Applied Mathematics Letters HyersâUlam stability ... - Semantic Scholar
Applied Mathematics Letters 22 (2009) 70–74
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Hyers–Ulam stability of linear partial differential equations of first order S.-M. Jung Mathematics Section, College of Science and Technology, Hong-Ik University, 339-701 Chochiwon, Republic of Korea
article
info
Article history: Received 21 February 2007 Received in revised form 11 February 2008 Accepted 27 February 2008
a b s t r a c t In this work, we will prove the Hyers–Ulam stability of linear partial differential equations of first order. © 2008 Elsevier Ltd. All rights reserved.
Keywords: Hyers–Ulam stability Partial differential equation Differential equation Inequality Approximation
1. Introduction Assume that X is a normed space over a scalar field K and that I is an open interval. Let a0 , a1 , . . . , an−1 be fixed elements of K. Assume that for a fixed function g : I → X and for any n-times differentiable function y : I → X satisfying the inequality
ky(n) (t) + an−1 y(n−1) (t) + · · · + a1 y0 (t) + a0 y(t) + g(t)k ≤ ε for all t ∈ I and for a given ε > 0, there exists a function y0 : I → X satisfying (n)
y0
(t) + an−1 y(0n−1) (t) + · · · + a1 y00 (t) + a0 y0 (t) + g(t) = 0
and ky(t) − y0 (t)k ≤ K (ε) for any t ∈ I, where K (ε) is an expression for ε with limε→0 K (ε) = 0. Then, we say that the above differential equation has the Hyers–Ulam stability. For more detailed definitions of the Hyers–Ulam stability, we refer the reader to [1–3]. Alsina and Ger [4] investigated the Hyers–Ulam stability of differential equations (see also [5,6]): If a differentiable function f : I → R is a solution of the differential inequality |y0 (t) − y(t)| ≤ ε, where I is an open subinterval of R, then there exists a solution f0 : I → R of the differential equation y0 (t) = y(t) such that |f (t) − f0 (t)| ≤ 3ε for any t ∈ I. The above result has been generalized by many mathematicians (Ref. [7,8]). Recently, the author [9] proved the following theorem concerning the Hyers–Ulam stability of a linear differential equation of first order: Theorem 1. Let X be a complex Banach space and let I = (a, b) be an open interval, where a, b ∈ R ∪ R t {±∞} are arbitrarily given with a < b. Assume that g : I → C and h : I → X are continuous functions such that g(t) and exp{R a g(u)du}h(t) are integrable t on (a, c) for each c ∈ I. Moreover, suppose ϕ : I → [0, ∞) is a function such that ϕ(t) exp{Re( a g(u)du)} is integrable on I, where Re z denotes the real part of a complex number z. If a continuously differentiable function y : I → X satisfies the differential inequality
ky0 (t) + g(t)y(t) + h(t)k ≤ ϕ(t)
E-mail address: [email protected]. 0893-9659/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.aml.2008.02.006
S.-M. Jung / Applied Mathematics Letters 22 (2009) 70–74
71
