Electronic Journal of Differential Equations, Vol. 2024 (2024), No. 24, pp. 1–26. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: 10.58997/ejde.2024.24 P-MEAN (µ1, µ2)-PSEUDO ALMOST PERIODIC PROCESSES AND APPLICATION TO INTEGRO-DIFFERENTIAL STOCHASTIC EVOLUTION EQUATIONS MOEZ AYACHI, SYED ABBAS Abstract. In this article, we investigate the existence and stability of p- mean (µ1, µ2)-pseudo almost periodic solutions for a class of non-autonomous integro-differential stochastic evolution equations in a real separable Hilbert space. Using stochastic analysis techniques and the contraction mapping prin- ciple, we prove the existence and uniqueness of p-mean (µ1, µ2)-pseudo almost periodic solutions. We also provide sufficient conditions for the stability of these solutions. Finally, we present three examples with numerical simulations to illustrate the significance of the main findings. 1. Introduction The concept of almost periodic functions was introduced in 1923 by Harald Bohr [13]. It plays an important role in describing the phenomena that are more or less periodic, which can be observed frequently in many fields, such as biology, celestial mechanics, dynamical population, engineering, and so on. For more details, we re- fer to [2, 19] and references therein. Almost periodic solutions refer to solutions of differential equations that oscillate over time, but not in a strictly periodic manner. In other words, their oscillations are not exactly periodic, but they exhibit some sort of repetitive behavior. Since the introduction of almost periodicity, several ex- tensions of this concept have been introduced, including pseudo-almost periodicity by Zhang [39], weighted pseudo-almost periodicity (WPAP) by Diagana [20, 21], and µ-pseudo almost periodicity (PAP) by Blot et al. [12]. A more general-class pseudo-almost periodicity called (µ1, µ2)-pseudo-almost periodicity was considered by Diagana et al. (see [22]). For more details about the (µ1, µ2)-p.a.p. functions and their applications in the qualitative theory of differential equations, we refer the reader to [3, 5, 22, 30, 31, 35, 37, 38]. Further, stochastic perturbations are unavoidable and omnipresent in both real and artificial systems. Accordingly, investigating the dynamical behaviors of the systems described by various types of stochastic perturbations is highly impor- tant. For more information about the elementary theories for stochastic differential equations, we refer to [25, 32]. The concept of almost periodicity is important in 2020 Mathematics Subject Classification. 34K50, 34K14, 34K20. Key words and phrases. P-mean (µ1, µ2)-pseudo almost periodic process; fixed-point theorem; integro-differential stochastic evolution equation; existence; stability. ©2024. This work is licensed under a CC BY 4.0 license. Submitted June 21, 2023 Published March 14, 2024. 1 2 M. AYACHI, S. ABBAS EJDE-2024/24 probability for investigating stochastic processes. Recently, Bezandry and Diagana initiated the concept of p-mean almost periodic processes and applied it to the study of the existence and uniqueness of square-mean almost periodic mild solu- tions to some classes of stochastic differential equations [8, 9, 11]. Since then, the concept has been generalized into square-mean pseudo-almost periodicity by Ch’erif [16], p-mean µ-pseudo almost periodicity by Diop et al. [23], and (µ1, µ2)-pseudo almost periodicity by Belmabrouk et al. [4]. Several works have focused on inves- tigating square-mean almost periodic processes, their various extensions, and their applications in stochastic differential equations, we refer to [7, 10, 14, 15, 27, 36, 40]. Stochastic integro-differential equations play a crucial role in the qualitative theory of differential equations due to their application in engineering, dynamical population, neural networks, biology, and so on. As a direct consequence, they have attracted an increasing amount of attention over the past few years. This article deals with the p-mean (µ1, µ2)-pseudo almost periodic ((µ1, µ2)-s.p.a.p. for short) mild solutions of the following non-autonomous integro-differential stochastic evolution equation in a real separable Hilbert space E : Z ′(t) = A(t)Z(t) + F1(t, Z(t)) + ∫ t −∞ Q(t− ζ)F2(ζ, Z(ζ))dζ + ∫ t −∞ R(t− ζ)G(ζ, Z(ζ))dW(ζ), ∀t ∈ R, (1.1) where A(t) : D(A(t)) ⊂ Lp(P, E) → Lp(P, E) is a family of densely defined closed linear operators satisfying the so-called “Acquistapace-Terrani” conditions, Q and R are convolution type kernels in L1(0,+∞) and L2(0,+∞), respectively, satisfying [24, Assumption 3.2], (W(t) : t ∈ R) is a K-valued Q-Brownian motion. Here F1, F2 : R × Lp(P, E) → Lp(P, E) and G : R × Lp(P, E) → Lp(P,L0 2) are jointly continuous functions satisfying some additional conditions. The spaces Lp(P, E), L0 2, and the Q-Brownian motion are defined in the next section. Equation (1.1) was studied in several special cases. For instance, Bezandry [6] investigated the existence and uniqueness of square-mean almost periodic mild solutions for equation (1.1) for p = 2 and A(t) = A. Li [27] addressed the problem of the existence, uniqueness, and asymptotic stability of square-mean almost periodic mild solutions of equation (1.1) in the case p = 2. More recently, Mbaye [29] considered the problem of the existence of square-mean µ-pseudo almost periodic mild solutions of equation (1.1) when A(t) = A. However, to the best of our knowledge, the existence, uniqueness, and