Electronic Journal of Differential Equations, Vol. 2022 (2022), No. 54, pp. 1–13. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu PERIODIC SOLUTIONS OF STOCHASTIC VOLTERRA EQUATIONS FENG CHEN Abstract. This article concerns the dynamical behavior of solutions to sto- chastic Volterra equations. We prove the existence of periodic solutions in distribution of stochastic Volterra equations. We use the Banach fixed point theorem and a Krasnoselski-Schaefer type fixed point theorem. 1. Introduction The dynamical behavior of solutions to stochastic differential equations (SDEs for short), such as long time behavior, ergodicity, and periodicity, has been studied in [11, 17, 31]. This paper concerns the existence of periodic solutions in the distribution of stochastic Volterra equations. Stochastic Volterra integral equations arise in many scientific areas such as mathematical finance, biology, etc, see [15, 18]. There are many well-known results concerning the dynamical behavior of the solutions with non-singular kernels and the Lipschitz coefficients, see [4, 29, 30]. The solutions of stochastic Volterra equations with singular kernel in finite-dimensional space have been studied in [10, 13, 34]. For the infinite-dimensional case, the existence, uniqueness, and large deviation estimate for stochastic Volterra equations in Banach spaces were studied in [37]. For more details about these topics and recent developments, we refer to [28] and references therein. It is well-known that periodic solutions are one of the fundamental problems in the qualitative theory of differential equations [24]. For the stochastic system, the existence of periodic solutions is also worth studying and discussing. Because of the effects of diffusion, we should study periodic solutions for SDEs in the distribution sense. In other words, obtaining periodic solutions in the probability or moment for SDEs is impossible, see [25]. Khasminskii [23] defined periodic solutions in the sense of periodic Markov processes and obtained the existence results via the Lyapunov method. Chen et al. [8] established Halanay’s criterion of SDEs and obtained periodic solutions in distribution. Ji et al. [20, 21] studied periodic proba- bility solutions to the Fokker-Planck equations. The existence of periodic solutions for semilinear SDEs has been studied in [27, 16] and references therein. Cheban and Liu [9] studied the problem of Poisson stability (in particular, stationarity, periodicity, quasi-periodicity, Bohr almost periodicity, Bohr almost automorphy, Birkhoff recurrence, Poisson stability) of solutions for semi-linear SDEs. Zhao and 2020 Mathematics Subject Classification. 