for all t ∈ I, then there exists a unique x ∈ X such that
Z t Z v Z t
y(t) − exp −
g ( u ) d u x − exp g ( u ) d u h ( v ) d v
a
a
t
Z
Z g(u)du
≤ exp −Re a
b
t
a
Z ϕ(v) exp Re
v
g(u)du dv
a
for every t ∈ I. Throughout this work, we will denote by R+ the set of all positive real numbers, i.e., R+ = (0, ∞), and by Re z the real part of a complex number z. In this work, we will prove the Hyers–Ulam stability of first-order linear partial differential equations of the form aux (x, y) + buy (x, y) + g(y)u(x, y) + h(y) = 0
(a ≤ 0, b > 0)
(1)
aux (x, y) + buy (x, y) + g(x)u(x, y) + h(x) = 0
(a > 0, b ≤ 0)
(2)
and
where g, h : R → C are continuous functions satisfying the conditions given in Theorems 2 and 3, respectively. +
2. Main results In the following theorem, we prove the Hyers–Ulam stability of a linear partial differential equation of first order (1). Theorem 2. Let u : R+ × R+ → C be a function which has continuous partial derivatives with respect to the first and second variables. Moreover, assume that u satisfies the following inequality:
|aux (x, y) + buy (x, y) + g(y)u(x, y) + h(y)| ≤ ε
(3)
for all x, y ∈ R+ and for some ε ≥ 0, where a < 0, b > 0 are constants and g, h : R+ → C are continuous functions. Furthermore, assume that g, h, and u satisfy Ry + (a) 0 g(t)dnt exists for all o y ∈ R ∪ {∞}; Ry R 1 w (b) 0 exp b 0 g(t)dt h(w)dw exists for all y ∈ R+ ∪ {∞}; n R∞ Rw o (c) 0 exp 1b Re 0 g(t)dt dw exists; (d) lim x→−∞ u(x, y) exists. y→+∞
Then, there exists a unique complex number θ such that Z Z Z 1 y 1 y 1 u(x, y) − exp − g ( t ) d t · θ − exp b
≤
ε b
1
b
0
Z
y
exp − Re b
0
g(t)dt
Z y
b Z
0
∞
exp
1 b
Re 0
w
0
w
g(t)dt h(w)dw
g(t)dt
dw
(4)
for all x, y ∈ R+ . Proof. We first introduce new coordinates (ξ, η) by a suitable change of axes: a
1
b
b
ξ = x − y and η =
y.
(5)
If we define u˜ (ξ, η) = u(ξ + aη, bη) = u(x, y), then it follows from (5) that
∂ξ ∂η + u˜ η (ξ, η) = u˜ ξ (ξ, η), ∂x ∂x ∂ξ ∂η a 1 uy (x, y) = u˜ ξ (ξ, η) + u˜ η (ξ, η) = − u˜ ξ (ξ, η) + u˜ η (ξ, η). ∂y ∂y b b ux (x, y) = u˜ ξ (ξ, η)
Hence, we have aux (x, y) + buy (x, y) = u˜ η (ξ, η),
and if we apply this equality to (3), we get
|˜uη (ξ, η) + g˜ (η)u˜ (ξ, η) + h˜ (η)| ≤ ε for all ξ, η ∈ R , where we define g˜ (η) = g(bη) = g(y) and h˜ (η) = h(bη) = h(y). +
(6)
S.-M. Jung / Applied Mathematics Letters 22 (2009) 70–74
72
If we set a
1
b
b
ξ = s − t and µ =
t,
(7)
then g˜ (µ) = g(bµ) = g(t) and it follows from (a) that Z 0
y
1
g˜ (µ)dµ =
by
Z
b
0
g(t)dt
exists for any y ∈ R+ .
(8)
Moreover, if we set a
1
b
b
ξ = v − w and ν =
w,
(9)
then we have h˜ (ν) = h(bν) = h(w) and it follows from (8) and (9) that Z ν Z w Z y Z 1 by 1 exp g˜ (µ)dµ h˜ (ν)dν = exp g(t)dt h(w)dw. 0
b
0
b
0
0
Hence, it follows from (b) that Z ν Z y exp g˜ (µ)dµ h˜ (ν)dν exists for all y ∈ R+ . 0
(10)
0
Analogously, it follows from (8) and (9) and (c) that Z 0
∞
ε exp Re
Z ν 0
g˜ (µ)dµ
dν =
ε b
∞
Z
exp
1
Z
b
0
w
Re 0
g(t)dt
dw
exists.
