On Ulam stability of a partial differential operator in Banach spaces

Diana Otrocol
(original), ISI/JCR, paper

Abstract

In this paper, we prove that, if \(\inf \limits_{x\in A}\left \vert f\left(x\right) \right \vert =m>0\), then the partial differential operator \(D\) defined by \(D\left( u\right) =\sum \limits_{k=1}^{n}f_{k}\frac{\partial u}{\partial x_{k}}-fu,\) where \(f,f_{i}\in C\left( A,\mathbb{R}\right),uC^{1}\left( A,X\right) ,\ i=1,\ldots,n\subset \mathbb{R}\) is an interval, \(A=I\times \mathbb{R}^{n-1}\) and \(X\) is a Banach space, is Ulam stable with the Ulam constant\(K=\frac{1}{m}\). Moreover, if \(\inf \limits_{x\in A}\left \vert f\left( x\right) \right \vert =0\), we prove that \(D\) is not generally Ulam stable.

Authors

Adela Novac
Department of Mathematics, Technical University of Cluj-Napoca, Cluj-Napoca, Romania

Diana Otrocol
Department of Mathematics, Technical University of Cluj-Napoca
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy

Dorian Popa
Department of Mathematics, Technical University of Cluj-Napoca, Romania

Keywords

Ulam stability; partial differential operator; gauge; Banach space

Paper coordinates

A. Novac, D. Otrocol, D. Popa, On Ulam stability of a partial differential operator in Banach spaces, Mathematics, 11 (2023) no. 11, art. no. 2488, https://doi.org/10.3390/math11112488

PDF

??

About this paper

Journal

Mathematics

Publisher Name

MDPI

DOI

https://doi.org/10.3390/math11112488

Print ISSN

2227-7390

Online ISSN

google scholar link

[1] Ulam, S.M. A Collection of Mathematical Problems; Interscience: New York, NY, USA, 1960. [Google Scholar]
[2] Hyers, D.H. On the stability of the linear functional equation. Proc. Matl. Acad. Sci. USA 1941, 27, 222–224. [Google Scholar] [CrossRef] [PubMed]
[3] Brzdek, J.; Popa, D.; Raşa, I.; Xu, B. Ulam Stability of Operators; Academic Press: Cambridge, MA, USA, 2018. [Google Scholar]
[4] Jung, S.-M. Hyers–Ulam Rassias Stability of Functional Equations in Mathematical Analysis; Hadronic Press: Palm Harbour, FL, USA, 2001. [Google Scholar]
[5] Baias, A.R.; Popa, D. On the best Ulam constant of the second order linear differential operator. Rev. De La Real Acad. De Cienc. Exactas 2020, 114, 23. [Google Scholar] [CrossRef]
[6] Fukutaka, R.; Onitsuka, M. Best constant in Hyers-Ulam stability of first order homogeneous linear differential equations with a periodic coefficient. J. Math. Anal. 2019, 473, 1432–1446. [Google Scholar] [CrossRef]
[7] Rus, I.A. Remarks on Ulam stability of the operatorial equations. Fixed Point Theory 2009, 10, 305–320. [Google Scholar]
[8] Jung, S.M. Hyers-Ulam stability of linear partial differential equations of first order. Appl. Math. Lett. 2009, 22, 70–74. [Google Scholar] [CrossRef]
[9] Lungu, N.; Popa, D. Hyers-Ulam stability of a first order partial differential equation. J. Math. Anal. Appl. 2012, 385, 86–91. [Google Scholar] [CrossRef]
[10] Marian, D.; Ciplea, S.A.; Lungu, N. Ulam-Hyers stability of a parabolic partial differential equation. Demonstr. Math. 2019, 52, 475–481. [Google Scholar] [CrossRef]
[11] Prastaro, A.; Rassias, T.M. Ulam stability in geometry of PDE’s. Nonlinear Funct. Anal. Appl. 2003, 8, 259–278. [Google Scholar]
[12] Brzdek, J.; Popa, D.; Raşa, I. Hyers–Ulam stability with respect to gauges. J. Math. Anal. Appl. 2017, 453, 620–628. [Google Scholar] [CrossRef]
[13] Vrabie, I.I. Differential Equations: An Introduction to Basic Concepts, Results and Applications; Word Scientific: Singapore, 2004. [Google Scholar]
[14] Corduneanu, C. Principles of Differential and Integral Equations; The Bronx: New York, NY, USA, 1977. [Google Scholar]
[15] Carter, S. On the Cobb-Douglas and all that’: The Solow-Simon correspondence over the neoclassical aggregate production function. J. Post Keynes. Econ. 2011, 34, 255–274. [Google Scholar] [CrossRef]
[16] Popa, D.; Ilca, M. On approximate Cobb-Douglas production functions. Carpathian J. Math. 2014, 30, 87–92. [Google Scholar]

On Ulam stability of a partial differential operator in Banach spaces

Adela Novac Technical University of Cluj-Napoca, Department of Mathematics, 28 Memorandumului Street, 400114 Cluj-Napoca, Romania adela.novac@math.utcluj.ro , Diana Otrocol Technical University of Cluj-Napoca, Department of Mathematics, 28 Memorandumului Street, 400114 Cluj-Napoca, Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, P.O.Box. 68-1, 400110 Cluj-Napoca, Romania Diana.Otrocol@math.utcluj.ro and Dorian Popa Technical University of Cluj-Napoca, Department of Mathematics, 28 Memorandumului Street, 400114 Cluj-Napoca, Romania Popa.Dorian@math.utcluj.ro
Abstract.

In this paper we prove that, if infx∈A|f​(x)|=m>0\inf\limits_{x\in A}|f(x)|=m>0 then, the partial differential operator DD defined by D​(u)=∑k=1nfk​∂u∂xk−f​uD(u)={\textstyle\sum\limits_{k=1}^{n}}f_{k}\frac{\partial u}{\partial x_{k}}-fu, where f,fi∈C​(A,ℝ),u∈C1​(A,X),i=1,…,n,I⊂ℝf,f_{i}\in C(A,\mathbb{R}),u\in C^{1}(A,X),\ i=1,\ldots,n,I\subset\mathbb{R} is an interval, A=I×ℝn−1A=I\times\mathbb{R}^{n-1}, XX is a Banach space, is Ulam stable with the Ulam constant K=1mK=\frac{1}{m}. Moreover, if infx∈A|f​(x)|=0\inf\limits_{x\in A}|f(x)|=0 we prove that DD is not generally Ulam stable.

Keywords: Ulam stability, partial differential operator, gauge, Banach space.

Mathematics Subject Classification: 39B82, 47A50.

1. Introduction

Ulam stability is one of the main topics in the theory of functional equations. Generally, a functional equation is called Ulam stable (or Hyers-Ulam stable) if for every approximate solution of the equation, there exists a solution of the equation near it. The problem of stability of functional equation was formulated by S. M. Ulam in a talk given at University of Wisconsin-Madison and concerns the equation of group homomorphism [15]. The first answer to Ulam’s problem was given by D. H. Hyers who proved that the Cauchy’s functional equation in Banach spaces is stable [7]. For this reason the stability of functional equation is called Ulam stability, or Hyers-Ulam stability. After Hyers’ result has been published a lot of papers dedicated to the study of Ulam stability, and the notion of Ulam stability was extended in various direction (see for more details the books [2], [8]). In the last years appeared many papers dedicated to the study of Ulam stability for differential equations and differential operators. For example the authors of [1] study the Ulam stability of a second order linear differential operator acting in a Banach space. They obtain a characterization of its Ulam stability and its best Ulam constant. Similar results were obtained by the authors of [6] for the first order linear differential equation with periodic coefficients. For various kind of operatorial equations, including partial and ordinary differential equations, results regarding their Ulam stability are presented in [14]. Recent results on Ulam stability of partial differential equations of order one are given in [9] and [10]. The stability of a parabolic partial differential equation is presented in [11] and an explicit form of the Ulam constant for the Laplace operator is obtained in [2] (see chapter 3). It seems that A. Prastaro and Th. M. Rassias published the first paper on Ulam stability of a partial differential equation [13]. By Ulam stability of an equation or of an operator is established a relation between a solution of a perturbed equation and an exact solution of it. If small perturbations of the equation produces small perturbation of the solution of it we say that the equation is Ulam stable. Ulam stability of an operator means Ulam stability of its associated equation. The goal of this paper is to give a result on Ulam stability for a linear and nonhomogeneous partial differential operator acting in Banach spaces and to obtain an explicit form of its Ulam constant. We improve and extend in this way some results for partial differential equations with constant coefficients given in [9] and for partial differential equations with nonconstant coefficients with two variables given in [10].

