Hyers-Ulam stability of a general linear partial differential equation

In this paper we will study Hyers-Ulam stability for a general linear partial differential equation of first order in a Banach space.


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

Main results
Lemma 2.1. Assume that the system of ordinary differential equations Proof. We consider a solution u of Eq. (1.1) and the change of coordinates where s ∈ [a, b), t = (t 2 , . . . , t n ) ∈ R n−1 , x = (x 2 , . . . , x n ) ∈ R n−1 . Then, omitting the arguments of u and v we have Let the function L be defined by Hence where F is an arbitrary function of class C 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: The function v is well defined since the integral Therefore G t is absolutely convergent. We remark that the integral exists and is negative, since Xn+1 X1 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.

Hyers-Ulam stability of a general 653
On the other hand v is a solution of (1.1) being of the form (2.2). We have: for every x 1 ∈ [a, b) and x ∈ R n−1 .

Uniqueness. Suppose that for a solution u of (1.2) there exist two solutions
Since v 1 = v 2 it follows that there exist x 0 = (x 02 . . . , x 0n ) such that .
Hence the uniqueness is proved. In what follows we consider the equation Let ε > 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 We consider a solution u of Eq. (2.8) and the change of coordinates ⎧ ⎨ ⎩ s = x t 2 = y − 1 and we obtain the linear equation The integrating factor is e −s 2 , hence Vol. 97 (2023)

Hyers-Ulam stability of a general 655
We obtain Let u 1 be a solution of inequality (2.9) and let g be defined by Then u 1 is given by We have m = inf x∈ [a,b) 2x 2 = 2a 2 > 0 and using Theorem 2.2 we get Since , a contradiction, hence the uniqueness is proved.
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