Electronic Journal of Differential Equations, Vol. 2024 (2024), No. 35, pp. 1–11. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu or https://ejde.math.unt.edu DOI: 10.58997/ejde.2024.35 MASSERA TYPE THEOREMS FOR ABSTRACT NON-AUTONOMOUS EVOLUTION EQUATIONS LAN-LING ZHENG, HUI-SHENG DING Abstract. We establish two fixed point theorems for affine maps in Banach spaces, with weaker assumptions than those in the literature. Then we estab- lish some Massera type results for abstract linear evolution equations without assuming the existence of bounded solutions, which is an indispensable condi- tion in the classical Massera theorem and in the earlier literature. As appli- cation, we present an existence result on periodic mild solutions to abstract nonautonomous semilinear evolution equations. 1. Introduction Let X be a Banach space. Our aim is to investigate the existence of periodic mild solutions to the nonautonomous evolution equations u′(t) = A(t)u(t) + g(t), t ≥ 0, (1.1) and u′(t) = A(t)u(t) + f(t, u(t)), t ≥ 0 (1.2) onX. g and f satisfy conditions specified later, and {A(t)}t≥0 satisfies the following assumptions: (A1) D(A(t)) = D ⊂ X for all t ≥ 0 and A(t) is not necessarily densely defined, i.e., D = X is not necessarily true; (A2) there exist M ≥ 1 and ω ∈ R such that (ω,+∞) ⊂ ρ(A(t)) for every t ≥ 0 and ∥ k∏ j=1 (λI −A(tj)) −1∥ ≤ M (λ− ω)k for every λ > ω and every finite sequence {tj}kj=1 with 0 ≤ t1 ≤ t2 ≤ · · · ≤ tk, where k = 1, 2, . . . ; (A3) the mapping t 7→ A(t)x is continuously differentiable in X for every x ∈ D; (A4) A(t+ 1) = A(t) for every t ≥ 0. The existence of periodic solutions to (1.1), (1.2) and their variants has been of great interest for many authors (cf. [1, 3, 4, 5, 7, 9, 8, 11, 12, 13, 14, 18, 19]). To establish the existence of periodic solutions to (1.2), one of the key steps is to consider first the existence of periodic solutions to the linear equation (1.1). It is 2020 Mathematics Subject Classification. 34C25, 35B10, 34G10, 47D06. Key words and phrases. Massera theorem; affine map; periodic solution; abstract evolution. ©2024. This work is licensed under a CC BY 4.0 license. Submitted October 8, 2023. Published June 6, 2024. 1 2 L.-L. ZHENG, H.-S. DING EJDE-2024/35 well-known that to find an initial value x0 ∈ X for a 1-periodic mild solution of (1.1), one only needs to solve the equation x0 = P (x0), where P : X → X;x 7→ U(1, 0)x+ ∫ 1 0 U(1, τ)g(τ)dτ stands for the Poincaré mapping and {U(t, s)}t≥s≥0 is the evolution system gener- ated by {A(t)}t≥0. Now the problem is transformed into a fixed point problem of the Poincaré mapping, an affine map. For fixed point theorems of Poincaré mapping, one of the celebrated