Let I=[a,b),a∈ℝ,b∈ℝ¯I=[a,b),\ a\in\mathbb{R},\ b\in\overline{\mathbb{R}} and A=I×ℝn−1A=I\times\mathbb{R}^{n-1}, XX a Banach space over ℝ\mathbb{R} and VV a vector space over ℝ.\mathbb{R}.

All over this paper by ∥⋅∥\left\|\cdot\right\| we denote the norm of the Banach space XX and by ∥⋅∥e\left\|\cdot\right\|_{e} the euclidian norm in ℝk.\mathbb{R}^{k}.

Definition 1.1.

A function ρV:V→[0,+∞]\rho_{V}:V\rightarrow[0,+\infty] is called a gauge on VV if the following properties hold:

  1. i)

    ρV​(x)=0⇔x=0;\rho_{V}(x)=0\Leftrightarrow x=0;

  2. ii)

    ρV​(λ​x)=|λ|​ρV​(x),∀x∈V,∀λ∈ℝ,λ≠0.\rho_{V}(\lambda x)=\left|\lambda\right|\rho_{V}(x),\ \forall x\in V,\forall\lambda\in\mathbb{R},\lambda\neq 0.

Let f,f1,f2,…,fn:A→ℝf,f_{1},f_{2},\ldots,f_{n}:A\rightarrow\mathbb{R} be continuous functions and D:C1​(A,X)→C​(A,X)D:C^{1}(A,X)\rightarrow C(A,X) defined by

(1) D​(u)=∑k=1nfk​∂u∂xk−f​u,u∈C1​(A,X).D(u)=\sum\limits_{k=1}^{n}f_{k}\frac{\partial u}{\partial x_{k}}-fu,\ u\in C^{1}(A,X).

Let φ∈C​(A,X)\varphi\in C(A,X) and define

(2) ∥φ∥∞=sup{∥φ(x)∥:x∈A}.\left\|\varphi\right\|_{\infty}=\sup\{\left\|\varphi(x)\right\|:x\in A\}.

Then ‖φ‖∞\left\|\varphi\right\|_{\infty} is a gauge on C​(A,X).C(A,X). Consider the same gauge on C1​(A,X).C^{1}(A,X).

Definition 1.2.

The operator DD is called Ulam stable if there exists K≥0K\geq 0 such that, for every ε>0\varepsilon>0 and every u∈C1​(A,X)u\in C^{1}(A,X) with the property

(3) ‖D​(u)‖∞≤ε,\left\|D(u)\right\|_{\infty}\leq\varepsilon,

there exists u0∈C1​(A,X)u_{0}\in C^{1}(A,X) such that D​(u0)=0D(u_{0})=0 and

(4) ‖u−u0‖∞≤K​ε.\left\|u-u_{0}\right\|_{\infty}\leq K\varepsilon.

The number KK is called an Ulam constant of D.D.

A function u∈C1​(A,X)u\in C^{1}(A,X) satisfying (3), for some positive ε\varepsilon, is called an approximate solution of the equation D​(u)=0.D(u)=0. So, we can reformulate the definition of DD as follows: the operator DD is said to be Ulam stable if for every approximate solution uu of the associated equation D​(u)=0D(u)=0 there exists an element u0∈ker⁡Du_{0}\in\ker D (i.e., a solution of D​(u)=0D(u)=0) close to uu (i.e., satisfying (4)).

Ulam stability of some linear operator acting in spaces endowed with gauges is studied in [2]. Since gauges are generalized norms or metrics, the results on characterization and on the best Ulam constant of such operators extend the results on Ulam stability given in normed spaces for linear operators (see [2], chapters 2 and 3). A detailed presentation of Ulam stability for some operators relative to gauges is given in [3]. Here the authors obtain Ulam stability characterization theorems and results regarding the reprezentation of the best Ulam constant. Concrete examples are obtained for differential operators. Results related to Ulam stability for partial differential operators are not found in [3]. Our goal is to give such results in the present paper.

2. Main result

The following result contains a representation of the solutions of the equation D​(u)=gD(u)=g.

Lemma 2.1.

Let g∈C​(A)g\in C(A) and suppose that f1≠0f_{1}\neq 0 on A.A. Let gk=fkf1, 2≤k≤n.g_{k}=\dfrac{f_{k}}{f_{1}},\ 2\leq k\leq n. Suppose that the system

(5) {x2′=g2​(t,x2,…,xn)⋮xn′=gn​(t,x2,…,xn),t∈I\left\{\begin{array}[c]{c}x_{2}^{\prime}=g_{2}(t,x_{2},\ldots,x_{n})\\ \vdots\\ x_{n}^{\prime}=g_{n}(t,x_{2},\ldots,x_{n})\end{array}\right.,\ t\in I

admits a global solution (φ2,…,φn):I→ℝn−1.(\varphi_{2},\ldots,\varphi_{n}):I\rightarrow\mathbb{R}^{n-1}. Then uu is a solution of the equation

(6) f1​(x)​∂u∂x1+⋯+fn​(x)​∂u∂xn−f​(x)​u=g​(x),x=(x1,…,xn)∈Af_{1}(x)\frac{\partial u}{\partial x_{1}}+\cdots+f_{n}(x)\frac{\partial u}{\partial x_{n}}-f(x)u=g(x),\ x=(x_{1},\ldots,x_{n})\in A

if and only if there exists a function F∈C1​(ℝn−1,X)F\in C^{1}(\mathbb{R}^{n-1},X) such that

(7) u(x1,…,xn)=e−L​(x1,x2−φ2​(x1),…,xn−φn​(x1))⋅\displaystyle u(x_{1},\ldots,x_{n})=e^{-L(x_{1},x_{2}-\varphi_{2}(x_{1}),\ldots,x_{n}-\varphi_{n}(x_{1}))}\cdot
⋅(∫ax1eL​(s,x2−φ2​(x1),…,xn−φn​(x1))​g​(s,x2−φ2​(x1)+φ2​(s),…,xn−φn​(x1)+φn​(s))f1​(s,x2−φ2​(x1)+φ2​(s),…,xn−φn​(x1)+φn​(s))ds+\displaystyle\quad\cdot\left(\int\nolimits_{a}^{x_{1}}\frac{e^{L(s,x_{2}-\varphi_{2}(x_{1}),\ldots,x_{n}-\varphi_{n}(x_{1}))}g(s,x_{2}-\varphi_{2}(x_{1})+\varphi_{2}(s),\ldots,x_{n}-\varphi_{n}(x_{1})+\varphi_{n}(s))}{f_{1}(s,x_{2}-\varphi_{2}(x_{1})+\varphi_{2}(s),\ldots,x_{n}-\varphi_{n}(x_{1})+\varphi_{n}(s))}ds+\right.
+F(x2−φ2(x1),…,xn−φn(x1))),\displaystyle\quad\left.+F(x_{2}-\varphi_{2}(x_{1}),\ldots,x_{n}-\varphi_{n}(x_{1}))\right),

where x=(x1,…,xn)∈A,x=(x_{1},\ldots,x_{n})\in A,\ and

L​(x1,x2,…,xn)=−∫ax1f​(s,x2+φ2​(s),…,xn+φn​(s))f1​(s,x2+φ2​(s),…,xn+φn​(s))​𝑑s.L(x_{1},x_{2},\ldots,x_{n})=-{\displaystyle\int\nolimits_{a}^{x_{1}}}\dfrac{f(s,x_{2}+\varphi_{2}(s),\ldots,x_{n}+\varphi_{n}(s))}{f_{1}(s,x_{2}+\varphi_{2}(s),\ldots,x_{n}+\varphi_{n}(s))}ds.
Proof.

