Ulam-Hyers-Rassias stability of neutral stochastic functional differential equations

Ulam-Hyers-Rassias stability of neutral stochastic functional differential equations Tom´as Caraballo Universidad de Sevilla, Dpto. Ecuaciones Diferenciales y An´alisis Num´erico, Facultad de Matem´aticas, c/ Tarfia s/n, 41012-Sevilla (Spain). [email protected] Lassaad Mchiri University of Sfax, Faculty of Sciences of Sfax, Department of Mathematics, Tunisia lassaadmc[email protected] Mohamed Rhaima University of Tunis Manar, Faculty of Sciences of Tunis, Department of Mathematics, Tunisia. [email protected]u.tn Abstract In this paper, by using the Gronwall inequality, we show two new results on the Ulam- Hyers and the Ulam-Hyers-Rassias stabilities of neutral stochastic functional differential equations. Two examples illustrating our results are exhibited. Mathematics Subject Classification (2010): 39B82, 39B52, 46H25. Keywords: Neutral Stochastic functional differential equations, Ulam-Hyers stability, Ulam- Hyers-Rassias stability, Gronwall inequality. 1 1 Introduction The stability problem of functional equations originated from a question of Stanislaw Ulam, posed in 1940, concerning the stability of group of homomorphisms. The question concerning the stability of homomorphisms is as follows: Let G1be a group, and let G2be a metric group with a metric dand a positive number . The problem is to analyze if there exists a positive number δsuch that for every g:G1→G2 satisfying the inequality d(g(xy), g(x)g(y)) ≤δ, ∀x, y ∈G1, there exists a homomorphism h:G1→G2such that d(g(x), h(x)) ≤, ∀x∈G1. In 1941, Donald H. Hyers [5] gave a partial answer to the question of Ulam in the context of Banach spaces in the case of δ-linear transformations, that is: Let F1,F2be two Banach spaces and let g:F1→F2be a linear transformation satisfying kg(x+y)−g(x)−g(y)k ≤ δ, ∀x, y ∈F1δ > 0. There exists a unique linear transformation L:F1→F2such that the limit L(x) = limn→+∞ g(2nx) 2n exists for each x∈F1and kg(x)−L(x)k ≤ δfor all x∈F1, that was the first significant breakthrough and a step towards more solutions in this area. Since then, a large number of papers have been published in connection with various generalizations of Ulam’s problem and Hyers’s theorem (see, for instance, [21], [4], [5], [6], [7], [8], [9], [10], [2], [17], [22], [12], and [3]). In 1978, Rassias [19] provided a generalized solution to the Ulam problem for approximate δ-linear transformations. In [18], Rassias obtained a generalization of the Hyers’s theorem. In 1994, Gavruta [1] obtained a generalization of Rassias’s Theorem for the unbounded Cauchy difference g(x+y)−g(x)−g(y) and stated the notion of generalized Ulam-Hyers-Rassias stability in the spirit of Rassias approach. To the best of our knowledge, there are a few papers about the Ulam-Hyers and the Ulam-Hyers-Rassias stability of stochastic differential equations in the literature (see [11, 16, 23]). In the literature, neutral stochastic functional differential equations attracted the attention of many researchers (see [13, 14, 15, 20] etc.). Therefore, it is important to generalize the research results of deterministic neutral functional differential equations to the stochastic case. The contents of the paper is as follows. In Section 2, we present some basic results and assumptions. Section 3 is devoted to show some sufficient conditions and assumptions ensuring the Ulam-Hyers and the Ulam-Hyers-Rassias stabilities of the solution of the system. Finally in Section 4, we analyze two examples to illustrate our results. 