Hyers-Ulam Stability of Nonhomogeneous Linear Differential Equations of Second Order

Abstract

The aim of this paper is to prove the stability in the sense of Hyers-Ulam of differential equation of second order 𝑦^′′+𝑝(𝑥)𝑦^′+𝑞(𝑥)𝑦+𝑟(𝑥)=0. That is, if 𝑓 is an approximate solution of the equation 𝑦^′′+𝑝(𝑥)𝑦^′+𝑞(𝑥)𝑦+𝑟(𝑥)=0, then there exists an exact solution of the equation near to 𝑓.

1. Introduction and Preliminaries

In 1940, Ulam [1] posed the following problem concerning the stability of functional equations: give conditions in order for a linear mapping near an approximately linear mapping to exist. The problem for the case of approximately additive mappings was solved by Hyers [2] when 𝐺₁ and 𝐺₂ are Banach spaces, and the result of Hyers was generalized by Rassias (see [3]). Since then, the stability problems of functional equations have been extensively investigated by several mathematicians (cf. [3–5]).

In connection with the stability of exponential functions, Alsina and Ger [6] remarked that the differential equation 𝑦^′=𝑦 has the Hyers-Ulam stability. More explicitly, they proved that if a differentiable function 𝑦∶𝐼→𝑅 satisfies |𝑦^′(𝑡)−𝑦(𝑡)|≤𝜀 for all 𝑡∈𝐼, then there exists a differentiable function 𝑔∶𝐼→𝑅 satisfying 𝑔^′(𝑡)=𝑔(𝑡) for any 𝑡∈𝐼 such that |𝑦(𝑡)−𝑔(𝑡)|≤3𝜀 for every𝑡∈𝐼.

The above result of Alsina and Ger has been generalized by Miura et al. [7], by Miura [8], and also by Takahasi et al. [9]. Indeed, they dealt with the Hyers-Ulam stability of the differential equation 𝑦^′(𝑡)=𝜆𝑦(𝑡), while Alsina and Ger investigated the differential equation 𝑦^′(𝑡)=𝑦(𝑡).

Furthermore, the result of Hyers-Ulam stability for first-order linear differential equations has been generalized by Miura et al. [10], by Takahasi et al. [11], and also by Jung [12]. They dealt with the nonhomogeneous linear differential equation of first order

𝑦^′+𝑝(𝑡)𝑦+𝑞(𝑡)=0. (1.1)

Jung [13] proved the generalized Hyers-Ulam stability of differential equations of the form

𝑡𝑦^′(𝑡)+𝛼𝑦(𝑡)+𝛽𝑡^𝑟𝑥₀=0 (1.2) and also applied this result to the investigation of the Hyers-Ulam stability of the differential equation

𝑡²𝑦^′′(𝑡)+𝛼𝑡𝑦^′(𝑡)+𝛽𝑦(𝑡)=0. (1.3) Recently, Wang et al. [14] discussed the Hyers-Ulam stability of the first-order nonhomogeneous linear differential equation

𝑝(𝑥)𝑦^′+𝑞(𝑥)𝑦+𝑟(𝑥)=0. (1.4) They proved the following theorem.

Theorem 1.1 (see [14]). Let 𝑝(𝑥),𝑞(𝑥), and 𝑟(𝑥) be continuous real functions on the interval 𝐼=(𝑎,𝑏) such that 𝑝(𝑥)≠0 and |𝑞(𝑥)|≥𝛿 for all 𝑥∈𝐼 and some 𝛿>0 independent of 𝑥∈𝐼. Then (1.4) has the Hyers-Ulam stability.

The aim of this paper is to investigate the Hyers-Ulam stability of the following linear differential equations of second order under some special conditions:

𝑦^′′+𝑝(𝑥)𝑦^′+𝑞(𝑥)𝑦+𝑟(𝑥)=0, (1.5) where 𝑦∈𝐶²(𝐼)=𝐶²(𝑎,𝑏),−∞<𝑎<𝑏<+∞.