result is by Chow and Hale [2] who proved that P has a fixed point if the range R(I −U(1, 0)) is closed and there exists z0 ∈ X such that supn∈N ∥Pnz0∥ < +∞. Recently, Zubelevich [20] made an important progress and got the result that P has a fixed point if X is reflexive and supn∈N ∥Pnz0∥ < +∞ for some z0 ∈ X. So, Zubelevich removed the closedness of R(I − U(1, 0)) in the case of X being reflexive. Very recently, Ezzinbi and Taoudi [6] also established several interesting results in this direction on locally convex spaces and ordered Banach spaces. It is needed to note that the boundedness of {Pnz0} is a key assumption in all the above literature. As one will see, in this paper, this key assumption is weakened, i.e., we establish two fixed point theorems for Poincaré mapping, where {Pnz0} is not necessarily bounded for some z0 ∈ X. Moreover, we give two examples, which satisfy our weakened assumptions but not the boundedness condition in the earlier literature [2, 20] (see Section 2). Based on our new fixed point theorems for Poincaré mapping, we discuss the existence of periodic solutions to (1.1) and obtain two Massera type theorems for (1.1). It is interesting to note that in our Massera type results, the existence of bounded solutions is not presupposed as in the classical Massera theorem. In fact, to the best of our knowledge, of all the latest Massera type results up until now, the existence of bounded solutions is an indispensable assumption (see, e.g., [6, 7, 9, 10] for some recent Massera type results). Moreover, compared with other type results on the existence of periodic solutions for (1.1), our Massera type theorems also have some improvements to some extent. For example, our Massera type theorems do not need the assumptions ω < 0 and Meω < 1 in [15]. As application of our Massera type theorems, in the last part of this paper, we establish an existence result on periodic mild solutions to (1.2). Throughout this paper, we denote by R the set of real numbers, by R+ the set of non-negative real numbers, by C the set of complex numbers, by N the set of positive integers, by Lp([a, b], X) (Lp loc(R+, X)) the space of all (locally) pth integrable functions, by C(R+, X) the space of all continuous functions from R+ to X, and P1(R+, X) the space of all 1-periodic functions from R+ to X. 2. Two fixed point theorems for affine maps We first present a fixed point theorem, where the boundedness assumption in the classical results by Chow and Hale is weakened. Theorem 2.1. Let X be a Banach space, B : X → X be a bounded linear operator, z ∈ X and P : X → X;x 7→ Bx + z. Assume that the range R(I − B) is closed EJDE-2024/35 MASSERA THEOREMS FOR