Pseudo almost periodic solutions for a class of differential equation with delays depending on state

Hou Yu Zhao

doi:10.1515/anona-2020-0049

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Pseudo almost periodic solutions for a class of differential equation with delays depending on state

Hou Yu Zhao

Published/Copyright: December 10, 2019

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From the journal Advances in Nonlinear Analysis Volume 9 Issue 1

Abstract

In this paper, the exponential dichotomy, and Tikhonov and Banach fixed point theorems are used to study the existence and uniqueness of pseudo almost periodic solutions of a class of iterative functional differential equations of the form

x′(t)=∑n=1k∑l=1∞Cl,n(t)(x[n](t))l+G(t),

where x^[n](t) denotes nth iterate of x(t).

Keywords: Iterative differential equation; pseudo almost periodic solutions; fixed point theorem

MSC 2010: 39B12; 39B82

1 Introduction

Delay differential equation of the form

x′(t)=f(t,x(t),x(t−τ1(t)),…,x(t−τk(t)))

has been discussed in [1, 2]. In particularly, the delay functions τ_j(t), j = 0, 1, …, k depend not only on unknown function, but also state, the delay functions τ_j(t, x(t)), j = 0, 1, …, k have been studied in many literatures. In [3], Cooke pointed out that it is highly desirable to establish the existence and stability properties of periodic solutions for equations of the form

x′(t)+ax(t−h(t,x(t)))=F(t),

in which the lag h(t, x(t)) implicitly involves x(t). Si and Wang [5] discussed the smooth solutions of equation of

x′(t)=λ1x(t)+λ2x[2](t)+…+λnx[n](t)+f(t),

where x^[1](t) = x(t), x^[2](t) = x(x(t)), x^[3](t) = x(x(x(t))), …, i.e., x^[i](t) denotes ith iterate of x(t), i = 1, 2, …, n. Later, by Schröder transformation, Liu and Si [5] considered the analytic solutions of the form

x′(t)=∑n=0k∑l=1∞Cl,n(t)(x[n](t))l+G(t),

where x^[n](t) denotes nth iterate of x(t), n = 0, 1, 2, …, k. Recently, In [6], Zhao and Fečkan studied the periodic solution of

x′(t)=∑n=1k∑l=1∞Cl,n(t)(x[n](t))l+G(t), (1.1)

where x^[n](t) denotes nth iterate of x(t), n = 1, 2, …, k. For some various properties of solutions for several iterative functional differential equations, we refer the interested reader to [7], [8, 9], [10], [11].

On the other hand, the existence of pseudo almost periodic solutions is among the most attractive topics in qualitative theory of differential equations due to their applications, especially in biology, economics and physics [12], [13, 14], [15], [16]. As pointed out by Ait Das and Ezzinbi [13], it is an interesting thing to study the pseudo almost periodic systems with delays. It is obviously that iterative functional differential equations are special type state-dependent delay differential equations. In [17], Liu pointed that the properties of the almost periodic functions do not always hold in the set of pseudo almost periodic functions and given an example: when x(t) is a pseudo almost periodic function, x(x(t)) may not be a pseudo almost periodic function. To the best of our knowledge, there are few results about pseudo almost periodic solutions for iterative functional differential equations except [17], [18] and [19].

In the present work, we propose an existence result for pseudo almost periodic solutions of Eq (1.1) by using Tikhonov fixed theorem. Uniqueness of the solution is achieved by Banach fixed point theorem.

This paper is organized as follows. In Section 2 we give some notes and establish the main existence result. In Section 3, we show that (1.1) has a unique pseudo almost periodic solution under some suitable conditions. Furthermore, we prove the stability depend on C_l,n and G. In Section 4, we present an example to illustrate the theory. Related problems are also studied in [20].

2 Existence result

In this section, the existence of pseudo almost periodic solutions of equation (1.1) will be studied. Throughout this paper, it will be assumed that

(H) C_1,1(t) : ℝ → (–∞, 0) is an almost periodic function, C_l,n(t) : ℝ → ℝ are pseudo almost periodic functions, and

C1,1+=supt∈RC1,1(t)<0,Cl,n+=supt∈R|Cl,n(t)|,G+=supt∈R|G(t)|,

where l = 1, 2, …; n = 2, …, k.

For convenience, we will use C(ℝ, ℝ) to denote the set of all continuous functions from ℝ to ℝ endowed with the usual metric d(f,g)=∑m=1∞2−m∥f−g∥m1+∥f−g∥m for ∥f – g∥_m = max_t∈[–m,m]|f(t) – g(t)|, so the topology on C(ℝ, ℝ) is the uniform convergence on each compact intervals of ℝ. We also consider the set BC(ℝ, ℝ) of all bounded and continuous functions from ℝ to ℝ with the norm ∥f∥ = sup_t∈ℝ|f(t)|, so the topology on BC(ℝ, ℝ) is the uniform convergence on ℝ. Denote by AP(ℝ, ℝ) the set of all almost periodic functions from ℝ to ℝ. Define the set

PAP0(R,R)={g∈BC(R,R)|limT→+∞12T∫−TT|g(t)|dt=0}.