Suppose that uu satisfies (6) and consider the change of coordinates:

(8) {t1=x1t2=x2−φ2​(x1)⋮tn=xn−φn​(x1)⇔{x1=t1x2=t2+φ2​(t1)⋮xn=tn+φn​(t1).\left\{\begin{array}[c]{l}t_{1}=x_{1}\\ t_{2}=x_{2}-\varphi_{2}(x_{1})\\ \vdots\\ t_{n}=x_{n}-\varphi_{n}(x_{1})\end{array}\right.\Leftrightarrow\left\{\begin{array}[c]{l}x_{1}=t_{1}\\ x_{2}=t_{2}+\varphi_{2}(t_{1})\\ \vdots\\ x_{n}=t_{n}+\varphi_{n}(t_{1})\end{array}\right..

Let

γ​(t)=(t1,t2+φ2​(t1),…,tn+φn​(t1)),t=(t1,…,tn).\gamma(t)=(t_{1},t_{2}+\varphi_{2}(t_{1}),\ldots,t_{n}+\varphi_{n}(t_{1})),\ t=(t_{1},\ldots,t_{n}).

Define the function ww by the relations

w​(t1,t2,…,tn)=u​(t1,t2+φ2​(t1),…,tn+φn​(t1))=u​(γ​(t))w(t_{1},t_{2},\ldots,t_{n})=u(t_{1},t_{2}+\varphi_{2}(t_{1}),\ldots,t_{n}+\varphi_{n}(t_{1}))=u(\gamma(t))

or

u​(x1,…,xn)=w​(x1,x2−φ2​(x1),…,xn−φn​(x1)).u(x_{1},\ldots,x_{n})=w(x_{1},x_{2}-\varphi_{2}(x_{1}),\ldots,x_{n}-\varphi_{n}(x_{1})).

Then

∂u∂x1\displaystyle\frac{\partial u}{\partial x_{1}} =∂w∂t1−∂w∂t2​φ2′​(t1)−…−∂w∂tn​φn′​(t1)\displaystyle=\frac{\partial w}{\partial t_{1}}-\frac{\partial w}{\partial t_{2}}\varphi_{2}^{\prime}(t_{1})-\ldots-\frac{\partial w}{\partial t_{n}}\varphi_{n}^{\prime}(t_{1})
∂u∂xk\displaystyle\frac{\partial u}{\partial x_{k}} =∂w∂tk,k=2,3,…,n.\displaystyle=\frac{\partial w}{\partial t_{k}},\ k=2,3,...,n.

Replacing in (6), we get

f1​(γ​(t))​(∂w∂t1−φ2′​(t1)​∂w∂t2−…−φn′​(t1)​∂w∂tn)+f2​(γ​(t))​∂w∂t2+…+fn​(γ​(t))​∂w∂tn−f​(γ​(t))​w=\displaystyle f_{1}(\gamma(t))\left(\frac{\partial w}{\partial t_{1}}\!-\!\varphi_{2}^{\prime}(t_{1})\frac{\partial w}{\partial t_{2}}\!-\!\ldots-\varphi_{n}^{\prime}(t_{1})\frac{\partial w}{\partial t_{n}}\right)\!+\!f_{2}(\gamma(t))\frac{\partial w}{\partial t_{2}}+\!\ldots\!+f_{n}(\gamma(t))\frac{\partial w}{\partial t_{n}}\!-\!f(\gamma(t))w=
=g​(γ​(t)),\displaystyle=g(\gamma(t)),

or

f1​(γ​(t))​∂w∂t1+(f2​(γ​(t))−f1​(γ​(t))​φ2′​(t1))​∂w∂t2+…+(fn​(γ​(t))−f1​(γ​(t))​φn′​(t1))​∂w∂tn−f​(γ​(t))​w=\displaystyle f_{1}(\gamma(t))\frac{\partial w}{\partial t_{1}}\!+\!\left(f_{2}(\gamma(t))-f_{1}(\gamma(t))\varphi_{2}^{\prime}(t_{1})\right)\frac{\partial w}{\partial t_{2}}\!+\!\ldots\!+\left(f_{n}(\gamma(t))\!-\!f_{1}(\gamma(t))\varphi_{n}^{\prime}(t_{1})\right)\frac{\partial w}{\partial t_{n}}\!-\!f(\gamma(t))w=
=g​(γ​(t)).\displaystyle=g(\gamma(t)).

Now, taking account of (5), i.e.,

φk′=gk=fkf1, 2≤k≤n,\varphi_{k}^{\prime}=g_{k}=\frac{f_{k}}{f_{1}},\ 2\leq k\leq n,

it follows

(9) ∂w∂t1−f​(γ​(t))f1​(γ​(t))​w=g​(γ​(t))f1​(γ​(t)),t=(t1,…,tn)∈I×ℝn−1.\frac{\partial w}{\partial t_{1}}-\frac{f(\gamma(t))}{f_{1}(\gamma(t))}w=\frac{g(\gamma(t))}{f_{1}(\gamma(t))},\ t=(t_{1},\ldots,t_{n})\in I\times\mathbb{R}^{n-1}.

Let the function LL be defined by

L​(t1,t2,…,tn)=−∫at1f​(s,t2+φ2​(s),…,tn+φn​(s))f1​(s,t2+φ2​(s),…,tn+φn​(s))​𝑑s,(t1,t2,…,tn)∈I×ℝn−1.L(t_{1},t_{2},\ldots,t_{n})=-{\displaystyle\int\nolimits_{a}^{t_{1}}}\frac{f(s,t_{2}+\varphi_{2}(s),\ldots,t_{n}+\varphi_{n}(s))}{f_{1}(s,t_{2}+\varphi_{2}(s),\ldots,t_{n}+\varphi_{n}(s))}ds,\ (t_{1},t_{2},\ldots,t_{n})\in I\times\mathbb{R}^{n-1}.

Then, multiplying equation (9) by eL​(t1,t2,…,tn)e^{L(t_{1},t_{2},\ldots,t_{n})} we get

∂∂t1​(w​(t1,t2,…,tn)⋅eL​(t1,t2,…,tn))=g​(t1,t2+φ2​(t1),…,tn+φn​(t1))f1​(t1,t2+φ2​(t1),…,tn+φn​(t1))​eL​(t1,t2,…,tn),\frac{\partial}{\partial t_{1}}\left(w(t_{1},t_{2},\ldots,t_{n})\cdot e^{L(t_{1},t_{2},\ldots,t_{n})}\right)=\frac{g(t_{1},t_{2}+\varphi_{2}(t_{1}),\ldots,t_{n}+\varphi_{n}(t_{1}))}{f_{1}(t_{1},t_{2}+\varphi_{2}(t_{1}),\ldots,t_{n}+\varphi_{n}(t_{1}))}e^{L(t_{1},t_{2},\ldots,t_{n})},

which leads to

w​(t1,t2,…,tn)⋅eL​(t1,t2,…,tn)=∫at1g​(s,t2+φ2​(s),…,tn+φn​(s))​eL​(s,t2,…,tn)f1​(s,t2+φ2​(s),…,tn+φn​(s))​𝑑s+F​(t2,…,tn),w(t_{1},t_{2},\ldots,t_{n})\cdot e^{L(t_{1},t_{2},\ldots,t_{n})}={\displaystyle\int\nolimits_{a}^{t_{1}}}\frac{g(s,t_{2}+\varphi_{2}(s),\ldots,t_{n}+\varphi_{n}(s))e^{L(s,t_{2},\ldots,t_{n})}}{f_{1}(s,t_{2}+\varphi_{2}(s),\ldots,t_{n}+\varphi_{n}(s))}ds+F(t_{2},\ldots,t_{n}),

where FF is an arbitrary function of class C1.C^{1}. Then

(10) w​(t1,t2,…,tn)=e−L​(t1,t2,…,tn)​(∫at1g​(s,t2+φ2​(s),…,tn+φn​(s))​eL​(s,t2,…,tn)f1​(s,t2+φ2​(s),…,tn+φn​(s))​𝑑s+F​(t2,…,tn))w(t_{1},t_{2},\ldots,t_{n})=e^{-L(t_{1},t_{2},\ldots,t_{n})}\left({\displaystyle\int\nolimits_{a}^{t_{1}}}\frac{g(s,t_{2}+\varphi_{2}(s),\ldots,t_{n}+\varphi_{n}(s))e^{L(s,t_{2},\ldots,t_{n})}}{f_{1}(s,t_{2}+\varphi_{2}(s),\ldots,t_{n}+\varphi_{n}(s))}ds+F(t_{2},\ldots,t_{n})\right)

Replacing t1,t2,…,tnt_{1},t_{2},\ldots,t_{n} from (8) in (10) the relation (7) is obtained.

