ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu
LAPLACE TRANSFORM AND GENERALIZED HYERS-ULAM STABILITY OF LINEAR DIFFERENTIAL EQUATIONS
QUSUAY H. ALQIFIARY, SOON-MO JUNG
Abstract. By applying the Laplace transform method, we prove that the linear differential equation
y(n)(t) +
n−1
X
k=0
αky(k)(t) =f(t)
has the generalized Hyers-Ulam stability, whereαkis a scalar,yandf aren times continuously differentiable and of exponential order.
1. Introduction
In 1940, Ulam [24] posed a problem concerning the stability of functional equa- tions: “Give conditions in order for a linear function near an approximately linear function to exist.” A year later, Hyers [5] gave an answer to the problem of Ulam for additive functions defined on Banach spaces: Let X1 and X2 be real Banach spaces andε >0. Then for every functionf :X1→X2 satisfying
kf(x+y)−f(x)−f(y)k ≤ε (x, y∈X1),
there exists a unique additive functionA:X1→X2with the property kf(x)−A(x)k ≤ε (x∈X1).
After Hyers’s result, many mathematicians have extended Ulam’s problem to other functional equations and generalized Hyers’s result in various directions (see [3, 6, 10, 18]). A generalization of Ulam’s problem was recently proposed by re- placing functional equations with differential equations: The differential equation ϕ(f, y, y0, . . . , y(n)) = 0 has Hyers-Ulam stability if for a givenε >0 and a function y such that |ϕ(f, y, y0, . . . , y(n))| ≤ε, there exists a solution ya of the differential equation such that |y(t)−ya(t)| ≤ K(ε) and limε→0K(ε) = 0. If the preceding statement is also true when we replace εand K(ε) by ϕ(t) and Φ(t), whereϕ,Φ are appropriate functions not depending on y andya explicitly, then we say that the corresponding differential equation has the generalized Hyers-Ulam stability (or Hyers-Ulam-Rassias stability).
2000Mathematics Subject Classification. 44A10, 39B82, 34A40, 26D10.
Key words and phrases. Laplace transform method; differential equations;
generalized Hyers-Ulam stability.
c
2014 Texas State University - San Marcos.
Submitted March 5, 2014. Published March 21, 2014.
1
Ob loza seems to be the first author who has investigated the Hyers-Ulam stability of linear differential equations (see [14, 15]). Thereafter, Alsina and Ger published their paper [1], which handles the Hyers-Ulam stability of the linear differential equationy0(t) =y(t): If a differentiable functiony(t) is a solution of the inequality
|y0(t)−y(t)| ≤ ε for any t ∈ (a,∞), then there exists a constant c such that
|y(t)−cet| ≤3εfor allt∈(a,∞).
Those previous results were extended to the Hyers-Ulam stability of linear dif- ferential equations of first order and higher order with constant coefficients in [12, 22, 23] and in [13], respectively. Furthermore, Jung has also proved the Hyers-Ulam stability of linear differential equations (see [7, 8, 9]). Rus investigated the Hyers-Ulam stability of differential and integral equations using the Gronwall lemma and the technique of weakly Picard operators (see [20, 21]). Recently, the Hyers-Ulam stability problems of linear differential equations of first order and sec- ond order with constant coefficients were studied by using the method of integral factors (see [11, 25]). The results given in [8, 11, 12] have been generalized by Cimpean and Popa [2] and by Popa and Ra¸sa [16, 17] for the linear differential equations ofnth order with constant coefficients.
Recently, Rezaei, Jung and Rassias have proved the Hyers-Ulam stability of linear differential equations by using the Laplace transform method (see [19]).
In this paper, by using the Laplace transform method, we prove that the linear differential equation of thenth order
y(n)(t) +
n−1
X
k=0
αky(k)(t) =f(t)
has the generalized Hyers-Ulam stability, whereαk is a scalar,yandf arentimes continuously differentiable and of exponential order, respectively.
2. Preliminaries
Throughout this paper,Fwill denote either the real fieldRor the complex field C. A functionf : (0,∞)→Fis said to be of exponential order if there are constants A, B∈Rsuch that
|f(t)| ≤AetB
for allt >0. For each functionf : (0,∞)→Fof exponential order, we define the Laplace transform off by
F(s) = Z ∞
0
f(t)e−stdt.
There exists a unique number −∞ ≤ σ < ∞ such that this integral converges if
<(s)> σand diverges if<(s)< σ, where<(s) denotes the real part of the (complex) numbers. The number σis called the abscissa of convergence and denoted byσf. It is well known that|F(s)| →0 as<(s)→ ∞. Furthermore,f is analytic on the open right half plane{s∈C:<(s)> σ}and we have
d
dsF(s) =− Z ∞
0
te−stf(t)dt (<(s)> σ).
