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

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., [1–4]). 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)=𝑓𝑥,𝑦,𝑢(𝑥,𝑦),𝜕𝜕𝑥𝑢(𝑥,𝑦),𝜕𝜕𝑦𝑢(𝑥,𝑦),0≤𝑥<𝑎,0≤𝑦<𝑏,(1.1) where 𝑓∈𝐶([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, 𝜑∈𝐶([0,𝑎)×[0,𝑏),ℝ₊) and (𝔹,|⋅|) be a real or complex Banach space.

We consider the following hyperbolic partial differential equation:𝜕²𝑢𝜕𝑥𝜕𝑦(𝑥,𝑦)=𝑓𝑥,𝑦,𝑢(𝑥,𝑦),𝜕𝜕𝑥𝑢(𝑥,𝑦),𝜕𝜕𝑦𝑢(𝑥,𝑦),0≤𝑥<𝑎,0≤𝑦<𝑏,(2.1) where 𝑓∈𝐶([0,𝑎)×[0,𝑏)×𝔹³,𝔹).

We also consider the following inequalities:||||𝜕²𝑣||||𝜕𝜕𝑥𝜕𝑦(𝑥,𝑦)−𝑓𝑥,𝑦,𝑣(𝑥,𝑦),𝜕𝜕𝑥𝑣(𝑥,𝑦),𝜕𝜕𝑣𝑦(𝑥,𝑦)||||≤𝜀,𝑥∈[0,𝑎),𝑦∈[0,𝑏),(2.2)₂𝑣||||𝜕𝜕𝑥𝜕𝑦(𝑥,𝑦)−𝑓𝑥,𝑦,𝑣(𝑥,𝑦),𝜕𝜕𝑥𝑣(𝑥,𝑦),𝜕𝜕𝑣𝑦(𝑥,𝑦)||||≤𝜑(𝑥,𝑦),𝑥∈[0,𝑎),𝑦∈[0,𝑏),(2.3)₂𝑣𝜕𝑥𝜕𝑦(𝑥,𝑦)−𝑓𝑥,𝑦,𝑣(𝑥,𝑦),𝜕𝜕𝑥𝑣(𝑥,𝑦),𝜕𝜕𝑣𝑦(𝑥,𝑦)||||≤𝜀𝜑(𝑥,𝑦),𝑥∈[0,𝑎),𝑦∈[0,𝑏).(2.4)

Definition 2.1. A function 𝑢 is a solution to (2.1) if 𝑢∈𝐶¹([0,𝑎)×[0,𝑏),𝔹),𝜕²𝑢𝜕𝑥𝜕𝑦∈𝐶([0,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 𝐶^1𝑓, 𝐶^2𝑓, and 𝐶^3𝑓>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 |||𝜕||𝑣𝑣(𝑥,𝑦)−𝑢(𝑥,𝑦)||≤𝐶^1𝑓𝜀,∀𝑥∈[0,𝑎),∀𝑦∈[0,𝑏),𝜕𝑥(𝑥,𝑦)−𝜕𝜕𝑥𝑢(𝑥,𝑦)|||≤𝐶^2𝑓||||𝜕𝑣𝜀,∀𝑥∈[0,𝑎),∀𝑦∈[0,𝑏),𝜕𝑦(𝑥,𝑦)−𝜕𝜕𝑦𝑢(𝑥,𝑦)||||≤𝐶^3𝑓𝜀,∀𝑥∈[0,𝑎),∀𝑦∈[0,𝑏).(2.6)

