Mild Solutions of Neutral Stochastic Partial Functional Differential Equations
This paper studies the existence and uniqueness of a mild solution for a neutral stochastic partial functional differential equation using a local Lipschitz condition. When the neutral term is zero and even in the deterministic special case, the result obtained here appears to be new. An example is included to illustrate the theory.
In this paper, a neutral stochastic partial functional differential equation is considered in a real separable Hilbert space of the form where .
A study of such class of (1.1) was initiated recently in Govindan . Equation (1.1) when has been well-studied; see Taniguchi et al. , Govindan , and the references cited therein. Using a global Lipschitz condition on the nonlinear terms and , existence and stability problems were addressed in  and also in  but with a different iteration procedure. For a motivation and details, we refer to .
In this note, our goal is to study the existence and uniqueness of the mild solution of (1.1) using a local Lipschitz condition. Even in the special case (when ), the result obtained here appears to be new. Taniguchi et al.  discussed this special case in when .
Adimy and Ezzinbi  studied the following neutral partial functional differential equation: where is a bounded linear operator from into (a Hilbert space) defined by , for , where the operator is given by and is of bounded variation and nonatomic at zero. Equation (1.3) has been well-studied, see Wu . The authors from  developed a basic theory on such equations and studied an existence result, among others, using a global Lipschitz condition on . For details, we refer to .
Clearly, (1.1) (when ) is more general than (1.3). Our main result (when ) proved using a local Lipschitz condition, therefore, is new in this case as well. We also refer to Ezzinbi et al.  for yet another class of deterministic neutral equations which is again a particular case of (1.1) wherein the authors study existence and regularity problems using global Lipschitz conditions.
The paper is organized as follows. In Section 2, we consider the formulation of the problem under study from  and the references therein. Section 3 is devoted to the main result on the existence and uniqueness of a mild solution of (1.1). An example is given in Section 4.
2. Mathematical Formulation
Let be real separable Hilbert spaces and be the space of bounded linear operators mapping into . For convenience, we will use the same notation to denote the norms in , and and use to denote innerproduct of and without any confusion. Let be a complete probability space with an increasing right continuous family of complete sub--algebras of . Let be a sequence of real-valued standard Brownian motions mutually independent defined on this probability space. Set where are nonnegative real numbers and is a complete orthonormal basis in . Let be an operator defined by . The above -valued stochastic process is called a -Wiener process. Now, we define the stochastic integral of a -valued -adapted process with respect to the -Wiener process .
Definition 2.1. Let be a -valued -adapted process such that for any . Then, we define the real-valued stochastic integral by where .
Definition 2.2. Let be an -valued function and let be a sequence . Then we define If , then is called -Hilbert-Schmidt operator and let denote the space of all -Hilbert-Schmidt operators from to .
Next, we define the -valued stochastic integral with respect to the -valued -Wiener process . See  and the references therein.
Definition 2.3. Let be a -adapted process satisfying . Then we define the -valued stochastic integral with respect to by where is the adjoint operator of .
A semigroup is said to be exponentially stable if there exist positive constants and such that , , where denotes the operator norm in . If , the semigroup is said to be a contraction. If is an analytic semigroup, see Pazy [8, page 60] with infinitesimal generator such that (the resolvent set of ) then it is possible to define the fractional power , for as a closed linear operator on its domain . Furthermore, the subspace is dense in and the expression defines a norm on . Let be the space of continuous functions with the norm .
For convenience of the reader, we will state the following lemmas that will be used in the sequel.
Lemma 2.4 (see ). Let be the infinitesimal generator of an analytic semigroup . If then,(a) for every and .(b)For every one has (c)For every the operator is bounded and (d)Let and then
Lemma 2.5 (see ). Let be the infinitesimal generator of an analytic semigroup of bounded linear operators in . Then, for any stochastic process which is strongly measurable with , and , the following inequality holds for : provided , where and is the Gamma function.
3. Existence and Uniqueness of a Solution
In this section, we establish the existence and uniqueness of a mild solution of (1.1) using local Lipschitz conditions.
We now make (1.1) precise: let be the infinitesimal generator of an analytic semigroup of bounded linear operators defined on . Let the functions , and be defined as follows: , where , and are Borel measurable; and for each are measurable with respect to the -algebra .
