1 Introduction

Some recent results regarding the Hyers-Ulam stability of a partial differential equation were formulated and proved by S. M. Jung and K. S. Lee [3], S. M. Jung [2], N. Lungu and D. Popa [10,11,12], I. A. Rus and N. Lungu [15], N. Lungu and S. Ciplea [7], N. Lungu and C. Craciun [8], N. Lungu and D. Marian [9]. I. A. Rus also studied the Hyers-Ulam stability for operatorial equations [16]. J. Brzdek, D. Popa, I. Rasa, B. Xu [1] presented a systematic approach to the subject of Hyers-Ulam stability. The first result proved on the Hyers-Ulam stability of partial differential equations is due to A. Prastaro and Th.M. Rassias [13]. Furthermore several results on the Hyers-Ulam stability of a variety of ordinary differential equations were formulated and proved by A. Prastaro and Th.M. Rassias [4,5,6, 14]. These authors studied the stability of a particular partial differential equation for functions of two variables.

In the following lines we deal with the Hyers-Ulam stability of the equation

$$\begin{aligned} \overset{n}{\underset{k=1}{\sum }}X_{k}\left( x \right) \frac{\partial u}{\partial x_{k}}=X_{n+1}\left( x \right) u\left( x \right) +X_{n+2}\left( x \right) , \end{aligned}$$
(1.1)

where \(\left( Y,\left\| \cdot \right\| \right) \) is a Banach space over the field \({\mathbb {R}}\), \(D \subset {\mathbb {R}}^n\), \(D=\left[ a,b\right) \times {\mathbb {R}}^{n-1},\) \(a\in {\mathbb {R}}, b\in {\mathbb {R}}\cup \left\{ +\infty \right\} , a<b, X_{k}\in C\left( D, {\mathbb {R}}\right) , k\in \left\{ 1,\ldots ,n\right\} , X_{n+2}\in C\left( D,Y\right) ,u\in C^{1}\left( D,Y\right) \) is the unknown function. Suppose that \(X_{1}\left( x \right) > 0\) for all \(x= \left( x_{1},x_{2},\ldots ,x_{n}\right) \in D \).

Definition 1.1

Equation (1.1) is said to be Hyers-Ulam stable if for every \(\varepsilon >0\) there exists \(\delta \left( \varepsilon \right) > 0\) such that for every \(u \in C^{1}\left( D,Y\right) \) satisfying

$$\begin{aligned} \left\| \overset{n}{\underset{k=1}{\sum }}X_{k}\left( x \right) \frac{\partial u}{\partial x_{k}}-X_{n+1}\left( x \right) u\left( x \right) -X_{n+2}\left( x \right) \right\| \le \varepsilon , x= \left( x_{1},x_{2},\ldots ,x_{n}\right) \in D,\nonumber \\ \end{aligned}$$
(1.2)

there exists a solution \(v\in C^{1}\left( D,Y\right) \) of Eq. (1.1) such that

$$\begin{aligned} \left\| u\left( x\right) -v\left( x\right) \right\| \le \delta \left( \varepsilon \right) , x= \left( x_{1},x_{2},\ldots ,x_{n}\right) \in D. \end{aligned}$$

2 Main results

Lemma 2.1

Assume that the system of ordinary differential equations

$$\begin{aligned} \frac{dx_{2}}{dx_{1}}=\frac{X_{2}\left( x_{1},x_{2},\ldots ,x_{n}\right) }{ X_{1}\left( x_{1},x_{2},\ldots ,x_{n}\right) },\cdots ,\frac{dx_{n}}{dx_{1}}= \frac{X_{n}\left( x_{1},x_{2},\ldots ,x_{n}\right) }{X_{1}\left( x_{1},x_{2},\ldots ,x_{n}\right) } \end{aligned}$$
(2.1)

admits a solution \(\varphi =\left( \varphi _{2},\cdots ,\varphi _{n}\right) :[a,b)\longrightarrow {\mathbb {R}}^{n-1}.\) Then \(u \in C^{1}\left( D,Y\right) \) is a solution of Eq. (1.1) if and only if there exists a function \(F\in C^{1}\left( I,Y\right) \) such that