Definition 2.3. Equation (2.1) is generalised Ulam-Hyers-Rassias stable if there exist the real numbers 𝐶^1𝑓,𝜑, 𝐶^2𝑓,𝜑, and 𝐶^3𝑓,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 |||𝜕||𝑣𝑣(𝑥,𝑦)−𝑢(𝑥,𝑦)||≤𝐶^1𝑓,𝜑𝜑(𝑥,𝑦),∀𝑥∈[0,𝑎),∀𝑦∈[0,𝑏),𝜕𝑥(𝑥,𝑦)−𝜕𝜕𝑥𝑢(𝑥,𝑦)|||≤𝐶^2𝑓,𝜑||||𝜕𝑣𝜑(𝑥,𝑦),∀𝑥∈[0,𝑎),∀𝑦∈[0,𝑏),𝜕𝑦(𝑥,𝑦)−𝜕𝜕𝑦𝑢(𝑥,𝑦)||||≤𝐶^3𝑓,𝜑𝜑(𝑥,𝑦),∀𝑥∈[0,𝑎),∀𝑦∈[0,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 𝑔∈𝐶([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)=𝑓𝑥,𝑦,𝑣(𝑥,𝑦),𝜕𝜕𝑥𝑣(𝑥,𝑦),𝜕𝜕𝑣𝑦(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 𝑔∈𝐶([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)=𝑓𝑥,𝑦,𝑣(𝑥,𝑦),𝜕𝜕𝑥𝑣(𝑥,𝑦),𝜕𝜕𝑣𝑦(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 𝑔∈𝐶([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)=𝑓𝑥,𝑦,𝑣(𝑥,𝑦),𝜕𝜕𝑥𝑣(𝑥,𝑦),𝜕𝜕𝑣𝑦(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)−^𝑥0^𝑦0||||𝑣𝑓𝑠,𝑡,𝑣(𝑠,𝑡),𝑣₁(𝑠,𝑡),𝑣₂(𝑠,𝑡)𝑑𝑠𝑑𝑡||||≤𝜀𝑥𝑦,₁||||𝑣(𝑥,𝑦)−𝑣₁(𝑥,0)−^𝑦0𝑓𝑥,𝑡,𝑣(𝑥,𝑡),𝑣₁(𝑥,𝑡),𝑣₂(𝑥,𝑡)𝑑𝑡||||≤𝜀𝑦,₂(𝑥,𝑦)−𝑣₂(0,𝑦)−^𝑥0𝑓𝑠,𝑦,𝑣(𝑠,𝑦),𝑣₁(𝑠,𝑦),𝑣₂(𝑠,𝑦)𝑑𝑠||||≤𝜀𝑥,(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)_𝑥0^𝑦0𝑓𝑠,𝑡,𝑣(𝑠,𝑡),𝑣₁(𝑠,𝑡),𝑣₂(𝑠,𝑡)𝑑𝑠𝑑𝑡+^𝑥0^𝑦0𝑣𝑔(𝑠,𝑡)𝑑𝑠𝑑𝑡₁𝑣(𝑥,𝑦)=𝑣₁(𝑥,0)+^𝑦0𝑓𝑥,𝑡,𝑣(𝑥,𝑡),𝑣₁(𝑥,𝑡),𝑣₂(𝑥,𝑡)𝑑𝑡+^𝑦0𝑔(𝑥,𝑡)𝑑𝑡₂(𝑥,𝑦)=𝑣₂(0,𝑦)+^𝑥0𝑓𝑠,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