The Uniqueness of the Periodic Solution for A Class of Differential Equations 1

In this paper we are concerned with a class of nonlinear dieren tial equations and obtaining the sucien t conditions for the uniqueness of the periodic solution by using Brouwer's xed point theory and the Sturm Theorem. Keywords. uniqueness, xed point, existence, control theory method, Sturm comparison theorem. AMS (MOS) subject classication: 34C25 where (t; x) 2 (0; 2 ) R and (t; x; z) 2 C 0 ((0; 2 ) R R), f 2 C 2 (0; 2 ) R , and (t; x; x 0 ); f(t; x) are 2 -periodic functions with respect to t.

It is easy to see that equation ( 1) is more general than the classical ordinary differential equation x = f (t, x) for all (t, x) ∈ [0, 2π] × R (2) During the past three decades, with the use of topological degree theory, general critical point theory, fixed point theory, boundary value condition theory and cross-ratio method, some profound results on the existence and the number of periodic solutions for equation (2) have been presented ( see references [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15] and the reference therein ).But none of these papers are concerned with the uniqueness of the periodic solutions for equation (1).However, when does the equations (1) or (2) have a unique 2π-periodic solution ?
In the present paper, using the Brouwer's fixed pointed theorem, the Sturm Theorem, and some results of the optimal control theory method, we are trying to obtain two theorems for the sufficient conditions which guarantee that equation (1) has a unique 2π-periodic solution.
(H2): Suppose that f tx , f xx : R 2 → R and [ f (t,x) µ(t,x,z) ] x : R 3 → R exist, and are 2π-periodic continuous with respect to t, where f tx denotes the partial derivative with respect to t and x, and [ f (t,x) µ(t,x,z) ] x denotes the derivative of the quotient f (t,x) µ(t,x,z) with respect to x. Suppose that there exist two positive real numbers L and M , one non-negative integer N , and two non-negative continuous functions u 1 (t) and u 2 (t), such that and where indicates "greater than or equal to" but not identically equal.(H3): Suppose that f x , f tx , and f xx are continuous and 2π-periodic with respect to t.
where k is the minimal positive integer suiting the inequality.Assume that there exists a β(x) ∈ C[0, 2π] such that Our main results are the following theorems: Theorem 1: If H(1) and H(2) hold, then equation (1) has a unique 2π-periodic solution.

The Proof of Main Results
Proof of Theorem 1.Consider the case G(t, x) < 0 for all (t, x) in S 1 and S 2 .Obviously, in the domain Ω 1 , we have f (t, x(t)) < 0 for any x(t) > x 0 (t).EJQTDE, 2000 No. 10, p. 2 In the domain Ω 2 , we have f (t, x(t)) > 0 for any x(t) < x 0 (t).In the compact set [0, 2π] × [x 1 , x 2 ], f (t, x) has the maximal value, denoted by m 1 , (m 1 < 0).Hence, we can choose some negative number k 1 such that k 1 > m1 L and the whole segment Similarly, in the compact set [0, 2π] × [x 3 , x 4 ], f (t, x) has the minimal value, denoted by m 2 , (m 2 > 0).Thus, we also can choose some positive number k 2 such that k 2 < m2 M and the whole segment and Letting Define an operator By the Brouwer's fixed point theorem, we obtain that T 1 has at least one fixed point in C[L 1 , L 2 ].That is, equation (1) has at least one 2π-periodic solution.
In the case G(t, x) > 0 for all (t, x) in S 1 and S 2 , the proof is similar to the above.In the domain Ω 1 , we have f (t, x(t)) > 0 for any x(t) > x 0 (t).In the domain Ω 2 , we have f (t, x(t)) < 0 for any x(t) < x 0 (t).In the compact set [0, 2π] × [x 1 , x 2 ], f (t, x) has the minimal value, denoted by n 1 , (n 1 > 0).Hence, we can choose some positive number such that k 3 < n1 M and the whole segment l 3 : x 3 (t) = k 3 t + x 1 is inside the set S 1 whenever 0 ≤ t ≤ 2π.Similarly, in the compact set [0, 2π] × [x 3 , x 4 ], f (t, x) has the maximal value, denoted by n 2 , (n 2 < 0).Thus, we also can choose some negative number k 4 such that k 4 > n2 L and the whole segment Letting L 1 = x 3 , L 2 = x 2 , L 3 = x 4 , and L 4 = x 1 , clearly, we can see that Define an operator we have that T 2 is continuous and maps By the Brouwer's fixed point theorem, we obtain that T 2 has at least one fixed point in C[L 1 , L 2 ].That is, equation ( 1) has at least one 2π-periodic solution.
Differentiating both sides of equation ( 1) with respect to t, we have where Here f t (t, x) denotes the partial derivative with respect to t, and F x (t, x) denotes the partial derivative with respect to x.By recalling the assumption (H2), we know that −u 1 (t) ≤ F x (t, x) ≤ −u 2 (t).
Compare equation ( 4) with By using the similar arguments as the above, and letting X 3 (τ )andX 4 (τ ) denote the solutions of the following initial value problems, respectively, we have This yields a contradiction with (12).(II).In the case N = 0, we can conclude that b(t) has zeros in R by the Sturm comparison theorem.Without the loss of generality, we assume that b(0) = 0. Due to the periodicity, b(t) has at least two zeros in (0, 2π).The rest of the proof is similar to that of the case (I), so it is omitted.
¿From the above proof, we know that equation (1) has at least a 2πperiodic solution.However, we know that every 2π-periodic solution of equation (1) must be 2π-periodic solution of equation (µ(t, x, x )x ) = F (t, x) .Therefore, we can conclude that under the assumptions (H1) and (H2) equation (1) has a unique 2π-periodic solution.The proof is complete.
Proof of Theorem 2. It is well-known that the following result of the optimal control theory can be widely applied.For the detailed proof, we may go back to [18].Some interesting applications of the optimal control theory method to several boundary value problems for ordinary differential equations can be found in [16][17][18][19].Let (k − 1) 2 < A < k 2 < B, where k is the minimal positive integer suiting the inequality.Suppose that u ∈ L[0, 2π] satisfying for λ = 1 2k .Then the periodic boundary value problem has a unique 2π-periodic solution for each 2π-periodic function f ∈ L[0, 2π].
To prove that equation ( 2) has at least one 2π-periodic solution, we can use the same arguments as that of theorem 1.To prove the uniqueness, by differentiating both sides of equation ( 2) with respect to t, we have EJQTDE, 2000 No. 10, p. 7 where F (t, x) = f t + f • f x and F x (t, x) = f tx + f xx • f + (f x ) 2 .Let X 1 (t) and X 2 (t) be any two 2π-periodic solutions of equation (13).Then b(t) = X 1 (t) − X 2 (t) is a 2π-periodic solution of ¿From the assumption, we see x 2 (t) + θx 1 (t)]dθx ≤ β(x).
By using the above result of optimal control theory, b(t) ≡ 0 for all t ∈ R. Therefore, equation ( 2) has a unique 2π-periodic solution.The proof is complete.

Conclusion
¿From Section 2, we can see that using the Sturm Theorem as well as the Brouwer's fixed pointed theorem is really an effective approach for equations (1) and ( 2).It is easily noted that even when a(t, x, x ) = 1, our conditions are different from all those in the previous references [1][2][3][4][5][6][7][8].We also can use this method to study the sublinear Duffing equations investigated in [20].