Báo cáo hóa học: " Research Article Fixed Points and Stability in Neutral Stochastic Differential Equations with Variable Delays" pot

Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2008, Article ID 407352, 11 pages doi:10.1155/2008/407352 Research Article Fixed Points and Stability in Neutral Stochastic Differential Equations with Variable Delays Meng Wu, 1 Nan-jing Huang, 1 and Chang-Wen Zhao 2 1 Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China 2 College of Business and Management, Sichuan University, Chengdu, Sichuan 610064, China Correspondence should be addressed to Nan-jing Huang, nanjinghuang@hotmail.com Received 4 April 2008; Accepted 9 June 2008 Recommended by Tomas Dom ´ ınguez Benavides We consider the mean square asymptotic stability of a generalized linear neutral stochastic differential equation with variable delays by using the fixed point theory. An asymptotic mean square stability theorem with a necessary and sufficient condition is proved, which improves and generalizes some results due to Burton, Zhang and Luo. Two examples are also given to illustrate our results. Copyright q 2008 Meng Wu et a l. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction Liapunov’s direct method has been successfully used to investigate stability properties of a wide variety of differential equations. However, there are many difficulties encountered in the study of stability by means of Liapunov’s direct method. Recently, Burton 1–4,Jung5, Luo 6,andZhang7 studied the stability by using the fixed point theory which solved the difficulties encountered in the study of stability by means of Liapunov’s direct method. Up till now, the fixed point theory is almost used to deal with the stability for deterministic differential equations, not for stochastic differential equations. Very recently, Luo 6 studied the mean square asymptotic stability for a class of linear scalar neutral stochastic differential equations. For more details of the stability concerned with the stochastic differential equations, we refer to 8, 9 and the references therein. Motivated by previous papers, in this paper, we consider the mean square asymptotic stability of a generalized linear neutral stochastic differential equation with variable delays by using the fixed point theory. An asymptotic mean square stability theorem with a necessary 2 Fixed Point Theory and Applications and sufficient condition is proved. Two examples is also given to illustrate our results. The results presented in this paper improve and generalize the main results in 1, 6, 7. 2. Main results Let Ω, F, {F t } t≥0 ,P be a complete filtered probability space and let Wt denote a one- dimensional standard Brownian motion defined on Ω, F, {F t } t≥0 ,P such that {F t } t≥0 is the natural filtration of Wt.Letat,bt, bt,ct,et,qt ∈ CR  ,R, and τt,δt ∈ CR  ,R   with t − τt →∞and t − δt →∞as t →∞.HereCS 1 ,S 2  denotes the set of all continuous functions φ : S 1 →S 2 with the supremum norm ·. In 2003, Burton 1 studied the equation x  t−btx  t − τt  2.1 and proved the following theorem. Theorem A Burton 1. Suppose that τtr and there exists a constant α<1 such that  t t−r   bs  r   ds   t 0   bs  r   e −  t s