60H15, 60H05, 60G22. Key words and phrases. Stochastic Volterra equations; periodic solutions; Banach fixed point theorem; Krasnoselski-Schaefer fixed point theorem. ©2022. This work is licensed under a CC BY 4.0 license. Submitted March 6, 2022. Published July 27, 2022. 1 2 F. CHEN EJDE-2022/54 Zheng [38] and Feng et al. [14] studied the existence of random periodic solutions of random dynamical systems. Based on the technique of upper and lower solutions and comparison principle, Ji et al. [19] obtained the existence of periodic solutions in distribution for stochastic differential equations. Via Wong-Zakai approxima- tions, Jiang and Li [22] obtained the existence of periodic solutions in distribution for stochastic dissipative systems. Lv et al. [26] obtained the existence of periodic solutions of the probability density function corresponding to the stochastic process by using the technique of deterministic partial differential equations. In this article, we consider the stochastic Volterra integral equation x(t) = a(t, x(t)) + ∫ t −∞ K(t, s)f(s, x(s))ds+ ∫ t −∞ K(t, s)g(s, x(s))dW (s), (1.1) where K : L2 loc(R× R,Rd×d) is a given kernel, a : R× Rd → Rd, f : R× Rd → Rd, g : R × Rd → Rd×m are continuous functions with additional properties which would be specified below, and W is m-dimensional Brownian motion. Our first goal is to use the Banach fixed point theorem to obtain the existence and uniqueness of periodic solutions in distribution for stochastic Volterra integral equations (1.1) under the Lipschitz conditions and integrable conditions. Banach fixed point theorem is an effective method to study periodic solutions in distribution for semilinear SDEs under Lipschitz conditions, for example, see [9, 27, 16]. Our second goal is to study the existence of periodic solutions in distribution for stochastic Volterra integral equations (1.1) under the Carathéodory condition, which is a non-Lipschitz condition. We prove the existence of such system by the Krasnoselski-Schaefer type fixed point theorem. Carathéodory condition is widely used in the field of mathematics physics. Taniguchi [33] proved the existence and uniqueness of SDEs under a relatively weak non-Lipschitz assumptions for drift term and diffusion term by successive approximation, which includes the well-known result of [36]. Many authors have studied the property of solutions for SDEs under Carathéodory conditions, see