Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 87, pp. 1–18. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: 10.58997/ejde.2025.87 ALMOST AUTOMORPHIC SOLUTIONS TO NON-AUTONOMOUS DYNAMIC EQUATIONS WITH STEPANOV-LIKE ALMOST AUTOMORPHIC FORCING TERMS ON TIME SCALES FENG-XIA ZHENG, HONG-XU LI Abstract. In this article, we generalize the concept of Stepanov-like almost automorphic func- tions on time scales and present some properties including the composition theorem. Based on it, some results on the existence and uniqueness of almost automorphic solution to non- autonomous dynamic equations with Stepanov-like almost automorphic forcing terms on time scales are established. In our results, we do not need to assume the uniform Lipschitz con- dition of the nonlinear forcing term and do not need to assume that the Green’s function is Bi-automorphic directly. Finally, an application to Lasota-Wazewska model on time scales is provided. 1. Introduction The theory of time scales was introduced by Stefan Hilger in his PhD thesis [14]. This theory unifies continuous and discrete analysis, which is a powerful tool for applications in population models, economics, quantum physics among others [2, 7, 9, 16, 17, 29]. The study of dynamic equations on time scales can avoid proving results twice. It can also be applied to investigate continuous-discrete hybrid process. Li and Wang [19, 20] introduced the concept of almost periodicity on time scales and after that, the concept of almost automorphy on time scales was introduced by Lizama et al. in [24]. Furthermore, many generalized types of almost periodicity and almost automorphy have been introduced on time scales, such as pseudo almost periodic function and pseudo almost automorphic function. The almost automorphic and almost periodic type solutions to dynamic equations on time scales have been widely investigated (see, e.g., [1, 15, 21, 23, 25, 26, 30, 31, 32] and the references therein). In 2015, Wang and Zhu [22] introduced Stepanov-like almost periodic functions on time scales avoiding Bochner transform. Then Tang and Li [27] introduced Stepanov-like almost periodicity on time scales by using Bochner-like transform and based on it, some results on the almost periodic solutions to the following non-autonomous semilinear dynamic equation u∆(t) = A(t)u(t) + f(t, u(t)), t ∈ T, (1.1) were presented, where T is a time scale and the nonlinear term is Stepanov-like almost periodic. Furthermore, Tang and Li [28] studied the Stepanov-like pseudo almost periodic functions on time scales, with applications to dynamic equations with delay. Subsequently, Es-saiydy and Zitane [13] extended this to weighted Stepanov-like pseudo almost periodicity on time scales, applying it to some classes of nonautonomous dynamic equations involving weighted Stepanov-like pseudo almost periodic forcing terms on time scales. On the other hand, [24] established the results on the almost automorphic solutions to dynamic equation (1.1), where the nonlinear term is almost automorphic. Es-saiydy and Zitane [11] introduced Stepanov-like (pseudo) almost automorphic 2020 Mathematics Subject Classification. 34C27, 34N05, 43A60. Key words and phrases. Non-autonomous dynamic equations; almost automorphic solutions; Stepanov-like almost automorphic; time scales. ©2025. This work is licensed under a CC BY 4.0 license. Submitted April 8, 2025. Published August 27, 2025. 1 2 F.-X. ZHENG, H.-X. LI EJDE-2025/87 functions on time scales by using Bochner-like transform and based on it, some results on the pseudo almost automorphic solutions to the semilinear dynamic equation u∆(t) = Au(t) + f(t, u(t)), t ∈ T, were obtained, where A is the generator of a C0-semigroup and the nonlinear term is Stepanov- like pseudo almost automorphic and continuous. However, we note that the convergence in the definition of Stepanov-like almost automorphy on time scales is uniform, which does not include the case of T = R. Lots of papers using this definition, such as [10, 12]. In addition, the composition theorems are established under uniform Lipschitz condition. Motivated by the above works, in this paper, we investigate the existence and uniqueness of almost automorphic solution to dynamic equation (1.1), where the nonlinear term is Stepanov-like almost automorphic. We generalize the concept of Stepanov-like almost automorphic functions on time scales including the case of T = R by using pointwise convergence, and present some basic properties for Stepanov-like almost automorphic functions on time scales. Especially, we construct the composition theorem of Stepanov-like almost automorphic functions. Then combining the composition theorem and the Banach fixed point theorem, some results on the existence and uniqueness of almost automorphic solution to dynamic equation (1.1) with Stepanov-like almost automorphic nonlinear term are established without assuming the uniform Lipschitz condition for the nonlinear forcing term. In addition, we do not assume that the Green’s function is Bi- automorphic directly. The paper is organized as follows. In Section 2, we present the concepts and properties of almost automorphic and Stepanov-like almost automorphic functions on time scales. In Section 3, we prove the composition theorem of Stepanov-like almost automorphic functions. Some sufficient conditions on the existence and uniqueness of almost automorphic solution to dynamic equation (1.1) with Stepanov-like almost automorphic nonlinear term are given in Section 4. Finally, we provide an application in Section 5. 