burdu  s s−r   bu  r   du ds ≤ α 2.2 for all t ≥ 0 and  ∞ 0 bsds  ∞. Then, for every continuous initial function φ : −r, 0 →R,the solution xtxt, 0,φ of 2.1 is bounded and tends to zero as t →∞. Recently, Zhang 7 studied the generalization of 2.1 as follows: x  t− n  j1 b j tx  t − τ j t  2.3 and obtained the following theorem. Theorem B Zhang 7. Suppose that τ j is differential, the inverse function g j t of t − τ j t exists, and there exists a constant α ∈ 0, 1 such that for t ≥ 0, lim inf t→∞  t 0 Qsds > −∞ and n  j1   t t−τ j t   b j  g j s    ds   t 0 e −  t s Qudu   b j s     τ  j s   ds   t 0 e −  t s Qudu   Qs    s s−τ j s   b j  g j v    dv ds  ≤ α, 2.4 where Qt  n j1 b j g j t. Then the zero solution of 2.3 is asymptotically stable if and only if  t 0 Qsds →∞,ast →∞. Very recently, Luo 6 considered the following neutral stochastic differential equation: d  xt − qtx  t − τt    atxtbtx  t − τt  dt   ctxtetx  t − δt  dWt 2.5 and obtained the following theorem. Meng Wu et al. 3 Theorem C Luo 6. Let τt be derivable. Assume that there exists a constant α ∈ 0, 1 and a continuous function ht : 0, ∞ → R such that for t ≥ 0, lim inf t→∞  t 0 hsds > −∞ and   qt     t t−τt   ashs   ds  t 0 e −  t s hudu    a  s−τs  h  s−τs  1−τ  s  bs−qshs   ds   t 0 e −  t s hudu   hs    s s−τs   auhu   du ds    t 0 e −2  t s hudu    cs      es    2 ds  1/2 ≤ α. 2.6 Then the zero solution of 2.5 is mean square asymptotically stable if and only if  t 0 hsds →∞, as t →∞. Now, we consider the generalization of 2.5: d  xt − n  j1 q j tx  t − τ j t    n  j1 b j tx  t − τ j t  dt  n  j1 c j tx  t − δ j t  dWt, 2.7 with the initial condition xsφs for s ∈  mt 0 ,t 0  , 2.8 where φ ∈ Cmt 0 ,t 0 ,R, b j t,c j t,q j t ∈ CR  ,R, τ j t,δ j t ∈ CR  ,R  , t − τ j t →∞, and t − δ j t →∞as t →∞and for each t 0 ≥ 0, m j t 0 min  inf  s − τ j s,s≥ t 0  , inf  s − δ j s,s≥ t 0  , m  t 0   min  m j  t 0  , 1 ≤ j ≤ n  . 2.9 Note that 2.7 becomes 2.5 for n  2, τ 1 t0,τ 2 tτt, b 1 tat,b 2 tbt, q 1 t 0,q 2 tqt, δ 1 t0,δ 2 tδt, c 1 tct, and c 2 tet. Thus, we know that 2.7 includes 2.1, 2.3,and2.5 as special cases. Our aim here is to generalize Theorems B and C to 2.7. Theorem 2.1. Suppose that τ j is differential, and there exist continuous functions h j t : 0, ∞ →R for j  1 ···n and a constant α ∈ 0, 1 such that for t ≥ 0 i lim inf t→∞  t 0 Hsds > −∞, ii n  j1   q j t    n  j1  t t−τ j t   h j s   ds  n  j1  t 0 e −  t s Hudu    h j  s − τ j s  1 − τ  j s   b j s−q j sHs   ds  n  j1  t 0 e −  t s Hudu   Hs    s s−τ j s   h j u   du ds  2 ⎛ ⎝  t 0 e −2  t s Hudu  n  j1   c j s    2 ds ⎞ ⎠ 1/2 ≤ α<1, 2.10 where Ht  n j1 h j t. 4 Fixed Point Theory and Applications Then the zero solution of 2.7 is mean square asymptotically stable if and only if  t 0 Hsds −→ ∞ as t −→ ∞. 2.11 Proof. For each t 0 ,denotebyS the Banach space of all F-adapted processes ψt, ω : mt 0 , ∞× Ω → R which are almost surely continuous in t with norm ψ S   E  sup s≥mt 0    ψs, ω   2  1/2 . 