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A Fixed Point Approach to the Stability of Differential Equations y
0= F (x, y)
Soon-Mo Jung
Mathematics Section, College of Science and Technology, Hong-Ik University, 339-701 Chochiwon, Republic of Korea
Abstract. Using a fixed point method, the Hyers-Ulam-Rassias stability will be proved for the differential equations of the formy0(x) =F(x, y(x)).
2000 Mathematics Subject Classification: Primary: 26D10, 47J99, 47N20; Sec- ondary: 34A40, 47E05, 47H10
Key words and phrases: Fixed point method, differential equation, Hyers-Ulam- Rassias stability, Hyers-Ulam stability.
1. Introduction
LetY be a normed space and letIbe an open interval. Assume that for any function f :I→Y satisfying the differential inequality
kan(x)y(n)(x) +an−1(x)y(n−1)(x) +· · ·+a1(x)y0(x) +a0(x)y(x) +h(x)k ≤ε for allx∈I and for someε≥0, there exists a solutionf0:I→Y of the differential equation
an(x)y(n)(x) +an−1(x)y(n−1)(x) +· · ·+a1(x)y0(x) +a0(x)y(x) +h(x) = 0 such that kf(x)−f0(x)k ≤K(ε) for anyx∈ I, where K(ε) is an expression of ε only. Then, we say that the above differential equation has the Hyers-Ulam stability.
If the above statement is also true when we replace ε and K(ε) by ϕ(x) and Φ(x), where ϕ,Φ :I →[0,∞) are functions not depending on f and f0 explicitly, then we say that the corresponding differential equation has the Hyers-Ulam-Rassias stability (or the generalized Hyers-Ulam stability).
We may apply these terminologies for other differential equations. For more detailed definitions of the Hyers-Ulam stability and the Hyers-Ulam-Rassias stability, refer to [5, 6].
Ob loza seems to be the first author who has investigated the Hyers-Ulam stability of linear differential equations (see [14, 15]). Here, we will introduce a result of
Communicated byRosihan M. Ali, Dato’.
Received:October 23, 2008; Accepted: February 16, 2009.
Alsina and Ger (see [1]): If a differentiable function f :I →Ris a solution of the differential inequality|y0(x)−y(x)| ≤ε, whereI is an open subinterval of R, then there exists a solutionf0:I→Rof the differential equationy0(x) =y(x) such that
|f(x)−f0(x)| ≤3εfor anyx∈I.
This result of Alsina and Ger has been generalized by Takahasi, Miura and Miya- jima: They proved in [17] that the Hyers-Ulam stability holds for the Banach space valued differential equationy0(x) =λy(x) (see also [11]).
Recently, Miura, Miyajima and Takahasi also proved the Hyers-Ulam stability of linear differential equations of first order, y0(x) +g(x)y(x) = 0, where g(x) is a continuous function, while the author proved the Hyers-Ulam stability of linear differential equations of other type (see [7, 8, 9, 12, 13]).
In this paper, for a bounded and continuous functionF(x, y), we will adopt the idea of C˘adariu and Radu [2, 3] and prove the Hyers-Ulam-Rassias stability as well as the Hyers-Ulam stability of the differential equations of the form
(1.1) y0(x) =F(x, y(x)).
2. Preliminaries
For a nonempty setX, we introduce the definition of the generalized metric on X.
A functiond:X×X →[0,∞] is called a generalized metric onX if and only ifd satisfies
(M1) d(x, y) = 0 if and only ifx=y;
(M2) d(x, y) =d(y, x) for allx, y∈X;
(M3) d(x, z)≤d(x, y) +d(y, z) for allx, y, z∈X.
We remark that the only one difference of the generalized metric from the usual metric is that the range of the former is permitted to include the infinity.
We now introduce one of fundamental results of fixed point theory. For the proof, we refer to [4]. This theorem will play an important rˆole in proving our main theorems.
Theorem 2.1. Let (X, d) be a generalized complete metric space. Assume that Λ :X →X is a strictly contractive operator with the Lipschitz constantL <1. If there exists a nonnegative integerk such that d(Λk+1x,Λkx)<∞for some x∈X, then the followings are true:
(a) The sequence{Λnx} converges to a fixed pointx∗ of Λ;
(b) x∗ is the unique fixed point of Λin
X∗={y∈X|d(Λkx, y)<∞}; (c) If y∈X∗, then
d(y, x∗)≤ 1
1−Ld(Λy, y).
