Table of Contents
ISRN Mathematical Analysis
Volume 2012 (2012), Article ID 609754, 10 pages
http://dx.doi.org/10.5402/2012/609754
Research Article

Ulam-Hyers-Rassias Stability of a Hyperbolic Partial Differential Equation

1Department of Mathematics, Technical University of Cluj-Napoca, Street C. Daicoviciu 15, 400020, Cluj-Napoca, Romania
2Department of Mathematics, Colfe's School, Horn Park Lane, London SE12 8AW, UK

Received 21 October 2011; Accepted 30 November 2011

Academic Editors: K. H. Kwon and J. V. Stokman

Copyright © 2012 Nicolaie Lungu and Cecilia Crăciun. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We consider a nonlinear hyperbolic partial differential equation in a general form. Using a Gronwall-type lemma we prove results on the Ulam-Hyers stability and the generalised Ulam-Hyers-Rassias stability of this equation.

1. Introduction

Results on Ulam stability for the functional equations are well known (see, e.g., [14]). In this paper we consider different types of Ulam stability of a nonlinear hyperbolic partial differential equation. Following results from Rus [5] and Rus and Lungu [6], we present some Ulam-Hyers and Ulam-Hyers-Rassias stabilities for the following equation: 𝜕2𝑢𝜕𝑥𝜕𝑦(𝑥,𝑦)=𝑓𝑥,𝑦,𝑢(𝑥,𝑦),𝜕𝑢𝜕𝑥(𝑥,𝑦),𝜕𝑢𝜕𝑦(𝑥,𝑦),0𝑥<𝑎,0𝑦<𝑏,(1.1) where 𝑓𝐶([0,𝑎)×[0,𝑏)×𝔹3,𝔹) and (𝔹,||) is a real or complex Banach space.

2. Preliminaries

In this section we present some definitions and results on different types of Ulam stability for a hyperbolic partial differential equation. We also present a Gronwall type lemma, which will be useful in the proof of the main results.

Let 𝑎,𝑏(0,], 𝜀>0, 𝜑𝐶([0,𝑎)×[0,𝑏),+) and (𝔹,||) be a real or complex Banach space.

We consider the following hyperbolic partial differential equation:𝜕2𝑢𝜕𝑥𝜕𝑦(𝑥,𝑦)=𝑓𝑥,𝑦,𝑢(𝑥,𝑦),𝜕𝑢𝜕𝑥(𝑥,𝑦),𝜕𝑢𝜕𝑦(𝑥,𝑦),0𝑥<𝑎,0𝑦<𝑏,(2.1) where 𝑓𝐶([0,𝑎)×[0,𝑏)×𝔹3,𝔹).

We also consider the following inequalities:||||𝜕2𝑣𝜕𝑥𝜕𝑦(𝑥,𝑦)𝑓𝑥,𝑦,𝑣(𝑥,𝑦),𝜕𝑣𝜕𝑥(𝑥,𝑦),𝜕𝑣||||[[||||𝜕𝜕𝑦(𝑥,𝑦)𝜀,𝑥0,𝑎),𝑦0,𝑏),(2.2)2𝑣(𝜕𝑥𝜕𝑦𝑥,𝑦)𝑓𝑥,𝑦,𝑣(𝑥,𝑦),𝜕𝑣(𝜕𝑥𝑥,𝑦),𝜕𝑣(||||[[||||𝜕𝜕𝑦𝑥,𝑦)𝜑(𝑥,𝑦),𝑥0,𝑎),𝑦0,𝑏),(2.3)2𝑣𝜕𝑥𝜕𝑦(𝑥,𝑦)𝑓𝑥,𝑦,𝑣(𝑥,𝑦),𝜕𝑣𝜕𝑥(𝑥,𝑦),𝜕𝑣||||[[𝜕𝑦(𝑥,𝑦)𝜀𝜑(𝑥,𝑦),𝑥0,𝑎),𝑦0,𝑏).(2.4)

Definition 2.1. A function 𝑢 is a solution to (2.1) if 𝑢𝐶1([[𝜕0,𝑎)×0,𝑏),𝔹),2𝑢([[𝜕𝑥𝜕𝑦𝐶0,𝑎)×0,𝑏),𝔹)(2.5) and 𝑢 satisfies (2.1).

From [5, 6] we have the following definitions and results.