2.12 Moreover, we set ψt, ωφt for t ∈ mt 0 ,t 0  and E|ψt, ω| 2 →0, as t →∞. At first, we suppose that 2.11 holds. Define an operator P : S →S by Pxtφt for t ∈ mt 0 ,t 0  and for t ≥ t 0 , Pxt  φt 0  − n  j1 q j t 0 φ  t 0 − τ j t 0   − n  j1  t 0 t 0 −τ j t 0  h j sφsds  e −  t t 0 Hudu  n  j1 q j tx  t − τ j t   n  j1  t t−τ j t h j sxsds   t t 0 e −  t s Hudu n  j1  h j  s − τ j s  1 − τ  j s   b j s − q j sHs  x  s − τ j s  ds −  t t 0 e −  t s Hudu Hs  n  j1  s s−τ j s h j uxudu  ds   t t 0 e −  t s Hudu  n  j1 c j sx  s − δ j s   dWs : 5  i1 I i t. 2.13 Now, we show the mean square continuity of P on t 0 , ∞.Letx ∈ S, T 1 > 0, and let |r| be sufficiently small. Then E   Px  T 1  r  − Px  T 1    2 ≤ 5 5  i1 E   I i  T 1  r  − I i  T 1    2 . 2.14 It is easy to verify that E   I i  T 1  r  − I i  T 1    2 −→ 0, as r −→ 0,i 1, 2, 3, 4. 2.15 Meng Wu et al. 5 It follows from the last term I 5 in 2.13 that E   I 5  T 1  r  − I 5  T 1    2  E       T 1 t 0 e −  T 1 s Hudu  e −  T 1 r T 1 Hudu − 1  n  j1 c j sx  s − δ j s  dWs   T 1 r T 1 e −  T 1 r s Hudu n  j1 c j sx  s − δ j s  dWs      2 ≤ 2E  T 1 t 0 e −2  T 1 s Hudu  e −  T 1 r T 1 Hudu − 1  2  n  j1   c j s   ·   x  s − δ j s     2 ds  2E  T 1 r T 1 e −2  T 1 r s Hudu  n  j1   c j s   ·   x  s − δ j s     2 ds −→ 0, as r −→ 0. 2.16 Therefore, P is mean square continuous on t 0 , ∞. Next, we verify that Px ∈ S. Since E|xt|→0, t −δ j t →∞as t →∞,foreach>0, there exists a T 1 >t 0 such that s ≥ T 1 implies E|xs| 2 <and E|xs − δ j s| 2 <. Thus, for t ≥ T 1 , the last term I 5 in 2.13 satisfies E   I 5 t   2 ≤ E  T 1 t 0 e −2  t s Hudu  n  j1 c j sx  s − δ j s   2 ds  E  t T 1 e −2  t s Hudu  n  j1 c j sx  s − δ j s   2 ds ≤ E  sup s≥mt 0    xs   2   T 1 t 0 e −2  t s Hudu  n  j1   c j s    2 ds    t T 1 e −2  t s Hudu  n  j1   c j s    2 ds. 2.17 By condition ii and 2.11, there exists T 2 >T 1 such that t ≥ T 2 implies E|I 5 t| 2 < α. 2.18 Thus, E|I 5 t| 2 →0, as t →∞. Similarly, we can show that E|I i t| 2 →0, i  1, 2, 3, 4, as t →∞. Thus, E|Pxt| 2 →0ast →∞. This yields Px ∈ S. Now we show that P : S →S is a contraction mapping. From ii, we can choose ε>0 such that α 2  ε<1. Thus, for each t 0 ≥ 0, we can find a constant L>0 such that  1  1 L   n  j1   q j t    n  j1  t t 0 e −  t s Hudu   Hs    s s−τ j s   h j u   du ds  n  j1  t t−τ j t   h j s   ds n  j1  t t 0 e −  t s Hudu   h j s−τ j s1−τ  j s b j s−q j sHs   ds  2  41  L  t t 0 e −2  t s Hudu  n  j1   c j s    2 ds ≤ α 2  ε<1. 2.19 6 Fixed Point Theory and Applications For any x, y ∈ S, it follows from 2.13, conditions i and ii, and Doob’s L p -inequality see 10 that e sup s≥mt 0    pxs − pys   2  e sup s≥t 0     n  j1 q j s  x  s − τ j s  − y  s − τ j s   n  j1  s s−τ j s h j v  xv − yv  dv   s t 0 e −  s v hudu n  j1  h j  v − τ j v  1 − τ  j v   b j v − q j vhv  ×  x  v − τ j v  − y  v − τ j v  dv −  s t 0 e −  s v hudu hv  n  j1  v v−τ j v h j u  xu − yu  du  dv   s t 0 e −  s v hudu  n  j1 c j v  x  v − δ j v  − y  v − δ j v   dwv     2 ≤  1  1 l  e sup s≥t 0  n  j1   q j s   ·   x  s − τ j s  − y  s − τ j s     n  j1  s s−τ j s   h j v   ·   xv − yv   dv   s t 0 e −  s v hudu n  j1   h j  v − τ j v  1 − τ  j v   b j v − q j vhv   ·   x  v − τ j v  − y  v − τ j v    dv   s t 0 e −  s v hudu hv  n  j1  v v−τ j v   h j u   ·   xu − yu   du  dv  2  41  l sup s≥t 0  e  s t 0 e −  s v hudu  n  j1   c j v   ·   x  v − δ j v  − y  v − δ j v     2 dv  ≤ e sup s≥mt 0    xs − ys   2 ·sup s≥t 0  1  1 l  n  j1   q j s    n  j1  s t 0 e −  s v hudu   hv    v v−τ j v   h j u   du ds  n  j1  s s−τ j s   h j v   dv  n  j1  s t 0 e −  s v hudu ×    h j  v − τ j v  1 − τ  j v   b j v − q j vhv   dv  2  41  l  s t 0 e −2  s v hudu  n  j1   c j v    2 dv  ≤  α 2  ε  e sup s≥mt 0    xs − ys   2 . 