(11)
In view of the inequality (6), the conditions (8), (10) and (11), together with Theorem 1, imply that for each fixed ξ ∈ R+ , there exists a unique complex number θ(ξ) such that Z η Z ν Z η u˜ (ξ, η) − exp − ˜ (ν)dν ˜ ˜ g (µ) d µ · θ(ξ) − exp g (µ) d µ h 0 0 0 Z η Z ∞ Z ν ≤ ε exp −Re g˜ (µ)dµ exp Re g˜ (µ)dµ dν (12) η
0
0
for all η ∈ R . According to a formula in the proof of [9, Theorem 1], it follows from (6) that Z η Z ν Z η z(η) = exp g˜ (µ)dµ u˜ (ξ, η) + exp g˜ (µ)dµ h˜ (ν)dν, +
0
0
0
and in view of (8) and (9), (a), (b), and (d), we conclude that
θ(ξ) = ηlim z(η) →∞ (
= lim exp η→∞
1
Z bη
b
0
)
1
g(t)dt u(ξ + aη, bη) + lim
η→∞ b
Z bη 0
exp
1
Z
b
0
w
g(t)dt h(w)dw
is a constant, say simply θ. We know that u˜ (ξ, η) = u(x, y) and it moreover follows from (5), (8) and (9) that Z η Z Z ν Z 1 y 1 w g˜ (µ)dµ = g(t)dt and g˜ (µ)dµ = g(t)dt. 0
b
0
b
0
0
Analogously, it follows from (9) that h˜ (ν) = h(bν) = h(w). Hence, applying the above arguments to the inequality (12) and taking y = bη and w = bν into account, we obtain the inequality (4). Remark 1. We can show by a tedious calculation that Z Z Z 1 y 1 y 1 u(x, y) = exp − g ( t ) dt θ − exp b
0
b
0
b
0
w
g(t)dt h(w)dw
is a solution of the partial differential equation (1). Analogously to Theorem 2, we will investigate the Hyers–Ulam stability of the partial differential equation (2).
S.-M. Jung / Applied Mathematics Letters 22 (2009) 70–74
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Theorem 3. Let u : R+ × R+ → C be a function which has continuous partial derivatives with respect to the first and second variables. Moreover, assume that u satisfies the following inequality:
|aux (x, y) + buy (x, y) + g(x)u(x, y) + h(x)| ≤ ε
(13)
for all x, y ∈ R+ and for some ε ≥ 0, where a > 0, b < 0 are constants and g, h : R+ → C are continuous functions. Furthermore, assume that g, h, and u satisfy g(t)dt exists for all y ∈ R+ ∪ {∞}; n R o (b0 ) 0 exp 1a 0w g(t)dt h(w)dw exists for all y ∈ R+ ∪ {∞}; n R R o (c0 ) 0∞ exp 1a Re 0w g(t)dt dw exists;
(a0 )
Ry 0
Ry
(d0 ) lim yx→+∞ u(x, y) exists. →−∞ Then, there exists a unique complex number θ such that Z Z Z 1 x 1 x 1 u(x, y) − exp − g(t)dt · θ − exp
≤
ε a
1
a 0 Z x
exp − Re a
0
a
g(t)dt
∞
Z
a Z
0
exp
x
1 a
w
0
w
Re 0
g(t)dt h(w)dw
g(t)dt
dw
(14)
for all x, y ∈ R+ . Proof. If we set v(x, y) = u(y, x) for all x, y ∈ R+ , then we have ux (x, y) = lim
h→0
uy (x, y) = lim
u(x + h, y) − u(x, y) h u(x, y + h) − u(x, y) h
h→0
= lim
h→0
= lim
v(y, x + h) − v(y, x) h v(y + h, x) − v(y, x) h
h→0
= vy (y, x), = vx (y, x).
So, we obtain aux (x, y) + buy (x, y) + g(x)u(x, y) + h(x) = avy (y, x) + bvx (y, x) + g(x)v(y, x) + h(x)
and it follows from (13) that
|bvx (y, x) + avy (y, x) + g(x)v(y, x) + h(x)| ≤ ε for any x, y ∈ R+ . If we exchange the roles of x and y in the above inequality, then we get
|bvx (x, y) + avy (x, y) + g(y)v(x, y) + h(y)| ≤ ε for all x, y ∈ R+ . In view of (a0 )–(d0 ), and Theorem 2, there exists a unique complex number θ such that Z w Z Z 1 1 y 1 y v(x, y) − exp − g(t)dt · θ − exp g(t)dt h(w)dw a 0 a 0 a 0 Z y Z ∞ Z w 1 ε 1 exp ≤ exp − Re g(t)dt Re g(t)dt dw a
a
0
a
y
0
for any x, y ∈ R+ . By exchanging the roles of x and y in the above inequality, we can easily verify the validity of inequality (14). Remark 2. By a tedious calculation we can show that Z Z Z 1 x 1 1 x u(x, y) = exp − g(t)dt θ − exp a
0
a
0
a
0
w
g(t)dt h(w)dw
is a solution of the partial differential equation (2). Remark 3. When the coefficient functions g and h are constants, the Hyers–Ulam stability of (1) or (2) was proved in [10]. But it is an open question whether the Hyers–Ulam stability is still true if the coefficient functions g and h in (1) or (2) are functions of two variables.