$$\begin{aligned}&u\left( x_{1},{\overline{x}}\right) =e^{-L\left( x_{1},{\overline{x}}-\varphi \left( x_{1}\right) \right) }\cdot \nonumber \\&\quad \left( \int \limits _{a}^{x_{1}}\frac{X_{n+2}\left( \theta ,\varphi \left( \theta \right) +{\overline{x}}-\varphi \left( x_{1}\right) \right) }{X_{1}\left( \theta ,\varphi \left( \theta \right) + {\overline{x}}-\varphi \left( x_{1}\right) \right) }e^{L\left( \theta , {\overline{x}}-\varphi \left( x_{1}\right) \right) }d\theta +F\left( \overline{ x}-\varphi \left( x_{1}\right) \right) \right) , \end{aligned}$$
(2.2)

for all \(x_{1}\in [a,b)\) and \({\overline{x}}=\left( x_{2},\ldots ,x_{n}\right) \in {\mathbb {R}}^{n-1},\) where

$$\begin{aligned} L\left( x_{1},{\overline{x}}\right) :=-\int _{a}^{x_{1}}\frac{X_{n+1}\left( \theta ,\varphi \left( \theta \right) +{\overline{x}}\right) }{X_{1}\left( \theta ,\varphi \left( \theta \right) +{\overline{x}}\right) }d\theta \end{aligned}$$

and

$$\begin{aligned} I:=\left\{ {\overline{x}}-\varphi \left( x_{1}\right) \mid x_{1}\in [a,b),{\overline{x}}\in {\mathbb {R}}^{n-1}\right\} . \end{aligned}$$

Proof

We consider a solution u of Eq. (1.1) and the change of coordinates

$$\begin{aligned} \left\{ \begin{array}{ll} s=x_{1} \\ t_{k}=x_{k}-\varphi _{k}\left( x_{1}\right) ,k\in \left\{ 2,\ldots ,n\right\} \end{array} \right. \Leftrightarrow \left\{ \begin{array}{ll} x_{1}=s \\ x_{k}=\varphi _{k}\left( s\right) +t_{k},k\in \left\{ 2,\ldots ,n\right\} . \end{array} \right. \nonumber \\ \end{aligned}$$
(2.3)

Define the function v by

$$\begin{aligned} v\left( s,{\overline{t}}\right) = u\left( s,\varphi \left( s\right) +{\overline{t}}\right) \Leftrightarrow u\left( x_{1},{\overline{x}}\right) = v\left( x_{1},{\overline{x}}-\varphi \left( x_{1}\right) \right) , \end{aligned}$$

where \(s\in [a,b),{\overline{t}}=\left( t_{2},\ldots ,t_{n}\right) \in {\mathbb {R}}^{n-1},{\overline{x}}=\left( x_{2},\ldots ,x_{n}\right) \in {\mathbb {R}}^{n-1}\). Then, omitting the arguments of u and v we have

$$\begin{aligned} \left\{ \begin{array}{ll} \frac{\partial u}{\partial x_{1}}=\frac{\partial v}{\partial s}-\varphi _{2}^{\prime }\left( s\right) \frac{\partial v}{\partial t_{2}}-\cdots -\varphi _{n}^{\prime }\left( s\right) \frac{\partial v}{\partial t_{n}} \\ \frac{\partial u}{\partial x_{k}}=\frac{\partial v}{\partial t_{k}}, k\in \left\{ 2,\ldots ,n\right\} \end{array} \right. \end{aligned}$$

and substituting in (1.1) we obtain that

$$\begin{aligned} \frac{\partial v}{\partial s} -\frac{X_{n+1}\left( s,\varphi \left( s\right) +{\overline{t}}\right) }{X_{1}\left( s,\varphi \left( s\right) +{\overline{t}}\right) } v\left( s,{\overline{t}}\right) =\frac{X_{n+2}\left( s,\varphi \left( s\right) +{\overline{t}}\right) }{X_{1}\left( s,\varphi \left( s\right) +{\overline{t}}\right) }. \end{aligned}$$
(2.4)

