Electronic Journal of Differential Equations, Vol. 2024 (2024), No. 70, pp. 1–26. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: 10.58997/ejde.2024.70 PRACTICAL STABILITY OF STOCHASTIC DIFFERENTIAL DELAY EQUATIONS DRIVEN BY G-BROWNIAN MOTION WITH GENERAL DECAY RATE TOMÁS CARABALLO, FATEN EZZINE, MOHAMED ALI HAMMAMI Abstract. This article is concerned with the quasi sure practical stability of nonlinear stochastic differential delay equations driven by G-Brownian motion (G-SDDEs) with a general decay rate. Sufficient conditions are established by constructing appropriate G-Lyapunov functionals. Moreover, we provide some numerical examples to demonstrate the effectiveness of the obtained results. 1. Introduction Since Peng [24, 25] set up the G-expectation and G-Brownian motion, many papers have been published on stochastic calculus based upon G-Brownian motion, see [11, 17] and the references therein. On that basis, Gao [13] and Peng [24] studied the existence and uniqueness of solution to G-stochastic differential equations (G-SDE) under a standard Lips- chitz condition. Moreover, Lin [19] obtained the existence and uniqueness of so- lution to G-SDE with reflecting boundary. Later on, several authors have been working on stochastic differential equations driven by G-Brownian motion, see [1, 13, 18, 19, 20, 25]. Stochastic models under G-framework proved to be powerful to analyze interesting applications in many branches of problems with uncertainty, risk measures, the superhedging in finance, etc. Many applied problems are modeled by non-delay systems. These are governed by the assumption that the future evolution of the system is determined just by the present state, being independent of the past states. In reality, such an assumption can be considered only as a first approximation to the real system. A more realistic model assumes that the evolution of the future states depends not only on the cur- rent state but also on the past history. Delay differential equations (DDEs) (also called hereditary systems, systems with aftereffect, functional differential equations, retarded differential equations) provide an appropriate model for physical processes whose time evolution depends on their history. Stochastic differential delay equa- tions (SDDEs, in short) have been widely investigated over the last decades, see [14, 21, 23]. 2020 Mathematics Subject Classification. 93E03, 60H10. Key words and phrases. G-Stochastic delay systems; quasi sure practical stability, decay function, G-Brownian motion, G-Lyapunov functional, G-Itô formula. ©2024. This work is licensed under a CC BY 4.0 license. Submitted March 29, 2024. Published November 11, 2024. 1 2 T. CARABALLO, F. EZZINE, M. A. HAMMAMI EJDE-2024/70 Recently, several works have been published on stochastic differential delay equa- tions driven by G-Brownian motions (G-SDDEs, in short). Young et al. [27] proved the existence and uniqueness of solution for a class of G-SDDEs. The problem of stability of G-SDDEs is more complicated, and there have been published only in a few works, see [22, 27, 29]. When the origin is not a trivial solution, we investigate the stability of the SDEs with respect to a small neighborhood of the origin. Several results on the stability of the nontrivial solution of stochastic systems are proposed in [6, 8, 9]. In the investigation of the asymptotic behavior of solutions to SDEs, one can find that a solution is asymptotically stable but may not necessarily be exponentially stable. Further, in the nonlinear and/or nonautonomous situations, it may happen that the stability cannot always be exponential but can be sub- or super-exponential, see [2, 3, 7]. For this reason, the main aim of this paper is to discuss the quasi sure practical stability with a general decay rate of G-SDDEs. Lyapunov’s technique is available to state sufficient conditions for the stability of solutions to SDDEs by using the construction of some Lyapunov functions or functionals. The latter method provides better conditions than using Lyapunov functions, although the construction of Lyapunov functionals is more complicated. Different works tackled the problem of the construction of Lyapunov functionals for a wide range of equations containing some hereditary properties, see [4, 5, 28]. The general method of Lyapunov functionals construction was proposed by Kol- manovskii and Shaikhet [15, 16, 28]. This approach has already been successfully used for functional differential equations, for difference equations with discrete time, for difference equations with continuous time, etc. Recently, the concept of practical stability with general decay rate of stochastic differential delay equations was introduced by Caraballo et al. [10]. Our main objective in this paper is to extend the results in [10] to the case of G-Brownian motion. Using the method of Lyapunov functionals and recently developed Itô calculus for SDDE driven by G-Brownian motion, we introduce and develop the practical stability with a general decay rate of stochastic differential equations with constant and time-varying delay driven by G-Brownian motion. To the best of our knowledge, no work has been done on the practical stability for delayed stochastic differential equations driven by G-Brownian motion in the literature. Motivated by these considerations, in this paper we will investigate the practical convergence to a small ball centered at the origin with a general decay rate in terms of the existence and construction of G-Lyapunov functionals. Fur- thermore, we construct G-Lyapunov functionals for stochastic differential equations with constant and time-varying delay driven by G-Brownian motions, to obtain suf- ficient conditions ensuring the practical convergence to a small ball centered at the origin with a general decay rate. The arrangement of the paper is presented as follows. In Section 2, we