A POWERED GRONWALL-TYPE INEQUALITY AND APPLICATIONS TO STOCHASTIC DIFFERENTIAL EQUATIONS

. In this paper we study a powered integral inequality involving a ﬁnite sum, which can be used to solve the inequalities with singular kernels. We present that the solution of the inequality is decided by a ﬁnite recursion, whose result is proved to be a continuous, bounded or asymptotic function. Mean- while, in order to overcome an obstacle from powers of integrals, we modify the method of monotonization into the powered monotonization. Furthermore, relying on the result and our technique of concaviﬁcation, we discuss a generalized stochastic integral inequality, and give an estimate of the mean square. In the end, as applications, we study uniform boundedness and continuous dependence of solutions for a class of stochastic diﬀerential equation in mean square.


1.
Introduction. Since Gronwall ([8]) and Bellman ([3]) investigated the basic form of integral inequalities, great efforts (see e.g. the monograph [18] and references therein) have been made to develop more complicated forms of integral inequalities separately towards the studies of existence, uniqueness, boundedness and stability of solutions and invariant manifolds for differential equations and integral equations. In 1956 Bihari ([4]) discussed the integral inequality u(t) ≤ α + t 0 f (s)φ(u(s))ds, t ≥ 0.
In 2000 his work was generalized by Lipovan ([13]) to the delay case, i.e., the time t is replaced with a differentiable delay b(t) satisfying 0 < b(t) ≤ t. In 2005 Agarwal, Deng and Zhang ( [1]) extended the result of [13] to the finite sum of integrals where the monotonicity of a is not required but φ i+1 is stronger monotone than φ i for each i. Later, by monotonizing the continuous functions φ i s to build a stronger monotone sequence, Wang ([22]) generalized (1) to the version of two variables, i.e. so-called Wendroff type. Singular integral inequality is another interesting topic. As indicated in [20], the integral t t0 f (t, s)u(s)ds is said to be singular if the kernel f (t, s) is singular at least on the line s = t; it is referred to be weakly singular if it is singular and t t0 |f (t, s)|ds < +∞ for all t 0 ≤ t < T ≤ +∞. The weakly singular integral inequality where β ≥ 0, 0 < γ < 1 and both a and u are nonnegative and locally integrable, was discussed in Henry's book [10]. An estimate of the unknown u was given in terms of Mittag-Leffler function ( [16]) by an iteration approach. Besides, another approach was given by Medved ([15]) in 1997, where the well-known Hölder's Inequality (see [9] or the Appendix) was applied to separate the weakly singular kernel (t − s) γ−1 from the integral. In 2002 Ma and Yang ( [14]) improved Medved's method and gave an estimate for a weakly singular integral inequality with the Erdélyi-Kober kernel s α−1 (t β − s β ) γ−1 (see [12, p.105]). Cheung, Ma and Tseng ( [5]) extended the result of [14] to Wendroff type.
In this paper we discuss the powered Gronwall-type inequality bi(t0) f i (t, s)φ i (u(s))ds} pi (2) for t 0 ≤ t < +∞, where n ∈ N, p i ≥ 1, and all a, b i , f i , φ i and u are nonnegative continuous functions for i = 1, . . . , n. This inequality contains n powers p i , which enable the inequality to be applicable to the integral inequalities with multiple singular kernels. In [5,14,15] integral inequalities with a uniform power were discussed by using the well-known Power Mean Inequality (see [9] or the Appendix), but it does not work in our case of different powers p i . In section 3 of this paper we monotonize the given functions to be a sequence of powered stronger nondecreasing functions rather than a sequence of stronger nondecreasing functions considered in [1], so as to estimate the unknown u by a continuous function which is defined by a finite recursion. In order to investigate the boundedness and asymptotics of the continuous function, we discuss the finite recursion in section 2 as preliminaries.
In section 4 we apply the result to a singular integral inequality, in which there are more than one singular kernels and the kernels are of more general form than those in [15] and [14]. We also apply the result in section 5 to a stochastic integral inequality, a nonlinear extension of the inequalities considered in [2,11]. Unlike [2,17,11,23], we need to concavify some nonlinear functions to deal with the mean square in the stochastic estimation. Finally, in section 6 we apply our results to investigate the uniform boundedness and continuous dependence of solutions on not only initial data but also given functions for a stochastic differential equation in mean square.
