Abstract
In this paper, we investigate the existence and uniqueness of solutions for mixed fractional q-difference boundary value problems involving the Riemann–Liouville and the Caputo fractional derivative. By using the Guo–Krasnoselskii fixed point theorem and Banach contraction mapping principle as well as Schaefer’s fixed point theorem, we obtain the main results. In addition, several examples are given to illustrate the main results.
Highlights
In recent years, the theory of fractional differential equations has become an important aspect of differential equations; see [1,2,3,4,5,6,7,8]
In 2011, Zhao and Sun et al [10] studied the existence of positive solutions for the nonlinear fractional differential equation boundary value problem
Dα0+ u(t) = λf u(t), 0 < t < 1, u(0) + u (0) = 0, u(1) + u (1) = 0, where 1 < α ≤ 2 is a real number, Dα0+ is the Caputo fractional derivative, λ > 0 and f : [0, +∞) → (0, +∞) is continuous, by using the properties of the Green function and Guo–Krasnoselskii fixed point theorem on cones, the eigenvalue intervals of the nonlinear fractional differential equation boundary value problem are considered, some sufficient conditions for the nonexistence and existence of at least one or two positive solutions for the boundary value problems are established
Summary
The theory of fractional differential equations has become an important aspect of differential equations; see [1,2,3,4,5,6,7,8]. In 2011, Zhao and Sun et al [10] studied the existence of positive solutions for the nonlinear fractional differential equation boundary value problem
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