# Hyers-Ulam Stability of the Delay Equationy'(t)=λy(t-τ)

Authors: Soon-Mo Jung; Janusz Brzdęk
Journal: Abstract and Applied Analysis
 (2010)

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## Abstract

We investigate the approximate solutionsy:[−τ,∞)→ℝ
of the delay differential equation
y'(t)=λy(t-τ)(t∈[0,∞))
with an initial condition, where
λ>0
and
τ>0
are real constants. We show that they can be “approximated” by solutions of the equation that are constant on the interval
[-τ,0]
and, therefore, have quite simple forms. Our results correspond to the notion of stability introduced by Ulam and Hyers.


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## Body

## 1. Introduction

While investigating real world phenomena we very often use equations. In general, it is well known that those equations are satisfied, however, with some error. Sometimes that error is neglected and it is believed that this will have only a minor influence on the final outcome. Since it is not always the case, it seems to be of interest to know when we can neglect the error, why, and to what extent.One of the tools for systematic treatment of the problem described above seems to be the notion of Hyers-Ulam stability and some ideas inspired by it. That notion has not actually been made very precise so far, and we still seek a better understanding of it (see, e.g., [1, 2]). But, roughly speaking, we might say that it is connected with the investigation of the following question:when is a function satisfying an equation with some “small” (in some sense) error “close” to a solution of that equation?The study of the stability problem for functional equations starts from the famous talk of Ulam and the partial solution of Hyers to Ulam's problem (see [3, 4]). Thereafter, Hyers and Ulam (see, e.g., [4–8]), but also several other authors (see, e.g., [9–12]), attempted to study the stability problem for various functional equations. In particular, we should mention here the well-known paper [13] of Th.M. Rassias, in which he actually rediscovered the result of Aoki [9] (cf. [14]), and which has significantly influenced research of numerous mathematicians (see [15–21] and the references therein).Assume thatX and Y are normed spaces and that I is an open subset of X. Let ℱ be a class of differentiable functions mapping I into Y. If for any real ε≥0 and any function y∈ℱ satisfying the differential inequality(1.1)sup⁡t∈I‖∑i=0nai(t)y(i)(t)+h(t)‖≤ε,
there exists a solution y0:I→Y of the differential equation(1.2)∑i=0nai(t)y(i)(t)+h(t)=0
such that(1.3)sup⁡t∈I‖y(t)-y0(t)‖≤K(ε),
(where K(ɛ) depends on ɛ only), then we say that the above differential equation is Hyers-Ulam stable in the class of function ℱ. We may use this terminology for other differential equations. For more detailed definitions of the Hyers-Ulam stability and some discussions and critiques of that subject, refer to [1, 2, 4, 13, 15, 16, 18–21].Obloza seems to be the first author who has investigated the Hyers-Ulam stability of linear differential equations (see [22, 23]) in the sense described above. Here, let us recall a result of Alsina and Ger (see [24]):If a differentiable functionf:I→ℝis a solution of the differential inequality|y′(t)-y(t)|≤ɛ, whereIis an open subinterval ofℝ, then there exists a solutionf0:I→ℝof the differential equationy′(t)=y(t)such that|f(t)-f0(t)|≤3ɛfor anyt∈I.An analogous result for the Banach space valued functions has been proved by Takahasi et al. [25]. For some further examples of investigations of such kind of stability of differential equations see also [26–35].In what follows,λ>0 and τ>0 stand for fixed real constants, unless clearly stated otherwise. Moreover, ℱ denotes the family of all continuous functions mapping the real interval [-τ,∞) into ℝ which are continuously differentiable in [0,∞).In this paper, motivated by the above-mentioned outcomes on Hyers-Ulam stability, we prove that a somewhat similar type of stability is valid for the delay differential equation(1.4)y′(t)=λy(t-τ)
in the class of functions ℱ with an initial condition. More precisely, for given real numbers ɛ1, ɛ2 and continuous functions ξ1,ξ2:[-τ,0]→ℝ with ɛ1<ɛ2 and ξ1(t)<ξ2(t) for t∈[-τ,0], we describe a behavior of solutions of the following problem (the delay differential inequality with an initial condition)(1.5)ɛ1≤y′(t)-λy(t-τ)≤ɛ2,t≥0,(1.6)ξ1(t)≤y(t)-y(0)≤ξ2(t),-τ≤t≤0,
and compare them with solutions of the delay equation (1.4) that are constant on the interval [-τ,0] (and therefore have quite simple forms). Functions satisfying those two inequalities may be considered to be approximate solutions of (1.4).