Now let uu be given by (7), we have to prove that uu is a solution of (1). Taking account of the change of coordinates (8) it is sufficient to prove that ww given by (10), satisfies (9). A simple calculation shows that ww is a solution of (9). ∎

The main result of this paper is given in the next theorem.

Theorem 2.2.

Let ε>0\varepsilon>0 be a given number, f1≠0f_{1}\neq 0 on AA and suppose that:

  1. i)

    the system (5) admits a global solution (φ2,…,φn):I→ℝn−1(\varphi_{2},\ldots,\varphi_{n}):I\rightarrow\mathbb{R}^{n-1};

  2. ii)

    infx∈A|f​(x)|=m>0\inf_{x\in A}\left|f(x)\right|=m>0.

Then for every u∈C1​(A,X)u\in C^{1}(A,X) satisfying ‖D​(u)‖∞≤ε,\left\|D(u)\right\|_{\infty}\leq\varepsilon, there exists a solution u0u_{0} of D​(u)=0D(u)=0 with the property

(11) ‖u−u0‖∞≤εm.\left\|u-u_{0}\right\|_{\infty}\leq\frac{\varepsilon}{m}.

Moreover, if L​(b,x2,…,xn):=limx1→bL​(x)=−∞,∀(x2,…,xn)∈ℝn−1,L(b,x_{2},\ldots,x_{n}):=\lim_{x_{1}\rightarrow b}L(x)=-\infty,\ \forall(x_{2},\ldots,x_{n})\in\mathbb{R}^{n-1}, then u0u_{0} is uniquely determined.

Proof.

Since f1f_{1} is continuous on the connected set AA and f1≠0f_{1}\neq 0 it follows that f1​(x)>0f_{1}(x)>0 or f1​(x)<0f_{1}(x)<0 for every x∈A.x\in A. We may suppose in what follows that f1​(x)>0f_{1}(x)>0, for every x∈A,x\in A, without loss of generality.

Existence. Let uu be a solution of (3) and put

∑k=1nfk​(x)​∂u​(x)∂xk−f​(x)​u​(x):=g​(x)\sum\limits_{k=1}^{n}f_{k}(x)\frac{\partial u(x)}{\partial x_{k}}-f(x)u(x):=g(x)

for every x=(x1,…,xn)∈A.x=(x_{1},\ldots,x_{n})\in A. Then, according to Lemma 2.1, we have:

u​(x1,…,xn)=\displaystyle u(x_{1},\ldots,x_{n})=
=e−L​(x1,x2−φ2​(x1),…,xn−φn​(x1))⋅\displaystyle=\!\!e^{-L(x_{1},x_{2}-\varphi_{2}(x_{1}),\ldots,x_{n}-\varphi_{n}(x_{1}))}\cdot
⋅(∫ax1eL​(s,x2−φ2​(x1),…,xn−φn​(x1))​g​(s,x2−φ2​(x1)+φ2​(s),…,xn−φn​(x1)+φn​(s))f1​(s,x2−φ2​(x1)+φ2​(s),…,xn−φn​(x1)+φn​(s))ds+\displaystyle\cdot\left(\int\nolimits_{a}^{x_{1}}\frac{e^{L(s,x_{2}-\varphi_{2}(x_{1}),\ldots,x_{n}-\varphi_{n}(x_{1}))}g(s,x_{2}-\varphi_{2}(x_{1})+\varphi_{2}(s),\ldots,x_{n}-\varphi_{n}(x_{1})+\varphi_{n}(s))}{f_{1}(s,x_{2}-\varphi_{2}(x_{1})+\varphi_{2}(s),\ldots,x_{n}-\varphi_{n}(x_{1})+\varphi_{n}(s))}ds+\right.
+F(x2−φ2(x1),…,xn−φn(x1)))\displaystyle\left.+F(x_{2}-\varphi_{2}(x_{1}),\ldots,x_{n}-\varphi_{n}(x_{1}))\right)

where F∈C1​(ℝn−1,X)F\in C^{1}(\mathbb{R}^{n-1},X).

We consider the function u0u_{0} defined by

u0(x1,…,xn)=e−L​(x1,x2−φ2​(x1),…,xn−φn​(x1))⋅\displaystyle u_{0}(x_{1},\ldots,x_{n})=\!\!e^{-L(x_{1},x_{2}-\varphi_{2}(x_{1}),\ldots,x_{n}-\varphi_{n}(x_{1}))}\cdot
⋅(∫abeL​(s,x2−φ2​(x1),…,xn−φn​(x1))​g​(s,x2−φ2​(x1)+φ2​(s),…,xn−φn​(x1)+φn​(s))f1​(s,x2−φ2​(x1)+φ2​(s),…,xn−φn​(x1)+φn​(s))ds+\displaystyle\cdot\left(\int\nolimits_{a}^{b}\frac{e^{L(s,x_{2}-\varphi_{2}(x_{1}),\ldots,x_{n}-\varphi_{n}(x_{1}))}g(s,x_{2}-\varphi_{2}(x_{1})+\varphi_{2}(s),\ldots,x_{n}-\varphi_{n}(x_{1})+\varphi_{n}(s))}{f_{1}(s,x_{2}-\varphi_{2}(x_{1})+\varphi_{2}(s),\ldots,x_{n}-\varphi_{n}(x_{1})+\varphi_{n}(s))}ds+\right.
+F(x2−φ2(x1),…,xn−φn(x1))),\displaystyle\left.+F(x_{2}-\varphi_{2}(x_{1}),\ldots,x_{n}-\varphi_{n}(x_{1}))\right),

for every x=(x1,…,xn)∈A.x=(x_{1},\ldots,x_{n})\in A. Since u0u_{0} is given by an integral on the noncompact interval [a,b),[a,b), first we have to prove that the function u0u_{0} is well defined, i.e., the improper integral is convergent.

We test the convergence of the integral

∫abeL​(s,x2−φ2​(x1),…,xn−φn​(x1))​g​(s,x2−φ2​(x1)+φ2​(s),…,xn−φn​(x1)+φn​(s))f1​(s,x2−φ2​(x1)+φ2​(s),…,xn−φn​(x1)+φn​(s))​𝑑s,{\displaystyle\int\nolimits_{a}^{b}}\frac{e^{L(s,x_{2}-\varphi_{2}(x_{1}),\ldots,x_{n}-\varphi_{n}(x_{1}))}g(s,x_{2}-\varphi_{2}(x_{1})+\varphi_{2}(s),\ldots,x_{n}-\varphi_{n}(x_{1})+\varphi_{n}(s))}{f_{1}(s,x_{2}-\varphi_{2}(x_{1})+\varphi_{2}(s),\ldots,x_{n}-\varphi_{n}(x_{1})+\varphi_{n}(s))}ds,

wherex=(x1,…,xn)∈A\ x=(x_{1},\ldots,x_{n})\in A or, according to (8), the convergence of the following integral

(12) ∫abeL​(s,t2,…,tn)​g​(s,t2+φ2​(s),…,tn+φn​(s))f1​(s,t2+φ2​(s),…,tn+φn​(s))​𝑑s.{\displaystyle\int\nolimits_{a}^{b}}e^{L(s,t_{2},\ldots,t_{n})}\dfrac{g(s,t_{2}+\varphi_{2}(s),\ldots,t_{n}+\varphi_{n}(s))}{f_{1}(s,t_{2}+\varphi_{2}(s),\ldots,t_{n}+\varphi_{n}(s))}ds.