S.-M. Jung / Applied Mathematics Letters 22 (2009) 70–74
74
3. An example Let u : R+ × R+ → R be a given function which has continuous partial derivatives with respect to the first and second variables. For a given constant c > 0, we define a continuous function g : R+ → R by −c (0 < t ≤ 1), g (t ) = −ct (t > 1). Assume that u satisfies
|aux (x, y) + buy (x, y) + g(y)u(x, y) + ky| ≤ ε for all x, y ∈ R+ and for some ε ≥ 0, where a ≤ 0, b > 0, k ≥ 0 are constants. If we set h(y) = ky, then we have Z y −cy (for 0 < y ≤ 1) c g(t)dt = − (1 + y2 ) (for y > 1), 0 2 2 b k bk c b Z w Z y − (for 0 < y ≤ 1) exp − y y + 1 c2 c c exp g(t)dt h(w)dw = b 2 b k c bk c b 0 0 1 − exp − − exp − (for y > 1) 1 + y2 c2 b c 2b and Z
∞
exp
0
1 b
Z
Re 0
w
g(t)dt
dw =
b c
< ∞.
Z c 1 − exp − + b
1
∞
c exp − 1 + w 2 dw 2b
According to Theorem 2, there exists a unique function θ : R+ → C such that a bk c k b u(x, y) − θ x − y − exp y − y+ 2 b c b c c Z ∞ ε c ε c c ≤ 1 − exp (y − 1) + exp y exp − 1 + w2 dw, c b b b 2b 1 for all x ∈ R+ and 0 < y ≤ 1, and a bk c c k u(x, y) − θ x − y − 1 − exp − exp 1 + y2 − 2 b c b 2b c Z ∞ ε c c 2 2 ≤ exp 1+y exp − 1+w dw b 2b 2b y for all x ∈ R+ and y > 1. Acknowledgments This work was supported by 2007 Hongik University Research Fund. The author is grateful to the referees for their helpful comments and suggestions. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
S. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific, New Jersey, London, 2002. S. Czerwik, Stability of Functional Equations of Ulam–Hyers–Rassias Type, Hadronic Press, Palm Harbor, 2003. D.H. Hyers, G. Isac, Th.M. Rassias, Stability of Functional Equations in Several Variables, Birkhäuser, Boston, 1998. C. Alsina, R. Ger, On some inequalities and stability results related to the exponential function, J. Inequal. Appl. 2 (1998) 373–380. M. Obłoza, Hyers stability of the linear differential equation, Rocznik Nauk.-Dydakt. Prace Mat. 13 (1993) 259–270. M. Obłoza, Connections between Hyers and Lyapunov stability of the ordinary differential equations, Rocznik Nauk.-Dydakt. Prace Mat. 14 (1997) 141–146. T. Miura, On the Hyers–Ulam stability of a differentiable map, Sci. Math. Japon. 55 (2002) 17–24. T. Miura, S.-M. Jung, S.-E. Takahasi, Hyers–Ulam–Rassias stability of the Banach space valued differential equations y0 = λy, J. Korean Math. Soc. 41 (2004) 995–1005. S.-M. Jung, Hyers–Ulam stability of linear differential equations of first order, II, Appl. Math. Lett. 19 (2006) 854–858. S.-M. Jung, K.-S. Lee, Hyers–Ulam stability of first order linear partial differential equations with constant coefficients, Math. Inequal. Appl. 10 (2007) 261–266.