3. Hyers-Ulam-Rassias stability
Recently, C˘adariu and Radu [2] applied the fixed point method to the investigation of the Jensen’s functional equation. Using such an idea, they could present a proof for the Hyers-Ulam stability of that equation (ref. [3, 10, 16]).
In this section, by using the idea of C˘adariu and Radu, we will prove the Hyers- Ulam-Rassias stability of the differential equation (1.1).
Theorem 3.1. For given real numbersa andbwith a < b, letI= [a, b] be a closed interval and choose a c ∈I. Let K and L be positive constants with0 < KL <1.
Assume that F : I×R → R is a continuous function which satisfies a Lipschitz condition
(3.1) |F(x, y)−F(x, z)| ≤L|y−z|
for any x ∈ I and y, z ∈ R. If a continuously differentiable function y : I → R satisfies
(3.2) |y0(x)−F(x, y(x))| ≤ϕ(x)
for allx∈I, whereϕ:I→(0,∞)is a continuous function with (3.3)
Z x
c
ϕ(τ)dτ
≤Kϕ(x)
for eachx∈I, then there exists a unique continuous functiony0:I→Rsuch that
(3.4) y0(x) =y(c) +
Z x
c
F(τ, y0(τ))dτ (consequently,y0 is a solution to(1.1))and
(3.5) |y(x)−y0(x)| ≤ K
1−KLϕ(x) for allx∈I.
Proof. Let us define a setX of all continuous functionsf :I→Rby (3.6) X={f :I→R|f is continuous}
and introduce a generalized metric onX as follows:
(3.7) d(f, g) = inf{C∈[0,∞]| |f(x)−g(x)| ≤Cϕ(x)∀x∈I}.
(We will here give a proof for the triangle inequality only. Assume that d(f, g) >
d(f, h) +d(h, g) would hold for somef, g, h∈X. Then, by (3.7), there would exist anx0∈I with
|f(x0)−g(x0)|>{d(f, h) +d(h, g)}ϕ(x0)
=d(f, h)ϕ(x0) +d(h, g)ϕ(x0)
≥ |f(x0)−h(x0)|+|h(x0)−g(x0)|, a contradiction.)
We assert that (X, d) is complete. Let {hn} be a Cauchy sequence in (X, d).
Then, for anyε >0, there exists an integerNε>0 such thatd(hm, hn)≤εfor all m, n≥Nε. It further follows from (3.7) that
(3.8) ∀ε >0∃Nε∈N∀m, n≥Nε ∀x∈I: |hm(x)−hn(x)| ≤εϕ(x).
If x is fixed, (3.8) implies that {hn(x)} is a Cauchy sequence in R. Since R is complete,{hn(x)}converges for eachx∈I. Thus, we can define a functionh:I→R by
h(x) = lim
n→∞hn(x).
If we letm increase to infinity, it then follows from (3.8) that
(3.9) ∀ε >0∃Nε∈N∀n≥Nε∀x∈I: |h(x)−hn(x)| ≤εϕ(x),
that is, sinceϕis bounded onI,{hn}converges uniformly toh. Hence,his contin- uous andh∈X.
Further if we consider (3.7) and (3.9), then we may conclude that
∀ε >0 ∃Nε∈N∀n≥Nε: d(h, hn)≤ε,
that is, the Cauchy sequence{hn}converges tohin (X, d). Hence, (X, d) is complete.
Now, an operator Λ :X →X is defined by (3.10) (Λf)(x) =y(c) +
Z x
c
F(τ, f(τ))dτ (x∈I)
for all f ∈X. (Indeed, according to the Fundamental Theorem of Calculus, Λf is continuously differentiable onI, sinceF andf are continuous functions. Hence, we may conclude that Λf ∈X.)