2. Preliminaries Let T be a time scale. The forward and backward jump operators σ, ρ : T → T are defined by σ(t) := inf{s ∈ T : s > t}, ρ(t) := sup{s ∈ T : s < t}, respectively. The graininess µ : T → [0,∞) is defined by µ(t) := σ(t) − t. Points that are called right-scattered, left-scattered and isolated if σ(t) > t, ρ(t) < t and σ(t) > t > ρ(t), respectively. Points that are called right-dense, left- dense and dense if σ(t) = t, ρ(t) = t and σ(t) = t = ρ(t), respectively. For a, b ∈ T, we define [a, b]T := {t ∈ T, a ≤ t ≤ b}, [a, b)T := {t ∈ T, a ≤ t < b}. Denote Tk = T − m if T has a left-scattered maximum m. Otherwise, Tk = T. Moreover, we assume that (X, ∥ · ∥), (Y, ∥ · ∥) are two Banach spaces. Definition 2.1 ([5, 6]). (i) A function f : T → X is said to be rd-continuous if it is right continuous at each right-dense point, and there exists a finite left limit at all left-dense points. Denote by Crd(T, X) the space of all such functions. (ii) A function f : T → X is said to be continuous if it is continuous at each right-dense point and each left-dense point. Denote by C(T, X) the space of all such functions. Definition 2.2 ([5, 6]). (i) A function p : T → R is said to be regressive if 1 + µ(t)p(t) ̸= 0, t ∈ Tk. We denote by R = R(T) = R(T, R) the space of regressive and rd-continuous functions. (ii) A matrix-valued function A : T → Rn×n is said to be regressive if I + µ(t)A(t) is invertible, t ∈ Tk. Denote by R = R(T) = R(T, Rn×n) the space of regressive and rd-continuous matrix-valued functions. Let p, q ∈ R(T, R). p⊕ q and ⊖p are defined as follows: (p⊕ q)(t) := p(t) + q(t) + µ(t)p(t)q(t), t ∈ Tk, EJDE-2025/87 ALMOST AUTOMORPHIC SOLUTIONS TO NON-AUTONOMOUS DYNAMIC EQUATIONS 3 (⊖p)(t) := −p(t) 1 + µ(t)p(t) , t ∈ Tk. Clearly, (R(T, R),⊕) is an Abelian group. For more details, see [5, 6]. Definition 2.3 ([5, 6]). Let f : T → X and t ∈ Tk. f∆(t) (provided it exists) is said to be the delta derivative of f at t if for any ε > 0, there is a neighborhood U of t such that ∥[f(σ(t))− f(s)]− f∆(t)[σ(t)− s]∥ ≤ |σ(t)− s|, s ∈ U. If u is delta differentiable, then u(σ(t)) = u(t) + µ(t)u∆(t). We remark that the definition of ∆-integrable functions on T is similar to normal Lebesgue integration, and all the theorems of normal Lebesgue integration theory are also true for ∆- integrals on T. For more details, see [3, 4, 5, 6, 8]. We denote by Lp loc(T, X) the space of all locally Lp ∆- integrable functions. Definition 2.4 ([19, 24]). A time scale T is said to be invariant under translations if Π := {α ∈ R : s± α ∈ T, s ∈ T} ≠ {0}. Lemma 2.5 ([27]). Let T be a time scale invariant under translation and K := inf{|α| : α ∈ Π, α ̸= 0}. Then K = 0 ⇔ T = R, K > 0 ⇔ T ̸= R, Π = { R, T = R, KZ, T ̸= R. In the following, we always let T be a time scale invariant under translation. Definition 2.6 ([24]). An rd-continuous function f : T → X is said to be almost automorphic (abbrev. as a.a.) if for any sequence {ξ′n}∞n=1 ⊂ Π, there exist a subsequence {ξn}∞n=1 of {ξ′n}∞n=1 and a function f̃ such that lim n→∞ f(t+ ξn) = f̃(t) is well defined for all t ∈ T, and lim n→∞ f̃(t− ξn) = f(t) for all t ∈ T. We denote by AA(T, X) the space of all such functions. Definition 2.7 ([22]). Let 1 ≤ p < ∞. A function f ∈ Lp loc(T, X) is said to be Stepanov-like bounded (abbrev. as Sp-bounded) if sup t∈T ( 1 K ∫ [t,t+K)T ∥f(s)∥p∆s )1/p < ∞, where K := { 1, T = R, K, T ̸= R, with K defined in Lemma 2.5. We denote by BSp(T, X) the space of all such functions, equipped with the norm ∥f∥Sp := sup t∈T ( 1 K ∫ [t,t+K)T ∥f(s)∥p∆s )1/p . Definition 2.8. A function f ∈ BSp(T, X) is said to be Stepanov-like almost automorphic (ab- breviated as Sp-a.a.) if for any sequence {ξ′n}∞n=1 ⊂ Π, there exist a subsequence {ξn}∞n=1 of {ξ′n}∞n=1 and a function f̃ ∈ Lp loc(T, X) such that lim n→∞ ( 1 K ∫ [t,t+K)T ∥f(s+ ξn)− f̃(s)∥p∆s )1/p = 0, lim n→∞ ( 1 K ∫ [t,t+K)T ∥f̃(s− ξn)− f(s)∥p∆s )1/p = 0 4 F.-X. ZHENG, H.