EVOLUTION EQUATIONS 3 and there exists x0 ∈ X such that lim n→+∞ ∥P nx0 n ∥ = 0. (2.1) Then P has a fixed point x ∈ X. Proof. Our proof starts with the observation that P has a fixed point if z ∈ R(I − B). Let xn = 1 n ∑n k=1 P kx0 for every n ∈ N, then by a direct calculation, we have lim n→+∞ (I −B)xn = lim n→+∞ [(I − P )xn + z] = lim n→+∞ 1 n [Px0 − Pn+1x0] + z = z. Combining this with R(I −B) is closed, we conclude that z ∈ R(I −B). □ Remark 2.2. In the case of R(I −B) = X, the conclusion holds without assump- tion (2.1). Moreover, Chow and Hale [2] obtained the same conclusion of Theorem 2.1 under the condition supn∈N ∥Pnx0∥ < +∞, which implies that (2.1) holds. However, the converse is not necessarily true. In fact, let B : l2 → l2; (x1, x2, . . . , xk, . . . ) 7→ ( 0, 4 √ 2x1, 4 √ 3 2 x2, . . . , 4 √ k + 1 k xk, . . . ) , and P : l2 → l2; (x1, x2, . . . , xk, . . . ) 7→ B(x1, x2, . . . , xk, . . . ) + (1, 0, 0, . . . ). By a direct calculation, we obtain that there exists x0 = (1, 0, 0, . . . ) ∈ l2 such that sup n∈N ∥Pnx0∥ ≥ sup n∈N 4 √ n+ 1 = +∞ and 0 ≤ lim n→+∞ ∥P nx0 n ∥ ≤ lim n→+∞ (n+ 1) 3 4 n = 0. If X have some special properties, then the condition that R(I − B) is closed can be removed. Theorem 2.3. Let X be a Banach space, B : X → X be a bounded linear operator, z ∈ X and P : X → X;x 7→ Bx+ z. Assume that there exists x0 ∈ X such that sup n∈N ∥ 1 n n∑ k=1 P kx0∥ < +∞, lim n→+∞ ∥P nx0 n ∥ = 0, (2.2) and one of the following conditions holds: (i) X is reflexive; (ii) there exists a separable Banach space Y such that X is the dual space of Y and B∗Y ⊂ Y , where B∗ is the dual operator of B and Y is considered as a subspace of Y ∗∗. Then P has a fixed point x ∈ X. Moreover, ∥x∥ ≤ sup n∈N ∥ 1 n n∑ k=1 P kx0∥. (2.3) 4 L.-L. ZHENG, H.-S. DING EJDE-2024/35 Proof. Firstly, we prove case (i). Let xn = 1 n n∑ k=1 P kx0 for n ∈ N. There exist {xnj} ⊂ {xn} and x ∈ X such that xnj converges weakly to x by using (2.2) and condition (i). So, for every x∗ ∈ X∗ lim j→+∞ ⟨x∗, Pxnj − Px⟩ = lim j→+∞ ⟨x∗, Bxnj −Bx⟩ = lim j→+∞ ⟨B∗x∗, xnj − x⟩ = 0, which means that Pxnj converges weakly to Px. On the other hand, by (2.2) and a direct calculation, we have lim j→+∞ Pxnj − xnj = lim j→+∞ 1 nj [ Px0 − Pnj+1x0 ] = 0, (2.4) where 0 is the zero member of X. Therefore, 0 ≤ lim j→+∞ |⟨x∗, Pxnj − x⟩| ≤ lim j→+∞ ( |⟨x∗, Pxnj − xnj ⟩|+ |⟨x∗, xnj − x⟩| ) = 0 for every x∗ ∈ X∗, which means that Pxnj converges weakly to x. From this and Pxnj converges weakly to Px, we obtain Px = x. In addition, (2.3) is obvious if x = 0. When x ̸= 0, we know that there exists x∗ ∈ X∗ such that ∥x∗∥ = 1 and < x∗, x >= ∥x∥. So ∥x∥ = ⟨x∗, x⟩ = ⟨x∗, x− xnj ⟩+ ⟨x∗, xnj ⟩ ≤ ⟨x∗, x− xnj ⟩+ ∥x∗∥∥xnj ∥ = ⟨x∗, x− xnj ⟩+ ∥ 1 nj nj∑ k=1 P kx0∥ ≤ ⟨x∗, x− xnj ⟩+ sup n∈N ∥ 1 n n∑ k=1 P kx0∥ for every j ∈ N. Combining this with xnj converges weakly to x, we obtain ∥x∥ ≤ lim j→+∞ [ ⟨x∗, x− xnj ⟩+ sup n∈N ∥ 1 n n∑ k=1 P kx0∥ ] = sup n∈N ∥ 1 n n∑ k=1 P kx0∥. Now, we prove case (ii). Let xn = 1 n n∑ k=1 P kx0 for n ∈ N. We conclude from (2.2) and condition (ii) that there exist {xnj } ⊂ {xn} and x ∈ X such that xnj weak∗-convergent to x, hence that lim j→+∞ ⟨Pxnj − Px, y⟩ = lim j→+∞ ⟨Bxnj −Bx, y⟩ = lim j→+∞ ⟨xnj − x,B∗y⟩ = 0 for every y ∈ Y , which means that Pxnj weak∗-convergent to Px. Furthermore, we obtain {Pxnj − xnj} weakly∗-convergent to 0 by (2.4). Then 0 ≤ lim j→+∞ |⟨Pxnj − x, y⟩| ≤ lim j→+∞ ( |⟨Pxnj − xnj , y⟩|+ |⟨xnj − x, y⟩| ) = 0 EJDE-2024/35 MASSERA THEOREMS FOR EVOLUTION EQUATIONS 5 for every y ∈ Y , which means that Pxnj weak∗-convergent to x. From this and Pxnj weak∗-convergent to Px, we have Px = x. Moreover, since xnj is weak∗- convergent to x, |⟨x, y⟩| = lim j→+∞ |⟨xnj , y⟩| ≤ lim sup j→+∞ ∥xnj∥∥y∥ = lim sup j→+∞ ∥xnj∥ ≤ sup n∈N ∥xn∥ for every y ∈ Y with ∥y∥ = 1. Therefore, ∥x∥ ≤ sup n∈N ∥xn∥ = sup n∈N ∥ 1 n n∑ k=1 P kx0∥. Hence, the conclusion is valid. □ Zubelevich [20] obtained the conclusion of the above theorem assuming that supn∈N ∥Pnx0∥ < +∞ and X is reflexive. We note that supn∈N ∥Pnx0∥ < +∞ implies sup n∈N ∥ 1 n n∑ k=1 P kx0∥ < +∞ and lim n→+∞ ∥P nx0 n ∥ = 0. However, the converse is not necessarily true. For example, let P : l2 → l2; (x1, x2, . . . , xk, . . . ) 7→ ( 0, 4 √ 2x1, 4 √ 3 2 x2, . . . , 4 √ k + 1 k xk, . . . ) . It is easy to see that P is a bounded linear operator and there exists x0 = (1, 0, . . . ) ∈ l2 such that sup n∈N ∥Pnx0∥ = sup n∈N 4 √ n+ 1 = +∞, lim n→+∞ ∥P nx0 n ∥ = lim n→+∞ 4 √ n+ 1 n = 0, sup n∈N ∥ 1 n n∑ k=1 P kx0∥ ≤ sup n∈N (n+ 1) 3 4 n ≤ 2. 3. Massera type theorems for linear evolution equations In this section, we consider the existence of 1-periodic mild solutions to (1.1) with the help of our new fixed point theorems for affine maps. Throughout this section, we assume that g ∈ P1(R+, X) ∩ L1 loc(R+, X). Next, we first recall the following definitions and results which are taken from [15, 17, 16]. Proposition 3.1 ([17]). Assume that {A(t)}t≥0 satisfies conditions (A1)–(A3) and t0 > 0. If z ∈ D and h ∈ L1([0, t0], X), then the limit u(t) := lim λ→0+ ( Uλ(t, 0)z + ∫ [ t λ ]λ 0 Uλ(t, r)h(r)dr ) (3.1) exists uniformly for t ∈ [0, t0], and u is a continuous function on [0, t0], where Uλ(t, 0) = [ t λ ]∏ j=1 (I − λA(jλ)) −1 for every t ∈ [0, t0] and λ > 0. The following theorem can be derived from (A1)–(A4) and Proposition 3.1 (see also [15, 16]). 6 L.-L. ZHENG, H.