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We investigate the spin fluctuations and the pairing symmetry inA$_{x}$Fe$_{2-y}$Se$_{2}$ by the fluctuation exchange approximation. Besidesthe on-site interactions, the next-nearest-neighbor antiferromagnetic coupling$J_{2}$ is also included. We find that both the itinerant and the localizednatures of electrons are important to describe the recent experimental resultsof the spin fluctuations and the pairing symmetry. In particular, a small$J_{2}$ coupling can change the pairing gap from the d-wave symmetry to thes-wave symmetry. We have also studied the real-space structures of the gapfunctions for different orbits in order to gain more insight on the nature ofthe pairing mechanism.

Spin fluctuations and pairing symmetry in A$_{x}$Fe$_{2-y}$Se$_{2}$: dual effect of the itinerant and the localized nature of electrons