Definition 2.2. Equation (2.1) is Ulam-Hyers stable if there exist the real numbers 𝐶1𝑓, 𝐶2𝑓, and 𝐶3𝑓>0 such that for any 𝜀>0 and for any solution 𝑣 to the inequality (2.2) there exists a solution 𝑢 to (2.1) with ||||𝑣(𝑥,𝑦)𝑢(𝑥,𝑦)𝐶1𝑓[[|||𝜀,𝑥0,𝑎),𝑦0,𝑏),𝜕𝑣(𝜕𝑥𝑥,𝑦)𝜕𝑢(|||𝜕𝑥𝑥,𝑦)𝐶2𝑓[[||||𝜀,𝑥0,𝑎),𝑦0,𝑏),𝜕𝑣𝜕𝑦(𝑥,𝑦)𝜕𝑢||||𝜕𝑦(𝑥,𝑦)𝐶3𝑓[[𝜀,𝑥0,𝑎),𝑦0,𝑏).(2.6)

Definition 2.3. Equation (2.1) is generalised Ulam-Hyers-Rassias stable if there exist the real numbers 𝐶1𝑓,𝜑, 𝐶2𝑓,𝜑, and 𝐶3𝑓,𝜑>0 such that for any 𝜀>0 and for any solution 𝑣 to the inequality (2.3) there exists a solution 𝑢 to (2.1) with ||||𝑣(𝑥,𝑦)𝑢(𝑥,𝑦)𝐶1𝑓,𝜑[[|||𝜑(𝑥,𝑦),𝑥0,𝑎),𝑦0,𝑏),𝜕𝑣(𝜕𝑥𝑥,𝑦)𝜕𝑢(|||𝜕𝑥𝑥,𝑦)𝐶2𝑓,𝜑[[||||𝜑(𝑥,𝑦),𝑥0,𝑎),𝑦0,𝑏),𝜕𝑣𝜕𝑦(𝑥,𝑦)𝜕𝑢||||𝜕𝑦(𝑥,𝑦)𝐶3𝑓,𝜑[[𝜑(𝑥,𝑦),𝑥0,𝑎),𝑦0,𝑏).(2.7)

Remark 2.4. A function 𝑣 is a solution to the inequality (2.2) if and only if, there exists a function 𝑔𝐶([0,𝑎)×[0,𝑏),𝔹), which depends on 𝑣, such that(i)For all 𝜀>0,|𝑔(𝑥,𝑦)|𝜀, for all 𝑥[0,𝑎), for all 𝑦[0,𝑏);(ii)For all 𝑥[0,𝑎), for all 𝑦[0,𝑏):𝜕2𝑣𝜕𝑥𝜕𝑦(𝑥,𝑦)=𝑓𝑥,𝑦,𝑣(𝑥,𝑦),𝜕𝑣𝜕𝑥(𝑥,𝑦),𝜕𝑣𝜕𝑦(𝑥,𝑦)+𝑔(𝑥,𝑦).(2.8)

Remark 2.5. A function 𝑣 is a solution to the inequality (2.3) if and only if, there exists a function 𝑔𝐶([0,𝑎)×[0,𝑏),𝔹), which depends on 𝑣, such that(i)|𝑔(𝑥,𝑦)|𝜑(𝑥,𝑦), for all 𝑥[0,𝑎), for all 𝑦[0,𝑏);(ii)For all 𝑥[0,𝑎), for all 𝑦[0,𝑏):𝜕2𝑣𝜕𝑥𝜕𝑦(𝑥,𝑦)=𝑓𝑥,𝑦,𝑣(𝑥,𝑦),𝜕𝑣𝜕𝑥(𝑥,𝑦),𝜕𝑣𝜕𝑦(𝑥,𝑦)+𝑔(𝑥,𝑦).(2.9)

Remark 2.6. A function 𝑣 is a solution to the inequality (2.4) if and only if, there exists a function 𝑔𝐶([0,𝑎)×[0,𝑏),𝔹), which depends on 𝑣, such that(i)For all 𝜀>0,|𝑔(𝑥,𝑦)|𝜀𝜑(𝑥,𝑦), for all 𝑥[0,𝑎), for all 𝑦[0,𝑏);(ii)For all 𝑥[0,𝑎), for all 𝑦[0,𝑏):𝜕2𝑣𝜕𝑥𝜕𝑦(𝑥,𝑦)=𝑓𝑥,𝑦,𝑣(𝑥,𝑦),𝜕𝑣𝜕𝑥(𝑥,𝑦),𝜕𝑣𝜕𝑦(𝑥,𝑦)+𝑔(𝑥,𝑦).(2.10)

