A NOTE ON RETARDED FUNCTIONAL DIFFERENTIAL EQUATIONS WITH ANTI-PERIODIC BOUNDARY CONDITIONS

In this work, we approach an anti-periodic boundary value problem for a class of retarded functional differential equations. We employ the method of lower and upper solutions to prove a uniqueness result for such problem. AMS Subject Classification: 39B99


Introduction
The ordinary differential equations are powerful theoretical representations of processes of evolution in which the variation rate of process state at each time depends on the state of the process at this time.However, there is a lot of real phenomena in which the variation rate of state in each moment depends not only on the state of the process at this moment, but also the history of phenomena states.Thus, for such cases, it makes convenient the use of other theoretical tools that describe such phenomena more appropriately.Such tools are the retarded functional differential equations.See [4], [5], [7], [8], [9] and [14].
This note deals with an anti-periodic boundary value problem for a class of retarded functional differential equations and it was inspired by the works [1], [2], [3], [6], [11] and [12].The functional differential inequalities generated by this problem are considered and a uniqueness result is established by the method of lower and upper solutions.The method of lower and upper solutions together with the monotone iterative technique has been used in the treatment of nonlinear differential equations, see [10], [11] and [13].

Let
Let us consider the retarded functional differential equation subject to the anti-periodic boundary condition where ⋆ f is a continuous real function defined on [0, T ] × C([−r, 0], R); ⋆ r ≥ 0 and T > 0; We denote by The following assumption will be fundamental to obtain our results: The next result establishes a relation between the lower and upper solutions of Problem (2.1)-( 2).(2.3) Let us consider the following three cases: Case 1: η ∈ (0, T ).Since u−v is continuously differentiable and reaches its maximum at η, it follows that (u − v) ′ (η) = 0. On the other hand, from the definition of the functions u and v, we have 0 But, if (2.6) holds, we can infer that u η ≤ v η on [−r, 0] and, in particular, u(η) ≤ v(η), which contradicts (2.4).
Case 2: η = 0.Then, u(0) − v(0) = Λ and, by (2.5), u(T ) − v(T ) ≤ Λ.However, from the definition of the functions u and v, we obtain Let K 0 be a positive integer such that, for k ≥ K 0 , one has u(t k )− v(t k ) > 0 (such K 0 exists by virtue of the continuity of u − v).Therefore, from the definition of the functions u and v, we conclude that This yields u(t k ) − v(t k ) ≤ 0, which does not occur.Case 3: η = T .
Here we can use an analogous argument to that used in the previous case.
The contradictions pointed in the three cases guarantee us that the assertion of the theorem is valid.
The last theorem shows a relation between the solution of Problem (2.1)-( 2) and the lower and upper solutions of the same problem when Condition (A) is satisfied.
Theorem 2.3.Let u and v be a lower solution and a upper solution of Problem (2.1)-(2), respectively, and x be a solution of the same problem.Assume that (A) holds.Then, Proof.At first, we will see that u(t) ≤ x(t) for all t ∈ [0, T ].Assume that there is τ ∈ [0, T ] such that u(τ ) > x(τ ) and denote by µ We consider the following three cases: β ∈ (0, T ), β = 0 and β = T .Analogously to the proof of Theorem ??lem1, we obtain contradictions that ensure the validity of the inequality u(t) ≤ x(t) for all t ∈ [0, T ].
The proof of the inequality x(t) ≤ v(t), for t ∈ [0, T ], is similar.

Definition 2 . 2 .
Functions u : [−r, T ] → R and v : [−r, T ] → R are said to be a pair of lower and upper related solutions for Problem (2.1)-(2) if they satisfy

Theorem 2 . 1 .
Let u and v be a lower solution and a upper solution of Problem (2.1)-(2), respectively.Assume that Condition (A) is fulfilled.Then, u(t) ≤ v(t), for all t ∈ [0, T ].