Mathematical Problems in Engineering

Mathematical Problems in Engineering / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 128479 | https://doi.org/10.1155/2011/128479

M. M. Pourpasha, Th. M. Rassias, R. Saadati, S. M. Vaezpour, "The Stability of Some Differential Equations", Mathematical Problems in Engineering, vol. 2011, Article ID 128479, 15 pages, 2011. https://doi.org/10.1155/2011/128479

The Stability of Some Differential Equations

Academic Editor: Alexander P. Seyranian
Received17 Oct 2011
Accepted17 Nov 2011
Published29 Dec 2011

Abstract

We generalize the results obtained by Jun and Min (2009) and use fixed point method to obtain the stability of the functional equation 𝑓(𝑥+ğœŽ(𝑦))=𝐹[𝑓(𝑥),𝑓(𝑦)], for a class of functions of a vector space into a Banach space where ğœŽ is an involution. Then we obtain the stability of the differential equations of the form ğ‘¦î…ž=𝐹[ğ‘ž(𝑥),𝑃(𝑥)𝑦(𝑥)].

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.

Let 𝑋 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 in𝑋∗=𝐽𝑔∈𝑋∶𝑑𝑘𝑓,𝑔<∞,(1.1)(c) if 𝑔∈𝑋∗, then𝑑𝑔,𝑓∗≤11−𝐿𝑑(𝐽𝑔,𝑔).(1.2)

2. Stability of the Generalized Functional Equations

The stability problem for a general equation of the form[]𝑓(𝐺(𝑥,𝑦))=𝐻𝑓(𝑥),𝑓(𝑦)(2.1) 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[]𝑓(𝑥+𝑦)=𝐹𝑓(𝑥),𝑓(𝑦).(2.2)

In this section, we generalized the Jun and Min’s results and use fixed point approach to obtain the stability of the functional equation[]𝑓(𝑥+ğœŽ(𝑦))=𝐹𝑓(𝑥),𝑓(𝑦)(2.3) for a class of functions of a vector space into a Banach space where ğœŽ is an involution.

Theorem 2.1. Let 𝑋 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 𝐹(𝐹(𝑢,𝑢),𝐹(𝑣,𝑣))=𝐹(𝐹(𝑢,𝑣),𝐹(𝑢,𝑣))(2.4) for all 𝑢,𝑣∶𝑋→𝑋 and there exists a real number 𝜅>0 with ‖(𝑢(𝑥),𝑢(ğœŽ(𝑥)))−(𝑣(𝑥),𝑣(ğœŽ(𝑥)))‖2≤𝜅‖𝑢(𝑥)−𝑣(𝑥)‖(2.5) for all 𝑢,𝑣∶𝑋→𝑋. Moreover, assume that 𝜑∶𝑋×𝑋→[0,∞) is a given function satisfying 𝜑(𝑥,ğœŽ(𝑦))≤𝜑(2𝑥,2𝑦)(2.6) for all 𝑥,𝑦∈𝑋. If 𝜅‖𝐹‖<1 and a function 𝑓∶𝑋→𝑌 satisfies the inequality []‖𝑓(𝑥+ğœŽ(𝑦))−𝐹𝑓(𝑥),𝑓(𝑦)‖≤𝜑(𝑥,𝑦)(2.7) for any 𝑥,𝑦∈𝑋, then there exists a unique solution 𝑓∗∶𝑋→𝑌 of (2.3) such that ‖𝑓(𝑥)−𝑓∗1(𝑥)‖≤1−𝜅‖𝐹‖𝜑(𝑥,𝑥).(2.8)