Abouagwa and Li [2], Xu, Pei, and Wu [35], Shao, Wang, and Yuan [32], for more details about these topics and recent developments. This article is organized as follows. In Section 2, we introduce some notation and definitions. In Section 3, we state and prove our main results by using the Banach fixed point theorem and Krasnoselski-Schaefer fixed point theorem. In Section 4, we provide an example to illustrate the applicability of our results. 2. Preliminaries Throughout this article, let (Ω,F ,Ft,P) be the complete probability space, with the filtration Ft. Let us denote by Rd×d the set of all d × d-dimensional ma- trices, equipped with the norm ‖ · ‖ defined by ‖M‖2 = Trace(MMT ) for all M ∈ Rd×d. The space Rd is equipped with the Euclidean norm, denoted by | · |. Let L2(P,Rd) stand for the space of all Rd-valued random variables X such that E|X|2 = ∫ Ω |X|2dP < ∞. For X ∈ L2(P,Rd), let ‖X‖2 = ( ∫ Ω |X|2dP)1/2. Denote by Cb(R, L2(P,Rd)) the Banach space of all bounded L2-continuous mappings from R to L2(P,Rd)) endowed with the norm ‖ · ‖∞ defined by ‖Y ‖∞ := supt∈R ‖Y (t)‖2 for all Y ∈ Cb(R, L2(P,Rd)). Let P(Rd) be the set of Borel probability measures on Rd. For a given Rd-valued random variable X, let pX be the distribution of X on Rd. For a given stochastic process, the law of X on Rd is denoted by µ : R→ P(Rd). EJDE-2022/54 PERIODIC SOLUTIONS OF STOCHASTIC VOLTERRA EQUATIONS 3 Following [8], we define ‖h‖∞ = sup x∈Rd |h(x)|, ‖h‖L = sup { |h(x)− h(y)| |x− y| ;x, y ∈ Rd, x 6= y } , ‖h‖BL = max{‖h‖∞, ‖h‖L}, dBL(µ, ν) = sup ‖h‖BL≤1 | ∫ hd(µ− ν)| for all µ, ν ∈ P(Rd) and all Lipschitz continuous real-valued functions h on Rd. It is known that (dBL,P(Rd)) is a complete metric space. Now we show the definition of the stochastic periodic solution in the distribution sense. Definition 2.1 ([8]). Let x(t) be a solution of (1.1). Suppose that a, K, f , and g are T -periodic with respect to t and x(t) satisfies the following conditions (i) px(t) = px(t+T ) for all t ∈ R+; (ii) There exists W̃ , which has the same distribution as W , such that x(t+ T ) is a solution of the equation x(t) = a(t, x(t)) + ∫ t −∞ K(t, s)f(s, x(s))ds+ ∫ t −∞ K(t, s)g(s, x(s))dW̃ (t). Then x(t) is said to be a stochastic periodic solution in distribution. We present the Krasnoselskii-Schaefer’s fixed point theorem, which is one of the key tools in our paper. Lemma 2.2 ([5]). Let S be a Banach space, and Ξ1,Ξ2 : S → S be two operators satisfying: Ξ1 is a contraction, and Ξ2 is completely continuous. Then, either the operator equation x = Ξ1x + Ξ2x has a solution, or the set Υ = {x ∈ S : λΞ1(xλ ) + λΞ2x = x} is unbounded for some λ ∈ (0, 1). The modified Bihari-type integral inequality was given in [7, 12] as follows. Sup- pose the l ≥ 0 and u ∈ BC(R,R+) satisfies u(t) ≤ l + ∫ R K1(t, s)γ1(s)γ3(u(s))ds+ ∫ t −∞ K2(t, s)γ2(s)γ3(u(s))ds (2.1) for all t ∈ R. Here K1,K2 ∈ C(R×R,R+), γ1, γ2 ∈ C(R,R+). Let γ3 ∈ C(R+,R+) be a nondecreasing function such that∫ b a γ3(r1/q)−qdr < +∞, ∫ +∞ a γ3(r1/q)−qdr = +∞, where q ≥ 1, 0 < a < b < +∞. Moreover, suppose that there exists a continuous function σ : R→ (0,∞) such that N1 = sup t∈R ∫ R K1(t, s)pσ(s)pds < +∞, (2.2) N2 = sup t∈R ∫ t −∞ K2(t, s)pσ(s)pds < +∞, (2.3) N3 = ∫ R σ(s)−q(γ1(s)q + γ2(s)q)ds < +∞, (2.4) 4 F. CHEN EJDE-2022/54 where 1 < p ≤ ∞, 1 ≤ q <∞ and 1 p + 1 q = 1. For a given ε > 0, let 0 < u0 < 3q/pεq be fixed and we define G(x) = ∫ x u0 γ3(r1/q)−qdr, x ≥ u0, H(x) = G(2x− 3q/pεq)−G(x)− ∫ R σ(s)−qP (s)ds, x ≥ u0 + 3q/pεq 2 , where P (x) = (3N1 + 3N2)q/p(γ1(x)q + γ2(x)q). If l > 0, define G,H with ε = l and assume that H is nondecreasing for x ≥ 3q/plq and H(c0) = 0 for some c0 > 3q/plq. If l = 0, assume that there exists ε0 > 0 such that G and H are defined with ε = ε0 and H is nondecreasing for x ≥ 3q/pεq0 and H(c0) = 0 for some c0 > 3q/pεq0. Lemma 2.3 ([7, Theorem 2.3]). Let 1 < p ≤ ∞, 1 ≤ q < ∞, and 1 p + 1 q = 1, and l ≥ 0. Suppose that u ∈ BC(R,R+) satisfies (2.1) for all t ∈ R, where γ1, γ2, γ3,K1,K2 are defined as above. Then ‖uq‖∞ ≤ G−1[G(c0) + (3N1 + 3N2)q/pN3]. 3. Existence and uniqueness of periodic solution of (1.1) In this section, we employ the Banach fixed point theorem on (1.1), and obtain a unique periodic solution. In addition to the continuous conditions on functions a, f , g, and K, we will need to impose some of the following assumptions. (H1) There exists T > 0, such that a(t + T ) = a(t), f(t + T , x) = f(t, x), g(t+ T , x) = g(t, x), and K(t+ T , s+ T ) = K(t, s) for all t, s ∈ R, x ∈ Rd. (H2) There exists a constant La > 0, such that ‖a(t, x)− a(t, y)‖ ≤ La‖x− y‖, where t ∈ R, x, y ∈ Rd. (H3) There exist constants Lf > 0 and Lg > 0, such that ‖f(t, x)− f(t, y)‖ ≤ Lf‖x− y‖, ‖g(t, x)− g(t, y)‖ ≤ Lg‖x− y‖, where t ∈ R, x, y ∈ Rd. (H4) There exists a constant C∗ > 0, such that sup t∈R ∫ t −∞ ‖K(t, s)‖pds ≤ C∗, where p ≥ 1. (H5) There exists a constant C̃ > 0, such that for all 0 ≤ u ≤ v ≤ T∫ u −∞ ‖K(u, s)−K(v, s)‖pds ≤ C̃|u− v|, where p ≥ 1. Theorem 3.1. Assume (H1)–(H5) hold , and that 3La + 3C∗(C∗ + 1)(L2 f + L2 g) < 1. Then (1.1) admits a unique continuous T -periodic solution