establish some preliminaries on sublinear expectations and G-Brownian motions. In Section 3, we state sufficient conditions for quasi sure practical stability of the G-SDDEs with a general decay rate by using G-Lyapunov’s functionals. In Section 4, we analyze the quasi sure practical stability with a general decay rate of stochastic differential equations with constant and time-varying delay by constructing suitable G-Lyapunov functionals. Moreover, we exhibit some examples to illustrate the theoretical findings. Finally, some conclusions appear in Section 5. EJDE-2024/70 PRACTICAL STABILITY OF STOCHASTIC DELAY EQUATIONS 3 2. Preliminaries This section reviews the basic concepts and notation within the G-framework which are needed in our analysis. The reader interested in a more detailed descrip- tion of the notions are referred, for instance, to [24, 25, 26]. Notation on G-stochastic calculus. Rn : Space of n-dimensional real column vectors, ⟨x, y⟩ : Scalar product of two vectors x, y ∈ Rn, If x ∈ Rn, ∥x∥ denotes its Euclidean norm, Ωt := {ω·∧t : ω ∈ Ω}, Ft = B(Ωt), B(Ω) : Borel σ-algebra of Ω, Cb,Lip(Rn) : the space of all bounded real-valued Lipschitz continuous functions, L0(Ω) : Space of all B(Ω)-measurable real functions, L0(Ωt) : Space of all B(Ωt)-measurable real functions, Bb(Ω): all bounded elements in L0(Ω),Bb(Ωt) := Bb(Ω) ∩ L0(Ωt), Lp G(Ω) : Banach space under the natural norm ∥x∥p = Ê (|x|p)1/p, Mp,0 G ([0, T ]) = { ζ := ζt(ω) = N−1∑ i=0 ζj1[ti,ti+1)(t), ∀N > 0, 0 = t0 < · · · < tN = T, ζi ∈ Lp G(ωti), i = 0, 1, 2, . . . , N − 1 } , Mp G([0, T ]) : Completion of Mp,0 G under ∥η∥Mp G = ( ∫ T 0 Ê ( |η(t)|p ) dt )1/p . Let Ω be a given set and let H be a linear space of real valued functions defined on Ω. We suppose that H satisfies b ∈ H for each constant b and ∥Y ∥ ∈ H if Y ∈ H. Definition 2.1. [24] A sublinear expectation Ê on H is a functional Ê : H → R satisfying the following properties: for all Y,Z ∈ H, (i) Monotonicity: if Y ≥ Z, then Ê(Y ) ≥ Ê(Z). (ii) Constant preserving: Ê(b) = b for all b ∈ R. (iii) Sub-additivity: Ê(Y + Z) ≤ Ê(Y ) + Ê(Z). (iv) Positive homogeneity: Ê(αY ) = αÊ(Y ) for α ≥ 0. The triple (Ω,H, Ê) is called a sublinear expectation space. Y ∈ H is called a random variable in (Ω,H, Ê). Y = (Y1, . . . , Yn), where Yj ∈ H is called an n-dimensional random vector in (Ω,H, Ê). Definition 2.2. [24] Weakly compact sets are defined to be sets which are compact with respect to the weak topology of a Banach space. The representation of a sublinear expectation can be expressed as a supremum of linear expectations. Theorem 2.3 ([25]). There exists a weakly compact family P of probability mea- sures defined on (Ω,B(Ω)), such that Ê(Y ) = sup p∈P Ep(Y ), Y ∈ L1 G(Ω). Definition 2.4 ([24]). In a sublinear expectation space (Ω,H, Ê), an n-dimensional random vector Z = (Z1, . . . , Zn) ∈ H is said to be independent from an m- dimensional random vector Y = (Y1, . . . , Ym) ∈ H under the sublinear expectation 4 T. CARABALLO, F. EZZINE, M. A. HAMMAMI EJDE-2024/70 Ê, if for any test function φ ∈ Cb,Lip(Rm+n) Ê(φ(Z, Y )) = Ê ( Ê (φ(z, Y )) |z=Z ) . Definition 2.5 ([24]). Let Y1 and Y2 be two n-dimensional random vectors defined on sublinear expectation spaces (Ω1,H1, Ê1) and (Ω2,H2, Ê2), respectively. They are called identically distributed, denoted by Y1 d = Y2, if Ê1(ψ(Y1)) = Ê2(ψ(Y2)), ∀ψ ∈ Cb,Lip(Rn). Ȳ is said to be an independent copy of Y , if Ȳ d = Y and Ȳ is independent from Y . Definition 2.6 ([24]). A random variable Y on a sublinear expectation space (Ω,H, Ê) is called G-normal distributed, denoted by Y ∼ N ( 0, [σ2, σ̄2] ) for a given pair 0 ≤ σ̄ ≤ σ, if for any c, d ≥ 0, cY + dỸ d = √ c2 + d2Y, where Ỹ is an independent copy of Y . Let Ω be the space of Rd-valued continuous paths (ωt)t≥0 with ω0 = 0. Further, we assume that Ω is a metric space equipped with the distance ϱ(ω1, ω2) := ∞∑ N=1 2−N ( max 0≤t≤N (∥ω1 t − ω2 t ∥) ∧ 1 ) , and consider the canonical process Bt(ω) = ωt, t ∈ [0,∞) for ω ∈ Ω; then for each fixed T ∈ [0,∞), we have L0 ip(ΩT ) := {ψ (Bt1 ,Bt2 , . . . ,Btn) : n ≥ 1, 0 ≤ t1 ≤ · · · ≤ tn ≤ T, ψ ∈ Cb,lip(Rd×n)}. Definition 2.7 ([24]). On the sublinear expectation space (Ω, L0 ip(ΩT ), Ê), the canonical process (Bt)t≥0 is called a G-Brownian motion, if the ensuing properties are satisfied: (i) B0 = 0; (ii) for t, s ≥ 0, the increment Bt+s − Bt d = √ sY , where Y is G-normal dis- tributed; (iii) for t, s ≥ 0, the increment Bt+s−Bt is independent from (Bt1 ,Bt2 , . . . ,Btn) for each n ∈ N, and 0 ≤ t1 ≤ t2 ≤ · · · ≤ tn ≤ t. Moreover, the sublinear expectation Ê(·) is called G-expectation. For σ̄2 = σ2 = 1, (Bt)t≥0 is the classical Brownian motion. For simplicity, let (Bt)t≥0 be a 1-dimensional G-Brownian motion. The letter G denotes the function G(b) := 1 2 Ê(bB2 1) = 1 2 (σ2b+ − σ2b−), b ∈ R, with σ2 := −Ê(−B2 1) ≤ Ê(B2 1) := σ2, 0 ≤ σ ≤ σ < ∞. Recall that b+ = max{0, b} and b− = −min{0, b}. Definition 2.8 ([24]). Let πN t , N = 1, 2, . . . , be a sequence of partitions of [0, t], (Bt)t≥0 be an n-dimensional G-Brownian motion. For each fixed b ∈ Rn, (Bb t)t≥0 EJDE-2024/70 PRACTICAL STABILITY OF STOCHASTIC DELAY EQUATIONS 5 is a 1-dimensional G-Brownian motion, we define ⟨Bb⟩t := ⟨b,Bt⟩ = lim µ(πN t )→0 N−1∑ i=0 (Bb tNj+1 − Bb tNj )2 = (Bb t) 2 − 2 ∫ t 0 Bb sdB b s. ⟨Bb⟩ is called the quadratic variation process of G-Brownian motion. Let b̄ ∈ Rn, we define the mutual variation process by ⟨Bb,Bb̄⟩t := 1 4 ( ⟨Bb +Bb̄⟩t − ⟨Bb − Bb̄⟩t ) = 1 4 ( ⟨Bb+b̄⟩t − ⟨Bb−b̄⟩t ) . Proposition 2.9 ([24]). Let (Bt)t≥0 be an n-dimensional G-Brownian motion on a sublinear expectation space (Ω,H, Ê). Then, (Bb t)t≥0 is a 1-dimensional G- Brownian motion for each b ∈ Rn, where Gb(β) = 1 2 ( σ2 bbT β + − σ2 −bbT β −) , σ2 bbT = 2G(bbT ) = Ê ( ⟨b,B1⟩2 ) , σ2 −bbT = −2G(−bbT ) = −Ê ( −⟨b,B1⟩2 ) . In particular, for each t, s ≥ 0, Bb t+s − Bb t d = N ( 0, [sσ2 −bbT , sσ 2 bbT ] ) . Definition 2.10 ([26]). For p ≥ 1 and T ∈ R+ fixed, we consider the type of simple processes, Mb,0([0, T ]) = { η := ηt(ω) = N−1∑ i=0 ξi1[ti,ti+1)(t), ∀N > 0, 0 = t0 < · · · < tN = T, ξi ∈ Bb(Ωti), i = 0, 1, 2, . . . , N − 1 } . For each p ≥ 1, we denote by Mp ⋆ ([0, T ]) the completion of Mb,0([0, T ]) under the norm: ∥η∥Mp([0,T ]) = ( Ê (∫ T 0 ∥ηt∥pdt ))1/p . Now, we introduce the natural Choquet capacity. Definition 2.11 ([24]). Let B(Ω) the Borel σ-algebra and P be a weakly compact collection of probability measures P defined on (Ω,B(Ω)), then the capacity Ĉ(·) associated to P is defined as follows: Ĉ(A) := sup P∈P P(A), A ∈ B(Ω). Definition 2.12 ([24]). A set A ⊂ B(Ω) is polar, if Ĉ(A) = 0. A property holds “quasi-surely” (q.s.), if it holds outside a polar set. Next we recall the following Borel-Cantelli lemma in the G-framework. Lemma 2.13 ([11]). Let {Ak} ⊂ B(Ω), such that ∞∑ k=1 Ĉ(Ak) <∞. Then, lim supk→∞ Ak is polar. 6 T. CARABALLO, F. EZZINE, M. A. HAMMAMI EJDE-2024/70 Lemma 2.14 ([30]). Let Bt be a one-dimensional G-Brownian motion, we suppose that there exist constants ϵ > 0 and ν > 0, such that Ê ( exp (ν2 2 (1 + ϵ) ∫ T 0 f2(s)d⟨B⟩s )) <∞. Then, for any T > 0 and η > 0, Ĉ ( sup 0≤t≤T (∫ t 0 f(s)dBs − ν 2 ∫ t 0 f2(s)d⟨B⟩s ) > η ) ≤ exp(−νη). 3. Practical stability of stochastic delay equation driven by G-Brownian motion Let τ > 0 and C([−τ, 0],Rn) denote the family of all continuous Rn-valued function φ defined on [−τ, 0] with the norm ∥φ∥ = sup−τ≤θ≤0 ∥φ(θ)∥. If x(t) is a continuous Rn-valued stochastic process on [−τ,∞), for every t ≥ 0 we define xt : [−τ, 0] → Rn by xt(θ) = x(t+θ),−τ ≤ θ ≤ 0, which is considered as C([−τ, 0],Rn)- valued stochastic process. Now, we consider the nonlinear stochastic differential delay equations driven by a G-Brownian motion in the form dx(t) = f(t, xt)dt+ h(t, xt)d⟨B⟩t + g(t, xt)dBt, t ≥ t0, (3.1) where Bt is a one-dimensional G-Brownian motion, with Bt ∼ N (0, [σ2t, σ̄2t]), and (⟨B⟩)t≥0 is the quadratic variation process of the G-Brownian, and f : [t0,∞) × C([−τ, 0],Rn) → Rn, g : [t0,∞)×C([−τ, 0],Rn) → Rn, h : [t0,∞)×C([−τ, 0],Rn) → Rn satisfy appropriate assumptions described below. To solve equation (3.1), we need to know an initial datum, so we assume that it is given as follows xt0 = ξ( in other words xt0(θ) = x(t0 + θ) = ξ(θ),−τ ≤ θ ≤ 0), (3.2) where ξ is a C([−τ, 0],Rn)-valued random variable. For the well-posedness of system (3.1), we impose the following hypotheses. (1) Linear growth condition: There exists a positive constant K1, such that for all φ ∈ C([−τ, 0],Rn), and all t ∈ [t0, T ], |f(t, φ)|2 + |h(t, φ)|2 + |g(t, φ)|2 ≤ K1(1 + |φ|2). (2) Lipschitz condition: There exists a positive constant K2, such that for all φ, φ̃ ∈ C([−τ, 0],Rn), and for all t ∈ [t0, T ], ∥f(t, φ)− f(t, φ̃)∥2 + ∥h(t, φ)− h(t, φ̃)∥2 + ∥g(t, φ)− g(t, φ̃)∥2 ≤ K2∥φ− φ̃∥2. Then, under these assumptions, the G-SDDE (3.1) with initial value (3.2) has a unique solution x(t), see [27] for details. The solution x(t) of (3.1) with initial value (3.2) satisfies the integral equation x(t) = ξ(0) + ∫ t t0 f(s, xs)ds+ ∫ t t0 h(s, xs)d⟨B⟩s + ∫ t t0 g(s, xs)dBs, q.s., x(t) = ξ(t− t0), t ∈ [t0 − τ, t0]. To calculate the stochastic differential of the process ϑ(t) = v(t, x(t)), where x(t) is a solution of the G-SDDE (3.1) and v : [0,∞)×Rn → R+, we define an operator L (called G-Lyapunov function) as Lv(t, x(t)) := vt(t, x(t)) + vxf(t, xt) EJDE-2024/70 PRACTICAL STABILITY OF STOCHASTIC DELAY EQUATIONS 7 +G ( ⟨vx(t, x(t)), 2h(t, xt)⟩+ ⟨vxx(t, x(t))g(t, xt), g(t, xt)⟩ ) , where vt(t, x) = ∂v ∂t (t, x) vx(t, x) = ( ∂v ∂x1 (t, x), . . . , ∂v ∂xn (t, x)); vxx(t, x) = ( ∂2v ∂xi∂xj (t, x) ) n×n . The G-Lyapunov function L can be implemented too for some functionals V (·, ·) : [0,∞)×C([−τ, 0],Rn) → R+. We assume that a functional V (t, φ) can be described in the form V (t, φ(0), φ(θ)), θ < 0, and for φ = xt, we put Vφ(t, x) = V (t, φ) = V (t, xt) = V (t, x, x(t+ θ)), θ < 0, x = φ(0) = x(t). (3.3) Let D represent the set of functionals for which the function Vφ(t, x), defined by (3.3), has a continuous derivative with respect to t and two continuous derivatives with respect to xi, i = 1, . . . , n. For functionals from D, the operator L of the G-SDDE (3.1) has the form LV (t, xt) = Vφt(t, x(t)) + Vφx(t, x(t))f(t, xt) +G ( ⟨Vφx(t, x(t)), 2h(t, xt)⟩ + ⟨Vφxx(t, x(t))g(t, xt), g(t, xt)⟩ ) . From the G-Itô formula it follows that for a functional V from D, dV (t, xt) = LV (t, xt)dt+ Vφx(t, x(t))g(t, xt)dBt. We assume that there exits t ∈ R+, such that f(t, 0) ̸= 0 or h(t, 0) ̸= 0 or g(t, 0) ̸= 0, i.e., the G-stochastic differential delay equation (3.1) does not have the trivial solution x ≡ 0. Now, we state the definition of practical exponential stability of a stochastic delay equation driven by G-Brownian motion (3.1) when the origin is no longer an equilibrium point. In this case we study the stability of solutions with respect to a small neighborhood of the origin. The study of the asymptotic behavior of solutions leads to investigate the sta- bility behavior of a small ball centered at the origin, Br := {x ∈ Rn : ∥x∥ ≤ r}, r > 0. Definition 3.1. (i) The ball Br := {x ∈ Rn : ∥x∥ ≤ r}, r > 0 is said to be quasi surely globally uniformly exponentially stable, if for each initial data ξ ∈ C([−τ, 0],Rn), such that 0 < ∥x(t, t0, ξ)∥ − r, for all t ≥ 0, lim sup t→∞ 1 t ln(∥x(t, t0, ξ)∥ − r) < 0, q.s. (ii) System (3.1) is said to be quasi surely practically uniformly exponentially stable, if there exists r > 0 such that Br is quasi surely uniformly exponentially stable. Next, we state the definition of practical convergence to the ball Br with a general decay function λ(t). 