# The Stability of Some Differential Equations

Authors: M. M. Pourpasha; Th. M. Rassias; R. Saadati; S. M. Vaezpour
Journal: Mathematical Problems in Engineering
 (2011)

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## Abstract

We generalize the results obtained by Jun and Min (2009)
and use fixed point method to obtain the stability of the functional equationf(x+σ(y))=F[f(x),f(y)], for a class of functions of a vector space
into a Banach space where σ is an involution. Then we obtain the stability of the differential equations of the form y′=F[q(x),P(x)y(x)].


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## Body

## 1. Introduction and Preliminary

The stability problem of functional equations originated from a question of Ulam [1] in 1940, concerning the stability of group homomorphisms.The stability concept that was introduced by Rassias’ theorem [2] in 1978 provided a large influence to a number of mathematicians to develop the notion of what is known today by the term Hyers-Ulam-Rassias stability of the linear mapping. Since then, the stability of several functional equations has been extensively investigated by several mathematicians, see [3–5]. They have many applications in Information Theory, Physics, Economic Theory, and Social and Behavior Sciences.In 1996, Isac and Rassias [6] were the first to use the fixed point methods to investigate the Hyers-Ulam-Rassias stability.LetX be a set. A function d:X×X→[0,∞] is called a generalized metric on X if and only if d satisfies(1)d(x,y)=0, if and only if x=y,(2)d(x,y)=d(y,x), for all x,y∈X,(3)d(x,z)≤d(x,y)+d(y,z), for all x,y,z∈X.Note that the only substantial difference of the generalized metric from the metric is that the range of generalized metric includes the infinity.We now introduce one of fundamental results of fixed point theory. For the proof, refer to [7]. For an extensive theory of fixed point theorems and other nonlinear methods, the reader is referred to the book of Hyers et al. [8].Theorem 1.1.
Let(X,d) be a generalized complete metric space. Assume that  J:X→X is a strictly contractive operator with the Lipschitz constant  0<L<1. If there exists a nonnegative integer  k such that  d(Jk+1f,Jkf)<∞ for some  f∈X, then the followings are true: (a)
the sequence{Jnf} converges to a fixed point f* of J,(b)
f* is the unique fixed point of J in(1.1)X*={g∈X:d(Jkf,g)<∞},(c)
ifg∈X*, then(1.2)d(g,f*)≤11-Ld(Jg,g).