## 2. An Auxiliary Theorem

Let us recall a description of a class of solutions of (1.4).Remark 2.1.
It is known that ifλ≠0 and τ>0 are real constants, then the general solution y:[-τ,∞)→ℝ of the delay differential equation (1.4), which is constant on the interval [-τ,0], is given by
(2.1)y(t)=α∑n=-1[t/τ]λn+1(t-nτ)n+1(n+1)!,t∈[-τ,∞),
where α is an arbitrary real number and [t/τ] denotes the greatest integer that is less than or equal to t/τ.The following theorem will be very useful in the sequel.Theorem 2.2.
Letz,ϕ∈ℱ. Assume that
(2.2)z′(t)-λz(t-τ)≤ϕ′(t)-λϕ(t-τ),t≥0,(2.3)z(t)-z(0)≤ϕ(t)-ϕ(0),t∈[-τ,0].
There exist unique solutions ẑ,ϕ̂∈ℱ of (1.4) satisfying the initial conditions
(2.4)ϕ̂(t)=ϕ(0),t∈[-τ,0),(2.5)ẑ(t)=z(0),t∈[-τ,0),
such that
(2.6)z(t)-ẑ(t)≤ϕ(t)-ϕ̂(t),t≥0.Proof.
Writey:=z-ϕ, y0:=z(0)-ϕ(0) and
(2.7)ŷ(t)=y0∑n=-1[t/τ]λn+1(t-nτ)n+1(n+1)!,t∈[-τ,∞).
Clearly,
(2.8)ŷ(t)=y0,t∈[-τ,0),
and by (2.2),
(2.9)y′(t)≤λy(t-τ),t≥0.
By the induction on [t/τ] we prove that, for each t≥0,
(2.10)y(t)≤ŷ(t).
Let[t/τ]=0 (i.e., 0≤t<τ). It follows from (2.3) and (2.9) that y′(t)≤λy0. If we integrate each term from 0 to t, then we get
(2.11)y(t)-y0≤λy0t,
which together with the definition of ŷ gives (2.10) (for t∈[0,τ)).
Now, take a nonnegative integerm and assume that inequality (2.10) is true for [t/τ]=m, that is,
(2.12)y(t)≤y0∑n=-1mλn+1(t-nτ)n+1(n+1)!,mτ≤t<mτ+τ.
We are to show that this is also the case for [t/τ]=m+1 (i.e., for mτ+τ≤t<mτ+2τ).
Due to (2.9), we have
(2.13)∫mτ+τty′(u)du≤λ∫mτ+τty(u-τ)du,
and hence
(2.14)y(t)≤λ∫mτt-τy(v)dv+y(mτ+τ)
for all mτ+τ≤t<mτ+2τ.
Substitutet-τ for t in (2.12) (with mτ+τ≤t<mτ+2τ) and integrate each term of the resulting inequalities from mτ+τ to t. Then, we have
(2.15)∫mτt-τy(v)dv≤y0∑n=0m+1λn(t-nτ)n+1(n+1)!-y0∑n=0m+1λn(mτ-nτ+τ)n+1(n+1)!
for any mτ+τ≤t<mτ+2τ. Moreover, since y is continuous in [0,∞),
(2.16)lim⁡t→mτ+τy(t)=y(mτ+τ)
and consequently (2.12) yields
(2.17)y(mτ+τ)≤y0∑n=-1mλn+1(mτ-nτ+τ)n+1(n+1)!.
Thus, from (2.15) and (2.17), we obtain
(2.18)λ∫mτt-τy(v)dv+y(mτ+τ)≤y0∑n=-1m+1λn+1(t-nτ)n+1(n+1)!
for any mτ+τ≤t<mτ+2τ. Hence, by (2.14) and the definition of ŷ, (2.10) holds true for [t/τ]=m+1 (i.e., mτ+τ≤t<mτ+2τ), as well.
Thus we have proved that (2.10) is valid for all t≥0. Define
(2.19)ϕ̂(t)=ϕ(0)∑n=-1[t/τ]λn+1(t-nτ)n+1(n+1)!,t≥-τ,ẑ(t)=z(0)∑n=-1[t/τ]λn+1(t-nτ)n+1(n+1)!,t≥-τ.
Clearly, (2.4) and (2.5) are valid. Moreover, ŷ=ẑ-ϕ̂ and hence, (2.6) follows from (2.10).
The uniqueness ofϕ̂ and ẑ follows from Remark 2.1.