-X. LI EJDE-2025/87 for all t ∈ T. We denote by SpAA(T, X) the space of all such functions. Remark 2.9. (i) Definition 2.8 is different from [10, 11, 12]. In fact, the convergence in the definition of Stepanov-like almost automorphy on R is pointwise (not uniform). So our Definition 2.8 includes the case of T = R. (ii) f ∈ BSp(hZ) if and only if f ∈ l∞(hZ). Indeed, we note that K = h if T = hZ. Then sup t∈hZ ( 1 h ∫ [t,t+h)T ∥f(s)∥p∆s )1/p = sup t∈hZ ∥f(t)∥ < ∞. (iii) f ∈ SpAA(hZ,X) if and only if f ∈ AA(hZ,X). Indeed, if f ∈ SpAA (hZ,X), for any se- quence {ξ′n}∞n=1 ⊂ Π, there exist a subsequence {ξn}∞n=1 of {ξ′n}∞n=1 and a function f̃ ∈ Lp loc(hZ,X) such that ( 1 h ∫ [t,t+h)T ∥f(s+ ξn)− f̃(s)∥p∆s )1/p → 0 as n → ∞. Then ∥f(t+ ξn)− f̃(t)∥ → 0 as n → ∞ since( 1 h ∫ [t,t+h)T ∥f(s+ ξn)− f̃(s)∥p∆s )1/p = ∥f(t+ ξn)− f̃(t)∥. (2.1) Similarly, we deduce that ∥f̃(t− ξn)− f(t)∥ → 0 as n → ∞ if( 1 h ∫ [t,t+h)T ∥f̃(s− ξn)− f(s)∥p∆s )1/p → 0 as n → ∞ for each t ∈ hZ. Conversely, if f ∈ AA(hZ,X), we can deduce that f ∈ SpAA(hZ,X) in view of (2.1). Definition 2.10. A function f : T × X → Y is said to be Sp-almost automorphic in t ∈ T for each x ∈ X if for any sequence {ξ′n}∞n=1 ⊂ Π, there exist a subsequence {ξn}∞n=1 of {ξ′n}∞n=1 and a function f̃(·, x) ∈ Lp loc(T, X) such that lim n→∞ ( 1 K ∫ [t,t+K)T ∥f(s+ ξn, x)− f̃(s, x)∥p∆s )1/p = 0 for (t, x) ∈ T×X, lim n→∞ ( 1 K ∫ [t,t+K)T ∥f̃(s− ξn, x)− f(s, x)∥p∆s )1/p = 0 for (t, x) ∈ T×X. We denote by SpAA(T×X,Y ) the space of all such functions. Definition 2.11. For each compact subset K ⊂ X, a function f : T × X → Y is said to be Sp-almost automorphic in t ∈ T uniformly in x ∈ K if for any sequence {ξ′n}∞n=1 ⊂ Π, there exist a subsequence {ξn}∞n=1 of {ξ′n}∞n=1 and a function f̃(·, x) ∈ Lp loc(T, X) such that lim n→∞ ( 1 K ∫ [t,t+K)T sup x∈K ∥f(s+ ξn, x)− f̃(s, x)∥p∆s )1/p = 0 for t ∈ T, lim n→∞ ( 1 K ∫ [t,t+K)T sup x∈K ∥f̃(s− ξn, x)− f(s, x)∥p∆s )1/p = 0 for t ∈ T. We denote by SpAAK(T×X,Y ) the space of all such functions. Lemma 2.12. The following statements hold: (i) Let f : T → X be a.a. Then f is bounded. (ii) Let {fn} be a sequence of a.a. functions such that limn→∞ fn(t) = f(t) converges uniformly for t ∈ T. Then f is a.a. (iii) Let f : T → X be a.a. and ϕ : X → Y be a continuous function. Then the composition function ϕ ◦ f : T → Y is a.a. (iv) Let f, g : T → X be a.a. Then fg defined by (fg)(t) = f(t)g(t) is a.a. (v) Let f : T → R be a.a. and f ̸= 0 on T. If 1 f is bounded, then 1 f is a.a. (vi) Let f : T → X be a.a. and g : T → X is Sp-a.a. Then fg is Sp-a.a. EJDE-2025/87 ALMOST AUTOMORPHIC SOLUTIONS TO NON-AUTONOMOUS DYNAMIC EQUATIONS 5 Proof. (i)-(v) have been established in [24]. It remains to prove (vi). Since f is a.a. and g is Sp-a.a., for any sequence {ξ′n}∞n=1 ⊂ Π, there exist a subsequence {ξn}∞n=1 of {ξ′n}∞n=1 and two functions f̃ , g̃ such that lim n→∞ f(t+ ξn) = f̃(t) for t ∈ T, lim n→∞ f̃(t− ξn) = f(t) for t ∈ T, (2.2) lim n→∞ ( 1 K ∫ [t,t+K)T ∥g(s+ ξn)− g̃(s)∥p∆s )1/p = 0 for t ∈ T, lim n→∞ ( 1 K ∫ [t,t+K)T ∥g̃(s− ξn)− g(s)∥p∆s )1/p = 0 for t ∈ T. (2.3) Using Minkowski’s inequality, we have( 1 K ∫ [t,t+K)T ∥f(s+ ξn)g(s+ ξn)− f̃(s)g̃(s)∥p∆s )1/p ≤ ( 1 K ∫ [t,t+K)T ∥(f(s+ ξn)− f̃(s))g̃(s)∥p∆s )1/p + ( 1 K ∫ [t,t+K)T ∥f(s+ ξn)(g(s+ ξn)− g̃(s))∥p∆s )1/p := In + Jn. Notice that In ≤ 2∥f∥∞∥g̃∥Sp . By Lebesgue’s dominated convergence theorem and (2.2), we obtain lim n→∞ In ≤ ( 1 K ∫ [t,t+K)T lim n→∞ ∥(f(s+ ξn)− f̃(s))g̃(s)∥p∆s )1/p = 0 for t ∈ T. Using (2.3) and the boundedness of f , we obtain lim n→∞ Jn ≤ lim n→∞ ( 1 K ∫ [t,t+K)T ∥g(s+ ξn)− g̃(s)∥p∆s )1/p ∥f∥∞ = 0 for t ∈ T. Thus, lim n→∞ ( 1 K ∫ [t,t+K)T ∥f(s+ ξn)g(s+ ξn)− f̃(s)g̃(s)∥p∆s )1/p = 0 for t ∈ T. Similarly, we can show that lim n→∞ ( 1 K ∫ [t,t+K)T ∥f̃(s− ξn)g̃(s− ξn)− f(s)g(s)∥p∆s )1/p = 0 for t ∈ T. That is, fg is Sp-a.a. □ 3. Composition theorem Lemma 3.1. Let K ⊂ X be compact and f ∈ SpAA(T × X,Y ) with f̃ the limit function in Definition 2.10. Assume that f satisfies the hypothesis (A1) There exists a nonnegative scalar function L ∈ SpAA(T, R) with L̃ the limit function in Definition 2.8 such that ∥f(t, x)− f(t, y)∥ ≤ L(t)∥x− y∥, x, y ∈ X, t ∈ T. (3.1) Then for x, y ∈ K, t ∈ T and a.e. s ∈ [t, t+K)T, ∥f̃(s, x)− f̃(s, y)∥ ≤ L̃(s)∥x− y∥, where L̃ ∈ BSp(T, R). 6 F.-X. ZHENG, H.-X. LI EJDE-2025/87 Proof. Since f ∈ SpAA(T×X,Y ) with f̃ the limit function in Definition 2.10, and L ∈ SpAA(T) with L̃ the limit function in Definition 2.8, it follows that for any sequence {ξ′n}∞n=1 ⊂ Π, there exists a subsequence {ξn}∞n=1 of {ξ′n}∞n=1 and two functions f̃ , L̃ such that lim n→∞ ( 1 K ∫ [t,t+K)T ∥f(s+ ξn, x)− f̃(s, x)∥p∆s )1/p = 0 for (t, x) ∈ T×X, lim n→∞ ( 1 K ∫ [t,t+K)T ∥L(s+ ξn)− L̃(s)∥p∆s )1/p = 0 for t ∈ T. Thus lim n→∞ f(s+ ξn, x) = f̃(s, x) for (t, x) ∈ T×X and a.e. s ∈ [t, t+K)T, (3.2) lim n→∞ L(s+ ξn) = L̃(s) for t ∈ T and a.e. s ∈ [t, t+K)T. (3.3) Then for x, y ∈ K, t ∈ T and a.e. s ∈ [t, t+K)T, we have ∥f̃(s, x)− f̃(s, y)∥ ≤ ∥f̃(s, x)−f(s+ξn, x)∥+∥f(s+ξn, x)−f(s+ξn, y)∥+∥f̃(s, y)−f(s+ξn, y)∥. Taking limits as n → ∞ on the above inequality, by (3.1), (3.2) and (3.3), we obtain that ∥f̃(s, x)− f̃(s, y)∥ ≤ lim n→∞ L(s+ ξn)∥x− y∥ = L̃(s)∥x− y∥. Moreover, by (3.3) and Fatou’s Lemma, we obtain for t ∈ T, 1 K ∫ [t,t+K)T ∥L̃(s)∥p∆s = 1 K ∫ [t,t+K)T ∥ lim n→∞ L(s+ ξn)∥p∆s = 1 K ∫ [t,t+K)T lim inf n→∞ ∥L(s+ ξn)∥p∆s ≤ lim inf n→∞ 1 K ∫ [t,t+K)T ∥L(s+ ξn)∥p∆s ≤ ∥L∥pSp , which means that L̃ ∈ BSp(T, R). □ Lemma 3.2. Let K ⊂ X be compact and f ∈ SpAA(T ×X,Y ). Assume that condition (A1) in Lemma 3.1 holds. Then f ∈ SpAAK(T×X,Y ). Proof. Since f ∈ SpAA(T × X,Y ), for any sequence {ξ′n}∞n=1 ⊂ Π, there exist a subsequence {ξn}∞n=1 of {ξ′n}∞n=1 and a function f̃ : T×X → Y such that lim n→∞ ( 1 K ∫ [t,t+K)T ∥f(s+ ξn, x)− f̃(s, x)∥p∆s Big)1/p = 0 for (t, x) ∈ T×X. (3.4) Thus lim n→∞ f(s+ ξn, x) = f̃(s, x) for (t, x) ∈ T×X and a.e. s ∈ [t, t+K)T, Moreover, by Lemma 3.1, there exists a nonnegative scalar function L̃ ∈ BSp(T, R) such that for all x, y ∈ K, t ∈ T and a.e. s ∈ [t, t+K)T, ∥f̃(s, x)− f̃(s, y)∥ ≤ L̃(s)∥x− y∥. (3.5) For ε > 0, there exists a finite subset {xi, i = 1, 2, . . . , k} such that K ⊂ ⋃k i=1 B(xi, ε), where B(xi, ε) denotes a neighborhood with xi ∈ K as the center and ε as the radius. Let t ∈ T, by (3.4), there exists an integer N = N(t, ε) such that( 1 K ∫ [t,t+K]T ∥f(s+ ξn, xi)− f̃(s, xi)∥p∆s )1/p < ε/k (3.6) EJDE-2025/87 ALMOST AUTOMORPHIC SOLUTIONS TO NON-AUTONOMOUS DYNAMIC EQUATIONS 7 for n > N , i = 1, 2, . . . , k. Let x ∈ K, there exists i ∈ {1, 2, . . . , k} such that x ∈ B(xi, ε). Combining (3.5) and (3.6), under condition (A1) in Lemma 3.1, we obtain that for n > N and a.e. s ∈ [t, t+K)T, sup x∈K ∥f(s+ ξn, x)− f̃(s, x)∥ ≤ sup x∈K ∥f(s+ ξn, x)− f(s+ ξn, xi)∥+ sup x∈K ∥f̃(s, xi)− f̃(s, x)∥+ max 1≤i≤k ∥f(s+ ξn, xi)− f̃(s, xi)∥ ≤ L(s+ ξn)ε+ L̃(s)ε+ k∑ i=1 ∥f(s+ ξn, xi)− f̃(s, xi)∥. (3.7) Since ( 1 K ∫ [t,t+K)T ∥L(s+ ξn)ε∥p∆s )1/p ≤ ∥L∥Spε for n ∈ N,( 1 K ∫ [t,t+K)T ∥L̃(s)ε∥p∆s )1/p ≤ ∥L̃∥Spε, ( 1 K ∫ [t,t+K)T k∑ i=1 ∥f(s+ ξn, xi)− f̃(s, xi)∥p∆s )1/p ≤ k∑ i=1 ε/k = ε for n > N, we deduce from (3.7) that for n > N ,( 1 K ∫ [t,t+K)T sup x∈K ∥f(s+ ξn, x)− f̃(s, x)∥p∆s )1/p ≤ (∥L∥Sp + ∥L̃∥Sp + 1)ε, which implies lim n→∞ ( 1 K ∫ [t,t+K)T sup x∈K ∥f(s+ ξn, x)− f̃(s, x)∥p∆s )1/p = 0 for t ∈ T. Similarly, we obtain lim n→∞ ( 1 K ∫ [t,t+K)T sup x∈K ∥f̃(s− ξn, x)− f(s, x)∥p∆s )1/p = 0 for t ∈ T. That is f ∈ SpAAK(T×X,Y ). □ Theorem 3.3. Let f ∈ SpAA(T×X,Y ) and condition (A1) in Lemma 3.1 holds. If x ∈ AA(T, X), then f(·, x(·)) ∈ SpAA(T, Y ). Proof. From condition (A1) in Lemma 3.1, it follows that ∥f(·, x(·))∥Sp ≤ ∥f(·, x(·))− f(·, 0)∥Sp + ∥f(·, 0)∥Sp ≤ ∥L∥Sp∥x∥∞ + ∥f(·, 0)∥Sp < ∞. That is, f(·, x(·)) is Sp-bounded. Since x ∈ AA(T, X), for any sequence {ξ′′n}∞n=1 ⊂ Π, there exist a subsequence {ξ′n}∞n=1 of {ξ′′n}∞n=1 and a function x̃ such that lim n→∞ x(t+ ξ′n) = x̃(t) for t ∈ T, lim n→∞ x̃(t− ξ′n) = x(t) for t ∈ T. (3.8) Let K = {x(t) : t ∈ T}, then K is compact and x̃(t) ∈ K for t ∈ T. By Lemma 3.2, we have f ∈ SpAAK(T×X,Y ), and a subsequence {ξn}∞n=1 of {ξ′n}∞n=1 and a function f̃ such that lim n→∞ ( 1 K ∫ [t,t+K)T sup x∈K ∥f(s+ ξn, x)− f̃(s, x)∥p∆s )1/p = 0 for t ∈ T, lim n→∞ ( 1 K ∫ [t,t+K)T sup x∈K ∥f̃(s− ξn, x)− f(s, x)∥p∆s )1/p = 0 for t ∈ T. (3.9) 8 F.-X. ZHENG, H.-X. LI EJDE-2025/87 Let t ∈ T, by Minkowski’s inequality and Lemma 3.1, we have( 1 K ∫ [t,t+K)T ∥f(s+ ξn, x(s+ ξn))− f̃(s, x̃(s))∥p∆s )1/p ≤ ( 1 K ∫ [t,t+K)T |f(s+ ξn, x(s+ ξn))− f̃(s, x(s+ ξn))∥p∆s )1/p + ( 1 K ∫ [t,t+K)T ∥f̃(s, x(s+ ξn))− f̃(s, x̃(s))∥p∆s )1/p ≤ ( 1 K ∫ [t,t+K)T sup x∈K ∥f(s+ ξn, x)− f̃(s, x)∥p∆s )1/p + ( 1 K ∫ [t,t+K)T L̃p(s)∥(x(s+ ξn)− x̃(s))∥p∆s )1/p := Jn + In. Notice that In ≤ 2∥x∥∞∥L̃∥Sp . By Lebesgue’s dominated convergence theorem and (3.8), we obtain lim n→∞ In ≤ ( 1 K ∫ [t,t+K)T lim n→∞ L̃p(s)∥(x(s+ ξn)− x̃(s))∥p∆s )1/p = 0. Meanwhile, (3.9) implies limn→∞ Jn = 0. That is, lim n→∞ ( 1 K ∫ [t,t+K)T ∥f(s+ ξn, x(s+ ξn))− f̃(s, x̃(s))∥p∆s )1/p = 0 for t ∈ T. Similarly, we obtain lim n→∞ ( 1 K ∫ [t,t+K)T ∥f̃(s− ξn, x̃(s− ξn))− f(s, x(s))∥p∆s )1/p = 0 for t ∈ T. Therefore, f(·, x(·)) ∈ SpAA(T, Y ). □ Remark 3.4. We note that to get the composition theorem 3.3, we do not assume the uniform Lipschitz condition. Instead, we use condition (A1) with a Lipschitz coefficient function L(t). 4. A.a. solutions Now we consider the non-autonomous dynamic equation u∆(t) = A(t)u(t) + f(t, u(t)), t ∈ T. (4.1) We first recall some concepts which will be used to obtain our main results. Definition 4.1 ([5, 6]). Let p ∈ R(T, R). The exponential function is defined as ep(t, s) = exp (∫ t s ξµ(τ)(p(τ))∆τ ) , s, t ∈ T, with ξh(z) = { 1 h log(1 + zh), h ̸= 0, z, h = 0, where Log is the principal logarithm function. Lemma 4.2 ([5, 6]). If p ∈ R and t, s, r ∈ T, then (i) ep(t, t) = 1. (ii) ep(σ(t), s) = (1 + µ(t)p(t))ep(t, s). (iii) ep(t, r)ep(r, s) = ep(t, s). (iv) e⊖p(s, t) = ep(t, s) = 1 ep(s,t) . (v) (ep(r, ·))△ = −pep(r, σ(·)) and∫ t s p(τ)ep(r, σ(τ))∆τ = ep(r, s)− ep(r, t). EJDE-2025/87 ALMOST AUTOMORPHIC SOLUTIONS TO NON-AUTONOMOUS DYNAMIC EQUATIONS 9 Lemma 4.3. Let α > 0, q > 1 be two constants and t, s ∈ T. Then (i) (e⊖α(t, s)) q ≤ e⊖(qα)(t, s), t ≥ s. (ii) For t ≥ σ(s), ∑ j≥1 Fj(t) is uniformly convergent. Moreover, ∑ j≥1 Fj(t) ≤ { 1 1−e−α ( 1−e−qα qα )1/q , T = R, 1 1−(1+qαK)−1/q ( eqαK−1 qα )1/q , T ̸= R, where Fj(t) = (∫ [t−jK,t−(j−1)K)T (e⊖α(t, σ(s))) q∆s )1/q . (iii) For t ≤ σ(s), ∑ j≥1 Ej(t) is uniformly convergent. Moreover, ∑ j≥1 Ej(t) ≤  1 1−e−α ( 1−e−qα qα )1/q , T = R, (1+qαK)−1/q 1−(1+qαK)−1/q ( eqαK−1 qα )1/q , T ̸= R, where Ej(t) = ( ∫ [t+(j−1)K,t+jK)T (e⊖α(σ(s), t)) q∆s )1/q . Proof. (i) If µ(τ) = 0, for τ ∈ [s, t]T, (e⊖α(t, s)) q = exp ( q ∫ t s (−α)∆τ ) = e⊖(qα)(t, s). If µ(τ) ̸= 0, for τ ∈ [s, t]T, according to the definition of exponential function, we have (e⊖α(t, s)) q = exp (∫ t s q µ(τ) log(1 + µ(τ)(⊖α))∆τ ) ≤ exp (∫ t s 1 µ(τ) log(1 + µ(τ)(⊖(qα)))∆τ ) = e⊖(qα)(t, s). (ii) Let Fj(t) = (∫ [t−jK,t−(j−1)K)T (e⊖α(t, σ(s))) q∆s )1/q . If T = R, by a simple computation, we obtain ∑ j≥1 Fj(t) = 1 1− e−α (1− e−qα qα )1/q . If T ̸= R, since σ(t) ≤ t+K, t ∈ T, exp (∫ t+K t 1 µ(τ) log(1 + µ(τ)qα)∆τ ) is decreasing in µ(τ) ∈ [0,K], τ ∈ [t, t + K]T, then according to the definition of exponential function, we have e⊖qα(t, t+K) = exp (∫ t t+K 1 µ(τ) log(1 + µ(τ)(⊖qα))∆τ ) ≤ exp (∫ t t+K ξµ(τ)=0(⊖qα)∆τ ) = eqαK. (4.2) Also, since exp ( − ∫ t t−K 1 µ(τ) log(1 + µ(τ)qα)∆τ ) 10 F.