## 2. Stability of the Generalized Functional Equations

The stability problem for a general equation of the form(2.1)f(G(x,y))=H[f(x),f(y)]
was investigated by Cholewa [9] in 1984. Indeed, Cholewa proved the superstability of the above equation under some additional assumptions on the functions and spaces involved.Recently, Jung and Min  [10] applied the fixed point method to the investigate the stability of functional equation(2.2)f(x+y)=F[f(x),f(y)].In this section, we generalized the Jun and Min’s results and use fixed point approach to obtain the stability of the functional equation(2.3)f(x+σ(y))=F[f(x),f(y)]
for a class of functions of a vector space into a Banach space where σ is an involution.Theorem 2.1.
LetX and  (Y,∥·∥) be a vector space over  K and a Banach space over  K, respectively. Let  (X×X,∥·∥2) be a Banach space over  K. Assume that  F:X×X→Y is a bounded linear transformation, whose norm is denoted by  ∥F∥, satisfying
(2.4)F(F(u,u),F(v,v))=F(F(u,v),F(u,v))
for all u,v:X→X and there exists a real number κ>0 with
(2.5)‖(u(x),u(σ(x)))-(v(x),v(σ(x)))‖2≤κ‖u(x)-v(x)‖
for all u,v:X→X. Moreover, assume that φ:X×X→[0,∞) is a given function satisfying
(2.6)φ(x,σ(y))≤φ(2x,2y)
for all x,y∈X. If κ∥F∥<1 and a function f:X→Y satisfies the inequality
(2.7)‖f(x+σ(y))-F[f(x),f(y)]‖≤φ(x,y)
for any x,y∈X, then there exists a unique solution f*:X→Y of (2.3) such that
(2.8)‖f(x)-f*(x)‖≤11-κ‖F‖φ(x,x).Proof.
First, we denote byX the set of all functions h:X→Y and by d the generalized metric on X defined as
(2.9)d(g,h)=inf⁡{C∈[0,∞):‖g(x)-h(x)‖≤CM′(x,y),∀x∈E1}.
By a similar method used at the proof of [4, Theorem 3.1], we can show that (X,d) is a generalized complete metric space. Now, let us define an operator J:X→X by
(2.10)(Jh)(x)=F[h(x2),h(σ(x2))]
for every x∈X. We assert that J is strictly contractive on X. Given g,h∈X, let C∈[0,∞] be an arbitrary constant with d(g,h)≤C, that is,
(2.11)‖g(x)-h(x)‖≤Cφ(x,y)
for each x∈X. By (2.5), (2.6), (2.10), and (2.11), we have
(2.12)‖Jg(x)-Jh(x)‖≤‖F[g(x2),g(σ(x2))]-F[h(x2),h(σ(x2))]‖≤‖F‖‖[g(x2),g(σ(x2))]-[h(x2),h(σ(x2))]‖≤‖F‖κ‖g(x2)-h(x2)‖≤‖F‖κCφ(x2,x2)≤‖F‖κCφ(x,y)
for every x∈X. Then, from (2.9) we have d(Jg,Jh)≤κ∥F∥d(g,h) for any g,h∈X, where κ∥F∥ is the Lipschitz constant with 0<κ∥F∥<1. Thus, J is strictly contractive.
Now, we claim thatd(Jf,f)≤∞. Replacing x/2 by x and σ(x/2) by y in (2.7), then it follows from (2.6) and (2.10) that
(2.13)‖f(x2+σ(σ(x2)))-F[f(x2),f(σ(x2))]‖≤φ(x2,σ(x2))‖f(x)-(Jf)(x)‖≤φ(x2,σ(x2))≤φ(x,x)
for every x∈X. Then,
(2.14)d(Jf,f)≤1≤∞.
Now, it follows from Theorem 1.1(a) that there exists a function f*:E1→E2 which is a fixed point of J, such that
(2.15)lim⁡n→∞d(Jnf,f*)=0.
From Theorem 1.1(c), we get
(2.16)d(Jnf,f*)≤11-κ‖F‖d(Jf,f)≤11-κ‖F‖′
which implies the validity of (2.8). According to Theorem 1.1(b), f* is the unique fixed point of J with d(f,f*)<∞.