Throughout this paper we denote𝑢1(𝑥,𝑦)=𝜕𝑢𝜕𝑥(𝑥,𝑦),𝑢2(𝑥,𝑦)=𝜕𝑢𝑣𝜕𝑦(𝑥,𝑦),1(𝑥,𝑦)=𝜕𝑣𝜕𝑥(𝑥,𝑦),𝑣2(𝑥,𝑦)=𝜕𝑣𝜕𝑦(𝑥,𝑦).(2.11)

Theorem 2.7. If 𝑣 is a solution to the inequality (2.2), then (𝑣,𝑣1,𝑣2) satisfies the following system of integral inequalities: ||||𝑣(𝑥,𝑦)𝑣(𝑥,0)𝑣(0,𝑦)+𝑣(0,0)𝑥0𝑦0𝑓𝑠,𝑡,𝑣(𝑠,𝑡),𝑣1(𝑠,𝑡),𝑣2||||||||𝑣(𝑠,𝑡)𝑑𝑠𝑑𝑡𝜀𝑥𝑦,1(𝑥,𝑦)𝑣1(𝑥,0)𝑦0𝑓𝑥,𝑡,𝑣(𝑥,𝑡),𝑣1(𝑥,𝑡),𝑣2(||||||||𝑣𝑥,𝑡)𝑑𝑡𝜀𝑦,2(𝑥,𝑦)𝑣2(0,𝑦)𝑥0𝑓𝑠,𝑦,𝑣(𝑠,𝑦),𝑣1(𝑠,𝑦),𝑣2||||(𝑠,𝑦)𝑑𝑠𝜀𝑥,(2.12) for all 𝑥[0,𝑎) and 𝑦[0,𝑏).

Proof. By Remark 2.4 we have that +𝑣(𝑥,𝑦)=𝑣(𝑥,0)+𝑣(0,𝑦)𝑣(0,0)𝑥0𝑦0𝑓𝑠,𝑡,𝑣(𝑠,𝑡),𝑣1(𝑠,𝑡),𝑣2(𝑠,𝑡)𝑑𝑠𝑑𝑡+𝑥0𝑦0𝑣𝑔(𝑠,𝑡)𝑑𝑠𝑑𝑡1(𝑥,𝑦)=𝑣1(𝑥,0)+𝑦0𝑓𝑥,𝑡,𝑣(𝑥,𝑡),𝑣1(𝑥,𝑡),𝑣2(𝑥,𝑡)𝑑𝑡+𝑦0𝑣𝑔(𝑥,𝑡)𝑑𝑡2(𝑥,𝑦)=𝑣2(0,𝑦)+𝑥0𝑓𝑠,𝑦,𝑣(𝑠,𝑦),𝑣1(𝑠,𝑦),𝑣2(𝑠,𝑦)𝑑𝑠+𝑥0𝑔(𝑠,𝑦)𝑑𝑠,(2.13) and we have the following inequalities: ||||𝑣(𝑥,𝑦)𝑣(𝑥,0)𝑣(0,𝑦)+𝑣(0,0)𝑥0𝑦0𝑓𝑠,𝑡,𝑣(𝑠,𝑡),𝑣1(𝑠,𝑡),𝑣2||||=||||(𝑠,𝑡)𝑑𝑠𝑑𝑡𝑥0𝑦0||||𝑔(𝑠,𝑡)𝑑𝑠𝑑𝑡𝑥0𝑦0||||||||𝑣𝑔(𝑠,𝑡)𝑑𝑠𝑑𝑡𝜀𝑥𝑦,1(𝑥,𝑦)𝑣1(𝑥,0)𝑦0𝑓𝑥,𝑡,𝑣(𝑥,𝑡),𝑣1(𝑥,𝑡),𝑣2||||(𝑥,𝑡)𝑑𝑡𝑦0||𝑔||||||𝑣(𝑥,𝑡)𝑑𝑡𝜀𝑦,2(𝑥,𝑦)𝑣2(0,𝑦)𝑥0𝑓𝑠,𝑦,𝑣(𝑠,𝑦),𝑣1(𝑠,𝑦),𝑣2(||||𝑠,𝑦)𝑑𝑠𝑥0||||𝑔(𝑠,𝑦)𝑑𝑠𝜀𝑥.(2.14)

The following two theorems are obtained in a similar fashion.