## 3. The Main Stability Results

Now, we present some corollaries that are immediate consequences of Theorem2.2. They contain stability results for (1.4).Corollary 3.1.
Lety,ψ1,ψ2∈ℱ. Assume that ψ1(0)=ψ2(0)=0,
(3.1)ψ1′(t)-λψ1(t-τ)≤y′(t)-λy(t-τ)≤ψ2′(t)-λψ2(t-τ),t≥0,(3.2)ψ1(t)≤y(t)-y(0)≤ψ2(t),t∈[-τ,0].
Then there is a unique solution ŷ∈ℱ of (1.4) such that ŷ(t)=y(0) for t∈[-τ,0) and
(3.3)ψ1(t)≤y(t)-ŷ(t)≤ψ2(t),t≥0.Proof.
Observe that (3.2) implies
(3.4)ψ1(t)-ψ1(0)≤y(t)-y(0)≤ψ2(t)-ψ2(0),t∈[-τ,0].
Hence, it is enough to use Theorem 2.2, first with z=ψ1, ϕ=y, and next with z=y, ϕ=ψ2, where
(3.5)ψ̂i(t)=ψi(0)∑n=-1[t/τ]λn+1(t-nτ)n+1(n+1)!=0,t∈[-τ,∞),
for i=1,2.Remark 3.2.
Observe that, forλ>0, the inequality in the proof of Corollary 3.1 implies that the estimation (3.3) is actually valid for all t≥-τ.
It is interesting whether analogues of Corollary3.1 and our further results can be obtained also for λ<0.The next corollary is a particular case of Corollary3.1 and corresponds to the classical Hyers-Ulam stability results.Corollary 3.3.
Lety,ψ2∈ℱ. Suppose that ψ2(0)=0,
(3.6)|y′(t)-λy(t-τ)|≤ψ2′(t)-λψ2(t-τ),t≥0,|y(t)-y(0)|≤ψ2(t),t∈[-τ,0].
Then there exists a unique solution ŷ∈ℱ of (1.4) satisfying the initial condition:
(3.7)ŷ(t)=y(0),t∈[-τ,0),
and such that
(3.8)|y(t)-ŷ(t)|≤ψ2(t),t≥0.Proof.
Clearly we have(3.9)-ψ2(t)+ψ2(0)≤y(t)-y(0)≤ψ2(t)-ψ2(0),t∈[-τ,0].
Hence from Theorem 2.2 with ψ1:=-ψ2, analogously as in the proof of Corollary 3.1, we get the statement.Note that the functionsψ1,ψ2 in Theorem 2.2 and Corollary 3.1 can be constant only if ψi(t)≡0, i=1,2. Therefore the case of problem (1.5), (1.6) is a bit more complicated. It is described by the following theorem (in particular, note that (1.6) implies (3.10)).Theorem 3.4.
Letɛ1,ɛ2 be real numbers with ɛ1≤ɛ2 and ξ1,ξ2∈ℱ be solutions of (1.4). Suppose that y∈ℱ satisfy inequality (1.5) and
(3.10)ξ1(t)-ξ1(0)≤y(t)-y(0)≤ξ2(t)-ξ2(0),t∈[-τ,0].
Then there exists unique solutions ŷ,ξ̂1,ξ̂2∈ℱ of (1.4) such that
(3.11)ŷ(t)=y(0),ξ̂i(t)=ξi(0)-ɛiλ,t∈[-τ,0),i=1,2,ξ1(t)-ξ̂1(t)-ɛ1λ≤y(t)-ŷ(t)≤ξ2(t)-ξ̂2(t)-ɛ2λ,t≥0.Proof.
Let(3.12)ψi:=ξi-ɛiλ,i=1,2.
Then
(3.13)ψi′(t)-λψi(t-τ)=ɛi,i=1,2.
Moreover (3.10) yields
(3.14)ψ1(t)-ψ1(0)≤y(t)-y(0)≤ψ2(t)-ψ2(0),t∈[-τ,0].
Consequently, by Theorem 2.2, there exists unique solutions ψ̂1,ψ̂2,ŷ∈ℱ of (1.4) such that ψ̂i(t)=ψi(0) and ŷ(t)=y(0) for t∈[-τ,0), i=1,2, and
(3.15)ψ1(t)-ψ̂1(t)≤y(t)-ŷ(t)≤ψ2(t)-ψ̂2(t),t≥0.
This and Remark 2.1 yield the statement.A particular case of Theorem3.4 is the subsequent corollary.Corollary 3.5.
Letɛ≥0 be a real constant, y,ξ∈ℱ, and ξ be a solution of (1.4). Suppose that |y′(t)-λy(t-τ)|≤ɛ for t≥0 and there is a real number u0 with |y(t)-u0|≤ξ(t) for t∈[-τ,0]. Then there are unique solutions ξ̂0,ŷ∈ℱ of (1.4) such that (3.7) holds,
(3.16)ξ̂0(t)=ξ(0)-ɛλ,t∈[-τ,0),
and, for each t≥0,
(3.17)|y(t)-ŷ(t)|≤ξ(t)-ξ̂0(t)-ɛλ=ξ(t)-ξ(0)-(ξ(0)-ɛλ)∑n=0[t/τ]λn+1(t-nτ)n+1(n+1)!.Proof.
It is enough to use Theorem3.4, with ɛ1:=-ɛ and ɛ2:=ɛ, and Remark 2.1.