8 T. CARABALLO, F. EZZINE, M. A. HAMMAMI EJDE-2024/70 Definition 3.2. Let λ(·) be a positive function defined for sufficiently large t > 0, such that λ(t) → ∞ as t → ∞. A solution x(·) to system (3.1) is said to decay to the ball Br quasi surely with decay function λ(t) and order at least γ > 0, if its generalized Lyapunov exponent is less than or equal to −γ, i.e., lim sup t→∞ ln(∥x(t, t0, ξ)∥ − r) lnλ(t) ≤ −γ, q.s. If in addition, 0 is a solution to system (3.1), the zero solution is said to be quasi surely practically asymptotically stable with decay function λ(t) and order at least γ, if every solution to system (3.1) tends to the ball Br quasi surely with decay function λ(t) and order at least γ, for all r > 0 sufficiently small. Replacing the decay function λ(t) by O(exp(t)) in the above definition leads to the quasi sure practical exponential stability. Our aim now is to study the practical stability of stochastic differential delay equations driven by G-Brownian motion with a general decay rate based upon the method of G-Lyapunov functionals. Theorem 3.3. Let V : R+×C([−τ, 0],Rn) → R+ be a functional from D. Assume that lnλ(t) is uniformly continuous on t ≥ 0 and there exists a constant δ ≥ 0, such that lim t→∞ ln ln t lnλ(t) ≤ δ. Let x(·) = x(·, 0, ξ) be a solution to (3.1) and assume that there exist constants q ∈ N⋆, m ≥ 0, b1 ≥ 0, b2 ∈ R, a non-increasing function φ1(t) > 0 and a continuous non-negative function φ2(t), such that for all t ≥ t0 ≥ 0, the following inequalities hold: (H1) λm(t)∥x(t)∥q ≤ V (t, xt). (H2) ∫ t t0 LV (s, xs)ds+ σ̄2 ∫ t t0 φ1(s)∥Vx(s, xs)g(s, xs)∥2ds, ≤ ∫ t t0 φ2(s)λ m(s)∥x(s)∥qds+ r(t), where r(·) is a continuous non-negative function. (H3) lim t→∞ sup ∫ t t0 φ2(s)ds lnλ(t) ≤ b2, lim t→∞ inf lnφ1(t) lnλ(t) ≥ −b1, lim t→∞ r(t) λm(t) = r̃ > 0. (H4) The solution x(t, t0, ξ) satisfies ∥x(t, t0, ξ)∥ > ( r(t) λm(t) )1/q , ∀t ≥ t0. Then lim sup t→∞ ln ( ∥x(t, t0, ξ)∥ − ( r(t) λm(t) )1/q) lnλ(t) ≤ − ( m− (b1 + (b2 + δ) ∨m) ) , q.s. Proof. Notice that λm(t)∥x(t)∥q − r(t) = λm(t) ( ∥x(t)∥q − r(t) λm(t) ) EJDE-2024/70 PRACTICAL STABILITY OF STOCHASTIC DELAY EQUATIONS 9 = λm(t) ( ∥x(t)∥q − (( r(t) λm(t) )1/q)q) . From the inequality a1 q − a2 q = (a1 − a2) ( a1 q−1 + a1 q−2a2 + a1 q−3a2 2 + · · ·+ a1 0a2 q−1 ) , it follows that λm(t)∥x(t)∥q − r(t) = λm(t) ( ∥x(t)∥q − (( r(t) λm(t) )1/q)q) = λm(t) ( ∥x(t)∥ − ( r(t) λm(t) )1/q)( ∥x(t)∥q−1 + ∥x(t)∥q−2 ( r(t) λm(t) )1/q + · · ·+ ( r(t) λm(t) ) q−1 q ) = λm(t) ( ∥x(t)∥ − ( r(t) λm(t) )1/q) q∑ k=1 ∥x(t)∥q−k ( r(t) λm(t) ) k−1 q . Since limt→∞ r(t) λm(t) = r̃ > 0, it follows that for 0 < r̃0 < r̃, there exits T̃ ≥ t0, such that r(t) λm(t) ≥ r̃0 for all t ≥ T̃ . As we are assuming that ∥x(t)∥ > ( r(t) λm(t) )1/q , for all t ≥ 0, we obtain q∑ k=1 ∥x(t)∥q−k ( r(t) λm(t) ) k−1 q = ∥x(t)∥q−1 + ∥x(t)∥q−2 ( r(t) λm(t) )1/q + · · ·+ ( r(t) λm(t) ) q−1 q ≥ r̃∗ = q (r̃0) (q−1)/q , ∀t ≥ T̃ ≥ t0. Hence, we see that λm(t)∥x(t)∥q − r(t) ≥ λm(t) ( ∥x(t)∥ − ( r(t) λm(t) )1/q) r̃∗, ∀t ≥ T̃ ≥ t0. This yields V (t, xt) ≥ λm(t)∥x(t)∥q ≥ λm(t)∥x(t)∥q − r(t) ≥ λm(t) ( ∥x(t)∥ − ( r(t) λm(t) )1/q) r̃∗. That is, r̃∗λm(t) ( ∥x(t)∥ − ( r(t) λm(t) )1/q) ≤ V (t, xt). Therefore, ln(r̃∗) +m lnλ(t) + ln ( ∥x(t)∥ − ( r(t) λm(t) )1/q) ≤ ln (V (t, xt)) , ∀t ≥ T̃ ≥ t0. Invoking the G-Itô formula, it follows that V (t, xt) = V (0, x0) + ∫ t t0 LV (s, xs)ds+ ∫ t t0 Vx(s, xs)g(s, xs)dBs. (3.4) 10 T. CARABALLO, F. EZZINE, M. A. HAMMAMI EJDE-2024/70 By using that lnλ(t) is uniformly continuous on t ≥ 0, we can obtain that, for each ε > 0, there exist two positive integers N = N(ε) and K1(ε), such that if K−1 2N ≤ t ≤ K 2N and K ≥ K1(ε), then | lnλ ( K 2N ) − lnλ(t)| ≤ ε. Thanks to Lemma 2.14, we deduce that Ĉ { ω : sup t0≤t≤ω ( M(t)− ν 2 ∫ t t0 ∥Vx(s, xs)g(s, xs)∥2d⟨B⟩s ) > η } ≤ exp(−νη), for any positive constants α, β and ω, with M(t) = ∫ t t0 Vx(s, xs)g(s, xs)dBs. For ε > 0, we set ν = 2φ1 ( K 2N ) , η = φ1 ( K 2N )−1 ln K − 1 2N , ω = K 2N , K = 2, 3, . . . . Applying the well-known Borel-Cantelli lemma (Lemma 2.13) for capacity, we can conclude that, for almost all ω ∈ Ω, there exists an integer K0 = K(ε, ω) > 0, such that M(t) ≤ φ1 ( K 2N )−1 ln K − 1 2N + φ1 ( K 2N ) ∫ t t0 ∥Vx(s, xs))g(s, xs)∥2d⟨B⟩s ≤ φ1 ( K 2N )−1 ln K − 1 2N + ∫ t t0 φ1(s)∥Vx(s, xs)g(s, xs)∥2d⟨B⟩s , for t0 ≤ t ≤ K 2N and K ≥ K0(ε, ω). Substituting the above inequality into (3.4), we obtain V (t, xt) ≤ V (0, x0) + φ1 ( K 2N )−1 ln K − 1 2N + ∫ t t0 LV (s, xs)ds + ∫ t t0 φ1(s)∥Vs(s, xs)g(s, xs)∥2d⟨B⟩s, for t0 ≤ t ≤ K 2N and K ≤ K0(ε, ω). From Peng [24, Chapter III], we have that for each 0 ≤ s ≤ t ≤ T , σ2(t− s) ≤ ⟨B⟩t − ⟨B⟩s ≤ σ̄2(t− s). Based on this fact, we deduce that V (t, xt) ≤ V (0, x0) + φ1 ( K 2N )−1 ln K − 1 2N + ∫ t t0 LV (s, xs)ds + σ̄2 ∫ t t0 φ1(s)∥Vs(s, xs)g(s, xs)∥2ds, for t0 ≤ t ≤ K/2N and K ≤ K0(ε, ω). It follows from conditions (H1) and (H2), that V (t, xt) ≤ V (0, x0) + φ1 ( K 2N )−1 ln K − 1 2N + r(t) + ∫ t t0 φ2(s)λ m(s)∥x(s)∥qds ≤ V (0, x0) + φ1 ( K 2N )−1 ln K − 1 2N + r(t) + ∫ t t0 φ2(s)V (s, xs)ds, EJDE-2024/70 PRACTICAL STABILITY OF STOCHASTIC DELAY EQUATIONS 11 for t0 ≤ t ≤ K 2N and K ≥ K0(ε, ω). Using Gronwall’s Lemma [12], we derive V (t, xt) ≤ ( V (0, x0) + φ1 ( K 2N )−1 ln K − 1 2N + r(t) ) exp (∫ t t0 φ2(s)ds ) . From (H3) we have that for any ε > 0, lim t→∞ sup ∫ t t0 φ2(s)ds lnλ(t) < b2 + ε, and limt→∞ inf lnφ1(t) lnλ(t) > −b1−ε. Thanks to the uniform continuity of lnλ(t), there exists a positive integer K1(ε), such that whenever t ≥ K1(ε), we have∫ t t0 φ2(s)ds ≤ (b2 + ε) lnλ(t), φ1 (K − 1 2N )−1 ≤ φ1(t) ≤ λ(t)b1+ε, for K−1 2N ≤ t ≤ K 2N and K ≥ K1(ε). Also observe that ln k − 1 2N ≤ ln t ≤ ln k 2N for k − 1 2N ≤ t ≤ k 2N . Therefore, for almost all ω ∈ Ω, we obtain lnV (t, xt) ≤ ln ( V (0, x0) + λ(t)b1+δ+2ε + r(t) ) + (b2 + ε) lnλ(t), for K−1 2N ≤ t ≤ K 2N and K ≥ K1(ε). Thus, we conclude that lim t→∞ sup lnV (t, xt) lnλ(t) ≤ (b1 + δ + 2ε) ∨m+ (b2 + ε) , q.s. Recall that for t ≥ T̃ ≥ t0 and q ∈ N⋆, we have ln ( ∥x(t)∥ − ( r(t) λm(t) )1/q) ≤ ln(V (t, xt))−m lnλ(t)− ln(r̃∗). Letting ε→ 0, lim t→∞ sup ln ( ∥x(t)∥ − ( ρ(t) λm(t) )1/q) lnλ(t) ≤ −(m− (b2 + (b1 + δ) ∨m)), q.s., as required. □ Next, we will infer the practical convergence toward the ball Br with a gen- eral decay rate of our stochastic differential delay equations driven by G-Brownian motion. Corollary 3.4. Let V : R+×C([−τ, 0],Rn) → R+ be a functional from D. Assume that lnλ(t) is uniformly continuous on t ≥ 0, and there exists a constant δ ≥ 0, such that lim t→∞ ln ln t lnλ(t) ≤ δ. Let x(·) = x(·, 0, ξ) be a solution to system (3.1) and assume that there exist con- stants q ∈ N⋆, m ≥ 0, b1 ≥ 0, b2 ∈ R, a non-increasing function φ1(t) > 0 and a continuous non-negative function φ2(t), such that, for all t ≥ t0 ≥ 0, and for any solution x(·) to Eq.