Theorem 2.8. If 𝑣 is a solution to the inequality (2.3), then (𝑣,𝑣1,𝑣2) satisfies the following system of integral inequalities: ||||𝑣(𝑥,𝑦)𝑣(𝑥,0)𝑣(0,𝑦)+𝑣(0,0)𝑥0𝑦0𝑓𝑠,𝑡,𝑣(𝑠,𝑡),𝑣1(𝑠,𝑡),𝑣2||||(𝑠,𝑡)𝑑𝑠𝑑𝑡𝑥0𝑦0||||𝑣𝜑(𝑠,𝑡)𝑑𝑠𝑑𝑡,1(𝑥,𝑦)𝑣1(𝑥,0)𝑦0𝑓𝑥,𝑡,𝑣(𝑥,𝑡),𝑣1(𝑥,𝑡),𝑣2||||(𝑥,𝑡)𝑑𝑡𝑦0||||𝑣𝜑(𝑥,𝑡)𝑑𝑡,2(𝑥,𝑦)𝑣2(0,𝑦)𝑥0𝑓𝑠,𝑦,𝑣(𝑠,𝑦),𝑣1(𝑠,𝑦),𝑣2||||(𝑠,𝑦)𝑑𝑠𝑥0𝜑(𝑠,𝑦)𝑑𝑠,(2.15) for all 𝑥[0,𝑎) and 𝑦[0,𝑏).

Theorem 2.9. If 𝑣 is a solution to the inequality (2.4), then (𝑣,𝑣1,𝑣2) satisfies the following system of integral inequalities: ||||𝑣(𝑥,𝑦)𝑣(𝑥,0)𝑣(0,𝑦)+𝑣(0,0)𝑥0𝑦0𝑓𝑠,𝑡,𝑣(𝑠,𝑡),𝑣1(𝑠,𝑡),𝑣2||||(𝑠,𝑡)𝑑𝑠𝑑𝑡𝜀𝑥0𝑦0||||𝑣𝜑(𝑠,𝑡)𝑑𝑠𝑑𝑡,1(𝑥,𝑦)𝑣1(𝑥,0)𝑦0𝑓𝑥,𝑡,𝑣(𝑥,𝑡),𝑣1(𝑥,𝑡),𝑣2||||(𝑥,𝑡)𝑑𝑡𝜀𝑦0||||𝑣𝜑(𝑥,𝑡)𝑑𝑡,2(𝑥,𝑦)𝑣2(0,𝑦)𝑥0𝑓𝑠,𝑦,𝑣(𝑠,𝑦),𝑣1(𝑠,𝑦),𝑣2||||(𝑠,𝑦)𝑑𝑠𝜀𝑥0𝜑(𝑠,𝑦)𝑑𝑠,(2.16) for all 𝑥[0,𝑎) and 𝑦[0,𝑏).

The following Gronwall type lemma is an important tool in proving the main results of this paper.

Lemma 2.10 (see [7], also [8]). One assumes that(i)𝑢,𝑣,𝐶(𝑛+,+); (ii)for any 𝑡𝑡0 one has 𝑢(𝑡)(𝑡)+𝑡𝑡0𝑣(𝑠)𝑢(𝑠)𝑑𝑠;(2.17)(iii)(𝑡) is positive and nondecreasing.Then, 𝑢(𝑡)(𝑡)exp𝑡𝑠𝑣(𝑟)𝑑𝑟,forany𝑡𝑡0.(2.18)

3. Ulam-Hyers Stability

In this section we present a result on the existence and uniqueness of the solution to (2.1) and derive a result on Ulam-Hyers stability for the same equation in the case of 𝑎< and 𝑏<.

Theorem 3.1. One assumes that(i)𝑎<, 𝑏<; (ii)𝑓𝐶([0,𝑎]×[0,𝑏]×𝔹3,𝔹);(iii)there exists 𝐿𝑓>0 such that ||𝑓𝑥,𝑦,𝑧1,𝑧2,𝑧3𝑓𝑥,𝑦,𝑡1,𝑡2,𝑡3||𝐿𝑓||𝑧max𝑖𝑡𝑖||,,𝑖=1,2,3(3.1) for all 𝑥[0,𝑎], 𝑦[0,𝑏] and 𝑧1,𝑧2,𝑧3,𝑡1,𝑡2,𝑡3𝔹.
Then,(a)for 𝜙𝐶1([0,𝑎],𝔹) and 𝜓𝐶1([0,𝑏],𝔹) (2.1) has a unique solution, which satisfies [],𝑢[];𝑢(𝑥,0)=𝜙(𝑥),𝑥0,𝑎(0,𝑦)=𝜓(𝑦),𝑦0,𝑏(3.2)(b)(2.1) is Ulam-Hyers stable.