-X. ZHENG, H.-X. LI EJDE-2025/87 is increasing in µ(τ) ∈ [0,K], τ ∈ [t, t + K]T, according to the definition of exponential function, we have e⊖(qα)(t, t−K) = exp ( ( ∫ t t−K 1 µ(τ) log(1 + µ(τ)(⊖qα))∆τ ) ≤ exp (∫ t t−K ξµ(τ)=K(⊖qα)∆τ ) = (1 + qαK)−1. (4.3) Using that e⊖(qα)(t− jK, t− (j − 1)K) = e⊖(qα)(t, t+K) ([28, Lemma 4.5]), we deduce from (4.2) that e⊖(qα)(t− jK, t− (j − 1)K) ≤ eqαK (4.4) Moreover, by Lemma 4.2 (iii) and (4.3), we have e⊖(qα)(t, t− jK) = (e⊖(qα)(t+K, t))j ≤ (1 + qαK)−j . (4.5) Combining (4.4) and (4.5), by (i) and Lemma 4.2 (iii) and (v), we obtain∑ j≥1 Fj(t) ≤ ∑ j≥1 (∫ [t−jK,t−(j−1)K)T e⊖(qα)(t, σ(s))∆s )1/q = ∑ j≥1 (1− eqαK ⊖(qα) (1 + qαK)−j )1/q ≤ 1 1− (1 + qαK)−1/q (eqαK − 1 qα )1/q . The proof of (iii) is similar to that for (ii); we omit it. □ Definition 4.4 ([19, 24]). Let A : T → Rn×n be a rd-continuous matrix-valued function, and X(t) be a fundamental matrix of the homogeneous equation of (4.1): u∆(t) = A(t)u(t), t ∈ T. (4.6) We say that (4.6) has an exponential dichotomy with parameters (α, c, P ) if there exists a projec- tion P , which is commutable with X(t), and two positive constants c, α such that ∥G(t, σ(s))∥ ≤ { ce⊖α(t, σ(s)), t ≥ σ(s), t, s ∈ T, ce⊖α(σ(s), t), t < σ(s), t, s ∈ T, (4.7) with G(t, σ(s)) = { X(t)PX−1(σ(s)), t ≥ σ(s), −X(t)(I − P )X−1(σ(s)), t < σ(s). (4.8) The matrix G is called the Green’s function of (4.6). If P = I, (4.6) is exponential stable with parameters (α, c, I) which means |Ψ(t, σ(s))| ≤ ce⊖α(t, σ(s)) for all t ≥ σ(s), t, s ∈ T, where Ψ(t, σ(s)) = X(t)X−1(σ(s)). Definition 4.5 ([24]). If u : T → Rn satisfies u(t) = ∫ T G(t, σ(s))f(s, u(s))∆s, t ∈ T with G(t, σ(s)) defined as (4.8), then u is called a solution of (4.1). Definition 4.6. A rd-continuous function G : T×T → Rn×n is said to be Bi-almost automorphic (abbreviated as Bi-a.a.) if for any sequence {ξ′n}∞n=1 ⊂ Π, there exist a subsequence {ξn}∞n=1 of {ξ′n}∞n=1 and a function G̃ such that lim n→∞ G(t+ ξn, s+ ξn) = G̃(t, s), lim n→∞ G̃(t− ξn, s− ξn) = G(t, s) for all (t, s) ∈ T2. Denote by BAA(T× T, Rn×n) the space of all such functions. EJDE-2025/87 ALMOST AUTOMORPHIC SOLUTIONS TO NON-AUTONOMOUS DYNAMIC EQUATIONS 11 Remark 4.7. Let G(t, s) = g(t − s) for some rd-continuous function g : T × T → Rn×n, then it is easy to verify that G ∈ BAA(T× T, Rn×n). Now we present some results on a.a. solutions to (4.1). We use the following assumptions: (A2) A ∈ R(T, Rn×n) ∩AA(T, Rn×n) such that {(I + µ(t)A(t))−1 : t ∈ T} is bounded. (A3) (4.6) has an exponential dichotomy with parameters (α, c, P ). (A4) (4.6) is exponentially stable with parameters (α, c, I). (A5) f ∈ SpAA(T×Rn, Rn) and satisfies condition (A1) in Lemma 3.1. Remark 4.8. Assume that (A2) and (A3) hold. Then G(t, σ(s)) is Bi-a.a. Indeed, if A is a rd-continuous matrix-valued functions, then A is ∆-integrable. By [24, Theorems 4.6,4.7,4.8], X(t) and X−1(t) are continuous which yields that X(t)X−1(σ(s)) is rd-continuous in t, s, that is, G(t, σ(s)) is rd-continuous. This together with [24, Lemma 5.5] leads to the conclusion. Lemma 4.9. If f ∈ SpAA(T, Rn) (1 < p < ∞) and there is a Bi a.a. function G(t, σ(s)) such that (4.7) holds, then the function u(t) = ∫ T G(t, σ(s))f(s)∆s belongs to AA(T, Rn). Proof. Let ϱ(t) = ∫ (−∞,t)T G(t, σ(s))f(s)∆s, t ∈ T, χ(t) = ∫ [t,∞)T G(t, σ(s))f(s)∆s, t ∈ T. Then u(t) = ϱ(t) + χ(t), t ∈ T. We only prove ϱ(t) ∈ AA(T, Rn), since χ(t) ∈ AA(T, Rn) can be obtained similarly. Let ϕj(t) = ∫ [t−jK,t−(j−1)K)T G(t, σ(s))f(s)∆s, for each t ∈ T and j = 1, 2, 3, . . . . Then ϱ(t) = ∑ j≥1 ϕj(t). Now we can complete the proof by the following 3 steps. Step 1. We prove that ϱ(t) = ∑ j≥1 ϕj(t) is uniformly convergent on T. Let q > 1 be such that 1 p + 1 q = 1, using Hölder inequality, it follows that for t ∈ T, j ∈ {1, 2, 3, . . . }, ∥ϕj(t)∥ ≤ ∫ [t−jK,t−(j−1)K)T ce⊖α(t, σ(s)) ∥f(s)∥∆s ≤ c (∫ [t−jK,t−(j−1)K)T (e⊖α(t, σ(s))) q∆s )1/q(∫ [t−jK,t−(j−1)K)T ∥f(s)∥p∆s )1/p ≤ c (∫ [t−jK,t−(j−1)K)T (e⊖α(t, σ(s))) q∆s )1/q K1/p∥f∥Sp . By Lemma 4.3 (ii), we know that∑ j≥1 (∫ [t−jK,t−(j−1)K)T (e⊖α(t, σ(s))) q∆s )1/q is uniformly convergent. Then we deduce that ϱ(t) = ∑ j≥1 ϕj(t) is uniformly convergent. Step 2. We prove that ϱ is rd-continuous on T. Let t ∈ T be any right-dense point. For any sequence {t+ hm} ⊂ T such that hm ≥ 0, hm → 0 as m → ∞, ∥ϕj(t+ hm)− ϕj(t)∥ → 0, j = 1, 2, 3, . . . . Indeed, for j = {1, 2, 3, . . . }, we deduce from (4.7) that ∥ϕj(t+ hm)− ϕj(t)∥ 12 F.-X. ZHENG, H.