On Ulam–Hyers Stability for a System of Partial Differential Equations of First Order

: The aim of this paper is to investigate generalized Ulam–Hyers stability and generalized Ulam–Hyers–Rassias stability for a system of partial differential equations of ﬁrst order. More precisely, we consider a system of two nonlinear equations of ﬁrst order with an unknown function of two independent variables, which satisfy the corresponding compatibility condition. The study method is that of differential inequalities of the Gronwall type.


Introduction
Ulam-Hyers stability is an important problem in functional equations theory, which was studied by many authors who can be found in the monograph [1].
The problem was posed by Ulam in 1940 (cf. [2,3]) in the following way. Let G be a group and H a metric group with metric d and f : G → H. For every ε > 0, ∃δ > 0 such that, if f : G → H verifies d ( f (xy) , f (x) f (y)) ≤ δ, x, y ∈ G, then ∃g : G → H a homomorphism with Firstly, Hyers [3] gaves an answer for additive Cauchy equation in Banach spaces, as follows. Let X 1 , X 2 be Banach spaces, δ > 0 and f : X 1 → X 2 such that There exists a unique additive mapping g : X 1 → X 2 verifying f (x) − g (x) ≤ δ, ∀x ∈ X 1 .
Ulam-Hyers stability of differential equations was firstly investigated by Alsina and Ger [4] in the following way. Let ε > 0, I ⊂ R be an open interval and f : The stability of differential linear equations of first order was studied in the papers [4,5]. Bernoulli equations were studied in [6] and Riccati equations in [7]. In [8], systems of first order linear differential equations were studied. Some results regarding linear differential equations of higher order were established in [9].
The first authors who studied Hyers-Ulam stability of partial differential equations were Prastaro and Rassias [10]. After that, a few results in this direction were given by other authors, regarding partial differential equations of first order [11,12], of order two [13], or of order three [14].
Recently, in a set of papers, Rus [13,15], has opened a new direction of study of the Ulam stability using Gronwall type inequalities and Picard operators technique. This will be used in this paper. Another direction of stability research is that in which results regarding fixed point theory are used [16].
In this paper, we will study the stability of a system of partial differential equations of order one, nonlinear. We mention that those are not studied yet in literature.
The general form of a system of n partial differential equations of first order for a function The normal form of this system is If the functions f i , i = 1, n admits continuous partial derivatives of order two, then ∂ 2 u These are the compatibility conditions and are not always identities. We will consider the system in the case of functions of two variables, denoted by x, y. Let u = u (x, y) .
Example 2. We consider the system In this case, the compatibility condition is reduced to the equation u [1 + u (1 + y)] = 0. Only two solutions are possible, namely u = 0 and u = − 1 1+y . Verifying, we get that u = 0 is the single solution.
Example 3. We consider the system The compatibility condition is not verified, hence the system has no solution.
From these examples, we see that Ulam stability for a system can be studied only for those which verify identical compatibility condition.
In the following lines, we will consider the system: We will study the problem (2) + (3) From ∂ 2 u ∂x∂y = ∂ 2 u ∂y∂x , the compatibility condition of the system is ∂g ∂x

Preliminaries
We will formulate the problem (2) + (3) as an integral equation, if condition (4) is satisfied. Since du = f dx + gdy, the problem (2) + (3) is equivalent with the integral equation Let ε > 0 and ϕ ∈ C D, R + , nondecreasing in x and y. Let the inequalities Definition 1. Equation (5) is generalized Ulam-Hyers stable if there exist the real numbers c 1 , c 2 such that, for each ε > 0 and for each solution v of (6), there exists a solution u of (5) with: Definition 2. Equation (5) is generalized Ulam-Hyers-Rassias stable with respect to ϕ if there exist the real numbers c 1 , c 2 such that for each solution v of (7) there exists a solution u of (5) with: |v (x, y) − u (x, y)| ≤ c 1 ϕ (x, y) + c 2 , ∀ (x, y) ∈ D.

Remark 1.
A function v is a solution of (6) if and only if there exists a function k ∈ C D, R such that (7) if and only if there exists a function k ∈ C D, R such that

Generalized Ulam-Hyers Stability
The first result is the following.
We remark that u (x, y) = xy 2 and v (x, y) = xy 2 + 3x 10 represent two surfaces. We have drawn these surfaces in Figure 1 and Figure 2, respectively, and, in Figure 3, both of them are overlapping. We can see that these surfaces are close to one another.

Conclusions
In this paper, we have studied generalized Ulam-Hyers and respectively Ulam-Hyers-Rassias stability of system (2), if condition (4) is satisfied, with initial condition (3). We have reduced this problem to Equation (5). We have used the method of differential inequalities, Gronwall types inequalities, and Picard operators technique. It is well known that these systems have many applications in the study of some partial differential equations of higher order, as in physics, chemistry, and other domains of science. Many problems from the domain Symmetry are modeled by differential equations and partial differential equations and those are approached in the stability point of view. The future study could be the stability of general system (1).