2.20 Meng Wu et al. 7 Therefore, P is contraction mapping with contraction constant α 2  ε. By the contraction mapping principle, P has a fixed point x ∈ S, which is a solution of 2.7 with xsφs on mt 0 ,t 0  and E|xt| 2 →0ast →∞. To obtain the mean square asymptotic stability, we need to show that the zero solution of 2.7 is mean square stable. Let >0 be given and choose δ>0andδ<satisfying the following condition: 4δK 2 1  Le 2  t 0 0 Hudu   α 2  ε  <, 2.21 where K  sup t≥0 {e −  t 0 Hsds }.Ifxtxt, t 0 ,φ is a solution of 2.7 with φ 2 <δ,then xtPxt defined in 2.13. We assume that E|xt| 2 <for all t ≥ t 0 . Notice that E|xt| 2  φ 2 <for t ∈ mt 0 ,t 0 .Ifthereexistst ∗ >t 0 such that E|xt ∗ | 2   and E|xt| 2 <for t ∈ mt 0 ,t ∗ ,then2.13 and 2.19 imply that E   x  t ∗    2 ≤ 1  Lφ 2  1  n  j1   q j  t 0     n  j1  t 0 t 0 −τ j t 0    h j s   ds  2 e −2  t ∗ t 0 Hudu    1  1 L   n  j1   q j  t ∗     n  j1  t ∗ t ∗ −τ j t ∗    h j s   ds   t ∗ t 0 e −  t ∗ s Hudu  n  j1  s s−τ j s   h j u   du    Hs   ds   t ∗ t 0 e −  t ∗ s Hudu n  j1   h j  s − τ j s  1 − τ  j s   b j s − q j sHs   ds  2    t ∗ t 0 e −2  t ∗ s Hudu  n  j1   c j s    2 ds ≤ 1  Lδ  1  n  j1   q j  t 0     n  j1  t 0 t 0 −τ j t 0    h j s   ds  2 e −2  t ∗ t 0 Hudu   α 2  ε  <, 2.22 which contradicts the definition of t ∗ . Thus, the zero solution of 2.7 is stable. It follows that the zero solution of 2.7 is mean square asymptotically stable if 2.11 holds. Conversely, we suppose that 2.11 fails. From i, there exists a sequence {t n } with t n →∞as n →∞such that lim n→∞  t n 0 Hudu  β,whereβ ∈ R. Then, we can choose a constant J>0 satisfying  t n 0 Hudu ∈ −J, J for all n ≥ 1. Denote ωs n  j1    h j  s − τ j s  1 − τ  j s   b j s − q j sHs      Hs    s s−τ j s   h j u   du 2.23 for all s ≥ 0. From ii,wehave  t n 0 e −  t n s Hudu ωsds ≤ α, 2.24 8 Fixed Point Theory and Applications which implies  t n 0 e  s 0 Hudu ωsds ≤ αe  t n 0 Hudu ≤ e J . 2.25 Therefore, the sequence {  t n 0 e  s 0 Hudu ωsds} has a convergent subsequence. Without loss of generality, we can assume that lim n→∞  t n 0 e  s 0 Hudu ωsds  γ 2.26 for some γ>0. Let k be an integer such that  t n t k e  s 0 Hudu ωsds < δ 0 8K 2.27 for all n ≥ k,whereδ 0 > 0satisfies8δ 0 K 2 e 2J α 2  ε < 1. Now we consider the solution xtxt, t k ,φ of 2.7 with φt k  2  δ 0 and φs 2 < δ 0 for s<t k . By the similar method in 2.22,wehaveE|xt| 2 < 1fort ≥ t k . We may choose φ so that Gt k  : φt k  − n  j1 q j t k φ  t k − τ j t k   − n  j1  t k t k −τ j t k  h j sφsds ≥ 1 2 δ 0 . 2.28 It follows from 2.13 and 2.28 with xtPxt that for n ≥ k, E      xt n  − n  j1 q j t n x  t n − τ j  t n  − n  j1  t n t n −τ j t n  h j sxsds      2 ≥ G 2 t k e −2  t n t k Hudu − 2Gt k e −  t n t k Hudu  t n t k e −  t n s Hudu ωsds ≥ δ 0 2 e −2  t n t k Hudu  δ 0 2 − 2K  t n t k e  s 0 Hudu ωsds  ≥ δ 2 0 8 e −2J > 0. 