nonlinear differential equations with variable coefficients

By applying the method of coincidence degree, some criteria are established for the existence of anti-periodic solutions for a class of fourth-order nonlinear differential equations with variable coefficients. Finally, an example isgiven to illustrate our result.


Introduction
In this paper, we should apply the method of coincidence degree to study the existence of anti-periodic solutions for a class of fourth-order nonlinear differential equations with variable coefficients in the form of u ′′′′ (t) − a(t)u ′′′ (t) − b(t)u ′′ (t) − c(t)u ′ (t) − g(t, u(t)) = e(t), (1.1) where is T -periodic in its first argument, and e ∈ C(R, R) is T -periodic with T 0 e(s) ds = 0.During the past thirty years, there has been a great deal of work on the problem of the periodic solutions of fourth-order nonlinear differential equations, which have been used to describe nonlinear oscillations [1][2][3][4][5], and fluid mechanical and nonlinear elastic mechanical phenomena [6][7][8][9][10][11][12].In [13], Bereanu discussed the existence of T -periodic solutions of the following fourth-order nonlinear differential equations: u ′′′′ (t) − pu ′′ (t) − g(t, u(t)) = e(t), which can be regarded as a special case of Eq. (1.1) with b(t) ≡ p and a(t) = c(t) ≡ 0.
Arising from problems in applied sciences, it is well-known that the existence of antiperiodic solutions plays a key role in characterizing the behavior of nonlinear differential equations as a special periodic solution and have been extensively studied by many authors during the past twenty years, see [14][15][16][17][18][19][20][21][22] and references therein.For example, anti-periodic trigonometric polynomials are important in the study of interpolation problems [23,24], and anti-periodic wavelets are discussed in [25].However, to the best of our knowledge, there are few papers to investigate the existence of anti-periodic solutions to Eq. (1.1) by applying the method of coincidence degree.
The main purpose of this paper is to establish sufficient conditions for the existence of T 2 -anti-periodic solutions to Eq. (1.1) by using the method of coincidence degree.The organization of this paper is as follows.In Section 2, we make some preparations.In Section 3, by using the method of coincidence degree, we establish sufficient conditions for the existence of T 2 -anti-periodic solutions to Eq. (1.1).An illustrative example is given in Section 4.

Preliminaries
For the readers' convenience, we first summarize a few concepts from [26].Let X and Y be Banach spaces.Let L : Dom L ⊂ X → Y be a linear mapping and N : X → Y be a continuous mapping.The mapping L will be called a Fredholm mapping of index zero if Im L is a closed subspace of Y and If L is a Fredholm mapping of index zero, then there exist continuous projectors P : X → X and Q : Y → Y such that Im P =Ker L and Im L=Ker Q=Im (I − Q).It follows that L| Dom L∩Ker P : (I − P )X → Im L is invertible and its inverse is denoted by K P .If Ω is a bounded open subset of X, the mapping N is called L-compact on X, if QN( Ω) is bounded and K P (I − Q)N : Ω → X is compact.Because Im Q is isomorphic to Ker L, there exists an isomorphism J : Im Q → Ker L.
The following fixed point theorem of coincidence degree is crucial in the arguments of our main results.Lemma 2.1.[26] Let X, Y be two Banach spaces, Ω ⊂ X be open bounded and symmetric with 0 ∈ Ω. Suppose that L : D EJQTDE, 2011 No. 12, p. 2 Example 2.1.The functions sin x and cos x are anti-periodic with anti-period π (as well as with anti-periods 3π, 5π, etc.).
We will adopt the following notations: where u is a T -periodic function.
Lemma 2.2.[27] For any u ∈ C 2 T one has that 3 Main result Theorem 3.1.Assume that the following conditions hold: Then Eq. (1.1) has at least one T 2 -anti-periodic solution.
EJQTDE, 2011 No. 12, p. 3 be two Banach spaces with the norms Define a linear operator L : D(L) ⊂ X → Y by setting where It is easy to see that In order to apply Lemma 2.1, we need to find an appropriate open bounded subset Ω in X. Corresponding to the operator equation Lx − Nx = λ(−Lx − N(−x)), λ ∈ (0, 1], we have and Suppose that u(t) ∈ X is an arbitrary T 2 -anti-periodic solution of system (3.1).Hence we have By using a similar argument as that in the proof of (3.2), we can easily obtain On the other hand, multiplying Eq. (3.1) by u and integrating it from 0 to T , it follows that ) and (H 3 ), we obtain , in which together with (H 2 ) and 0 ≤ δ < 1 imply that there exists a positive constant M 1 satisfying Therefore, from (3.2), (3.3) and (3.7), we can choose a constant M 2 such that Therefore, there exists a positive constant M 4 such that Assume that min s∈[0,T ] [b(s) − 3 2 a ′ (s)] ≥ 0. In view of (3.6), we have Set M 7 = max{M 3 , M 5 , M 6 }.Together with Lemma 2.3, there exists a positive constant M 8 satisfying It is clear that Ω satisfies all the requirements in Lemma 2.1 and condition (H) is satisfied.In view of all the discussions above, we conclude from Lemma 2.1 that Eq. (1.1) has at least one T 2 -anti-periodic solution.This completes the proof.
Remark 3.1.From the proof of Theorem 3.1, we can see that the delay term τ (t) in Eq. (3.9) has no effect on the result in Theorem 3.1.So the result in Theorem 3.1 also holds for Eq.(3.9).Proof.When 0 ≤ r < (2π) −3/2 , it is easy to verify that (H 2 ) holds.Furthermore, it suffices to remark that the function g(t, u) ≡ | sin t|u  uniformly with respect to t ∈ R. Hence (H 3 ) and (H 4 ) hold and the result follows from Theorem 3.1.This completes the proof.

3 − 2 0
Ker L = {0} and Im L = u ∈ Y : T 0 u(s) ds = 0 ≡ Y. Thus dim Ker L = 0 = codim Im L, and L is a linear Fredholm operator of index zero.Define the continuous projector P : X → Ker L and the averaging projector Q : Y → Y by P u(t) = Qu(t) = 1 T T 0 u(s) ds ≡ 0. Hence Im P = Ker L and Ker Q = Im L. Denoting by L −1 P : Im L → D(L) ∩ Ker P the inverse of L| D(L)∩KerP , we have 6T 2 t + 12T t 2 − 16t 3 192 T u(s) ds.Clearly, QN and L −1 P (I − Q)N are continuous.Using the Arzela-Ascoli theorem, it is not difficult to show that QN( Ω), L −1 P (I − Q)N( Ω) are relatively compact for any open bounded set Ω ⊂ X.Therefore, N is L-compact on Ω for any open bounded set Ω ⊂ X.