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In the intelligent communication field, deep learning (DL) has attracted much attention due to its strong fitting ability and data-driven learning capability. Compared with the typical DL feedforward network structures, an enhancement structure with direct data feedback have been studied and proved to have better performance than the feedfoward networks. However, due to the above simple feedback methods lack sufficient analysis and learning ability on the feedback data, it is inadequate to deal with more complicated nonlinear systems and therefore the performance is limited for further improvement. In this paper, a novel multi-agent feedback enabled neural network (MAFENN) framework is proposed, which make the framework have stronger feedback learning capabilities and more intelligence on feature abstraction, denoising or generation, etc. Furthermore, the MAFENN framework is theoretically formulated into a three-player Feedback Stackelberg game, and the game is proved to converge to the Feedback Stackelberg equilibrium. The design of MAFENN framework and algorithm are dedicated to enhance the learning capability of the feedfoward DL networks or their variations with the simple data feedback. To verify the MAFENN framework's feasibility in wireless communications, a multi-agent MAFENN based equalizer (MAFENN-E) is developed for wireless fading channels with inter-symbol interference (ISI). Experimental results show that when the quadrature phase-shift keying (QPSK) modulation scheme is adopted, the SER performance of our proposed method outperforms that of the traditional equalizers by about 2 dB in linear channels. When in nonlinear channels, the SER performance of our proposed method outperforms that of either traditional or DL based equalizers more significantly, which shows the effectiveness and robustness of our proposal in the complex channel environment.