Proof. First, we denote by 𝑋 the set of all functions â„Žâˆ¶ğ‘‹â†’ğ‘Œ and by d the generalized metric on 𝑋 defined as 𝑑[(𝑔,ℎ)=inf𝐶∈0,∞)∶‖𝑔(𝑥)−ℎ(𝑥)â€–â‰¤ğ¶ğ‘€î…ž(𝑥,𝑦),∀𝑥∈𝐸1.(2.9) 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 î‚ƒâ„Žî‚€ğ‘¥(ğ½â„Ž)(𝑥)=𝐹2î‚î‚€ğœŽî‚€ğ‘¥,ℎ2(2.10) for every 𝑥∈𝑋. We assert that 𝐽 is strictly contractive on 𝑋. Given 𝑔,â„Žâˆˆğ‘‹, let 𝐶∈[0,∞] be an arbitrary constant with 𝑑(𝑔,ℎ)≤𝐶, that is, ‖𝑔(𝑥)−ℎ(𝑥)‖≤𝐶𝜑(𝑥,𝑦)(2.11) for each 𝑥∈𝑋. By (2.5), (2.6), (2.10), and (2.11), we have ‖‖‖‖𝐹𝑔𝑥𝐽𝑔(𝑥)âˆ’ğ½â„Ž(𝑥)‖≤2î‚î‚€ğœŽî‚€ğ‘¥,𝑔2î‚ƒâ„Žî‚€ğ‘¥î‚î‚î‚„âˆ’ğ¹2î‚î‚€ğœŽî‚€ğ‘¥,ℎ2‖‖‖‖‖‖𝑔𝑥≤‖𝐹‖2î‚î‚€ğœŽî‚€ğ‘¥,𝑔2âˆ’î‚ƒâ„Žî‚€ğ‘¥î‚î‚î‚„2î‚î‚€ğœŽî‚€ğ‘¥,ℎ2‖‖‖‖‖‖𝑔𝑥≤‖𝐹‖𝜅2î‚î‚€ğ‘¥âˆ’â„Ž2‖‖‖𝑥≤‖𝐹‖𝜅𝐶𝜑2,𝑥2≤‖𝐹‖𝜅𝐶𝜑(𝑥,𝑦)(2.12) for every 𝑥∈𝑋. Then, from (2.9) we have 𝑑(𝐽𝑔,ğ½â„Ž)≤𝜅‖𝐹‖𝑑(𝑔,ℎ) for any 𝑔,â„Žâˆˆğ‘‹, where 𝜅‖𝐹‖ is the Lipschitz constant with 0<𝜅‖𝐹‖<1. Thus, 𝐽 is strictly contractive.
Now, we claim that 𝑑(𝐽𝑓,𝑓)≤∞. Replacing 𝑥/2 by 𝑥 and ğœŽ(𝑥/2) by 𝑦 in (2.7), then it follows from (2.6) and (2.10) that ‖‖‖𝑓𝑥2î‚€ğœŽî‚€ğ‘¥+ğœŽ2𝑓𝑥−𝐹2î‚î‚€ğœŽî‚€ğ‘¥,𝑓2‖‖‖𝑥≤𝜑2𝑥,ğœŽ2𝑥‖𝑓(𝑥)−(𝐽𝑓)(𝑥)‖≤𝜑2𝑥,ğœŽ2≤𝜑(𝑥,𝑥)(2.13) for every 𝑥∈𝑋. Then, 𝑑(𝐽𝑓,𝑓)≤1≤∞.(2.14) 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 limğ‘›â†’âˆžğ‘‘î€·ğ½ğ‘›ğ‘“,𝑓∗=0.(2.15) From Theorem 1.1(c), we get 𝑑𝐽𝑛𝑓,𝑓∗≤111−𝜅‖𝐹‖𝑑(𝐽𝑓,𝑓)≤1âˆ’ğœ…â€–ğ¹â€–î…ž(2.16) 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 ‖‖(𝐽𝑛𝑓)(𝑥+ğœŽ(𝑦))−𝐹(𝐽𝑛𝑓)(𝑥),(𝐽𝑛𝑓‖‖≤))(𝑦)(𝜅‖𝐹‖𝑛𝜑(𝑥,𝑥)(2.17) for all 𝑛∈𝑁 and 𝑥,𝑦∈𝑋. Indeed, it follows from (2.4), (2.5), (2.6), (2.7), and (2.10) that ‖[]‖=‖‖‖𝐹𝑓(𝐽𝑓)(𝑥+ğœŽ(𝑦))−𝐹(𝐽𝑓)(𝑥),(𝐽𝑓)(𝑦)𝑥+ğœŽ(𝑦)2î‚¶î‚µğœŽî‚µ,𝑓𝑥+ğœŽ(𝑦)2𝐹𝑓𝑥−𝐹2î‚î‚€ğœŽî‚€ğ‘¥,𝑓2𝑓𝑦,𝐹2î‚î‚€ğœŽî‚€ğ‘¦,𝑓2‖‖‖‖‖‖𝑓≤‖𝐹‖𝑥+ğœŽ(𝑦)2î‚¶î‚µğœŽî‚µ,𝑓𝑥+ğœŽ(𝑦)2−𝐹𝑓𝑥2î‚î‚€ğœŽî‚€ğ‘¥,𝑓2𝑓𝑦,𝐹2î‚î‚€ğœŽî‚€ğ‘¦,𝑓2‖‖‖‖‖‖𝑓≤‖𝐹‖𝜅𝑥+ğœŽ(𝑦)2−𝐹𝑓𝑥2𝑦,𝑓2‖‖‖𝑥≤‖𝐹‖𝜅𝜑2,𝑦2≤‖𝐹‖𝜅𝜑(𝑥,𝑥)(2.18) for any 𝑥,𝑦∈𝑋. Then, it follows from (2.4), (2.5), (2.6), (2.10), and (2.17) that ‖‖𝐽𝑛+1𝑓𝐽(𝑥+ğœŽ(𝑦))−𝐹𝑛+1𝑓𝐽(𝑥),𝑛+1𝑓‖‖=‖‖‖𝐹𝐽(𝑦)𝑛𝑓𝑥+ğœŽ(𝑦)2,ğ½ğ‘›ğ‘“î‚µğœŽî‚µğ‘¥+ğœŽ(𝑦)2𝐹𝐽−𝐹𝑛𝑓𝑥2,ğ½ğ‘›ğ‘“î‚€ğœŽî‚€ğ‘¥2𝐽,𝐹𝑛𝑓𝑦2,ğ½ğ‘›ğ‘“î‚€ğœŽî‚€ğ‘¦2‖‖‖‖‖‖𝐽≤‖𝐹‖𝜅𝑛𝑓𝑥+ğœŽ(𝑦)2−𝐹𝐽𝑛𝑓𝑥2,𝐽𝑛𝑓𝑦2‖‖‖≤(‖𝐹‖𝜅)𝑛+1𝜑𝑥2,𝑦2≤(‖𝐹‖𝜅)𝑛+1𝜑(𝑥,𝑥)(2.19) for all 𝑛∈𝑁, which proves the validity of (2.17).
Finally, we prove that 𝑓∗(𝑥+ğœŽ(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 ‖‖𝑓∗𝑓(𝑥+ğœŽ(𝑦))−𝐹∗(𝑥),𝑓∗‖‖(𝑦)=limğ‘›â†’âˆžâ€–â€–â€–ğ½ğ‘›ğ‘“î‚µğ‘¥+ğœŽ(𝑦)2−𝐹𝐽𝑛𝑓𝑥2,𝐽𝑛𝑓𝑦2‖‖‖≤limğ‘›â†’âˆž(‖𝐹‖𝜅)𝑛𝜑(𝑥,𝑥)=0(2.20) for all 𝑥,𝑦∈𝑋, which implies that 𝑓∗ is a solution of  (2.7).