The Laplace transform off is sometimes denoted byL(f). It is well known thatL is linear and one-to-one.
Conversely, letf(t) be a continuous function whose Laplace transformF(s) has the abscissa of convergenceσf, then the formula for the inverse Laplace transforms yields
f(t) = 1 2πi lim
T→∞
Z α+iT
α−iT
F(s)estds= 1 2π
Z ∞
−∞
e(α+iy)tF(α+iy)dy
for any real constantα > σf, where the first integral is taken along the vertical line
<(s) =αand converges as an improper Riemann integral and the second integral is used as an alternative notation for the first integral (see [4]). Hence, we have
L(f)(s) = Z ∞
0
f(t)e−stdt (<(s)> σf) L−1(F)(t) = 1
2π Z ∞
−∞
e(α+iy)tF(α+iy)dy (α > σf).
The convolution of two integrable functionsf, g: (0,∞)→Fis defined by (f∗g)(t) :=
Z t
0
f(t−x)g(x)dx.
ThenL(f∗g) =L(f)L(g).
Lemma 2.1 ([19]). Let P(s) = Pn
k=0αksk and Q(s) = Pm
k=0βksk, where m, n are nonnegative integers with m < n andαk, βk are scalars. Then there exists an infinitely differentiable function g: (0,∞)→Fsuch that
L(g) =Q(s)
P(s) (<(s)> σP) and
g(i)(0) =
(0 fori∈ {0,1, . . . , n−m−2}, βm/αn fori=n−m−1
whereσP = max{<(s) :P(s) = 0}.
Lemma 2.2 ([19]). Given an integer n > 1, let f : (0,∞) → F be a continuous function and letP(s) be a complex polynomial of degreen. Then there exists ann times continuously differentiable functionh: (0,∞)→F such that
L(h) = L(f)
P(s) (<(s)>max{σP, σf}),
whereσP = max{<(s) :P(s) = 0} andσf is the abscissa of convergence forf. In particular, it holds that h(i)(0) = 0for every i∈ {0,1, . . . , n−1}.
3. Main Results
LetFdenote eitherRorC. In the following theorem, using the Laplace transform method, we investigate the generalized Hyers-Ulam stability of the linear differential equation of first order
y0(t) +αy(t) =f(t). (3.1)
Theorem 3.1. Letαbe a constant inFand letϕ: (0,∞)→(0,∞)be an integrable function. If a continuously differentiable function y : (0,∞) → F satisfies the inequality
|y0(t) +αy(t)−f(t)| ≤ϕ(t) (3.2)
for allt >0, then there exists a solutionyα: (0,∞)→Fof the differential equation (3.1)such that
|y(t)−yα(t)| ≤e−<(α)t Z t
0
e<(α)xϕ(x)dx
for any t >0.
Proof. If we define a functionz: (0,∞)→Fbyz(t) :=y0(t) +αy(t)−f(t) for each t >0, then
L(y)−y(0) +L(f)
s+α = L(z)
s+α. (3.3)
If we setyα(t) :=y(0)e−αt+ (E−α∗f)(t), whereE−α(t) =e−αt, thenyα(0) =y(0) and
L(yα) = y(0) +L(f)
s+α = yα(0) +L(f)
s+α . (3.4)
Hence, we get
L yα0(t) +αyα(t)
=sL(yα)−yα(0) +αL(yα) =L(f).
SinceLis a one-to-one operator, it holds that y0α(t) +αyα(t) =f(t).
Thus,yα is a solution of (3.1).
Moreover, by (3.3) and (3.4), we obtainL(y)− L(yα) =L(E−α∗z). Therefore, we have
y(t)−yα(t) = (E−α∗z)(t). (3.5) In view of (3.2), it holds that
|z(t)| ≤ϕ(t) (3.6)
for allt >0, and it follows from the definition of convolution, (3.5), and (3.6) that
|y(t)−yα(t)|=|(E−α∗z)(t)|
=
Z t
0
E−α(t−x)z(x)dx
≤ Z t
0
e−α(t−x) ϕ(x)dx
≤e−<(α)t Z t
0
e<(α)xϕ(x)dx
for allt >0. (We remark that Rt
0e<(α)xϕ(x)dxexists for each t >0 providedϕis
an integrable function.)