stability of p-mean (µ1, µ2)-s.p.a.p. mild solutions of equation (1.1) is an untreated topic, which is the main motivation of this work. Acquistapace-Terrani, which is discussed in this work, is an important condition that ensures the existence of a unique evolution family that is necessary for the cor- responding integral form of a given differential equation. The equation considered in this work is very general in nature, and several other equations can be derived as special cases of it. Moreover, the concept of (µ1, µ2)-pseudo almost periodicity is a very general concept and can be applicable to situations where other kinds of functions, such as almost periodic and pseudo-almost periodic, cannot be used to describe the underlying dynamics. The conditions obtained for existence are very general in nature, and other conditions can be obtained as a special case. For EJDE-2024/24 P-MEAN (µ1, µ2)-PSEUDO ALMOST PERIODIC PROCESSES 3 example, one can choose the second component as zero in order to get the corre- sponding result for almost periodic. Similar results can be obtained by adjusting the equation and space. Moreover, stability is also established, and the condition obtained holds for any p; the particular case when p = 2 is given special emphasis. Application to various fields along with numerical graphs makes this work more useful for researchers, especially those who are interested in application. This work is structured as follows. In Section 2, we present some basic notations and definitions. In Section 3, we establish some sufficient conditions to support the existence, uniqueness, and stability of the p-mean (µ1, µ2)-s.p.a.p. mild solution on R of equation (1.1). In the last section, three examples with numerical simulations are presented for better illustrations and to validate the analytical findings. 2. Preliminaries This section introduces relevant notation, definitions and preliminary facts that are needed for the study. 2.1. Q-Brownian motion. Let βn(t), n = 1, 2, 3, . . . be a sequence of real val- ued standard Brownian motions mutually independent on (Ω,F ,P). Set W(t) :=∑∞ n=1 √ ξnβn(t)en, t ≥ 0, where ξn ≥ 0 (n ≥ 1), are non-negative real numbers and (en)n≥1 is an orthonormal basis in the Hilbert space K. Let Q be a non- negative symmetric operator with finite trace defined by Q(en) = ξnen, such that Tr[Q] := ∑∞ n=1 ξn < ∞. It is well known that E[W(t)] = 0 and, for all t ≥ s ≥ 0, the distribution ofW(t)−W(s) is a Gaussian distributionN (0, (t−s)Q). The above mentioned K-valued stochastic processes (W(t))t≥0 is called Q-Brownian motion. Note that a K-valued Q-Brownian motion (W(t))t∈R can be obtained as follows: let {Wi(t) : t ∈ R+}, i = 1, 2, be independent K-valued Q-Brownian motion. Then W(t) = { W1(t), if 0 ≤ t, W2(−t), if 0 ≥ t, is a K-valued Q-Brownian motion with R as time parameters. Let Ft = σ{W(u) : u ≤ t}. To define stochastic integrals with respect to the Q-Brownian motion W, let us denote K0 = Q 1 2K. Now define L0 2 := {φ ∈ L(K0, E) : Tr[φQφ∗] < ∞} the space of Hilbert-Schmidt operators fromK0 to E equipped with the norm ∥φ∥2L0 2 := Tr[φQφ∗] for any φ ∈ L0 2. The next result is a particular case of [33, Lemma 2.2]. Lemma 2.1 ([11, 33]). For any p ≥ 2 and for any arbitrary L0 2-valued predictable processes Ψ(t), t ∈ [0, T ], there exists a constant Cp > 0 such that E [ sup s∈[0,t] ∥∥∫ s 0 Ψ(s)dW(s) ∥∥p] ≤ Cp.E [ ∫ t 0 ∥Ψ(s)∥2L0 2 ds ]p/2 . 2.2. P-mean almost periodic stochastic processes. Assume that (E , ∥ ·∥) and (K, ∥ · ∥K) are real separable Hilbert spaces. (Ω,F ,P) is supposed to be a complete probability space. Let p ≥ 2, and denote Lp(P, E) as the collection of all strongly measurable E-valued random variables Y satisfying E∥Y ∥p < +∞, where the ex- pectation E is defined by E[Y ] := ∫ Ω Y (ω)dP(ω). Note that Lp(P, E) is a Banach space when it is equipped with a norm ∥Y ∥Lp := [E∥Y ∥p]1/p . 4 M. AYACHI, S. ABBAS EJDE-2024/24 Definition 2.2 ([11]). A stochastic processes Z : R → Lp(P, E) is said to be stochastically bounded if there exists a constant C > 0 such that E∥Z(t)∥p ≤ C, for all t ∈ R. The process Z is said to be stochastically continuous if limt→s E∥Z(t)− Z(s)∥p = 0. We denote SBC (R, Lp(P, E)) the collection of all stochastically bounded and continuous processes from R into Lp(P, E). Then (SBC (R, Lp(P, E)) , ∥ · ∥∞) is a Banach space, where ∥Z∥∞ := sup t∈R [E∥Z(t)∥p]1/p. Definition 2.3 ([11]). A continuous stochastic processes Z : R → Lp(P, E) is said to be p-mean almost periodic processes, if for any ϵ > 0 we can find l = l(ϵ) > 0 such that for all ϱ ∈ R, there exists r ∈ [ϱ, ϱ+ l] satisfying E∥Z(t+ r)− Z(t)∥p < ϵ, ∀t ∈ R. We denote the collection of all such stochastic processes by SAP(R, Lp(P, E)). Proposition 2.4 ([11]). The following properties hold for the stochastic processes SAP(R, Lp(P, E)): (1) (SAP(R, Lp(P, E)), ∥ · ∥∞) is a Banach space. (2) SAP(R, Lp(P, E)) is invariant by translation. (3) SAP(R, Lp(P, E)) ⊂ SBC (R, Lp(P, E)) is a closed subspace. 