Contents lists available at ScienceDirect
Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml
Hyers–Ulam stability of linear partial differential equations of first order S.-M. Jung Mathematics Section, College of Science and Technology, Hong-Ik University, 339-701 Chochiwon, Republic of Korea
article
info
Article history: Received 21 February 2007 Received in revised form 11 February 2008 Accepted 27 February 2008
a b s t r a c t In this work, we will prove the Hyers–Ulam stability of linear partial differential equations of first order. © 2008 Elsevier Ltd. All rights reserved.
Keywords: Hyers–Ulam stability Partial differential equation Differential equation Inequality Approximation
1. Introduction Assume that X is a normed space over a scalar field K and that I is an open interval. Let a0 , a1 , . . . , an−1 be fixed elements of K. Assume that for a fixed function g : I → X and for any n-times differentiable function y : I → X satisfying the inequality
ky(n) (t) + an−1 y(n−1) (t) + · · · + a1 y0 (t) + a0 y(t) + g(t)k ≤ ε for all t ∈ I and for a given ε > 0, there exists a function y0 : I → X satisfying (n)
y0
(t) + an−1 y(0n−1) (t) + · · · + a1 y00 (t) + a0 y0 (t) + g(t) = 0
and ky(t) − y0 (t)k ≤ K (ε) for any t ∈ I, where K (ε) is an expression for ε with limε→0 K (ε) = 0. Then, we say that the above differential equation has the Hyers–Ulam stability. For more detailed definitions of the Hyers–Ulam stability, we refer the reader to [1–3]. Alsina and Ger [4] investigated the Hyers–Ulam stability of differential equations (see also [5,6]): If a differentiable function f : I → R is a solution of the differential inequality |y0 (t) − y(t)| ≤ ε, where I is an open subinterval of R, then there exists a solution f0 : I → R of the differential equation y0 (t) = y(t) such that |f (t) − f0 (t)| ≤ 3ε for any t ∈ I. The above result has been generalized by many mathematicians (Ref. [7,8]). Recently, the author [9] proved the following theorem concerning the Hyers–Ulam stability of a linear differential equation of first order: Theorem 1. Let X be a complex Banach space and let I = (a, b) be an open interval, where a, b ∈ R ∪ R t {±∞} are arbitrarily given with a < b. Assume that g : I → C and h : I → X are continuous functions such that g(t) and exp{R a g(u)du}h(t) are integrable t on (a, c) for each c ∈ I. Moreover, suppose ϕ : I → [0, ∞) is a function such that ϕ(t) exp{Re( a g(u)du)} is integrable on I, where Re z denotes the real part of a complex number z. If a continuously differentiable function y : I → X satisfies the differential inequality
ky0 (t) + g(t)y(t) + h(t)k ≤ ϕ(t)
E-mail address: [email protected]. 0893-9659/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.aml.2008.02.006
S.-M. Jung / Applied Mathematics Letters 22 (2009) 70–74
71
for all t ∈ I, then there exists a unique x ∈ X such that