Let the function L be defined by

$$\begin{aligned} L\left( s, {\overline{t}}\right) =-\int _{a}^{s}\frac{X_{n+1}\left( \theta ,\varphi \left( \theta \right) +{\overline{t}}\right) }{X_{1}\left( \theta ,\varphi \left( \theta \right) +{\overline{t}}\right) }d\theta , \end{aligned}$$

\(s\in \left[ a,b\right) ,{\overline{t}}=\left( t_{2},\ldots ,t_{n}\right) \in I. \)

The integrating factor of Eq. (2.4) is \(e^{L\left( s,{\overline{t}}\right) }\) and we have

$$\begin{aligned} \frac{\partial }{\partial s}\left( e^{L\left( s,{\overline{t}}\right) }v\right) =\frac{X_{n+2}\left( s,\varphi \left( s\right) +{\overline{t}}\right) }{X_{1}\left( s,\varphi \left( s\right) +{\overline{t}}\right) }\cdot e^{L\left( s,{\overline{t}}\right) }. \end{aligned}$$

Hence

$$\begin{aligned}&v\left( s,{\overline{t}}\right) =e^{-L\left( s,{\overline{t}}\right) } \cdot \nonumber \\&\quad \cdot \left( \int \limits _{a}^{s}\frac{X_{n+2}\left( \theta ,\varphi \left( \theta \right) +{\overline{t}}\right) }{X_{1}\left( \theta ,\varphi \left( \theta \right) +{\overline{t}}\right) }e^{L\left( \theta ,{\overline{t}}\right) }d\theta +F\left( {\overline{t}}\right) \right) , \end{aligned}$$
(2.5)

where F is an arbitrary function of class \(C^{1}.\)

Substituting \(s,t_{2},\ldots ,t_{n}\) from (2.3) in (2.5) the relation (2.2) is obtained. Conversely, every function v given by (2.5) is a solution of Eq. (2.4). Hence u given by (2.2) is a solution of Eq. (1.1). \(\square \)

Theorem 2.2

Let \(\varepsilon >0\) be a given number. Suppose that the system (2.1) admits a solution \( \left( \varphi _{2},\ldots ,\varphi _{n}\right) ,\varphi _{k}:\left[ a,b\right) \rightarrow {\mathbb {R}}, k\in \left\{ 2,\ldots ,n\right\} \) and \(\underset{ x\in D}{\inf }\ X_{n+1}\left( x\right) :=m>0.\) Then for every function \(u\in C^{1}\left( D,Y\right) \) satisfying the relation (1.2) there exists a solution \(v\in C^{1}\left( D,Y\right) \) of (1.1) such that

$$\begin{aligned} \left\| u\left( x \right) -v\left( x \right) \right\| \le \frac{\varepsilon }{m}, x \in D. \end{aligned}$$

Moreover if \(L\left( b,{\overline{x}}\right) :=\underset{x_{1}\rightarrow b}{\lim }L\left( x_{1},{\overline{x}}\right) =-\infty , \) for all \({\overline{x}}=\left( x_{2},\ldots ,x_{n}\right) \in {\mathbb {R}}^{n-1}, \) then v is uniquely determined.

Proof

Existence. Let u be a solution of inequality (1.2) and let g be defined by

$$\begin{aligned} \overset{n}{\underset{k=1}{\sum }}X_{k}\left( x \right) \frac{\partial u}{\partial x_{k}} -X_{n+1}\left( x \right) u\left( x \right) -X_{n+2}\left( x \right) :=g\left( x \right) , x \in D. \end{aligned}$$