)−^𝑥0^𝑦0=||||𝑓𝑠,𝑡,𝑣(𝑠,𝑡),𝑣₁(𝑠,𝑡),𝑣₂(𝑠,𝑡)𝑑𝑠𝑑𝑡_𝑥0||||^𝑦0𝑔(𝑠,𝑡)𝑑𝑠𝑑𝑡||||≤^𝑥0^𝑦0||||𝑣||𝑔(𝑠,𝑡)||𝑑𝑠𝑑𝑡≤𝜀𝑥𝑦,₁||||𝑣(𝑥,𝑦)−𝑣₁(𝑥,0)−^𝑦0𝑓𝑥,𝑡,𝑣(𝑥,𝑡),𝑣₁(𝑥,𝑡),𝑣₂(𝑥,𝑡)𝑑𝑡||||≤^𝑦0||𝑔(𝑥,𝑡)||𝑑𝑡≤𝜀𝑦,₂(𝑥,𝑦)−𝑣₂(0,𝑦)−^𝑥0𝑓𝑠,𝑦,𝑣(𝑠,𝑦),𝑣₁(𝑠,𝑦),𝑣₂(𝑠,𝑦)𝑑𝑠||||≤^𝑥0||𝑔(𝑠,𝑦)||𝑑𝑠≤𝜀𝑥.(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)−^𝑥0^𝑦0≤𝑓𝑠,𝑡,𝑣(𝑠,𝑡),𝑣₁(𝑠,𝑡),𝑣₂(𝑠,𝑡)𝑑𝑠𝑑𝑡_𝑥0||||^𝑦0||||𝑣𝜑(𝑠,𝑡)𝑑𝑠𝑑𝑡,₁||||𝑣(𝑥,𝑦)−𝑣₁(𝑥,0)−^𝑦0𝑓𝑥,𝑡,𝑣(𝑥,𝑡),𝑣₁(𝑥,𝑡),𝑣₂(𝑥,𝑡)𝑑𝑡||||≤^𝑦0𝜑(𝑥,𝑡)𝑑𝑡,₂(𝑥,𝑦)−𝑣₂(0,𝑦)−^𝑥0𝑓𝑠,𝑦,𝑣(𝑠,𝑦),𝑣₁(𝑠,𝑦),𝑣₂(𝑠,𝑦)𝑑𝑠||||≤^𝑥0𝜑(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)−^𝑥0^𝑦0≤𝜀_𝑥0𝑓𝑠,𝑡,𝑣(𝑠,𝑡),𝑣₁(𝑠,𝑡),𝑣₂(𝑠,𝑡)𝑑𝑠𝑑𝑡||||^𝑦0||||𝑣𝜑(𝑠,𝑡)𝑑𝑠𝑑𝑡,₁||||𝑣(𝑥,𝑦)−𝑣₁(𝑥,0)−^𝑦0𝑓𝑥,𝑡,𝑣(𝑥,𝑡),𝑣₁(𝑥,𝑡),𝑣₂(𝑥,𝑡)𝑑𝑡||||≤𝜀^𝑦0𝜑(𝑥,𝑡)𝑑𝑡,₂(𝑥,𝑦)−𝑣₂(0,𝑦)−^𝑥0𝑓𝑠,𝑦,𝑣(𝑠,𝑦),𝑣₁(𝑠,𝑦),𝑣₂(𝑠,𝑦)𝑑𝑠||||≤𝜀^𝑥0𝜑(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,ℎ∈𝐶(ℝ^𝑛+,ℝ₊); (ii)for any 𝑡≥𝑡₀ one has 𝑢(𝑡)≤ℎ(𝑡)+^𝑡𝑡₀𝑣(𝑠)𝑢(𝑠)𝑑𝑠;(2.17)(iii)ℎ(𝑡) is positive and nondecreasing.Then, 𝑢(𝑡)≤ℎ(𝑡)exp^𝑡𝑠𝑣(𝑟)𝑑𝑟,forany𝑡≥𝑡₀.(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)𝑓∈𝐶([0,𝑎]×[0,𝑏]×𝔹³,𝔹);(iii)there exists 𝐿_𝑓>0 such that ||𝑓𝑥,𝑦,𝑧₁,𝑧₂,𝑧₃−𝑓𝑥,𝑦,𝑡₁,𝑡₂,𝑡₃||≤𝐿_𝑓max||𝑧_𝑖−𝑡_𝑖||,𝑖=1,2,3,(3.1) for all 𝑥∈[0,𝑎], 𝑦∈[0,𝑏] and 𝑧₁,𝑧₂,𝑧₃,𝑡₁,𝑡₂,𝑡₃∈𝔹.