Z t Z v Z t
y(t) − exp −
g ( u ) d u x − exp g ( u ) d u h ( v ) d v
a
a
t
Z
Z g(u)du
≤ exp −Re a
b
t
a
Z ϕ(v) exp Re
v
g(u)du dv
a
for every t ∈ I. Throughout this work, we will denote by R+ the set of all positive real numbers, i.e., R+ = (0, ∞), and by Re z the real part of a complex number z. In this work, we will prove the Hyers–Ulam stability of first-order linear partial differential equations of the form aux (x, y) + buy (x, y) + g(y)u(x, y) + h(y) = 0
(a ≤ 0, b > 0)
(1)
aux (x, y) + buy (x, y) + g(x)u(x, y) + h(x) = 0
(a > 0, b ≤ 0)
(2)
and
where g, h : R → C are continuous functions satisfying the conditions given in Theorems 2 and 3, respectively. +
2. Main results In the following theorem, we prove the Hyers–Ulam stability of a linear partial differential equation of first order (1). Theorem 2. Let u : R+ × R+ → C be a function which has continuous partial derivatives with respect to the first and second variables. Moreover, assume that u satisfies the following inequality:
|aux (x, y) + buy (x, y) + g(y)u(x, y) + h(y)| ≤ ε
(3)
for all x, y ∈ R+ and for some ε ≥ 0, where a < 0, b > 0 are constants and g, h : R+ → C are continuous functions. Furthermore, assume that g, h, and u satisfy Ry + (a) 0 g(t)dnt exists for all o y ∈ R ∪ {∞}; Ry R 1 w (b) 0 exp b 0 g(t)dt h(w)dw exists for all y ∈ R+ ∪ {∞}; n R∞ Rw o (c) 0 exp 1b Re 0 g(t)dt dw exists; (d) lim x→−∞ u(x, y) exists. y→+∞
Then, there exists a unique complex number θ such that Z Z Z 1 y 1 y 1 u(x, y) − exp − g ( t ) d t · θ − exp b
≤
ε b
1
b
0
Z
y
exp − Re b
0
g(t)dt
Z y
b Z
0
∞
exp
1 b
Re 0
w
0
w
g(t)dt h(w)dw
g(t)dt
dw
(4)
for all x, y ∈ R+ . Proof. We first introduce new coordinates (ξ, η) by a suitable change of axes: a
1
b
b
ξ = x − y and η =
y.
(5)
If we define u˜ (ξ, η) = u(ξ + aη, bη) = u(x, y), then it follows from (5) that
∂ξ ∂η + u˜ η (ξ, η) = u˜ ξ (ξ, η), ∂x ∂x ∂ξ ∂η a 1 uy (x, y) = u˜ ξ (ξ, η) + u˜ η (ξ, η) = − u˜ ξ (ξ, η) + u˜ η (ξ, η). ∂y ∂y b b ux (x, y) = u˜ ξ (ξ, η)
Hence, we have aux (x, y) + buy (x, y) = u˜ η (ξ, η),
and if we apply this equality to (3), we get
|˜uη (ξ, η) + g˜ (η)u˜ (ξ, η) + h˜ (η)| ≤ ε for all ξ, η ∈ R , where we define g˜ (η) = g(bη) = g(y) and h˜ (η) = h(bη) = h(y). +
(6)
S.-M. Jung / Applied Mathematics Letters 22 (2009) 70–74
72
If we set a
1
b
b
ξ = s − t and µ =
t,
(7)
then g˜ (µ) = g(bµ) = g(t) and it follows from (a) that Z 0
y
1
g˜ (µ)dµ =
by
Z
b
0
g(t)dt
exists for any y ∈ R+ .
(8)
Moreover, if we set a
1
b
b
ξ = v − w and ν =
w,
(9)
then we have h˜ (ν) = h(bν) = h(w) and it follows from (8) and (9) that Z ν Z w Z y Z 1 by 1 exp g˜ (µ)dµ h˜ (ν)dν = exp g(t)dt h(w)dw. 0
b
0
b
0
0
Hence, it follows from (b) that Z ν Z y exp g˜ (µ)dµ h˜ (ν)dν exists for all y ∈ R+ . 0
(10)
0
Analogously, it follows from (8) and (9) and (c) that Z 0
∞
ε exp Re
Z ν 0
g˜ (µ)dµ
dν =
ε b
∞
Z
exp
1
Z
b
0
w
Re 0
g(t)dt
dw
exists.