We now assert that(2.17)‖(Jnf)(x+σ(y))-F[(Jnf)(x),(Jnf)(y)]‖≤(κ‖F‖)nφ(x,x)
for all n∈N and x,y∈X. Indeed, it follows from (2.4), (2.5), (2.6), (2.7), and (2.10) that
(2.18)‖(Jf)(x+σ(y))-F[(Jf)(x),(Jf)(y)]‖=‖F[f(x+σ(y)2),f(σ(x+σ(y)2))]-F[F[f(x2),f(σ(x2))],F[f(y2),f(σ(y2))]]F[f(x+σ(y)2),f(σ(x+σ(y)2))]‖≤‖F‖‖[f(x+σ(y)2),f(σ(x+σ(y)2))]-[F[f(x2),f(σ(x2))],F[f(y2),f(σ(y2))]][f(x+σ(y)2),f(σ(x+σ(y)2))]‖≤‖F‖κ‖f(x+σ(y)2)-[F[f(x2),f(y2)]]‖≤‖F‖κφ(x2,y2)≤‖F‖κφ(x,x)
for any x,y∈X. Then, it follows from (2.4), (2.5), (2.6), (2.10), and (2.17) that
(2.19)‖(Jn+1f)(x+σ(y))-F[(Jn+1f)(x),(Jn+1f)(y)]‖=‖F[Jnf(x+σ(y)2),Jnf(σ(x+σ(y)2))]-F[F[Jnf(x2),Jnf(σ(x2))],F[Jnf(y2),Jnf(σ(y2))]][Jnf(x+σ(y)2),Jnf(σ(x+σ(y)2))]‖≤‖F‖κ‖Jnf(x+σ(y)2)-[F[Jnf(x2),Jnf(y2)]]‖≤(‖F‖κ)n+1φ(x2,y2)≤(‖F‖κ)n+1φ(x,x)
for all n∈N, which proves the validity of (2.17).
Finally, we prove thatf*(x+σ(y))=F[f*(x),f*(y)] for any x,y∈X. Since F is continuous as a bounded linear transformation, it follows from (2.15) and (2.17) that
(2.20)‖f*(x+σ(y))-F[f*(x),f*(y)]‖=lim⁡n→∞‖Jnf(x+σ(y)2)-[F[Jnf(x2),Jnf(y2)]]‖≤lim⁡n→∞(‖F‖κ)nφ(x,x)=0
for all x,y∈X, which implies that f* is a solution of  (2.7).Corollary 2.2.
LetX and  (Y,∥·∥) be a vector space over  K and a Banach space over  K, respectively, and let  (Y×Y,∥·∥2) be a Banach space over  K. Assume that  F:Y×Y→Y is a bounded linear transformation, whose norm is denoted by  F, satisfying condition (2.4) and that there exists a real number  κ>0 satisfying condition (2.5). If  κ∥F∥<1 and a function  f:X→Y satisfies the inequality
(2.21)‖f(x+y)-F[f(x),f(y)]‖≤ϵ(‖x‖p+‖y‖p)
for  all x,y∈X and for some nonnegative real constants θ and p, then there exists a unique solution f*:X→Y of 1.2 such that
(2.22)‖f(x)-f*(x)‖≤2θ1-κ‖F‖‖x‖p
for all x∈X.Example 2.3.
Assume thatX=Y=ℂ, and consider the Banach spaces (ℂ,|·|) and (ℂ×ℂ,|·|2),where we define |(u(t),v(t))|2=|u|2+|v|2 for all u,v:ℂ→ℂ.Let A and B be fixed complex numbers with |A|+|B|<1/2, and let F:ℂ×ℂ→ℂ be a linear transformation defined by
(2.23)F(u(t),v(t))=Au(t)+Bv(t).
Then it is easy to show that F satisfies condition (2.13).
Ifu and v are complex numbers satisfying |(u(t),v(t))|2≤1 for all t, then
(2.24)|F(u,v)|≤|A||u|+|B||v|≤|A|+|B|.
Thus, we get
(2.25)‖F‖≤sup⁡{|F(u,v)|:u,v∈Cwith|(u,v)|2≤1}≤|A|+|B|,
which implies the boundedness of the linear transformation F.
On the other hand, we obtain(2.26)‖(u(x),u(σ(x)))-(v(x),v(σ(x)))‖2≤2‖u(x)-v(x)‖
for any u,v∈ℂ, then we have
(2.27)‖F‖κ≤2(|A|+|B|)≤1.
If the function f:ℂ→ℂ satisfies the inequality
(2.28)‖f*(x+σ(y))-F[f*(x),f*(y)]‖
for all x,y∈ℂ and for some ε>0, then Corollary 2.2 (with θ=ε/2 and p=0) implies that there exists a unique function f*:ℂ→ℂ such that
(2.29)‖f*(x+σ(y))=F[f*(x),f*(y)]‖
for all x,y∈ℂ and
(2.30)|f*(x)-f(x)|≤ε1-2(|A|+|B|)
for any x∈ℂ.


