A Fixed Point Approach to the Stability of an Integral Equation Related to the Wave Equation

Abstract

We will apply the fixed point method for proving the generalized Hyers-Ulam stability of the integral equation (1/2𝑐)∫^{𝑥+𝑐𝑡}_{𝑥−𝑐𝑡}𝑢(𝜏,𝑡₀)𝑑𝜏=𝑢(𝑥,𝑡) which is strongly related to the wave equation.

1. Introduction

In 1940, Ulam [1] gave a wide ranging talk before the mathematics club of the University of Wisconsin in which he discussed a number of important unsolved problems. Among those was the question concerning the stability of group homomorphisms:  Let 𝐺₁ be a group and let 𝐺₂ be a metric group with the metric 𝑑(⋅,⋅). Given 𝜀>0, does there exist a 𝛿>0 such that if a function ℎ∶𝐺₁→𝐺₂ satisfies the inequality 𝑑(ℎ(𝑥𝑦),ℎ(𝑥)ℎ(𝑦))<𝛿 for all 𝑥,𝑦∈𝐺₁, then there exists a homomorphism 𝐻∶𝐺₁→𝐺₂ with 𝑑(ℎ(𝑥),𝐻(𝑥))<𝜀 for all 𝑥∈𝐺₁?

The case of approximately additive functions was solved by Hyers [2] under the assumption that 𝐺₁ and 𝐺₂ are the Banach spaces. Indeed, he proved that each solution of the inequality ‖𝑓(𝑥+𝑦)−𝑓(𝑥)−𝑓(𝑦)‖≤𝜀, for all 𝑥 and 𝑦, can be approximated by an exact solution, say an additive function. In this case, the Cauchy additive functional equation, 𝑓(𝑥+𝑦)=𝑓(𝑥)+𝑓(𝑦), is said to have the Hyers-Ulam stability.

Rassias [3] attempted to weaken the condition for the bound of the norm of the Cauchy difference as follows: 𝑓𝑥+𝑦−𝑓(𝑥)−𝑓𝑦≤𝜀‖𝑥‖^𝑝+𝑦^𝑝 (1) and proved the Hyers theorem. That is, Rassias proved the generalized Hyers-Ulam stability (or Hyers-Ulam-Rassias stability) of the Cauchy additive functional equation. (Aoki [4] has provided a proof of a special case of Rassias’ theorem just for the stability of the additive function. Aoki did not prove the stability of the linear function, which was implied by Rassias’ theorem.) Since then, the stability of several functional equations has been extensively investigated [5–12].

The terminologies generalized Hyers-Ulam stability, Hyers-Ulam-Rassias stability, and Hyers-Ulam stability can also be applied to the case of other functional equations, differential equations, and various integral equations.

Let 𝑐 and 𝑡₀ be fixed real numbers with 𝑐>0. For any differentiable function ℎ∶ℝ×ℝ→ℂ, the function defined as 𝑢()1𝑥,𝑡∶=2𝑐^{𝑥+𝑐𝑡}_{𝑥−𝑐𝑡}ℎ𝜏,𝑡₀𝑑𝜏 (2) is a solution of the wave equation 𝑢_𝑡𝑡(𝑥,𝑡)=𝑐²𝑢_𝑥𝑥(𝑥,𝑡), (3) as we see 𝑢_𝑡()=1𝑥,𝑡𝜕2𝑐𝜕𝑡^{𝑥+𝑐𝑡}_{𝑥−𝑐𝑡}ℎ𝜏,𝑡₀=1𝑑𝜏2ℎ𝑥+𝑐𝑡,𝑡₀+12ℎ𝑥−𝑐𝑡,𝑡₀,𝑢_𝑡𝑡()=𝑐𝑥,𝑡2ℎ_𝑥𝑥+𝑐𝑡,𝑡₀−𝑐2ℎ_𝑥𝑥−𝑐𝑡,𝑡₀,𝑢_𝑥()=1𝑥,𝑡ℎ2𝑐𝑥+𝑐𝑡,𝑡₀−1ℎ2𝑐𝑥−𝑐𝑡,𝑡₀,𝑢_𝑥𝑥()=1𝑥,𝑡ℎ2𝑐^𝑥𝑥+𝑐𝑡,𝑡₀−1ℎ2𝑐^𝑥𝑥−𝑐𝑡,𝑡₀, (4) from which we know that 𝑢(𝑥,𝑡) satisfies the wave equation (3).