(11)
In view of the inequality (6), the conditions (8), (10) and (11), together with Theorem 1, imply that for each fixed ξ ∈ R+ , there exists a unique complex number θ(ξ) such that Z η Z ν Z η u˜ (ξ, η) − exp − ˜ (ν)dν ˜ ˜ g (µ) d µ · θ(ξ) − exp g (µ) d µ h 0 0 0 Z η Z ∞ Z ν ≤ ε exp −Re g˜ (µ)dµ exp Re g˜ (µ)dµ dν (12) η
0
0
for all η ∈ R . According to a formula in the proof of [9, Theorem 1], it follows from (6) that Z η Z ν Z η z(η) = exp g˜ (µ)dµ u˜ (ξ, η) + exp g˜ (µ)dµ h˜ (ν)dν, +
0
0
0
and in view of (8) and (9), (a), (b), and (d), we conclude that
θ(ξ) = ηlim z(η) →∞ (
= lim exp η→∞
1
Z bη
b
0
)
1
g(t)dt u(ξ + aη, bη) + lim
η→∞ b
Z bη 0
exp
1
Z
b
0
w
g(t)dt h(w)dw
is a constant, say simply θ. We know that u˜ (ξ, η) = u(x, y) and it moreover follows from (5), (8) and (9) that Z η Z Z ν Z 1 y 1 w g˜ (µ)dµ = g(t)dt and g˜ (µ)dµ = g(t)dt. 0
b
0
b
0
0
Analogously, it follows from (9) that h˜ (ν) = h(bν) = h(w). Hence, applying the above arguments to the inequality (12) and taking y = bη and w = bν into account, we obtain the inequality (4). Remark 1. We can show by a tedious calculation that Z Z Z 1 y 1 y 1 u(x, y) = exp − g ( t ) dt θ − exp b
0
b
0
b
0
w
g(t)dt h(w)dw
is a solution of the partial differential equation (1). Analogously to Theorem 2, we will investigate the Hyers–Ulam stability of the partial differential equation (2).
S.-M. Jung / Applied Mathematics Letters 22 (2009) 70–74
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Theorem 3. Let u : R+ × R+ → C be a function which has continuous partial derivatives with respect to the first and second variables. Moreover, assume that u satisfies the following inequality:
|aux (x, y) + buy (x, y) + g(x)u(x, y) + h(x)| ≤ ε
(13)
for all x, y ∈ R+ and for some ε ≥ 0, where a > 0, b < 0 are constants and g, h : R+ → C are continuous functions. Furthermore, assume that g, h, and u satisfy g(t)dt exists for all y ∈ R+ ∪ {∞}; n R o (b0 ) 0 exp 1a 0w g(t)dt h(w)dw exists for all y ∈ R+ ∪ {∞}; n R R o (c0 ) 0∞ exp 1a Re 0w g(t)dt dw exists;
(a0 )
Ry 0
Ry
(d0 ) lim yx→+∞ u(x, y) exists. →−∞ Then, there exists a unique complex number θ such that Z Z Z 1 x 1 x 1 u(x, y) − exp − g(t)dt · θ − exp
≤
ε a
1
a 0 Z x
exp − Re a
0
a
g(t)dt
∞
Z
a Z
0
exp
x
1 a
w
0
w
Re 0
g(t)dt h(w)dw
g(t)dt
dw
(14)
for all x, y ∈ R+ . Proof. If we set v(x, y) = u(y, x) for all x, y ∈ R+ , then we have ux (x, y) = lim
h→0
uy (x, y) = lim
u(x + h, y) − u(x, y) h u(x, y + h) − u(x, y) h
h→0
= lim
h→0
= lim
v(y, x + h) − v(y, x) h v(y + h, x) − v(y, x) h
h→0
= vy (y, x), = vx (y, x).
So, we obtain aux (x, y) + buy (x, y) + g(x)u(x, y) + h(x) = avy (y, x) + bvx (y, x) + g(x)v(y, x) + h(x)
and it follows from (13) that
|bvx (y, x) + avy (y, x) + g(x)v(y, x) + h(x)| ≤ ε for any x, y ∈ R+ . If we exchange the roles of x and y in the above inequality, then we get
|bvx (x, y) + avy (x, y) + g(y)v(x, y) + h(y)| ≤ ε for all x, y ∈ R+ . In view of (a0 )–(d0 ), and Theorem 2, there exists a unique complex number θ such that Z w Z Z 1 1 y 1 y v(x, y) − exp − g(t)dt · θ − exp g(t)dt h(w)dw a 0 a 0 a 0 Z y Z ∞ Z w 1 ε 1 exp ≤ exp − Re g(t)dt Re g(t)dt dw a
a
0
a
y
0
for any x, y ∈ R+ . By exchanging the roles of x and y in the above inequality, we can easily verify the validity of inequality (14). Remark 2. By a tedious calculation we can show that Z Z Z 1 x 1 1 x u(x, y) = exp − g(t)dt θ − exp a
0
a
0
a
0
w
g(t)dt h(w)dw
is a solution of the partial differential equation (2). Remark 3. When the coefficient functions g and h are constants, the Hyers–Ulam stability of (1) or (2) was proved in [10]. But it is an open question whether the Hyers–Ulam stability is still true if the coefficient functions g and h in (1) or (2) are functions of two variables.