2 Preliminaries and definitions Let {Ω,F,(Ft)t≥0,P}be a complete probability space with a filtration satisfying the usual conditions, i.e., the filtration is continuous on the right and F0contains all P-zero sets. W(t) 2 is an m-dimensional Brownian motion defined on the probability space. Let L2([a, b],Rn) be the family of Rn-valued Ft-adapted processes {f(t)}a≤t≤bsuch that Rb a|f(t)|2dt < ∞a.s. and M2([a, b],Rn) the family of processes {f(t)}a≤t≤bin L2([a, b],Rn) such that ERb a|f(t)|2dt < ∞. Let C([−τ, 0]; Rn) denote the family of functions ϕfrom [−τ, 0] to Rnthat are right-continuous and have limits on the left. C([−τ, 0]; Rn) is equipped with the norm kϕk= sup−τ≤s≤0|ϕ(s)| and |x|=√xTxfor any x∈Rn. Denote Cb F0([−τ, 0]; Rn) to be the family of all F0-measurable bounded C([−τ, 0]; Rn)-valued random variables ξ={ξ(θ) : −τ≤θ≤0}. Let L2 Ft([−τ, 0]; Rn), t≥0, denote the family of all Ft-measurable, C([−τ, 0]; Rn)-valued random variables ϕ={ϕ(θ) : −τ≤θ≤0}such that sup−τ≤θ≤0E|ϕ(θ)|2<∞. Consider the following neutral stochastic functional differential equation for 0 ≤t0< T fixed: d[x(t)−G(xt)] = f(t, xt)dt +g(t, xt)dW(t), t0≤t≤T, (2.1) with the initial condition xt0=ξ∈L2 Ft0([−τ, 0]; Rn),(2.2) and recall that, given x∈C([t0, T ]; Rn), for each t∈[t0, T] we denote by xt(·) the function in C([t0−τ, 0]; Rn) defined as xt(θ) = x(t+θ),−τ≤θ≤0. We assume that f: [t0, T]×C([−τ, 0]; Rn)−→ Rn, g : [t0, T]×C([−τ, 0]; Rn)−→ Rn×m, G:C([−τ, 0]; Rn)−→ Rn. Using the definition of Itˆo’s stochastic differential and integrating the two sides of equation (2.1) from t0to t, we have x(t)−G(xt) = x(t0)−G(xt0) + Zt t0 f(s, xs)ds +Zt t0 g(s, xs)dW(s), t0≤t≤T, (2.3) We will establish some assumptions ensuring the existence and uniqueness of a solution, denoted by x(t;t0, ξ), for equation (2.1). H1: (Uniform Lipschitz condition): Assume that there exists a constant L > 0 such that |f(t, ϕ1)−f(t, ϕ2)|2∨|g(t, ϕ1)−g(t, ϕ2)|2≤L||ϕ1−ϕ2||2, for all t∈[t0, T] and ϕ1,ϕ2∈C([−τ, 0]; Rn), where the notation a∨bdefine the maximum of a and b. H2: (Linear growth condition): Assume that there exists a constant α > 0 such that for all (t, ϕ)∈[t0, T]×C([−τ, 0]; Rn) |f(t, ϕ)|2∨|g(t, ϕ)|2≤α(1 + ||ϕ||2), H3: Assume that there is a constant β∈[0,1 2) such that |G(ϕ1)−G(ϕ2)| ≤ β||ϕ1−ϕ2||,(2.4) for all ϕ1,ϕ2∈C([−τ, 0]; Rn). 3 Definition 2.1. An Rn-valued stochastic processes x(t) on t0−τ≤t≤Tis called a solution to equation (2.1) with initial condition (2.2) if the following conditions are satisfied: (i) it is continuous and {x(t)}t0≤t≤Tis Ft-adapted. (ii) {f(t, xt)}∈L1([t0, T],Rn) and {g(t, xt)}∈L2([t0, T],Rn×m). (iii) xt0=ξand (2.3) holds for every t0≤t≤T. Theorem 2.1. If assumptions H1−H3are satisfied, then there exists a unique solution x∈ M2([t0−τ, T ],Rn)to equation (2.1) with initial condition (2.2). Proof. See [14]. 