Proof. (a) This is a known result (see, e.g, [9, 10]).
(b) Let 𝑣 be a solution to the inequality (2.2) and let 𝑢 be the unique solution to (2.1), which satisfies the following conditions: [],𝑢[].𝑢(𝑥,0)=𝑣(𝑥,0),𝑥0,𝑎(0,𝑦)=𝑣(0,𝑦),𝑦0,𝑏(3.3) From Theorem 2.7, the hypothesis (iii), and Gronwall Lemma 2.10, it follows that ||||||||𝑣(𝑥,𝑦)𝑢(𝑥,𝑦)𝑣(𝑥,𝑦)𝑣(𝑥,0)𝑣(0,𝑦)+𝑣(0,0)𝑥0𝑦0𝑓𝑠,𝑡,𝑣(𝑠,𝑡),𝑣1(𝑠,𝑡),𝑣2||||+(𝑠,𝑡)𝑑𝑠𝑑𝑡𝑥0𝑦0||𝑓𝑠,𝑡,𝑣(𝑠,𝑡),𝑣1(𝑠,𝑡),𝑣2(𝑠,𝑡)𝑓𝑠,𝑡,𝑢(𝑠,𝑡),𝑢1(𝑠,𝑡),𝑢2||(𝑠,𝑡)𝑑𝑠𝑑𝑡𝜀𝑥𝑦+𝐿𝑓𝑥0𝑦0max𝑖{1,2,3}||𝑣𝑖(𝑠,𝑡)𝑢𝑖||𝐿(𝑠,𝑡)𝑑𝑠𝑑𝑡𝜀𝑎𝑏exp𝑓𝑎𝑏=𝐶1𝑓𝜀,where𝐶1𝑓𝐿=𝑎𝑏exp𝑓.𝑎𝑏(3.4) Similarly we have ||𝑣1(𝑥,𝑦)𝑢1||||||𝑣(𝑥,𝑦)1(𝑥,𝑦)𝑣1(𝑥,0)𝑦0𝑓𝑥,𝑡,𝑣(𝑥,𝑡),𝑣1(𝑥,𝑡),𝑣2||||+(𝑥,𝑡)𝑑𝑡𝑦0||𝑓𝑥,𝑡,𝑣(𝑥,𝑡),𝑣1(𝑥,𝑡),𝑣2(𝑥,𝑡)𝑓𝑥,𝑡,𝑢(𝑥,𝑡),𝑢1(𝑥,𝑡),𝑢2||(𝑥,𝑡)𝑑𝑡𝜀𝑦+𝐿𝑓𝑦0max𝑖{1,2,3}||𝑣𝑖(𝑥,𝑡)𝑢𝑖||𝐿(𝑥,𝑡)𝑑𝑡𝜀𝑏exp𝑓𝑏=𝐶2𝑓𝜀,where𝐶2𝑓𝐿=𝑏exp𝑓𝑏,||𝑣2(𝑥,𝑦)𝑢2||||||𝑣(𝑥,𝑦)2(𝑥,𝑦)𝑣2(0,𝑦)𝑥0𝑓𝑠,𝑦,𝑣(𝑠,𝑦),𝑣1(𝑠,𝑦),𝑣2||||+(𝑠,𝑦)𝑑𝑠𝑥0||𝑓𝑠,𝑦,𝑣(𝑠,𝑦),𝑣1(𝑠,𝑦),𝑣2(𝑠,𝑦)𝑓𝑠,𝑦,𝑢(𝑠,𝑦),𝑢1(𝑠,𝑦),𝑢2||(𝑠,𝑦)𝑑𝑠𝜀𝑥+𝐿𝑓𝑥0max𝑖{1,2,3}||𝑣𝑖(𝑠,𝑦)𝑢𝑖(||𝐿𝑠,𝑦)𝑑𝑠𝜀𝑎exp𝑓𝑎=𝐶3𝑓𝜀,where𝐶3𝑓𝐿=𝑎exp𝑓𝑎.(3.5)

Remark 3.2. In general, if 𝑎= or 𝑏=, then (2.1) is not Ulam-Hyers stable.

