Existence of positive solutions of nonlinear fractional q -difference equation with parameter
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Existence of positive solutions of nonlinear fractional q-difference equation with parameter Xinhui Li, Zhenlai Han* and Shurong Sun *
Correspondence: [email protected] School of Mathematical Sciences, University of Jinan, Jinan, Shandong 250022, P.R. China
Abstract In this paper, we study the boundary value problem of a class of nonlinear fractional q-difference equations with parameter involving the Riemann-Liouville fractional derivative. By means of a fixed point theorem in cones, some positive solutions are obtained. As applications, some examples are presented to illustrate our main results. MSC: 39A13; 34B18; 34A08 Keywords: fractional q-difference equations; boundary value problems; fixed point theorem in cones; positive solutions
1 Introduction The q-difference calculus is an interesting and old subject that many researchers devote their time to studying. The q-difference calculus or quantum calculus were first developed by Jackson [, ], while basic definitions and properties can be found in the papers [, ]. The q-difference calculus describes many phenomena in various fields of science and engineering []. The origin of the fractional q-difference calculus can be traced back to the works in [, ] by Al-Salam and by Agarwal. The q-difference calculus is a necessary part of discrete mathematics. More recently, there has been much research activity concerning the fractional q-difference calculus [– ]. Relevant theory about fractional q-difference calculus has been established [], such as q-analogues of integral and difference fractional operators properties as Mittag-Leffler function [], q-Laplace transform, q-Taylor’s formula [, ], just to mention some. It is not only the requirements of the fractional q-difference calculus theory but also its the broad application. Apart from this old history of q-difference equations, the subject has received a considerable interest of many mathematicians and from many aspects, theoretical and practical. Specifically, q-difference equations have been widely used in mathematical physical problems, dynamical system and quantum models [], q-analogues of mathematical physical problems including heat and wave equations [], sampling theory of signal analysis [, ]. What is more, the fractional q-difference calculus plays an important role in quantum calculus. As generalizations of integer order q-difference, fractional q-difference can describe physical phenomena much better and more accurately. Perhaps due to the development of © 2013 Li et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Li et al. Advances in Difference Equations 2013, 2013:260 http://www.advancesindifferenceequations.com/content/2013/1/260
fractional differential equations [–], an interest has been observed in studying boundary value pr
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