We have

‖eL​(s,t2,…,tn)​g​(s,t2+φ2​(s),…,tn+φn​(s))f1​(s,t2+φ2​(s),…,tn+φn​(s))​d​s‖≤\displaystyle\left\|e^{L(s,t_{2},\ldots,t_{n})}\dfrac{g(s,t_{2}+\varphi_{2}(s),\ldots,t_{n}+\varphi_{n}(s))}{f_{1}(s,t_{2}+\varphi_{2}(s),\ldots,t_{n}+\varphi_{n}(s))}ds\right\|\leq
≤eL​(s,t2,…,tn)f1​(s,t2+φ2​(s),…,tn+φn​(s))​‖g​(s,t2+φ2​(s),…,tn+φn​(s))‖\displaystyle\leq\frac{e^{L(s,t_{2},\ldots,t_{n})}}{f_{1}(s,t_{2}+\varphi_{2}(s),\ldots,t_{n}+\varphi_{n}(s))}\left\|g(s,t_{2}+\varphi_{2}(s),\ldots,t_{n}+\varphi_{n}(s))\right\|
≤eL​(s,t2,…,tn)f1​(s,t2+φ2​(s),…,tn+φn​(s))​ε\displaystyle\leq\frac{e^{L(s,t_{2},\ldots,t_{n})}}{f_{1}(s,t_{2}+\varphi_{2}(s),\ldots,t_{n}+\varphi_{n}(s))}\varepsilon
=ε​eL​(s,t2,…,tn)f​(s,t2+φ2​(s),…,tn+φn​(s))⋅f​(s,t2+φ2​(s),…,tn+φn​(s))f1​(s,t2+φ2​(s),…,tn+φn​(s))\displaystyle=\frac{\varepsilon e^{L(s,t_{2},\ldots,t_{n})}}{f(s,t_{2}+\varphi_{2}(s),\ldots,t_{n}+\varphi_{n}(s))}\cdot\frac{f(s,t_{2}+\varphi_{2}(s),\ldots,t_{n}+\varphi_{n}(s))}{f_{1}(s,t_{2}+\varphi_{2}(s),\ldots,t_{n}+\varphi_{n}(s))}
≤εm⋅∂∂s​(e−L​(s,t2,…,tn)),(s,t2,…,tn)∈I×ℝn−1.\displaystyle\leq\frac{\varepsilon}{m}\cdot\frac{\partial}{\partial s}\left(e^{-L(s,t_{2},\ldots,t_{n})}\right),\ (s,t_{2},\ldots,t_{n})\in I\times\mathbb{R}^{n-1}.

Since

∫abεm​∂∂s​(e−L​(s,t2,…,tn))​𝑑s=εm​(1−eL​(b,t2,…,tn))≤εm,{\displaystyle\int\nolimits_{a}^{b}}\dfrac{\varepsilon}{m}\dfrac{\partial}{\partial s}\left(e^{-L(s,t_{2},\ldots,t_{n})}\right)ds=\dfrac{\varepsilon}{m}\left(1-e^{L(b,t_{2},\ldots,t_{n})}\right)\leq\dfrac{\varepsilon}{m},

it follows, in view of the comparison test, that the integral (12) is absolutely convergent, therefore the function u0u_{0} is well defined. On the other hand u0u_{0} is a solution of (1) being of the form (7).

We have:

‖u​(x)−u0​(x)‖=\displaystyle\left\|u(x)-u_{0}(x)\right\|=
=∥e−L​(x1,x2−φ2​(x1),…,xn−φn​(x1))∫x1b−eL​(s,x2−φ2​(x1),…,xn−φn​(x1))⋅\displaystyle=\left\|e^{-L(x_{1},x_{2}-\varphi_{2}(x_{1}),\ldots,x_{n}-\varphi_{n}(x_{1}))}\int\nolimits_{x_{1}}^{b}-e^{L(s,x_{2}-\varphi_{2}(x_{1}),\ldots,x_{n}-\varphi_{n}(x_{1}))}\cdot\right.
⋅g​(s,x2−φ2​(x1)+φ2​(s),…,xn−φn​(x1)+φn​(s))f1​(s,x2−φ2​(x1)+φ2​(s),…,xn−φn​(x1)+φn​(s))ds∥\displaystyle\quad\left.\cdot\frac{g(s,x_{2}-\varphi_{2}(x_{1})+\varphi_{2}(s),\ldots,x_{n}-\varphi_{n}(x_{1})+\varphi_{n}(s))}{f_{1}(s,x_{2}-\varphi_{2}(x_{1})+\varphi_{2}(s),\ldots,x_{n}-\varphi_{n}(x_{1})+\varphi_{n}(s))}ds\right\|
≤e−L​(x1,x2−φ2​(x1),…,xn−φn​(x1))​∫x1bε​eL​(s,x2−φ2​(x1),…,xn−φn​(x1))f1​(s,x2−φ2​(x1)+φ2​(s),…,xn−φn​(x1)+φn​(s))​𝑑s\displaystyle\leq e^{-L(x_{1},x_{2}-\varphi_{2}(x_{1}),\ldots,x_{n}-\varphi_{n}(x_{1}))}\int\nolimits_{x_{1}}^{b}\frac{\varepsilon e^{L(s,x_{2}-\varphi_{2}(x_{1}),\ldots,x_{n}-\varphi_{n}(x_{1}))}}{f_{1}(s,x_{2}-\varphi_{2}(x_{1})+\varphi_{2}(s),\ldots,x_{n}-\varphi_{n}(x_{1})+\varphi_{n}(s))}ds
≤e−L​(x1,x2−φ2​(x1),…,xn−φn​(x1))​∫x1bε​eL​(s,x2−φ2​(x1),…,xn−φn​(x1))f1​(s,x2−φ2​(x1)+φ2​(s),…,xn−φn​(x1)+φn​(s))\displaystyle\leq e^{-L(x_{1},x_{2}-\varphi_{2}(x_{1}),\ldots,x_{n}-\varphi_{n}(x_{1}))}\int\nolimits_{x_{1}}^{b}\frac{\varepsilon e^{L(s,x_{2}-\varphi_{2}(x_{1}),\ldots,x_{n}-\varphi_{n}(x_{1}))}}{f_{1}(s,x_{2}-\varphi_{2}(x_{1})+\varphi_{2}(s),\ldots,x_{n}-\varphi_{n}(x_{1})+\varphi_{n}(s))}
⋅f​(s,t2+φ2​(s),…,tn+φn​(s))f​(s,t2+φ2​(s),…,tn+φn​(s))​d​s\displaystyle\quad\cdot\dfrac{f(s,t_{2}+\varphi_{2}(s),\ldots,t_{n}+\varphi_{n}(s))}{f(s,t_{2}+\varphi_{2}(s),\ldots,t_{n}+\varphi_{n}(s))}ds
≤εm​e−L​(x1,x2−φ2​(x1),…,xn−φn​(x1))​∫x1bf​(s,t2+φ2​(s),…,tn+φn​(s))​eL​(s,x2−φ2​(x1),…,xn−φn​(x1))f1​(s,x2−φ2​(x1)+φ2​(s),…,xn−φn​(x1)+φn​(s))​𝑑s\displaystyle\leq\frac{\varepsilon}{m}e^{-L(x_{1},x_{2}-\varphi_{2}(x_{1}),\ldots,x_{n}-\varphi_{n}(x_{1}))}\int\nolimits_{x_{1}}^{b}\dfrac{f(s,t_{2}+\varphi_{2}(s),\ldots,t_{n}+\varphi_{n}(s))e^{L(s,x_{2}-\varphi_{2}(x_{1}),\ldots,x_{n}-\varphi_{n}(x_{1}))}}{f_{1}(s,x_{2}-\varphi_{2}(x_{1})+\varphi_{2}(s),\ldots,x_{n}-\varphi_{n}(x_{1})+\varphi_{n}(s))}ds
≤εm​e−L​(x1,x2−φ2​(x1),…,xn−φn​(x1))​∫x1b∂∂s​(−eL​(s,x2−φ2​(x1),…,xn−φn​(x1)))​𝑑s\displaystyle\leq\frac{\varepsilon}{m}e^{-L(x_{1},x_{2}-\varphi_{2}(x_{1}),\ldots,x_{n}-\varphi_{n}(x_{1}))}\int\nolimits_{x_{1}}^{b}\frac{\partial}{\partial s}\left(-e^{L(s,x_{2}-\varphi_{2}(x_{1}),\ldots,x_{n}-\varphi_{n}(x_{1}))}\right)ds
=εm​e−L​(x1,x2−φ2​(x1),…,xn−φn​(x1))​(eL​(x1,x2−φ2​(x1),…,xn−φn​(x1))−eL​(b,x2−φ2​(x1),…,xn−φn​(x1)))\displaystyle=\frac{\varepsilon}{m}e^{-L(x_{1},x_{2}-\varphi_{2}(x_{1}),\ldots,x_{n}-\varphi_{n}(x_{1}))}\left(e^{L(x_{1},x_{2}-\varphi_{2}(x_{1}),\ldots,x_{n}-\varphi_{n}(x_{1}))}-e^{L(b,x_{2}-\varphi_{2}(x_{1}),\ldots,x_{n}-\varphi_{n}(x_{1}))}\right)
=εm​(1−eL​(b,x2−φ2​(x1),…,xn−φn​(x1))−L​(x1,x2−φ2​(x1),…,xn−φn​(x1)))≤εm,∀(x1,…,xn)∈A,\displaystyle=\frac{\varepsilon}{m}\left(1-e^{L(b,x_{2}-\varphi_{2}(x_{1}),\ldots,x_{n}-\varphi_{n}(x_{1}))-L(x_{1},x_{2}-\varphi_{2}(x_{1}),\ldots,x_{n}-\varphi_{n}(x_{1}))}\right)\leq\frac{\varepsilon}{m},\ \forall(x_{1},\ldots,x_{n})\in A,