We prove that Λ is strictly contractive onX. For anyf, g∈X, letCf g∈[0,∞]
be an arbitrary constant withd(f, g)≤Cf g, that is, by (3.7), we have
(3.11) |f(x)−g(x)| ≤Cf gϕ(x)
for anyx∈I. It then follows from (3.1), (3.3), (3.7), (3.10) and (3.11) that
|(Λf)(x)−(Λg)(x)|=
Z x
c
{F(τ, f(τ))−F(τ, g(τ))}dτ
≤
Z x
c
|F(τ, f(τ))−F(τ, g(τ))|dτ
≤L
Z x
c
|f(τ)−g(τ)|dτ
≤LCf g
Z x
c
ϕ(τ)dτ
≤KLCf gϕ(x)
for allx∈I, that is,d(Λf,Λg)≤KLCf g. Hence, we can conclude thatd(Λf,Λg)≤ KLd(f, g) for anyf, g∈X, where we note that 0< KL <1.
It follows from (3.6) and (3.10) that for an arbitrary g0 ∈ X, there exists a constant 0< C <∞with
|(Λg0)(x)−g0(x)|=
y(c) + Z x
c
F(τ, g0(τ))dτ−g0(x)
≤Cϕ(x)
for all x∈I, sinceF(x, g0(x)) and g0(x) are bounded on I and minx∈I ϕ(x)>0.
Thus, (3.7) implies that
d(Λg0, g0)<∞.
Therefore, according to Theorem 2.1 (a), there exists a continuous function y0: I → Rsuch that Λng0 →y0 in (X, d) and Λy0 =y0, that is, y0 satisfies equation (3.4) for everyx∈I.
We will now verify that{g∈X|d(g0, g)<∞}=X. For anyg∈X, sincegand g0are bounded onIand minx∈I ϕ(x)>0, there exists a constant 0< Cg<∞such that
|g0(x)−g(x)| ≤Cgϕ(x)
for anyx∈I. Hence, we haved(g0, g)<∞for allg∈X, that is,{g∈X|d(g0, g)<
∞} = X. Hence, in view of Theorem 2.1 (b), we conclude that y0 is the unique continuous function with the property (3.4).
On the other hand, it follows from (3.2) that
−ϕ(x)≤y0(x)−F(x, y(x))≤ϕ(x)
for allx∈I. If we integrate each term in the above inequality fromc tox, then we obtain
y(x)−y(c)− Z x
c
F(τ, y(τ))dτ
≤
Z x
c
ϕ(τ)dτ for anyx∈I. Thus, by (3.3) and (3.10), we get
|y(x)−(Λy)(x)| ≤
Z x
c
ϕ(τ)dτ
≤Kϕ(x)
for eachx∈I, which implies that
(3.12) d(y,Λy)≤K.
Finally, Theorem 2.1 (c) together with (3.12) implies that d(y, y0)≤ 1
1−KLd(Λy, y)≤ K 1−KL, which means that the inequality (3.5) holds true for allx∈I.
In the last theorem, we have investigated the Hyers-Ulam-Rassias stability of the differential equation (1.1) defined on a bounded and closed interval. We will now prove the theorem for the case of unbounded intervals. More precisely, Theorem 3.1 is also true ifI is replaced by an unbounded interval such as (−∞, b],R, or [a,∞), as we see in the following theorem.
Theorem 3.2. For given real numbers aandb, letI denote either(−∞, b] orRor [a,∞). Set either c=aforI= [a,∞) orc=b forI= (−∞, b]or c is a fixed real number ifI=R. LetK andLbe positive constants with0< KL <1. Assume that F :I×R→R is a continuous function which satisfies a Lipschitz condition (3.1) for all x∈I and all y, z ∈R. If a continuously differentiable function y : I →R satisfies the differential inequality (3.2) for all x ∈ I, where ϕ : I → (0,∞) is a continuous function satisfying the condition(3.3)for any x∈I, then there exists a unique continuous functiony0:I→Rwhich satisfies(3.4) and(3.5) for allx∈I.
Proof. We will give the proof for the caseI=Ronly. The other cases can similarly be proved.
For anyn∈N, we defineIn= [c−n, c+n]. (We setIn = [b−n, b] forI= (−∞, b]
andIn = [a, a+n] forI= [a,∞).) According to Theorem 3.1, there exists a unique continuous functionyn:In→Rsuch that
(3.13) yn(x) =y(c) +
Z x
c
F(τ, yn(τ))dτ and
(3.14) |y(x)−yn(x)| ≤ K
1−KLϕ(x) for allx∈In. The uniqueness ofyn implies that ifx∈In, then (3.15) yn(x) =yn+1(x) =yn+2(x) =· · · .