Then,(a)for 𝜙∈𝐶¹([0,𝑎],𝔹) and 𝜓∈𝐶¹([0,𝑏],𝔹) (2.1) has a unique solution, which satisfies 𝑢𝑢((𝑥0,,0𝑦))==𝜙𝜓((𝑥𝑦)),,∀∀𝑥𝑦∈∈[[00,,𝑏𝑎]];,(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: 𝑢𝑢((𝑥0,,0𝑦))==𝑣𝑣((𝑥0,,𝑦0)),,∀∀𝑦𝑥∈∈[[00,,𝑏𝑎]].,(3.3) From Theorem 2.7, the hypothesis (iii), and Gronwall Lemma 2.10, it follows that ||𝑣(≤𝑥,||||𝑦𝑣)(−𝑥,𝑢𝑦)(𝑥−,𝑦𝑣)(||𝑥,0)−𝑣(0,𝑦)+𝑣(0,0)−_𝑥0^𝑦0+𝑓𝑠,𝑡,𝑣(𝑠,𝑡),𝑣₁(𝑠,𝑡),𝑣₂(𝑠,𝑡)𝑑𝑠𝑑𝑡_𝑥0||||^𝑦0≤𝜀𝑥𝑦+𝐿||𝑓_𝑓𝑠,𝑡,𝑣(𝑠,𝑡),𝑣₁(𝑠,𝑡),𝑣₂(𝑠,𝑡)−𝑓𝑠,𝑡,𝑢(𝑠,𝑡),𝑢₁(𝑠,𝑡),𝑢₂(𝑠,𝑡)||𝑑𝑠𝑑𝑡^𝑥0^𝑦0≤𝜀𝑎𝑏exp𝐿_𝑓max_{𝑖∈{1,2,3}}||𝑣_𝑖(𝑠,𝑡)−𝑢_𝑖(𝑠,𝑡)||𝑑𝑠𝑑𝑡𝑎𝑏=𝐶^1𝑓𝜀,where𝐶^1𝑓∶=𝑎𝑏exp𝐿_𝑓𝑎𝑏.(3.4) Similarly we have ||𝑣₁≤(𝑥||||,𝑣𝑦)−𝑢₁(𝑥,𝑦)₁||+(𝑥,𝑦)−𝑣₁(𝑥,0)−^𝑦0𝑓𝑥,𝑡,𝑣(𝑥,𝑡),𝑣₁(𝑥,𝑡),𝑣₂(𝑥,𝑡)𝑑𝑡_𝑦0||||≤𝜀𝑦+||𝑓𝐿_𝑓𝑥,𝑡,𝑣(𝑥,𝑡),𝑣₁(𝑥,𝑡),𝑣₂(𝑥,𝑡)−𝑓𝑥,𝑡,𝑢(𝑥,𝑡),𝑢₁(𝑥,𝑡),𝑢₂(𝑥,𝑡)||𝑑𝑡^𝑦0≤𝜀𝑏exp𝐿_𝑓max_{𝑖∈{1,2,3}}||𝑣_𝑖(𝑥,𝑡)−𝑢_𝑖(𝑥,𝑡)||𝑑𝑡||𝑣₂𝑏=𝐶^2𝑓𝜀,where𝐶^2𝑓∶=𝑏exp𝐿_𝑓𝑏,≤(𝑥||||,𝑣𝑦)−𝑢₂(𝑥,𝑦)₂||+(𝑥,𝑦)−𝑣₂(0,𝑦)−^𝑥0𝑓𝑠,𝑦,𝑣(𝑠,𝑦),𝑣₁(𝑠,𝑦),𝑣₂(𝑠,𝑦)𝑑𝑠_𝑥0||||≤𝜀𝑥+||𝑓𝐿_𝑓𝑠,𝑦,𝑣(𝑠,𝑦),𝑣₁(𝑠,𝑦),𝑣₂(𝑠,𝑦)−𝑓𝑠,𝑦,𝑢(𝑠,𝑦),𝑢₁(𝑠,𝑦),𝑢₂(𝑠,𝑦)||𝑑𝑠^𝑥0≤𝜀𝑎exp𝐿_𝑓max_{𝑖∈{1,2,3}}||𝑣_𝑖(𝑠,𝑦)−𝑢_𝑖(𝑠,𝑦)||𝑑𝑠𝑎=𝐶^3𝑓𝜀,where𝐶^3𝑓∶=𝑎exp𝐿_𝑓𝑎.