in distribution. EJDE-2022/54 PERIODIC SOLUTIONS OF STOCHASTIC VOLTERRA EQUATIONS 5 Proof. Firstly, we show that (Sϕ)(t) is continuous with respect to t. Consider the nonlinear operator S defined on Cb(R, L2(P,Rd)): (Sϕ)(t) = a(t, ϕ(t)) + ∫ t −∞ K(t, s)f(s, ϕ(s))ds+ ∫ t −∞ K(t, s)g(s, ϕ(s))dW (s). If ϕ(t) is an L2 bounded solution of (1.1), one obtains that E‖(Sϕ)(u)− (Sϕ)(v)‖2 ≤ 5E‖a(u, ϕ(u))− a(v, ϕ(v))‖2 + 5E ∥∥∫ u −∞ (K(u, s)−K(v, s))f(s, ϕ(s))ds ∥∥2 + 5E ∥∥∫ v u K(v, s)f(s, ϕ(s))ds ∥∥2 + 5E ∥∥∫ u −∞ (K(u, s)−K(v, s))g(s, ϕ(s))dW (s) ∥∥2 + 5E ∥∥∫ v u K(v, s)g(s, ϕ(s))dW (s) ∥∥2 := 5I1 + 5I1 + 5I2 + 5I3 + 5I4 + 5I5, where −∞ < u ≤ v ≤ T . Since a(t) is continuous and T -periodic with respect to t, we have I1 ≤ C‖u− v‖2, where C is a constant. Since f(t, ϕ(t)) is Lipschitz in ϕ and periodic in t and is L2 bounded, we have sup t∈R ‖f(t, ϕ(t))‖ ≤ sup t∈R ‖f(t, ϕ(t))− f(t, 0)‖+ sup t∈R ‖f(t, 0)‖ ≤ Lf‖ϕ‖+ sup t∈R ‖f(t, 0)‖ < +∞. Since supt∈R ‖f(t, ϕ(t))‖ = supt∈R(E‖f(t, ϕ(t))‖2)1/2, there exists M1 such that sup s∈R E‖f(s, ϕ(s))‖2 < M1, sup s∈R E‖g(s, ϕ(s))‖2 < M2, where M1 and M2 are constants. By Cauchy-Schwarz’s inequality, we have I2 ≤ ∫ u −∞ ‖K(u, s)−K(v, s)‖dsE ∫ u −∞ ‖K(u, s)−K(v, s)‖‖f(s, ϕ(s))‖2ds ≤ sup s∈R E‖f(s, ϕ(s))‖2 (∫ u −∞ ‖K(u, s)−K(v, s)‖ds )2 ≤M1C̃ 2|u− v|2. Since K is continuous, there exists a constant C̄ such that if t, s ∈ R, then ‖K(t, s)‖ ≤ C̄, a.e. I3 ≤M1C̄ 2|u− v|2. By Itô’s isometry formula, we have I4 ≤ ∫ u −∞ ‖K(u, s)−K(v, s)‖2E‖g(s, ϕ(s))‖2ds 6 F. CHEN EJDE-2022/54 ≤ sup s∈R E‖g(s, ϕ(s))‖2 ∫ u −∞ ‖K(u, s)−K(v, s)‖2ds ≤M2C̃|u− v|. Similarly, I5 ≤M2C̄ 2|u− v|. Then E‖(Sϕ)(u)− (Sϕ)(v)‖2 ≤ C(|u− v|2 + |u− v|), where C is a constant. This shows that (Sϕ)(t) is continuous in t. Secondly, we show that S is a contraction mapping on Cb(R, L2(P,Rd)). For ϕ,ψ ∈ Cb(R, L2(P,Rd)) and each t ∈ R, we have E‖(Sϕ)(t)− (Sψ)(t)‖2 = 3E‖a(t, ϕ)− a(t, ψ)‖2 + 3E ∥∥∫ t −∞ K(t, s)(f(s, ϕ(s))− f(s, ψ(s)))ds ∥∥2 + 3E ∥∥∫ t −∞ K(t, s)(g(s, ϕ(s))− g(s, ψ(s)))dW (s) ∥∥2 ≤ 3L2 a‖ϕ− ψ‖2 + 3 ∫ t −∞ ‖K(t, s)‖ds ×E ∫ t −∞ ‖K(t, s)‖ ‖(f(s, ϕ(s))− f(s, ψ(s))‖2ds + 3E ∫ t −∞ ‖K(t, s)(g(s, ϕ(s))− g(s, ψ(s)))‖2ds ≤ 3L2 a‖ϕ− ψ‖2 + 3 (∫ t −∞ ‖K(t, s)‖ds )2 L2 f sup s∈R E‖ϕ(s)− ψ(s)‖2 + 3 ∫ t −∞ ‖K(t, s)‖2dsL2 g sup s∈R E‖ϕ(s)− ψ(s)‖2 ≤ (3La + 3C∗(C∗ + 1)(L2 f + L2 g)) sup s∈R E‖ϕ(s)− ψ(s)‖2. Since 3La + 3C∗(C∗ + 1)(L2 f + L2 g) < 1, S is a contraction on Cb(R, L2(P,Rd)). Then there exists a unique v ∈ Cb(R, L2(P,Rd)) such that Sv = v, which is the unique solution to (1.1). Thirdly, we show that v(t) is a T -periodic