so

‖u−u0‖∞≤εm.\left\|u-u_{0}\right\|_{\infty}\leq\frac{\varepsilon}{m}.

The existence is proved.

Uniqueness.

Suppose thatL​(b,x2,…,xn)=−∞\ L(b,x_{2},\ldots,x_{n})=-\infty and for a solution uu of (3) there exist two solutions u1,u2u_{1},u_{2} of D​(u)=0D(u)=0, u1≠u2u_{1}\neq u_{2}, with the property (11). Then uk,k=1,2,u_{k},k=1,2, are given by

uk​(x1,…,xn)=\displaystyle u_{k}(x_{1},\ldots,x_{n})=
=e−L​(x1,x2−φ2​(x1),…,xn−φn​(x1))⋅\displaystyle=e^{-L(x_{1},x_{2}-\varphi_{2}(x_{1}),\ldots,x_{n}-\varphi_{n}(x_{1}))}\cdot
⋅(∫ax1f​(s,x2−φ2​(x1)+φ2​(s),…,xn−φn​(x1)+φn​(s))​eL​(s,x2−φ2​(x1),…,xn−φn​(x1))f1​(s,x2−φ2​(x1)+φ2​(s),…,xn−φn​(x1)+φn​(s))ds+\displaystyle\quad\cdot\left(\int\nolimits_{a}^{x_{1}}\frac{f(s,x_{2}-\varphi_{2}(x_{1})+\varphi_{2}(s),\ldots,x_{n}-\varphi_{n}(x_{1})+\varphi_{n}(s))e^{L(s,x_{2}-\varphi_{2}(x_{1}),\ldots,x_{n}-\varphi_{n}(x_{1}))}}{f_{1}(s,x_{2}-\varphi_{2}(x_{1})+\varphi_{2}(s),\ldots,x_{n}-\varphi_{n}(x_{1})+\varphi_{n}(s))}ds+\right.
+Fk(x2−φ2(x1),…,xn−φn(x1))),k=1,2,F1≠F2.\displaystyle\quad\left.+F_{k}(x_{2}-\varphi_{2}(x_{1}),\ldots,x_{n}-\varphi_{n}(x_{1}))\right),\ k=1,2,\ F_{1}\neq F_{2}.

We have

‖u1​(x1,…,xn)−u2​(x1,…,xn)‖=\displaystyle\left\|u_{1}(x_{1},\ldots,x_{n})-u_{2}(x_{1},\ldots,x_{n})\right\|=
=‖u1​(x1,…,xn)−u​(x1,…,xn)‖+‖u​(x1,…,xn)−u2​(x1,…,xn)‖\displaystyle=\left\|u_{1}(x_{1},\ldots,x_{n})-u(x_{1},\ldots,x_{n})\right\|+\left\|u(x_{1},\ldots,x_{n})-u_{2}(x_{1},\ldots,x_{n})\right\|
≤2​εm,∀(x1,…,xn)∈A,\displaystyle\leq\frac{2\varepsilon}{m},\ \forall(x_{1},\ldots,x_{n})\in A,

which is equivalent to

(13) e−L​(x1,x2−φ2​(x1),…,xn−φn​(x1))​‖F1​(x2−φ2​(x1),…,xn−φn​(x1))−F2​(x2−φ2​(x1),…,xn−φn​(x1))‖\displaystyle e^{-L(x_{1},x_{2}-\varphi_{2}(x_{1}),\ldots,x_{n}-\varphi_{n}(x_{1}))}\left\|F_{1}(x_{2}-\varphi_{2}(x_{1}),\ldots,x_{n}-\varphi_{n}(x_{1}))-F_{2}(x_{2}-\varphi_{2}(x_{1}),\ldots,x_{n}-\varphi_{n}(x_{1}))\right\|
≤2​εm,∀(x1,…,xn)∈A.\displaystyle\leq\frac{2\varepsilon}{m},\ \forall(x_{1},\ldots,x_{n})\in A.

Since u1≠u2u_{1}\neq u_{2} it follows that there exists (x~2,…,x~n)∈ℝn−1(\widetilde{x}_{2},\ldots,\widetilde{x}_{n})\in\mathbb{R}^{n-1} such that F1​(x~2,…,x~n)≠F2​(x~2,…,x~n)F_{1}(\widetilde{x}_{2},\ldots,\widetilde{x}_{n})\neq F_{2}(\widetilde{x}_{2},\ldots,\widetilde{x}_{n}). For

{x2−φ2(x1)=:x~2⋮xn−φn(x1)=:x~n\left\{\begin{array}[c]{c}x_{2}-\varphi_{2}(x_{1})=:\widetilde{x}_{2}\\ \vdots\\ x_{n}-\varphi_{n}(x_{1})=:\widetilde{x}_{n}\end{array}\right.

the relation (13) becomes

(14) e−L​(x1,x~2,…,x~n)​‖F1​(x~2,…,x~n)−F2​(x~2,…,x~n)‖≤2​εm,∀x1∈I.e^{-L(x_{1},\widetilde{x}_{2},\ldots,\widetilde{x}_{n})}\left\|F_{1}(\widetilde{x}_{2},\ldots,\widetilde{x}_{n})-F_{2}(\widetilde{x}_{2},\ldots,\widetilde{x}_{n})\right\|\leq\frac{2\varepsilon}{m},\ \forall x_{1}\in I.

Now letting x1→bx_{1}\rightarrow b in (14) it follows ∞≤\infty\leq 2​εm\dfrac{2\varepsilon}{m}, a contradiction. Uniqueness is proved. ∎

A consequence of the stability result given in Theorem 2.2 is presented in the next corollary.

Corollary 2.3.

LetF=(f2,…,fn):A→ℝn−1,f1≠0\ F=(f_{2},\ldots,f_{n}):A\rightarrow\mathbb{R}^{n-1},\ f_{1}\neq 0 on A.A.\ Suppose that there exists two continuous functions α,β:I→[0,+∞)\alpha,\beta:I\rightarrow[0,+\infty) such that

(15) ‖F​(t,x)‖e≤(α​(t)​‖x‖e+β​(t))​|f1​(t,x)|\left\|F(t,x)\right\|_{e}\leq\left(\alpha(t)\left\|x\right\|_{e}+\beta(t)\right)\left|f_{1}(t,x)\right|

for every (t,x)∈I×ℝn−1(t,x)\in I\times\mathbb{R}^{n-1} and infx∈A|f​(x)|=m,m>0.\inf_{x\in A}\left|f(x)\right|=m,\ m>0. Then the operator DD given by (1) is Ulam stable with the Ulam constant L=1m.L=\frac{1}{m}.

Proof.