For anyx∈R, let us definen(x)∈Nas
n(x) = min{n∈N|x∈In}.
Moreover, we define a functiony0:R→Rby
(3.16) y0(x) =yn(x)(x),
and we assert thaty0 is continuous. For an arbitraryx1∈R, we choose the integer n1=n(x1). Then,x1 belongs to the interior ofIn1+1and there exists anε >0 such thaty0(x) =yn1+1(x) for all xwithx1−ε < x < x1+ε. Sinceyn1+1is continuous atx1, so isy0. That is,y0is continuous atx1 for anyx1∈R.
We will now show thaty0 satisfies (3.4) and (3.5) for allx∈R. For an arbitrary x∈R, we choose the integern(x). Then, it holds thatx∈In(x)and it follows from (3.13) and (3.16) that
y0(x) =yn(x)(x) =y(c) + Z x
c
F(τ, yn(x)(τ))dτ =y(c) + Z x
c
F(τ, y0(τ))dτ, where the last equality holds true because n(τ) ≤ n(x) for any τ ∈ In(x) and it follows from (3.15) and (3.16) that
yn(x)(τ) =yn(τ)(τ) =y0(τ).
Sincex∈In(x) for everyx∈R, by (3.14) and (3.16), we have
|y(x)−y0(x)|=|y(x)−yn(x)(x)| ≤ K
1−KLϕ(x) for anyx∈R.
Finally, we show thaty0is unique. Letz0:R→Rbe another continuous function which satisfies (3.4) and (3.5), with z0 in place of y0, for allx ∈R. Suppose xis an arbitrary real number. Since the restrictions y0|I
n(x)(=yn(x)) and z0|I
n(x) both satisfy (3.4) and (3.5) for all x ∈ In(x), the uniqueness of yn(x) = y0|I
n(x) implies that
y0(x) = y0|I
n(x)(x) =z0|I
n(x)(x) =z0(x), as required.
4. Hyers-Ulam stability
In the following theorem, we prove the Hyers-Ulam stability of the differential equa- tion (1.1) defined on a finite and closed interval.
Theorem 4.1. Given c ∈R andr > 0, let I denote a closed ball of radius r and centered at c, that is, I ={x∈R| c−r≤x≤c+r} and let F : I×R→R be a continuous function which satisfies a Lipschitz condition (3.1) for all x∈ I and y, z ∈ R, where L is a constant with 0 < Lr < 1. If a continuously differentiable function y:I→Rsatisfies the differential inequality
(4.1) |y0(x)−F(x, y(x))| ≤ε
for all x ∈ I and for some ε ≥ 0, then there exists a unique continuous function y0:I→R satisfying equation(3.4) (y0 is a solution to(1.1))and
(4.2) |y(x)−y0(x)| ≤ r
1−Lrε for any x∈I.
Proof. First, we define a setX of all continuous functionsf :I→Rby X={f :I→R|f is continuous}
and introduce a generalized metric onX as follows:
d(f, g) = inf{C∈[0,∞]| |f(x)−g(x)| ≤C ∀x∈I}.
It is then obvious that (X, d) is a generalized complete metric space (see the proof of Theorem 3.1).
Let us define an operator Λ :X →X by
(4.3) (Λf)(x) =y(c) +
Z x
c
F(τ, f(τ))dτ (x∈I)
for anyf ∈X. (It is true that Λf ∈X because Λf is continuously differentiable in view of the Fundamental Theorem of Calculus.)
We now assert that Λ is strictly contractive onX. Forf, g∈X, letCf g∈[0,∞]
be an arbitrary constant withd(f, g)≤Cf g, that is, let us assume that
(4.4) |f(x)−g(x)| ≤Cf g
for allx∈I. Moreover, it follows from (3.1), (4.3) and (4.4) that
|(Λf)(x)−(Λg)(x)|=
Z x
c
{F(τ, f(τ))−F(τ, g(τ))}dτ
≤
Z x
c
|F(τ, f(τ))−F(τ, g(τ))|dτ
≤L
Z x
c
|f(τ)−g(τ)|dτ
≤LrCf g
for each x ∈ I, that is, d(Λf,Λg) ≤ LrCf g. Thus, it follows that d(Λf,Λg) ≤ Lrd(f, g) for all f, g∈X and we note that 0< Lr <1.
