The Stability of Some Differential Equations

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+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)].

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(𝐽^𝑘+1𝑓,𝐽^𝑘𝑓)<∞ for some 𝑓∈𝑋, then the followings are true: (a) the sequence {𝐽^𝑛𝑓} converges to a fixed point 𝑓^∗ of 𝐽,(b)𝑓^∗ is the unique fixed point of 𝐽 in𝑋^∗=𝑔∈𝑋∶𝑑𝐽^𝑘𝑓,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 form𝑓(𝐺(𝑥,𝑦))=𝐻[𝑓(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 equation𝑓(𝑥+𝑦)=𝐹[𝑓(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 equation𝑓(𝑥+𝜎(𝑦))=𝐹[𝑓(𝑥),𝑓(𝑦)](2.3) for a class of functions of a vector space into a Banach space where 𝜎 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(𝜎(𝑥)))−(𝑣(𝑥),𝑣(𝜎(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,𝜎(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+𝜎(𝑦))−𝐹[𝑓(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^∗(𝑥)‖≤11−𝜅‖𝐹‖𝜑(𝑥,𝑥).(2.8)

Proof. First, we denote by 𝑋 the set of all functions ℎ∶𝑋→𝑌 and by d the generalized metric on 𝑋 defined as 𝑑(𝑔,ℎ)=inf𝐶∈[0,∞)∶‖𝑔(𝑥)−ℎ(𝑥)‖≤𝐶𝑀^(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 (𝐽ℎ)(𝑥)=𝐹ℎ𝑥2,ℎ𝜎𝑥2(2.10) 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, ‖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 ‖𝐽𝑔(𝑥)−𝐽ℎ(𝑥)‖≤‖‖‖𝐹𝑔𝑥≤‖𝐹‖‖‖‖𝑔2𝑥,𝑔𝜎𝑥2−𝐹ℎ𝑥2,ℎ𝜎𝑥2‖‖‖≤‖𝐹‖𝜅‖‖‖𝑔𝑥2,𝑔𝜎𝑥2−ℎ𝑥2,ℎ𝜎𝑥2‖‖‖≤‖𝐹‖𝜅𝐶𝜑2𝑥−ℎ𝑥2‖‖‖≤‖𝐹‖𝜅𝐶𝜑(𝑥2,𝑦,)𝑥2(2.12) 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 that 𝑑(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 ‖‖‖𝑓𝑥2+𝜎𝜎𝑥2−𝐹𝑓𝑥2‖𝑓(,𝑥𝑓)−𝜎(𝐽𝑓𝑥2)(𝑥)‖‖‖‖≤≤𝜑𝜑𝑥𝑥2,𝜎𝑥2≤𝜑(𝑥2,𝑥,𝜎)𝑥2(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 lim_𝑛→∞𝑑𝐽^𝑛𝑓,𝑓^∗=0.(2.15) From Theorem 1.1(c), we get 𝑑𝐽^𝑛𝑓,𝑓^∗≤11−𝜅‖𝐹‖𝑑(𝐽𝑓,𝑓)≤1−𝜅1‖𝐹‖^(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 ‖‖(𝐽^𝑛𝑓)(𝑥+𝜎(𝑦))−𝐹(𝐽^𝑛𝑓)(𝑥),(𝐽^𝑛𝑓)(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 ‖(=𝐽𝑓‖‖‖)𝐹(𝑥𝑓+𝜎(𝑥𝑦+))𝜎−(𝐹𝑦)[(𝐽𝑓)(𝑥),(𝐽𝑓)(𝑦)]‖−𝐹𝐹𝑓𝑥2,𝑓𝜎𝑥+2𝜎(𝑦)≤‖𝐹‖‖‖‖𝑓2𝑥+,𝑓𝜎(𝑦𝜎)𝑥2,𝐹𝑓2𝑦,𝑓𝜎2𝑦‖‖‖−𝐹𝑓𝑥2,𝑓𝜎𝑥+2𝜎(𝑦)≤‖𝐹‖𝜅‖‖‖𝑓2𝑥,𝑓+𝜎𝜎(𝑦)𝑥2,𝐹𝑓2𝑦,𝑓𝜎2𝑦‖‖‖≤‖𝐹‖𝜅𝜑𝑥2−𝐹𝑓𝑥2,𝑓2𝑦‖‖‖≤‖𝐹‖𝜅𝜑(𝑥2,𝑥,2𝑦)(2.18) for any 𝑥,𝑦∈𝑋. Then, it follows from (2.4), (2.5), (2.6), (2.10), and (2.17) that ‖‖=𝐽‖‖‖𝐹^𝑛+1𝑓𝐽(𝑥+𝜎(𝑦))−𝐹𝐽^𝑛+1𝑓(𝑥),𝐽^𝑛+1𝑓(𝑦)_𝑛‖‖𝑓𝑥+𝜎(𝑦)2,𝐽^𝑛𝑓𝜎𝑥+𝜎(𝑦)−𝐹𝐹𝐽_𝑛2𝑓𝑥2,𝐽^𝑛𝑓𝜎𝑥2,𝐹𝐽^𝑛𝑓𝑦2,𝐽^𝑛𝑓𝜎𝑦≤‖𝐹‖𝜅‖‖‖𝐽_𝑛2‖‖‖𝑓𝑥+𝜎(𝑦)2−𝐹𝐽^𝑛𝑓𝑥2,𝐽^𝑛𝑓𝑦≤(‖𝐹‖𝜅)_𝑛+12‖‖‖𝜑𝑥≤(‖𝐹‖𝜅)_𝑛+12,2𝑦𝜑(𝑥,𝑥)(2.19) for all 𝑛∈𝑁, which proves the validity of (2.17).
Finally, we prove that 𝑓^∗(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 ‖‖𝑓^∗=(𝑥l+im𝜎_𝑛→∞(𝑦))−𝐹𝑓^∗(𝑥),𝑓^∗(𝑦)‖‖‖‖‖𝐽_𝑛𝑓𝑥+𝜎(𝑦)2−𝐹𝐽^𝑛𝑓𝑥2,𝐽^𝑛𝑓𝑦≤lim_𝑛→∞2‖‖‖(‖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(𝑥+𝑦)−𝐹[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 ||𝐹(𝑢,𝑣)||≤||𝐴|||𝑢|+||𝐵|||𝑣|≤||𝐴||+||𝐵||.(2.24) Thus, we get ‖𝐹‖≤sup||𝐹(𝑢,𝑣)||∶𝑢,𝑣∈ℂwith||(𝑢,𝑣)||₂≤1≤||𝐴||+||𝐵||,(2.25) which implies the boundedness of the linear transformation 𝐹.
On the other hand, we obtain ‖(𝑢(𝑥),𝑢(𝜎(𝑥)))−(𝑣(𝑥),𝑣(𝜎(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^∗(𝑥+𝜎(𝑦))−𝐹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^∗(𝑥+𝜎(𝑦))=𝐹𝑓^∗(𝑥),𝑓^∗(𝑦)‖‖(2.29) for all 𝑥,𝑦∈ℂ and ||𝑓^∗(𝑥)−𝑓(𝑥)||≤𝜀1−√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 inequality‖‖𝑎_𝑛(𝑥)𝑦^(𝑛)(𝑥)+𝑎_𝑛−1(𝑥)𝑦^(𝑛−1)(𝑥)+⋯+𝑎₁(𝑥)𝑦^(𝑥)+𝑎₀(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 equation𝑎_𝑛(𝑥)𝑦^(𝑛)(𝑥)+𝑎_𝑛−1(𝑥)𝑦^(𝑛−1)(𝑥)+⋯+𝑎₁(𝑥)𝑦^(𝑥)+𝑎₀(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 |𝑦^(𝑡)−𝑦(𝑡)|≤𝜀, where 𝐼 is an open subinterval of ℝ, then there exists a differentiable function 𝑦₀∶𝐼→ℝ satisfying 𝑦^0(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 function 𝐹(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 form𝑦^(𝑥)=𝐹(𝑞(𝑥),𝑝(𝑥)𝑦(𝑥)).(3.3)