2.29 If the zero solution of 2.7 is mean square asymptotic stable, then E|xt| 2  E|xt, t k ,φ| 2 →0ast →0. Since t n − τ j t n  →∞, t n − δ j t n  →∞ as n →∞ and condition ii and 2.11 hold, E      xt n  − n  j1 q j t n x  t n − τ j  t n  − n  j1  t n t n −τ j t n  h j sxsds      2 −→ 0, as n −→ ∞, 2.30 which contradicts 2.29. Therefore, 2.11 is necessary for Theorem 2.1. This completes the proof. Remark 2.2. Theorem 2.1 still holds if condition ii is satisfied for t ≥ t a for some t a ∈ R  . Meng Wu et al. 9 Remark 2.3. Theorem 2.1 improves Theorem C under different conditions. Corollary 2.4. Suppose that τ j is differential, the inverse function g j t of t − τ j t exists, and there exists a constant α ∈ 0, 1 such that for t ≥ 0, lim inf t→∞  t 0 Qsds > −∞ and n  j1   q j t    n  j1  t t−τ j t   b j  g j s    ds  n  j1  t 0 e −  t s Qudu   b j sτ  j s − q j sQs   ds  n  j1  t 0 e −  t s Qudu   Qs    s s−τ j s   b j  g j u    du ds2 ⎛ ⎝  t 0 e −2  t s Qudu  n  j1   c j s    2 ds ⎞ ⎠ 1/2 ≤ α<1, 2.31 where Qt  n j1 − b j g j t. Then the zero solution of 2.7 is mean square asymptotically stable if and only if  t 0 Qsds →∞as t →∞. Remark 2.5. When h j t−b j g j t for j  1 ···n, Theorem 2.1 reduces to Corollary 2.4.Onthe other hand, we choose q j t ≡ c j t ≡ 0andb j ≡−b j for j  1 ···n,thenCorollary 2.4 reduces to Theorem B. 3. Two examples In this section, we give two examples to illustrate applications of Theorem 2.1 and Corollary 2.4. Example 3.1. Consider the following linear neutral stochastic delay differential equation: d  xt− xt − t/2 1000    − xt−t/2 1616t − 3sin t4 4848t x  t − t 4  dt   xt 24 √ 3  4t − xt−sin t 12 √ 34t  dWt. 3.1 Then the zero solution of 3.1 is mean square asymptotically stable. Proof. Choosing h 1 t1/8  16t and h 2 t7/48  64t in Theorem 2.1,wehave Ht 1 8  16t  7 48  64t , 11 48  64t ≤ Ht ≤ 13 48  64t , 2  j1  t t−τ j t   h j s   ds   t t/2 1 8  16s ds   t 3t/4 7 48  64s ds −→ 0.07479, as t −→ ∞, 2  j1  t 0 e −  t s Hudu   Hs    s s−τ j s   h j u   du ds ≤  t 0 e −  t s 11/4864udu 13 48  64s ·0.07479 ds ≤ 0.08839, 2 ⎛ ⎝  t 0 e −2  t s Hudu  2  j1   c j s    2 ds ⎞ ⎠ 1/2 ≤ 2   t 0 e −  t s 11/2432udu 1 824  32s ds  1/2 ≤ 0.21320, 2  j1  t 0 e −  t s Hudu   h j s − τ j s1 − τ  j s  b j s − q j sHs   ds ≤  t 0 e −  t s 11/4864udu  0.013 48  64s  17 144  192s  ds ≤ 0.013 11  17 33  0.51634. 3.2 10 Fixed Point Theory and Applications It easy to check that  ∞ 0 Hsds  ∞.Letα  0.001  0.07479  0.08839  0.21320  0.51634. Then, α  0.89372 < 1andthezerosolutionof3.1 is mean square asymptotically stable by Theorem 2.1. Example 3.2. Consider the following delay differential equation: x  t− 1 6  4t x  t − t 3  − 1 12  4t x  t − 2 3 t  . 3.3 Then the zero solution of 3.3 is asymptotically stable. Proof. Choosing h 1 th 2 t1/4  4t in Theorem 2.1,wehaveHt1/2  2t and 2  j1  t t−τ j t   h j s   ds   t 2/3t 1 4  4s ds   t t/3 1 4  4s ds −→ 1 2 ln 3 − 1 4 ln 2  0.37602, as t −→ ∞, 2  j1  t 0 e −  t s Hudu   Hs    s s−τ j s   h j u   du ds ≤  t 0 e −  t s 1/22udu 1 2  2s ·0.37602 ds ≤ 0.37602. 3.4 Notice that q j tc j t ≡ 0and 2  j1    h j  s − τ j s  1 − τ  j s   b j s − q j sHs        3 12  8s · 2 3 − 1 6  4s          3 12  4s · 1 3 − 1 12  4s      0. 