Corollary 3.2. Letαbe a constant inFand letϕ: (0,∞)→(0,∞)be an integrable function such that
Z t
0
e<(α)(x−t)ϕ(x)dx≤Kϕ(t) (3.7)
for allt >0 and for some positive real constantK. If a continuously differentiable function y: (0,∞)→Fsatisfies the inequality (3.2)for all t >0, then there exists a solution yα: (0,∞)→Fof the differential equation (3.1)such that
|y(t)−yα(t)| ≤Kϕ(t) for any t >0.
In the following remark, we show that there exists an integrable function ϕ : (0,∞)→(0,∞) satisfying the condition (3.7).
Remark 3.3. Letαbe a constant in Fwith<(α)>−1. If we defineϕ(t) =Aet for allt >0 and for someA >0, then we have
Z t
0
e<(α)(x−t)ϕ(x)dx= Z t
0
e<(α)(x−t)Aexdx
= 1
1 +<(α)
Aet−Ae−<(α)t
≤ 1
1 +<(α)ϕ(t) for eacht >0.
Now, we apply the Laplace transform method to the proof of the generalized Hyers-Ulam stability of the linear differential equation of second order
y00(t) +βy0(t) +αy(t) =f(t). (3.8) Theorem 3.4. Let α andβ be constants in F such that there exist a, b∈ F with a+b=−β,ab=α, anda6=b. Assume thatϕ: (0,∞)→(0,∞)is an integrable function. If a twice continuously differentiable functiony: (0,∞)→F satisfies the inequality
|y00(t) +βy0(t) +αy(t)−f(t)| ≤ϕ(t) (3.9) for allt >0, then there exists a solutionyc : (0,∞)→Fof the differential equation (3.8)such that
|y(t)−yc(t)| ≤ e<(a)t
|a−b|
Z t
0
e−<(a)xϕ(x)dx+ e<(b)t
|a−b|
Z t
0
e−<(b)xϕ(x)dx
for allt >0.
Proof. If we define a functionz: (0,∞)→Fbyz(t) :=y00(t) +βy0(t) +αy(t)−f(t) for eacht >0, then we have
L(z) = s2+βs+α
L(y)−[sy(0) +βy(0) +y0(0)]− L(f). (3.10) In view of (3.10), a functiony0: (0,∞)→Fis a solution of (3.8) if and only if
s2+βs+α
L(y0)−sy0(0)−[βy0(0) +y00(0)] =L(f). (3.11) Now, sinces2+βs+α= (s−a)(s−b), (3.10) implies that
L(y)−sy(0) + [βy(0) +y0(0)] +L(f)
(s−a)(s−b) = L(z)
(s−a)(s−b). (3.12) If we set
yc(t) :=y(0)aeat−bebt
a−b + [βy(0) +y0(0)]Ea,b(t) + (Ea,b∗f)(t), (3.13) whereEa,b(t) := eata−b−ebt, thenyc(0) =y(0). Moreover, since
y0c(t) =y(0)a2eat−b2ebt
a−b + [βy(0) +y0(0)]aeat−bebt a−b + d
dt(Ea,b∗f)(t), (Ea,b∗f)(t) = eat
a−b Z t
0
e−axf(x)dx− ebt a−b
Z t
0
e−bxf(x)dx,
we have
y0c(0) =y(0)a2−b2
a−b + [βy(0) +y0(0)]a−b a−b
= (a+b)y(0) +βy(0) +y0(0)
=y0(0).
It follows from (3.13) that
L(yc) = syc(0) + [βyc(0) +yc0(0)] +L(f)
(s−a)(s−b) . (3.14)
Now, (3.11) and (3.14) imply that yc is a solution of (3.8). Applying (3.12) and (3.14) and considering the facts thatyc(0) =y(0),yc0(0) =y0(0), andL(Ea,b∗z) =
L(z)
(s−a)(s−b), we obtain L(y)− L(yc) = L(Ea,b ∗z) or equivalently, y(t)−yc(t) = (Ea,b∗z)(t).
In view of (3.9), it holds that|z(t)| ≤ϕ(t), and it follows from the definition of the convolution that
|y(t)−yc(t)|=|(Ea,b∗z)(t)|
≤ e<(a)t
|a−b|
Z t
0
e−<(a)xϕ(x)dx+ e<(b)t
|a−b|
Z t
0
e−<(b)xϕ(x)dx
for any t > 0. We remark that Rt
0e−<(a)xϕ(x)dx and Rt
0e−<(b)xϕ(x)dx exist for
anyt >0 providedϕis an integrable function.
