  This paper establishes a version of Nevanlinna theory based on Jackson
difference operator $D_{q}f(z)=\frac{f(qz)-f(z)}{qz-z}$ for meromorphic
functions of zero order in the complex plane $\mathbb{C}$. We give the
logarithmic difference lemma, the second fundamental theorem, the defect
relation, Picard theorem and five-value theorem in sense of Jackson
$q$-difference operator. By using this theory, we investigate the growth of
entire solutions of linear Jackson $q$-difference equations
$D^{k}_{q}f(z)+A(z)f(z)=0$ with meromorphic coefficient $A,$ where $D^k_q$ is
Jackson $k$-th order difference operator, and estimate the logarithmic order of
some $q$-special functions.