2.3. P-mean (µ1, µ2)-pseudo almost periodic processes. LetB be the Lebesgue σ-field of R and N be the set of all positive measures µ on B satisfying µ(R) = +∞ and µ([r, s]) < +∞ for any r, s ∈ R (r < s). To establish our results, we need the following assumptions: (A1) Let µ1, µ2 ∈ N such that lim supm→+∞ µ1([−m,m]) µ2([−m,m]) < +∞. (A2) For all s ∈ R, there exists α > 0 and a bounded interval I of R such that µ1 ({c+ s : c ∈ C}) ≤ αµ1(C), C ∈ B satisfies C ∩ I = ∅. Definition 2.5. Let µ1, µ2 ∈ N. A stochastic processes Z : R → Lp(P, E) is said to be p-mean (µ1, µ2)-ergodic, if Z ∈ SBC (R, Lp(P, E)) and satisfies lim m→+∞ 1 µ2([−m,m]) ∫ m −m E∥Z(t)∥pdµ1(t) = 0. The collection of all such stochastic processes is denoted by SO (R, Lp(P, E), µ1,2). Proposition 2.6. If µ1, µ2 ∈ N satisfy (A1), then (SO (R, Lp(Ω,H), µ1,2) , ∥ · ∥∞) is a Banach space. Proof. It is easy to see that SO (R, Lp(P, E), µ1,2) is a vector subspace of the Banach space SBC (R, Lp(P, E)). To complete the proof, we need to prove that the space SO(R, Lp(P, E), µ1,2) is closed in SBC (R, Lp(P, E)). Let (Zn)n be a sequence in SO (R, Lp(P, E), µ1,2) such that limn→+∞ ∥Zn − Z∥∞ = 0. Since µ2(R) = +∞, it follows that µ2([−m,m]) > 0 for m sufficiently large. Then, by using the inequality∫ m −m E∥Z(t)∥pdµ1(t) ≤ 2p−1 ∫ m −m E∥Zn(t)− Z(t)∥pdµ1(t) + 2p−1 ∫ m −m E∥Zn(t)∥pdµ1(t), EJDE-2024/24 P-MEAN (µ1, µ2)-PSEUDO ALMOST PERIODIC PROCESSES 5 we have 1 µ2([−m,m]) ∫ m −m E∥Z(t)∥pdµ1(t) ≤ 2p−1µ1([−m,m]) µ2([−m,m]) ∥Zn − Z∥p∞ + 2p−1 1 µ2([−m,m]) ∫ m −m E∥Zn(t)∥pdµ1(t). Thus, in view of (A1), and the fact that (Zn)n ∈ SO (R, Lp(P, E), µ1,2), we obtain lim sup m→+∞ 1 µ2([−m,m]) ∫ m −m E∥Z(t)∥pdµ1(t) ≤ 2p−1cst.∥Zn − Z∥p∞, for all n ∈ N. Finally, since limn→+∞ ∥Zn − Z∥p∞ = 0, we conclude that lim m→+∞ 1 µ2([−m,m]) ∫ m −m E∥Z(t)∥pdµ1(t) = 0. The proof is complete. □ Proposition 2.7 ([4]). Let µ1, µ2 ∈ N, J be a bounded interval (eventually J = ∅). Moreover, suppose that (A1) hold and Z ∈ SBC (R, Lp(P, E)). Then the following assertions are equivalent: (1) Z ∈ SO (R, Lp(P, E), µ1,2). (2) limm→+∞ 1 µ2([−m,m]\J) ∫ [−m,m]\J mathbbE∥Z(t)∥pdµ1(t) = 0. (3) For any υ > 0, limm→+∞ µ1{t∈[−m,m]\J:E∥Z(t)∥p>υ} µ2{t∈[−m,m]\J} = 0. Definition 2.8. Let µ1, µ2 ∈ N. A continuous stochastic processes Z : R → Lp(P, E) is said to be p-mean (µ1, µ2)-pseudo almost periodic processes (p-mean (µ1, µ2)-s.p.a.p. for short) if it can be expressed as Z = Za + Ze, where Za ∈ SAP(R, Lp(P, E)) and Ze ∈ SO (R, Lp(P, E), µ1,2). We denote the collection of all such stochastic processes by SPAP (R, Lp(P, E), µ1,2) Theorem 2.9 ([4]). Let µ1, µ2 ∈ N satisfy (A2), then SPAP (R, Lp(P, E), µ1,2) is invariant by translation. Theorem 2.10 ([4]). Let µ1, µ2 ∈ N and Z ∈ SPAP (R, Lp(P, E), µ1,2) be such that Z = Za +Ze, where Za ∈ SAP(R, Lp(P, E)) and Ze ∈ SO (R, Lp(P, E), µ1,2). If SPAP(R, Lp(P, E), µ1,2) is invariant by translation, then {Za(t) : t ∈ R} ⊂ {Z(t) : t ∈ R}. Theorem 2.11. Let µ1, µ2 ∈ N satisfy (A2). Then the decomposition of a p- mean (µ1, µ2)-s.p.a.p. stochastic processes in the form Z = Za + Ze, where Za ∈ SAP(R, Lp(P, E)) and Ze ∈ SO (R, Lp(P, E), µ1,2) is unique. Proof. Suppose that Z = Za 1 + Ze 1 = Za 2 + Ze 2 , where Za 1 , Z a 2 ∈ SAP(R, Lp(P, E)) and Ze 1 , Z e 2 ∈ SO (R, Lp(P, E), µ1,2), then 0 = (Z1 − Za 2 ) + (Ze 1 − Ze 2) ∈ SAP (R, Lp(P, E), µ1,2) , where Za 1 − Za 2 ∈ SAP(R, Lp(P, E)) and Ze 1 − Ze 2 ∈ SO (R, Lp(P, E), µ1,2). Then from Theorem 2.10, we obtain (Za 1 − Za 2 ) (R) ⊂ {0}. Consequently Za 1 = Za 2 and Ze 1 = Ze 2 . □ Remark 2.12. Za and Ze in definition 2.8 are called the p-mean almost periodic component and the p-mean (µ1, µ2)-ergodic perturbation of the stochastic processes Z respectively. 6 M. AYACHI, S. ABBAS EJDE-2024/24 Theorem 2.13 ([4]). If µ1, µ2 ∈ N satisfy (A1) and (A2), then (SAP(R, Lp(P, E), µ1,2), ∥ · ∥∞) is a Banach space. Let (E1, ∥ · ∥1), (E2, ∥ · ∥2) be Banach spaces, and Lp(P, E1), Lp(P, E2) be corre- sponding Lp-spaces. Consider the following spaces of stochastic processes SAP (R× Lp(P, E1), Lp(P, E2)) = {F (·, Y ) ∈ SAP (R, Lp(P, E2)) for any Y ∈ Lp(P, E1)} , SO (R× Lp(P, E1), Lp(P, E2), µ1,2) = {F (·, Y ) ∈ SO (R, Lp(P, E2), µ1,2) for any Y ∈ Lp(P, E1)} . Definition 2.14. Let µ1, µ2 ∈ N. A stochastically continuous processes F : R × Lp(P, E1) → Lp(P, E2) is said to be p-mean (µ1, µ2)-pseudo almost periodic in t ∈ R for any Y ∈ Lp(P, E1), if it can be expressed as F = F a + F e, where F a ∈ SAP(R×Lp(P, E1), Lp(P, E2)) and F e ∈ SO (R× Lp(P, E1), Lp(P, E2), µ1,2). We denote the collection of all such stochastically continuous processes by SPAP (R× Lp(P, E1), Lp(P, E2), µ1,2) . Theorem 2.15. Let µ1, µ2 ∈ N satisfy (A2). Suppose that F ∈ SPAP(R × Lp(P, E1), Lp(P, E2), µ1,2) satisfies Lipschitz condition in the second variable, that is, there exists L > 0 such that for any Y1, Y2 ∈ Lp(P, E1) and for all t ∈ R E∥F (t, Y1)− F (t, Y2)∥p2 ≤ L.E∥Y1 − Y2∥p1. (2.1) Then F (·, Z(·)) ∈ SPAP (R, Lp(P, E2), µ1,2) for any Z ∈ PAP (R, Lp(P, E1), µ1,2). Proof. From Definitions 2.8 and 2.14, let F = F a + F e and Z = Za + Ze, where F a ∈ SAP(R × Lp(P, E1), Lp(P, E2)), F e ∈ SO(R × Lp(P, E1), Lp(P, E2), µ1,2), Za ∈ SAP(R, Lp(P, E1), µ1,2), and Ze ∈ SO(R, Lp(P, E1), µ1,2). Then, we can write F (t, Z(t)) = F a(t, Za(t)) + [F (t, Z(t))− F (t, Za(t))] + [F (t, Za(t))− F a(t, Za(t))] = F a(t, Za(t)) + [F (t, Z(t))− F (t, Za(t))] + F e(t, Za(t)). First, we claim that F a(·, Za(·)) ∈ SAP (R, Lp(P, E2)). In fact, since F ∈ SPAP (R× Lp(P, E1), Lp(P, E2), µ1,2), then for all Y ∈ Lp(P, E1), we have F (·, Y ) ∈ SPAP (R, Lp(P, E2), µ1,2) . Hence, we can write F (·, Y ) = F a(·, Y )+F e(·, Y ) with F a(·, Y ) ∈ SAP(R, Lp(P, E2)) and F e(·, Y ) ∈ SO(R, Lp(P, E2), µ1,2). Since ∥F a(·, Y1)−F a(·, Y2)∥2 is almost peri- odic component of the p-mean (µ1, µ2)-s.p.a.p. function ∥F (·, Y1)−F (·, Y2)∥2, by us- ing Theorem 2.10, we deduce that ∥F a(·, Y1)−F a(·, Y2)∥∞ ≤ ∥F (·, Y1)−F (·, Y2)∥∞, which implies that, for any t ∈ R and Y1, Y2 ∈ Lp(P, E1) E∥F a(t, Y1)− F a(t, Y2)∥p2 ≤ E∥F (t, Y1)− F (t, Y2)∥p2 ≤ L.E∥Y1 − Y2∥p1. Let us define K = {Za(t) : t ∈ R}. Since Za belongs to SAP (R, Lp(P, E1)), it follows that K is a compact set. Therefore, using [11, Theorem 4.4], we deduce that G(·, Za(·)) ∈ SAP (R, Lp(P, E2)). EJDE-2024/24 P-MEAN (µ1, µ2)-PSEUDO ALMOST PERIODIC PROCESSES 7 Next, we claim that F (·, Z(·)) − F (·, Za(·)) ∈ SO (R, Lp(P, E2), µ1,2). By using the Lipschitz condition, we obtain lim m→∞ 1 µ2([−m,m]) ∫ m −m E∥F (t, Z(t))− F (t, Za(t))∥p2dµ1(t) ≤ lim m→∞ 1 µ2([−m,m]) ∫ m −m L.E∥Z(t)− Za(t)∥p1dµ1(t) ≤ lim m→∞ 1 µ2([−m,m]) ∫ m −m L.E∥Ze(t)∥p1dµ1(t) = 0. Finally, it remains to show that F e(·, Za(·)) ∈ SO (R, Lp(P, E2), µ1,2). Indeed, in the view of (2.1) and Theorem 2.10, it follows that E∥F e(t, Y1)− F e(t, Y2)∥p2 = E∥F (t, Y1)− F a(t, Y1)− F (t, Y2) + F a(t, Y2)∥p2 ≤ 2p−1E∥F (t, Y1)− F (t, Y2)∥p2 + 2p−1E∥F a(t, Y1)− F a(t, Y2)∥p2 ≤ 2pL.E∥Y1 − Y2∥p1. (2.2) Since K = {Za(t) : t ∈ R} is compact, for ϵ > 0, there exist Y1, . . . , Yk ∈ K, such that K ⊂ ∪k i=1B ( Yi, ϵ 22p−1ωL ) , where ω := lim sup m→+∞ µ1 ([−m,m]) µ2 ([−m,m]) < +∞, B ( Yi, ϵ 2p−1ωL ) := { Y ∈ K : E∥Yi − Y ∥p1 ≤ ϵ 22p−1ωL } . By using (2.2) along with the above result, we obtain K ⊂ ∪k i=1 { Y ∈ K : ∀t ∈ R,E∥F e(t, Y )− F e(t, Yi)∥p2 ≤ ϵ 2p−1ω } . Let t ∈ R and Y ∈ K, then there exists i∗ ∈ {1, . . . , k} such that E∥F e(t, Y )− F e(t, Yi∗)∥ p 2 ≤ ϵ 2p−1ω . Therefore, E∥F e(t, Za(t))∥p2 ≤ 2p−1E∥F e(t, Za(t))− F e(t, Yi∗)∥ p 2 + 2p−1E∥F e(t, Yi∗)∥ p 2 ≤ ϵ ω + 2p−1E∥F e(t, Yi∗)∥ p 2 ≤ ϵ ω + 2p−1 k∑ i=1 E∥F e(t, Yi)∥p2. (2.3) Since for all i ∈ {1, . . . , k}, lim m→∞ 1 µ2([−m,m]) ∫ m −m E∥F e(t, Yi)∥p2dµ1(t) = 0, by using (A1) and (2.3), we obtain lim sup m→∞ 1 µ2([−m,m]) ∫ m −m E∥F e(t, Za(t))∥p2dµ1(t) ≤ ϵ, for all ϵ > 0, 8 M. AYACHI, S. ABBAS EJDE-2024/24 which further implies lim m→∞ 1 µ2([−m,m]) ∫ m −m E∥F e(t, Za(t))∥p2dµ1(t) = 0. Therefore, H(·, Za(·)) ∈ SO (R, Lp(P, E2), µ1,2). The proof is complete. □ 3. Application to integro-differential stochastic evolution equations In this section, we establish some sufficient conditions to ensure the existence, uniqueness and stability of p-mean (µ1, µ2)-s.p.a.p. mild solution of (1.1). Definition 3.1. An Ft progressively measurable processes (Z(t))t∈R is called mild solution of (1.1) if it satisfies the following stochastic integral equation Z(t) = U(t, a)Z(a) + ∫ t a U(t, s)F1(s, Z(s))ds + ∫ t a U(t, s) ∫ s a Q(s− ζ)F2(ζ, Z(ζ))dζds + ∫ t a U(t, s) ∫ s a R(s− ζ)G(ζ, Z(ζ))dW(ζ)ds (3.1) for all t ≥ a and each a ∈ R. The Acquistapace-Terreni conditions (ATC, for short), which was firstly intro- duced in [1], play an important role in the study of non-autonomous evolution equations. We state it below for the readers’ convenience. Definition 3.2. A family of closed linear operators A(t) for t ∈ R on a Banach space (E , ∥ · ∥) with domain D(A(t)) (possibly not densely defined) satisfies ATC, if there exist constants w > 0, γ ∈ (π2 , π), K1,K2 ≥ 0 and ν1, ν2 ∈ (0, 1] with ν1 + ν2 > 1 such that Sγ ∪ {0} ⊂ ρ(A(t)− w), ∥R(λ,A(t)− w)∥ ≤ K2 1 + |λ| , ∥(A(t)− w)R(λ,A(t)− w)[R(w,A(t))−R(w,A(s))]∥ ≤ K2.|t− s|ν1 |λ|−ν2 , for all t, s ∈ R, λ ∈ Sγ := {λ ∈ C− {0} : |argλ| ≤ γ}. Lemma 3.3. Let A(t) be a family of closed linear operators which satisfies ATC. Then there exists a unique evolution family {U(t, s)}−∞ 0 such that ∥U(t, s)∥ ≤ Me−κ(t−s), for all t, s ∈ R, such that t ≥ s. (A4) R(w,A(·)) ∈ AP (R,L(Lp(P, E))), for w in Definition 3.2. EJDE-2024/24 P-MEAN (µ1, µ2)-PSEUDO ALMOST PERIODIC PROCESSES 9 (A5) The processes Fi : R × Lp(P, E) → Lp(P, E) (i = 1, 2) and G : R × Lp(P, E) → Lp(P,L0 2) are p-mean (µ1, µ2)-pseudo almost periodic in t ∈ R for any Y ∈ Lp(P, E). Moreover, F1, F2, and G are Lipschitz in the follow- ing sense : there exists Li > 0 (i = 1, 2, 3) such that E∥Fi(t, Y1)− Fi(t, Y2)∥p ≤ Li.E∥Y1 − Y2∥p, i = 1, 2, E∥G(t, Y1)−G(t, Y2)∥pL0 2 ≤ L3.E∥Y1 − Y2∥p, for all stochastic processes Y1, Y2 ∈ Lp(P, E) and t ∈ R. The next Lemma, which can be seen as an immediate consequence of [28, Propo- sition 4.4] is essential to study the existence of p-mean (µ1, µ2)-s.p.a.p. mild solu- tions. Lemma 3.4. Suppose that (A3), (A4) hold. Then, for any ϵ > 0 and h > 0, there exists l = l(ϵ) > 0 such that every interval of length l contains at least a number r satisfying ∥U(t+ r, s+ r)− U(t, s)∥ ≤ ϵe− κ 2 (t−s), for all t− s ≥ h. Lemma 3.5. Let µ1, µ2 ∈ N satisfy (A1)–(A4) hold. Furthermore, assume that Z ∈ SPAP (R, Lp(P, E), µ1,2). Then the function Γ : t 7→ ∫ t −∞ U(t, s)Z(s)ds, belongs to SPAP (R, Lp(P, E), µ1,2). Proof. For Z ∈ SPAP (R, Lp(P, E), µ1,2), there exist Za ∈ SAP (R, Lp(P, E)) and Ze ∈ SO (R, Lp(P, E), µ1,2), such that Z = Za + Ze. Consequently, we can write Γ(t) = Γ1(t) + Γ2(t) = ∫ t −∞ U(t, s)Za(s)ds+ ∫ t −∞ U(t, s)Ze(s)ds. First, we show that Γ1 ∈ SAP (R, Lp(P, E)). In fact, since Za ∈ SAP (R, Lp(P, E)), so according to Lemma 3.4, for a given ϵ > 0, one can find l(ϵ) > 0 such that for any interval of length l(ϵ) contains at least a number r with the property that ∥U(t+ r, s+ r)− U(t, s)∥ ≤ ϵe− κ 2 (t−s), for all t− s ≥ ϵ, (3.2) and E∥Za(t+ r)− Za(t)∥p < η, for all t ∈ R, (3.3) where η = η(ϵ) → 0 as ϵ → 0. By using (3.2), (3.3), Hölder’s inequality, and that |x+ y + z|p ≤ 3p−1(|z|p + |y|p + |z|p), it follows that E∥Γ1(t+ r)− Γ1(t)∥p = E ∥∥∫ t+r −∞ U(t+ r, s)Za(s)ds− ∫ t −∞ U(t, s)Za(s)ds ∥∥p = E ∥∥∫ +∞ 0 U(t+ r, t+ r − s)Za(t+ r − s)ds− ∫ +∞ 0 U(t, t− s)Za(t− s)ds ∥∥p ≤ 3p−1E [ ∫ +∞ 0 ∥U(t+ r, t+ r − s)∥ · ∥Za(t+ r − s)− Za(t− s)∥ds ]p 10 M. AYACHI, S. ABBAS EJDE-2024/24 + 3p−1E [ ∫ ϵ 0 ∥U(t+ r, t+ r − s)− U(t, t− s))∥ · ∥Za(t− s)∥ds ]p + 3p−1E [ ∫ +∞ ϵ ∥U(t+ r, t+ r − s)− U(t, t− s))∥ · ∥Za(t− s)∥ds ]p ≤ 3p−1MpE [ ∫ +∞ 0 e−κs∥Za(t+ r − s)− Za(t− s)∥ds ]p + 3p−1MpE [ ∫ ϵ 0 2e−κs∥Za(t− s)∥ds ]p + 3p−1ϵpE [ ∫ +∞ ϵ e− κ 2 s∥Za(t− s)∥ds ]p ≤ 3p−1Mp [ ∫ +∞ 0 e−κsds ]p−1 ∫ +∞ 0 e−κsE∥Za(t+ r − s)− Za(t− s)∥pds + 3p−12pMp [ ∫ ϵ 0 e−κsds ]p−1 ∫ ϵ 0 e−κsE∥Xa(t− s)∥pds + 3p−1ϵp [ ∫ +∞ ϵ e− κ 2 sds ]p−1 ∫ +∞ ϵ e− κ 2 sE∥Za(t− s)∥pds ≤ 3p−1Mp [ ∫ +∞ 0 e−κsds ]p sup s∈R E∥Za(t+ r − s)− Za(t− s)∥p + 3p−12pMp [ ∫ ϵ 0 e−κsds ]p sup s∈R E∥Za(t− s)∥p + 3p−1ϵp [ ∫ +∞ ϵ e− κ 2 sds ]p sup s∈R E∥Za(t− s)∥p ≤ 3p−1Mp(1/κ)pη(ϵ) + 3p−12pMp∥Z∥p∞ϵp + 3p−1(2/κ)p∥Z∥p∞ϵp, which implies that Γ1 ∈ SAP (R, Lp(P, E)). Next, we check that Γ2 ∈ SO (R, Lp(P, E), µ1,2), that is lim m→+∞ 1 µ2([−m,m]) ∫ m −m E∥Γ2(t)∥pdµ1(t) = 0. Let us denote A := {s : s ≤ t} and B := {v : v ≥ 0}. Applying Hölder’s inequality and Fubini’s theorem, for m > 0, we obtain 1 µ2([−m,m]) ∫ m −m E∥Γ2(t)∥pdµ1(t) = 1 µ2([−m,m]) ∫ m −m E ∥∥ ∫ t −∞ U(t, s)Ze(s)ds ∥∥pdµ1(t) ≤ 1 µ2([−m,m]) ∫ m −m E [ ∫ t −∞ ∥U(t, s)∥∥Ze(s)∥ds ]p dµ1(t) ≤ Mp µ2([−m,m]) ∫ m −m {[∫ t −∞ e−κ(t−s)ds ]p−1 ∫ t −∞ e−κ(t−s)E∥Ze(s)∥pds } dµ1(t) ≤ Mp κp−1 1 µ2([−m,m]) ∫ m −m {∫ +∞ −∞ e−κ(t−s)E∥Ze(s)∥pχAds } dµ1(t) ≤ Mp κp−1 1 µ2([−m,m]) ∫ ∞ −∞ {∫ +m −m e−κ(t−s)E∥Ze(s)∥pχAdµ1(t) } ds. EJDE-2024/24 P-MEAN (µ1, µ2)-PSEUDO ALMOST PERIODIC PROCESSES 11 By making change of variables v = t− s, it follows that 1 µ2([−m,m]) ∫ m −m E∥Γ2(t)∥pdµ1(t) ≤ Mp κp−1 1 µ2([−m,m]) ∫ −∞ +∞ {∫ +m −m e−κvE∥Ze(t− v)∥pχBdµ1(t) } (−dv) ≤ Mp κp−1 1 µ2([−m,m]) ∫ +∞ 0 {∫ +m −m e−κvE∥Ze(t− v)∥pdµ1(t) } dv ≤ Mp κp−1 ∫ +∞ 0 { e−κv µ2([−m,m]) ∫ +m −m E∥Ze(t− v)∥pdµ1(t) } dv. One can see that∣∣∣ e−κv µ2([−m,m]) ∫ +m −m E∥Ze(t− v)∥pdµ1(t) ∣∣∣ ≤ e−κv∥Xe∥p∞ µ1([−m,m]) µ2([−m,m]) , for all v ≥ 0. Since µ1 and µ2 satisfy (A2), from Theorem 2.9, we have [t 7→ Xe(t− v)] ∈ SO (R, Lp(P, E), µ1,2) . Then, by using (A2) and the Lebesgue dominate convergence theorem, we obtain lim m→+∞ Mp κp−1 ∫ +∞ 0 { e−κv µ2([−m,m]) ∫ +m −m E∥Ze(t− v)∥pdµ1(t) } dv = 0, which implies that lim m→+∞ 1 µ2([−m,m]) ∫ m −m E∥Γ2(t)∥pdµ1(t). Finally, we obtain Γ = Γ1 + Γ2 ∈ SPAP (R, Lp(P, E), µ1,2). □ Lemma 3.6. Let µ1, µ2 ∈ N satisfy (A1), (A2) and Z ∈ SPAP (R, Lp(P, E), µ1,2). Then the function Λ : t 7→ ∫ t −∞ Q(t− ζ)Z(ζ)dζ, belongs to SPAP (R, Lp(P, E), µ1,2). Proof. For Z ∈ SPAP (R, Lp(P, E), µ1,2), there exist Za ∈ SAP (R, Lp(P, E)) and Ze ∈ SO (R, Lp(P, E), µ1,2), such that Z = Za + Ze. Hence, we can write Λ(t) = Λ1(t) + Λ2(t) = ∫ t −∞ Q(t− ζ)Za(ζ)dζ + ∫ t −∞ Q(t− ζ)Ze(ζ)dζ = ∫ +∞ 0 Q(ζ)Za(t− ζ)dζ + ∫ +∞ 0 Q(ζ)Ze(t− ζ)dζ. First, let us show that Λ1 ∈ SAP (R, Lp(P, E)). Since Za ∈ SAP (R, Lp(P, E)), so for any given ϵ > 0, one can find l(ϵ) > 0 such that for any interval of length l(ϵ) contains at least a number r such that E∥Za(t+ r)− Za(t)∥p < ϵ ∥Q∥pL1(0,+∞) , for all t ∈ R. Now by using Hölder’s inequality, we obtain E∥Λ1(t+ r)− Λ1(t)∥p 12 M. AYACHI, S. ABBAS EJDE-2024/24 = E ∥∥ ∫ +∞ 0 Q(ζ)Za(t+ r − ζ)dζ − ∫ +∞ 0 Q(ζ)Za(t− ζ)dζ ∥∥p ≤ E [ ∫ +∞ 0 ∥Q(ζ)∥∥Za(t+ r − ζ)− Za(t− ζ)∥dζ ]p ≤ [ ∫ +∞ 0 ∥Q(ζ)∥dζ ]p−1 ∫ +∞ 0 ∥Q(ζ)∥E∥Za(t+ r − ζ)− Za(t− ζ)∥pdζ ≤ ∥Q∥pL1(0,+∞) sup ζ∈R E∥Za(t+ r − ζ)− Za(t− ζ)∥p ≤ ϵ, which implies that Λ1 ∈ SAP (R, Lp(P, E)). Next, we check that Λ2 ∈ SO (R, Lp(P, E), µ1,2); that is lim m→+∞ 1 µ2([−m,m]) ∫ m −m E∥Λ2(t)∥pdµ1(t) = 0. By using Hölder’s inequality and Fubini’s theorem, for m > 0, we obtain 1 µ2([−m,m]) ∫ m −m E∥Λ2(t)∥pdµ1(t) = 1 µ2([−m,m]) ∫ m −m E big∥ ∫ +∞ 0 Q(ζ)Ze(t− ζ)dζ ∥∥pdµ1(t) ≤ 1 µ2([−m,m]) ∫ m −m E [ ∫ +∞ 0 ∥Q(ζ)∥ · ∥Ze(t− ζ)∥dζ ]p dµ1(t) ≤ 1 µ2([−m,m]) ∫ m −m {[∫ +∞ 0 ∥Q(ζ)∥dζ ]p−1 ∫ +∞ 0 ∥Q(ζ)∥E∥Ze(t− ζ)∥pdζ } dµ1(t) ≤ ∥Q∥p−1 L1(0,+∞) ∫ +∞ 0 { ∥Q(ζ)∥ µ2([−m,m]) ∫ m −m E∥Ze(t− ζ)∥pdµ1(t) } dζ. Since ∣∣∣ ∥Q(ζ)∥ µ2([−m,m]) ∫ m −m E∥Xe(t− ζ)∥pdµ1(t) ∣∣∣ ≤ ∥Q(ζ)∥∥Ze∥p∞ µ1([−m,m]) µ2([−m,m]) , for all ζ ≥ 0, by using the Lebesgue dominate convergence theorem, (A1), and that the space SO(R, Lp(P, E), µ1,2) is invariant by translation, we obtain lim m→+∞ ∫ +∞ 0 { ∥Q(ζ)∥ µ2([−m,m]) ∫ m −m E∥Ze(t− ζ)∥pdµ1(t) } dζ = 0, which implies that lim m→+∞ 1 µ2([−m,m]) ∫ m −m E∥Λ2(t)∥pdµ1(t). Finally, we obtain Λ = Λ1 + Λ2 ∈ SPAP (R, Lp(P, E), µ1,2). □ Lemma 3.7. Let µ1, µ2 ∈ N satisfy (A1), (A2) and Z ∈ SPAP ( R, Lp(P,L0 2), µ1,2 ) . Then the function ∆ : t 7→ ∫ t −∞ R(t− ζ)Z(ζ)dW(ζ), belongs to SPAP (R, Lp(P, E), µ1,2). EJDE-2024/24 P-MEAN (µ1, µ2)-PSEUDO ALMOST PERIODIC PROCESSES 13 Proof. Let us assume that Z ∈ SPAP ( R, Lp(P,L0 2), µ1,2 ) . Then, there exist Za ∈ SAP(R, Lp(P,L0 2)) and Ze ∈ SO(R, Lp(P,L0 2), µ1,2), such that Z = Za + Ze. Hence, we can write ∆(t) = ∆1(t) + ∆2(t) = ∫ t −∞ R(t− ζ)Za(ζ)dW(ζ) + ∫ t −∞ R(t− ζ)Ze(ζ)dW(ζ). First, let us show that ∆1 ∈ SAP (R, Lp(P, E)). Since Za ∈ SAP ( R, Lp(P,L0 2) ) , for given ϵ > 0, one can find l(ϵ) > 0 such that for any interval of length l(ϵ) contains at least r with the property that E∥Za(t+ r)− Za(t)∥pL0 2 < ϵ Cp.