-X. LI EJDE-2025/87 = ∥∥∥∫ [t−(j−1)K,t+hm−(j−1)K)T G(t+ hm, σ(s))f(s)∆s − ∫ [t−jK,t+hm−jK)T G(t+ hm, σ(s))f(s)∆s + ∫ [t−jK,t−(j−1)K)T (G(t+ hm, σ(s))−G(t, σ(s)))f(s)∆s ∥∥∥ ≤ c ∫ [t−(j−1)K,t+hm−(j−1)K)T ∥f(s)∥∆s+ c ∫ [t−jK,t+hm−jK)T ∥f(s)∥∆s + ∥∥∫ [t−jK,t−(j−1)K)T (G(t+ hm, σ(s))−G(t, σ(s)))f(s)∆s ∥∥ := I1m + I2m + Jm. Note that I1m ≤ ch1/q m (∫ [t−(j−1)K,t+hm−(j−1)K)T ∥f(s)∥p∆s )1/p ≤ ch1/q m K1/p∥f∥Sp by using Hölder inequality. So we have I1m → 0 as hm → 0. Similarly, we have I2m → 0 as hm → 0. Now we show that Jm → 0 as hm → 0. Since G(t, σ(s)) is Bi-a.a., then G(t, σ(s)) is rd-continuous. Thus ∥G(t+ hm, σ(s))−G(t, σ(s)))∥ → 0 as hm → 0, which yields ∥(G(t+ hm, σ(s))−G(t, σ(s)))f(s)∥ → 0 (4.9) as hm → 0. From (4.7), it follows that ∥(G(t+ hm, σ(s))−G(t, σ(s)))f(s)∥ ≤ 2c∥f(s)∥, t, s ∈ T. Then Jm ≤ 2c ∫ [t−jK,t−(j−1)K)T ∥f(s)∥∆s ≤ 2cK∥f∥Sp . By Lebesgue’s dominated convergence theorem and (4.9), we obtain Jm → 0 as hm → 0. That is, ϕj is right continuous for each right dense point t ∈ T. Similarly, we can prove that the left limit of ϕj exists at each left-dense point t ∈ T. So ϕj is rd-continuous for t ∈ T, j = 1, 2, 3, . . . . Therefore, the uniform limit ϱ = ∑ j≥1 ϕj is rd-continuous on T. Step 3. We prove that ϱ ∈ AA(T, Rn). Since f ∈ SpAA(T, Rn) and G ∈ BAA(T× T, Rn), for any sequence {t′n}∞n=1 ⊂ Π, there exist a subsequence {tn}∞n=1 of {t′n}∞n=1 and two functions f̃ , G̃ such that ( 1 K ∫ [t,t+K)T ∥f(tn + s)− f̃(s)∥pds )1/p → 0 as n → ∞,( 1 K ∫ [t,t+K)T ∥f̃(s− tn)− f(s)∥pds )1/p → 0 as n → ∞, (4.10) for each t ∈ T. And ∥G(t+ tn, σ(s) + tn)− G̃(t, σ(s))∥ → 0 as n → ∞, ∥G̃(t− tn, σ(s)− tn)−G(t, σ(s))∥ → 0 as n → ∞, (4.11) for each t, s ∈ T. Let j ∈ {1, 2, 3, . . . }. We set ϕ̃j(t) = ∫ [t−jK,t−(j−1)K)T G̃(t, σ(s))f̃(s)∆s, t ∈ T. Then using that σ(s+ tn) = σ(s) + tn ([25, Lemma 3.3]), we deduce from (4.7) that ∥ϕj(t+ tn)− ϕ̃j(t)∥ ≤ ∫ [t−jK,t−(j−1)K)T ∥G(t+ tn, σ(s) + tn)(f(s+ tn)− f̃(s))∥∆s EJDE-2025/87 ALMOST AUTOMORPHIC SOLUTIONS TO NON-AUTONOMOUS DYNAMIC EQUATIONS 13 + ∫ [t−jK,t−(j−1)K)T ∥(G(t+ tn, σ(s) + tn)− G̃(t, σ(s)))f̃(s)∥∆s ≤ c ∫ [t−jK,t−(j−1)K)T ∥f(s+ tn)− f̃(s)∥∆s + ∫ [t−jK,t−(j−1)K)T ∥(G(t+ tn, σ(s) + tn)− G̃(t, σ(s)))f̃(s)∥∆s ≤ cK ( 1 K ∫ [t−jK,t−(j−1)K)T ∥f(s+ tn)− f̃(s)∥p∆s )1/p + ∫ [t−jK,t−(j−1)K)T ∥(G(t+ tn, σ(s) + tn)− G̃(t, σ(s)))f̃(s)∥∆s := In + Jn. Using (4.10), we obtain In → 0 as n → ∞ for t ∈ T. From (4.7) and (4.11), it follows that ∥(G(t+ tn, σ(s) + tn)− G̃(t, σ(s)))f̃(s)∥ ≤ 2c∥f̃(s)∥. Then Jn ≤ 2c ∫ [t−jK,t−(j−1)K)T ∥f̃(s)∥∆s ≤ 2cK∥f̃∥Sp . By Lebesgue’s dominated convergence theorem and (4.11), we obtain Jn → 0 as n → ∞ for t ∈ T. Thus ∥ϕj(t+ tn)− ϕ̃j(t)∥ → 0 as n → ∞ for t ∈ T. Similarly, we can easily get that ∥ϕ̃j(t− tn)− ϕj(t)∥ → 0 as n → ∞ for t ∈ T. That is, ϕj ∈ AA(T, Rn) for j = 1, 2, 3, . . . . So the uniform limit ϱ = ∑ j≥1 ϕj is almost automorphic by Lemma 2.12 (ii). □ Theorem 4.10. Assume that (A2)–(A4) hold. If ∥L∥Sp <  ( 2c 1−e−α )−1( 1−e−qα qα )−1/q , T = R,( (1+qαK)−1/q+1 1−(1+qαK)−1/q c(eqαK−1) qα )−1 , T ̸= R, (4.12) where 1 p + 1 q = 1, then equation (4.1) has a unique a.a. solution given by u(t) = ∫ T G(t, σ(s))f(s, u(s))∆s, t ∈ T (4.13) with G(t, σ(s)) defined in (4.8). Proof. Let u ∈ AA(T, Rn). From (A5), it follows that f(·, u(·)) ∈ SpAA(T, Rn) by Theorem 3.3. From [27, Theorem 3], we know that u∆(t) = A(t)u(t) + f(t), t ∈ T (4.14) has a unique solution satisfying (4.13). Consider operator Γ defined on AA (T, Rn) as (Γu)(t) = ∫ T G(t, σ(s))f(s, u(s))∆s, t ∈ T. By Remark 4.8, we know that G(t, σ(s)) is Bi-a.a. under conditions (A2) and (A3). Then we deduce from Lemma 4.9 that Γ : AA(T, Rn) → AA(T, Rn). If T = R, by using Hölder inequality and Lemma 4.3 (ii) and (iii), we obtain that for any u, v ∈ AA(R,Rn), ∥Γu(t)− Γv(t)∥ ≤ ∫ t −∞ ce−α(t−s)∥f(s, u(s))− f(s, v(s))∥ds 14 F.-X. ZHENG, H.-X. LI EJDE-2025/87 + ∫ ∞ t ce−α(s−t)∥f(s, u(s))− f(s, v(s))∥ds ≤ c (∫ t −∞ e−α(t−s)L(s)ds+ ∫ ∞ t e−α(s−t)L(s)ds ) ∥u− v∥∞ = c ∑ k≥1 ∫ t−k+1 t−k e−α(t−s)L(s)ds∥u− v∥∞ + c ∑ k≥1 ∫ t+k t+k−1 e−α(s−t)L(s)ds∥u− v∥∞ ≤ c ∑ k≥1 (∫ t−k+1 t−k e−qα(t−s)ds ) )1/q∥L∥Sp∥u− v∥∞ + c ∑ k≥1 (∫ t+k t+k−1 e−qα(s−t)ds )1/q ∥L∥Sp∥u− v∥∞ ≤ 2c 1− e−α (1− e−qα qα )1/q ∥L∥Sp∥u− v∥∞, t ∈ R. If T ̸= R, by using Hölder inequality and Lemma 4.3 (ii) and (iii), we obtain that for any u, v ∈ AA(T, Rn), ∥Γu(t)− Γv(t)∥ ≤ ∫ (−∞,t)T c e⊖α(t, σ(s))∥f(s, u(s))− f(s, v(s))∥∆s + ∫ [t,∞)T c e⊖α(σ(s), t)∥f(s, u(s))− f(s, v(s))∥∆s ≤ c (∫ (−∞,t)T e⊖α(t, σ(s))L(s)∆s+ ∫ [t,∞)T e⊖α(σ(s), t)L(s)∆s ) ∥u− v∥∞ = c ∑ j≥1 ∫ [t−jK,t−(j−1)K)T e⊖α(t, σ(s))L(s)∆s∥u− v∥∞ + c ∑ j≥1 ∫ [t+(j−1)K,t+jK)T e⊖α(σ(s), t)L(s)∆s∥u− v∥∞ ≤ c ∑ j≥1 (∫ [t−jK,t−(j−1)K)T (e⊖α(t, σ(s))) q∆s )1/q K1/p∥L∥Sp∥u− v∥∞ + c ∑ j≥1 (∫ [t+(j−1)K,t+jK)T (e⊖α(σ(s), t)) q∆s )1/q K1/p∥L∥Sp∥u− v∥∞ ≤ (1 + qαK)−1/q + 1 1− (1 + qαK)−1/q c ( eqαK − 1 ) qα ∥L∥Sp∥u− v∥∞, t ∈ T. Hence, the mapping Γ is a contraction by assumption (4.12). By Banach fixed point theorem, Γ has a unique fixed point in AA(T, Rn). Thus equation (4.1) has a unique a.a. solution. □ If (4.6) is exponentially stable, from the proof of Theorem 4.10, we obtain the following result immediately. Corollary 4.11. Assume that (A2), (A3), (A5) hold. If ∥L∥Sp <  ( c 1−e−α )−1( 1−e−qα qα )−1/q , T = R,( 1 1−(1+qαK)−1/q c(eqαK−1) qα )−1 , T ̸= R, EJDE-2025/87 ALMOST AUTOMORPHIC SOLUTIONS TO NON-AUTONOMOUS DYNAMIC EQUATIONS 15 where 1 p + 1 q = 1, then equation (4.1) has a unique a.a. solution given by u(t) = ∫ (−∞,t)T Ψ(t, σ(s))f(s, u(s))∆s, t ∈ T. Remark 4.12. (i) Theorem 4.10 shows that we extend the result of [24, Theorem 6.3]. With- out assuming the uniform Lipschitz condition of the nonlinear forcing term, the existence and uniqueness of almost automorphic solution to dynamic equation (4.1) with Stepanov-like almost automorphic nonlinear term are established in our results. Moreover, we do not need to assume that the Green’s function is Bi-a.a. directly in view of Remark 4.8. (ii) Comparing with [27, Theorem 4] and [28, Theorem 4.10], based on Lemma 4.3, the con- traction conditions in Theorem 4.10 and Corollary 4.11 are different. 