Abstract
In this paper we will study Hyers-Ulam stability for a general linear partial differential equation of first order in a Banach space.
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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
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
there exists a solution \(v\in C^{1}\left( D,Y\right) \) of Eq. (1.1) such that
2 Main results
Lemma 2.1
Assume that the system of ordinary differential equations
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
for all \(x_{1}\in [a,b)\) and \({\overline{x}}=\left( x_{2},\ldots ,x_{n}\right) \in {\mathbb {R}}^{n-1},\) where
and
Proof
We consider a solution u of Eq. (1.1) and the change of coordinates
Define the function v by
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
and substituting in (1.1) we obtain that
Let the function L be defined by
\(s\in \left[ a,b\right) ,{\overline{t}}=\left( t_{2},\ldots ,t_{n}\right) \in I. \)
The integrating factor of Eq. (2.4) is \(e^{L\left( s,{\overline{t}}\right) }\) and we have
Hence
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
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
According to Lemma 2.1 we have:
where \(x_{1}\in [a,b)\) and \({\overline{x}}\in {\mathbb {R}}^{n-1}\).
Let v be defined by
where \(x_{1}\in [a,b)\) and \({\overline{x}}\in {\mathbb {R}}^{n-1}\).
The function v is well defined since the integral
\({\overline{t}}=\left( t_{2},\ldots ,t_{n} \right) \in I\) is convergent. Indeed, since \(\left\| g\right\| \le \varepsilon ,\) we have
Therefore \(G \left( {\overline{t}} \right) \) is absolutely convergent.
We remark that the integral
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:
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
given by
\( i\in \left\{ 1,2\right\} , x_{1}\in [a,b)\) and \({\overline{x}}\in R^{n-1}.\) We have
hence
Since \(v_{1}\ne v_{2}\) it follows that there exist \(\overline{x_0}=\left( x_{02} \ldots , x_{0n} \right) \) such that
For \({\overline{x}}=\varphi \left( x_{1}\right) + \overline{x_0},\) in (2.6) we get
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
\(x\in \left[ a, b\right) , a>0, a<b, y, z\in {\mathbb {R}}, D=\left[ a,b\right) \times {\mathbb {R}}^{2}, u\in C^{1}\left( D,Y\right) .\)
Let \(\varepsilon >0 \) be a given number. For every solution \(u_{1}\) of the inequality
there exists a solution \(v_{1}\) of Eq. (2.8) such that
Indeed, the characteristic system is
Let
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
Define the function v by
and we obtain the linear equation
The integrating factor is \(e^{-s^{2}},\) hence
We obtain
hence
Let \(u_{1}\) be a solution of inequality (2.9) and let g be defined by
Then \(u_{1}\) is given by
where \(t_{2}=y-\frac{1}{x^{2}},t_{3}=z-\frac{1}{x}.\)
Let \(v_{1}\) be defined by
We have \(m=\underset{x\in [a,b)}{\inf }\left( 2x^{2}\right) =2a^{2}>0\) and using Theorem 2.2 we get
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
given by
where \(t_{2}=y-\frac{1}{x^{2}},t_{3}=z-\frac{1}{x}.\) We have
which is equivalent to
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
If \(x \rightarrow \infty \) we get \(\infty \le \frac{ \varepsilon }{a^{2}},\) a contradiction, hence the uniqueness is proved.
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Ciplea, S.A., Lungu, N., Marian, D. et al. Hyers-Ulam stability of a general linear partial differential equation. Aequat. Math. 97, 649–657 (2023). https://doi.org/10.1007/s00010-023-00960-3
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DOI: https://doi.org/10.1007/s00010-023-00960-3