Corollary 3.5. Let αand β be constants in F such that there exista, b∈F with a+b=−β,ab=α, anda6=b. Assume thatϕ: (0,∞)→(0,∞)is an integrable function for which there exists a positive real constant K with
Z t
0
e<(a)(t−x)+e<(b)(t−x)
ϕ(x)dx≤Kϕ(t) (3.15)
for allt >0. If a twice continuously differentiable functiony: (0,∞)→Fsatisfies the inequality(3.9)for allt >0, then there exists a solution yc: (0,∞)→Fof the differential equation(3.8)such that
|y(t)−yc(t)| ≤ K
|a−b|ϕ(t) for allt >0.
We now show that there exists an integrable functionϕ: (0,∞)→(0,∞) which satisfies the condition (3.15).
Remark 3.6. Letα and β be constants in F such that there exista, b ∈F with a+b=−β,ab=α,<(a)<1,<(b)<1, anda6=b. If we defineϕ(t) =Aetfor all t >0 and for someA >0, then we get
Z t
0
e<(a)(t−x)+e<(b)(t−x) ϕ(x)dx
= Z t
0
e<(a)(t−x)+e<(b)(t−x) Aexdx
= A
1− <(a)
et−e<(a)t
+ A
1− <(b)
et−e<(b)t
≤ 1
1− <(a)+ 1 1− <(b)
ϕ(t)
for allt >0.
Similarly, we apply the Laplace transform method to investigate the generalized Hyers-Ulam stability of the linear differential equation ofnth order
y(n)(t) +
n−1
X
k=0
αky(k)(t) =f(t) (3.16) Theorem 3.7. Let α0, α1, . . . , αn be scalars in F with αn = 1, where n is an integer larger than 1. Assume that ϕ: (0,∞)→(0,∞)is an integrable function of exponential order. If anntimes continuously differentiable functiony: (0,∞)→F satisfies the inequality
y(n)(t) +
n−1
X
k=0
αky(k)(t)−f(t)
≤ϕ(t) (3.17)
for all t > 0, then there exist real constants M > 0 and σg and a solution yc : (0,∞)→Fof the differential equation (3.16)such that
|y(t)−yc(t)| ≤M Z t
0
eα(t−x)ϕ(x)dx
for allt >0andα > σg.
Proof. Applying integration by parts repeatedly, we derive L y(k)
=skL(y)−
k
X
j=1
sk−jy(j−1)(0)
for any integer k > 0. Using this formula, we can prove that a function y0 : (0,∞)→Fis a solution of (3.16) if and only if
L(f) =
n
X
k=0
αkskL(y0)−
n
X
k=1
αk k
X
j=1
sk−jy0(j−1)(0)
=
n
X
k=0
αkskL(y0)−
n
X
j=1 n
X
k=j
αksk−jy(j−1)0 (0)
=Pn,0(s)L(y0)−
n
X
j=1
Pn,j(s)y0(j−1)(0),
(3.18)
wherePn,j(s) :=Pn
k=jαksk−j forj∈ {0,1, . . . , n}.
Let us define a functionz: (0,∞)→Fby z(t) :=y(n)(t) +
n−1
X
k=0
αky(k)(t)−f(t) (3.19) for allt >0. Then, similarly as in (3.18), we obtain
L(z) =Pn,0(s)L(y)−
n
X
j=1
Pn,j(s)y(j−1)(0)− L(f).
Hence, we get
L(y)− 1 Pn,0(s)
Xn
j=1
Pn,j(s)y(j−1)(0) +L(f)
= L(z)
Pn,0(s). (3.20) Letσf be the abscissa of convergence forf, lets1, s2, . . . , sn be the roots of the polynomial Pn,0(s), and let σP = max{<(sk) :k ∈ {1,2, . . . , n}}. For any s with
<(s)>max{σf, σP}, we set G(s) := 1
Pn,0(s) Xn
j=1
Pn,j(s)y(j−1)(0) +L(f)
. (3.21)
By Lemma 2.2, there exists anntimes continuously differentiable functionf0such that
L(f0) = L(f)
Pn,0(s) (3.22)
for allswith<(s)>max{σf, σP} and
f0(i)(0) = 0 (3.23)
for anyi∈ {0,1, . . . , n−1}.
Forj∈ {1,2, . . . , n}, we note that Pn,j(s) Pn,0(s) = 1
sj − Pj−1
k=0αksk
sjPn,0(s) (3.24)
for everys with<(s)>max{0, σP}. Applying Lemma 2.1 for the case ofQ(s) = Pj−1
k=0αksk andP(s) =sjPn,0(s), we can find an infinitely differentiable function gj such that
L(gj) = Pj−1
k=0αksk
sjPn,0(s) (3.25)
andgj(k)(0) = 0 for k∈ {0,1, . . . , n−1}.