## 3. Stability of the Generalized Differential Equations

LetY be a normed space, and let I be an open interval. Assume that for any function y:I→Y satisfying the differential inequality(3.1)‖an(x)y(n)(x)+an-1(x)y(n-1)(x)+⋯+a1(x)y′(x)+a0(x)y(x)+h(x)‖≤ε
for all x∈I and for some ε≥0, there exists a solution y0:I→Y of the differential equation(3.2)an(x)y(n)(x)+an-1(x)y(n-1)(x)+⋯+a1(x)y′(x)+a0(x)y(x)+h(x)=0
such that ∥y(x)-y0(x)∥≤K(ε) for any x∈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 replaceε and K(ε) by φ(x) and Φ(x), where φ,Φ:I→[0,∞) are functions not depending on y and y0 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 [11, 12].In 1998, Alsina and Ger investigated the Hyers-Ulam stability of differential equations. They proved in [13] that if a differentiable function y:I→ℝ satisfies the differential inequality |y′(t)-y(t)|≤ε, where I is an open subinterval of ℝ, then there exists a differentiable function y0:I→ℝ satisfying y0′(t)=y0(t) and |y0(t)-y(t)|≤3ε for any t∈I.Alsina and Ger’s results have been generalized by Takahasi et al. [14]. They proved that the Hyers-Ulam stability holds for the Banach space-valued differential equation y′(x)=λy(x) (see also [15]).Recently, Takahasi et al. also proved the Hyers-Ulam stability of linear differential equations of first order,y′(x)+g(x)y(x)=0, where g(x) is a continuous function, and they also proved the Hyers-Ulam stability of linear differential equations of other types  (see [16–18]).In this section, for a bounded and continuous functionF(x,y), we will adopt the idea of Cădariu and Radu [19, 20] and prove the Hyers-Ulam-Rassias stability as well as the Hyers-Ulam stability of the differential equations of the form(3.3)y′(x)=F(q(x),p(x)y(x)).Theorem 3.1.
For given real numbersa and  b with  a<b, let  I=[a,b] be a closed interval and choose  c∈I. Let  K and  L be positive constants with  0<KL<1. Assume that  F:I×ℝ→ℝ is a continuous function which satisfies a Lipschitz condition
(3.4)|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
(3.5)|y′(x)-F(q(x),p(x)y(x))|≤φ(x)
for all x∈I, where p(x),q(x)  are  continuous functions in which |p(x)|≤c and φ:I→(0,∞) is a continuous function with
(3.6)|∫cxφ(t)dt|≤Kφ(x)
for each x∈I, then there exists a unique continuous function y0:I→ℝ such that
(3.7)y0(x)=y(c)+∫cxF(q(x),p(x)y(x))dt
(consequently, y0 is a solution to (2.15)) and
(3.8)|y(x)-y0(x)|≤K1-KLφ(x)
for all x∈I.Proof.
Let us define a setX of all continuous functions f:I→ℝ by
(3.9)X={f:I→R∣fiscontinuous}
and introduce a generalized metric on X as follows:
(3.10)d(f,g)=inf⁡{C∈[0,∞):|f(x)-g(x)|≤Cφ(x),∀x∈I}.
By a similar method used at the proof of [4, Theorem 3.1], we assert that (X,d) is complete. Let {hn} be a Cauchy sequence in (X,d).
Then, for anyε>0, there exists an integer Nε>0 such that d(hm,hn)≤ε for all m,n∈Nε. It further follows from (3.10) that
(3.11)∀ε>0∃Nε∈N∀m,n∈Nε∀x∈I:|hm(x)-hn(x)|≤εφ(x).Equation (3.11) implies that {hn(x)} is a Cauchy sequence in ℝ. Since ℝ is complete, {hn(x)} converges for each x∈I. Thus, we can define a function h:I→ℝ by
(3.12)h(x)=lim⁡n→∞hn(x).
Let m increase to infinity, then by (3.11) we have