A function φ ∈ BC(ℝ, ℝ) is called pseudo almost periodic if it can be expressed as φ = h + g, where h ∈ AP(ℝ, ℝ) and g ∈ PAP₀(ℝ, ℝ). The collection of such functions will be denoted by PAP(ℝ, ℝ). In particular, (PAP(ℝ, ℝ), ∥⋅∥) is a Banach space [21]. For M, L > 0, define

BC(M,L)={φ∈C(R,R)||φ(t)|≤M,|φ(t2)−φ(t1)|≤L|t2−t1|,forallt,t1,t2∈R},BS(M,L)=S(R,R)∩BC(M,L),S∈{PAP,AP,PAP0}.

From [17], it is easy to see that B_S(M, L), S ∈ {PAP, AP, PAP₀, C} are closed convex and bounded subsets of BC(ℝ, ℝ) and C(ℝ, ℝ), respectively. Furthermore, by the Arzelá-Ascoli theorem, the subset B_C(M, L) is compact in C(ℝ, ℝ).

On the other hand, the subset B_C(M, L) is not precompact in BC(ℝ, ℝ), since a sequence {MsechL(t+n)M}n=1∞∈BC(M,L) has no a convergent subsequence in BC(ℝ, ℝ). As in [19], we have following Theorem.

Theorem 2.1

The subsets B_S(M, L), S ∈ {PAP, AP, PAP₀, C} are not precompact in BC(ℝ, ℝ). They are precompact in C(ℝ, ℝ). Furthermore, B_AP(M, L) and B_PAP₀(M, L) are just precompact in C(ℝ, ℝ), while B_C(M, L) is compact in C(ℝ, ℝ).

So (non-)compactness of B_PAP(M, L) in C(R, R) is still open.

Finally, we recall Tikhonov fixed point theorem.

Theorem 2.2

(Tikhonov) Let Ω be a non-empty compact convex subset of a locally convex topological vector space X. Then any continuous function A : Ω → Ω has a fixed point.

Now we apply Theorem 2.2. Let X be either PAP(ℝ, ℝ) or BC(ℝ, ℝ) but fixed. If φ ∈ X, from Corollary 5.4 in [21], it follows φ^[2] ∈ X. Consider an auxiliary equation

x′(t)=C1,1(t)x(t)+∑n=2kC1,n(t)φ[n](t)+∑n=1k∑l=2∞Cl,n(t)(φ[n](t))l+G(t). (2.2)

where G ∈ PAP(ℝ, ℝ) and C_1,1(t) < 0. It is easy to see that the linear equation

x′(t)=C1,1(t)x(t)

admits an exponential dichotomy on ℝ, by Theorem 2.3 in [15], we know that (2.1) has exactly one solution

xφ(t)=∫−∞te∫stC1,1(u)du(∑n=2kC1,n(s)φ[n](s)+∑n=1k∑l=2∞Cl,n(s)(φ[n](s))l+G(s))ds

in X.

Let A : B_C(M, L) → BC(ℝ, ℝ) be defined by

(Aφ)(t)=∫−∞te∫stC1,1(u)du(∑n=2kC1,n(s)φ[n](s)+∑n=1k∑l=2∞Cl,n(s)(φ[n](s))l+G(s))ds,t∈R. (2.3)

Note A : B_PAP(M, L) → PAP(ℝ, ℝ).

Lemma 2.1

For any φ, ψ ∈ B_C(M, L), t₁, t₂ ∈ ℝ, the following inequality hold,

||φ[n]−ψ[n]||≤∑j=0n−1Lj∥φ−ψ∥,n=1,2,…. (2.4)

Proof

It can be obtained by direct calculation by the definition of B_C(M, L).□

Lemma 2.2

Operator A : B_C(M, L) → C(ℝ, ℝ) is continuous.

Proof

Let φ_j → φ₀ as j → ∞ for φ_j ∈ B_C(M, L), j ∈ ℕ₀ = ℕ ∪ {0} uniformly on any compact interval [–m, m], m ∈ ℕ of ℝ. Set

hj(s)=∑n=2kC1,n(s)φj[n](s)+∑n=1k∑l=2∞Cl,n(s)(φj[n](s))l+G(s),j∈N0.

Then h_j → h₀ uniformly on [–m, m]. Next we have

e∫s−mC1,1(u)duhj(s)≤(M∑n=2kC1,n++∑n=1k∑l=2∞Cl,n+Ml+G+)e−C1,1+(s+m),s∈[−∞,−m].