Corollary 2.2. Let 𝑋 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 [𝑓]‖𝑓(𝑥+𝑦)−𝐹(𝑥),𝑓(𝑦)‖≤𝜖‖𝑥‖𝑝+‖𝑦‖𝑝(2.21) 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 ‖𝑓(𝑥)−𝑓∗(𝑥)‖≤2𝜃1−𝜅‖𝐹‖‖𝑥‖𝑝(2.22) for all 𝑥∈𝑋.

Example 2.3. Assume that 𝑋=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 𝐹(𝑢(𝑡),𝑣(𝑡))=𝐴𝑢(𝑡)+𝐵𝑣(𝑡).(2.23) Then it is easy to show that 𝐹 satisfies condition (2.13).
If 𝑢 and 𝑣 are complex numbers satisfying |(𝑢(𝑡),𝑣(𝑡))|2≤1 for all 𝑡, then ||||≤||𝐴||||𝐵||||𝐴||+||𝐵||𝐹(𝑢,𝑣)|𝑢|+|𝑣|≤.(2.24) Thus, we get ||𝐹||||||‖𝐹‖≤sup(𝑢,𝑣)∶𝑢,𝑣∈ℂwith(𝑢,𝑣)2≤||𝐴||+||𝐵||≤1,(2.25) which implies the boundedness of the linear transformation 𝐹.
On the other hand, we obtain ‖(𝑢(𝑥),𝑢(ğœŽ(𝑥)))−(𝑣(𝑥),𝑣(ğœŽ(𝑥)))‖2≤√2‖𝑢(𝑥)−𝑣(𝑥)‖(2.26) for any 𝑢,𝑣∈ℂ, then we have √‖𝐹‖𝜅≤2||𝐴||+||𝐵||≤1.(2.27) If the function 𝑓∶ℂ→ℂ satisfies the inequality ‖‖𝑓∗𝑓(𝑥+ğœŽ(𝑦))−𝐹∗(𝑥),𝑓∗‖‖(𝑦)(2.28) 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 ‖‖𝑓∗𝑓(𝑥+ğœŽ(𝑦))=𝐹∗(𝑥),𝑓∗‖‖(𝑦)(2.29) for all 𝑥,𝑦∈ℂ and ||𝑓∗||≤𝜀(𝑥)−𝑓(𝑥)√1−2||𝐴||+||𝐵||(2.30) for any 𝑥∈ℂ.