solution in distribution to (1.1). v(t+ T ) = a(t+ T , v(t+ T )) + ∫ t+T −∞ K(t+ T , s)f(s, v(s))ds + ∫ t+T −∞ K(t+ T , s)g(s, v(s))dW (s) = a(t+ T , v(t+ T )) + ∫ t −∞ K(t+ T , s+ T )f(s+ T , v(s+ T ))ds + ∫ t −∞ K(t+ T , s+ T )g(s+ T , v(s+ T ))dW̃ (s), (3.1) where W̃ (t) = W (t+ T )−W (T ). EJDE-2022/54 PERIODIC SOLUTIONS OF STOCHASTIC VOLTERRA EQUATIONS 7 We consider the process v̂(t), which satisfies the equation v̂(t) = a(t+ T , v̂(t)) + ∫ t −∞ K(t+ T , s+ T )f(s+ T , v̂(s))ds + ∫ t −∞ K(t+ T , s+ T )g(s+ T , v̂(s))dW (s). (3.2) Note that v(t+ T ) has the same distribution as v̂(t). By (H1), we have E‖v̂(t)− v(t)‖2 ≤ 3E‖a(t+ T , v̂(t))− a(t, v(t))‖2 + 3E ∥∥∫ t −∞ K(t+ T , s+ T )f(s+ T , v̂(s))ds− ∫ t −∞ K(t, s)f(s, v(s))ds ∥∥2 + 3E ∥∥ ∫ t −∞ K(t+ T , s+ T )g(s+ T , v̂(s))dW (s) − ∫ t −∞ K(t, s)g(s, v(s))dW (s) ∥∥2 ≤ 3E‖a(t+ T , v̂(t))− a(t, v(t))‖2 + 3E ∥∥ ∫ t −∞ K(t, s)(f(s+ T , v̂(s))− f(s, v(s)))ds ∥∥2 + 3E ∥∥ ∫ t −∞ K(t, s)(g(s+ T , v̂(s))− g(s, v(s)))dW (s) ∥∥2 ≤ 3E‖a(t+ T , v̂(t))− a(t, v(t))‖2 + 3 (∫ t −∞ K(t, s)ds )2 sup s∈R E‖f(s+ T , v̂(s))− f(s, v(s))‖2 + 3 ∫ t −∞ ‖K(t, s)‖2ds sup s∈R E‖g(s+ T , v̂(s))− g(s, v(s))‖2 = 0. Since v(t+ T ) has the same distribution with v̂(t), v(t) is a T -periodic solution in distribution to (1.1). This completes the proof. � The second result is established by using Krasnoselskii-Schaefer type fixed point theorem with the following Carathéodory conditions on f and g. (H6) There exist functions α(t) : R→ R+ and Γ(v) : R+ → R+ such that (1) Γ(v) is continuous monotone non-decreasing concave and satisfy∫ b a Γ(r1/q)−qdr < +∞, ∫ +∞ a Γ(r1/q)−qdr = +∞, where p ≥ 1, 0 < a < b < +∞. (2) For any positive t ∈ R and x ∈ Rd, the functions f(t, x) and g(t, x) are continuous in x and satisfy ‖f(t, x)‖2 ≤ α(t)Γ(‖x‖2), ‖g(t, x)‖2 ≤ α(t)Γ(‖x‖2). (3) There exists a constant M∗ > 0, such that sup t∈R ∫ t −∞ ‖K(t, s)‖pα(s)ds ≤M∗, 8 F. CHEN EJDE-2022/54 where p ≥ 1. (4) There exists a constant M̃ > 0, such that for all 0 ≤ u ≤ v ≤ T ,∫ u −∞ |(K(u, s)−K(v, s))α(s)|pds ≤ M̃ |u− v|. Theorem 3.2. Suppose (H1), (H2), (H4)–(H6) hold, and suppose 3L2 a < 1. Then (1.1) has at least one continuous T -periodic solution in distribution. Proof. We split the operator S in Theorem 3.1 into two parts: (S1ϕ)(t) = a(t, ϕ(t)), (3.3) (S2ϕ)(t) = ∫ t −∞ K(t, s)f(s, ϕ(s))ds+ ∫ t −∞ K(t, s)g(s, ϕ(s))dW (s). (3.4) Step 1. We show that S1 is a contraction mapping. For u, v ∈ Cb(R, L2(P,Rd)), we have E‖(S1u)(t)− (S1v)(t)‖2 = E‖a(t, u)− a(t, v)‖2 ≤ L2 aE‖u− v‖2. (3.5) Since 3L2 a < 1, S1 is a contraction mapping. Step 