Let , , and be a strongly monotone and Lipschitz mapping. A Krasnoselskii-type sequence is constructed and proved to converge strongly to the unique solution of . Furthermore, our technique of proo f is of independent interest.

To investigate dynamical behavior of the Hopfield neural network model when its dimension becomes increasingly large, a Hopfield-type lattice system is developed as the infinite dimensional extension of the classical Hopfield model. The existence of global attractors is established for both the lattice system and its finite dimensional approximations. Moreover, the global attractors for the finite dimensional approximations are shown to converge to the attractor for the infinite dimensional lattice system upper semi-continuously.

A Hopfield neural lattice model is developed as the infinite dimensional extension of the classical finite dimensional Hopfield model. In addition, random external inputs are considered to incorporate environmental noise. The resulting random lattice dynamical system is first formulated as a random ordinary differential equation on the space of square summable biinfinite sequences. Then the existence and uniqueness of solutions, as well as long term dynamics of solutions are investigated.