3. Stability of the Generalized Differential Equations

Let 𝑌 be a normed space, and let 𝐼 be an open interval. Assume that for any function 𝑦∶𝐼→𝑌 satisfying the differential inequalityâ€–â€–ğ‘Žğ‘›(𝑥)𝑦(𝑛)(𝑥)+ğ‘Žğ‘›âˆ’1(𝑥)𝑦(𝑛−1)(𝑥)+⋯+ğ‘Ž1(𝑥)ğ‘¦î…ž(𝑥)+ğ‘Ž0‖‖(𝑥)𝑦(𝑥)+ℎ(𝑥)≤𝜀(3.1) for all 𝑥∈𝐼 and for some 𝜀≥0, there exists a solution 𝑦0∶𝐼→𝑌 of the differential equationğ‘Žğ‘›(𝑥)𝑦(𝑛)(𝑥)+ğ‘Žğ‘›âˆ’1(𝑥)𝑦(𝑛−1)(𝑥)+⋯+ğ‘Ž1(𝑥)ğ‘¦î…ž(𝑥)+ğ‘Ž0(𝑥)𝑦(𝑥)+ℎ(𝑥)=0(3.2) 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 replace 𝜀 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 |ğ‘¦î…ž(𝑡)−𝑦(𝑡)|≤𝜀, where 𝐼 is an open subinterval of ℝ, then there exists a differentiable function 𝑦0∶𝐼→ℝ satisfying ğ‘¦î…ž0(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 ğ‘¦î…ž(𝑥)=𝜆𝑦(𝑥) (see also [15]).

Recently, Takahasi et al. also proved the Hyers-Ulam stability of linear differential equations of first order, ğ‘¦î…ž(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 function 𝐹(𝑥,𝑦), we will adopt the idea of C̆adariu and Radu [19, 20] and prove the Hyers-Ulam-Rassias stability as well as the Hyers-Ulam stability of the differential equations of the formğ‘¦î…ž(𝑥)=𝐹(ğ‘ž(𝑥),𝑝(𝑥)𝑦(𝑥)).(3.3)

Theorem 3.1. For given real numbers ğ‘Ž and 𝑏 with ğ‘Ž<𝑏, let 𝐼=[ğ‘Ž,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 ||||||||𝐹(𝑥,𝑦)−𝐹(𝑥,𝑧)≤𝐿𝑦−𝑧(3.4) for any 𝑥∈𝐼 and 𝑦,𝑧∈ℝ. If a continuously differentiable function 𝑦∶𝐼→ℝ satisfies ||ğ‘¦î…ž||(𝑥)−𝐹(ğ‘ž(𝑥),𝑝(𝑥)𝑦(𝑥))≤𝜑(𝑥)(3.5) for all 𝑥∈𝐼, where 𝑝(𝑥),ğ‘ž(𝑥) are continuous functions in which |𝑝(𝑥)|≤𝑐 and 𝜑∶𝐼→(0,∞) is a continuous function with ||||𝑥𝑐𝜑||||(𝑡)𝑑𝑡≤𝐾𝜑(𝑥)(3.6) for each 𝑥∈𝐼, then there exists a unique continuous function 𝑦0∶𝐼→ℝ such that 𝑦0(𝑥)=𝑦(𝑐)+𝑥𝑐𝐹(ğ‘ž(𝑥),𝑝(𝑥)𝑦(𝑥))𝑑𝑡(3.7) (consequently, 𝑦0 is a solution to (2.15)) and ||𝑦(𝑥)−𝑦0||≤𝐾(𝑥)𝜑1−𝐾𝐿(𝑥)(3.8) for all 𝑥∈𝐼.