Theorem 3.1. For given real numbers 𝑎 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 ||𝐹(𝑥,𝑦)−𝐹(𝑥,𝑧)||≤𝐿||𝑦−𝑧||(3.4) for any 𝑥∈𝐼 and 𝑦,𝑧∈ℝ. If a continuously differentiable function 𝑦∶𝐼→ℝ satisfies ||𝑦^(𝑥)−𝐹(𝑞(𝑥),𝑝(𝑥)𝑦(𝑥))||≤𝜑(𝑥)(3.5) 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 ||||^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(𝑞(𝑥),𝑝(𝑥)𝑦(𝑥))𝑑𝑡(3.7) (consequently, 𝑦₀ is a solution to (2.15)) and ||𝑦(𝑥)−𝑦₀(𝑥)||≤𝐾1−𝐾𝐿𝜑(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,𝑔)=inf𝐶∈[0,∞)∶||𝑓(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 {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. It further follows from (3.10) that ∀𝜀>0∃𝑁_𝜀∈ℕ∀𝑚,𝑛∈𝑁_𝜀∀𝑥∈𝐼∶||ℎ_𝑚(𝑥)−ℎ_𝑛(𝑥)||≤𝜀𝜑(𝑥).(3.11)Equation (3.11) implies that {ℎ_𝑛(𝑥)} is a Cauchy sequence in ℝ. Since ℝ is complete, {ℎ_𝑛(𝑥)} converges for each 𝑥∈𝐼. Thus, we can define a function ℎ∶𝐼→ℝ by ℎ(𝑥)=lim_𝑛→∞ℎ_𝑛(𝑥).(3.12) Let 𝑚 increase to infinity, then by (3.11) we have ∀𝜀>0∃𝑁_𝜀∈ℕ∀𝑛∈𝑁_𝜀∀𝑥∈𝐼∶||ℎ(𝑥)−ℎ_𝑛(𝑥)||≤𝜀𝜑(𝑥).(3.13)
Since 𝜑 is bounded on 𝐼, {ℎ_𝑛} converges uniformly to ℎ. Hence, ℎ is continuous and ℎ∈𝑋.
Further, considering (3.10) and (3.13), then ∀𝜀>0∃𝑁_𝜀∈ℕ∀𝑛∈𝑁_𝜀∶𝑑ℎ,ℎ_𝑛≤𝜀.(3.14) Then, the Cauchy sequence {ℎ_𝑛} converges to ℎ in (𝑋,𝑑). Hence, (𝑋,𝑑) is complete.
Now, define the operator 𝐽∶𝑋→𝑋 by (𝐽𝑓)(𝑥)=𝑦(𝑐)+^𝑥𝑐𝐹(𝑞(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 𝐶_𝑓𝑔∈[0,∞] be an arbitrary constant with 𝑑(𝑓,𝑔)≤𝐶_𝑓𝑔, then, by (2.15), we have ||𝑓(𝑥)−𝑔(𝑥)||≤𝐶_𝑓𝑔𝜑(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(𝑞(𝑥),𝑝(𝑥)𝑓(𝑥))−𝐹(𝑞(𝑥),𝑝(𝑥)𝑔(𝑥))}𝑑𝑡_𝑥𝑐||||≤𝐿||||||_𝑥𝑐𝐹(𝑞(𝑥),𝑝(𝑥)𝑓(𝑥))−𝐹(𝑞(𝑥),𝑝(𝑥)𝑔(𝑥))||𝑑𝑡||||≤𝐿𝐶_𝑓𝑔||𝑓(𝑡)−𝑔(𝑡)||𝑑𝑡||||||||^𝑥𝑐≤𝐾𝐿𝐶_𝑓𝑔𝜑(𝑡)𝑑𝑡||||𝜑(𝑥)(3.17) for all 𝑥∈𝐼. Then, 𝑑(𝐽𝑓,𝐽𝑔)≤𝐾𝐿𝐶_𝑓𝑔. 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 min_𝑥∈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 𝐽^𝑛𝑔₀→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 min_𝑥∈𝐼𝜑(𝑥)>0, there exists a constant 0<𝐶_𝑔<1 such that ||𝑔(𝑥)−𝑔₀(𝑥)||≤𝐶_𝑔𝜑(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)≤𝑦^(𝑥)−𝐹(𝑞(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(𝑞(𝑥),𝑝(𝑥)𝑦(𝑥))𝑑𝑡||||≤||||^𝑥𝑐𝜑(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≤11−𝐾𝐿𝑑(𝐽𝑦,𝑦)≤1−𝐾𝐾𝐿,(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 [𝑎,∞).