Let G=(g2,…,gn),gk=fkf1, 2≤k≤n.G=(g_{2},\ldots,g_{n}),\ g_{k}=\dfrac{f_{k}}{f_{1}},\ 2\leq k\leq n. Then by (15) we get

‖G​(t,x)‖e≤α​(t)​‖x‖e+β​(t),(t,x)∈A,\left\|G(t,x)\right\|_{e}\leq\alpha(t)\left\|x\right\|_{e}+\beta(t),\ (t,x)\in A,

and taking account of Theorem 2.4.5 ([16], see also [5]) we conclude that the system (5) admits a global solution φ:I→ℝn−1.\varphi:I\rightarrow\mathbb{R}^{n-1}. Now the conclusion of the corollary is a consequence of Theorem 2.2. ∎

Remark 2.4.

The conclusion of the Corollary 2.3 holds true if in the right hand of the relation (15) f1f_{1} is replaced by fp, 1≤p≤n,f_{p},\ 1\leq p\leq n,\ with fp≠0f_{p}\neq 0 on A.A.

Example 2.5.

Let a1,…,an∈ℝ,a1≠0,A=[0,+∞)×ℝn−1,f:A→ℝa_{1},\ldots,a_{n}\in\mathbb{R},\ a_{1}\neq 0,\ A=[0,+\infty)\times\mathbb{R}^{n-1},f:A\rightarrow\mathbb{R} be a continuous function with infx∈A|f​(x)|=m,m>0.\inf_{x\in A}\left|f(x)\right|=m,\ m>0. Then the operator with constant coefficients D:C1​(A,X)→C​(A,X)D:C^{1}(A,X)\rightarrow C(A,X)

D​(u)=∑k=1nak​∂v∂xk−f​u,u∈C1​(A,X)D(u)=\sum\limits_{k=1}^{n}a_{k}\frac{\partial v}{\partial x_{k}}-fu,\ \ u\in C^{1}(A,X)

is Ulam stable with the Ulam constant L=1m.L=\frac{1}{m}.

Proof.

Clearly the system (5) associated with the operator DD has a global solution φ:[0,+∞)→ℝn−1,\varphi:[0,+\infty)\rightarrow\mathbb{R}^{n-1}, so the conclusion follows according to Theorem 2.2. ∎

Example 2.6.

Let A=I×ℝ,f1,f2,f∈C​(A,ℝ)A=I\times\mathbb{R},\ f_{1},f_{2},f\in C(A,\mathbb{R}) and D:C1​(A,X)→C​(A,X),D:C^{1}(A,X)\rightarrow C(A,X),

D​(u)=f1​∂u∂x1+f2​∂u∂x2−f​u.D(u)=f_{1}\frac{\partial u}{\partial x_{1}}+f_{2}\frac{\partial u}{\partial x_{2}}-fu.

If inf(x1,x2)∈A|f​(x1,x2)|=m,m>0\inf\limits_{(x_{1},x_{2})\in A}\left|f(x_{1},x_{2})\right|=m,m>0 and there exist two continuous functions α,β:I→[0,∞)\alpha,\beta:I\rightarrow[0,\infty) such that

(16) |f2​(x1,x2)f1​(x1,x2)|≤α​(x1)​|x2|+β​(x2),\left|\frac{f_{2}(x_{1},x_{2})}{f_{1}(x_{1},x_{2})}\right|\leq\alpha(x_{1})\left|x_{2}\right|+\beta(x_{2}),

for all (x1,x2)∈A,(x_{1},x_{2})\in A, then DD is Ulam stable with the Ulam constant L=1mL=\frac{1}{m}.

Proof.

The condition (16) leads to the existence of a global solution of the system (5) which in this case is the equation

x2′=f2​(x1,x2)f1​(x1,x2).x_{2}^{\prime}=\frac{f_{2}(x_{1},x_{2})}{f_{1}(x_{1},x_{2})}.

The conclusion follows in view of Corollary 2.3. ∎

3. A nonstability result

We show in what follows that if the condition infx∈A​|f​(x)|=m,m>0,\underset{x\in A}{\inf}\left|f(x)\right|=m,\ m>0, is not satisfied, then the operator DD defined by (1) is not generally Ulam stable. In this respect we have the following nonstability result for the case f=0f=0.

Theorem 3.1.

Let I=[a,∞),A=I×ℝn−1,n≥2.I=[a,\infty),\ A=I\times\mathbb{R}^{n-1},n\geq 2. Suppose that the system (5) admits a solution (φ2,…,φn):I→ℝn−1(\varphi_{2},\ldots,\varphi_{n}):I\rightarrow\mathbb{R}^{n-1} and there exists p∈{1,…,n}p\in\{1,\ldots,n\} such that fp≠0f_{p}\neq 0 on AA and supx∈A​|fp​(x)|<+∞.\underset{x\in A}{\sup}\left|f_{p}(x)\right|<+\infty. Then the operator D~:C1​(A,X)→C​(A,X)\widetilde{D}:C^{1}(A,X)\rightarrow C(A,X)

D~​(u)=∑k=1nfk​∂u∂xk,u∈C1​(A,X),\widetilde{D}(u)={\textstyle\sum\limits_{k=1}^{n}}f_{k}\frac{\partial u}{\partial x_{k}},\ \ \ u\in C^{1}(A,X),

is not Ulam stable.

Proof.

Without loss of generality we may suppose that p=1p=1 and let sup|f1​(x)|=λ,λ≠∞.\sup\left|f_{1}(x)\right|=\lambda,\lambda\neq\infty. Suppose that D~\widetilde{D} is Ulam stable with the Ulam constant L>0L>0. Let ε>0\varepsilon>0 and u∈C1​(A,X)u\in C^{1}(A,X) a solution of the equationD~​(u)=ε​θ,θ∈X,‖θ‖=1.\ \widetilde{D}(u)=\varepsilon\theta,\ \theta\in X,\left\|\theta\right\|=1. Then, according to Lemma 2.1, uu has the form

u​(x)\displaystyle u(x) =∫ax1ε​θf1​(s,x2−φ2​(x1)+φ2​(s),…,xn−φn​(x1)+φn​(s))​𝑑s\displaystyle={\displaystyle\int\nolimits_{a}^{x_{1}}}\frac{\varepsilon\theta}{f_{1}(s,x_{2}-\varphi_{2}(x_{1})+\varphi_{2}(s),\ldots,x_{n}-\varphi_{n}(x_{1})+\varphi_{n}(s))}ds
+F​(x2−φ2​(x1),…,xn−φn​(x1)),\displaystyle\quad+F(x_{2}-\varphi_{2}(x_{1}),\ldots,x_{n}-\varphi_{n}(x_{1})),

where F∈C1​(ℝn−1,X),x=(x1,…,xn)∈A.F\in C^{1}(\mathbb{R}^{n-1},X),\ x=(x_{1},\ldots,x_{n})\in A.

Since ‖D~​(u)‖∞=ε,\left\|\widetilde{D}(u)\right\|_{\infty}=\varepsilon, it follows, in view of the stability of D~\widetilde{D}, that there exists u0∈ker⁡D~\ u_{0}\in\ker\widetilde{D} such that

(17) ‖u−u0‖∞≤L​ε.\left\|u-u_{0}\right\|_{\infty}\leq L\varepsilon.

Now, taking into account again Lemma 2.1 we get

u0​(x)=G​(x2−φ2​(x1),…,xn−φn​(x1)),x=(x1,…,xn)∈A,u_{0}(x)=G(x_{2}-\varphi_{2}(x_{1}),\ldots,x_{n}-\varphi_{n}(x_{1})),\ x=(x_{1},\ldots,x_{n})\in A,

for some G∈C1​(ℝn−1,X).G\in C^{1}(\mathbb{R}^{n-1},X).

Let z​(x)=u​(x)−w​(x),x∈Az(x)=u(x)-w(x),\ x\in A and let xk=φk​(x1), 2≤k≤n.x_{k}=\varphi_{k}(x_{1}),\ 2\leq k\leq n.