(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 𝑙_𝑓∈𝐶¹([0,∞)×[0,∞),ℝ₊) such that ||𝑓𝑥,𝑦,𝑧₁,𝑧₂,𝑧₃−𝑓𝑥,𝑦,𝑡₁,𝑡₂,𝑡₃||≤𝑙_𝑓(𝑥,𝑦)max||𝑧_𝑖−𝑡_𝑖||,𝑖=1,2,3,(4.1) for all 𝑥,𝑦∈[0,∞);(iii)there exist 𝜆^1𝜑,𝜆^2𝜑,𝜆^3𝜑>0 such that ^𝑥0^𝑦0𝜑(𝑠,𝑡)𝑑𝑠𝑑𝑡≤𝜆^1𝜑𝜑(𝑥,𝑦),∀𝑥,𝑦∈[0,∞),_𝑦0𝜑(𝑥,𝑡)𝑑𝑡≤𝜆^2𝜑𝜑(𝑥,𝑦),∀𝑥,𝑦∈[0,∞),_𝑥0𝜑(𝑠,𝑦)𝑑𝑠≤𝜆^3𝜑𝜑(𝑥,𝑦),∀𝑥,𝑦∈[0,∞);(4.2)(iv)𝜑∶ℝ₊×ℝ₊→ℝ₊ 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)=𝑓𝑥,𝑦,𝑢(𝑥,𝑦),𝑢¹𝑢𝑢((𝑥0,,0𝑦))==𝑣𝑣((𝑥0,,𝑦0)()𝑥,,,𝑦∀∀)𝑦𝑥,𝑢∈∈₂(𝑥[[00,,𝑦,∞∞))),.,∀𝑥,𝑦∈[0,∞),(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)+^𝑥0^𝑦0𝑢𝑓𝑠,𝑡,𝑢(𝑠,𝑡),𝑢₁(𝑠,𝑡),𝑢₂(𝑠,𝑡)𝑑𝑠𝑑𝑡,₁𝑢(𝑥,𝑦)=𝑣₁(𝑥,0)+^𝑦0𝑓𝑥,𝑡,𝑢(𝑥,𝑡),𝑢₁(𝑥,𝑡),𝑢₂(𝑥,𝑡)𝑑𝑡,₂(𝑥,𝑦)=𝑣₂(0,𝑦)+^𝑥0𝑓𝑠,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)−^𝑥0^𝑦0≤𝑓𝑠,𝑡,𝑣(𝑠,𝑡),𝑣₁(𝑠,𝑡),𝑣₂(𝑠,𝑡)𝑑𝑠𝑑𝑡_𝑥0||||^𝑦0||||𝑣𝜑(𝑠,𝑡)𝑑𝑠𝑑𝑡≤𝜆^1𝜑𝜑(𝑥,𝑦),∀𝑥,𝑦∈[0,∞),₁≤(𝑥,𝑦)−𝑣₁(𝑥,0)−^𝑦0𝑓𝑥,𝑡,𝑣(𝑥,𝑡),𝑣₁(𝑥,𝑡),𝑣₂(𝑥,𝑡)𝑑𝑡_𝑦0||||||||𝑣𝜑(𝑥,𝑡)𝑑𝑡≤𝜆^2𝜑𝜑(𝑥,𝑦),∀𝑥,𝑦∈[0,∞),₂≤(𝑥,𝑦)−𝑣₂(0,𝑦)−^𝑥0𝑓𝑠,𝑦,𝑣(𝑠,𝑦),𝑣₁(𝑠,𝑦),𝑣₂(𝑠,𝑦)𝑑𝑠_𝑥0||||𝜑(𝑠,𝑦)𝑑𝑠≤𝜆^3𝜑𝜑(𝑥,𝑦),∀𝑥,𝑦∈[0,∞).