-S. DING EJDE-2024/35 Proposition 3.2. Assume that {A(t)}t≥0 satisfies conditions (A1)–(A3). Then the limit U(t, s)z = lim λ→0+ Uλ(t, s)z exists for every z ∈ D and (t, s) ∈ △, where Uλ(t, s) = [ t λ ]∏ j=[ sλ ]+1 (I − λA(jλ))−1 and △ = {(t, s)|t ≥ s ≥ 0}. Also, the families {U(t, s)}t≥s≥0 and {Uλ(t, s)}t≥s≥0 satisfy the following properties: (i) Uλ(t, t)z = z and Uλ(t, r)Uλ(r, s)z = Uλ(t, s)z for every z ∈ D, λ > 0 and t ≥ r ≥ s ≥ 0; (ii) for every λ > 0 and (t, s) ∈ △, ∥Uλ(t, s)∥ ≤ M ( 1 1− λw ) t−s λ +1 ; (3.2) (iii) U(t, s) : D → D is a bounded linear operator for every (t, s) ∈ △; (iv) U(t, t)z = z and U(t, r)U(r, s)z = U(t, s)z for every z ∈ D and t ≥ r ≥ s ≥ 0; (v) for every z ∈ D and (t, s) ∈ △, ∥U(t, s)z∥ ≤ Meω(t−s)∥z∥; (3.3) (vi) for every h ∈ L1 loc(R+, X) and (t, s) ∈ △, U(t, s) lim λ→0+ ∫ s 0 Uλ(s, r)h(r)dr = lim λ→0+ ∫ s 0 Uλ(t, r)h(r)dr. (3.4) (vii) for every h ∈ L1 loc(R+, X) and t ≥ 0, lim λ→0+ ∫ [ t λ ]λ 0 Uλ(t, r)h(r)dr = lim λ→0+ ∫ t 0 Uλ(t, r)h(r)dr; (3.5) In addition, if {A(t)}t≥0 satisfies the condition (A4), then (viii) for every z ∈ D, k ∈ N and (t, s) ∈ △, U 1 k (t+ 1, s+ 1)z = U 1 k (t, s)z; (3.6) (ix) for every z ∈ D and (t, s) ∈ △, U(t+ 1, s+ 1)z = U(t, s)z. (3.7) Definition 3.3 ([15]). A continuous function u : [0,+∞) → D is called a mild solution of (1.2) if u satisfies u(t) = U(t, s)u(s) + lim λ→0+ ∫ t s Uλ(t, r)f(r, u(r))dr for every (t, s) ∈ △. Then, a continuous function u : [0,+∞) → D is called a mild solution of (1.1) if u satisfies u(t) = U(t, s)u(s) + lim λ→0+ ∫ t s Uλ(t, r)g(r)dr (3.8) for every (t, s) ∈ △. EJDE-2024/35 MASSERA THEOREMS FOR EVOLUTION EQUATIONS 7 Remark 3.4. Let g ∈ L1 loc(R+, X). By Proposition 3.1 and (3.5), we note that u is continuous if u satisfies (3.8). Remark 3.5. Let t0 > 0. According to [17, Theorem 4.2], if g ∈ W 1,1([0, t0], X), x ∈ D, and A(0)x+ g(0) ∈ D, then u : [0, t0] → D; t 7→ U(t, 0)x+ lim λ→0+ ∫ t 0 Uλ(t, r)g(r)dr is a classical solution to (1.1) on [0, t0], where W 1,1([0, t0], X) = { u ∈ L1([0, t0], X) : u(t) = u0 + ∫ t 0 v(s)ds for some u0 ∈ X and v ∈ L1([0, t0], X), t ∈ [0, t0] } . Using Theorems 2.1 and 2.3, we obtain the following results. Theorem 3.6. Assume that there exists a mild solution u0 of (1.1) such that lim n→+∞ ∥u0(n) n ∥ = 0 and R(I −U(1, 0)) is closed, where I is the identity map and R(I −U(1, 0)) is the range of I − U(1, 0). Then (1.1) has a 1-periodic mild solution u satisfying ∥u∥ ≤ Me|ω| ( ∥u(0)∥+ ∫ 1 0 ∥g(σ)∥dσ ) (3.9) Furthermore, the 1-periodic mild solution of (1.1) is unique if lim t→+∞ ∥U(t, 0)x∥ = 0 for every x ∈ D with sup t≥0 ∥U(t, 0)x∥ < +∞. (3.10) Proof. Let P : D → D;x 7→ U(1, 0)x+ lim λ→0+ ∫ 1 0 Uλ(1, r)g(r)dr. Take X = D, B = U(1, 0), z = limλ→0+ ∫ 1 0 Uλ(1, r)g(r)dr, and