5. Application Consider the following Lasota-Wazewska model on time scales: u∆(t) = −β(t)u(σ(t)) + η(t)e−γ(t)u(t), t ∈ T, (5.1) where u represents the number of red blood cells, β > 0 is the rate of death of a red blood cell, while η > 0 and γ > 0 are the parameters related to the rate of production of a red blood cell. For more details about this model, see [18]. If p ∈ R, then the dynamic equation u∆(t) = p(t)u(t) is called regressive. Lemma 5.1 ([5]). Let u∆(t) = p(t)u(t) be regressive and t0 ∈ T. Then ep(t, t0) is a solution to the initial value problem u∆(t) = p(t)u(t), u(t0) = 1 on T. Lemma 5.2. Let β : T → R such that β = inft∈T β(t) > 0. The equation u∆(t) = (⊖β)(t)u(t) (5.2) is exponential stable. Proof. Let X(t) be a fundamental matrix of (5.2). It is clear that X(t)X−1(s) = e⊖β(t, s) by Lemma 5.1. If µ(τ) = 0, for τ ∈ [s, t]T, e⊖β(t, s) = exp ( − ∫ t s β(τ)∆τ ) ≤ exp ( − ∫ t s β∆τ ) = e⊖β(t, s). If µ(τ) ̸= 0, for τ ∈ [s, t]T, by Definition 4.1, we have e⊖β(t, s) = exp (∫ t s 1 µ(τ) log(1 + µ(τ)(⊖β)(τ))∆τ ) ≤ exp (∫ t s 1 µ(τ) log(1 + µ(τ)(⊖β))∆τ ) = e⊖β(t, s). So |X(t)X−1(σ(s))| ≤ e⊖β(t, σ(s)) for all t ≥ σ(s), t, s ∈ T. That is, (5.2) admits exponential stable with parameters (β, 1, I). □ Theorem 5.3. Suppose that β, γ are positive a.a. with β = inft∈T β(t) > 0, γ = supt∈T γ(t) > 0, η is positive Sp-a.a. for 1 < p < ∞. If ∥η∥Spγ <  ( 1 1−e−β )−1( 1−e−qβ qβ )−1/q , T = R,( 1 1−(1+qβK)−1/q (eqβK−1) qβ )−1 , T ̸= R, 16 F.-X. ZHENG, H.-X. LI EJDE-2025/87 where 1 p + 1 q = 1, then equation (5.1) has a unique a.a. solution given by u(t) = ∫ (−∞,t)T e⊖β(t, s)η(s)e −γ(s)u(s)∆s, t ∈ T. Proof. Using the formula u(σ(t)) = u(t) + µ(t)u∆(t) and the definition of ⊖β, (5.1) can be trans- formed into the equation u∆(t) = (⊖β)(t)u(t) + 1 1 + µ(t)β(t) η(t)e−γ(t)u(t), t ∈ T. By a simple computations, we have 1 + µ(t)(⊖β)(t) = 1 1 + µ(t)β(t) > 0. That is, ⊖β ∈ R(T, R). Note that µ is non-negative a.p. ([24, Theorem 3.4]). By Lemma 2.12 (iv)-(v), it is easy to get that ⊖β is a.a. since β is positive a.a. Moreover, {(1 + µ(t)(⊖β)(t))−1 : t ∈ T} = {1 + µ(t)β(t) : t ∈ T} is bounded. That is, ((A2) holds. Then it follows from Lemma 5.2 that u∆(t) = (⊖β)(t)u(t) is exponential stable with parameters (β, 1, I). That is, (A4) holds. Let f(t, u) = 1 1+µ(t)β(t)η(t)e −γ(t)u. Since β, µ and γ are a.a. and η is Sp-a.a., by Lemma 2.12 (iii)-(vi), we deduce that f(·, u) ∈ SpAA(T, R). and |f(t, u)− f(t, v)| ≤ L(t)|u− v| with L(t) = 1 1+µ(t)β(t)η(t)γ(t). Clearly, L ∈ SpAA(T) by Lemma 2.12 (iii)-(vi). That is, (A5) holds. Moreover, ∥L∥Sp ≤ ∥η∥Spγ. Thus all conditions of Corollary 4.11 are satisfied, which yields that equation (5.1) has a unique a.a. solution given by u(t) = ∫ (−∞,t)T e⊖β(t, σ(s)) 1 1 + µ(s)β(s) η(s)e−γ(s)u(s)∆s = ∫ (−∞,t)T e⊖β(t, s)η(s)e −γ(s)u(s)∆s. □ Example 5.4. Consider the equation u∆(t) = −(2.1 + sin( √ 3t))u(σ(t)) + η(t)e−(1.1+cos( √ 2t))u(t), t ∈ T, (5.3) with η(t) = { 0.25| sin 1 2+cos t+cos( √ 2t) |, t ∈ (n− 0.02, n+ 0.02), n ∈ Z, 0, otherwise. Obviously, β(t) = 2.1 + sin( √ 3t) and γ(t) = 1.1 + cos( √ 2t) are positive a.a. with β = 1.1 > 0, γ = 2.1 > 0, η is positive Sp-a.a. for p = 2. Note that ∥η∥Spγ < 0.05 ∗ 2.1 = 0.105. If T = R, then q = 2 and ( 1 1− e−β )−1(1− e−qβ qβ )−1/q ≈ 1.0494 > ∥η∥Spγ. If T = Z, then K = 1, q = 2 and( 1 1− (1 + qβK)−1/q (eqβK − 1) qβ )−1 ≈ 0.1209 > ∥η∥Spγ. By Theorem 5.3, equation (5.3) has a unique a.a. solution (see Figures 1 and 2). EJDE-2025/87 ALMOST AUTOMORPHIC SOLUTIONS TO NON-AUTONOMOUS DYNAMIC EQUATIONS 17 Figure 1. T = R. Curve of the a.a. solution of (5.3) with initial value u(0) = 0.05. Figure 2. T = Z. Curve of the a.a. solution of (5.3) with initial value u(0) = 0.05. Acknowledgment. We would like to express our sincere gratitude to the editors and referees for their valuable comments. This work is supported by a Grant of NNSF of China (No. 11971329). References [1] A. A. H. Ahmed, B. Hazarika; Aymptotically almost automorphic solution of abstract integro-dynamic equa- tion. J. Nonlinear Convex A., 25 (2024) (9):2269–2293. [2] F. M. Atici, D. C. Biles, A. Lebedinsky; An application of time scales to economics. Math. Comput. Model., 43 (2006): 718–726. [3] M. Bohner, G. S. Guseinov; Multiple integration on time scales. Dynam. Systems Appl., 14 (2005): 579–606. [4] M. Bohner, G. S. Guseinov; Multiple Lebesgue integration on time scales. Adv. Difference Equ., 2006 (2006): 1–12. [5] M. Bohner, A. Peterson; Dynamic equations on time scales: An introduction with applications. Birkhäuser, Boston, 2001. [6] M. Bohner, A. Peterson; Advances in Dynamic Equations on Time Scales. Birkhäuser, Boston, 2003. 