Let the following assumptions hold a.s.: (H1) is the infinitesimal generator of an analytic semigroup of bounded linear operators in and that the semigroup is a contraction, (H2) The functions and are continuous and that there exist positive constants such that for all and . Under this assumption, we may suppose that there exists a positive constant such that (H3)The function is continuous and that there exists a positive constant such that for all and . Under this assumption, we may suppose that there exists a positive constant such that (H4) is continuous in the quadratic mean sense:
Definition 3.1. An -valued stochastic process is called a mild solution of the problem (1.1)-(1.2) if(i) is -adapted with , a.s.,(ii), a.s., (iii) satisfies the integral equation Next, assume that is a fixed time. Let be the subspace of all continuous processes which belong to the space with the norm , where . See .
Let the past process , such that .
In the rest of the paper, we will restrict to the interval .
Theorem 3.2. Suppose that the assumptions (H1)–(H4) are satisfied. Then, there exists a time such that (1.1) has a unique mild solution. Further, if , then .
Define a map on :
To prove this theorem, we need some lemmas. The first one establishes the continuity of the map defined on and taking values in thereby showing that it is a well-defined map in the space . The second one then shows that maps into itself. See [2, 9, 10].
Lemma 3.3. For arbitrary is continuous on in the sense.
Proof. Let , and . Consider
By virtue of closedness of and the fact that commutes with on , we have by Lemma 2.4 and the assumption (H3) that
Next, using Lemmas 2.4 and 2.5 and assumption (H3), we obtain
where we chose , such that . Next, by assumption (H2):
Hence, using similar arguments as in Ahmed [9, Theorem 6.3.2, pages 206–209], one can find constants and depending on the parameters , such that
Lastly, for the stochastic integral term , again exploiting Lemma 2.4 and assumption (H2), we obtain wherein we used Da Prato and Zabczyk [11, Theorem 6.10, page 160] or Lemma 2.4 . Arguing as before, we find constants and such that Similar estimates hold for Thus letting , thanks to (H4) the desired continuity follows from all the foregoing estimates.
Lemma 3.4. maps into itself, that is, .
Proof. Let and assume that . Then
We now estimate each term on the R.H.S. of (3.17): By Lemma 2.4 and assumption (H3), we have Next, using assumption (H3) and Lemma 2.5, we have and by assumption (H2) and Lemma 2.4, we get Lastly, by [11, Theorem 6.10] and assumption (H2), we have Consequently, , implying that maps into itself.
Next, assume that , that is, . Then and therefore
Proof of Theorem 3.2. Let and assume that . Then for any fixed , we have
Now choosing sufficiently small, we can find a positive number such that
for any . Hence, by the Banach fixed point theorem, has a unique fixed point and this fixed point is the unique mild solution of (1.1) on . Next, we continue the solution for , see Ahmed  and Govindan . For notational convenience, set . For , where , we say that a function is a continuation of to the interval if(a), and(b) .The terminology mild continuation applied to is justified by the observation that if we define a new function on by setting
and , then is a mild solution of (1.1) on . The existence and uniqueness of the mild continuation is demonstrated exactly as above with only some minor changes. The details are therefore omitted. Repeating this procedure, one continues the solution till the time where is the maximum interval of the existence and uniqueness of a solution. For finite, as . If not, then there exists a sequence converging to and a finite positive number such that for all . Taking sufficiently large so that is infinitesimally close to , one can use the previous arguments to extend the solution beyond , which is a contradiction.
Next, assume that . In that case, This completes the proof.
4. An Example
Consider the neutral stochastic partial functional differential equation with finite delays , and : where is a standard one-dimensional Wiener process, are continuous functions and .
Take . Define by with domain are absolutely continuous, , . Then where , , is the orthonormal set of eigenvectors of .
It is well known that is the infinitesimal generator of an analytic semigroup in and is given by that satisfies , and hence is a contraction semigroup.
The author thanks the referees, particularly, one of the referees profusely for his/her painstaking efforts for reading the paper very carefully and for sending a long report with comments, suggestions, and guidance that led to a significant improvement of the paper. The author also wishes to thank Professor V. V. Anh, Member, Editorial Board for his comments and for summarizing the reports. Finally, the author gratefully acknowledges the financial support received from CONACYT, COFAA-IPN, and SIP-IPN.
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