Since

∫−∞−me−C1,1+(s+m)ds=−1C1,1+<∞,

we can apply the Lebesgue dominated convergence theorem to obtain (Aφ_j)(–m) → (Aφ₀)(–m). From (2.2),

xj′(t)−x0′(t)=C1,1(t)(xj(t)−x0(t))+hj(t)−h0(t) (2.5)

for x_j(t) = (Aφ_j)(t), k ∈ ℕ₀. Integrating the both sides of (2.5) from –m to t, we have

xj(t)−x0(t)=xj(−m)−x0(−m)+∫−mtC1,1(s)(xj(s)−x0(s))ds+∫−mt(hj(s)−h0(s))ds

and

for any t ∈ [–m, m]. Then Gronwall’s inequality implies

∥xj−x0∥m≤|xj(−m)−x0(−m)|+2m∥hj−h0∥me−2C1,1+m,

which means

∥Aφj−Aφ0∥m≤|Aφj(−m)−Aφ0(−m)|+2m∥hj−h0∥me−2C1,1+m.

Hence Aφ_j(t) → Aφ₀(t) uniformly on t ∈ [–m, m]. Since m ∈ ℕ is arbitrarily, we get Aφ_j → Aφ₀ in C(ℝ, ℝ), i.e., A : B_C(M, L) → C(ℝ, ℝ) is continuous. This proves the continuity of A.□

Of course, then A : B_PAP(M, L) → C(ℝ, ℝ) is continuous as well.

Theorem 2.3

Suppose (H) holds, then Eq. (1.1) has a solution

φ∈BPAP(M,L)¯⊂C(R,R)

for the constants M and L satisfy

∑n=1k∑l=1∞Cl,n+Ml+G+≤0, (2.6)

2(M∑n=2kC1,n++∑n=1k∑l=2∞Cl,n+Ml+G+)≤L. (2.7)

For any φ ∈ B_C(M, L), t, t₁, t₂ ∈ ℝ, from (2.6) we have

Without loss of generality, assume t₂ ≥ t₁, using (2.7)

|(Aφ)(t2)−(Aφ)(t1)|≤|∫−∞t2e∫st2C1,1(u)du(∑n=2kC1,n(s)φ[n](s)+∑n=1k∑l=2∞Cl,n(s)(φ[n](s))l+G(s))ds−∫−∞t1e∫st1C1,1(u)du(∑n=2kC1,n(s)φ[n](s)+∑n=1k∑l=2∞Cl,n(s)(φ[n](s))l+G(s))ds|≤∑n=2k[|∫−∞t1e∫st1C1,1(u)du(e∫t1t2C1,1(u)du−1)C1,n(s)φ[n](s)ds|+|∫t1t2e∫st2C1,1(u)duC1,n(s)φ[n](s)ds|]+∑n=1k∑l=2∞[|∫−∞t1e∫st1C1,1(u)du(e∫t1t2C1,1(u)du−1)Cl,n(s)(φ[n](s))l)ds|+|∫t1t2e∫st2C1,1(u)duCl,n(s)(φ[n](s))lds|]+[|∫−∞t1e∫st1C1,1(u)du(e∫t1t2C1,1(u)du−1)G(s)ds|+|∫t1t2e∫st2C1,1(u)duG(s)ds|]≤2M∑n=2kC1,n+|t2−t1|+2∑n=1k∑l=2∞Cl,n+Ml|t2−t1|+2G+|t2−t1|≤L|t2−t1|.

This shows that Aφ ∈ B_C(M, L). Hence A : B_C(M, L) → B_C(M, L) and also A : B_PAP(M, L) → B_PAP(M, L). By Lemma 2.2 we get A : B_PAP(M, L) → B_PAP(M, L). So all conditions of Tikhonov fixed theorem are satisfied for A with Ω = B_PAP(M, L) and X = C(ℝ, ℝ). Thus there exists a fixed point φ in B_PAP(M, L) of A. This is equivalent to say that φ is a solution of (1.1) in B_PAP(M, L). This completes the proof.□

3 Uniqueness and stability

In this section, uniqueness and stability of (1.1) will be proved.

In addition to the assumption of Theorem 2.3, suppose that

∑n=2k∑j=0n−1LjC1,n++∑n=1k∑l=2∞∑j=0n−1LjCl,n+lMl−1<−C1,1+, (3.8)

then Eq. (1.1) has unique solution in B_PAP(M, L).

For operator A from B_PAP(M, L) into B_PAP(M, L), where we consider B_PAP(M, L) in BC(ℝ, ℝ). For φ, ψ ∈ B_PAP(M, L), by (2.3)and (2.4)

where Γ=−1C1,1+(∑n=2k∑j=0n−1LjC1,n++∑n=1k∑l=2∞∑j=0n−1LjCl,n+lMl−1).