Proof. Let us define a set 𝑋 of all continuous functions 𝑓∶𝐼→ℝ by 𝑋={𝑓∶𝐼→ℝ∣𝑓iscontinuous}(3.9) and introduce a generalized metric on 𝑋 as follows: 𝑑[||𝑓||(𝑓,𝑔)=inf𝐶∈0,∞)∶(𝑥)−𝑔(𝑥)≤𝐶𝜑(𝑥),∀𝑥∈𝐼.(3.10) By a similar method used at the proof of [4, Theorem 3.1], we assert that (𝑋,𝑑) is complete. Let {â„Žğ‘›} be a Cauchy sequence in (𝑋,𝑑).
Then, for any 𝜀>0, there exists an integer 𝑁𝜀>0 such that 𝑑(â„Žğ‘š,â„Žğ‘›)≤𝜀 for all 𝑚,𝑛∈𝑁𝜀. It further follows from (3.10) that ∀𝜀>0∃𝑁𝜀∈ℕ∀𝑚,𝑛∈𝑁𝜀||â„Žâˆ€ğ‘¥âˆˆğ¼âˆ¶ğ‘š(𝑥)âˆ’â„Žğ‘›||(𝑥)≤𝜀𝜑(𝑥).(3.11)Equation (3.11) implies that {â„Žğ‘›(𝑥)} is a Cauchy sequence in ℝ. Since ℝ is complete, {â„Žğ‘›(𝑥)} converges for each 𝑥∈𝐼. Thus, we can define a function â„Žâˆ¶ğ¼â†’â„ by ℎ(𝑥)=limğ‘›â†’âˆžâ„Žğ‘›(𝑥).(3.12) Let 𝑚 increase to infinity, then by (3.11) we have ∀𝜀>0∃𝑁𝜀∈ℕ∀𝑛∈𝑁𝜀||âˆ€ğ‘¥âˆˆğ¼âˆ¶â„Ž(𝑥)âˆ’â„Žğ‘›||(𝑥)≤𝜀𝜑(𝑥).(3.13)
Since 𝜑 is bounded on 𝐼, {â„Žğ‘›} converges uniformly to ℎ. Hence, ℎ is continuous and â„Žâˆˆğ‘‹.
Further, considering (3.10) and (3.13), then ∀𝜀>0âˆƒğ‘ğœ€âˆˆâ„•âˆ€ğ‘›âˆˆğ‘ğœ€î€·âˆ¶ğ‘‘â„Ž,â„Žğ‘›î€¸â‰¤ğœ€.(3.14) Then, the Cauchy sequence {â„Žğ‘›} converges to ℎ in (𝑋,𝑑). Hence, (𝑋,𝑑) is complete.
Now, define the operator 𝐽∶𝑋→𝑋 by (𝐽𝑓)(𝑥)=𝑦(𝑐)+𝑥𝑐𝐹(ğ‘ž(𝑥),𝑝(𝑥)𝑓(𝑥))𝑑𝑡𝑥∈𝐼(3.15) 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 ||||𝑓(𝑥)−𝑔(𝑥)≤𝐶𝑓𝑔𝜑(𝑥)(3.16) for any 𝑥∈𝐼. It then follows from (3.4), (3.6), (3.10), (3.15), and (3.16) that ||||≤||||(𝐽𝑓)(𝑥)−(𝐽𝑔)(𝑥)𝑥𝑐||||≤||||{𝐹(ğ‘ž(𝑥),𝑝(𝑥)𝑓(𝑥))−𝐹(ğ‘ž(𝑥),𝑝(𝑥)𝑔(𝑥))}𝑑𝑡𝑥𝑐||||||||||||𝐹(ğ‘ž(𝑥),𝑝(𝑥)𝑓(𝑥))−𝐹(ğ‘ž(𝑥),𝑝(𝑥)𝑔(𝑥))𝑑𝑡≤𝐿𝑥𝑐||𝑓||||||(𝑡)−𝑔(𝑡)𝑑𝑡≤𝐿𝐶𝑓𝑔||||𝑥𝑐||||𝜑(𝑡)𝑑𝑡≤𝐾𝐿𝐶𝑓𝑔𝜑(𝑥)(3.17) 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 ||𝐽𝑔0(𝑥)−𝑔0||=||||(𝑥)𝑦(𝑐)+𝑥𝑐𝐹𝑝(𝑡),𝑔0(𝑡)𝑑𝑡−𝑔0||||(𝑥)≤𝐶𝜑(𝑥)(3.18) 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 𝑑𝐽𝑔0,𝑔0<∞.(3.19) 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 ||𝑔(𝑥)−𝑔0||(𝑥)≤𝐶𝑔𝜑(𝑥)(3.20) 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 −𝜑(𝑥)â‰¤ğ‘¦î…ž(𝑥)−𝐹(ğ‘ž(𝑥),𝑝(𝑥)𝑦(𝑥))≤𝜑(𝑥)(3.21) for all 𝑥∈𝐼. If we integrate each term in the above inequality from 𝑐 to 𝑥, then we obtain ||||𝑦(𝑥)−𝑦(𝑐)−𝑥𝑐𝐹||||≤||||(ğ‘ž(𝑥),𝑝(𝑥)𝑦(𝑥))𝑑𝑡𝑥𝑐𝜑||||(𝑡)𝑑𝑡(3.22) for any 𝑥∈𝐼. Thus, by (3.6) and (3.15), we get ||||𝑦(𝑥)−𝐽𝑦(𝑥)≤𝐾𝜑(𝑥)(3.23) for each 𝑥∈𝐼, which implies that 𝑑(𝐽𝑦,𝑦)≤𝐾.(3.24) Finally, Theorem 1.1(c) and (3.24) implys that 𝑑𝐽𝑦,𝑦0≤1𝐾1−𝐾𝐿𝑑(𝐽𝑦,𝑦)≤1−𝐾𝐿,(3.25) 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 Theorem 3.1 is also true if 𝐼 is replaced by an unbounded interval such as (−∞,𝑏], ℝ, or [ğ‘Ž,∞).

