In this paper we consider the following equation involving uniformly elliptic nonlocal operator in the upper half space { A 2 s u ( x )= f ( u ( x )) , x ∈ R + n , u ( x ) > 0 , x ∈ R + n , u ( x )= 0 , x ∉ R + n . We first develop a narrow region principle for antisymmetric functions in unbounded domains, in which we assume that u has polynomial growth instead of the usual decay condition u→0 at infinity, which is the improvement of the partial result of Wu-Qu-Zhang-Zhang[Math Meth Appl Sci. 2023;46:3721-3740]. Then we obtain the monotonicity of positive solutions in the upper half space by a direct method of moving planes, which extends the monotonic result of Li[Adv. Nonlinear Stud., 24 (2024), 451-462]. Based on the monotonicity results, we prove the existence of positive bounded solutions for uniformly elliptic nonlocal equation on the whole space. We believe that these methods will also be helpful for equations involving other non local operators.