A NOTE ON PERIODIC SOLUTIONS OF FUNCTIONAL DIFFERENTIAL EQUATIONS

The existence of periodic solution for a certain functional differential equation with quasibounded nonlinearity is established.

defined by x t(0) x(t+0) for all Assume that the equation x'(t) has no nontrivial T-periodic solutions.Also, without loss of generality we assume T>_r.
Fennell [2]  uniformly in t.It is the purpose of this note to generalize Fennell's result by relaxing this requirement.We shall see that the limit in (1.3) can be allowed to be positive.
Using the mapping f in (1.1), we give the following definition.The function S.H. CHANG f is said to be quasibounded with respect to if the number fl rain (max lf(ll)l ),! (1.4) OtT is finite; in this case, Ill is called the quasinorm of f.In recent years, equations with quasibounded nonlinearities have been studied extensively.We shall show that if f is quasibounded and has a quasinorm smaller than a certain positive number then Eq. (1.1) has at least one T-periodic solution.Our proof uses a technique general- izing that used in [2].
Under the assumption for (1.2), the functional differential equation x'(t) e(t,x t) + h(t), (2.1) where L is the same as in (i.I) and h: [0,) (2.2) THEOREM.If, in addition to the given assumptions for the equation (I.i), f is quasibounded with respect to and has a quasinorm fl l/K, where K is given by (2.2), then (I.I) has at least one T-periodic solution.
Let X be the Banach space of continuous T-periodic functions from [-r,) into R n with the supremum norm.For each Now, define a mapping P: X +X by P x(,()), i.e., P is the unique r T-periodic solution of x'(t) L(t,x t) + f(t,t).
Then P is a continuous mapping.

Since
Ifl I/K, there exists e 0 such that Ifl + e < I/K.Then by the definition of quasiboundedness (1.4) there exists O(e) 0 such that whenever II II -> .()and O -< <-T.
Then let M max{KN, O(e) and D { e X: II II M}.
We claim that (i) P(D) D and (ii) P(D) is relatively compact.
Using the inequality (2.3), we obtain that II PII max IP(t) K max If(S,s) I.
O_<t<T O<s<T Now for D and O s -< T, if II sll < O(e) then Klf(S,s) <-KN M and if II sJl p(e) then Klf(S,s) < JJsJi -< IIJJ <-M.Thus iJ PJJ -< whenever # D. This proves (i). (ii) can be established by using an argument similar to that used in [2].By Schauder's fixed point theorem ([3], or see [I, p. 131]) there exists D such that P , which completes the proof of the theorem.COROLLARY (FENNELL [2]).If, in addition to the given assumptions for the equation (i.I), f satisfies the condition (1.3), then (I.i) has at least one T-periodic solution.
PROOF.The condition (1.3) implies that If O.
has established the existence of T-periodic solution for the equation n is continuous and T-periodic, has a unique T-periodic solution.Let x(@,h): F-r,m) R n denote the solution of (2.1) with initial value C Let U: C C be the operator defined by r r rU XT(,O).Then U is completely continuous and the T-periodic solution of(2.1)