3 Main results In this section, we discuss the Ulam-Hyers and the Ulam-Hyers-Rassias stability of equation (2.1) under the assumptions H1-H3. Definition 3.1. Equation (2.1) is Ulam-Hyers stable with respect to if there exists a constant C > 0 such that for each  > 0 and for each solution y∈ M2([t0−τ, T ],Rn), with yt0=ξ, of the following inequation: E|y(t)−G(yt)−(y(t0)−G(yt0)) −Zt t0 f(s, ys)ds −Zt t0 g(s, ys)dW(s)|2≤, ∀t∈[t0−τ, T ],(3.1) there exists a solution x∈ M2([t0−τ, T],Rn) of (2.1), with xt0=ξ, such that E|y(t)−x(t)|2≤C, ∀t∈[t0−τ, T ]. Definition 3.2. Equation (2.1) is generalized Ulam-Hyers-Rassias stable with respect to z(·)∈C([t0−τ, T ]; Rn) if there exists a constant M > 0 such that for each solution y∈ M2([t0−τ, T ],Rn), with yt0=ξ, satisfying E|y(t)−G(yt)−(y(t0)−G(yt0))−Zt t0 f(s, ys)ds−Zt t0 g(s, ys)dW(s)|2≤z(t),∀t∈[t0−τ, T ],(3.2) there exists a solution x(t)∈ M2([t0−τ, T],Rn) of (2.1), with xt0=ξ, such that E|y(t)−x(t)|2≤Mz(t), ∀t∈[t0−τ, T ]. We cite now a technical lemma before proving our main theorems. 4 Lemma 3.1. For any x,y≥0and 0<  < 1, we have: (x+y)2≤x2 +y2 1−. Proof. See [14]. Theorem 3.2. Under assumptions H1−H3, equation (2.1) is Ulam-Hyers stable. Proof. By Theorem 2.1, there exists a unique solution x∈ M2([t0−τ, T ],Rn) to equation (2.1) with initial condition (2.2). Let y∈ M2([t0−τ, T ],Rn), with yt0=xt0=ξ, be a solution to (3.1), then for t∈[t0−τ, t0], we have E|y(t)−x(t)|2= 0. By the triangle inequality, equation (2.3) and inequality (3.1), we have for t∈[t0, T] |y(t)−x(t)|2 ≤2 y(t)−G(yt)−(y(t0)−G(yt0)) −Zt t0 f(s, ys)ds −Zt t0 g(s, ys)dW(s) 2 , +2  (G(yt)−G(xt)) + Zt t0f(s, ys)−f(s, xs)ds +Zt t0g(s, ys)−g(s, xs)dW(s) 2 , ≤2L(t)+2J(t), where L(t) =  y(t)−G(yt)−(y(t0)−G(yt0)) −Zt t0 f(s, ys)ds −Zt t0 g(s, ys)dW(s) 2 and J(t) =  (G(yt)−G(xt)) + Zt t0f(s, ys)−f(s, xs)ds +Zt t0g(s, ys)−g(s, xs)dW(s) 2 . We denote by I(t) = Zt t0f(s, ys)−f(s, xs)ds +Zt t0g(s, ys)−g(s, xs)dW(s). Then, J(t) = |(G(yt)−G(xt)) + I(t)|2. Thus, by Lemma 3.1, we obtain for any 0 < β < 1 2 |y(t)−x(t)|2≤2L(t)+2|(G(yt)−G(xt)) + I(t)|2, ≤2L(t) + 2 β|G(yt)−G(xt)|2+2 1−β|I(t)|2, ≤2L(t)+2βkyt−xtk2+2 1−β|I(t)|2. 5 Therefore, taking supremum and expectation, we have Esup t0≤s≤t|y(s)−x(s)|2≤2+ 2βEsup t0≤s≤t|y(s)−x(s)|2+2 1−βEsup t0≤s≤t|I(s)|2, which implies Esup t0≤s≤t|y(s)−x(s)|2≤2 1−2β+2 (1 −β)(1 −2β)Esup t0≤s≤t|I(s)|2. On the other hand, by the H¨older inequality, the Burkh¨older–Davis–Gundy inequality and the Lipschitz condition, one has Esup t0≤s≤t|I(s)|2≤2E Zt t0 f(s, ys)−f(s, xs)ds 2!+ 2E Zt t0 g(s, ys)−g(s, xs)dW(s) 2! ≤2(t−t0)EZt t0|f(s, ys)−f(s, xs)|2ds +8EZt t0|g(s, ys)−g(s, xs)|2ds ≤2L(T−t0)Zt t0 Ekys−xsk2ds + 8LZt t0 Ekys−xsk2ds = 2L(T−t0+ 4) Zt t0 Ekys−xsk2ds ≤2L(T−t0+ 4) Zt t0 Esup t0≤r≤s|y(r)−x(r)|2ds. Therefore, Esup t0≤s≤t|y(s)−x(s)|2≤2 1−2β+4L(T−t0+ 4) (1 −β)(1 −2β)Zt t0 Esup t0≤r≤s|y(r)−x(r)|2ds. The Gronwall inequality implies now, for all t∈[t0, T] and 0 < β < 1 2, Esup t0≤s≤t|y(s)−x(s)|2≤2 1−2βexp 4L(T−t0+ 4) (1 −β)(1 −2β)(T−t0) ≤2 1−2βexp 4LT(T+ 4) (1 −β)(1 −2β), which implies that E|y(t)−x(t)|2≤M, where M=2 1−2βexp 4LT (T+4) (1−β)(1−2β),∀t∈[t0−τ, T ]. The proof is complete. 