(3.13)∀ε>0∃Nε∈N∀n∈Nε∀x∈I:|h(x)-hn(x)|≤εφ(x).
Sinceφ is bounded on I, {hn} converges uniformly to h. Hence, h is continuous and h∈X.
Further, considering (3.10) and (3.13), then
(3.14)∀ε>0∃Nε∈N∀n∈Nε:d(h,hn)≤ε.
Then, the Cauchy sequence {hn} converges to h in (X,d). Hence, (X,d) is complete.
Now, define the operatorJ:X→X by
(3.15)(Jf)(x)=y(c)+∫cxF(q(x),p(x)f(x))dtx∈I
for all f∈X. (Indeed, according to the Fundamental Theorem of Calculus, Jf is continuously differentiable on I, since F and f are continuous functions. Hence, we may conclude that Jf∈X.) We prove that J is strictly contractive on X. For any f,g∈X, let Cfg∈[0,∞] be an arbitrary constant with d(f,g)≤Cfg, then, by (2.15), we have
(3.16)|f(x)-g(x)|≤Cfgφ(x)
for any x∈I. It then follows from (3.4), (3.6), (3.10), (3.15), and (3.16) that
(3.17)|(Jf)(x)-(Jg)(x)|≤|∫cx{F(q(x),p(x)f(x))-F(q(x),p(x)g(x))}dt|≤|∫cx|F(q(x),p(x)f(x))-F(q(x),p(x)g(x))|dt|≤L|∫cx|f(t)-g(t)|dt|≤LCfg|∫cxφ(t)dt|≤KLCfgφ(x)
for all x∈I. Then, d(Jf,Jg)≤KLCfg. Hence, we can conclude that d(Jf,Jg)≤KLd(f,g) for any f,g∈X (note that 0<KL<1). It follows from (3.9) and (3.15) that for an arbitrary g0∈X, there exists a constant 0<C<1 with
(3.18)|(Jg0)(x)-g0(x)|=|y(c)+∫cxF(p(t),g0(t))dt-g0(x)|≤Cφ(x)
for all x∈I, since F(x,g0(x)) and g0(x) are bounded on I and minx∈Iφ(x)>0. Thus, (3.10) implies that
(3.19)d(Jg0,g0)<∞.
Therefore, according to Theorem 1.1(a), there exists a continuous function y0:I→ℝ such that Jng0→y0 in (X,d) and Jy0=y0, that is, y0 satisfies (3.7) for every x∈I. For any g∈X, since g and g0 are bounded on I and minx∈Iφ(x)>0, there exists a constant 0<Cg<1 such that
(3.20)|g(x)-g0(x)|≤Cgφ(x)
for any x∈I. Hence, we have d(g0,g)<∞ for all g∈X, that is, {g∈X∣d(g0,g)<∞}=X. Hence, in view of Theorem 1.1(b), we conclude that y0 is the unique continuous function with the property (3.7).
On the other hand, it follows from (3.5) that
(3.21)-φ(x)≤y′(x)-F(q(x),p(x)y(x))≤φ(x)
for all x∈I. If we integrate each term in the above inequality from c to x, then we obtain
(3.22)|y(x)-y(c)-∫cxF(q(x),p(x)y(x))dt|≤|∫cxφ(t)dt|
for any x∈I. Thus, by (3.6) and (3.15), we get
(3.23)|y(x)-Jy(x)|≤Kφ(x)
for each x∈I, which implies that
(3.24)d(Jy,y)≤K.
Finally, Theorem 1.1(c) and (3.24) implys that
(3.25)d(Jy,y0)≤11-KLd(Jy,y)≤K1-KL,
which means that inequality (3.24) holds true for all x∈I.Now, we prove the last theorem for unbounded intervals. Also we show that Theorem3.1 is also true if I is replaced by an unbounded interval such as (-∞,b], ℝ, or [a,∞).Theorem 3.2.
For given real numbersa and  b, let  I denote either  (-∞,b],  ℝ, or  [a,∞). Set either  c=a for  I=[a,∞) or  c=b for  I=(-∞,b] or  c is a fixed real number if  I=ℝ. Let  K and  L be positive constants with  0<KL<1. Assume that  F:I×ℝ→ℝ is a continuous function which satisfies Lipschitz condition (3.4) for any  x∈I and  y,z∈ℝ. If a continuously differentiable function  y:I→ℝ satisfies
(3.26)|y′(x)-F(q(x),p(x)y(x))|≤φ(x)
for all x∈I, where p(x) is a continuous function and φ:I→(0,∞) is a continuous function satisfying condition (3.6) for each x∈I, then there exists a unique continuous function y0:I→ℝ such that