4. Generalised Ulam-Hyers-Rassias Stability

In this section we prove the generalised Ulam-Hyers-Rassias stability of the hyperbolic partial differential equation (2.1). We consider (2.1) and the inequality (2.3) in the case 𝑎= and 𝑏=.

Theorem 4.1. One assumes that(i)𝐶([0,)×[0,)×𝔹3,𝔹);(ii)there exists 𝑙𝑓𝐶1([0,)×[0,),+) such that ||𝑓𝑥,𝑦,𝑧1,𝑧2,𝑧3𝑓𝑥,𝑦,𝑡1,𝑡2,𝑡3||𝑙𝑓||𝑧(𝑥,𝑦)max𝑖𝑡𝑖||,,𝑖=1,2,3(4.1) for all 𝑥,𝑦[0,);(iii)there exist 𝜆1𝜑,𝜆2𝜑,𝜆3𝜑>0 such that 𝑥0𝑦0𝜑(𝑠,𝑡)𝑑𝑠𝑑𝑡𝜆1𝜑[𝜑(𝑥,𝑦),𝑥,𝑦0,),𝑦0𝜑(𝑥,𝑡)𝑑𝑡𝜆2𝜑[𝜑(𝑥,𝑦),𝑥,𝑦0,),𝑥0𝜑(𝑠,𝑦)𝑑𝑠𝜆3𝜑[𝜑(𝑥,𝑦),𝑥,𝑦0,);(4.2)(iv)𝜑+×++ is increasing.Then (2.1) (𝑎= and 𝑏=) is generalised Ulam-Hyers-Rassias stable.