3.5 It is easy to see that all the conditions of Theorem 2.1 hold for α  0.376020.37602  0.75204 < 1. Thus, Theorem 2.1 implies that the zero solution of 3.3 is asymptotically stable. However, Theorem B cannot be used to verify that the zero solution of 3.3 is asymptotically stable. In fact, b 1 t1/6  4t, b 2 t1/12  4t, b 1 g 1 t  1/6  6t, b 2 g 2 t  1/12  12t,and|Qt|  1/4  4t.Ast →∞, 2  j1  t t−τ j t   b j  g j s    ds ≤  t 2/3t 1 6  6s ds   t t/3 1 12  12s ds −→ 1 4 ln 3 − 1 6 ln 2  0.15913. 3.6 Notice that 2  j1   b j sτ  j s − q j sQs    1 18  12s  1 18  6s ≤ 1 4  4s . 3.7 It follows from 3.7 that 2  j1  t 0 e −  t s Qudu   b j sτ  j s − q j sQs   ds ≤  t 0 e −  t s 1/44udu 1 4  4s ds ≤ 1. 3.8 [...]... Burton and T Furumochi, Fixed points and problems in stability theory for ordinary and functional differential equations, ” Dynamic Systems and Applications, vol 10, no 1, pp 89–116, 2001 5 S.-M Jung, “A fixed point approach to the stability of a Volterra integral equation,” Fixed Point Theory and Applications, vol 2007, Article ID 57064, 9 pages, 2007 6 J Luo, Fixed points and stability of neutral stochastic. .. differential equations, ” Journal of Mathematical Analysis and Applications, vol 334, no 1, pp 431–440, 2007 7 B Zhang, Fixed points and stability in differential equations with variable delays,” Nonlinear Analysis, vol 63, no 5–7, pp e233–e242, 2005 8 V B Kolmanovskii and L E Shaikhet, “Matrix Riccati equations and stability of stochastic linear systems with nonincreasing delays,” Functional Differential Equations, ... 20060610005 References 1 T A Burton, Stability by fixed point theory or Liapunov theory: a comparison,” Fixed Point Theory, vol 4, no 1, pp 15–32, 2003 2 T A Burton, “Liapunov functionals, fixed points, and stability by Krasnoselskii’s theorem,” Nonlinear Studies, vol 9, no 2, pp 181–190, 2002 3 T A Burton, Fixed points and stability of a nonconvolution equation,” Proceedings of the American Mathematical... no 3-4, pp 279–293, 1997 9 K Liu, Stability of In nite Dimensional Stochastic Differential Equation with Applications, vol 135 of Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2006 10 I Karatzas and S E Shreve, Brownian Motion and Stochastic Calculus, vol 113 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 2nd edition,...Meng Wu et al 11 From 3.6 , we obtain 2 j 1 t t e− s Q u du Q s 0 s b j gj u s−τj s du ds ≤ t 0 t e− s 1 1/ 4 4u du 4 4s · 0.15913 ds ≤ 0.15913 3.9 Combining 3.6 , 3.8 , and 3.9 , we see that the condition 2.4 of Theorem B does not hold with α 1.31825 Acknowledgement This work was supported by the National Natural Science Foundation of China 10671135 and Specialized Research Fund for the Doctoral Program . Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2008, Article ID 407352, 11 pages doi:10.1155/2008/407352 Research Article Fixed Points and Stability in Neutral Stochastic Differential. point approach to the stability of a Volterra integral equation,” Fixed Point Theory and Applications, vol. 2007, Article ID 57064, 9 pages, 2007. 6 J. Luo, Fixed points and stability of neutral. 3679–3687, 2004. 4 T. A. Burton and T. Furumochi, Fixed points and problems in stability theory for ordinary and functional differential equations, ” Dynamic Systems and Applications, vol. 10, no.