Conversely, we know that every solution 𝑢∶ℝ×ℝ→ℂ of the wave equation (3) can be expressed by 𝑢(𝑥,𝑡)=𝑓(𝑥+𝑐𝑡)+𝑔(𝑥−𝑐𝑡), (5) where 𝑓,𝑔∶ℝ×ℝ→ℂ are arbitrary twice differentiable functions. If these 𝑓(𝑥,𝑡) and 𝑔(𝑥,𝑡) satisfy 12𝑐^{𝑥+𝑐𝑡}_{𝑥−𝑐𝑡}𝑓(𝜏)(),1𝑑𝜏=𝑓𝑥+𝑐𝑡2𝑐^{𝑥+𝑐𝑡}_{𝑥−𝑐𝑡}𝑔(𝜏)𝑑𝜏=𝑔(𝑥−𝑐𝑡) (6) for all 𝑥,𝑡∈ℝ, then 𝑢(𝑥,𝑡) expressed by (5) satisfies the integral equation (7). These facts imply that the integral equation (7) is strongly connected with the wave equation (3).

Cădariu and Radu [13] applied the fixed point method to the investigation of the Cauchy additive functional equation. Using such a clever idea, they could present another proof for the Hyers-Ulam stability of that equation [14–19].

In this paper, we introduce the integral equation: 12𝑐^{𝑥+𝑐𝑡}_{𝑥−𝑐𝑡}𝑢𝜏,𝑡₀𝑑𝜏=𝑢(𝑥,𝑡), (7) which may be considered as a special form of (2), and prove the generalized Hyers-Ulam stability of the integral equation (7) by using ideas from [13, 15, 19, 20]. More precisely, assume that 𝜑(𝑥,𝑡) is a given function and 𝑢(𝑥,𝑡) is an arbitrary and continuous function which satisfies the integral inequality: 12𝑐^{𝑥+𝑐𝑡}_{𝑥−𝑐𝑡}𝑢𝜏,𝑡₀𝑑𝜏−𝑢(𝑥,𝑡)≤𝜑(𝑥,𝑡). (8) If there exist a function 𝑢₀(𝑥,𝑡) and a constant 𝐶>0 such that 12𝑐^{𝑥+𝑐𝑡}_{𝑥−𝑐𝑡}𝑢₀𝜏,𝑡₀𝑑𝜏=𝑢₀(),𝑢()𝑥,𝑡𝑥,𝑡−𝑢₀(𝑥,𝑡)≤𝐶𝜑(𝑥,𝑡), (9) then we say that the integral equation (7) has the generalized Hyers-Ulam stability.

2. Preliminaries

For a nonempty set 𝑋, we introduce the definition of the generalized metric on 𝑋. A function 𝑑∶𝑋×𝑋→[0,∞] is called a generalized metric on 𝑋 if and only if 𝑑 satisfies (𝑀₁)𝑑(𝑥,𝑦)=0 if and only if 𝑥=𝑦; (𝑀₂)𝑑(𝑥,𝑦)=𝑑(𝑦,𝑥) for all 𝑥,𝑦∈𝑋; (𝑀₃)𝑑(𝑥,𝑧)≤𝑑(𝑥,𝑦)+𝑑(𝑦,𝑧) for all 𝑥,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 [21]. This theorem will play an important role in proving our main theorems.

Theorem 1. Let (𝑋,𝑑) be a generalized complete metric space. Assume that Λ∶𝑋→𝑋 is a strictly contractive operator with the Lipschitz constant 𝐿<1. If there exists a nonnegative integer 𝑘 such that 𝑑(Λ^𝑘+1𝑥,Λ^𝑘𝑥)<∞ for some 𝑥∈𝑋, then the following are true: (a)the sequence {Λ^𝑛𝑥} converges to a fixed point 𝑥^∗ of Λ; (b)𝑥^∗ is the unique fixed point of Λ in 𝑋^∗=𝑦∈𝑋∣𝑑Λ^𝑘𝑥,𝑦<∞; (10)(c)if 𝑦∈𝑋^∗, then 𝑑𝑦,𝑥^∗≤1𝑑.1−𝐿Λ𝑦,𝑦 (11)

3. The Generalized Hyers-Ulam Stability

In the following theorem, for given real numbers 𝑎, 𝑏, 𝑐, and 𝑡₀ satisfying 𝑐>0, 𝑡₀>0, and 𝑎+𝑐𝑡₀<𝑏−𝑐𝑡₀, let 𝐼∶=[𝑎,𝑏], 𝑇∶=(0,𝑡₀], and 𝐼₀∶=[𝑎+𝑐𝑡₀,𝑏−𝑐𝑡₀] be finite intervals. Assume that 𝐿 and 𝑀 are positive constants with 0<𝐿<1. Moreover, let 𝜑∶𝐼×𝑇→(0,1] be a continuous function satisfying 12𝑐^{𝑥+𝑐𝑡}_{𝑥−𝑐𝑡}𝜑𝜏,𝑡₀𝑑𝜏≤𝐿𝜑(𝑥,𝑡) (12) for all 𝑥∈𝐼₀ and 𝑡∈𝑇.