Proof. Let 𝑣 be a solution to the inequality (2.3). Denote by 𝑢 the unique solution to the Darboux problem: 𝜕2𝑢𝜕𝑥𝜕𝑦(𝑥,𝑦)=𝑓𝑥,𝑦,𝑢(𝑥,𝑦),𝑢1(𝑥,𝑦),𝑢2[[[(𝑥,𝑦),𝑥,𝑦0,),𝑢(𝑥,0)=𝑣(𝑥,0),𝑥0,),𝑢(0,𝑦)=𝑣(0,𝑦),𝑦0,).(4.3) If 𝑢 is a solution to the Darboux problem (4.3), then (𝑢,𝑢1,𝑢2) is a solution to the following system: 𝑢(𝑥,𝑦)=𝑣(𝑥,0)+𝑣(0,𝑦)𝑣(0,0)+𝑥0𝑦0𝑓𝑠,𝑡,𝑢(𝑠,𝑡),𝑢1(𝑠,𝑡),𝑢2𝑢(𝑠,𝑡)𝑑𝑠𝑑𝑡,1(𝑥,𝑦)=𝑣1(𝑥,0)+𝑦0𝑓𝑥,𝑡,𝑢(𝑥,𝑡),𝑢1(𝑥,𝑡),𝑢2𝑢(𝑥,𝑡)𝑑𝑡,2(𝑥,𝑦)=𝑣2(0,𝑦)+𝑥0𝑓𝑠,𝑦,𝑢(𝑠,𝑦),𝑢1(𝑠,𝑦),𝑢2(𝑠,𝑦)𝑑𝑠.(4.4) From Theorem 2.8 and the hypothesis (iii), it follows that ||||𝑣(𝑥,𝑦)𝑣(𝑥,0)𝑣(0,𝑦)+𝑣(0,0)𝑥0𝑦0𝑓𝑠,𝑡,𝑣(𝑠,𝑡),𝑣1(𝑠,𝑡),𝑣2||||(𝑠,𝑡)𝑑𝑠𝑑𝑡𝑥0𝑦0𝜑(𝑠,𝑡)𝑑𝑠𝑑𝑡𝜆1𝜑[||||𝑣𝜑(𝑥,𝑦),𝑥,𝑦0,),1(𝑥,𝑦)𝑣1(𝑥,0)𝑦0𝑓𝑥,𝑡,𝑣(𝑥,𝑡),𝑣1(𝑥,𝑡),𝑣2||||(𝑥,𝑡)𝑑𝑡𝑦0𝜑(𝑥,𝑡)𝑑𝑡𝜆2𝜑[||||𝑣𝜑(𝑥,𝑦),𝑥,𝑦0,),2(𝑥,𝑦)𝑣2(0,𝑦)𝑥0𝑓𝑠,𝑦,𝑣(𝑠,𝑦),𝑣1(𝑠,𝑦),𝑣2||||(𝑠,𝑦)𝑑𝑠𝑥0𝜑(𝑠,𝑦)𝑑𝑠𝜆3𝜑[𝜑(𝑥,𝑦),𝑥,𝑦0,).(4.5) Using (4.5) gives us ||||||||𝑣(𝑥,𝑦)𝑢(𝑥,𝑦)𝑣(𝑥,𝑦)𝑣(𝑥,0)𝑣(0,𝑦)+𝑣(0,0)𝑥0𝑦0𝑓𝑠,𝑡,𝑣(𝑠,𝑡),𝑣1(𝑠,𝑡),𝑣2||||+(𝑠,𝑡)𝑑𝑠𝑑𝑡𝑥0𝑦0||𝑓𝑠,𝑡,𝑣(𝑠,𝑡),𝑣1(𝑠,𝑡),𝑣2(𝑠,𝑡)𝑓𝑠,𝑡,𝑢(𝑠,𝑡),𝑢1(𝑠,𝑡),𝑢2||(𝑠,𝑡)𝑑𝑠𝑑𝑡𝜆1𝜑𝜑(𝑥,𝑦)+𝑥0𝑦0𝑙𝑓(𝑠,𝑡)max𝑖{1,2,3}||𝑣𝑖(𝑠,𝑡)𝑢𝑖||(𝑠,𝑡)𝑑𝑠𝑑𝑡.(4.6) From Lemma 2.10 it follows that ||||𝑣(𝑥,𝑦)𝑢(𝑥,𝑦)𝜆1𝜑exp00𝑙𝑓[||||(𝑠,𝑡)𝑑𝑠𝑑𝑡𝜑(𝑥,𝑦),𝑥,𝑦0,)or𝑣(𝑥,𝑦)𝑢(𝑥,𝑦)𝐶1𝑓,𝜑[𝜑(𝑥,𝑦),𝑥,𝑦0,),(4.7) where 𝐶1𝑓,𝜑=𝜆1𝜑exp00𝑙𝑓(𝑠,𝑡)𝑑𝑠𝑑𝑡.
Similarly, we have ||𝑣1(𝑥,𝑦)𝑢1||(𝑥,𝑦)𝜆2𝜑𝜑(𝑥,𝑦)+𝑦0𝑙𝑓(𝑥,𝑡)max𝑖{1,2,3}||𝑣𝑖(𝑥,𝑡)𝑢𝑖||(𝑥,𝑡)𝑑𝑡,(4.8) and from Lemma 2.10 we get ||𝑣1(𝑥,𝑦)𝑢1||(𝑥,𝑦)𝜆2𝜑exp0𝑙𝑓[||𝑣(𝑥,𝑡)𝑑𝑡𝜑(𝑥,𝑦),𝑥,𝑦0,)or1(𝑥,𝑦)𝑢1||(𝑥,𝑦)𝐶2𝑓,𝜑[𝜑(𝑥,𝑦),𝑥,𝑦0,),(4.9) where 𝐶2𝑓,𝜑=𝜆2𝜑exp0𝑙𝑓(𝑥,𝑡)𝑑𝑡.
Also, ||𝑣2(𝑥,𝑦)𝑢2||(𝑥,𝑦)𝜆3𝜑𝜑(𝑥,𝑦)+𝑥0𝑙𝑓(𝑠,𝑦)max𝑖{1,2,3}||𝑣𝑖(𝑠,𝑦)𝑢𝑖||(𝑠,𝑦)𝑑𝑠.(4.10) By using Lemma 2.10 we obtain ||𝑣2(𝑥,𝑦)𝑢2||(𝑥,𝑦)𝜆3𝜑exp0𝑙𝑓[||𝑣(𝑠,𝑦)𝑑𝑠𝜑(𝑥,𝑦),𝑥,𝑦0,)or2(𝑥,𝑦)𝑢2||(𝑥,𝑦)𝐶3𝑓,𝜑[𝜑(𝑥,𝑦),𝑥,𝑦0,),(4.11) where 𝐶3𝑓,𝜑=𝜆3𝜑exp0𝑙𝑓(𝑠,𝑦)𝑑𝑠.
So, (2.1) is generalised Ulam-Hyers-Rassias stable.

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