## 4. Some Immediate Consequences

Clearly, ifξ(0)=0, then (3.17) has the following simpler form:(4.1)|y(t)-ŷ(t)|≤ξ(t)+ɛλ∑n=0[t/τ]λn+1(t-nτ)n+1(n+1)!.
If ξ(t)≡0 (i.e., y(t)=u0 for t∈[-τ,0]) and λ=τ=1, then the inequality (3.17) takes the following form:(4.2)|y(t)-y0∑n=-1[t](t-n)n+1(n+1)!|≤ɛ∑n=0[t](t-n)n+1(n+1)!,t≥0.From Theorem3.4 we can derive an estimation of solutions of the equation(4.3)y′(t)=λ(y(t-τ)+d(t-τ)),t∈[0,∞),
where d∈ℱ is fixed. Namely we have the following result.Corollary 4.1.
Letξ1,ξ2∈ℱ be solutions of (1.4) and y∈ℱ be a solution of (4.3) satisfying condition (3.10). Let ɛ1:=inft∈[0,∞)d′(t) and ɛ2:=supt∈[0,∞)d′(t). Then, for each t≥0,
(4.4)ξ1(t)-ξ1(0)+(y(0)+d(0)+ɛ1λ-ξ1(0))∑n=0[t/τ]λn+1(t-nτ)n+1(n+1)!≤y(t)+d(t)≤ξ2(t)-ξ2(0)+(y(0)+d(0)+ɛ2λ-ξ2(0))∑n=0[t/τ]λn+1(t-nτ)n+1(n+1)!.Proof.
Assume thatɛ1>-∞. Write ϕ:=y+d and z:=ξ1-(ɛ1/λ). Then ϕ′(t)-λϕ(t-τ)=ϕ′(t) for t≥0. Consequently (2.2) holds true. Further, (3.10) implies (2.3). Now, it is enough to use Theorem 2.2.
Ifɛ2<∞, we use Theorem 2.2 with z:=y+d and ϕ:=ξ2-(ɛ2/λ).We end the paper with some estimation of solutions of a more general equation(4.5)y′(t)=λy(t-τ)+b(Ty(t)),t∈[-τ,∞),
where Z is a nonempty set, I=[-τ,∞), and 𝒯 is an operator that maps the family of function ℝI into the family of function ZI, and b:Z→ℝ.Corollary 4.2.
Letξ1,ξ2∈ℱ be solutions of (1.4) and y∈ℱ be a solution of (4.5) satisfying condition (3.10). Let ɛ1=inft∈ℝb(t) and ɛ2=supt∈ℝb(t). Then, for each t≥0,
(4.6)ξ1(t)-ξ1(0)+(y(0)+ɛ1λ-ξ1(0))∑n=0[t/τ]λn+1(t-nτ)n+1(n+1)!≤y(t)≤ξ2(t)-ξ2(0)+(y(0)+ɛ2λ-ξ2(0))∑n=0[t/τ]λn+1(t-nτ)n+1(n+1)!.Proof.
Note thatɛ1≤y′(t)-λy(t-τ)=b(𝒯y(t))≤ɛ2 for t≥0. So it is enough to use Theorem 2.2 analogously as in the proof of Corollary 4.1, first with ϕ=y and z:=ξ1-(ɛ1/λ), and next with z:=y and ϕ:=ξ2-(ɛ2/λ).



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