2. We prove that the operator S2 is completely continuous. The proof is divided into four steps. Step 2.1. We show that S2 maps bounded sets Br = {v : ‖v(t)‖22 ≤ r} into bounded sets. Indeed, we will show that there exists a positive constant q such that for each w ∈ Br, one has that E‖(S2w)(t)‖2 ≤ q. For each w ∈ Br, we have E‖(S2w)(t)‖2 ≤ 2E ∥∥∫ t −∞ K(t, s)f(s, w(s))ds ∥∥2 + 2E ∥∥ ∫ t −∞ K(t, s)g(s, w(s))dW (s) ∥∥2 ≤ 2 ∫ t −∞ ‖K(t, s)‖dsE ∫ t −∞ ‖K(t, s)‖‖f(s, w(s))‖2ds + 2E ∫ t −∞ ‖K(t, s)‖2‖g(s, w(s))‖2ds ≤ Γ(r) (∫ t −∞ ‖K(t, s)‖ds ∫ t −∞ ‖K(t, s)‖α(s)ds+ ∫ t −∞ ‖K(t, s)‖2α(s)ds ) ≤ Γ(r)M∗(C∗ + 1) := q. Then we have for every w ∈ Br, we have E‖(S2w)(t)‖2 ≤ q. Step 2.2. We prove that S2 maps bounded sets w ∈ Br into equicontinuous sets. For t, t+ h ∈ R, w ∈ Br, we have E‖(S2w)(t+ h)− (S2w)(t)‖2 ≤ 4E ∥∥∫ t −∞ (K(t+ h, s)−K(t, s))f(s, w(s))ds ∥∥2 + 4E ∥∥∫ t+h t K(t+ h, s)f(s, w(s))ds ∥∥2 + 4E ∥∥∫ t −∞ (K(t+ h, s)−K(t, s))g(s, w(s))dW (s) ∥∥2 EJDE-2022/54 PERIODIC SOLUTIONS OF STOCHASTIC VOLTERRA EQUATIONS 9 + 4E ∥∥ ∫ t+h t K(t+ h, s)g(s, w(s))dW (s) ∥∥2 ≤ 4Γ(r) ∫ t −∞ ‖K(t+ h, s)−K(t, s)‖ds ∫ t −∞ ‖K(t+ h, s)−K(t, s)‖α(s)ds + 4Γ(r) ∫ t+h t ‖K(t+ h, s)‖ds ∫ t+h t ‖K(t+ h, s)‖α(s)ds + 4Γ(r) ∫ t −∞ ‖(K(t+ h, s)−K(t, s))α(s)‖2ds + 4Γ(r) ∫ t+h t ‖K(t+ h, s)α(s)‖2ds ≤ 4Γ(r)(C̃M̃h2 + C̄M̄h2 + M̃h+ M̄h). We deduce that the right-hand side of the above inequality is independent of w ∈ Br and tends to zero as h→ 0. Thus, the set {S2w : w ∈ Br} is equicontinuous. Step 2.3. We show that S2 is continuous. Let {wn}∞n=0 ⊂ Br, with wn → w in Br. By assumption (H6), we have f(t, wn(t))→ f(t, w(t)), g(t, wn(t))→ g(t, w(t)) (n→∞), for each t ∈ R, and since E‖f(t, wn(t))− f(t, w(t))‖2 ≤ 2α(t)Γ(r), E‖g(t, wn(t))− g(t, w(t))‖2 ≤ 2α(t)Γ(r). Then, by the dominated convergence theorem, we obtain E‖S2wn − S2w‖2 ≤ 2E ∥∥∫ t −∞ K(t, s)(f(t, wn(t))− f(t, w(t))ds ∥∥2 + 2E ∥∥ ∫ t −∞ K(t, s)(g(t, wn(t))− g(t, w(t))dW (s) ∥∥2 ≤ 2 (∫ t −∞ ‖K(t, s)‖ds )2 ∫ t −∞ E‖f(t, wn(t))− f(t, w(t))‖2ds + 2 ∫ t −∞ ‖K(t, s)‖2ds ∫ t −∞ E‖g(t, wn(t))− g(t, w(t))‖2ds → 0, as n→∞. Thus, S2 is continuous. Step 2.4. S2 maps Br into a precompact set in Br. Let t ∈ R be fixed and be a real number 0 < ε < t. For w ∈ Br, define the operator (S∗2w)(t) = K(ε) ∫ t−ε −∞ K(t− ε, s)f(s, w(s))ds+K(ε) ∫ t−ε −∞ K(t− ε, s)g(s, w(s))dW (s) = ∫ t−ε −∞ K(t, s)f(s, w(s))ds+ ∫ t−ε −∞ K(t, s)g(s, w(s))dW (s). SinceK(t) is a compact operator, the set {S∗2w)(t), w ∈ Br} is relatively compact for every ε, 0 < ε < t. Moreover, for every w ∈ Br, we have E‖(S2w)(t)− (S∗2w)(t)‖2 10 