2. Finite recursion. Consider the following finite recursion of first order POWERED GRONWALL-TYPE INEQUALITY AND APPLICATIONS TO SDE   7209 for t 0 ≤ t < +∞, where f i s, b i s, p i s and n are given as in (2) For each x : implying that the right hand side of (3), i.e., , is well defined for all t ∈ [t 0 , +∞). In the following lemma we will use the notation C b ([t 0 , +∞), R) to present the set of all bounded functions in C([t 0 , +∞), R), which is a Banach space equipped with the norm x := sup t∈[t0,+∞) |x(t)|. We need the following assumptions: Lemma 1. Suppose that f i s, b i s, p i s, W i s and n are given as in (3). If (P1) holds, On the other hand, the continuity of both b i and f i implies the continuity of the function F i (t) := bi(t) t0 In order to prove the continuity of the operator J i , consider an arbitrary function x * ∈ C b ([t 0 , +∞), R + ) and an arbitrary ε > 0. By the continuity of W i , there exists δ > 0 such that if |u − u * | < δ then In the case (C1), from (6) we have where we note that W i is continuous and strictly increasing and lim t→+∞ +∞ t0 f i (t, s)ds = 0. By induction we can prove that 0 ≤ lim sup t→+∞ x n (t) ≤ lim sup t→+∞ x 0 (t) = 0.
Further, we give the continuous dependence of x n (t) on the initial value x 0 (t) in (5). For convenience, let Proof. The assertion given in the case (P2) in the proof of Lemma 2 shows that the composition W −1 On the other hand, since |W i (u 1 ) − W i (u 2 )| ≥ L 1 |u 1 − u 2 | for all u 1 , u 2 ∈ R + , we get By (8) and (9) we obtain By induction we can prove (7) and completes the proof.
3. Powered Gronwall-type inequality. In this section, consider inequality (2), where those given functions a, b i , f i , φ i were assumed to be continuous and nonnegative. We simply suppose that for all i = 1, . . . , n, Otherwise, one can monotonize them as done in [1]. For the functions φ i (i = 1, ..., n), we need not only a monotonization for each but also a reduction to a sequence of strongly nondecreasing functions for them. Define w i s (i = 1, ..., n) recursively by where p i ≥ 1 is given in (2). We can easily check that the sequence {w i } satisfies: (i): every w i is nondecreasing; (ii): for every i = 1, . . . , n − 1 the ratio w pi+1 i+1 /w pi i is nondecreasing, i.e., w pi i ∝ w pi+1 i+1 as denoted in [19] and [1]. For convenience, such a sequence {w i } is said to be strongly nondecreasing in power of the sequence {p i }. The case that p i ≡ 1 for all i = 1, . . . , n is exactly the strongly nondecreasing one considered in [1,22]. We will amplify the 'bad' φ i s to be the 'good' w i s in the proof of the main theorem. The procedure of amplification is referred to as a powered strong monotonization. Let where x 0 > 0 is an arbitrarily given constant. We will remark that the choice of x 0 does not affect our result just after Theorem 1.
Note that we assume φ i (s) > 0 for all s ≥ 0 in (H3). Otherwise, we can amplify φ i a little to beφ with an arbitrarily chosen constant ε > 0 in (2). Clearly, Thus, the above defreezing process (12) makes the hypothesis (H3) reasonable. Those φ i s having no zeros guarantee that w i s, appearing in the denominators of (10) and (11), have no zeros and therefore (10) and (11) are meaningful for all s, x ∈ R + .
Proof of Lemma 5. We claim that If the claimed (13) is true, then w pi i (s)/s is nonincreasing in s ∈ R + . Clearly, w pi i (s)/s > 0 for all s ∈ R + by (H3) and (10). Then, there exists a constant C > 0 such that w pi This proves the lemma. In order to prove (13), we note that w 1 (s) = max τ ∈[0,s] {φ 1 (τ )} by (10), which implies that w 1 ∈ C(R + , R + \{0}) is nondecreasing, as shown in Figure 1. As introduced in [28], an interior point x * in an interval I is called a fort of a mapping F : I → I if F is not strictly monotone in any small neighborhood of x * . Let S * (w 1 ) consist of such forts s * of w 1 that w 1 is strictly increasing in either a left halfneighborhood (s * − δ, s * ] or a right half-neighborhood [s * , s * + δ) with a sufficiently small δ > 0. There are at most countably many such small half neighborhoods because they can be chosen so small not to intersect each other but each of them contains a rational. Hence, S * (w 1 ) is a countable set. Let S * (w 1 ) := {s k : k = 1, ..., ς}, where ς denotes the cardinality of S * (w 1 ) and can be equal to ∞, and use it to partition R + into where s 0 := 0 and s ς+1 := +∞. Then, for each k = 0, ..., ς, the restriction w 1 | [s k ,s k+1 ) is nondecreasing as shown in Figure 2, i.e., w 1 | [s k ,s k+1 ) is either strictly increasing or flat; otherwise, there is a points ∈ S * ∩ (s k , s k+1 ) as shown in Figure  3, a contradiction to the choice of S * (w 1 ). If w 1 | [s k ,s k+1 ) is strictly increasing, then Otherwise Clearly, s/w p1 1 (s k ) is nondecreasing. It follows from (15) that (13) for i = 1 also holds on [s k , s k+1 ).
