ON EXISTENCE OF PERIODIC SOLUTION TO CERTAIN NONLINEAR THIRD ORDER DIFFERENTIAL

In this paper, it is investigated the existence of periodic solutions to the nonlinear third order differential equation : x000 + c2(t)x 00 + c1(t)x 0 + f(t, x) = p(t, x, x0, x00). The Leray-Schauder principle is used to show the existence of periodic solutions of this equation. Subjclass : [AMS] 34C25, 65L06.


Introduction
With respect to our observations, up to now, the problem of existence of periodic solutions for various nonlinear second and third order differential equations have been studied in the literature by only a few authors, see, for example, the papers of Mehri & Shadman [1], Shadman & Mehri [2] and Tunç & C ¸inar [3].
Meanwhile, it is also worth mentioning that, in 1995, Shadman & Mehri [2] discussed the existence of periodic solutions to nonlinear third order differential equation of the form: x 000 + c 2 (t)x 00 + c 1 (t)x 0 + f (t, x) = e(t).
In [2], the authors used Leray-Schauder principle to show the existence of periodic solutions of this equation.

Main Result
Our main result is the following theorem.
(ii) |p(t, x, x 0 , x 00 )| ≤ |e(t)|, for all t, x, x 0 and x 00 , and Then equation (1.1) possesses a solution satisfying the boundary conditions, Proof.First, we show an estimate on the magnitude of the solutions of problem: for all t, and we also use Wirtinger's inequalities in the following from: where Now, we suppose that x(t) is a solution of the problem given by (2.2).In the light of the assumptions of the theorem, we have from (2.2) that; Hence, by using the Minkowski's inequality, we get It can also be seen from Wirtinger's inequality (2.3) that Cemil Tunç Hence, On the other hand, since then, clearly, it follows ´3 .

Cemil Tunç
x 000 + c 2 (t)x 00 + c 1 (t)x 0 = 0 has no non-trivial solution which satisfies the boundary conditions (2.1), This guarantees the existence of a Greens function for the problem (2.2).Clearly, the problem (2.2) is equivalent to the following: In view of the sphere it follows for arbitrary R > ρ that equation (2.9) has no solution on the sphere S R .Now, by noting Leary-Schauder principle and the complete continuity of the operator ´#ds, one can conclude that equation (2.9) has at least a solution in the open ball {x : As a result, there exists a solution of equation (2.9) in the closed ball B ρ .Therefore, this fact implies that the problem (2.2) has, at the least, a solution for µ = 1 .2 Remark.When we take p(t, x, y, z) = e(t) in equation ( 1 Corollary.In addition to the assumptions of theorem, we assume that the following conditions hold: (iv) The functions c 1 (t) and c 2 (t) are ω-periodic, that is, c 1 (t + ω) ≡ c 1 (t) and c 2 (t + ω) ≡ c 2 (t).
Then equation (1.1) has a 2ω-periodic solution with zero mean.
Proof : Let x(t) be a 2ω-periodic extension of solution x(t) defined as follows: Clearly, x(t) ∈ C 2 [−ω, ω].Next, in the light of the above assumptions imposed on the functions c 1 (t), c 2 (t), f(t, x) and p(t, x, y, z), it can be easily shown that x(t) is a solution of equation (1.1) which satisfies the periodic boundary condition: x(i) (ω) = x(i) (−ω), (i = 0, 1, 2).This fact shows that the solution x(t) has a zero mean value.The proof of corollary is now complete.2

Subject to the fact
.1), then the conditions of theorem reduce those of Shadman and Mehri [2, Theorem 1].It is also clear that our result generalizes that of Shadman and Mehri [2, Theorem 1].