x0 = u0(0) in Theorem 2.1. Note that P k(x0) = u0(k) for every k ∈ N. By Theorem 2.1, P has a fixed point x ∈ D. Let u : [0,+∞) → X; t 7→ U(t, 0)x+ lim λ→0+ ∫ t 0 Uλ(t, r)g(r)dr. (3.11) By using (3.4), Remark 3.4 and (iv) in Proposition 3.1, we have U(t, s)u(s) + lim λ→0+ ∫ t s Uλ(t, r)g(r)dr = U(t, s) [ U(s, 0)x+ lim λ→0+ ∫ s 0 Uλ(s, r)g(r)dr ] + lim λ→0+ ∫ t s Uλ(t, r)g(r)dr = U(t, 0)x+ U(t, s) lim λ→0+ ∫ s 0 Uλ(s, r)g(r)dr + lim λ→0+ ∫ t s Uλ(t, r)g(r)dr = U(t, 0)x+ lim λ→0+ ∫ s 0 Uλ(t, r)g(r)dr + lim λ→0+ ∫ t s Uλ(t, r)g(r)dr = U(t, 0)x+ lim λ→0+ ∫ t 0 Uλ(t, r)g(r)dr = u(t) 8 L.-L. ZHENG, H.-S. DING EJDE-2024/35 for every (t, s) ∈ △, which means that u is a mild solution to (1.1). Moreover, by g ∈ P1(R+, X), (3.4), (3.6), (3.7) and (iv) in Proposition 3.2, we have u(t) = U(t, 0)x+ lim λ→0+ ∫ t 0 Uλ(t, r)g(r)dr = U(t, 0)Px+ lim k→+∞ ∫ t 0 U 1 k (t, r)g(r)dr = U(t, 0) [ U(1, 0)x+ lim λ→0+ ∫ 1 0 Uλ(1, r)g(r)dr ] + lim k→+∞ ∫ t 0 U 1 k (t+ 1, r + 1)g(r + 1)dr = U(t+ 1, 1) [ U(1, 0)x+ lim λ→0+ ∫ 1 0 Uλ(1, r)g(r)dr ] + lim k→+∞ ∫ t+1 1 U 1 k (t+ 1, r)g(r)dr = U(t+ 1, 0)x+ lim λ→0+ ∫ 1 0 Uλ(t+ 1, r)g(r)dr + lim λ→0+ ∫ t+1 1 Uλ(t+ 1, τ)g(τ)dτ = U(t+ 1, 0)x+ lim λ→0+ ∫ t+1 0 Uλ(t+ 1, r)g(r)dr = u(t+ 1) for every t ≥ 0, which means that u is 1-periodic. In addition, let u1 and u2 be two 1-periodic mild solutions to (1.1). Note that u1(0)− u2(0) ∈ D and sup t≥0 ∥U(t, 0)(u1(0)− u2(0))∥ = sup t≥0 ∥u1(t)− u2(t)∥ < +∞. So, limt→+∞ ∥u1(t) − u2(t)∥ = limt→+∞ ∥U(t, 0)(u1(0) − u2(0))∥ = 0 by (3.10). Combining this with u1−u2 ∈ P1(R+, X), we obtain u1 = u2. Thus, the 1-periodic mild solution to (1.1) is unique. Furthermore, we obtain (3.9) by using (3.2) and (3.3). □ Theorem 3.7. Assume that there exists a mild solution u0 of (1.1) such that sup n∈N ∥ 1 n n∑ k=1 u0(k)∥ < +∞, lim n→+∞ ∥u0(n) n ∥ = 0, and one of the following conditions holds: (i) X is reflexive; (ii) there exists a separable Banach space Y such that D is the dual space of Y and U∗(1, 0)Y ⊂ Y , where U∗(1, 0) is the dual operator of U(1, 0) and Y is considered as a subspace of Y ∗∗. Then (1.1) has a 1-periodic mild solution u satisfying ∥u∥ ≤ Me|ω| ( sup n∈N ∥ 1 n n∑ k=1 u0(k)∥+ ∫ 1 0 ∥g(σ)∥dσ ) . (3.12) Furthermore, the 1-periodic mild solution of (1.1) is unique if (3.10) holds. Proof. By using Theorem 2.3, similar to the proof of Theorem 3.6, one can show that (1.1) has a 1-periodic mild solution u given by (3.11) and u satisfies (3.9). In EJDE-2024/35 MASSERA THEOREMS FOR EVOLUTION EQUATIONS 9 addition, according to (2.3), we have ∥u(0)∥ = ∥x∥ ≤ sup n∈N ∥ 1 n n∑ k=1 P k(x0)∥ = sup n∈N ∥ 1 n n∑ k=1 u0(k)∥. Combining this with (3.9), we obtain (3.12). □ 4. Application to semilinear evolution equations