Thus

∥φ−ψ∥≤Γ∥φ−ψ∥.

By (3.8), we know Γ < 1 and the fixed point φ must be unique.□

The unique solution obtained in Theorem 3.1 depends continuously on the given functions C_l,n(t) and G(t), for l = 1, 2, …, ; n = 1, …, k.

Proof

Under the assumptions of Theorem 3.1, if any functions C_l,n(t), C͠_l,n(t) and G(t), G͠(t) are given, and satisfy (2.6) and (2.7) for l = 1, …, ∞; n = 1, …, k. Then there are two unique corresponding functions x(t) and x͠(t) in B_PAP(M, L) such that

x(t)=(Ax)(t)=∫−∞te∫stC1,1(u)du(∑n=2kC1,n(s)x[n](s)+∑n=1k∑l=2∞Cl,n(s)(x[n](s))l+G(s))ds

and

x~(t)=(A~x~)(t)=∫−∞te∫stC~1,1(u)du(∑n=2kC~1,n(s)x~[n](s)+∑n=1k∑l=2∞C~l,n(s)(x~[n](s))l+G~(s))ds.

Then

∥x−x~∥=∥Ax−A~x~∥≤∥Ax−Ax~∥+∥Ax~−A~x~∥,

i.e.,

∥x−x~∥≤∥Ax~−A~x~∥1−Γ. (3.9)

Note

|(Ax~)(t)−(A~x~)(t)|≤|∫−∞te∫stC1,1(u)du(∑n=2kC1,n(s)x~[n](s)+∑n=1k∑l=2∞Cl,n(s)(x~[n](s))l+G(s))ds−∫−∞te∫stC~1,1(u)du(∑n=2kC~1,n(s)x~[n](s)+∑n=1k∑l=2∞C~l,n(s)(x~[n](s))l+G~(s))ds|≤|∫−∞t(e∫stC1,1(u)du−e∫stC~1,1(u)du)(∑n=2kC1,n(s)x~[n](s)+∑n=1k∑l=2∞Cl,n(s)(x~[n](s))l+G(s))ds|+∑n=2k|∫−∞te∫stC~1,1(u)du(C1,n(s)−C~1,n(s))x~[n](s)ds|+∑n=1k∑l=2∞|∫−∞te∫stC~1,1(u)du(Cl,n(s)−C~l,n(s))(x~[n](s))lds|+|∫−∞te∫stC~1,1(u)du(G(s)−G~(s))ds|≤1C1,1+C~1,1+(M∑n=2kC1,n++∑n=1k∑l=2∞Cl,n+Ml+G+)∥C1,1−C~1,1∥−MC~1,1+∑n=2k∥(C1,n−C~1,n∥−1C~1,1+∑n=1k∑l=2∞Ml∥Cl,n−C~l,n∥−1C~1,1+∥G−G~∥≤Δ(∑l=1∞∑n=1k∥Cl,n−C~l,n∥+∥G−G~∥), (3.10)

where

Δ=−1C~1,1+max{−1C1,1+(M∑n=2kC1,n++∑n=1k∑l=2∞Cl,n+Ml+G+),1,M,M2,M3,…}.

Using (3.9) and (3.10), we have

∥x−x~∥≤Δ1−Γ(∑l=1∞∑n=1k∥Cl,n−C~l,n∥+∥G−G~∥).

This completes the proof. □

4 Example

In this section, an example is provided to illustrate that the assumptions of Theorem 2.3 do not self-contradict.

Example 4.1

Now, we will show that the conditions in Theorem 2.3 do not self-contradict. Consider the following equation:

x′(t)=−∑l=1∞|sin⁡t+sin⁡3t|+8100l(x(t))l+∑l=1∞|cos⁡t+cos⁡5t|+1100l(x[2](t))l+1100(sin⁡t+sin⁡2t), (4.11)

−110≤C1,1+≤−225,Cs,1+≤10100l,Cl,2+≤3100l,G+≤150,s=2,3,…,l=1,2,….

∑n=12∑l=1∞Cl,n+Ml+G+≤−712475<0

and

2(MC1,2++∑n=12∑l=2∞Cl,n+Ml+G+)≤2542475<1=L.

then (2.6) and (2.7) are satisfied. By Theorem 2.3, equation (4.11) has a solution in B_PAP(1, 1).

Acknowledgement

This work was partially supported by the National Natural Science Foundation of China (Grant No. 11501069, 11671061), Science and Technology Research Program of Chongqing Municipal Education Commission (Grant No. KJQN201800502, KJQN201900525), Foundation of youth talent of Chongqing Normal University (02030307-00039)

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Received: 2019-02-18

Accepted: 2019-10-04

Published Online: 2019-12-10

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Keywords for this article

Iterative differential equation; pseudo almost periodic solutions; fixed point theorem

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