(3.1), defined in the future, assumptions (H1)− (H3) hold, and the following assumption is also satisfied (H4’) There exists r̃′ > r̃ > 0, such that the solution x(t, t0, ξ) satisfies ∥x(t, t0, ξ)∥ > ( r̃′ )1/q , ∀t ≥ t0. 12 T. CARABALLO, F. EZZINE, M. A. HAMMAMI EJDE-2024/70 Then lim sup t→∞ ln ( ∥x(t, t0, ξ)∥ − ( r̃′ )1/q) lnλ(t) ≤ −γ, q.s., where γ = m− (b2 + (b1 + δ) ∨m). In particular, if m > (b2 + (b1 + δ) ∨m), the solution to system (3.1) tends to the ball Br, with r = ( r̃′ )1/q quasi surely with decay function λ(t) and order at least γ. Remark 3.5. Notice that the condition m > b2 + (b1 + δ) ∨ m (or equivalently γ > 0) in the corollary holds in the next cases: • If b1 + δ ≤ m, then the condition becomes m > b2 + m. Therefore, this needs b2 < 0. • If b1 + δ > m, then the condition turns into m > b2 + b1 + δ which also needs b2 < 0. As a conclusion, to ensure that γ is positive requires that b2 < 0, and this implies that when b1 + δ ≤ m, then γ > 0, and when b2 + δ > m, then b2 must be smaller than m− b1 − δ. Proof of Corollary 3.4. By Theorem 3.3, it follows that lim sup t→∞ ln ( ∥x(t)∥ − ( r(t) λm(t) )1/q) lnλ(t) ≤ −γ, q.s. Since, we have limt→∞ r(t) λm(t) = r̃ < r̃′, there exists T̃ ≥ t0 such that r(t) λm(t) ≤ r̃′, for all t ≥ T̃ ≥ t0. Consequently, lim sup t→∞ ln ( ∥x(t)∥ − ( r̃′ )1/q) lnλ(t) ≤ lim sup t→∞ ln ( ∥x(t)∥ − ( r(t) λm(t) )1/q) lnλ(t) ≤ −γ, q.s., where γ = m − (b2 + (b1 + δ) ∨m). Hence, if m > b2 + (b1 + δ) ∨ m, then the solution to system (3.1) tends to the ball Br, with r = ( r̃′ )1/q quasi surely with decay function λ(t) and order at least γ. □ We analyze the following example to show how the previous theorem can be implemented. Example 3.6. Consider the following one-dimensional stochastic differential delay equation with constant time delay driven by G-Brownian motion. dx(t) = − β + 1 2(1 + t) x(t)dt+ 1 1 + t x(t− τ)d⟨B⟩t + (1 + t)− 1 2 dBt, t ≥ 0, x(t) = ξ(t), t ∈ [−τ, 0], (3.5) where β ∈ R+, Bt is a one-dimensional G-Brownian motion with Bt ∼ N (0, [ 13 , 1 2 ]) and τ is a positive constant. For Φ ∈ C([−τ, 0],R) and t ≥ 0, we define f(t,Φ) = − β + 1 2(1 + t) Φ(0), g(t,Φ) = (1 + t)− 1 2 , h(t,Φ) = 1 1 + t Φ(−τ) . EJDE-2024/70 PRACTICAL STABILITY OF STOCHASTIC DELAY EQUATIONS 13 Now, we aim at investigating the practical stability with a general decay rate of system (3.5) by using a G-Lyapunov functional. Consider the functional V (t, xt) := (1 + t)∥x(t)∥2 + 1 4 ∫ t t−τ ∥x(u)∥2du. Then, it is easy to check that for arbitrary α > 1 and φ1(t) = β 4(1+t)α , we obtain∫ t 0 LV (s, xs)ds+ ∫ t 0 β 16(1 + s)α ∥Vx(s, xs)g(s, xs)∥2ds ≤ ∫ t 0 ∥x(s)∥2ds+ ∫ t 0 −(β + 1)∥x(s)∥2ds+ 2 ∫ t 0 ∥x(s)∥∥x(s− τ)∥d⟨B⟩s + ∫ t 0 d⟨B⟩s + ∫ t 0 ∥x(s)∥2ds− ∫ t 0 ∥x(s− τ)∥2ds+ ∫ t 0 β 4(1 + s)α−2 ∥x(s)∥2ds ≤ ∫ t 0 ∥x(s)∥2ds+ ∫ t 0 −(β + 1)∥x(s)∥2ds+ 2 ∫ t 0 1 4 ∥x(s)∥∥x(s− τ)∥ds+ ∫ t 0 1 4 ds + ∫ t 0 ∥x(s)∥2ds− ∫ t 0 ∥x(s− τ)∥2ds+ ∫ t 0 β 4(1 + s)α−2 ∥x(s)∥2ds ≤ ∫ t 0 ∥x(s)∥2ds+ ∫ t 0 −(β + 1)∥x(s)∥2ds+ ∫ t 0 1 4 ∥x(s)∥2ds + ∫ t 0 1 4 ∥x(s− τ)∥2ds+ ∫ t 0 1 4 ds + ∫ t 0 ∥x(s)∥2ds− ∫ t 0 ∥x(s− τ)∥2ds+ ∫ t 0 β 4(1 + s)α−2 ∥x(s)∥2ds. That is, ∫ t 0 LV (s, xs)ds+ ∫ t 0 β 16(1 + s)α ∥Vx(s, xs)g(s, xs)∥2ds, ≤ 1 4 t+ ∫ t 0 ( 1 2 − β 1 + s + β (1 + s)α−1 ) (1 + s)∥x(s)∥2ds. Hence, we see that φ2(t) = β (1 + t)α−1 + 1 2 − β 1 + t , r(t) = 1 4 t. Taking λ(t) = (1 + t) and doing easy computations, we can check that δ = 0, b1 = α, b2 = 1 2 − β, r̃ = 1 4 , m = 1. Finally, using Corollary 3.4 we deduce that lim t→∞ sup ln ( ∥x(t)∥ − 1 4 ) ln(1 + t) ≤ −γ, q.s., where γ = β−α+ 1 2 . Hence, the solution to system (3.5) tends to the ball Br quasi surely with decay function λ(t) = (1 + t), r = 1 4 and order at least γ whenever β > α− 1 2 . 14 T. CARABALLO, F. EZZINE, M. A. HAMMAMI EJDE-2024/70 4. Practical stability In this section we construct G-Lyapunov functionals for practical stability of stochastic delay differential equations driven by G-Brownian motion. Corollary 3.4 shows that the quasi sure practical stability with a general de- cay rate of G-SDDEs (3.1) can be reduced to the construction of appropriate G-Lyapunov functionals. In the following, we propose a procedure to construct G-Lyapunov functionals for G-SDDEs, which consists of four steps. Step 1: Let us represent (3.1) in the form dz(t, xt) = (f1(t, x(t)) + f2(t, xt)) dt+ (h1(t, x(t)) + h2(t, xt)) d⟨B⟩t + (g1(t, x(t)) + g2(t, xt)) dBt, (4.1) where z(t, xt) is some functional of xt, the functions f1(t, x(t)), h1(t, x(t)) and g1(t, x(t)), depend on t and x(t) only and do not depend on the previous val- ues x(t + θ), θ < 0, of the solution. Assume that there exists t ∈ R+, such that f1(t, ·) ̸= 0 or h1(t, ·) ̸= 0 or g1(t, ·) ̸= 0. Step 2: Consider the auxiliary differential equation without memory dy(t) = f1(t, y(t))dt+ h1(t, y(t))d⟨B⟩t + g1(t, y(t))dBt. (4.2) Assume that (4.2) is quasi sure practical stable with a general decay rate and there exists a G-Lyapunov function v(t, y(t)), which satisfies the conditions of Corollary 3.4. Step 3: A G-Lyapunov functional V (t, xt) for (3.1) is constructed in the form V = V1 + V2, where V1(t, xt) = v(t, z(t, xt)). Here the argument y of the function v(t, y) is replaced on the functional z(t, xt) from the left-hand side of (4.1). Step 4: Usually, the functional V1(t, xt) almost fulfills the conditions of Corollary 3.4. To fully satisfy these conditions, it is necessary