S.-M. Jung / Applied Mathematics Letters 22 (2009) 70–74
74
3. An example Let u : R+ × R+ → R be a given function which has continuous partial derivatives with respect to the first and second variables. For a given constant c > 0, we define a continuous function g : R+ → R by −c (0 < t ≤ 1), g (t ) = −ct (t > 1). Assume that u satisfies
|aux (x, y) + buy (x, y) + g(y)u(x, y) + ky| ≤ ε for all x, y ∈ R+ and for some ε ≥ 0, where a ≤ 0, b > 0, k ≥ 0 are constants. If we set h(y) = ky, then we have Z y −cy (for 0 < y ≤ 1) c g(t)dt = − (1 + y2 ) (for y > 1), 0 2 2 b k bk c b Z w Z y − (for 0 < y ≤ 1) exp − y y + 1 c2 c c exp g(t)dt h(w)dw = b 2 b k c bk c b 0 0 1 − exp − − exp − (for y > 1) 1 + y2 c2 b c 2b and Z
∞
exp
0
1 b
Z
Re 0
w
g(t)dt
dw =
b c
< ∞.
Z c 1 − exp − + b
1
∞
c exp − 1 + w 2 dw 2b
According to Theorem 2, there exists a unique function θ : R+ → C such that a bk c k b u(x, y) − θ x − y − exp y − y+ 2 b c b c c Z ∞ ε c ε c c ≤ 1 − exp (y − 1) + exp y exp − 1 + w2 dw, c b b b 2b 1 for all x ∈ R+ and 0 < y ≤ 1, and a bk c c k u(x, y) − θ x − y − 1 − exp − exp 1 + y2 − 2 b c b 2b c Z ∞ ε c c 2 2 ≤ exp 1+y exp − 1+w dw b 2b 2b y for all x ∈ R+ and y > 1. Acknowledgments This work was supported by 2007 Hongik University Research Fund. The author is grateful to the referees for their helpful comments and suggestions. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
S. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific, New Jersey, London, 2002. S. Czerwik, Stability of Functional Equations of Ulam–Hyers–Rassias Type, Hadronic Press, Palm Harbor, 2003. D.H. Hyers, G. Isac, Th.M. Rassias, Stability of Functional Equations in Several Variables, Birkhäuser, Boston, 1998. C. Alsina, R. Ger, On some inequalities and stability results related to the exponential function, J. Inequal. Appl. 2 (1998) 373–380. M. Obłoza, Hyers stability of the linear differential equation, Rocznik Nauk.-Dydakt. Prace Mat. 13 (1993) 259–270. M. Obłoza, Connections between Hyers and Lyapunov stability of the ordinary differential equations, Rocznik Nauk.-Dydakt. Prace Mat. 14 (1997) 141–146. T. Miura, On the Hyers–Ulam stability of a differentiable map, Sci. Math. Japon. 55 (2002) 17–24. T. Miura, S.-M. Jung, S.-E. Takahasi, Hyers–Ulam–Rassias stability of the Banach space valued differential equations y0 = λy, J. Korean Math. Soc. 41 (2004) 995–1005. S.-M. Jung, Hyers–Ulam stability of linear differential equations of first order, II, Appl. Math. Lett. 19 (2006) 854–858. S.-M. Jung, K.-S. Lee, Hyers–Ulam stability of first order linear partial differential equations with constant coefficients, Math. Inequal. Appl. 10 (2007) 261–266.