According to Lemma 2.1 we have:

$$\begin{aligned} u\left( x_{1},{\overline{x}}\right) =&\,e^{-L\left( x_{1},{\overline{x}}-\varphi \left( x_{1}\right) \right) }\cdot \nonumber \\&\quad \left( \int _{a}^{x_{1}}\frac{X_{n+2}\left( \theta ,\varphi \left( \theta \right) +{\overline{x}}-\varphi \left( x_{1}\right) \right) }{X_{1}\left( \theta ,\varphi \left( \theta \right) + {\overline{x}}-\varphi \left( x_{1}\right) \right) }e^{L\left( \theta , {\overline{x}}-\varphi \left( x_{1}\right) \right) }d\theta \right. \\&\quad {+}\left. \int _{a}^{x_{1}}\frac{g\left( \theta ,\varphi \left( \theta \right) {+}{\overline{x}}{-}\varphi \left( x_{1}\right) \right) }{X_{1}\left( \theta ,\varphi \left( \theta \right) {+} {\overline{x}}{-}\varphi \left( x_{1}\right) \right) }e^{L\left( \theta , {\overline{x}}{-}\varphi \left( x_{1}\right) \right) }d\theta {+}F\left( \overline{ x}{-}\varphi \left( x_{1}\right) \right) \right) , \end{aligned}$$

where \(x_{1}\in [a,b)\) and \({\overline{x}}\in {\mathbb {R}}^{n-1}\).

Let v be defined by

$$\begin{aligned} v\left( x_{1},{\overline{x}}\right) =&\,e^{-L\left( x_{1},{\overline{x}}-\varphi \left( x_{1}\right) \right) }\cdot \nonumber \\&\quad \left( \int _{a}^{x_{1}}\frac{X_{n+2}\left( \theta ,\varphi \left( \theta \right) +{\overline{x}}-\varphi \left( x_{1}\right) \right) }{X_{1}\left( \theta ,\varphi \left( \theta \right) + {\overline{x}}-\varphi \left( x_{1}\right) \right) }e^{L\left( \theta , {\overline{x}}-\varphi \left( x_{1}\right) \right) }d\theta \right. \\&\quad {+}\left. \int _{a}^{b}\frac{g\left( \theta ,\varphi \left( \theta \right) {+}{\overline{x}}{-}\varphi \left( x_{1}\right) \right) }{X_{1}\left( \theta ,\varphi \left( \theta \right) {+} {\overline{x}}{-}\varphi \left( x_{1}\right) \right) }e^{L\left( \theta , {\overline{x}}{-}\varphi \left( x_{1}\right) \right) }d\theta {+}F\left( \overline{ x}{-}\varphi \left( x_{1}\right) \right) \right) , \end{aligned}$$

where \(x_{1}\in [a,b)\) and \({\overline{x}}\in {\mathbb {R}}^{n-1}\).

The function v is well defined since the integral

$$\begin{aligned} G \left( {\overline{t}} \right) :=\int _{a}^{b}\frac{g \left( \theta ,\varphi \left( \theta \right) +{\overline{t}} \right) }{X_{1}\left( \theta ,\varphi \left( \theta \right) +{\overline{t}} \right) }e^{L\left( \theta ,{\overline{t}}\right) }d\theta , \end{aligned}$$

\({\overline{t}}=\left( t_{2},\ldots ,t_{n} \right) \in I\) is convergent. Indeed, since \(\left\| g\right\| \le \varepsilon ,\) we have

$$\begin{aligned}&\left\| G \left( {\overline{t}} \right) \right\| \le \int _{a}^{b}\left\| \frac{g \left( \theta ,\varphi \left( \theta \right) +{\overline{t}} \right) }{X_{1}\left( \theta ,\varphi \left( \theta \right) +{\overline{t}} \right) }e^{L\left( \theta ,{\overline{t}}\right) }\right\| d\theta \\&\quad \le \int _{a}^{b} \frac{\varepsilon }{X_{n+1}\left( \theta ,\varphi \left( \theta \right) +{\overline{t}} \right) }\cdot \frac{X_{n+1}\left( \theta ,\varphi \left( \theta \right) +{\overline{t}} \right) }{X_{1}\left( \theta ,\varphi \left( \theta \right) +{\overline{t}} \right) }e^{L\left( \theta ,{\overline{t}}\right) } d\theta \le \\&\quad \le \frac{\varepsilon }{m}\int _{a}^{b} \frac{\partial }{\partial \theta }\left( - e^{L\left( \theta ,{\overline{t}}\right) }\right) d\theta =\frac{\varepsilon }{m}\left( 1-e^{L\left( b,{\overline{t}} \right) }\right) \le \frac{\varepsilon }{m}. \end{aligned}$$

Therefore \(G \left( {\overline{t}} \right) \) is absolutely convergent.

