. Anal. 2011, Article ID 292860, 15 p. (2011; Zbl 1216.39012)
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References:

[1] V. Kac and P. Cheung, Quantum Calculus, Springer, New York, NY, USA, 2002. · Zbl 0986.05001
[2] F. H. Jackson, “On q-definite integrals,” Quarterly Journal, vol. 41, pp. 193-203, 1910. · JFM 41.0317.04
[3] G. Gasper and M. Rahman, Basic Hypergeometric Series, Cambridge University Press, Cambridge, UK, 1990. · Zbl 0695.33001
[4] M. H. Annaby and Z. S. Mansour, “q-Taylor and interpolation series for Jackson q-difference operators,” Journal of Mathematical Analysis and Applications, vol. 344, no. 1, pp. 472-483, 2008. · Zbl 1149.40001 · doi:10.1016/j.jmaa.2008.02.033
[5] A. Dobrogowska and A. Odzijewicz, “Second order q-difference equations solvable by factorization method,” Journal of Computational and Applied Mathematics, vol. 193, no. 1, pp. 319-346, 2006. · Zbl 1119.39017 · doi:10.1016/j.cam.2005.06.009
[6] M. El-Shahed and H. A. Hassan, “Positive solutions of q-Difference equation,” Proceedings of the American Mathematical Society, vol. 138, no. 5, pp. 1733-1738, 2010. · Zbl 1201.39003 · doi:10.1090/S0002-9939-09-10185-5
[7] E. O. Ayoola, “Quantum stochastic differential inclusions satisfying a general lipschitz condition,” Dynamic Systems and Applications, vol. 17, no. 3-4, pp. 487-502, 2008. · Zbl 1202.81130
[8] A. Belarbi and M. Benchohra, “Existence results for nonlinear boundary-value problems with integral boundary conditions,” Electronic Journal of Differential Equations, vol. 2005, no. 06, pp. 1-10, 2005. · Zbl 1075.34015
[9] M. Benaïm, J. Hofbauer, and S. Sorin, “Stochastic approximations and differential inclusions, Part II: applications,” Mathematics of Operations Research, vol. 31, no. 4, pp. 673-695, 2006. · Zbl 1284.62511 · doi:10.1287/moor.1060.0213
[10] Y. K. Chang, W. T. Li, and J. J. Nieto, “Controllability of evolution differential inclusions in Banach spaces,” Nonlinear Analysis, Theory, Methods and Applications, vol. 67, no. 2, pp. 623-632, 2007. · Zbl 1128.93005 · doi:10.1016/j.na.2006.06.018
[11] S. K. Ntouyas, “Neumann boundary value problems for impulsive differential inclusions,” Electronic Journal of Qualitative Theory of Differential Equations, no. 22, pp. 1-13, 2009. · Zbl 1195.34041
[12] J. Simsen and C. B. Gentile, “Systems of p-Laplacian differential inclusions with large diffusion,” Journal of Mathematical Analysis and Applications, vol. 368, no. 2, pp. 525-537, 2010. · Zbl 1191.35163 · doi:10.1016/j.jmaa.2010.02.006
[13] G. V. Smirnov, Introduction to the Theory of Differential Inclusions, American Mathematical Society, Providence, RI, USA, 2002. · Zbl 0992.34001
[14] N. Apreutesei and G. Apreutesei, “A Trotter-Kato type result for a second order difference inclusion in a Hilbert space,” Journal of Mathematical Analysis and Applications, vol. 361, no. 1, pp. 195-204, 2010. · Zbl 1181.47066 · doi:10.1016/j.jmaa.2009.08.065
[15] F. M. Atici and D. C. Biles, “First order dynamic inclusions on time scales,” Journal of Mathematical Analysis and Applications, vol. 292, no. 1, pp. 222-237, 2004. · Zbl 1064.34009 · doi:10.1016/j.jmaa.2003.11.053
[16] A. Cernea and C. Georgescu, “Necessary optimality conditions for differential-difference inclusions with state constraints,” Journal of Mathematical Analysis and Applications, vol. 334, no. 1, pp. 43-53, 2007. · Zbl 1124.49018 · doi:10.1016/j.jmaa.2006.12.020
[17] Y. K. Chang and W. T. Li, “Existence results for second-order dynamic inclusion with m-point boundary value conditions on time scales,” Applied Mathematics Letters, vol. 20, no. 8, pp. 885-891, 2007. · Zbl 1154.34310 · doi:10.1016/j.aml.2006.09.004
[18] K. Deimling, Multivalued Differential Equations, Walter De Gruyter, Berlin, Germany, 1992. · Zbl 0760.34002
[19] S. Hu and N. Papageorgiou, Handbook of Multivalued Analysis, Theory I, Kluwer Academic, Dodrecht, The Netherlands, 1997. · Zbl 0887.47001
[20] J. P. Aubin and A. Cellina, Differential Inclusions, Springer, Heidelberg, Germany, 1984. · Zbl 0538.34007
[21] J. P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhäauser, Boston, Mass, USA, 1990. · Zbl 0713.49021
[22] A. Lasota and Z. Opial, “An application of the Kakutani-Ky Fan theorem in the theory of ordinary differential equations,” Bulletin de l’Académie Polonaise des Sciences. Série des Sciences Mathématiques, Astronomiques et Physiques, vol. 13, pp. 781-786, 1965. · Zbl 0151.10703
[23] A. Granas and J. Dugundji, Fixed Point Theory, Springer, New York, NY, USA, 2003. · Zbl 1025.47002
[24] A. Bressan and G. Colombo, “Extensions and selections of maps with decomposable values,” Studia Mathematica, vol. 90, pp. 69-86, 1988. · Zbl 0677.54013
[25] T. A. Laz\var, A. Petru\csel, and N. Shahzad, “Fixed points for non-self operators and domain invariance theorems,” Nonlinear Analysis, Theory, Methods and Applications, vol. 70, no. 1, pp. 117-125, 2009. · Zbl 1183.47052 · doi:10.1016/j.na.2007.11.037
[26] R. Wegrzyk, “Fixed point theorems for multifunctions and their applications to functional equations,” Dissertationes Mathematicae, vol. 201, pp. 1-28, 1982. · Zbl 0521.54034
[27] H. Covitz and S. B. Nadler Jr., “Multi-valued contraction mappings in generalized metric spaces,” Israel Journal of Mathematics, vol. 8, no. 1, pp. 5-11, 1970. · Zbl 0192.59802 · doi:10.1007/BF02771543
[28] C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Mathematics 580, Springer, Heidelberg, Germany, 1977. · Zbl 0346.46038
[29] B. Ahmad and S. K. Ntouyas, “Some existence results for boundary value problems of fractional differential inclusions with non-separated boundary conditions,” Electronic Journal of Qualitative Theory of Differential Equations, vol. 71, pp. 1-17, 2010. · Zbl 1206.26005
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