The result of Lemma 5, i.e., W i (+∞) = +∞, is required in the beginning of section 2. It guarantees that those lemmas given in section 2 is applicable with W i s defined in (11) by w i s and therefore by φ i s. In particular, by Lemma 2, the result .., n. So, we are ready to give the following main result. Theorem 1. Suppose that (H1)-(H3) hold and u(t) satisfies (2) for t 0 ≤ t < +∞. Then where J i and W i are defined in (4) and (11) respectively for i = 1, ..., n.
Remark that, as indicated in [1], different choices of x 0 in W i do not affect the result of Theorem 1. In fact, for positive constant y 0 = x 0 , let Proof of Theorem 1. First of all, we monotonize functions φ i s in (2). In view of [1,22], we employ the procedure of powered strong monotonization, mentioned just before (11), to amplify the functions φ i (i = 1, . . . , n) into the functions w i (i = 1, . . . , n) defined in (11). Then, if u satisfies inequality (2), we have for an arbitrarily chosen T ∈ [t 0 , +∞). In what follows, we solve u from (17), regarded as the auxiliary inequality of (2). Let and define a 0 (T, t) := a(T ) and a i (T, t) : Clearly, for each nonnegative function implying that the right hand side of (19) is well defined for all t ∈ [t 0 , +∞) as in Lemma 2.
In order to prove (19) inductively, assume that (19) holds for n = m. Let u satisfy the inequality Then it can be written as where f i (T, s)w i (u(s))ds} pi , a nonnegative and nondecreasing function on [t 0 , +∞). One can compute the derivative in the integrand of which we have Then Integrating both sides of the above inequality from t 0 to t, we get for all t ∈ [t 0 , T ]. Let ξ(t) := W 1 (ζ(t) + a(T )). Then the above inequality can be rewritten as Since we know that each ϕ i+1 (W −1 1 (u)) (i = 1, . . . , m) is continuous, nondecreasing and positive on [0, +∞) and ϕ pi i (W −1 1 (u)) ∝ ϕ pi+1 i+1 (W −1 1 (u)) (i = 2, . . . , m). It follows that (22) is of the same form as (17) for n = m and fulfills the inductive assumption. Therefore, where γ 0 (T, t) :=J 1 a 0 (T, t), Thus, we get from (21) and (23) that In order to simplify (24), we claim that W −1 1 (γ i (T, t)) = a i+1 (T, t) for i = 1, . . . , m. It is easy to verify that i.e., the claim is true for i = 0. Assume that the claim holds for i = k. Then by the inductive assumption, (T, t), which proves the claim by induction. It follows from (24) and the claim that

Thus, the assertion (19) is proved by induction.
Finally, we observe (19) and (18) and that t ≤ T . Letting t = T , we have , ∀T ∈ [t 0 , +∞). Since T is chosen arbitrarily, (16) is proved and the proof of Theorem 1 is completed.
Our Theorem 1 improves some results obtained in [1,6,22,26]. Look at [1] for instance, where the unknown function u is estimated by but a n (t) is determined recursively by In contrast, the result (16) of our Theorem 1, i.e., u(t) ≤ (W −1 n • J n ) • · · · • (W −1 1 • J 1 )a(t), is simply dependent on a(t). One can use Lemmas 2-4, properties of those operators J i , to give boundedness, asymptotics and dependence for the unknown u, as done in section 6 for a stochastic differential equation. Besides, [1] gives the existence interval [t 0 , T 1 ) for the unknown u, where T 1 is determined by , i = 1, ..., n, but in contrast our existence interval obtained in Theorem 1 is [t 0 , +∞).