∥R∥pL2(0,+∞) , for all t ∈ R. Let W(ζ) = W(ζ + r)−W(r) for each ζ ∈ R. Then, W is also a Brownian motion and has the same distribution as W. By making change of variable v = ζ − r, and using Lemma 2.1, we obtain E∥∆1(t+ r)−∆1(t)∥p = E ∥∥ ∫ t+r −∞ R(t+ r − ζ)Xa(ζ)dW(ζ)− ∫ t −∞ R(t− ζ)Za(ζ)dW(ζ) ∥∥p = E ∥∥ ∫ t −∞ R(t− v)Xa(v + r)dW(v)− ∫ t −∞ R(t− v)Za(v)dW(v) ∥∥p = E ∥∥ ∫ t −∞ R(t− v) [ Za(v + r)− Za(v) ] dW(v) ∥∥p ≤ CpE [ ∫ t −∞ ∥R(t− v)∥2∥Za(v + r)− Za(v)∥2L0 2 dv ]p/2 ≤ CpE [ ∫ t −∞ ∥R(t− v)∥2 2 q ∥R(t− v)∥2 2 p ∥Za(v + r)− Za(v)∥2L0 2 dv ]p/2 , where q > 0 solves 1 p/2 + 1 q/2 = 2 p + 2 q = 1. Then, from Hölder’s inequality, we obtain E∥∆1(t+ r)−∆1(t)∥p ≤ Cp [ ∫ t −∞ ∥R(t− v)∥2dv ] p−2 2 ∫ t −∞ ∥R(t− v)∥2E∥Za(v + r)− Za(v)∥pL0 2 dv ≤ Cp∥R∥pL2(0,+∞) sup v∈R E∥Za(v + r)− Za(v)∥pL0 2 ≤ ϵ. Which implies that ∆1 ∈ SAP (R, Lp(P, E)). Next, we check that ∆2 ∈ SO (R, Lp(P, E), µ1,2). Let us denote A := {ζ : ζ ≤ t} and B := {v : v ≥ 0}. By using Hölder’s inequality and Fubini’s theorem, we obtain, for m > 0; 1 µ2([−m,m]) ∫ m −m E∥∆2(t)∥pdµ1(t) = 1 µ2([−m,m]) ∫ m −m E ∥∥ ∫ t −∞ R(t− ζ)Ze(ζ)dW(ζ) ∥∥pdµ1(t) 14 M. AYACHI, S. ABBAS EJDE-2024/24 ≤ 1 µ2([−m,m]) Cp ∫ m −m E [ ∫ t −∞ ∥R(t− ζ)∥2∥Ze(ζ)∥2L0 2 dζ ]p/2 dµ1(t) ≤ 1 µ2([−m,m]) Cp ∫ m −m {[∫ t −∞ ∥R(t− ζ)∥2dζ ] p−2 2 × ∫ t −∞ ∥R(t− ζ)∥2E∥Ze(ζ)∥pL0 2 dζ } dµ1(t) ≤ 1 µ2([−m,m]) Cp∥R∥p−2 L2(0,+∞) ∫ m −m {∫ +∞ −∞ ∥R(t− ζ)∥2E∥Ze(ζ)∥pL0 2 χAdζ } dµ1(t) ≤ 1 µ2([−m,m]) Cp∥R∥p−2 L2(0,+∞) ∫ +∞ −∞ {∫ m −m ∥R(t− ζ)∥2E∥Ze(ζ)∥pL0 2 χAdµ1(t) } dζ. By making change of variables v = t− ζ, we obtain 1 µ2([−m,m]) ∫ m −m E∥∆2(t)∥pdµ1(t) ≤ 1 µ2([−m,m]) Cp∥R∥p−2 L2(0,+∞) ∫ +∞ −∞ {∫ m −m ∥R(v)∥2E∥Ze(t− v)∥pL0 2 χBdµ1(t) } dv = Cp∥R∥p−2 L2(0,+∞) ∫ +∞ 0 { ∥R(v)∥2 µ2([−m,m]) ∫ m −m E∥Ze(t− v)∥pL0 2 dµ1(t) } dv. Since, for all v ≥ 0∣∣∣ ∥R(v)∥2 µ2([−m,m]) ∫ m −m E∥Xe(t− v)∥pL0 2 dµ1(t) ∣∣∣ ≤ ∥R(v)∥2∥Ze∥p∞ µ1([−m,m]) µ2([−m,m]) , by using the Lebesgue dominate convergence theorem, (A1), and that the space SO(R, Lp(P,L0 2), µ1,2) is invariant by translation, we obtain lim m→+∞ ∫ +∞ 0 { ∥R(v)∥2 µ2([−m,m]) ∫ m −m E∥Ze(t− v)∥pL0 2 dµ1(t) } dv = 0, which implies that lim m→+∞ 1 µ2([−m,m]) ∫ m −m E∥∆2(t)∥pdµ1(t). Finally ∆ = ∆1 +∆2 ∈ SPAP (R, Lp(P, E), µ1,2). □ Theorem 3.8. Let µ1, µ2 ∈ N satisfy (A1), (A2). Suppose that (A3)–(A5) hold. Then (1.1) has a unique p-mean (µ1, µ2)-s.p.a.p. mild solution, which can be ex- plicitly expressed as Z(t) = ∫ t −∞ U(t, s)F1(s, Z(s))ds+ ∫ t −∞ U(t, s) ∫ s −∞ Q(s− ζ)F2(ζ, Z(ζ))dζds + ∫ t −∞ U(t, s) ∫ s −∞ R(s− ζ)G(ζ, Z(ζ))dW(ζ)ds, t ∈ R, whenever Θp := Mp(1/κ)p [ L1 + L2∥Q∥pL1(0,+∞) + CpL3∥R∥pL2(0,+∞) ] < (1/3)p−1, (3.4) for p > 2, and Θ2 := M2(1/κ)2 [ L1 + L2∥Q∥2L1(0,+∞) + L3∥R∥2L2(0,+∞) ] < 1/3, (3.5) EJDE-2024/24 P-MEAN (µ1, µ2)-PSEUDO ALMOST PERIODIC PROCESSES 15 for p = 2. Proof. First of all, it is not difficult to see that the stochastic processes Z(t) = ∫ t −∞ U(t, s)F1(s, Z(s))ds+ ∫ t −∞ U(t, s) ∫ s −∞ Q(s− ζ)F2(ζ, Z(ζ))dζds + ∫ t −∞ U(t, s) ∫ s −∞ R(s− ζ)G(ζ, Z(ζ))dW(ζ)ds is well defined and satisfies (3.1). Hence it is a mild solution of (1.1). Let us consider a nonlinear operator Γ given by (ΓZ)(t) = ∫ t −∞ U(t, s)F1(s, Z(s))ds+ ∫ t −∞ U(t, s) ∫ s −∞ Q(s− ζ)F2(ζ, Z(ζ))dζds + ∫ t −∞ U(t, s) ∫ s −∞ R(s− ζ)G(ζ, Z(ζ))dW(ζ)ds. We prove that Γ is a self mapping from SPAP (R, Lp(P, E), µ1,2) to itself and it is a contraction. Hence, according to the Banach contraction principle, we can conclude that Γ has a unique fixed point Z∗ ∈ SPAP (R, Lp(P, E), µ1,2), that is ΓZ ∗ = Z∗ which satisfies (3.1). Let us first show that Γ is a self mapping. Let Z ∈ SPAP (R, Lp(P, E), µ1,2). Then, using (A5) and Theorem 2.15, we can easily see that G(·, Z(·)) belongs to SPAP ( R, Lp(P,L0 2), µ1,2 ) , and F1(·, Z(·)), F2(·, Z(·)) ∈ SPAP (R, Lp(P, E), µ1,2). Consequently, in view of Lemmas 3.5, 3.6 and 3.7, we can conclude that ΓZ ∈ SPAP (R, Lp(P, E), µ1,2). Now, to complete the proof, we have to check that Γ is a strict contraction mapping on SPAP(R, Lp(P, E), µ1,2). In fact, for each t ∈ R, and Z1, Z2 ∈ SPAP(R, Lp(P, E), µ1,2), we have E∥(ΓZ1)(t)− (ΓZ2)(t)∥p ≤ 3p−1E∥ ∫ t −∞ U(t, s) [F1(s, Z1(s))− F1(s, Z2(s))] ds∥p + 3p−1E∥ ∫ t −∞ U(t, s) ∫ s −∞ Q(s− ζ) [F2(ζ, Z1(ζ))− F2(ζ, Z2(ζ))] dζds∥p + 3p−1E∥ ∫ t −∞ U(t, s) ∫ s −∞ R(s− ζ) [G(ζ, Z1(ζ))−G(ζ, Z2(ζ))] dW(ζ)ds∥p ≤ 3p−1E [ ∫ t −∞ ∥U(t, s)∥∥F1(s, Z1(s))− F1(s, Z2(s))∥ds ]p + 3p−1E [ ∫ t −∞ ∥U(t, s)∥∥ ∫ s −∞ Q(s− ζ)[F2(ζ, Z1(ζ))− F2(ζ, Z2(ζ))]dζ∥ds ]p + 3p−1E [ ∫ t −∞ ∥U(t, s)∥∥ ∫ s −∞ R(s− ζ)[G(ζ, Z1(ζ))−G(ζ, Z2(ζ))]dW(ζ)ds∥ ]p . Now, we evaluate the first term of the right-hand side with the help of Hölder’s inequality as follows 3p−1E [ ∫ t −∞ ∥U(t, s)∥∥F1(s, Z1(s))− F1(s, Z2(s))∥ds ]p ≤ 3p−1MpE [ ∫ t −∞ e−κ(t−s)∥F1(s, Z1(s))− F1(s, Z2(s))∥ds ]p 16 M. AYACHI, S. ABBAS EJDE-2024/24 ≤ 3p−1Mp [ ∫ t −∞ e−κ(t−s)ds ]p−1 ∫ t −∞ e−κ(t−s)E∥F1(s, Z1(s))− F1(s, Z2(s))∥pds ≤ 3p−1MpL1 [ ∫ t −∞ e−κ(t−s)ds ]p−1 ∫ t −∞ e−κ(t−s)E∥Z1(s)− Z2(s)∥pds ≤ 3p−1MpL1 [ ∫ t −∞ e−κ(t−s)ds ]p−1[ ∫ t −∞ e−κ(t−s)ds ] sup s∈R E∥Z1(s)− Z2(s)∥p ≤ 3p−1MpL1(1/κ) p∥Z1 − Z2∥p∞. Furthermore, by using the Hölder’s inequality for the second term, we have 3p−1E [ ∫ t −∞ ∥U(t, s)∥∥ ∫ s −∞ Q(s− ζ)[F2(ζ, Z1(ζ))− F2(ζ, Z2(ζ))]dζ∥ds ]p ≤ 3p−1MpE [ ∫ t −∞ e−κ(t−s)∥ ∫ s −∞ Q(s− ζ)[F2(ζ, Z1(ζ))− F2(ζ, Z2(ζ))]dζ∥ds ]p ≤ 3p−1Mp [ ∫ t −∞ e−κ(t−s)ds ]p−1 ∫ t −∞ e−κ(t−s)E∥ ∫ s −∞ Q(s− ζ)[F2(ζ, Z1(ζ)) − F2(ζ, Z2(ζ))]dζ∥pds ≤ 3p−1Mp(1/κ)p−1 ∫ t −∞ e−κ(t−s)E [ ∫ s −∞ ∥Q(s− ζ)∥ ∥F2(ζ, Z1(ζ)) − F2(ζ, Z2(ζ))∥dζ ]p ds ≤ 3p−1Mp(1/κ)p−1 ∫ t −∞ e−κ(t−s) [ ∫ s −∞ ∥Q(s− ζ)∥dζ ]p−1 ∫ s −∞ ∥Q(s− ζ)∥ × E∥F2(ζ, Z1(ζ))− F2(ζ, Z2(ζ))∥pdζds ≤ 3p−1MpL2(1/κ) p−1 ∫ t −∞ e−κ(t−s) [ ∫ s −∞ ∥Q(s− ζ)∥dζ ]p−1 ∫ s −∞ ∥Q(s− ζ)∥ × E∥Z1(ζ)− Z2(ζ)∥pdζds ≤ 3p−1MpL2(1/κ) p−1∥Q∥pL1(0,+∞) [ ∫ t −∞ e−κ(t−s)ds ] sup ζ∈R E∥Z1(ζ)− Z2(ζ)∥p ≤ 3p−1MpL2(1/κ) p∥Q∥pL1(0,+∞)∥Z1 − Z2∥p∞. For the third term, we use Hölder’s