6 Theorem 3.3. Assume that hypotheses H2and H3are satisfied and that there exists ψ∈ L1([t0−τ, +∞),R+)such that, for all ϕ1, ϕ2∈C([−τ, 0],Rn)and t∈[t0, T ], |f(t, ϕ1)−f(t, ϕ2)|2∨|g(t, ϕ1)−g(t, ϕ2)|2≤ψ(t)kϕ1−ϕ2k2. Then, (a) Equation (2.1) has a unique solution xwhich belongs to M2([t0−τ, T ],Rn). (b) Equation (2.1) is generalized Ulam-Hyers-Rassias stable with respect to nondecreasing functions z(·)∈C([t0−τ, T ]; Rn). Proof. (a) Uniqueness: Let x1(t) and x2(t) be two solutions of equation (2.1). Using Lemma 2.4 in [14, page 204], both solutions belong to M2([t0−τ, T ],Rn). Moreover, we have x1(t)−x2(t) = G(x1 t)−G(x2 t) + K(t), where K(t) = Zt t0f(s, x1 s)−f(s, x2 s)ds +Zt t0g(s, x1 s)−g(s, x2 s)dW(s). By Lemma 3.1 and assumption H3, x1(t)−x2(t) 2=G(x1 t)−G(x2 t) + K(t) 2 ≤1 βG(x1 t)−G(x2 t) 2+1 1−β|K(t)|2 ≤β x1 t−x2 t  2+1 1−β|K(t)|2. Then, Esup t0≤s≤tx1(s)−x2(s) 2≤βEsup t0≤s≤tx1(s)−x2(s) 2+1 1−βEsup t0≤s≤t|K(s)|2. It follows that Esup t0≤s≤tx1(s)−x2(s) 2≤1 (1 −β)2Esup t0≤s≤t|K(s)|2. On the other hand, Esup t0≤s≤t|K(s)|2≤2 (T−t0+ 4) Zt t0 ψ(s)E x1 s−x2 s  2ds ≤2 (T−t0+ 4) Zt t0 ψ(s)Esup t0≤r≤sx1(r)−x2(r) 2ds. 7 Thus, Esup t0≤s≤tx1(s)−x2(s) 2≤2(T−t0+ 4) (1 −β)2Zt t0 ψ(s)Esup t0≤r≤sx1(r)−x2(r) 2ds By the Gronwall inequality, Esup t0≤t≤Tx1(t)−x2(t) 2= 0, which implies that x1(t) = x2(t) for t0≤t≤T, and therefore for all t0−τ≤t≤T, almost surely. Existence: The proof of the existence is similar to [14], but we take δ=β+2(T−t0+ 4)λ 1−β, where λis a positive constant such that Z+∞ t0−τ ψ(t)dt ≤λ. (b) Using equation (2.3), x(t) is a solution of equation (2.1). By condition (3.2), for z∈C([t0−τ, T ],Rn), we have E y(t)−G(yt)−(y(t0)−G(yt0)) −Zt t0 f(s, ys)ds −Zt t0 g(s, ys)dW(s) 2 ≤z(t),∀t∈[t0−τ, T ]. Moreover, we know that, for t∈[t0−τ, t0], E|y(t)−x(t)|2= 0. Proceeding as in the previous theorem when t∈[t0, T], we have |y(t)−x(t)|2≤2L(t)+2J(t). Then, for any 0 < β < 1 2and t∈[t0, T] |y(t)−x(t)|2≤2L(t)+2βkyt−xtk2+2 1−β|I(t)|2. Thus, taking again supremum and expectation, we have Esup t0≤s≤t|y(s)−x(s)|2≤2z(t)+2βEsup t0≤s≤t|y(s)−x(s)|2+2 1−βEsup t0≤s≤t|I(s)|2. Therefore, Esup t0≤s≤t|y(s)−x(s)|2≤2 1−2βz(t) + 2 (1 −β)(1 −2β)Esup t0≤s≤t|I(s)|2. 8 On the other hand, we can see that Esup t0≤s≤t|I(s)|2≤2(T−t0+ 4) Zt t0 ψ(s)Ekys−xsk2ds ≤2(T−t0+ 4) Zt t0 ψ(s)Esup t0≤r≤s|y(r)−x(r)|2ds ≤2(T+ 4) Zt t0 ψ(s)Esup t0≤r≤s|y(r)−x(r)|2ds. Then, Esup t0≤s≤t|y(s)−x(s)|2≤2 1−2βz(t) + 4(T+ 4) (1 −β)(1 −2β)Zt t0 ψ(s)Esup t0≤r≤s|y(r)−x(r)|2ds. Using the Gronwall lemma, Esup t0≤s≤t|y(s)−x(s)|2≤2 1−2βz(t) exp 4(T+ 4) (1 −β)(1 −2β)Zt t0 ψ(s)ds ≤2 1−2βz(t) exp 4(T+ 4) (1 −β)(1 −2β)Z+∞ −τ ψ(s)ds ≤Cz(t), for all t∈[t0, T], where C=2 1−2βexp λ4(T+ 4) (1 −β)(1 −2β). Finally, for all t∈[t0−τ, T ], we have E|y(t)−x(t)|2≤Cz(t). The proof is complete. 4 Examples In this section we will show two examples to illustrate the applicability and interest of our abstract results. Example 1: Consider the following neutral stochastic functional differential system for each  > 0 and for t∈[t0−τ, T ]            d[x(t)−G(xt)] = f(t, xt)dt +g(t, xt)dW(t), E|x(t)−G(xt)−(x(t0)−G(xt0)) −Rt t0f(s, xs)ds −Rt t0g(s, xs)dW(s)|2≤, xt0=ξ, (4.1) 9