18 F.-X. ZHENG, H.-X. LI EJDE-2025/87 [7] D. Brigo, F. Mercurio; Discrete time vs continuous time stock-price dynamics and implications for option pricing. Financ. Stoch., 4 (2004): 147–159. [8] A. Cabada, D. R. Vivero; Expression of the Lebesgue ∆-integral on time scales as a usual Lebesgue integral; application to the calculus of ∆-antiderivatives. Math. Comput. Modelling, 43 (2006): 194–207. [9] F. B. Christiansen, T. M. Fenchel; Theories of populations in biological communities, volume 20. Springer- Verlag, Berlin, 1977. [10] S. Dhama, S. Abbas; Existence and stability of weighted pseudo almost automorphic solution of dynamic equation on time scales with weighted Stepanov-like (Sp) pseudo almost automorphic coefficients. Qual. Theory Dyn. Syst., 19 (2020) (1): 46. [11] M. Es-Saiydy, M. Zarhouni, M. Zitane; Stepanov-like pseudo almost automorphy on time scales: New devel- opments and applications. Asia Pac. J. Math., 9 (2022): 9. [12] M. Es-saiydy, M. Zitane; Stepanov-like pseudo almost automorphic dynamics of QVRNNs with mixed delays on time scales via a direct method. Asia Pac. J. Math., 7 (2020): 32. [13] M. Es-saiydy, M. Zitane; Weighted Stepanov-Like Pseudo Almost Periodicity on Time Scales and Applications. Differ. Equat. Dyn. Sys., 31 (2023) (4): 869–893. [14] S. Hilger. Ein maßkettenkalkül mit anwendung auf zentrumsmannigfaltigkeiten. PhD thesis, Universität Würzburg, 1988. [15] M. Hu, P. L. Xie; Almost periodic solutions of neutral delay functional differential equations on time scales. Bull. Malays. Math. Sci. Soc., 38 (2015) (1): 317–331. [16] S. Keller; Asymptotisches Verhalten invarianter Faserbündel bei Diskretisierung und Mittelwertbildung im Rahmen der Analysis auf Zeitskalen. PhD thesis, Universität Augsburg, 1999. [17] I. Klapper, H. Qian; Remarks on discrete and continuous large-scale models of DNA dynamics. Biophys. J., 74 (1998) (5): 2504–2514. [18] A. Lasota, M. Ważewska-Czyżewska; Mathematical problems of the dynamics of a system of red blood cells. Ann. Polish Math. Soc. Ser. III, Appl. Math., 17 (1976): 23–40. [19] Y. K. Li, C. Wang; Almost periodic functions on time scales and applications. Discrete Dyn. Nat. Soc., 2011 (2011): 1095–1114. [20] Y. K. Li, C. Wang. Uniformly almost periodic functions and almost periodic solutions to dynamic equations on time scales. Abstr. Appl. Anal., 2011 (2011): 1–22. [21] Y. K. Li, C. Wang; Pseudo almost periodic functions and pseudo almost periodic solutions to dynamic equations on time scales. Adv. Difference Equ., 2012. [22] Y. K. Li, P. Wang; Almost periodic solution for neutral functional dynamic equations with stepanov-almost periodic terms on time scales. Discrete Contin. Dyn. Syst. Ser. S, 10 (2017) (3): 463–473. [23] Y. K. Li, L. L. Zhao. Weighted pseudo-almost periodic functions on time scales with applications to cellular neural networks with discrete delays. Math. Methods Appl. Sci., 40 (2017) (6): 1905–1921. [24] C. Lizama, J. G. Mesquita; Almost automorphic solutions of dynamic equations on time scales. J. Funct. Anal., 265 (2013): 2267–2311. [25] C. Lizama, J. G. Mesquita; Asymptotically almost automorphic solutions of dynamic equations on time scales. Topol. Methods Nonlinear Anal., 54 (2019) (1):59–80. [26] C. H. Tang, H. X. Li; The connection between pseudo almost periodic functions defined on time scales and on the real line. Bull. Aust. Math. Soc., 95 (2017) (3): 482–494. [27] C. H. Tang, H. X. Li; Bochner-like transform and Stepanov almost periodicity on time scales with applications. Symmetry, 10 (2018): 566. [28] C. H. Tang, H. X. Li; Stepanov-like pseudo almost periodic functions on time scales and applications to dynamic equations with delay. Open Math., 16:826–841, 2018. [29] C. C. Tisdell and A. Zaidi. Basic qualitative and quantitative results for solutions to nonlinear, dynamic equations on time scales with an application to economic modelling. Nonlinear Anal., 68 (2008) (11): 3504– 3524. [30] C. Wang, Y. K. Li; Weighted pseudo almost automorphic functions with applications to abstract dynamic equations on time scales. Ann. Polon. Math., 108 (2013) (3): 225–240. [31] Q. L. Wang, Z. J. Liu; Existence and stability of positive almost periodic solutions for a competitive system on time scales. Math. Comput. Simulation, 138 (2017): 65–77. [32] Z. J. Yao; Uniqueness and global exponential stability of almost periodic solution for hematopoiesis model on time scales. J. Nonlinear Sci. Appl., 8 (2015) (2): 142–152. Feng-Xia Zheng School of Science, Xihua University, Chengdu 610039, Sichuan, China Email address: zhengfengxiade@163.com Hong-Xu Li (corresponding author) Department of Mathematics, Sichuan University, Chengdu 610064, Sichuan, China Email address: hoxuli@scu.edu.cn 1. Introduction 2. Preliminaries 3. Composition theorem 4. A.a. solutions 5. Application Acknowledgment References