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We study the periodic solutions of the second-order differential equations of the form where the functions, , and are periodic of period in the variable t.

In this paper we shall study the existence of periodic solutions of the second-order differential equation of the form

where the dot denotes derivative with respect to the time t, and the functions

We note that the second-order differential Equation (1), when

In a recent paper [

Here we study the periodic solutions of the second-order differential Equation (1) when

Theorem 1. We define the functions

where

Assume that the functions

the differential Equation (1) has a

Theorem 1 is proved in section 3 using the averaging theory described in section 2. Two applications of Theorem 1 are the following.

Corollary 1. We consider the differential Equation (1) with

Corollary 2. We consider the differential Equation (1) with

Corollaries 1 and 2 are also proved in section 3.

Theorem 2. Assuming that

and setting

with

Assume that

Theorem 2 is proved in section 4. Two applications of Theorem 2 are the following.

Corollary 3. We consider the differential Equation (1) with

Corollary 4. We consider the differential Equation (1) with

Corollaries 3 and 4 are also proved in section 4.

We state the results from the averaging method that we shall use for proving the results of this work.

We consider differential systems of the form

where

has a submanifold of dimension n of T-periodic solutions, i.e. of periodic solutions of period T.

We denote by

where

By assumption there exists an open set V such that

Theorem 3. We suppose that there is an open and bounded set V with

If there is

Theorem 3 is due to Malkin [

Proof of Theorem 1. Introducing the variable

Doing the rescaling

System (10) with

Note that all these periodic orbits have period

The fundamental matrix solution

Now we compute the function

By Theorem 3 each zero

Going back through the change of variables for every periodic solution

Proof of Corollary 1. We must apply Theorem 1 with

We compute the functions

System

Proof of Corollary 2. We apply Theorem 1 with

Computing the functions

System

Proof of Theorem 2. As in the proof of Theorem 1, the second-order differential Equation (1) can be written as the first order differential system (9). Doing the rescaling

System (11) with

The solution

Note that these periodic orbits have period

The fundamental matrix solution

We compute the function

By Theorem 3, each zero

Going back through the change of variables for every periodic solution

of the differential Equation (1) for

Proof of Corollary 3. We apply Theorem 2 with

We compute the functions

System

Proof of Corollaryc 4. We apply Theorem 2 with

We compute the functions

System

The second author is partially supported by a MINECO grant MTM2013-40998-P, an AGAUR grant number 2014SGR568, and the grants FP7-PEOPLE-2012-IRSES 318999 and 316338.

ZeynebBouderbala,JaumeLlibre,AmarMakhlouf, (2016) Periodic Solutions of a Class of Second-Order Differential Equation. Applied Mathematics,07,227-232. doi: 10.4236/am.2016.73021