We denote by 𝑋 the set of all functions 𝑓∶𝐼×𝑇→ℂ with the following properties: (a)𝑓(𝑥,𝑡) is continuous for all 𝑥∈𝐼₀ and 𝑡∈𝑇; (b)𝑓(𝑥,𝑡)=0 for all 𝑥∈𝐼⧵𝐼₀ and 𝑡∈𝑇; (c)|𝑓(𝑥,𝑡)|≤𝑀𝜑(𝑥,𝑡) for all 𝑥∈𝐼₀ and 𝑡∈𝑇. Moreover, we introduce a generalized metric on 𝑋 as follows: 𝑑[]∣𝑓()(()𝑓,𝑔∶=inf𝐶∈0,∞𝑥,𝑡−𝑔𝑥,𝑡)≤𝐶𝜑𝑥,𝑡∀𝑥∈𝐼₀,𝑡∈𝑇. (13)

Theorem 2. If a function 𝑢∈𝑋 satisfies the integral inequality: 12𝑐^{𝑥+𝑐𝑡}_{𝑥−𝑐𝑡}𝑢𝜏,𝑡₀𝑑𝜏−𝑢(𝑥,𝑡)≤𝜑(𝑥,𝑡) (14) for all 𝑥∈𝐼₀ and 𝑡∈𝑇, then there exists a unique function 𝑢₀∈𝑋 which satisfies 12𝑐^{𝑥+𝑐𝑡}_{𝑥−𝑐𝑡}𝑢₀𝜏,𝑡₀𝑑𝜏=𝑢₀(),𝑢()𝑥,𝑡𝑥,𝑡−𝑢₀(≤1𝑥,𝑡)𝜑()1−𝐿𝑥,𝑡 (15)(16) for all 𝑥∈𝐼₀ and 𝑡∈𝑇.