Theorem 3.2. For given real numbers 𝑎 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 ||𝑦^(𝑥)−𝐹(𝑞(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(𝑥)||≤𝐾1−𝐾𝐿𝜑(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 𝐼_𝑛=[𝑐−𝑛,𝑐+𝑛]. (We set 𝐼_𝑛=[𝑏−𝑛,𝑏] for 𝐼=(−∞,𝑏] and 𝐼_𝑛=[𝑎,𝑎+𝑛] for 𝐼=[𝑎,∞)). By Theorem 3.1, there exists a unique continuous function 𝑦_𝑛∶𝐼_𝑛→ℝ such that 𝑦_𝑛(𝑥)||=𝑦𝑦(𝑐)+^𝑥𝑐𝐹𝑞(𝑥),𝑝(𝑥)𝑦_𝑛(𝑡)𝑑𝑡(3.28)_𝑛(𝑥)−𝑦(𝑥)||≤𝐾1−𝐾𝐿𝜑(𝑥)(3.29) for all 𝑥∈𝐼_𝑛. The uniqueness of 𝑦_𝑛 implies that, if 𝑥∈𝐼_𝑛, then 𝑦_𝑛(𝑥)=𝑦_𝑛+1(𝑥)=𝑦_𝑛+2(𝑥)=⋯.(3.30) For any 𝑥∈ℝ, define 𝑛(𝑥)∈ℕ as 𝑛(𝑥)=min𝑛∈𝑁∣𝑥∈𝐼_𝑛.(3.31) Moreover, define a function 𝑦₀∶ℝ→ℝ by 𝑦₀(𝑥)=𝑦_𝑛(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 𝐼_𝑛1+1 and there exists an 𝜀>0 such that 𝑦₀(𝑥)=𝑦_𝑛1+1(𝑥) for all 𝑥 with 𝑥₁−𝜀<𝑥<𝑥₁+𝜀. Since 𝑦_𝑛1+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∈𝐼_𝑛(𝑥) and it follows from (3.28) and (3.32) that 𝑦₀(𝑥)=𝑦_𝑛(𝑥)(𝑥)=𝑦(𝑐)+^𝑥𝑐𝐹𝑞(𝑥),𝑝(𝑥)𝑦_𝑛(𝑥)(𝑡)𝑑𝑡=𝑦(𝑐)+^𝑥𝑐𝐹𝑞(𝑥),𝑝(𝑥)𝑦₀(𝑡)𝑑𝑡(3.33) since 𝑛(𝑡)≤𝑛(𝑥) for any 𝑡∈𝐼_𝑛(𝑥). Then, from (3.30) and (3.32) we have 𝑦_𝑛(𝑥)(𝑡)=𝑦_𝑛(𝑡)(𝑡)=𝑦₀(𝑡).(3.34) Since 𝑥∈𝐼_𝑛(𝑥) for every 𝑥∈ℝ, by (3.29) and (3.32), we have ||𝑦₀(𝑥)−𝑦(𝑥)||=||𝑦_𝑛(𝑥)(𝑥)−𝑦(𝑥)||≤𝐾1−𝐾𝐿𝜑(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)}(=𝑦_𝑛(𝑥)) and 𝑧₀∣_𝐼𝑛(𝑥) both satisfy (3.7) and (3.8) for all 𝑥∈𝐼_𝑛(𝑥), the uniqueness of 𝑦_𝑛(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 ||𝑦^(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(𝑥)||≤𝑟1−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 ||𝑦^(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(𝑡)𝑑𝑡||||=𝑥²2+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 ||𝑦𝑦₀(𝑥)=𝑦(0)+^𝑥0{𝐿𝑦(𝑡)−𝑝(𝑡)}𝑑𝑡₀(𝑥)−𝑦(𝑥)||≤𝐾1−𝐾𝐿(𝑥+𝜀)(3.41) for any 𝑥∈𝐼.

Example 3.5. Let 𝑎 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 ||𝑦^(𝑥)−𝐿𝑦(𝑥)−𝑝(𝑥)||≤𝑎^𝑥(3.42) for all 𝑥∈𝐼. If we set 𝜑(𝑥)=𝑎^𝑥, then we have ||||^𝑥𝑐𝜑(𝑡)𝑑𝑡||||≤1ln𝑎𝜑(𝑥)(3.43) for any 𝑥∈𝐼. By Theorem 3.2, there exists a unique continuous function 𝑦₀∶𝐼→ℝ with ||𝑦𝑦₀(𝑥)=𝑦(0)+^𝑥0{𝐿𝑦(𝑡)−𝑝(𝑡)}𝑑𝑡₀(𝑥)−𝑦(𝑥)||≤1ln𝑎−𝐿𝑎^𝑥(3.44) for any 𝑥∈𝐼.