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The coexistence of multiple stable equilibria in recurrent neural networks is an important dynamic characteristic for associative memory and other applications. In this paper, the existence and local Mittag-Leffler stability of multiple equilibria are investigated for a class of fractional-order recurrent neural networks with discontinuous and nonmonotonic activation functions. By using Brouwer's fixed point theory, several conditions are established to ensure the existence of 5ⁿ equilibria, in which all the coεmponents of 4ⁿ equilibria are located in the continuous intervals of the activation functions. and some of the components of 5ⁿ - 4ⁿ equilibria are located at some discontinuous points of the activation functions. The introduction of discontinuous activation functions makes the neural networks have more equilibria than those with continuous activation functions. Furthermore, some criteria are proposed to ensure local Mittag-Leffler stability of 3ⁿ equilibria. The introduction of nonmonotonic activation functions makes the neural networks have more stable equilibria than those with monotonic activation functions. Two examples are given to illustrate the effectiveness of the results.