Proof. First, we show that (𝑋,𝑑) is complete. Let {ℎ_𝑛} be a Cauchy sequence in (𝑋,𝑑). Then, for any 𝜀>0 there exists an integer 𝑁_𝜀>0 such that 𝑑(ℎ_𝑚,ℎ_𝑛)≤𝜀 for all 𝑚,𝑛≥𝑁_𝜀. In view of (13), we have ∀𝜀>0,∃𝑁_𝜀∈ℕ,∀𝑚,𝑛≥𝑁_𝜀,∀𝑥∈𝐼₀ℎ,∀𝑡∈𝑇:_𝑚(𝑥,𝑡)−ℎ_𝑛(𝑥,𝑡)≤𝜀𝜑(𝑥,𝑡). (17) If 𝑥 and 𝑡 are fixed, (17) implies that {ℎ_𝑛(𝑥,𝑡)} is a Cauchy sequence in ℂ. Since ℂ is complete, {ℎ_𝑛(𝑥,𝑡)} converges for any 𝑥∈𝐼₀ and 𝑡∈𝑇. Thus, considering (b), we can define a function ℎ∶𝐼×𝑇→ℂ by ℎ(𝑥,𝑡)∶=lim_𝑛→∞ℎ_𝑛(𝑥,𝑡),(𝑥∈𝐼,𝑡∈𝑇). (18) Since 𝜑 is bounded on 𝐼₀×𝑇, (17) implies that {ℎ_𝑛|_𝐼0×𝑇} converges uniformly to ℎ|_𝐼0×𝑇 in the usual topology of ℂ. Hence, ℎ is continuous and |ℎ| is bounded on 𝐼₀×𝑇 with an upper bound 𝑀𝜑(𝑥,𝑡); that is, ℎ∈𝑋. (It has not been proved yet that {ℎ_𝑛} converges to ℎ in (𝑋,𝑑).)
If we let 𝑚 increase to infinity, it follows from (17) that ∀𝜀>0,∃𝑁_𝜀∈ℕ,∀𝑛≥𝑁_𝜀,∀𝑥∈𝐼₀ℎ(),∀𝑡∈𝑇:𝑥,𝑡−ℎ_𝑛(𝑥,𝑡)≤𝜀𝜑(𝑥,𝑡). (19) By considering (13), we get ∀𝜀>0,∃𝑁_𝜀∈ℕ,∀𝑛≥𝑁_𝜀∶𝑑ℎ,ℎ_𝑛≤𝜀. (20) This implies that the Cauchy sequence {ℎ_𝑛} converges to ℎ in (𝑋,𝑑). Hence, (𝑋,𝑑) is complete.
We now define an operator Λ∶𝑋→𝑋 by ()1Λℎ)(𝑥,𝑡∶=∫2𝑐^{𝑥+𝑐𝑡𝑥−𝑐𝑡}ℎ𝜏,𝑡₀𝑑𝜏𝑥∈𝐼₀,0(),𝑡∈𝑇otherwise (21) for all ℎ∈𝑋. Then, according to the fundamental theorem of calculus, Λℎ is continuous on 𝐼₀×𝑇. Furthermore, it follows from (12), (c), and (21) that ≤1|(Λℎ)(𝑥,𝑡)|2𝑐^{𝑥+𝑐𝑡}_{𝑥−𝑐𝑡}ℎ𝜏,𝑡₀≤1𝑑𝜏2𝑐^{𝑥+𝑐𝑡}_{𝑥−𝑐𝑡}𝑀𝜑𝜏,𝑡₀()()𝑑𝜏≤𝑀𝐿𝜑𝑥,𝑡<𝑀𝜑𝑥,𝑡 (22) for any 𝑥∈𝐼₀ and 𝑡∈𝑇. Hence, we conclude that Λℎ∈𝑋.
We assert that Λ is strictly contractive on 𝑋. Given any 𝑓,𝑔∈𝑋, let 𝐶_𝑓𝑔∈[0,∞] be an arbitrary constant with 𝑑(𝑓,𝑔)≤𝐶_𝑓𝑔. That is, 𝑓(𝑥,𝑡)−𝑔(𝑥,𝑡)≤𝐶_𝑓𝑔𝜑(𝑥,𝑡) (23) for all 𝑥∈𝐼₀ and 𝑡∈𝑇. Then, it follows from (12), (21), and (23) that ()−(=1Λ𝑓𝑥,𝑡Λ𝑔𝑥,𝑡)2𝑐^{𝑥+𝑐𝑡}_{𝑥−𝑐𝑡}𝑓𝜏,𝑡₀−𝑔𝜏,𝑡₀≤1𝑑𝜏2𝑐^{𝑥+𝑐𝑡}_{𝑥−𝑐𝑡}𝑓𝜏,𝑡₀−𝑔𝜏,𝑡₀≤𝐶𝑑𝜏_𝑓𝑔2𝑐^{𝑥+𝑐𝑡}_{𝑥−𝑐𝑡}𝜑𝜏,𝑡₀𝑑𝜏≤𝐿𝐶_𝑓𝑔𝜑(𝑥,𝑡) (24) for all 𝑥∈𝐼₀ and 𝑡∈𝑇. That is, 𝑑(Λ𝑓,Λ𝑔)≤𝐿𝐶_𝑓𝑔. Hence, we may conclude that 𝑑(Λ𝑓,Λ𝑔)≤𝐿𝑑(𝑓,𝑔) for any 𝑓,𝑔∈𝑋 and we note that 0<𝐿<1.
We prove that the distance between the first two successive approximations of Λ is finite. Let ℎ₀∈𝑋 be given. By (b), (c), and (13) and from the fact that Λℎ₀∈𝑋, we have Λℎ₀(𝑥,𝑡)−ℎ₀(𝑥,𝑡)≤Λℎ₀(𝑥,𝑡)+ℎ₀(()𝑥,𝑡)≤2𝑀𝜑𝑥,𝑡 (25) for any 𝑥∈𝐼₀ and 𝑡∈𝑇. Thus, (13) implies that 𝑑Λℎ₀,ℎ₀≤2𝑀<∞. (26)
Therefore, it follows from Theorem 1(a) that there exists a 𝑢₀∈𝑋 such that Λ^𝑛ℎ₀→𝑢₀ in (𝑋,𝑑) and Λ𝑢₀=𝑢₀.
In view of (c) and (13), it is obvious that {𝑓∈𝑋∣𝑑(ℎ₀,𝑓)<∞}=𝑋, where ℎ₀ was chosen with the property (26). Now, Theorem 1(b) implies that 𝑢₀ is the unique element of 𝑋 which satisfies (Λ𝑢₀)(𝑥,𝑡)=𝑢₀(𝑥,𝑡) for any 𝑥∈𝐼₀ and 𝑡∈𝑇.
Finally, Theorem 1(c), together with (13) and (14), implies that 𝑑𝑢,𝑢₀≤1𝑑()≤11−𝐿Λ𝑢,𝑢,1−𝐿 (27) since (14) means that 𝑑(Λ𝑢,𝑢)≤1. In view of (13), we can conclude that (16) holds for all 𝑥∈𝐼₀ and 𝑡∈𝑇.

Remark 3. Even though condition (12) seems to be strict, the condition can be satisfied provided that 𝑎 and 𝑏 are chosen so that |𝑏−𝑎| is small enough and 𝑐 is a large number.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The author would like to express his cordial thanks to the referees for useful remarks. This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (no. 2013R1A1A2005557).