Theorem 3.2. For given real numbers ğ‘Ž and 𝑏, let 𝐼 denote either (−∞,𝑏], ℝ, or [ğ‘Ž,∞). Set either 𝑐=ğ‘Ž for 𝐼=[ğ‘Ž,∞) 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 ||ğ‘¦î…ž||(𝑥)−𝐹(ğ‘ž(𝑥),𝑝(𝑥)𝑦(𝑥))≤𝜑(𝑥)(3.26) 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 ||𝑦(𝑥)−𝑦0||≤𝐾(𝑥)𝜑1−𝐾𝐿(𝑥)(3.27) for all 𝑥∈𝐼.

Proof. We prove for 𝐼=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=[ğ‘Ž,ğ‘Ž+𝑛] for 𝐼=[ğ‘Ž,∞)). By Theorem 3.1, there exists a unique continuous function 𝑦𝑛∶𝐼𝑛→ℝ such that 𝑦𝑛(𝑥)=𝑦(𝑐)+ğ‘¥ğ‘ğ¹î€·ğ‘ž(𝑥),𝑝(𝑥)𝑦𝑛||𝑦(𝑡)𝑑𝑡(3.28)𝑛||≤𝐾(𝑥)−𝑦(𝑥)1−𝐾𝐿𝜑(𝑥)(3.29) for all 𝑥∈𝐼𝑛. The uniqueness of 𝑦𝑛 implies that, if 𝑥∈𝐼𝑛, then 𝑦𝑛(𝑥)=𝑦𝑛+1(𝑥)=𝑦𝑛+2(𝑥)=⋯.(3.30) For any 𝑥∈ℝ, define 𝑛(𝑥)∈ℕ as 𝑛(𝑥)=min𝑛∈𝑁∣𝑥∈𝐼𝑛.(3.31) Moreover, define a function 𝑦0∶ℝ→ℝ by 𝑦0(𝑥)=𝑦𝑛(𝑥)(𝑥),(3.32) 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 that 𝑦0 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 𝑦0(𝑥)=𝑦𝑛(𝑥)(𝑥)=𝑦(𝑐)+ğ‘¥ğ‘ğ¹î€·ğ‘ž(𝑥),𝑝(𝑥)𝑦𝑛(𝑥)(𝑡)𝑑𝑡=𝑦(𝑐)+ğ‘¥ğ‘ğ¹î€·ğ‘ž(𝑥),𝑝(𝑥)𝑦0(𝑡)𝑑𝑡(3.33) since 𝑛(𝑡)≤𝑛(𝑥) for any 𝑡∈𝐼𝑛(𝑥). Then, from (3.30) and (3.32) we have 𝑦𝑛(𝑥)(𝑡)=𝑦𝑛(𝑡)(𝑡)=𝑦0(𝑡).(3.34) Since 𝑥∈𝐼𝑛(𝑥) for every 𝑥∈ℝ, by (3.29) and (3.32), we have ||𝑦0||=||𝑦(𝑥)−𝑦(𝑥)𝑛(𝑥)||≤𝐾(𝑥)−𝑦(𝑥)𝜑1−𝐾𝐿(𝑥)(3.35) for any 𝑥∈ℝ.
Finally, we show that 𝑦0 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 𝑦0(𝑥)=𝑦0∣𝐼𝑛(𝑥)(𝑥)=𝑧0∣𝐼𝑛(𝑥)(𝑥)=𝑧0(𝑥)(3.36) as required.