to calculate LV1(t, xt) and estimate it. Then, we choose the additional functional V2(t, xt) in a standard way. The representation (4.1) is not unique. This fact allows, using different repre- sentations of the type of (4.1) or different ways to estimate LV1(t, xt), to construct different G-Lyapunov functionals and, as a result, to obtain different sufficient con- ditions for the practical stability with general decay rate. The above procedure is a general method of Lyapunov functionals construction, which was proposed by Kolmanovskii and Shaikhet [15, 16, 28], and it has already been successfully used for functional differential equations, for difference equations with discrete time, for difference equations with continuous time. This method is used here for stochastic differential equations with delay driven by G-Brownian motion. Our interest now is to investigate the quasi sure practical stability with a general decay rate of stochastic differential equations with a constant and time- varying delay driven by G-Brownian motion exploiting the method of Lyapunov functionals construction. Now we construct G-Lyapunov functionals for stochastic differential equations with constant delay driven by G-Brownian motion. Consider the following stochas- tic differential equation with constant delay driven by G-Brownian motion, dx(t) = (F (t, x(t)) + f(t, x(t), x(t− τ1))) dt + h(t, x(t), x(t− τ2))d⟨B⟩t + g(t, x(t), x(t− τ3))dBt, x(t) = ξ(t− t0), t ∈ (t0 − τ̃ , t0), (4.3) EJDE-2024/70 PRACTICAL STABILITY OF STOCHASTIC DELAY EQUATIONS 15 where τ̄ = max[τ1, τ2], τ̃ = max[τ̄ , τ3], F : R+ × Rn → ×Rn, f : [t0,∞)× Rn × Rn → Rn, g : [t0,∞)× Rn × Rn → Rn×m, h : [t0,∞)× Rn × Rn → Rn×m. Here Bt is an m-dimensional G-Brownian motion, ⟨B⟩t≥0 is the quadratic variation process of the G-Brownian B. Remark that (4.3) is a particular case of (3.1). We will apply the method described above to construct G-Lyapunov functionals for (4.3), and, as a consequence, to deduce sufficient conditions ensuring the quasi sure practical stability with decay function λ(t), where λ(·) ∈ C1(R+). Theorem 4.1. Assume that lnλ(t) is uniformly continuous on t ≥ 0, there exists a constant δ ≥ 0 such that lim t→∞ ln ln t lnλ(t) ≤ δ. Let ψ(t) be a continuous non-negative function, and r(t) a non-negative continuous differentiable function such that for all t ≥ t0 ≥ 0, the following inequalities hold: (1) 2⟨x, F (t, x)⟩ ≤ (ψ(t)− U)∥x∥2 + r′(t) λm(t) , U > 0, ∥f̃(t,Φ)∥ ≤ a1∥Φ(−τ1)∥, ∥h̃(t,Φ)∥ ≤ a2∥Φ(−τ2)∥, ∥g̃(t,Φ)∥ ≤ a3∥Φ(−τ3)∥, ∥Φ(0)g̃(t,Φ)∥ ≤ a4∥Φ(−τ3)∥, (4.4) where f̃(t,Φ) = f(t,Φ(0),Φ(−τ1)), g̃(t,Φ) = g(t,Φ(0),Φ(−τ3)), h̃(t,Φ) = h(t,Φ(0),Φ(−τ2)). (2) lim t→∞ sup ∫ t t0 ψ(s)ds lnλ(t) ≤ a, a ∈ R, lim t→∞ sup t lnλ(t) = C ≥ 0, lim t→∞ r(t) λm(t) = r̃ > 0. (3) There exists r̃′ ≥ r̃ > 0, such that the solution x(t, t0, ξ) satisfies ∥x(t, t0, ξ)∥ > ( r̃′ )1/2 , ∀t ≥ t0. Then lim t→∞ ln ( ∥x(t, t0, ξ)∥ − ( r̃′ )1/2) lnλ(t) ≤ −γ, q.s., where γ = UC − ( m+ a+ δ + (2a1 + 2σ̄2a2 + ā)C ) , ā = σ̄2(a23 + a24). In particular, if UC > m + (a + δ) + (2a1 + 2σ̄2a2 + ā)C, then the solution to system (4.3) tends to the ball Br, with r = (r̃′)1/2 quasi surely, with decay function λ(t) and order at least γ. Proof. Based upon the procedure of G-Lyapunov functionals construction, we con- sider the auxiliary equation without memory of the type (4.2) as ẏ(t) = F (t, y(t)). (4.5) 16 T. CARABALLO, F. EZZINE, M. A. HAMMAMI EJDE-2024/70 Our interest now is to prove that the solution to system (4.5) tends to the ball Br, with r = (r̃′)1/2 quasi surely with decay function λ(t). We consider the function v(t, y) = λm(t)∥y∥2, m ≥ 0 as a Lyapunov function for Eq.(4.5). Then, we have to prove that v(t, y) satisfies all conditions of Corollary 3.4. Based upon (4.4), we have∫ t t0 vs(s, y(s))ds+ ∫ t t0 vx(s, y(s))F (s, y(s))ds ≤ ∫ t t0 mλ′(s)λm−1(s)∥y(s)∥2ds+ ∫ t t0 2λm(s)⟨y(s), F (s, y(s)⟩ds ≤ ∫ t t0 mλ′(s)λm−1(s)∥y(s)∥2ds+ ∫ t t0 ( λm(s) (ψ(s)− U) ∥y(s)∥2 + r′(s) ) ds ≤ ∫ t t0 ( m λ′(s) λ(s) + ψ(s)− U ) λm(s)∥y(s)∥2ds+ r(t)− r(t0). That is, ∫ t t0 vs(s, y(s))ds+ ∫ t t0 vx(s, y(s))F (s, y(s))ds, ≤ ∫ t t0 ( m λ′(s) λ(s) + ψ(s)− U ) λm(s)∥y(s)∥2ds+ r(t), we set φ2(t) = mλ′(t) λ(t) + ψ(t)− U . Based on assumption (A2), we obtain lim t→∞ sup ∫ t t0 φ2(s)ds lnλ(t) ≤ m+ a− UC. In view of Corollary 3.4, we deduce that lim t→∞ sup ln ( ∥y(t)∥ − (r̃′)1/2 ) lnλ(t) ≤ −γ, q.s., where γ = UC − (a+ δ ∨m). Hence, if UC > (a+ δ ∨m), the solution to system (4.4) tends to the ball Br, with r = (r̃′)1/2 quasi surely with decay function λ(t) and order at least γ. Now we construct a G-Lyapunov functional V for (4.3) in the form V = V1 + V2, where V1(t, xt) = λm(t)∥x(t)∥2. Considering φ1(t) = 1 4λm(t) for t ≥ 0, we obtain∫ t t0 LV1(s, xs)ds+ σ̄2 ∫ t t0 φ1(s)∥V1x(s, xs)g̃(s, x(s), x(s− τ3))∥2ds = ∫ t t0 mλ′(s)λm−1(s)∥x(s)∥2ds+ ∫ t t0 2λm(s)⟨F (s, x(s)), x(s)⟩ds + ∫ t t0 2λm(s)⟨f̃(s, x(s), x(s− τ1)), x(s)⟩ds + ∫ t t0 2λm(s)⟨h̃(s, x(s), x(s− τ2)), x(s)⟩d⟨B⟩s EJDE-2024/70 PRACTICAL STABILITY OF STOCHASTIC DELAY EQUATIONS 17 + ∫ t t0 λm(s)∥g̃(s, x(s), x(s− τ3))∥2d⟨B⟩s + ∫ t t0 σ̄2λm(s)∥x(s)g̃(s, x(s), x(s− τ3))∥2ds. Based on Peng [24, Chapter III], we have that for each 0 ≤ s ≤ t ≤ T , σ2(t− s) ≤ ⟨B⟩t − ⟨B⟩s ≤ σ̄2(t− s). Then ∫ t t0 LV1(s, xs)ds+ ∫ t t0 σ̄2φ1(s)∥V1x(s, xs)g̃(s, x(s), x(s− τ))∥2ds = ∫ t t0 mλ(s)λm−1(s)∥x(s)∥2ds+ ∫ t t0 2λm(s)⟨F (s, x(s)), x(s)⟩ds + ∫ t t0 2λm(s)⟨f̃(s, x(s), x(s− τ1)), x(s)⟩ds + ∫ t t0 2σ̄2λm(s)⟨h̃(s, x(s), x(s− τ2)), x(s)⟩ds + ∫ t t0 σ̄2λm(s)∥g̃(s, x(s), x(s− τ3))∥2ds + ∫ t t0 σ̄2λm(s)∥x(s)g̃(s, x(s), x(s− τ3))∥2ds. Taking into account assumption (4.4),∫ t t0 LV1(s, xs)ds+ ∫ t t0 1 4λm(s) ∥V1x(s, xs)g̃(s, x(s), x(s− τ))∥2ds ≤ ∫ t t0 λm(s) ( m λ′(s) λ(s) + ψ(s)− U ) ∥x(s)∥2ds + ∫ t t0 2a1λ m(s)∥x(s)∥ ∥x(s− τ1)∥ds+ ∫ t t0 2σ̄2a2λ m(s)∥x(s)∥ ∥x(s− τ2)∥ds + ∫ t t0 σ̄2a23λ m(s)∥x(s− τ3)∥2ds+ ∫ t t0 σ̄2a24λ m(s)∥x(s− τ3)∥2ds+ r(t) ≤ ∫ t t0 λm(s) (( m λ′(s) λ(s) + ψ(s)− U ) + a1 + σ̄2a2 ) ∥x(s)∥2ds + ∫ t t0 a1λ m(s)∥x(s− τ1)∥2ds+ ∫ t t0 σ̄2a2λ m(s)∥x(s− τ2)∥2ds + ∫ t t0 āλm(s)∥x(s− τ3)∥2ds+ r(t), where ā = σ̄2(a23 + a24). Let V2(t, xt) = a1 ∫ t t−τ1 λm(u+ τ1)∥x(u)∥2du+ σ̄2a2 ∫ t t−τ2 λm(u+ τ2)∥x(u)∥2du + ā ∫ t t−τ3 λm(u+ τ3)∥x(u)∥2du. 18 T. CARABALLO, F. EZZINE, M. A. HAMMAMI EJDE-2024/70 Hence, we obtain∫ t t0 LV2(s, xs)ds = a1 ∫ t t0 λm(s+ τ1)∥x(s)∥2ds− a1 ∫ t t0 λm(s)∥x(s− τ1)∥2ds + σ̄2a2 ∫ t t0 λm(s+ τ2)∥x(s)∥2ds− σ̄2a2 ∫ t t0 λm(s)∥x(s− τ2)∥2ds + ā ∫ t t0 λm(s+ τ3)∥x(s)∥2ds− ā ∫ t t0 λm(s)∥x(s− τ3)∥2ds ≃ a1 ∫ t t0 λm(s)∥x(s)∥2ds− a1 ∫ t t0 λm(s)∥x(s− τ1)∥2ds + σ̄2a2 ∫ t t0 λm(s)∥x(s)∥2ds− σ̄2a2 ∫ t t0 λm(s)∥x(s− τ2)∥2ds + ā ∫ t t0 λm(s)∥x(s)∥2ds− ā ∫ t t0 λm(s)∥x(s− τ3)∥2ds. For V = V1 + V2, we have∫ t t0 LV (s, xs)ds+ ∫ t t0 σ̄2 4λm(s) ∥Vx(s, xs)g̃(s, x(s), x(s− τ3))∥2ds ≤ ∫ t t0 λm(s) ( m λ′(s) λ(s) + ψ(s) + 2a1 + 2σ̄2a2 + ā− U ) ∥x(s)∥2ds+ r(t). Thus, φ2(t) = m λ′(t) λ(t) + ψ(t) + 2a1 + 2σ̄2a2 + ā− U, φ1(t) = 1 4λm(t) . Hence, we arrive at lim t→∞ sup ∫ t t0 φ2(s)ds lnλ(t) ≤ m+ a+ (2a1 + 2σ̄2a2 + ā− U)C, lim t→∞ inf lnφ1(t) lnλ(t) ≥ −m. Therefore, Corollary 3.4 allows us to conclude that lim t→∞ ln ( ∥x(t, t0, ξ)∥ − ( r̃′ )1/2) lnλ(t) ≤ −γ, q.s., where γ = UC − ( m+ a+ δ + (2a1 + 2σ̄2a2 + ā) ) . Consequently, if UC > m + (a+ δ)+(2a1+2σ̄2a2+ ā)C, the solution to system (4.3) tends to the ball Br, with r = (r̃′)1/2 quasi surely with decay function λ(t). □ Now we construct G-Lyapunov functionals for stochastic differential equations with time-varying delay driven by G-Brownian motion. We consider the follow- ing stochastic differential equation with time-varying delay driven by G-Brownian EJDE-2024/70 PRACTICAL STABILITY OF STOCHASTIC DELAY EQUATIONS 19 motion, dx(t) = [F (t, x(t)) + f(t, x(t), x(t− τ1(t)))] dt + h(t, x(t), x(t− τ2(t)))d⟨B⟩t + g(t, x(t), x(t− τ3(t)))dBt, τ1(t) ∈ [0, τ10], τ2(t) ∈ [0, τ20], τ3(t) ∈ [0, τ30], τ̄ = max[τ10, τ20], τ = max[τ̄ , τ30], x(t) = ξ(t− t0), t ∈ [t0 − τ, t0], (4.6) where F : R+ × Rn → ×Rn, f : [t0,∞)× Rn × Rn → Rn, h : [t0,∞)× Rn × Rn → Rn×m, g : [t0,∞)× Rn × Rn → Rn×m. Here Bt is an m-dimensional G-Brownian motion, ⟨B⟩t≥0 is the quadratic variation process of the G-Brownian motion B. Notice that (4.6) is a particular case of (3.1). Now, we apply the procedure of constructing G-Lyapunov functionals for (4.6), to state sufficient conditions ensuring the quasi sure practical uniform exponential stability, with decay function λ(t) = exp(t). The construction of G-Lyapunov functionals for general decay functions will be analyzed elsewhere. Theorem 4.2. Let ϕ1(t) be a continuous non-negative function, ϕ2(t), ϕ3(t) > 0 non-increasing functions and r(t) a continuous non-negative differentiable function such that, for all t ≥ t0 ≥ 0, (H3) holds, and the following assumptions as well, (1) 2⟨x, F (t, x)⟩ ≤ (ϕ1(t)− U)∥x∥2 + r′(t) exp(mt) , U > 0, ∥f̃(t,Φ)| ≤ ϕ2(t)∥Φ(−τ1(t))∥, ∥h̃(t,Φ)∥ ≤ ϕ3(t)∥Φ(−τ2(t))∥, ∥g̃(t,Φ)∥ ≤ c4∥Φ(−τ3(t))∥, ∥Φ(0)g̃(t,Φ)∥ ≤ c5∥Φ(−τ3(t))∥, (4.7) where f̃(t,Φ) = f(t,Φ(0),Φ(−τ1(t))), h̃(t,Φ) = h(t,Φ(0),Φ(−τ2(t))), g̃(t,Φ) = g(t,Φ(0),Φ(−τ3(t))), and τ1(t) ∈ [0, τ10 ], τ̇1(t) ≤ τ1 ≤ 1, τ2(t) ∈ [0, τ20], τ̇2(t) ≤ τ2 ≤ 1, τ3(t) ∈ [0, τ30], τ̇3(t) ≤ τ3 ≤ 1. (4.8) (2) lim t→∞ sup ∫ t t0 ϕ1(s)ds t ≤ c1, c1 > 0, lim t→∞ sup ∫ t t0 ϕ2(s)ds t ≤ c2, c2 > 0, lim t→∞ sup ∫ t t0 ϕ3(s)ds t ≤ c3, c3 > 0, 20 T. CARABALLO, F. EZZINE, M. A. HAMMAMI EJDE-2024/70 lim t→∞ r(t) exp(mt) = r̃, r̃ > 0. Then lim t→∞ ln ( ∥x(t, t0, ξ)∥ − (r̃′ )1/2 ) lnλ(t) ≤ −γ, q.s., where γ = U − ( m+ c1 + ( 1 + exp(mτ10) 1− τ1 ) c2 + σ̄2 ( 1 + exp(mτ20) 1− τ2 ) c3 + c̄ exp(mτ30) 1− τ3 ) , c̄ = σ̄2 ( c24 + c25 ) . In particular, if U > m+ c1 + ( 1 + exp(mτ10) 1− τ1 ) c2 + σ̄2 ( 1 + exp(mτ20) 1− τ2 ) c3 + c̄ exp(mτ30) 1− τ3 , the solution to (4.6) tends to the ball Br, with r = (r̃′)1/2 quasi surely uniformly practically exponentially stable, i.e., with decay function λ(t) = exp(t), and order at least γ. Proof. Proceeding as in the proof of Theorem 4.1, we consider the auxiliary equation without memory of the type (4.2), ẏ(t) = F (t, y(t)). (4.9) We have to prove that the solution to (4.9) tends to the ball Br, with r = (r̃′)1/2 and decay function λ(t). To this end, we consider the function v(t, y) = exp(mt)∥y∥2 with m ≥ 0 as a Lyapunov function for (4.9). Therefore, we prove that v(t, y) satisfies all conditions of Corollary 3.4. Using (4.7), we have∫ t t0 vs(s, y(s))ds+ ∫ t t0 vx(s, y(s))F (s, y(s))ds, ≤ ∫ t t0 (m+ ϕ1(s)− U) exp(ms)∥y(s)∥2ds+ r(t). Thus, setting φ2(t) = m+ ϕ1(t)− U , by Corollary 3.4, we obtain lim t→∞ sup ln ( ∥y(t)∥ − (r̃′)1/2 ) t ≤ −γ, q.s., where γ = U − (c1 +m), then if U > c1 +m, and the solution to (4.9) tends to the ball Br, with r = (r̃′)1/2 practically uniformly exponentially stable with order at least γ = U − (c1 +m). Based on this procedure, now we construct a G-Lyapunov functional V for (4.6) in the form V = V1 + V2, where V1(t, xt) = exp(mt)∥x(t)∥2. Consider φ1(t) = 1 4 exp(mt) , t ≥ 0, we then deduce∫ t t0 LV1(s, xs)ds+ σ̄2 ∫ t t0 φ1(s)∥V1x(s, xs)g̃(s, xs)∥2ds = ∫ t t0 exp(ms)∥x(s)∥2ds+ ∫ t t0 2 exp(ms)⟨F (s, x(s)), x(s)⟩ds + ∫ t t0 2 exp(ms)⟨f̃(s, xs), x(s)⟩ds+ ∫ t t0 2 exp(ms)⟨h̃(s, xs), x(s)⟩d⟨B⟩s EJDE-2024/70 PRACTICAL STABILITY OF STOCHASTIC DELAY EQUATIONS 21 + ∫ t t0 exp(ms)∥g̃(s, xs)∥2d⟨B⟩s + ∫ t t0 σ̄2 exp(ms)∥x(s)G̃(s, xs)∥2ds ≤ ∫ t t0 exp(ms)∥x(s)∥2ds+ ∫ t t0 2 exp(ms)⟨F (s, x(s)), x(s)⟩ds + ∫ t t0 2 exp(ms)⟨f̃(s, xs), x(s)⟩ds+ ∫ t t0 2σ̄2 exp(ms)⟨h̃(s, xs), x(s)⟩ds + ∫ t t0 σ̄2 exp(ms)∥g̃(s, xs)∥2ds+ ∫ t t0 σ̄2 exp(ms)∥x(s)G̃(s, xs)∥2ds. Taking into account assumption (4.7), we have∫ t t0 LV1(s, xs)ds+ ∫ t t0 σ̄2 4 exp(ms) ∥V1x(s, xs)g̃(s, xs)∥2ds ≤ ∫ t t0 exp(ms)(m+ ϕ1(s)− U) ∥x(s)∥2ds+ r(t) + ∫ t t0 2ϕ2(s) exp(ms)∥x(s)∥ ∥x(s− τ1(s))∥ds + ∫ t t0 2σ̄2ϕ3(s) exp(ms)∥x(s)∥∥x(s− τ2(s))∥ds + ∫ t t0 σ̄2c24 exp(ms)∥x(s− τ3(s))∥2ds+ ∫ t t0 σ̄2c25 exp(ms)∥x(s− τ3(s))∥2ds ≤ ∫ t t0 exp(ms) ( (m+ ϕ1(s)− U) + ϕ2(s) + σ̄2ϕ3(s) ) ∥x(s)∥2ds + ∫ t t0 ϕ2(s) exp(ms)∥x(s− τ1(s))∥ds+ ∫ t t0 σ̄2ϕ3(s) exp(ms)∥x(s− τ2(s))∥2ds + ∫ t t0 c̄ exp(ms)∥x(s− τ3(s))∥2ds+ r(t), where c̄ = σ̄2(c24 + c25). Let V2(t, xt) = 1 1− τ1 ∫ t t−τ1(t) exp(m(u+ τ10))ϕ2(u)∥x(u)∥2du + σ̄2 1− τ2 ∫ t t−τ2(t) exp(m(u+ τ20))ϕ3(u)∥x(u)∥2du + c̄ 1− τ3 ∫ t t−τ3(t) exp(m(u+ τ30))∥x(u)∥2du. Hence,∫ t t0 LV2(s, xs)ds = 1 1− τ1 ∫ t t0 exp(m(s+ τ10))ϕ2(s)∥x(s)∥2ds − 1 1− τ1 ∫ t t0 (1− τ̇1(s)) exp(m(s− τ1(s) + τ10))ϕ2(s− τ1(s))∥x(s− τ1(s))∥2ds 22 T. CARABALLO, F. EZZINE, M. A. HAMMAMI EJDE-2024/70 + σ̄2 1− τ2 ∫ t t0 exp(m(s+ τ20))ϕ3(s)∥x(s)∥2ds − σ̄2 1− τ2 ∫ t t0 (1− τ̇2(s)) exp(m(s− τ2(s) + τ20))ϕ3(s− τ2(s))∥x(s− τ2(s))∥2ds + c̄ 1− τ3 ∫ t t0 exp(m(s+ τ30))∥x(s)∥2ds − c̄ 1− τ3 ∫ t t0 (1− τ̇3(s)) exp(m(s− τ3(s) + τ30))∥x(s− τ3(s))∥2ds ≤ 1 1− τ1 ∫ t t0 exp(m(s+ τ10))ϕ2(s)∥x(s)∥2ds − 1 1− τ1 ∫ t t0 (1− τ1) exp(ms) exp(m(τ10 − τ1(s)))ϕ2(s− τ1(s))∥x(s− τ1(s))∥2ds + σ̄2 1− τ2 ∫ t t0 exp(m(s+ τ20))ϕ3(s)∥x(s)∥2ds − σ̄2 1− τ2 ∫ t t0 (1− τ2) exp(ms) exp(m(τ20 − τ2(s)))ϕ3(s− τ2(s))∥x(s− τ2(s))∥2ds + c̄ 1− τ3 ∫ t t0 exp(m(s+ τ30))∥x(s)∥2ds − c̄ 1− τ3 ∫ t t0 (1− τ3) exp(ms) exp(m(τ30 − τ2(s)))∥x(s− τ3(s))∥2ds. In other words,∫ t t0 LV2(s, xs)ds ≤ 1 1− τ1 ∫ t t0 exp(m(s+ τ10))ϕ2(s)∥x(s)∥2ds − ∫ t t0 exp(ms)ϕ2(s− τ1(s))∥x(s− τ1(s))∥2ds + σ̄2 1− τ2 ∫ t t0 exp(m(s+ τ20))ϕ3(s)∥x(s)∥|2ds − σ̄2 ∫ t t0 exp(ms)ϕ3(s− τ2(s))∥x(s− τ2(s))∥2ds + c̄ 1− τ3 ∫ t t0 exp(m(s+ τ30))∥x(s)∥2ds − c̄ ∫ t t0 exp(ms)∥x(s− τ3(s))∥2ds. For V = V1 + V2, it follows that∫ t t0 LV (s, xs)ds+ ∫ t t0 σ̄2 4 exp(ms) ∥Vx(s, xs)g̃(s, xs∥2ds ≤ ∫ t t0 exp(ms) ( m+ ϕ1(s)− U + ( 1 + exp(mτ10) 1− τ1 ) ϕ2(s) + σ̄2 ( 1 + exp(mτ20) 1− τ2 ) ϕ3(s) + c̄ exp(mτ30) 1− τ3 ) ∥x(s)∥2ds+ r(t). EJDE-2024/70 PRACTICAL STABILITY OF STOCHASTIC DELAY EQUATIONS 23 Then, we obtain φ2(t) = m+ ϕ1(s)− U + ( 1 + exp(mτ10) 1− τ1 ) ϕ2(s) + σ̄2 ( 1 + exp(mτ20) 1− τ2 ) ϕ3(s) + c̄ exp(mτ30) 1− τ3 , φ1(t) = 1 4 exp(mt) . It follows that lim t→∞ sup ∫ t t0 φ2(s)ds t ≤ m+ c1 − U + ( 1 + exp(mτ10) 1− τ1 ) c2 + σ̄2 ( 1 + exp(mτ20) 1− τ2 ) c3 + c̄ exp(mτ30) 1− τ3 , lim t→∞ inf lnφ1(t) t ≥ −m. Eventually, based upon Corollary 3.4, we deduce that lim t→∞ ln ( ∥x(t, t0, ξ)∥ − (r̃′ )1/2 ) lnλ(t) ≤ −γ, q.s., where γ = U − ( m+ c1 − U + ( 1 + exp(mτ10) 1− τ1 ) c2 + σ̄2 ( 1 + exp(mτ20) 1− τ2 ) c3 + c̄ exp(mτ30) 1− τ3 ) . Hence, if U > m + c1 + ( 1 + exp(mτ10) 1−τ1 ) c2 + σ̄2 ( 1 + exp(mτ20) 1−τ2 ) c3 + c̄ exp(mτ30) 1−τ3 , the solution to (4.6) tends to the ball Br, with r = (r̃′)1/2 quasi surely uniformly practically exponentially stable with decay function λ(t) = exp(t), and order at least γ. □ Now we present an illustrative example that implements the previous result. Example 4.3. Consider the one-dimensional stochastic differential equation with time-varying delay driven by G-Brownian motion, dx(t) = (1 2 (α+ exp(−t)− U)x(t) + 1 2(1 + ∥x(t)∥) + 1 t+ 1 x(t− τ1(t)) ) dt, + cos(t)x(t− τ2(t))d⟨B⟩t + g(x(t)) x(t− τ3(t)) 1 + ∥x(t)∥ dBt, t ≥ 0, (4.10) where x(t) = ξ(t) and t ∈ [−τ, 0], with the conditions τ1(t) ∈ [0, τ10], τ̇1(t) ≤ τ10 ≤ 1, τ2(t) ∈ [0, τ20], τ̇2(t) ≤ τ20 ≤ 1, τ3(t) ∈ [0, τ30], τ̇3(t) ≤ τ30 ≤ 1. Here α,U ∈ R+, g(·) : R → R is a bounded Lipshitz continuous function, such that g(0) ̸= 0, and ∥g(x)∥ ≤ l, l > 0. Bt is a one-dimensional G-Brownian motion with Bt ∼ N (0, [ 12 , 1]), and τ̄ = max[τ10, τ20], τ = max[τ̄ , τ30]. 24 T. CARABALLO, F. EZZINE, M. A. HAMMAMI EJDE-2024/70 Now we set this problem in our formulation by taking F (t, x) = 1 2 (α+ exp(−t)− U)x+ 1 2(1 + ∥x(t)∥) , f̃(t,Φ) = 1 1 + t Φ(−τ1(t)), h̃(t,Φ) = cos(t)Φ(−τ2(t)), g̃(t,Φ) = g(Φ(0)) Φ(−τ3(t)) 1 + ∥Φ(0)∥ , x ∈ R, Φ ∈ C([−τ, 0],R). For m = 2, we can check that 2⟨x, F (t, x)⟩ ≤ (α+ exp(−t)− U)∥x∥2 + exp(t) exp(2t) , ∥f̃(t,Φ)∥ ≤ 1 t+ 1 ∥Φ(−τ1(t))∥, ∥h̃(t,Φ)∥ ≤ ∥Φ(−τ2(t))∥, ∥g̃(t,Φ)∥ ≤ l∥Φ(−τ3(t))∥, ∥Φ(0)g̃(t,Φ)∥ ≤ l∥Φ(−τ3(t))∥. Therefore, ϕ1(t) = (α+ exp(−t)), ϕ2(t) = 1 1 + t , ϕ3(t) = 1, r(t) = exp(t). Then, we can choose constants in Theorem 4.2 as follows: c1 = α, c2 = 0, c3 = 1, c4 = c5 = l, r̃ = 1. Finally, Theorem 4.2 allows us to conclude that lim t→∞ sup ln(∥x(t)∥ − 1) t ≤ −γ, q.s. where γ = U − ( α+ 3 + exp(2τ20) 1− τ2 + 2l2 exp(2τ30) 1− τ3 ) . Hence, if U > α+ 3 + exp(2τ20) 1− τ2 + 2l2 exp(2τ30) 1− τ3 , we deduce that the solution to (4.10) is quasi surely practically exponentially stable, i.e., with decay function λ(t) = exp(t), and order at least γ. 5. Conclusion This article studies the practical convergence to a small ball centered at the origin with a general decay rate of G-SDDES. By using G-Lyapunov functionals, some sufficient conditions of practical stability with a general decay rate of G-SDDEs is stated. Meanwhile, we construct suitable Lyapunov functionals for G-SDDES with constant and time-varying delay to obtain sufficient conditions ensuring the practical exponential stability with a general decay rate. Finally, some examples to illustrate the effectiveness of the proposed techniques are presented. EJDE-2024/70 PRACTICAL STABILITY OF STOCHASTIC DELAY EQUATIONS 25 Acknowledgments. 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Zhang, Z.Chen; Exponential stability for stochastic differential equation driven by G- Brownian motion, Applied Mathematics Letters, 25 (2012), 1906–1910. Tomás Caraballo Depto. Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Uni- versidad de Sevilla, c/ Tarfia s/n, 41012-Sevilla, Spain. Department of Mathematics, Wenzhou University, Wenzhou, Zhejiang Province, 325035, China Email address: caraball@us.es Faten Ezzine University of Sfax, Faculty of Sciences of Sfax, Department of Mathematics, Tunisia Email address: ezzinefaten94fss@gmail.com Mohamed Ali Hammami University of Sfax, Faculty of Sciences of Sfax, Department of Mathematics, Tunisia Email address: MohamedAli.Hammami@fss.rnu.tn 1. Introduction 2. Preliminaries Notation on G-stochastic calculus 3. Practical stability of stochastic delay equation driven by G-Brownian motion 4. Practical stability 5. Conclusion Acknowledgments References