Then

z​(x1,φ2​(x1),…,φn​(x1))=∫ax1ε​θf1​(s,φ2​(s),…,φn​(s))​𝑑s+F​(0,…,0)−G​(0,…,0),z(x_{1},\varphi_{2}(x_{1}),\ldots,\varphi_{n}(x_{1}))={\displaystyle\int\nolimits_{a}^{x_{1}}}\frac{\varepsilon\theta}{f_{1}(s,\varphi_{2}(s),\ldots,\varphi_{n}(s))}ds+F(0,\ldots,0)-G(0,\ldots,0),

and

‖z​(x1,φ2​(x1),…,φn​(x1))−F​(0,…,0)+G​(0,…,0)‖=\displaystyle\left\|z(x_{1},\varphi_{2}(x_{1}),\ldots,\varphi_{n}(x_{1}))-F(0,\ldots,0)+G(0,\ldots,0)\right\|=
=∫ax1ε|f1​(s,φ2​(s),…,φn​(s))|​𝑑s≥\displaystyle={\displaystyle\int\nolimits_{a}^{x_{1}}}\frac{\varepsilon}{\left|f_{1}(s,\varphi_{2}(s),\ldots,\varphi_{n}(s))\right|}ds\geq
≥ελ​(x1−a),∀x1∈[a,∞).\displaystyle\geq\frac{\varepsilon}{\lambda}(x_{1}-a),\forall x_{1}\in[a,\infty).

Therefore, denoting F​(0,…,0)−G​(0,…,0)=h,h∈XF(0,\ldots,0)-G(0,\ldots,0)=h,\ h\in X, we get

‖z​(x1,φ2​(x1),…,φn​(x1))‖\displaystyle\left\|z(x_{1},\varphi_{2}(x_{1}),\ldots,\varphi_{n}(x_{1}))\right\| ≥‖z​(x1,φ2​(x1),…,φn​(x1))−h‖−‖h‖\displaystyle\geq\left\|z(x_{1},\varphi_{2}(x_{1}),\ldots,\varphi_{n}(x_{1}))-h\right\|-\left\|h\right\|
≥ελ​(x1−a)−‖h‖,∀x1∈[a,∞).\displaystyle\geq\frac{\varepsilon}{\lambda}(x_{1}-a)-\left\|h\right\|,\forall x_{1}\in[a,\infty).

Hence

limx1→∞​‖z​(x1,φ2​(x1),…,φn​(x1))‖=+∞,\underset{x_{1}\rightarrow\infty}{\lim}\left\|z(x_{1},\varphi_{2}(x_{1}),\ldots,\varphi_{n}(x_{1}))\right\|=+\infty,

a contradiction to (17). The theorem is proved. ∎

Example 3.2.

Let a1,a2,…,an∈ℝ,(a1,…,an)≠(0,…,0),I=[a,∞),A=I×ℝn−1,a_{1},a_{2},\ldots,a_{n}\in\mathbb{R},(a_{1},\ldots,a_{n})\neq(0,\ldots,0),\ I=[a,\infty),A=I\times\mathbb{R}^{n-1}, and D:C1​(A,X)→C​(A,X)D:C^{1}(A,X)\rightarrow C(A,X)

D​(u)=∑k=1nak​∂u∂xk,u∈C1​(A,X).D(u)={\textstyle\sum\limits_{k=1}^{n}}a_{k}\frac{\partial u}{\partial x_{k}},\ u\in C^{1}(A,X).

Then DD is not Ulam stable.

Proof.

Suppose a1≠0a_{1}\neq 0 without loss of generality. Then the system (5) admits in this case a global solution and the operator DD is not Ulam stable according to Theorem 3.1. ∎

4. An application

One of the best known forms for the study of production functions in economics is Cobb-Douglas production function (see [4]). C.W. Cobb, a mathematician and P.H. Douglass, an economist, obtained a function which provides a relation between labor, capital and the production function. Cobb proposed a relation of the form

(18) Q=A​Lα​KβQ=AL^{\alpha}K^{\beta}

where QQ is the production function, LL is the labor, KK is the capital and A,α,βA,\alpha,\beta are some positive constants. Remark that QQ is a homogeneous function of degree α+β,\alpha+\beta, so it satisfies the Euler’s partial differential equation

(19) L​∂Q∂L+K​∂Q∂K=(α+β)​Q.L\frac{\partial Q}{\partial L}+K\frac{\partial Q}{\partial K}=(\alpha+\beta)Q.

Over the time, the Cobb-Douglas production functions have received appreciations and criticism from many scientists (see [12]). That’s why we think that the consideration of the notion of approximate production function is justified. We call approximate production function a function which satisfies approximately the equation (19) in the sense defined in this paper. In this respect the study of Ulam stability of the equation (19) is important. We have the following result.

Theorem 4.1.

Let ε>0\varepsilon>0 and suppose that an approximate production function Q​(L,K)Q(L,K) satisfies the relation

|L​∂Q∂L+K​∂Q∂K−(α+β)​Q|≤ε,\left|L\frac{\partial Q}{\partial L}+K\frac{\partial Q}{\partial K}-(\alpha+\beta)Q\right|\leq\varepsilon,

for all L,K>0.L,K>0. Then there exists a production function Q¯​(L,K)\overline{Q}(L,K), i.e., Q¯\overline{Q} satisfies the equation (19) such that

|Q​(L,K)−Q¯​(L,K)|≤εα+β,\left|Q(L,K)-\overline{Q}(L,K)\right|\leq\frac{\varepsilon}{\alpha+\beta},

for all L,K>0.L,K>0.

Proof.

The proof follows as a simple consequence of Theorem 2.2. ∎

We conclude that the Cobb-Douglas equation is Ulam stable, so small perturbations of it produces small perturbations of the production function.

Some other practical applications of Ulam stability for the partial differential operator DD can be obtained for some processes governed by first order semilinear partial differential equations as conservation laws, trafic flow, linear transport equation, etc.

References

  • [1] Baias, A.R., Popa, D., On the best Ulam constant of the second order linear differential operator. Revista de la Real Academia de Ciencias Exactas 2020 114 (23), https://doi.org/10.1007/s13398-019-00776-4.
  • [2] Brzdek, J., Popa, D., Raşa, I., and Xu, B., Ulam stability of operators, Academic Press, 2018.
  • [3] Brzdek, J., Popa, D., Raşa, I., Hyers–Ulam stability with respect to gauges, Journal of Mathematical Analysis and Applications, 2017 453 (1), 620–628, https://doi.org/10.1016/j.jmaa.2017.04.022.
  • [4] Carter, S., On the Cobb-Douglas and all that’: The Solow-Simon correspondence over the neoclassical aggregate production function, Journal of Post Keynesian Economics, 2011 34(2), 255–274.
  • [5] Corduneanu, C., Principles of differential and integral equations, Chelsea Publishing Company The Bronx, New York, 1977.
  • [6] Fukutaka, R., Onitsuka, M., Best constant in Hyers-Ulam stability of first order homogeneous linear differential equations with a periodic coefficient. Journal of Mathematical Analysis 2019, 473(2), 1432–1446, https://doi.org/10.1016/j.jmaa.2019.01.030.
  • [7] Hyers, D. H., On the stability of the linear functional equation. Proc. Matl. Acad. Sci. USA 1941, 27, 222–224.
  • [8] Jung, S.-M., Hyers–Ulam Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press, Palm Harbour, 2001.
  • [9] Jung, S. M., Hyers-Ulam stability of linear partial differential equations of first order. Appl. Math. Lett., 2009, 22, 70–74.
  • [10] Lungu, N., Popa, D., Hyers-Ulam stability of a first order partial differential equation. J. Math. Anal. Appl., 2012, 385, 86–91.
  • [11] Marian, D., Ciplea, S.A., Lungu, N., Ulam-Hyers stability of a parabolic partial differential equation. Demonstratio Mathematica, 2019, 52(1) 475–481, https://doi.org/10.1515/dema-2019-0040.
  • [12] Popa, D., Ilca, M., On approximate Cobb-Douglas production functions. Carpathian J. Math., 2014, 30(1), 87–92.
  • [13] Prastaro, A., Rassias, Th.M., Ulam stability in geometry of PDE’s. Nonlinear Functional Analysis and Applications, 202, 8(2), 259–278.
  • [14] Rus, I. A., Remarks on Ulam stability of the operatorial equations. Fixed Point Theory, 2009, 10, 305–320.
  • [15] Ulam, S. M., A Collection of Mathematical Problems, Interscience, New York, 1960.
  • [16] Vrabie, I. I., Differential equations. An introduction to basic concepts, results and applications, Word Scientific, 2004.
2023

Related Posts