Corollary 3.3. Given 𝑐∈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 ||ğ‘¦î…ž||(𝑥)−𝐹(𝑥,𝑦(𝑥))≤𝜀(3.37) for all 𝑥∈𝐼 and for some 𝜀≥0, then there exists a unique continuous function 𝑦0∶𝐼→ℝ satisfying (3.7) and ||𝑦(𝑥)−𝑦0||≤𝑟(𝑥)1−𝑟𝐿𝜀(3.38) for any 𝑥∈𝐼.

Example 3.4. We choose positive constants 𝐾 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 ||ğ‘¦î…ž||(𝑥)−𝐿𝑦(𝑥)−𝑝(𝑥)≤𝑥+𝜀(3.39) 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 ||||𝑥𝑐𝜑||||=𝑥(𝑡)𝑑𝑡22+𝜀𝑥≤𝐾𝜑(𝑥)(3.40) 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 𝑦0(𝑥)=𝑦(0)+𝑥0||𝑦{𝐿𝑦(𝑡)−𝑝(𝑡)}𝑑𝑡0||≤𝐾(𝑥)−𝑦(𝑥)1−𝐾𝐿(𝑥+𝜀)(3.41) for any 𝑥∈𝐼.

Example 3.5. Let ğ‘Ž be a constant greater than 1 and choose a constant 𝐿 with 0<𝐿<lnğ‘Ž. Given an interval 𝐼=[0,1) and a polynomial 𝑝(𝑥), suppose 𝑦∶𝐼→ℝ is a continuously differentiable function satisfying ||ğ‘¦î…ž||(𝑥)−𝐿𝑦(𝑥)−𝑝(𝑥)â‰¤ğ‘Žğ‘¥(3.42) for all 𝑥∈𝐼. If we set 𝜑(𝑥)=ğ‘Žğ‘¥, then we have ||||𝑥𝑐𝜑||||≤1(𝑡)𝑑𝑡𝜑lnğ‘Ž(𝑥)(3.43) for any 𝑥∈𝐼. By Theorem 3.2, there exists a unique continuous function 𝑦0∶𝐼→ℝ with 𝑦0(𝑥)=𝑦(0)+𝑥0||𝑦{𝐿𝑦(𝑡)−𝑝(𝑡)}𝑑𝑡0||≤1(𝑥)−𝑦(𝑥)ğ‘Žlnğ‘Žâˆ’ğ¿ğ‘¥(3.44) for any 𝑥∈𝐼.