For the sake of convenience, all the integrals in the rest of the work will be viewed as existing. We say that (1.5) has the Hyers-Ulam stability if there exists a constant 𝐾>0 with the following property: for every 𝜀>0,𝑦∈𝐶²(𝐼), if

|𝑦^′′+𝑝(𝑥)𝑦^′+𝑞(𝑥)𝑦+𝑟(𝑥)|≤𝜀, (1.6) then there exists some 𝑧∈𝐶²(𝐼) satisfying

𝑧^′′+𝑝(𝑥)𝑧^′+𝑞(𝑥)𝑧+𝑟(𝑥)=0 (1.7) such that |𝑦(𝑥)−𝑧(𝑥)|<𝐾𝜀. We call such 𝐾 a Hyers-Ulam stability constant for (1.5).

2. Main Results

Now, the main results of this work are given in the following theorems.

Theorem 2.1. If a twice continuously differentiable function 𝑦∶𝐼→𝑅 satisfies the differential inequality
|𝑦^′′+𝑝(𝑥)𝑦^′+𝑞(𝑥)𝑦+𝑟(𝑥)|≤𝜀 (2.1) for all 𝑡∈𝐼 and for some 𝜀>0, and the Riccati equation 𝑢^′(𝑥)+𝑝(𝑥)𝑢(𝑥)−𝑢²(𝑥)=𝑞(𝑥) has a particular solution, then there exists a solution 𝑣∶𝐼→𝑅 of (1.5) such that |𝑦(𝑥)−𝑣(𝑥)|≤𝐾𝜀, (2.2) where 𝐾>0 is a constant.

Proof. Let 𝜀>0 and 𝑦∶𝐼→𝑅 be a continuously differentiable function such that |𝑦^′′+𝑝(𝑥)𝑦^′+𝑞(𝑥)𝑦+𝑟(𝑥)|≤𝜀. (2.3)
We will show that there exists a constant 𝐾 independent of 𝜀 and 𝑣 such that |𝑦−𝑣|<𝐾𝜀 for some 𝑣∈𝐶²(𝐼) satisfying 𝑣^′′+𝑝(𝑥)𝑣^′+𝑞(𝑥)𝑣+𝑟(𝑥)=0.
Assume that 𝑐(𝑥) is a particular solution of Riccati equation 𝑢^′(𝑥)+𝑝(𝑥)𝑢(𝑥)−𝑢²(𝑥)=𝑞(𝑥); if we set
𝑔(𝑥)=𝑦^′(𝑥)+𝑐(𝑥)𝑦(𝑥),𝑑(𝑥)=𝑝(𝑥)−𝑐(𝑥), (2.4) then 𝑔^′(𝑥)=𝑦^′′(𝑥)+𝑐(𝑥)𝑦^′(𝑥)+𝑐^′(𝑥)𝑦(𝑥), (2.5) thus |𝑔^′(𝑥)+𝑑(𝑥)𝑔(𝑥)+𝑟(𝑥)|=|𝑦^′′(𝑥)+(𝑐(𝑥)+𝑑(𝑥))𝑦^′(𝑥)+(𝑐^′(𝑥)+𝑑(𝑥)𝑐(𝑥))𝑦(𝑥)+𝑟(𝑥)|=|𝑦^′′+𝑝(𝑥)𝑦^′+𝑞(𝑥)𝑦+𝑟(𝑥)|≤𝜀. (2.6)
Using the similar technique in [14], we can prove
𝑤(𝑥)=exp{^𝑥_𝑎(−𝑑(𝑠))𝑑𝑠}[(𝑔(𝑏)−𝜀)exp{−^𝑏_𝑎−(−𝑑(𝑥))𝑑𝑠}_𝑏𝑥−𝑟(𝑠)exp{−^𝑠_𝑎(−𝑑(𝑡))𝑑𝑡}𝑑𝑠]=exp{^𝑥_𝑎(−𝑑(𝑠))𝑑𝑠}[(𝑔(𝑏)−𝜀)exp{^𝑏_𝑎(𝑑(𝑥))𝑑𝑠}+^𝑏_𝑥𝑟(𝑠)exp{^𝑠_𝑎(𝑑(𝑡))𝑑𝑡}𝑑𝑠] (2.7) satisfying 𝑤^′(𝑥)+𝑑(𝑥)𝑤(𝑥)+𝑟(𝑥)=0, (2.8) and there exists an 𝐿>0 such that |𝑔(𝑥)−𝑤(𝑥)|≤𝐿𝜀. (2.9)
By 𝑔(𝑥)=𝑦^′(𝑥)+𝑐(𝑥)𝑦(𝑥), we get
|𝑦^′(𝑥)+𝑐(𝑥)𝑦(𝑥)−𝑤(𝑥)|≤𝜀. (2.10)
Using the same technique as above, we know that
𝑣(𝑥)=exp{^𝑥_𝑎(−𝑐(𝑠))𝑑𝑠}[(𝑦(𝑏)−𝜀)exp{−^𝑏_𝑎−(−𝑐(𝑥))𝑑𝑠}_𝑏𝑥𝑤(𝑠)exp{−^𝑠_𝑎(−𝑐(𝑡))𝑑𝑡}𝑑𝑠]=exp{^𝑥_𝑎(−𝑐(𝑠))𝑑𝑠}[(𝑦(𝑏)−𝜀)exp{^𝑏_𝑎(𝑐(𝑠))𝑑𝑠}−^𝑏_𝑥𝑤(𝑠)exp{^𝑠_𝑎(𝑐(𝑡))𝑑𝑡}𝑑𝑠] (2.11) satisfying 𝑣^′(𝑥)+𝑐(𝑥)𝑣(𝑥)−𝑤(𝑥)=0, (2.12) and there exists a 𝐾>0 such that |𝑦(𝑥)−𝑣(𝑥)|≤𝐾𝜀. (2.13) The desired conclusion is proved.