Remark that the method used in [5,6,14,15,21] for a single power does not work in our case. In fact, when p i ≡ p for i = 1, . . . , n, the inequality (27) is of a single power, i.e., q i ≡ q for i = 1, . . . , n, which includes those powered inequalities considered in [5,6,14,15,21]. In this case, λ = q in (28). Thus, the power q outside the integrals in (27) is eliminated and the method of monotonization used in [1,22] is applicable. However, this method is not efficient to our case of different powers.

5.
Application to stochastic inequality. The development of stochastic differential equations demands results on stochastic integral inequalities (see e.g. [2,11,23]). Let (Ω, F, P ) be a probability space with a nonempty set Ω, a σ-algebra F on Ω consisting of subsets of Ω, and a probability measure P . A random variable X(ω) on (Ω, F, P ) is an F-measurable function X : Ω → R n , i.e., X −1 (U ) := {ω ∈ Ω|X(ω) ∈ U } ∈ F for each Borel set U ⊂ R n . The expectation of a random variable X(ω) is defined by the integral EX(ω) := Ω X(ω)dP (ω) provided that Ω |X(ω)|dP (ω) < +∞. Let {B t (ω) : t ≥ 0} be a 1-dimensional standard Brownian motion in Ω. Suppose that α(ω) is a random variable independent of the σ-algebra generated by all B s (t 0 ≤ s ≤ t) and satisfies that E|α(ω)| 2 < +∞, F t is the σ-algebra generated by all B s (0 ≤ s ≤ t) and α(ω), and M 2 ω [t 0 , T ] consists of all measurable and F t -adapted stochastic processes G such that T t0 EG 2 (s, ω)ds < +∞. In 1946 Itô ([11]) considered the stochastic integral inequality system where β, γ > 0 are constant and G ∈ M 2 ω [t 0 , T ]. He obtained the estimate of mean square u(t, ω). It is worthy mentioning that, unlike deterministic ones, one cannot amplify the integrand G in the first inequality of (30) with the second inequality of (30) because the Itô integral t t0 G(s, ω)dB s , being a stochastic process, does not satisfy the triangle inequality with respect to the absolute value, as indicated in many monographs e.g. [7,17]. In 2005 Amano ([2]) discussed an extension of (30), where α(ω) was replaced by a given stochastic process.
Before we state our result, we need the notations For amplification in inequalities, we need to modify φ i and ψ j to be concave functions, which are defined bỹ Corollary 2. Suppose that (A1)-(A4) hold and u(t, ω) satisfies (31). Then the mean square Eu 2 (t, ω) has the estimate for all t ∈ [t 0 , T ].
We first claim that lim sup In fact,φ i ,ψ j ∈ C 1 (R + , R + \{0}) as known in (A4), which implies that φ i , ψ j ∈ C 1 (R + , R + \{0}) and further implies that φ i , ψ j ∈ C(R + \{0}, R). Moreover,φ i (s) ∝ s, i.e.,φ i (s)/s is nonincreasing, as known in (A4). It follows that which givesφ i (s) ≤φ i (s)/s. Thus, we have where the definition (32) which proves the first inequality of claimed (44). Similarly, we can prove the second inequality and complete the proof of the claimed (44). Next, we need the following lemma but leave its proof to the end of this section.
We end this section with the following proof.
6. Applications to a stochastic equation. In this section we end this paper with applications to the stochastic differential equation dx(t, ω) = F (t, x(t, ω))dt + G(t, x(t, ω))dB t , for all t ∈ [0, T ], f, g ∈ C([0, T ], R + ) andφ,ψ ∈ C 1 (R + , R + ) and α, B t , F t and M 2 ω [0, T ] are defined in the beginning of Section 5. An earlier work was given by Itô ([11]) in 1946 for the existence and uniqueness for solutions of equation (50) in the special case that f, g are both constant andφ(x) =ψ(x) = |x|, which is equivalent to the Lipschitz conditions for F and G. Later, in 1971 Yamada and Watanabe ( [24,25]) generally considered the condition (51), where f, g are both constant,φ is a strictly increasing and concave nonnegative function andψ is a strictly increasing positive function such that x0 0 dx/ψ 2 (x) = +∞ for some x 0 > 0 and gave the uniqueness for solutions of equation (50).