(4.5) Using (4.5) gives us ||𝑣(≤𝑥,||||𝑦𝑣)(−𝑥,𝑢𝑦)(𝑥−,𝑦𝑣)(||𝑥,0)−𝑣(0,𝑦)+𝑣(0,0)−_𝑥0^𝑦0+𝑓𝑠,𝑡,𝑣(𝑠,𝑡),𝑣₁(𝑠,𝑡),𝑣₂(𝑠,𝑡)𝑑𝑠𝑑𝑡_𝑥0||||^𝑦0≤𝜆_1𝜑||𝑓𝑠,𝑡,𝑣(𝑠,𝑡),𝑣₁(𝑠,𝑡),𝑣₂(𝑠,𝑡)−𝑓𝑠,𝑡,𝑢(𝑠,𝑡),𝑢₁(𝑠,𝑡),𝑢₂(𝑠,𝑡)||𝑑𝑠𝑑𝑡𝜑(𝑥,𝑦)+^𝑥0^𝑦0𝑙_𝑓(𝑠,𝑡)max_{𝑖∈{1,2,3}}||𝑣_𝑖(𝑠,𝑡)−𝑢_𝑖(𝑠,𝑡)||𝑑𝑠𝑑𝑡.(4.6) From Lemma 2.10 it follows that ||𝑣(𝑥,𝑦)−𝑢(𝑥,𝑦)||≤𝜆^1𝜑exp^∞0^∞0||𝑣(𝑥,𝑦)−𝑢(𝑥,𝑦)||≤𝐶_1𝑓,𝜑𝑙_𝑓(𝑠,𝑡)𝑑𝑠𝑑𝑡𝜑(𝑥,𝑦),∀𝑥,𝑦∈[0,∞)or𝜑(𝑥,𝑦),∀𝑥,𝑦∈[0,∞),(4.7) where 𝐶^1𝑓,𝜑∶=𝜆^1𝜑exp∫^∞0∫^∞0𝑙_𝑓(𝑠,𝑡)𝑑𝑠𝑑𝑡.
Similarly, we have ||𝑣₁(𝑥,𝑦)−𝑢₁(𝑥,𝑦)||≤𝜆^2𝜑𝜑(𝑥,𝑦)+^𝑦0𝑙_𝑓(𝑥,𝑡)max_{𝑖∈{1,2,3}}||𝑣_𝑖(𝑥,𝑡)−𝑢_𝑖(𝑥,𝑡)||𝑑𝑡,(4.8) and from Lemma 2.10 we get ||||𝑣𝑣₁(𝑥,𝑦)−𝑢₁(𝑥,𝑦)||≤𝜆^2𝜑exp^∞0𝑙_𝑓(𝑥,𝑡)𝑑𝑡𝜑(𝑥,𝑦),∀𝑥,𝑦∈[0,∞)or₁(𝑥,𝑦)−𝑢₁(𝑥,𝑦)||≤𝐶^2𝑓,𝜑𝜑(𝑥,𝑦),∀𝑥,𝑦∈[0,∞),(4.9) where 𝐶^2𝑓,𝜑∶=𝜆^2𝜑exp∫^∞0𝑙_𝑓(𝑥,𝑡)𝑑𝑡.
Also, ||𝑣₂(𝑥,𝑦)−𝑢₂(𝑥,𝑦)||≤𝜆^3𝜑𝜑(𝑥,𝑦)+^𝑥0𝑙_𝑓(𝑠,𝑦)max_{𝑖∈{1,2,3}}||𝑣_𝑖(𝑠,𝑦)−𝑢_𝑖(𝑠,𝑦)||𝑑𝑠.(4.10) By using Lemma 2.10 we obtain ||||𝑣𝑣₂(𝑥,𝑦)−𝑢₂(𝑥,𝑦)||≤𝜆^3𝜑exp^∞0𝑙_𝑓(𝑠,𝑦)𝑑𝑠𝜑(𝑥,𝑦),∀𝑥,𝑦∈[0,∞)or₂(𝑥,𝑦)−𝑢₂(𝑥,𝑦)||≤𝐶^3𝑓,𝜑𝜑(𝑥,𝑦),∀𝑥,𝑦∈[0,∞),(4.11) where 𝐶^3𝑓,𝜑∶=𝜆^3𝜑exp∫^∞0𝑙_𝑓(𝑠,𝑦)𝑑𝑠.
So, (2.1) is generalised Ulam-Hyers-Rassias stable.