Theorem 2.2. Let 𝑝(𝑥),𝑞(𝑥), and 𝑟(𝑥) be continuous real functions on the interval 𝐼=(𝑎,𝑏) such that 𝑝(𝑥)≠0 and 𝑦₀(𝑥) is a nonzero bounded particular solution 𝑝(𝑥)𝑦^′′+𝑞(𝑥)𝑦^′+𝑟(𝑥)𝑦=0. If 𝑦∶𝐼→𝑅 is a twice continuously differentiable function, which satisfies the differential inequality |𝑝(𝑥)𝑦^′′+𝑞(𝑥)𝑦^′+𝑟(𝑥)𝑦|≤𝜀 (2.14) for all 𝑡∈𝐼 and for some 𝜀>0, then there exists a solution 𝑣∶𝐼→𝑅 such that |𝑦(𝑥)−𝑣(𝑥)|≤𝐾𝜀, (2.15) where 𝐾>0 is a constant, and 𝑣 satisfies 𝑝(𝑥)𝑣^′′+𝑞(𝑥)𝑣^′+𝑟(𝑥)𝑣=0.

Proof. Setting 𝑦(𝑥)=𝑦₀(𝑥)∫^𝑥_𝑎𝑧(𝑠)𝑑𝑠, we obtain
𝑦^′(𝑥)=𝑦^′₀(𝑥)^𝑥_𝑎𝑧(𝑠)𝑑𝑠+𝑦₀(𝑥)𝑧(𝑥) (2.16) and also 𝑦^′′(𝑥)=𝑦₀^′′(𝑥)^𝑥_𝑎𝑧(𝑠)𝑑𝑠+2𝑦^′₀(𝑥)𝑧(𝑥)+𝑦₀(𝑥)𝑧^′(𝑥) (2.17)
By a simple calculation, we see that
|𝑝(𝑥)𝑦₀(𝑥)𝑧^′(𝑥)+[2𝑝(𝑥)𝑦^′₀(𝑥)+𝑞(𝑥)𝑦₀(𝑥)]𝑧(𝑥)|=|𝑝(𝑥)𝑦^′′(𝑥)+𝑞(𝑥)𝑦^′(𝑥)+𝑟(𝑥)𝑦(𝑥)|≤𝜀. (2.18)
Without loss of generality, we may assume that 𝑝(𝑥)𝑦₀(𝑥)>0. Using the similar technique in [14], we know that
𝑧₁(𝑥)=exp{−^𝑥_𝑎2𝑝(𝑠)𝑦^′₀(𝑠)+𝑞(𝑠)𝑦₀(𝑠)𝑑𝑠}[(𝑧(𝑏)−𝜀)exp{^𝑏_𝑎2𝑝(𝑠)𝑦^′₀(𝑠)+𝑞(𝑠)𝑦₀(𝑠)𝑑𝑠}] (2.19) satisfies 𝑝(𝑥)𝑦₁(𝑥)𝑧^′₁(𝑥)+[2𝑝(𝑥)𝑦^′₁(𝑥)+𝑞(𝑥)𝑦₁(𝑥)]𝑧₁(𝑥)=0 (2.20) and also |𝑧(𝑥)−𝑧₁(𝑥)|≤𝐿𝜀 (2.21) for some 𝐿>0.
From the inequalities −𝐿𝜀≤𝑧(𝑥)−𝑧₁(𝑥)≤𝐿𝜀 and the similar technique in [14], we further get that
𝑧₂(𝑥)=(𝑦(𝑏)𝑦₁(𝑏)−𝜀)−^𝑏_𝑥𝑧₁(𝑠)𝑑𝑠 (2.22) satisfies 𝑧₂(𝑥)−𝑧₁(𝑥)=0 (2.23) and also |𝑧(𝑥)−𝑧₂(𝑥)|≤𝑄𝜀 (2.24) for some 𝑄>0.
Consequently, we have
|𝑦(𝑥)−𝑧₂(𝑥)𝑦₀(𝑥)|≤𝑀𝜀 (2.25) for some positive constant 𝑀.
Define 𝑣(𝑥)=𝑧₂(𝑥)𝑦₀(𝑥). It then follows from the above inequality that |𝑧(𝑥)−𝑣(𝑥)|≤𝑀𝜀 holds for every 𝑥∈𝐼. We can easily verify that 𝑣 satisfies 𝑝(𝑥)𝑣^′′+𝑞(𝑥)𝑣^′+𝑟(𝑥)𝑣=0. This completes the proof of our theorem.