F. CHEN EJDE-2022/54 ≤ 2E ∥∥∫ t t−ε K(t, s)f(s, w(s))ds ∥∥2 + 2E ∥∥∫ t t−ε K(t, s)g(s, w(s))dW (s) ∥∥2 ≤ 2ε ∫ t t−ε ‖K(t, s)‖2E‖f(s, w(s))‖2ds+ 2 ∫ t t−ε ‖K(t, s)‖2E‖g(s, w(s))‖2ds ≤ 2εΓ(r) ∫ t t−ε ‖K(t, s)‖2α(s)ds+ 2Γ(r) ∫ t t−ε ‖K(t, s)‖2α(s)ds. Thus, E‖(S2w)(t) − (S∗2w)(t)‖2 → 0 as ε → 0. Hence the set {S∗2w)(t), w ∈ Br} is precompact, and by Arzela-Ascoli theorem, the operator S2 is completely continuous. Step 3. We show that the set Υ = {u ∈ S : λS1(uλ ) + λS2u = u, λ ∈ (0, 1)} is bounded. Indeed, for u ∈ Υ and λ ∈ (0, 1), one has E‖u(t)‖2 ≤ 3E‖λa(t, u λ )‖2 + 3E ∥∥λ ∫ t −∞ K(t, s)f(s, ϕ(s))ds ∥∥2 + 3E ∥∥λ ∫ t −∞ K(t, s)g(s, ϕ(s))dW (s) ∥∥2 ≤ 3L2 aE‖u(t)‖2 + 3 ∫ t −∞ ‖K(t, s)‖ds ∫ t −∞ ‖K(t, s)‖α(s)Γ(E‖u(s)‖2)ds + 3 ∫ t −∞ ‖K(t, s)‖2α(s)Γ(E‖u(s)‖2)ds. Hence, E‖u(t)‖2 ≤ 3C∗ 1− 3L2 a ∫ t −∞ ‖K(t, s)‖α(s)Γ(E‖u(s)‖2)ds + 3 1− 3L2 a ∫ t −∞ ‖K(t, s)‖2α(s)Γ(E‖u(s)‖2)ds. By Lemma 2.3, it follows that Υ is bounded. In concluding, by Krasnoselskii-Schaefer’s fixed point theorem, S has a fixed point. Then (1.1) has a continuous solution. Step 4. We show that v(t) is a T -periodic solution in distribution to (1.1). Using the same argument of (3.1)-(3.2), and (H1) in Theorem 3.1, we have E‖v̂(t)− v(t)‖2 = 0. Since v(t+ T ) has the same distribution with v̂(t), v(t) is a T -periodic solution of (1.1) in distribution. This completes the proof. � 4. Application Consider the following stochastic Volterra equation x(t) = ax(t) sin t+ ∫ t −∞ K(t, s)σ(s, x(s))dW (s), (4.1) where 0 < a < 1 2 ,K(t, s) = e−w(t−s), w > 0, σ(t, x) = cos t ∑∞ k=1 σk(t, x), σ2k+1(t, x) = b2k+1 sin kpx, σ2k(t, x) = b2k cos kpx, p > 0, b2k = O(k−(p+ 1 2 )), b2k+1 = O(k−(p+ 1 2 )). Then by [3, 6] we have ‖σ(t, x)− σ(t, y)‖2 EJDE-2022/54 PERIODIC SOLUTIONS OF STOCHASTIC VOLTERRA EQUATIONS 11 = | cos t|2 ∞∑ k=1 (b22k(cos kpx− cos kpy)2 + b22k+1(sin kpx− sin kpy)2) ≤ C ∞∑ k=1 k−(2p+1)((cos kpx− cos kpy)2 + (sin kpx− sin kpy)2) ≤ C ∞∑ k=1 k−(2p+1) sin2 k p(x− y) 2 ≤ Γ(|x− y|2) for all |x− y| is sufficiently small, where Γ(x) =  0, x = 0, Cx(log 1 x )1/2, 0 < x ≤ δ, Cδ(log 1 x )1/2, x > δ is a concave non-decreasing continuous function on R+ satisfying ∫ +∞ 0+ x Γ(x)dx = +∞ and δ ∈ (0, 1) is sufficiently small. It is easy to get that (H1)− (H2), (H4)− (H6) hold, and 3L2 a < 1. By Theorem 3.2, (4.1) admits a continuous 2π-periodic solution in distribution. Acknowledgments. The author wants to thank the anonymous reviewers for their careful reading and for their comments which greatly improved this article. 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