Almost automorphic evolution equations with compact almost automorphic solutions

$({\mathrm R},{\mathcal B})$-Multidimensional-almost automorphic functions with applications to integral and partial differential equations

“…Then it is said that f (•) is almost automorphic if and only if for every real sequence (b n ) there exist a subsequence (a n ) of (b n ) and a map g : R → X such that lim n→∞ f t + a n = g(t) and lim n→∞ g t − a n = f (t), (1.1) pointwise for t ∈ R. If this is the case, then f ∈ C b (R : X) and the limit function g(•) is bounded on R but not necessarily continuous on R. Furthermore, if the convergence of limits appearing in (1.1) is uniform on compact subsets of R, then it is said that f (•) is compactly almost automorphic. Recall that an almost automorphic function f (•) is compactly almost automorphic if and only if it is uniformly continuous [21,Lemma 3.7]. Concerning two-parameter almost automorphic functions, we want only to recall that the authors of [19] and [29] have used the following notion: A jointly continuous function F : R × X → Y is said to be almost automorphic if and only if for every sequence of real numbers (s ′ n ) there exists a subsequence (s n ) such that G(t; x) := lim n→∞ F t + s n ; x is well defined for each t ∈ R and x ∈ X, and…”

Pseudo compact almost automorphy of neutral type Clifford-valued neural networks with mixed delays

“…This is because, as the extension of real-valued, complex-valued and quaternion-valued neural networks, Clifford-valued neural networks are superior to real-valued, complex-valued and quaternion-valued ones in many aspects. Especially when dealing with high-dimensional data, multi-level data and valuable problems involving spatial geometric transformation, Clifford neural networks have more advantages [10][11][12][13]. As we all know, neural networks have been successfully applied in many fields, such as associative memory, pattern recognition, optimal control, signal processing, assistant decision-making, artificial intelligence, and so on.…”

“…Also, all of these applications are closely related to their dynamics. However, there are few results about the dynamics of Clifford-valued neural networks [13][14][15][16][17][18][19]. In addition, Clifford-valued neural networks refer to the neural networks whose state variables, connection weights and external inputs are all Clifford numbers.…”

“…Lemma 2.2. [13] A function f ∈ KAA(R, X) if and only if it is almost automorphic and uniformly continuous.…”

“…Figure 3. Curves of x 12p (t) and x 13 p (t) of system (1) with the initial values (x12 1 (0), x 12 2 (0)) T = (−0.04, 0.04) T , (0.08, −0.07) T and (x13 1 (0), x 13 2 (0)) T = (0.07, −0.06) T , (−0.02, 0.03) T .…”