Positive Solutions for Nonlinear Caputo Type Fractional q-Difference Equations with Integral Boundary Conditions

Since Al-Salam [1] and Agarwal [2] introduced the fractional q-difference calculus, the theory of fractional q-difference calculus itself and nonlinear fractional q-difference equation boundary value problems have been extensively investigated by many researchers. For some recent developments on fractional q-difference calculus and boundary value problems of fractional q-difference equations, see [3–16] and the references therein. For example, authors [17–20] considered some anti-periodic boundary value problems of nonlinear fractional q-difference equations. By applying the generalized Banach contraction principle, the monotone iterative method, and the Krasnoselskii’s fixed point theorem. In [21], the authors investigated Caputo q-fractional initial value problems independently of the paper [3] where some open problems raised there. In [22], Mittag-Leffler stabilitry of q-fractional systems was investigated. In [23,24], some important q-fractional inequalities were proved. Those inequalities are necessary for the development of q-fractional systems. Zhao et al. [25] showed some existence results of positive solutions to nonlocal q-integral boundary value problems of nonlinear fractional q-derivative equations. Under different conditions, Graef and Kong [26,27] investigated the existence of positive solutions for boundary value problems with fractional q-derivatives in terms of different ranges of λ, respectively. By applying some standard fixed point theorems, Agarwal et al. [28] and Ahmad et al. [29] showed some existence results for sequential q-fractional integrodifferential equations with q-antiperiodic boundary conditions and nonlocal four-point boundary conditions, respectively. In [30], by applying a mixed monotone method and the Guo-Krasnoselskii fixed point theorem, Zhao and Yang obtained the existence and uniqueness of positive solutions for the singular coupled integral boundary value problem of nonlinear higher-order fractional q-difference equations.