Let
fj(t) := tj−1
(j−1)!−gj(t) (3.26)
forj∈ {1,2, . . . , n}. Then we have fj(i)(0) =
(0 fori∈ {0,1, . . . , j−2, j, j+ 1, . . . , n−1},
1 fori=j−1. (3.27)
If we define
yc(t) :=
n
X
j=1
y(j−1)(0)fj(t) +f0(t), then the conditions (3.23) and (3.27) imply that
yc(i)(0) =y(i)(0) (3.28)
for everyi∈ {0,1, . . . , n−1}. Moreover, it follows from (3.21)–(3.28) that L(yc) =
n
X
j=1
y(j−1)(0)L(fj) +L(f0)
=
n
X
j=1
y(j−1)(0)1
sj − L(gj)
+ L(f) Pn,0(s)
= 1
Pn,0(s) Xn
j=1
Pn,j(s)y(j−1)(0) +L(f)
(3.29)
for eachswith <(s)>max{0, σf, σP}.
Now, (3.18) implies that yc is a solution of (3.16). Moreover, by (3.20) and (3.29), we have
L(y)− L(yc) = L(z)
Pn,0(s). (3.30)
Applying Lemma 2.1 for the case of Q(s) = 1 and P(s) = Pn,0(s), we find an infinitely differentiable functiong: (0,∞)→Fsuch that
L(g) = 1
Pn,0(s) (3.31)
which implies that g(t) =L−1
1 Pn,0(s)
= 1 2π
Z ∞
−∞
e(α+iy)t 1
Pn,0(α+iy)dy for any real constantα > σg. Moreover, it holds that
|g(t−x)| ≤ 1 2π
Z ∞
−∞
e(α+iy)(t−x)
1
|Pn,0(α+iy)|dy
≤ 1 2π
Z ∞
−∞
eα(t−x) 1
|Pn,0(α+iy)|dy
≤ 1
2πeα(t−x) Z ∞
−∞
1
|Pn,0(α+iy)|dy
≤M eα(t−x)
(3.32)
for allα > σg, where
M = 1 2π
Z ∞
−∞
1
|Pn,0(α+iy)|dy <∞,
because n is an integer larger than 1. By (3.17) and (3.19), it also holds that
|z(t)| ≤ϕ(t) for allt >0.
In view of (3.30), (3.31), and (3.32), we obtain
L(y)− L(yc) =L(g)L(z) =L(g∗z).
Consequently, we havey(t)−yc(t) = (g∗z)(t) for anyt >0. Hence, it follows from (3.17), (3.19), and (3.32) that
|y(t)−yc(t)|=|(g∗z)(t)| ≤ Z t
0
|g(t−x)||z(x)|dx≤M Z t
0
eα(t−x)ϕ(x)dx
for allt >0 and for any real constantα > σg, which completes the proof.
Corollary 3.8. Let α0, α1, . . . , αn be scalars in F with αn = 1, where n is an integer larger than1. Assume that there exist real constantsαandK >0such that a function ϕ: (0,∞)→(0,∞)satisfies
Z t
0
eα(t−x)ϕ(x)dx≤Kϕ(t)
for all t >0. Moreover, assume that the constant σg given in Theorem 3.7 is less thanα. If an ntimes continuously differentiable function y : (0,∞)→F satisfies the inequality (3.17) for all t >0, then there exist a real constants M > 0 and a solution yc: (0,∞)→F of the differential equation(3.16) such that
|y(t)−yc(t)| ≤KM ϕ(t) for allt >0.
Remark 3.9. Assume thatα <1. If we define ϕ(t) = Aet for all t > 0 and for someA >0, then we get
Z t
0
eα(t−x)ϕ(x)dx= Z t
0
eα(t−x)Aexdx= A
1−α et−eαt
≤ 1 1−αϕ(t) for allt >0.
Acknowledgements. This research was supported by Basic Science Research Pro- gram through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (No. 2013R1A1A2005557).
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Qusuay H. Alqifiary
Department of Mathematics, University of Belgrade, Belgrade, Serbia.
University of Al-Qadisiyah, Al-Diwaniya, Iraq E-mail address:[email protected]
Soon-Mo Jung
Mathematics Section, College of Science and Technology, Hongik University, 339–701 Sejong, Korea
E-mail address:[email protected]