Analogously to the proof of Theorem 3.1, we can show that eachg0∈X satisfies the propertyd(Λg0, g0)<∞. Therefore, Theorem 2.1 (a) implies that there exists a continuous function y0:I→Rsuch that Λng0→y0in (X, d) asn→ ∞, and such thaty0= Λy0, that is,y0satisfies equation (3.4) for anyx∈I.
Ifg∈X, theng0 andgare continuous functions defined on a compact intervalI.
Hence, there exists a constantC >0 with
|g0(x)−g(x)| ≤C
for all x ∈ I. This implies that d(g0, g) < ∞ for every g ∈ X, or equivalently, {g ∈ X | d(g0, g) < ∞} = X. Therefore, according to Theorem 2.1 (b), y0 is a unique continuous function with the property (3.4). Furthermore, it follows from (4.1) that
−ε≤y0(x)−F(x, y(x))≤ε
for allx∈I. If we integrate each term of the above inequality fromc tox, then we have
|(Λy)(x)−y(x)| ≤εr for anyx∈I, that is, it holds thatd(Λy, y)≤εr.
It now follows from Theorem 2.1 (c) that d(y, y0)≤ 1
1−Lrd(Λy, y)≤ r 1−Lrε, which implies the validity of (4.2) for eachx∈I.
It is an open problem, whether the differential equation (1.1) has the Hyers-Ulam stability when the relevant domain is an infinite interval.
5. Examples
In this section, we show that there certainly exist functions y(x) which satisfy all the conditions given in Theorems 3.1, 3.2 and 4.1.
Example 5.1. We choose positive constantsKandLwithKL <1. For a positive number ε <2K, let I= [0,2K−ε] be a closed interval. Given a polynomialp(x), we assume that a continuously differentiable functiony:I→Rsatisfies
|y0(x)−Ly(x)−p(x)| ≤x+ε
for all x ∈ I. If we set F(x, y) = Ly+p(x) and ϕ(x) = x+ε, then the above inequality has the identical form with (3.2). Moreover, we obtain
Z x
0
ϕ(τ)dτ
=1
2x2+εx≤Kϕ(x)
for each x∈I, since Kϕ(x)−x2/2−εx≥0 for allx∈I. According to Theorem 3.1, there exists a unique continuous functiony0:I→Rsuch that
y0(x) =y(0) + Z x
0
{Ly0(τ) +p(τ)}dτ and
|y(x)−y0(x)| ≤ K
1−KL(x+ε) for anyx∈I.
Example 5.2. Let a be a constant greater than 1 and choose a constant L with 0< L <lna. Given an intervalI= [0,∞) and a polynomialp(x), supposey:I→R is a continuously differentiable function satisfying
|y0(x)−Ly(x)−p(x)| ≤ax for allx∈I. If we setϕ(x) =ax, then we have
Z x
0
ϕ(τ)dτ
≤ 1 lnaϕ(x)
for any x∈ I. In view of Theorem 3.2, there exists a unique continuous function y0:I→Rwith
y0(x) =y(0) + Z x
0
{Ly0(τ) +p(τ)}dτ and
|y(x)−y0(x)| ≤ 1 lna−Lax for eachx∈I.
Example 5.3. Let r and L be positive constants with 0 < Lr < 1 and define a closed intervalI={x∈R|c−r≤x≤c+r}for some real numberc. Assume that a continuously differentiable functiony:I→Rsatisfies
|y0(x)−Ly(x)−p(x)| ≤ε
for all x∈ I and for someε ≥0, where p(x) is a polynomial. Then, by Theorem 4.1, there exists a unique continuous functiony0:I→Rsuch that
y0(x) =y(c) + Z x
c
{Ly0(τ) +p(τ)}dτ and
|y(x)−y0(x)| ≤ r 1−Lrε for allx∈I.
Acknowledgement. This work was supported by a grant from the National Re- search Foundation of Korea funded by the Korean Government (No. 2009-0071206).
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