inequality and Lemma 2.1 to obtain 3p−1E [ ∫ t −∞ ∥U(t, s)∥∥ ∫ s −∞ R(s− ζ)[G(ζ, Z1(ζ)) −G(ζ, Z2(ζ))]dW(ζ)∥ds ]p ≤ 3p−1MpE [ ∫ t −∞ e−κ(t−s)∥ ∫ s −∞ R(s− ζ)[G(ζ, Z1(ζ))−G(ζ, Z2(ζ))]dW(ζ)∥ds ]p ≤ 3p−1Mp [ ∫ t −∞ e−κ(t−s)ds ]p−1 ∫ t −∞ e−κ(t−s)E∥ ∫ s −∞ R(s− ζ)[G(ζ, Z1(ζ)) −G(ζ, Z2(ζ))]dW(ζ)∥pds ≤ 3p−1MpCp(1/κ) p−1 ∫ t −∞ e−κ(t−s)E [ ∫ s −∞ ∥R(s− ζ)∥2∥G(ζ, Z1(ζ)) EJDE-2024/24 P-MEAN (µ1, µ2)-PSEUDO ALMOST PERIODIC PROCESSES 17 −G(ζ, Z2(ζ))∥2L0 2 dζ ]p/2 ds ≤ 3p−1MpCp(1/κ) p−1 ∫ t −∞ e−κ(t−s) [ ∫ s −∞ ∥R(s− ζ)∥2dζ ] p−2 2 ∫ s −∞ ∥R(s− ζ)∥2 × E∥G(ζ, Z1(ζ))−G(ζ, Z2(ζ))∥pL0 2 dζds ≤ 3p−1MpCpL3(1/κ) p−1 ∫ t −∞ e−κ(t−s) [ ∫ s −∞ ∥R(s− ζ)∥2dζ ] p−2 2 ∫ s −∞ ∥R(s− ζ)∥2 × E∥Z1(ζ)− Z2(ζ)∥pdζds ≤ 3p−1MpCpL3(1/κ) p−1 ∫ t −∞ e−κ(t−s) [ ∫ s −∞ ∥R(s− ζ)∥2dζ ] p 2 dζds × sup ζ∈R E∥Z1(ζ)− Z2(ζ)∥p ≤ 3p−1MpCpL3(1/κ) p∥R∥pL2(0,+∞)∥Z1 − Z2∥p∞. Therefore, for each t ∈ R, we can deduce that E∥(ΓZ1)(t)− (ΓZ2)(t)∥p ≤ 3p−1Mp(1/κ)p [ L1 + L2∥Q∥pL1(0,+∞) + CpL3∥R∥pL2(0,+∞) ] ∥Z1 − Z1∥p∞. Hence, ∥ΓZ1 − ΓZ2∥p∞ ≤ 3p−1Θp∥Z1 − Z2∥p∞. For the case p = 2, by using the same arguments used to prove [27, Theorem 3.3], we obtain ∥ΓZ1 − ΓZ1∥2∞ ≤ 3Θ2∥Z1 − Z2∥2∞. Which implies that Γ is a contraction. Hence, by the Banach contraction principle, we can deduce that Γ has a unique fixed point Z∗ ∈ SPAP (R, Lp(P, E), µ1,2), which correspond to the unique p-mean (µ1, µ2)-s.p.a.p. mild solution on R of (1.1). This completes the proof. □ Finally, we investigate the stability of solution of (1.1) obtained in the previous Theorem. First, we recall the definition of stability. Definition 3.9. The unique p-mean (µ1, µ2)-s.p.a.p. mild solution Z∗(t) of (1.1) is said to be stable in p-mean sense, if for arbitrary ϵ > 0, there exists η > 0 such that E∥Z(t)− Z∗(t)∥p < ϵ, for all t ≥ 0, whenever E∥Z(0) − Z∗(0)∥p < η, where Z(t) stands for a solution of(1.1), with initial value Z(0). Theorem 3.10. Let µ1, µ2 ∈ N satisfy (A1), (A2). Suppose that (A3)–(A5) hold. Then (1.1) has a unique p-mean (µ1, µ2)-s.p.a.p. mild solution which is stable pro- vided that Θp := Mp(1/κ)p [ L1 + L2∥Q∥pL1(0,+∞) + CpL3∥R∥pL2(0,+∞) ] < (1/4)p−1, (3.6) for p > 2, and Θ2 := M2(1/κ)2 [ L1 + L2∥Q∥2L1(0,+∞) + L3∥R∥2L2(0,+∞) ] < 1/4, (3.7) for p = 2. 18 M. AYACHI, S. ABBAS EJDE-2024/24 Proof. Note that condition (3.7) (resp. (3.6)) implies condition (3.5) (resp. (3.4)). Then, from Theorem 3.8, we know that (1.1) has a unique p-mean (µ1, µ2)-s.p.a.p. mild solution Z∗, whose integral form is given by (3.1). Let Z(t) be an arbitrary solution of (1.1) with initial value Z(0). Then E∥Z(t)− Z∗(t)∥p = E ∥∥U(t, 0) [Z(0)− Z∗(0)] + ∫ t 0 U(t, s) [F1(s, Z(s))− F1(s, Z ∗(s))] ds + ∫ t 0 U(t, s) ∫ s 0 Q(s− ζ)[F2(ζ, Z(ζ))− F2(ζ, Z ∗(ζ))]dζds + ∫ t 0 U(t, s) ∫ s 0 R(s− ζ)[G(ζ, Z(ζ))−G(ζ, Z∗(ζ))]dW(ζ)ds ∥∥p. Now assume that p > 2. With the help of Hölder’s inequality, for any t ≥ 0, we have E∥Z(t)− Z∗(t)∥p ≤ 4p−1E∥U(t, 0) [Z(0)− Z∗(0)] ∥p + 4p−1E∥ ∫ t 0 U(t, s)[F1(s, Z(s))− F1(s, Z ∗(s))]ds∥p + 4p−1E∥ ∫ t 0 U(t, s) ∫ s 0 Q(s− ζ)[F2(ζ, Z(ζ))− F2(ζ, Z ∗(ζ))]dζds∥p + 4p−1E∥ ∫ t 0 U(t, s) ∫ s 0 R(s− ζ)[G(ζ, Z(ζ))−G(ζ, Z∗(ζ))]dW(ζ)ds∥p ≤ 4p−1∥U(t, 0)∥pE∥Z(0)− Z∗(0)∥p + 4p−1E [ ∫ t 0 ∥U(t, s)∥∥F1(s, Z(s))− F1(s, Z ∗(s))∥ds ]p + 4p−1E [ ∫ t 0 ∥U(t, s)∥ · ∥ ∫ s 0 Q(s− ζ)[F2(ζ, Z(ζ))− F2(ζ, Z ∗(ζ))]dζ∥ds ]p + 4p−1E [ ∫ t 0 ∥U(t, s)∥ · ∥ ∫ s 0 R(s− ζ)[G(ζ, Z(ζ))−G(ζ, Z∗(ζ))]dW(ζ)∥ds ]p ≤ 4p−1Mpe−κptE∥Z(0)− Z∗(0)∥p + 4p−1MpE [ ∫ t 0 e−κ(t−s)∥F1(s, Z(s))− F1(s, Z ∗(s))∥ds ]p + 4p−1MpE [ ∫ t 0 e−κ(t−s)∥ ∫ s 0 Q(s− ζ)(F2(ζ, Z(ζ))− F2(ζ, Z ∗(ζ)))dζ∥ds ]p + 4p−1MpE [ ∫ t 0 e−κ(t−s)∥ ∫ s 0 R(s− ζ)(G(ζ, Z(ζ))−G(ζ, Z∗(ζ)))dW(ζ)∥ds ]p ≤ 4p−1Mpe−κptE∥Z(0)− Z∗(0)∥p + 4p−1Mp [ ∫ t 0 e−κ(t−s)ds ]p−1 ∫ t 0 e−κ(t−s)E∥F1(s, Z(s))− F1(s, Z ∗(s))∥pds + 4p−1Mp [ ∫ t 0 e−κ(t−s)ds ]p−1 ∫ t 0 e−κ(t−s)E∥ ∫ s 0 Q(s− ζ)(F2(ζ, Z(ζ)) EJDE-2024/24 P-MEAN (µ1, µ2)-PSEUDO ALMOST PERIODIC PROCESSES 19 − F2(ζ, Z ∗(ζ)))dζ∥pds + 4p−1Mp [ ∫ t 0 e−κ(t−s)ds ]p−1 ∫ t 0 e−κ(t−s)E∥ ∫ s 0 R(s− ζ)(G(ζ, Z(ζ)) −G(ζ, Z∗(ζ)))dW(ζ)∥pds ≤ 4p−1MpE∥Z(0)− Z∗(0)∥p + 4p−1Θp sup t∈R E∥Z(t)− Z∗(t)∥p, which implies that sup t∈R E∥Z(t)−Z∗(t)∥p ≤ 4p−1MpE∥Z(0)−Z∗(0)∥p + 4p−1Θp sup t∈R E∥Z(t)−Z∗(t)∥p. That is sup t∈R E∥Z(t)− Z∗(t)∥p ≤ 4p−1Mp 1− 4p−1Θp E∥Z(0)− Z∗(0)∥p. So, for all t > 0, we obtain E∥Z(t)− Z∗(t)∥p ≤ 4p−1Mp 1− 4p−1Θp E∥Z(0)− Z∗(0)∥p. For ϵ > 0, choosing 0 < η < ϵ 1−4p−1Θp 4p−1Mp , we obtain E∥Z(0)− Z∗(0)∥p < η =⇒ E∥Z(t)− Z∗(t)∥p ≤ ϵ, for all t > 0, According to Definition 3.9, we can conclude that (1.1) has a unique p-mean (µ1, µ2)-s.p.a.p. mild solution which is stable in p-mean sense. Now assume that p = 2. By help of the Cauchy-Schwartz inequality, the Ito’s isometry and the Fubini’s theorem, we have E∥Z(t)− Z∗(t)∥2 ≤ 4E∥U(t, 0) [Z(0)− Z∗(0)] ∥2 + 4E∥ ∫ t 0 U(t, s) [F1(s, Z(s))− F1(s, Z ∗(s))] ds∥2 + 4E∥ ∫ t 0 U(t, s) ∫ s 0 Q(s− ζ) [F2(ζ, Z(ζ))− F2(ζ, Z ∗(ζ))] dζds∥2 + 4E∥ ∫ t 0 U(t, s) ∫ s 0 R(s− ζ) [G(ζ, Z(ζ))−G(ζ, Z∗(ζ))] dW(ζ)ds∥2 ≤ 4MpE∥Z(0)− Z∗(0)∥2 + 4Θ2 × sup t∈R E∥Z(t)− Z∗(t)∥2, Thus, for t > 0, we obtain E∥Z(t)− Z∗(t)∥2 ≤ 4M2 1− 4Θ2 E∥Z(0)− Z∗(0)∥2. For ϵ > 0, choosing 0 < η < ϵ 1−4Θ2 4M2 , we obtain E∥Z(0)− Z∗(0)∥2 < η =⇒ E∥Z(t)− Z∗(t)∥2 ≤ ϵ, for all t > 0. According to Definition 3.9, we can conclude that (1.1) has a unique square-mean (µ1, µ2)-s.p.a.p. mild solution which is stable in square-mean sense. This completes the proof. □ 20 M. AYACHI, S. ABBAS EJDE-2024/24 4. Examples of applications In this section, we provide some examples to illustrate the practical usefulness of of our results established in the