Jun 16, 2016 - Abstract. In this paper, we prove the existence and uniqueness of T-periodic solution for the third-order neutral functional differential equation.
International Journal of Mathematical Analysis Vol. 10, 2016, no. 17, 817 - 831 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijma.2016.6345
On the Existence and Uniqueness of Periodic Solution for a Third-Order Neutral Functional Differential Equation Samuel A. Iyase Department of Mathematics, Covenant University, Ota PMB 1023, Canaanland, Ota, Ogun State. Nigeria Olawale J. Adeleke Department of Mathematics, Covenant University, Ota PMB 1023, Canaanland, Ota, Ogun State. Nigeria c 2016 Samuel A. Iyase and Olawale J. Adeleke. This article is distributed Copyright under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract In this paper, we prove the existence and uniqueness of T -periodic solution for the third-order neutral functional differential equation 000
(x(t) − cx(t − τ )) +f (x0 (t))x00 (t)+g(x(t))x0 (t)+h(t, x(t−r(t))) = e(t) The proof is based on Mawhin’s coincidence degree theory and analysis techniques. An example is also provided.
Mathematics Subject Classification: 34C25 Keywords: Existence and uniqueness, third-order neutral functional differential equation, coincidence degree, Mawhin’s continuation theorem
1
Introduction
The existence and uniqueness of periodic solutions of third order functional differential equations have been widely investigated and are still being investigated due to their applications in various practical problems. See [1, 3, 5, 11,
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Samuel A. Iyase and Olawale J. Adeleke
13, 14] and the references therein. In [14], Zhang discussed the properties of the neutral operator (Ax)(t) = x(t) − cx(t − δ) where c and δ are constants. In [12], the authors established criteria for existence of periodic solutions to the neutral first order functional equation 0
(x(t) + cx(t − τ )) + g(t, x(t − σ)) = p(t) with
T
Z
p(t)dt = 0 0
Xin and Cheng [13] in a recent article investigated the existence of positive periodic solution to the third-order neutral differential equation 000
(x(t) − c(t)x (t − δ(t))) = a(t)x(t) − λb(t)f (x(t − r(t))) Motivated by the above facts, we consider in this paper, sufficient conditions for the existence and uniqueness of T -periodic solution to the third-order neutral functional differential equation 000
(x(t) − cx(t − τ )) +f (x0 (t))x00 (t)+g(x(t))x0 (t)+h(t, x(t−r(t))) = e(t) (1.1) where r, g, e : R −→ R are T -periodic and h : R × R −→ R is T -periodic in the first argument, c and τ are constants with |c| < 1. f, g : R −→ R are continuous functions and T -periodic with T > 0. We shall use some inequality techniques and Mawhin’s continuation theorem of coincidence degree theory to establish our results. The rest of this paper is organized as follows. In section 2 we state some preliminary results and other technical details. In section 3 we will prove the main existence results and in section 4 we shall establish uniqueness results. We will give an example of an application in section 5.
2
Preliminary Results
We state here the necessary definitions, Lemmas and other technical details necessary in establishing our results. we denote Z kxkk = { 0
T
1
|x(t)|k dt} 2 ,
kxk∞ = max |x(t)|, t∈[0,T ]
Let X = {x ∈ C 2 (R, R), x(t + T ) = x(t)} Y = {y ∈ C(R, R), y(t + T ) = y(t)}
kek∞ = max |e(t)| t∈[0,T ]
819
Periodic solution of a neutral FDE
Clearly X and Y are Banach spaces with the norms 0
00
kxk = max{kxk∞ , kx k∞ , kx k∞ },
kyk = kyk∞
Define the operator A by A : X −→ X,
(Ax)(t) = x(t) − cx(t − τ )
and the operator L by 000
L : D(L) ⊂ X −→ Y, where
Lx = (Ax) (t) 000
D(L) = {x ∈ X; x (t) ∈ C(R, R)} We also define the nonlinear operator N : X −→ Y by setting 0
00
0
N x = −f (x (t))x (t) − g(x(t))x (t) − h(t, x(t) − r(t)) + e(t) Then (1.1) can be written as Lx = N x. It is easy to show that KerL = R and RT ImL = {y ∈ Y : 0 y(t)dt = 0}. Thus L is a Fredholm operator of index zero. Let the projections P : X −→ KerL and Q : Y −→ Y be defined as follows Z Z 1 T 1 T x(s)ds, Qy = y(s)ds Px = T 0 T 0 Then ImP = KerL and KerQ = ImL. Let LP = L |D(L)∩KerP : D(L) ∩ KerP −→ ImL. Then LP has continuous inverse L−1 P on ImL defined by Z Z T t T −1 −1 (t − s)y(s)ds + (t − s)y(s)ds (LP y)(t) = A − T 0 0 Thus N is L−compact on every open bounded subset Ω of X. We also mention here that from the definition of the operator A, it is easy to see that 00 0 000 000 (Ax) (t) = Ax (t) = Ax (t) Lemma 2.1 (9) . If |c| < 1, then A has continuous inverse on X and kxk∞ (1) kA−1 xk ≤ |1−|c|| for all x ∈ X RT RT 1 |f (t)|dt for all f ∈ X (2) 0 |(A−1 f )(t)|dt ≤ |1−|c|| 0 RT R T 2 1 (3) 0 |(A−1 f )(t)|2 dt ≤ (1−|c|) f (t)dt for all f ∈ X 2 0 In what follows, we shall use the following continuation theorem of coincidence degree.
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Samuel A. Iyase and Olawale J. Adeleke
Lemma 2.2 (2) . Let X and Y be two Banach spaces. Suppose that L : D(L) ⊂ X −→ Y is a Fredholm operator with index zero and N : X −→ Y is L− compact on Ω, where Ω is an open bounded subset in X. In addition, if the following conditions hold (a) Lx 6= N x, for all (x, λ) ∈ ∂Ω × (0, 1) (b) QN x 6= 0 for all x ∈ KerL ∩ ∂Ω (c) deg{QN, Ω ∩ KerL, 0} = 6 0 Then Lx = N x has at least one solution in D(L) ∩ Ω We list the following assumptions H1: There exists non-negative constants c1 , c2 and c3 such that (1) |f (x)| ≤ c1 for all x ∈ R (2) |g(x)| ≤ c2 , |g(x1 ) − g(x2 )| ≤ c3 |x1 − x2 | for all x, x1 , x2 ∈ R H2: There exists positive constants m0 , and d such that (1) |h(t, x1 ) − h(t, x2 )| ≤ m0 |x1 − x2 | for all t, x1 , x2 ∈ R (2) x{h(t, x) − e(t)} < 0 for all t ∈ R and |x| ≥ d
3
Main Results Theorem 3.1 Suppose that assumptions (H1)-(H2) hold with |c| < 1 and |c|(1 + |c|) + 41 c1 T 2 + 18 m0 T 3
