## Positive solutions of $$q$$-difference equation.(English)Zbl 1201.39003

Authors’ abstract: We investigate the existence of positive solutions of the $$q$$-difference equation $$-D_q^2u(t)=a(t) f(u(t))$$ with some boundary conditions by applying a fixed point theorem in cones.

### MSC:

 39A13 Difference equations, scaling ($$q$$-differences) 39A12 Discrete version of topics in analysis 34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations 34B27 Green’s functions for ordinary differential equations 39A22 Growth, boundedness, comparison of solutions to difference equations
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### References:

 [1] M. H. Annaby and Z. S. Mansour, \?-Taylor and interpolation series for Jackson \?-difference operators, J. Math. Anal. Appl. 344 (2008), no. 1, 472 – 483. · Zbl 1149.40001 [2] L. H. Erbe and Haiyan Wang, On the existence of positive solutions of ordinary differential equations, Proc. Amer. Math. Soc. 120 (1994), no. 3, 743 – 748. · Zbl 0802.34018 [3] George Gasper and Mizan Rahman, Basic hypergeometric series, 2nd ed., Encyclopedia of Mathematics and its Applications, vol. 96, Cambridge University Press, Cambridge, 2004. With a foreword by Richard Askey. · Zbl 1129.33005 [4] Da Jun Guo and V. Lakshmikantham, Nonlinear problems in abstract cones, Notes and Reports in Mathematics in Science and Engineering, vol. 5, Academic Press, Inc., Boston, MA, 1988. · Zbl 0661.47045 [5] F.H. Jackson, On $$q$$-definite integrals. Quart. J. Pure and Appl. Math., 41: 193-203, 1910. · JFM 41.0317.04 [6] M. A. Krasnosel$$^{\prime}$$skiĭ, Positive solutions of operator equations, Translated from the Russian by Richard E. Flaherty; edited by Leo F. Boron, P. Noordhoff Ltd. Groningen, 1964. [7] Man Kam Kwong, On Krasnoselskii’s cone fixed point theorem, Fixed Point Theory Appl. (2008), Art. ID 164537, 18. · Zbl 1203.47029 [8] S. D. Marinković, P. M. Rajković and M. Stanković, The Linear $$q$$-Differential Equation and $$q$$-Holonomic Functions, XVII Conference on Applied Mathematics, D. Herceg, H. Zarin, eds.; Department of Mathematics and Informatics, Novi Sad, 2007, 13-20.
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