Firstly we study the growth of meromorphic solutions of linear difference equation of the form$$A_k\left(z\right)f(z+c_k)+\cdots +A_1\left(z\right)f(z+c_1)+A_0\left(z\right)f\left(z\right)=F\left(z\right),$$ where $A_k\left(z\right),...,A_0\left(z\right)$ and $F\left(z\right)$ are meromorphic functions of finite logarithmic order, $c_i$$(i=1,......$

Let f(z) be a finite order transcendental meromorphic function such that λ(1/f(z)) < σ(f(z)), and let c ∈ ℂ∖0 be a constant such that f(z+c) ≢ f(z) + c. We mainly prove that $max\tau \left(f\left(z\right)\right),\tau \left({\Delta }_{c}f\left(z\right)\right)=max\tau \left(f\right(z\left)\right),\tau \left(f\right(z+c\left)\right)=max\tau \left({\Delta }_{c}f\left(z\right)\right),\tau \left(f(z+c)\right)=\sigma \left(f\left(z\right)\right)$, where τ(g(z)) denotes the exponent of convergence of fixed points of the meromorphic function g(z), and σ(g(z)) denotes the order of growth of g(z).