Let , for and all . Define the following Picard sequence
If we can give the approximate solutions by means of Picard iteration, then the existence-and-uniqueness theorem of the solutions for INSFDEs at phase space can be discussed next.
Theorem 1 If (B1), (B2) and (B3) hold, then there exists a unique solution to system (1) with initial value (2).
To show Theorem 1, first of all, let us prove two useful lemmas.
Lemma 7 Under the assumptions (B1), (B2) and (B3), for all , there exist positive constants , , such that
Proof Obviously, . By induction, . In fact, from Hölder’s inequality and the elementary inequality , we have
Taking the exception on both sides, and by Hölder’s inequality and Lemma 4, thus we get
Let . According to the property of a concave function, always exists maximum on the bounded closed interval , without loss of generality, taking as . The line L always lies on the curve , and line L passes through the point with the slope . Hence, for all , , it then follows
where and are positive constants. Thus, (4) can be simplified as
According to the fact , one can see that
Again, noting that
for any .
By (7), (8) and Doob’s martingale inequality, (6) becomes
where , . Consequently,
since k is arbitrary, by means of Gronwall’s inequality, we derive that
On the other hand, by the elementary inequality , one gets
From (B1), (B2), (B3), Hölder’s inequality and Jensen inequality for the concave function , it yields that
where . From (11), (14) can be simplified as
where . The proof is complete. □
For all , define
For , define the recursive function
We choose such that
In fact, from the monotonicity of a concave function and the relationship between , and , (19) becomes , that is,
From (5), and noting the fact that , we can get
There exists a positive
for all , .
Proof According to the definition of the function , we have
From (19), we also have
Thus we derive that for all .
By induction, assume that , holds for some . Now we check Lemma 8 is valid for . In fact,
From (17), one gets that
The proof is complete. □
Proof for the uniqueness of Theorem 1 Let and be any two solutions of system (1) with the initial data (2). Thus, we have
By the elementary inequality , Hölder’s inequality and Lemma 4, one can find that
From (B1), (B2), the definition of the norm at phase space and (7), then we have
Lemma 5 and Lemma 6 yield , . The proof of uniqueness is complete. □
Proof for the existence of Theorem 1 Note that is continuous on . For each , is decreasing on , and for each t, is also a decreasing sequence. From the dominated convergence theorem, we can define the function as
For all , Lemma 5 and Lemma 6 imply that .
By Lemma 8, for , we have as , that is, as . From the completeness of , the assumptions (B1), (B2), (B3) and the property of the function to , it then follows that for all ,
Then, for all ,
which demonstrates that is one solution of system (1) with initial data (2) on , where . By iteration, the existence of solutions to system (1) on can be obtained. The proof is complete. □
From Theorem 1, the existence and uniqueness of solutions to system (1) are defined on a finite interval . If all additions of the existence and uniqueness theorem are satisfied on every finite subinterval of , then system (1) will have a unique solution on the entire interval . So, we have the following corollary.
Corollary 1 Assume that for each real number , there exist positive constants and such that for all and all , it follows that
where is defined in (B1). Then the system
has a unique global solution ; moreover, .
The proof is similar to that in Theorem 1, we omit it here.