Next, we establish the existence of 1-periodic mild solutions for (1.2) by our Massera type results obtained in the previous section. For convenience, we list some assumptions: (A5) f(·, x) ∈ P1(R+, X) ∩ L1 loc(R+, X) for every x ∈ X; (A6) there exist ϱ > 0 and L > 0 such that ∥f(t, x1)− f(t, x2)∥ ≤ L∥x1 − x2∥ for every t ≥ 0 and x1, x2 ∈ X with ∥x1∥, ∥x2∥ ≤ ϱ. From now on, we denote γ = ∫ 1 0 ∥f(σ,0)∥dσ. Theorem 4.1. Assume that (a) there exists α > 0 with Me|ω|(α+ 1) < min { 1 L , ϱ ϱL+ γ } (4.1) such that (1.1) has a mild solution ug 0 satisfies sup n∈N ∥ 1 n n∑ k=1 ug 0(k)∥ ≤ α ∫ 1 0 ∥g(σ)∥dσ and lim n→+∞ ∥u g 0(n) n ∥ = 0 (4.2) for every g ∈ P1(R+, X) ∩ L1 loc(R+, X); (b) limt→+∞ ∥U(t, 0)x∥ = 0 for every x ∈ D with sup t≥0 ∥U(t, 0)x∥ < +∞; (c) one of the following conditions holds: (i) X is reflexive; (ii) there exists a separable Banach space Y such that D is the dual space of Y and U∗(1, 0)Y ⊂ Y , where U∗(1, 0) is the dual operator of U(1, 0) and Y is considered as a subspace of Y ∗∗. Then (1.2) has a 1-periodic mild solution u. Proof. Let Bϱ = {v ∈ P1(R+, X) ∩ C(R+, X)| supt∈R+ ∥v(t)∥ ≤ ϱ} and T : Bϱ → Bϱ; v 7→ uv, where uv is the 1-periodic mild solution of (1.1) with g = f(·, v(·)) by Theorem 3.7. Step 1. We show that Tv ∈ Bϱ for every v ∈ Bϱ. Let v ∈ Bϱ. Then we know from Theorem 3.7 that (1.1) has a unique 1-periodic mild solution uv satisfying ∥uv∥ ≤ Me|ω| [ sup n∈N ∥ 1 n n∑ k=1 ug 0(k)∥+ ∫ 1 0 ∥g(σ)∥dσ ] , Combining this with (4.2), (A5) and (A6), we obtain ∥uv∥ ≤ Me|ω|(α+ 1) ∫ 1 0 ∥f(σ, v(σ))∥dσ, 10 L.-L. ZHENG, H.-S. DING EJDE-2024/35 ≤ Me|ω|(α+ 1) ( L sup t∈R+ ∥v(t)∥+ ∫ 1 0 ∥f(σ,0)∥dσ ) ≤ Me|ω|(α+ 1)(Lϱ+ γ) ≤ ϱ. So, Tv = uv ∈ Bϱ by (4.1). Step 2.We show that T is a contraction. Let v1, v2 ∈ Bϱ. Then ∥Tv1 − Tv2∥ = sup t∈R+ ∥uv1(t)− uv2(t)∥ = sup t∈[0,1] ∥U(t, 0)(uv1(0)− uv2(0)) + lim λ→0+ ∫ t 0 Uλ(t, r)(f(r, v1(r))− (f(r, v2(r)))dr∥ ≤ Me|ω|(α+ 1)L sup t∈R+ ∥v1(t)− v2(t)∥ ≤ Me|ω|(α+ 1)L∥v1 − v2∥, where uv1and uv2 are the corresponding 1-periodic mild solution to (1.1) with g1 = f(·, v1(·)) and g2 = f(·, v2(·)), respectively. By (4.1), we conclude that T is a contraction. 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Lan-Ling Zheng School of Mathematics and Statistics, Jiangxi Normal University, Nanchang, Jiangxi 330022, China Email address: 202150000054@jxnu.edu.cn Hui-Sheng Ding (corresponding author) School of Mathematics and Statistics, Jiangxi Normal University, Nanchang, Jiangxi 330022, China Email address: dinghs@mail.ustc.edu.cn 1. Introduction 2. Two fixed point theorems for affine maps 3. Massera type theorems for linear evolution equations 4. Application to semilinear evolution equations Acknowledgments References