(3.27)|y(x)-y0(x)|≤K1-KLφ(x)
for all x∈I.Proof.
We prove forI=ℝ. The other cases can be proved similarly. For any n∈ℕ, we define In=[c-n,c+n]. (We set In=[b-n,b] for I=(-∞,b] and In=[a,a+n] for I=[a,∞)). By Theorem 3.1, there exists a unique continuous function yn:In→ℝ such that
(3.28)yn(x)=y(c)+∫cxF(q(x),p(x)yn(t))dt(3.29)|yn(x)-y(x)|≤K1-KLφ(x)
for all x∈In. The uniqueness of yn implies that, if x∈In, then
(3.30)yn(x)=yn+1(x)=yn+2(x)=⋯.
For any x∈ℝ, define n(x)∈ℕ as
(3.31)n(x)=min⁡{n∈N∣x∈In}.
Moreover, define a function y0:ℝ→ℝ by
(3.32)y0(x)=yn(x)(x),
and we assert that y0 is continuous. For an arbitrary x1∈ℝ, we choose the integer n1=n(x1). Then, x1 belongs to the interior of In1+1 and there exists an ε>0 such that y0(x)=yn1+1(x) for all x with x1-ε<x<x1+ε. Since yn1+1 is continuous at x1, so is y0. That is, y0 is continuous at x1 for any x1∈ℝ.
We will now show thaty0 satisfies (3.8) for all x∈ℝ. For an arbitrary x∈ℝ, we choose the integer n(x). Then, it holds that x∈In(x) and it follows from (3.28) and (3.32) that
(3.33)y0(x)=yn(x)(x)=y(c)+∫cxF(q(x),p(x)yn(x)(t))dt=y(c)+∫cxF(q(x),p(x)y0(t))dt
since n(t)≤n(x) for any t∈In(x). Then, from (3.30) and (3.32) we have
(3.34)yn(x)(t)=yn(t)(t)=y0(t).
Since x∈In(x) for every x∈ℝ, by (3.29) and (3.32), we have
(3.35)|y0(x)-y(x)|=|yn(x)(x)-y(x)|≤K1-KLφ(x)
for any x∈ℝ.
Finally, we show thaty0 is unique. Let z0:ℝ→ℝ be another continuous function which satisfies (3.8), with z0 in place of y0, for all x∈ℝ. Suppose x is an arbitrary real number. Since the restrictions y0∣In(x)(=yn(x)) and z0∣In(x) both satisfy (3.7) and (3.8) for all x∈In(x), the uniqueness of yn(x)=y0∣In(x) implies that
(3.36)y0(x)=y0∣In(x)(x)=z0∣In(x)(x)=z0(x)
as required.Corollary 3.3.
Givenc∈ℝ and  r>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 (3.4) for all  x∈I and  y,z∈R, where  L is a constant with  0<Lr<1. If a continuously differentiable function  y:I→ℝ satisfies the differential inequality
(3.37)|y′(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 (3.7) and
(3.38)|y(x)-y0(x)|≤r1-rLε
for any x∈I.Example 3.4.
We choose positive constantsK and L with KL<1. For a positive number ε<2K, let I=[0,2K-ε] be a closed interval. Given a polynomial p(x), we assume that a continuously differentiable function y:I→ℝ satisfies
(3.39)|y′(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. Moreover, we obtain
(3.40)|∫cxφ(t)dt|=x22+εx≤Kφ(x)
for each x∈I, since Kφ(x)-x2/2-εx≥0 for all x∈I. By Theorem 3.1, there exists a unique continuous function y0:I→ℝ such that
(3.41)y0(x)=y(0)+∫0x{Ly(t)-p(t)}dt|y0(x)-y(x)|≤K1-KL(x+ε)
for any x∈I.Example 3.5.
Leta be a constant greater than 1 and choose a constant L with 0<L<lna. Given an interval I=[0,1) and a polynomial p(x), suppose y:I→ℝ is a continuously differentiable function satisfying
(3.42)|y′(x)-Ly(x)-p(x)|≤ax
for all x∈I. If we set φ(x)=ax, then we have
(3.43)|∫cxφ(t)dt|≤1ln⁡aφ(x)
for any x∈I. By Theorem 3.2, there exists a unique continuous function y0:I→ℝ with
(3.44)y0(x)=y(0)+∫0x{Ly(t)-p(t)}dt|y0(x)-y(x)|≤1ln⁡a-Lax
for any x∈I.



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*Source: 128479-2011-12-29.xml*