EXISTENCE OF PERIODIC SOLUTIONS FOR A CLASS OF EVEN ORDER DIFFERENTIAL EQUATIONS WITH DEVIATING ARGUMENT

Using Mawhin’s continuation theorem we establish the existence of periodic solutions for a class of even order differential equations with deviating argument.

For the sake of completeness, we first state Mawhin's continuation theorem [3].Let X and Y be two Banach space and L : DomL ⊂ X −→ Y is a linear mapping and N : X −→ Y is a continuous mapping.The mapping L will be called a Fredholm mapping of index zero if dimKerL = codimImL < +∞, and ImL is closed in Y.If L is a Fredholm mapping of index zero, there exist continuous projectors P : X −→ X and Q : Y −→ Y such that ImP = KerL and ImL = KerQ = Im(I − Q).It follows that L| DomL∩KerP : (I − P )X −→ ImL has an inverse which will be denoted by K P .
If Ω is an open and bounded subset of X, the mapping N will be called L−compact on Ω if QN(Ω) is bounded and K P (I − Q)N(Ω) is compact.Since ImQ is isomorphic to KerL, there exists an isomorphism J : ImQ −→ KerL.The following theorem is called Mawhin's continuation theorem (see [3]).

Main Result
Now we make the following assumptions on a i (t): (iii) There exists a positive constant r with m 0 > r, such that with A− 2M 0 +m 0 +r 2(m 0 −r) B > 0 and 1 − A * > 0, where and Our main result is the following theorem.
Theorem 2.1 Under the assumptions (i), (ii) and (iii), if and then Eq.( 1) has at least one T −periodic solution.In order to prove the main theorem we need some preliminaries.Set Define the operators L : X −→ Y and N : X −→ Y , respectively, by and Clearly, and is closed in Y. Thus L is a Fredholm mapping of index zero.Let us define P : X → X and Q : Y → Y /Im(L), respectively, by for x = x(t) ∈ X and for y = y(t) ∈ Y .It is easy to see that ImP = KerL and ImL = KerQ = Im(I − Q).
In order to prove our main result, we need the following Lemmas [6,7].The first result follows from [6 and Remark 2.1] and the second from [7].