We remark that the integral

$$\begin{aligned} L\left( b, {\overline{t}}\right) =\underset{x_{1}\rightarrow b}{\lim }L\left( x_{1},{\overline{t}}\right) =-\int _{a}^{b} \frac{X_{n+1}\left( \theta ,\varphi \left( \theta \right) +{\overline{t}} \right) }{X_{1}\left( \theta ,\varphi \left( \theta \right) +{\overline{t}} \right) }e^{L\left( \theta ,{\overline{t}}\right) } d\theta \end{aligned}$$

exists and is negative, since \(\frac{X_{n+1}}{X_{1}}\) is a positive function, hence L is decreasing with respect to \(x_1\) on [a, b), therefore admits left and right limits at every point.

On the other hand v is a solution of (1.1) being of the form (2.2). We have:

$$\begin{aligned}&\left\| u\left( x_{1},{\overline{x}}\right) -v\left( x_{1},{\overline{x}}\right) \right\| = \\&\quad \left\| e^{-L\left( x_{1},{\overline{x}}-\varphi \left( x_{1}\right) \right) }\left( -\int _{x_{1}}^{b}\frac{g \left( \theta ,\varphi \left( \theta \right) +{\overline{x}}-\varphi \left( x_{1}\right) \right) }{X_{1}\left( \theta ,\varphi \left( \theta \right) +{\overline{x}}-\varphi \left( x_{1}\right) \right) } \cdot e^{L\left( \theta ,{\overline{x}}-\varphi \left( x_{1}\right) \right) }d\theta \right) \right\| \\&\quad \le e^{-L\left( x_{1},{\overline{x}}-\varphi \left( x_{1}\right) \right) }\cdot \\&\quad \cdot \int _{x_{1}}^{b} \frac{\varepsilon }{X_{n+1}\left( \theta ,\varphi \left( \theta \right) +{\overline{t}} \right) }\cdot \frac{X_{n+1}\left( \theta ,\varphi \left( \theta \right) +{\overline{t}} \right) }{X_{1}\left( \theta ,\varphi \left( \theta \right) +{\overline{t}} \right) }e^{L\left( \theta ,{\overline{t}}\right) } d\theta \\&\quad \le \frac{\varepsilon }{m} e^{-L\left( x_{1},{\overline{x}}-\varphi \left( x_{1}\right) \right) }\int _{x_{1}}^{b} \frac{\partial }{\partial \theta }\left( -e^{L\left( \theta ,{\overline{x}}-\varphi \left( x_{1}\right) \right) }\right) d\theta \\&\quad =\frac{\varepsilon }{m}\left( 1-e^{L\left( b,{\overline{x}}-\varphi \left( x_{1}\right) \right) -L\left( x_{1},{\overline{x}}-\varphi \left( x_{1}\right) \right) }\right) \le \frac{\varepsilon }{m}, \end{aligned}$$

for every \(x_{1}\in [a,b)\) and \({\overline{x}}\in R^{n-1}\).

Uniqueness. Suppose that for a solution u of (1.2) there exist two solutions \(v_{1},v_{2},\) \(v_{1}\ne v_{2}\) of (1.1) such that

$$\begin{aligned} \left\| u\left( x_{1},{\overline{x}}\right) -v_{i}\left( x_{1},{\overline{x}}\right) \right\| \le \frac{\varepsilon }{m}, \end{aligned}$$

given by

$$\begin{aligned}&v_{i}\left( x_{1},{\overline{x}}\right) =e^{-L\left( x_{1},{\overline{x}}-\varphi \left( x_{1}\right) \right) }\cdot \nonumber \\&\quad \left( \int _{a}^{x_{1}}\frac{X_{n+2}\left( \theta ,\varphi \left( \theta \right) +{\overline{x}}-\varphi \left( x_{1}\right) \right) }{X_{1}\left( \theta ,\varphi \left( \theta \right) + {\overline{x}}-\varphi \left( x_{1}\right) \right) }e^{L\left( \theta , {\overline{x}}-\varphi \left( x_{1}\right) \right) }d\theta +F_{i}\left( \overline{ x}-\varphi \left( x_{1}\right) \right) \right) , \end{aligned}$$