We can prove the following corollaries by using an analogous argument. Hence, we omit the proofs.

Corollary 2.3. Let 𝑝(𝑥),𝑞(𝑥), and 𝑟(𝑥) be continuous real functions on the interval 𝐼=(𝑎,𝑏) such that 𝑝(𝑥)≠0 and 𝑟²+𝑝(𝑥)𝑟+𝑞(𝑥)=0. If 𝑦∶𝐼→𝑅 is a twice continuously differentiable function, which satisfies the differential inequality
|𝑝(𝑥)𝑦^′′+𝑞(𝑥)𝑦^′+𝑟(𝑥)𝑦|≤𝜀 (2.26) for all 𝑡∈𝐼 and for some 𝜀>0, then there exists a solution 𝑣∶𝐼→𝑅 such that |𝑦(𝑥)−𝑣(𝑥)|≤𝐾𝜀, (2.27) where 𝐾>0 is a constant, and 𝑣 satisfies 𝑝(𝑥)𝑣^′′+𝑞(𝑥)𝑣^′+𝑟(𝑥)𝑣=0.

Corollary 2.4. Let 𝑝(𝑥),𝑞(𝑥),𝑟(𝑥), and 𝑠(𝑥) be continuous real functions on the interval 𝐼=(𝑎,𝑏) such that 𝑝(𝑥)≠0 and 𝑦₀(𝑥) is a nonzero bounded particular solution 𝑝(𝑥)𝑦^′′′+𝑞(𝑥)𝑦^′′+𝑟(𝑥)𝑦^′+𝑠(𝑥)𝑦=0. If 𝑦∶𝐼→𝑅 is a twice continuously differentiable function, which satisfies the differential inequality
|𝑝(𝑥)𝑦^′′′+𝑞(𝑥)𝑦^′′+𝑟(𝑥)𝑦^′+𝑠(𝑥)𝑦|≤𝜀 (2.28) for all 𝑡∈𝐼 and for some 𝜀>0, then there exists a solution 𝑣∶𝐼→𝑅 such that |𝑦(𝑥)−𝑣(𝑥)|≤𝐾𝜀, (2.29) where 𝐾>0 is a constant, and 𝑣 satisfies 𝑝(𝑥)𝑣^′′′+𝑞(𝑥)𝑣^′′+𝑟(𝑥)𝑣^′+𝑠(𝑥)𝑣=0.

Acknowledgments

The work of this paper was supported by the National Natural Science Foundation of China (10871213). The authors would like to thank the referee for a number of valuable suggestions regarding a previous version of this paper.