References

  1. S. M. Ulam, A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics, no. 8, Interscience, New York, NY, USA, 1960.
  2. T. M. Rassias, “On the stability of the linear mapping in Banach spaces,” Proceedings of the American Mathematical Society, vol. 72, no. 2, pp. 297–300, 1978. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  3. L. Cădariu and V. Radu, “On the stability of the Cauchy functional equation: a fixed point approach,” Grazer Mathematische Berichte, vol. 346, pp. 43–52, 2004. View at: Google Scholar | Zentralblatt MATH
  4. S.-M. Jung and Z.-H. Lee, “A fixed point approach to the stability of quadratic functional equation with involution,” Fixed Point Theory and Applications, vol. 2008, Article ID 732086, 11 pages, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  5. M. M. Pourpasha, J. M. Rassias, R. Saadati, and S. M. Vaezpour, “A fixed point approach to the stability of Pexider quadratic functional equation with involution,” journal of Inequalities and Applications, vol. 2010, Article ID 839639, 18 pages, 2010. View at: Google Scholar | Zentralblatt MATH
  6. G. Isac and T. M. Rassias, “Stability of ϕ-additive mappings: applications to nonlinear analysis,” International Journal of Mathematics and Mathematical Sciences, vol. 19, no. 2, pp. 219–228, 1996. View at: Publisher Site | Google Scholar
  7. J. B. Diaz and B. Margolis, “A fixed point theorem of the alternative, for contractions on a generalized complete metric space,” Bulletin of the American Mathematical Society, vol. 74, pp. 305–309, 1968. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  8. D. H. Hyers, G. Isac, and T. M. Rassias, Stability of Functional Equations in Several Variables, Progress in Nonlinear Differential Equations and their Applications, Birkhäuser Boston, Boston, Mass, USA, 1998.
  9. P. W. Cholewa, “The stability problem for a generalized Cauchy type functional equation,” Académie de la République Populaire Roumaine. Revue Roumaine de Mathématiques Pures et Appliquées, vol. 29, no. 6, pp. 457–460, 1984. View at: Google Scholar | Zentralblatt MATH
  10. S.-M. Jung and S. Min, “A fixed point approach to the stability of the functional equation f(x+y)=F[f(x),f(y)],” Fixed Point Theory and Applications, vol. 2009, Article ID 912046, 8 pages, 2009. View at: Publisher Site | Google Scholar
  11. S.-M. Jung, “Hyers-Ulam stability of linear differential equations of first order. II,” Applied Mathematics Letters, vol. 19, no. 9, pp. 854–858, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  12. S.-M. Jung, “Hyers-Ulam stability of linear differential equations of first order. III,” Journal of Mathematical Analysis and Applications, vol. 311, no. 1, pp. 139–146, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  13. C. Alsina and R. Ger, “On some inequalities and stability results related to the exponential function,” Journal of Inequalities and Applications, vol. 2, no. 4, pp. 373–380, 1998. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  14. S.-E. Takahasi, T. Miura, and S. Miyajima, “On the Hyers-Ulam stability of the Banach space-valued differential equation,” Bulletin of the Korean Mathematical Society, vol. 39, no. 2, pp. 309–315, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  15. S.-E. Takahasi, H. Takagi, T. Miura, and S. Miyajima, “The Hyers-Ulam stability constants of first order linear differential operators,” Journal of Mathematical Analysis and Applications, vol. 296, no. 2, pp. 403–409, 2004. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  16. T. Miura, “On the Hyers-Ulam stability of a differentiable map,” Scientiae Mathematicae Japonicae, vol. 55, no. 1, pp. 17–24, 2002. View at: Google Scholar
  17. T. Miura, S. Miyajima, and S.-E. Takahasi, “A characterization of Hyers-Ulam stability of first order linear differential operators,” Journal of Mathematical Analysis and Applications, vol. 286, no. 1, pp. 136–146, 2003. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  18. T. Miura, S.-e. Takahasi, and H. Choda, “On the Hyers-Ulam stability of real continuous function valued differentiable map,” Tokyo Journal of Mathematics, vol. 24, no. 2, pp. 467–476, 2001. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  19. L. Cădariu and V. Radu, “Fixed points and the stability of Jensen's functional equation,” Journal of Inequalities in Pure and Applied Mathematics, vol. 4, no. 1, article 4, 7 pages, 2003. View at: Google Scholar | Zentralblatt MATH
  20. L. Cădariu and V. Radu, “On the stability of the Cauchy functional equation: a fixed point approach,” in Iteration Theory (ECIT '02), vol. 346, pp. 43–52, Karl-Franzens-Universitaet Graz, Graz, Austria, 2004. View at: Google Scholar | Zentralblatt MATH

Copyright © 2011 M. M. Pourpasha et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


More related articles

 PDF Download Citation Citation
 Download other formatsMore
 Order printed copiesOrder
Views471
Downloads348
Citations

Related articles

We are committed to sharing findings related to COVID-19 as quickly as possible. We will be providing unlimited waivers of publication charges for accepted research articles as well as case reports and case series related to COVID-19. Review articles are excluded from this waiver policy. Sign up here as a reviewer to help fast-track new submissions.