\( i\in \left\{ 1,2\right\} , x_{1}\in [a,b)\) and \({\overline{x}}\in R^{n-1}.\) We have

$$\begin{aligned}&\left\| v_{1}\left( x_{1},{\overline{x}}\right) -v_{2}\left( x_{1},{\overline{x}}\right) \right\| \le \left\| v_{1}\left( x_{1},{\overline{x}}\right) -u\left( x_{1},{\overline{x}}\right) \right\| +\left\| u\left( x_{1},{\overline{x}}\right) -v_{2}\left( x_{1},{\overline{x}}\right) \right\| \\&\quad \le \frac{2\varepsilon }{m}, x_{1}\in [a,b), {\overline{x}}\in R^{n-1}, \end{aligned}$$

hence

$$\begin{aligned} e^{-L\left( x_{1},{\overline{x}}-\varphi \left( x_{1}\right) \right) } \cdot \left\| F_{1}\left( {\overline{x}}-\varphi \left( x_{1}\right) \right) -F_{2}\left( {\overline{x}}-\varphi \left( x_{1}\right) \right) \right\| \le \frac{2\varepsilon }{m}. \end{aligned}$$
(2.6)

Since \(v_{1}\ne v_{2}\) it follows that there exist \(\overline{x_0}=\left( x_{02} \ldots , x_{0n} \right) \) such that

$$\begin{aligned} F_{1}\left( \overline{x_0} \right) \ne F_{2}\left( \overline{x_0} \right) . \end{aligned}$$

For \({\overline{x}}=\varphi \left( x_{1}\right) + \overline{x_0},\) in (2.6) we get

$$\begin{aligned} e^{-L\left( x_{1},\overline{x_0}\right) } \cdot \left\| F_{1}\left( \overline{x_0}\right) -F_{2}\left( \overline{x_0} \right) \right\| \le \frac{2\varepsilon }{m}, x_{1}\in \left[ a,b\right) . \end{aligned}$$
(2.7)

If \(x_{1}\rightarrow b \) in (2.7), we have \(\infty \le \frac{2\varepsilon }{m},\) a contradiction. Hence the uniqueness is proved. \(\square \)

Example 2.3

In what follows we consider the equation

$$\begin{aligned} x\frac{\partial u}{\partial x}-2y\frac{\partial u}{\partial y}-z\frac{ \partial u}{\partial z}=2x^{2}u+\left( x+1\right) yz, \end{aligned}$$
(2.8)

\(x\in \left[ a, b\right) , a>0, a<b, y, z\in {\mathbb {R}}, D=\left[ a,b\right) \times {\mathbb {R}}^{2}, u\in C^{1}\left( D,Y\right) .\)

Let \(\varepsilon >0 \) be a given number. For every solution \(u_{1}\) of the inequality

$$\begin{aligned} \left\| x\frac{\partial u}{\partial x}-2y\frac{\partial u}{\partial y}-z \frac{\partial u}{\partial z}-2x^{2}u-\left( x+1\right) yz\right\| \le \varepsilon \end{aligned}$$
(2.9)

there exists a solution \(v_{1}\) of Eq. (2.8) such that

$$\begin{aligned} \left\| u_{1} \left( x,y,z\right) -v_{1}\left( x,y,z\right) \right\| \le \frac{ \varepsilon }{2a^{2}}. \end{aligned}$$

Indeed, the characteristic system is

$$\begin{aligned} \frac{dx}{x}=\frac{dy}{-2y}=\frac{dz}{-z}. \end{aligned}$$
(2.10)

Let

$$\begin{aligned} \left\{ \begin{array}{ll} y=\frac{1}{x^{2}} \\ z=\frac{1}{x} \end{array} \right. \end{aligned}$$

be solutions of system (2.10). We have \(\varphi =\left( \varphi _{2},\varphi _{3}\right) :[a,b)\longrightarrow {\mathbb {R}}^{2}, \varphi _{2} (x) =\frac{1}{x^{2}},\varphi _{3} (x) =\frac{1}{x}.\)

