Positive Periodic Solutions of Third-Order Ordinary Differential Equations with Delays

and Applied Analysis 3 Let 0 < M < (2π/√3ω)3. Then, the solution of (8)Φ(t) > 0 for every t ∈ [0, ω]. If h ∈ C+ ω (R) and h(t) ̸ ≡ 0, by (9), the ω-periodic solution u = Ph of (7) is positive. We will show that the ω-periodic solution has stronger positivity. Let

In recent years, the existence of periodic solutions for first-order and second-order delay differential equations has been researched by many authors; see [1][2][3][4][5] for the firstorder equations and see [6][7][8][9][10][11][12] for the second-order ones.In some practice models, only positive periodic solutions are significant.In [3,8,9,11,12], the authors obtained the existence of positive periodic solutions for some first-order and second-order delay differential equations by using Krasnoselskii's fixed-point theorem of cone mapping.But, few people consider the existence of positive periodic solutions for third-order delay differential equations.
The third-order delay differential equations have their important physical contexts, for example, which can be formulated from the problem of the wave solution of the Korteweg-de Vries (KdV) equation with time delay.Recently, Zhao and Xu [13] pointed out that the KdV equation with time delay has more actual significance and they considered the solitary wave solution of the following KdV equation with time delay:   (, ) +  (,  − )   (, ) +   (,  − ) −   (, ) = 0, where  is a given constant and   (,  − ) means the backward diffusion with time delay.They looked for a wave solution (, ) = ( + ) with  > 0 and from (2) obtained the following third-order delay ordinary differential equation of the profile : () +  ( − ) +   ( − ) −   () = 0,  ∈ R.
Equation ( 3) is a special form of (1).If we look for a periodic wave solution of (2), we need to discuss the existence of the periodic solution of the delay ordinary differential equation (3).Hence, the existence problem of periodic solutions of the general third-order delay differential equation ( 1) is a significant topic.

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Abstract and Applied Analysis Some theorems and methods of nonlinear functional analysis have been applied to research on this problem, such as the methods of topological degree and Leray-Schauder fixedpoint theorem [14,19], the upper and lower solutions method and monotone iterative technique [15][16][17], the implicit function theorem [18], and Mawhin coincidence degree theory [20].Especially, in recent years, the fixed-point theorem of Krasnoselskii's cone expansion or compression type has been availably applied to some special third-order periodic boundary problems of ordinary differential equations, and some results of existence and multiplicity of positive periodic solutions have been obtained; see [21,22].In [21], Chu and Zhou considered the periodic boundary value problem for the third-order equation where  ∈ (0, 1/ √ 3) is a constant and  ∈ ([0, 2]×(0, ∞)).
Using the Krasnoselskii's fixed-point theorem in cones, they obtained the existence results of positive solutions.Their results extended the one obtained by the Schauder fixedpoint theorem in [19].In [22], by the Krasnoselskii's fixedpoint theorem in cones, Feng established some existence and multiplicity results of positive periodic solutions for the thirdorder equation where  and  are positive constants and satisfy certain conditions.In [23], the present author extended and improved the results in [9,10] to the general third-order equation that nonlinearity  explicitly contains derivative terms   () and   ().However, all of these works are on the third-order equations without delays and the argument methods are not applicable to the delay equation (1).Motivated by the facts mentioned above, we research the existence of positive periodic solutions of the third-order delay equation (1).We will use the fixed-point index theory in cones in a meticulous way to obtain the essential conditions on the existence of positive periodic solutions of (1).Our main results will be given in Section 3. Some preliminaries to discuss (1) are presented in Section 2.
Let  > 0 be a constant.For ℎ ∈   (R), we consider the existence of -periodic solution of the linear third-order differential equation It is easy to verify that the linear third-order boundary value problem has a unique solution.We denote the solution by Φ().By [16, Lemma 2.1], the -periodic solution of ( 7) can be expressed by Φ.By [16, Lemma 2.1] or a direct calculation, we easily obtain the following lemma.
Proof.Let ℎ ∈  +  (R) and let  = ℎ.For every  ∈ R, from (9), it follows that and, therefore, Using ( 9) again, we obtain that For every  ∈ R, since we have Hence,  ∈ .Now, we consider the periodic solution problem of the linear third-order differential equation with variable coefficient Let  ∈   (R) be a positive -periodic function and satisfy the assumption and set Then, 0 <  ≤  < (2/ √ 3) 3 , and the conclusion of Lemma 3 holds.For (22), we have the following lemma: Proof.Let  and  be the positive constants defined by (24) and let  :   (R) →   (R) be the -periodic solution operator of (7) given by (9).By Lemma 3, ( +  (R)) ⊂  +  (R), and  :   (R) →   (R) is a positive linear bounded operator.We rewrite (22) into the form of Then, it is easy to see that the -periodic solution problem of ( 22) is equivalent to the operator equation in Banach space   (R) where  is the identity operator in   (R) and  :   (R) →   (R) is the product operator defined by which is a positive linear bounded operator.We prove that the norm of  ∘  in L(  (R),   (R)) satisfies ‖ ∘ ‖ < 1.
By the definition of operator  and Lemma 4, the positive -periodic solution of ( 1) is equivalent to the nonzero fixed point of .We will find the nonzero fixed point of  by using the fixed-point index theory in cones.
We recall some concepts and conclusions on the fixedpoint index in [15,16].Let  be a Banach space and  ⊂  be a closed convex cone in .Assume Ω is a bounded open subset of  with boundary Ω and  ∩ Ω ̸ = 0. Let  :  ∩ Ω →  be a completely continuous mapping.If  ̸ =  for any  ∈ ∩Ω, then the fixed-point index  (, ∩Ω, ) has definition.One important fact is that if  (,  ∩ Ω, ) ̸ = 0, then  has a fixed point in  ∩ Ω.The following two lemmas in [24,25] are needed in our argument.In next section, we will use Lemma 6 and Lemma 7 to discuss the existence of positive -periodic solutions of (1).