preceding section. We can use W(t)dt = dB(t), where the derivative is taken in a stochastic sense. It is evident that periodicity is very idealistic situation, and several systems may not show exactly periodic behavior. To capture such behavior, other generalized functions may be used, such as almost periodic, pseudo-almost periodic, etc. This paper analyzes even a more general class. of functions that can capture the not-so usual dynamics of several systems Example 4.1. Let us consider the generalized stochastic equation with time vary- ing coefficient with Dirichlet boundary conditions, ∂ ∂t u(t, x) = −a(t) ∂2 ∂x2 u(t, x) + f(t, x) + b(t)u(t, x) + h dB(t) dt , u(t, 0) = u(t, π) = 0, u(0, x) = u0(x), x ∈ (0, π), t ∈ R+. (4.1) Let us define E = L2(0, π) and A(t)v = −a(t)Av, Av = v ′′ with D(A) = H2(0, π)∩ H1 0 (0, π). It is well known that A generates an analytic semigroup T (t) : t ≥ 0. Thus U(t, s) = e− ∫ t s a(ξ)dξT (t − s). As the semigroup T (t) is bounded, we obtain the condition ∥U(t, s)∥ ≤ Me−a(t−s). Let u(t, x) = u(t)x, after these setup, the above problem can be written in the abstract form du = (A(t)u+ F1(t, u))dt+ hdB(t) or du dt = (A(t)u+ F1(t, u)) + hW(t). We assume that the functions f , b and h are (µ1, µ2) s.p.a.p. In particular, we can consider a(t) = c1e −t + | sin t|, b(t) = c2 1 1 + t2 + c4(sin t+ sin √ 2t). We see that E∥F1(t, x)− F1(t, y)∥ ≤ E∥f(t, x)− f(t, y)∥+ |b(t)|E∥x− y∥. So under the assumption that f is Lipschitz, we obtain F1 Lipschitz, here L1 = Lf +(2c4+c2). Also k = (1+c1). Thus we can always choose constants c1, c2, c4, so that the required condition of Theorem holds. Hence the existence and uniqueness of (µ1, µ2)-s.p.a.p. solution is ensured. Example 4.2 (Stochastic logistic model with distributed delay). Let us consider the stochastic logistic model with distributed delay and time varying rate, dx(t) = ( x(t)(−a(t)− b(t)x(t)) + ∫ t −∞ k(t− s)x(s)ds ) dt+ σdB(t). (4.2) We assume that a(·), b(·) are (µ1, µ2)-s.p.a.p. and k(·) ∈ L1([0,∞)). In this case, we can see A(t) = a(t), F1(t, x(t)) = −b(t)x2(t), Q(t) = k(t), F2(t, x(t)) = x(t), R(t) = σ, G = 1. In this case, we can see that U(t, s) = e− ∫ t s a(ξ)dξ. To satisfies the required condition, we assume that a(t) is positive and bounded. Hence ∥U(t, s)∥ ≤ e−a(t−s) for some positive constant a. We choose a(t) = c1e −t + | sin t|, b(t) = c2 1 1 + t2 + c4(sin t+ sin √ 2t), k(t) = e−t. EJDE-2024/24 P-MEAN (µ1, µ2)-PSEUDO ALMOST PERIODIC PROCESSES 21 We can see in Figures 1 and 2 that the solution is (µ1, µ2)-s.p.a.p. One can see that fluctuations are less when drift coefficient σ is small. 0 1 2 3 4 5 6 7 8 9 10 t -2 0 2 4 6 8 10 12 x (t ) c1 = 2, c2 = 1, c4 = 1, σ = 3 0 1 2 3 4 5 6 7 8 9 10 t 0 0.2 0.4 0.6 0.8 1 1.2 x (t ) c1 = 2, c2 = 1, c4 = 1, σ = 0.2 Figure 1. Numerical solutions of (4.2) for different parameters. Example 4.3 (Stochastic cellular neural networks with distributed delays). Let us consider the system of stochastic differential equations with distributed delays dxi(t) = ( − ci(t)xi(t) + n∑ j=1 bij ∫ t −∞ kij(t− s)gj(xj(s))ds+ Ii ) dt+ σidB(t). (4.3) In model (4.3), n correspond to the number of neurons in the network. For i, j = 1, . . . , n; xi represent the ith neuron state, ci(t) represent the decay rate, 22 M. AYACHI, S. ABBAS EJDE-2024/24 0 1 2 3 4 5 6 7 8 9 10 t -6 -4 -2 0 2 4 6 8 x (t ) c1 = 0.2, c2 = 0.1, c4 = 1, σ = 3 0 1 2 3 4 5 6 7 8 9 10 t 0 0.5 1 1.5 x (t ) c1 = 0.2, c2 = 0.1, c4 = 1, σ = 0.2 Figure 2. Numerical solutions of (4.2) for different parameters. gj represent the activation function of the jth neuron, Ii represent the external input in the ith neuron, bij represent the connection weight and the kernel function kij ∈ L1([0,∞)). For more details on neural networks, we refer to [18, 34] and references therein. In this equation, we suppose that ci(·) are positive bounded function. Also, we choose ci(t) = aie −t + | sin t|, gj(xj(t)) = ( ai 1 1 + t2 + c4((sin t+ sin √ 2t) ) xj(t), and kij(t) = e−t. The corresponding plots for i = 1, 2 are depicted in Figures 3 and 4. One can clearly see that the graphs show (µ1, µ2)-s.p.a.p. behavior. In this case also, one can see that fluctuations are less when drift coefficients σ1, σ2 are small. 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Votsia; Pseudo almost periodic solutions of infinite class under the light of measure theory and applications, Mathematical Methods in the Applied Sciences, 44(6) (2021), 4580-4597. [39] C. Zhang; Almost periodic type functions and ergodicity, Science Press Beijing, BeijingKluwer Academic Publishers, Dordrecht, 2003. xii+355 pp. ISBN:7-03-010489-7. [40] H. Zhou, W. Jiang; Existence and stability of positive almost periodic solution for stochastic Lasota-Wazewska model, Journal of Applied Mathematics and Computing, 47(1) (2015), 61- 71. Moez Ayachi Laboratory of Mathematics and Applications (LR17ES11), Faculty of Sciences of Gabes, University of Gabes, Gabes 6072, Tunisia Email address: Moez.Ayachi@fsg.rnu.tn Syed Abbas (corresponding author) School of Mathematical and Statistical Sciences, Indian Institute of Technology Mandi, Mandi, H.P., 175005, India Email address: abbas@iitmandi.ac.in 1. Introduction 2. Preliminaries 2.1. Q-Brownian motion 2.2. P-mean almost periodic stochastic processes 2.3. P-mean (1,2)-pseudo almost periodic processes 3. Application to integro-differential stochastic evolution equations 4. Examples of applications Acknowledgements References