We consider a solution u of Eq. (2.8) and the change of coordinates

$$\begin{aligned} \left\{ \begin{array}{ll} s=x \\ t_{2}=y-\frac{1}{x^{2}} \\ t_{3}=z-\frac{1}{x} \end{array} \right. . \end{aligned}$$

Define the function v by

$$\begin{aligned} v\left( s,t_{2},t_{3}\right) =u\left( s,t_{2}+\frac{1}{x^{2}},t_{3}+\frac{1}{ x}\right) \Leftrightarrow u\left( x,y,z\right) =v\left( x,y-\frac{1}{x^{2}},z-\frac{1}{x}\right) \end{aligned}$$

and we obtain the linear equation

$$\begin{aligned} \frac{\partial v}{\partial s}-2sv=\frac{s+1}{s}\left( t_{2}t_{3}+\frac{ t_{2}s^{2}+t_{3}s+1}{s^{3}}\right) . \end{aligned}$$

The integrating factor is \(e^{-s^{2}},\) hence

$$\begin{aligned} e^{-s^{2}}\frac{\partial v}{\partial s}-2se^{-s^{2}}v= & {} e^{-s^{2}}\frac{s+1}{s} \left( t_{2}t_{3}+\frac{t_{2}s^{2}+t_{3}s+1}{s^{3}}\right) , \\ \frac{\partial }{\partial s}\left( e^{-s^{2}}v\right)= & {} e^{-s^{2}}\frac{s+1}{s }\left( t_{2}t_{3}+\frac{t_{2}s^{2}+t_{3}s+1}{s^{3}}\right) . \end{aligned}$$

We obtain

$$\begin{aligned} e^{-s^{2}}v= & {} \int _{a}^{s}e^{-\theta ^{2}}\frac{\theta +1}{\theta }\left( t_{2}t_{3}+\frac{t_{2}\theta ^{2}+t_{3}\theta +1}{\theta ^{3}}\right) d\theta +F\left( t_{2},t_{3}\right) \\ v\left( s,t_{2},t_{3}\right)= & {} e^{s^{2}}\left[ \int _{a}^{s}e^{-\theta ^{2}} \frac{\theta +1}{\theta }\left( t_{2}t_{3}+\frac{t_{2}\theta ^{2}+t_{3}\theta +1}{\theta ^{3}}\right) d\theta +F\left( t_{2},t_{3}\right) \right] , \end{aligned}$$

hence

$$\begin{aligned} v\left( s,t_{2},t_{3}\right) =e^{s^{2}}\int _{a}^{s}e^{-\theta ^{2}}\frac{ \left( \theta +1\right) \left( t_{2}t_{3}\theta ^{3}+t_{2}\theta ^{2}+t_{3}\theta +1\right) }{\theta ^{4}}d\theta +F\left( t_{2},t_{3}\right) e^{s^{2}}. \end{aligned}$$

Let \(u_{1}\) be a solution of inequality (2.9) and let g be defined by

$$\begin{aligned} x\frac{\partial u}{\partial x}-2y\frac{\partial u}{\partial y}-z\frac{ \partial u}{\partial z}-2x^{2}u-\left( x+1\right) yz:=g\left( x,y,z\right) . \end{aligned}$$

Then \(u_{1}\) is given by

$$\begin{aligned}&u_{1}\left( x,y,z\right) =\\&\quad =e^{x^{2}}\int _{a}^{x}e^{-\theta ^{2}}\frac{\left( \theta +1\right) \left( t_{2}t_{3}\theta ^{3}+t_{2}\theta ^{2}+t_{3}\theta +1\right) +g\left( \theta ,t_{2}+\frac{1}{\theta ^{2}},t_{3}+\frac{1}{ \theta }\right) }{\theta ^{4}}d\theta \\&\quad \qquad \,+F\left( t_{2},t_{3}\right) e^{x^{2}}, \end{aligned}$$

where \(t_{2}=y-\frac{1}{x^{2}},t_{3}=z-\frac{1}{x}.\)

Let \(v_{1}\) be defined by

$$\begin{aligned} v_{1}\left( x,y,z\right) =&\,e^{x^{2}}\int _{a}^{x}e^{-\theta ^{2}}\frac{\left( \theta +1\right) \left( t_{2}t_{3}\theta ^{3}+t_{2}\theta ^{2}+t_{3}\theta +1\right) }{\theta ^{4}}d\theta \\&\quad +e^{x^{2}}\int _{a}^{b}e^{-\theta ^{2}}\frac{g\left( \theta ,t_{2}+\frac{1}{\theta ^{2}},t_{3}+\frac{1}{ \theta }\right) }{\theta ^{4}}d\theta +F\left( t_{2},t_{3}\right) e^{x^{2}}. \end{aligned}$$

We have \(m=\underset{x\in [a,b)}{\inf }\left( 2x^{2}\right) =2a^{2}>0\) and using Theorem 2.2 we get

$$\begin{aligned} \left\| u_{1} \left( x,y,z\right) -v_{1}\left( x,y,z\right) \right\| \le \frac{ \varepsilon }{2a^{2}}, \left( x,y,z\right) \in D. \end{aligned}$$

If \(b=\infty \), then \(v_{1}\) is unique. For uniqueness we consider two solutions \(v_{1},v_{2},v_{1}\ne v_{2}\) of (2.8) such that

$$\begin{aligned} \left\| u_{1} \left( x,y,z\right) -v_{i}\left( x,y,z\right) \right\| \le \frac{ \varepsilon }{2a^{2}}, i\in \left\{ 1,2\right\} , \left( x,y,z\right) \in D, \end{aligned}$$

given by

$$\begin{aligned} v_{i}\left( x,y,z\right) =&\,e^{x^{2}}\int _{a}^{x}e^{-\theta ^{2}}\frac{\left( \theta +1\right) \left( t_{2}t_{3}\theta ^{3}+t_{2}\theta ^{2}+t_{3}\theta +1\right) }{\theta ^{4}}d\theta \\&\quad {+}e^{x^{2}}\int _{a}^{b}e^{-\theta ^{2}}\frac{g\left( \theta ,t_{2}{+}\frac{1}{\theta ^{2}},t_{3}{+}\frac{1}{ \theta }\right) }{\theta ^{4}}d\theta {+}F_{i}\left( t_{2},t_{3}\right) e^{x^{2}}, i\in \left\{ 1,2\right\} , \end{aligned}$$

where \(t_{2}=y-\frac{1}{x^{2}},t_{3}=z-\frac{1}{x}.\) We have

$$\begin{aligned}&\left\| v_{1} \left( x,y,z\right) -v_{2}\left( x,y,z\right) \right\| \le \left\| v_{1} \left( x,y,z\right) -u_{1}\left( x,y,z\right) \right\| \\&\quad +\left\| u_{1} \left( x,y,z\right) -v_{2}\left( x,y,z\right) \right\| \\&\quad \le \frac{ \varepsilon }{a^{2}}, \left( x,y,z\right) \in D, \end{aligned}$$

which is equivalent to

$$\begin{aligned} e^{x^{2}}\left\| F_{1}\left( y-\frac{1}{x^{2}}, z-\frac{1}{x}\right) -F_{2}\left( y-\frac{1}{x^{2}}, z-\frac{1}{x}\right) \right\| \le \frac{ \varepsilon }{a^{2}}. \end{aligned}$$

Since \(v_{1}\ne v_{2}\), there exist \(y_{0},z_{0}\) such that \(F_{1}\left( y_{0}, z_{0}\right) \ne F_{2}\left( y_{0}, z_{0}\right) .\) For \(y=\frac{1}{x^2}+y_{0}\), \(z=\frac{1}{x}+z_{0}\) we get

$$\begin{aligned} e^{x^{2}}\left\| F_{1}\left( y_{0}, z_{0}\right) - F_{2}\left( y_{0}, z_{0}\right) \right\| \le \frac{ \varepsilon }{a^{2}}. \end{aligned}$$

If \(x \rightarrow \infty \) we get \(\infty \le \frac{ \varepsilon }{a^{2}},\) a contradiction, hence the uniqueness is proved.