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

Authors: Nicolaie Lungu; Cecilia Crăciun
Journal: ISRN Mathematical Analysis
 (2012)

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## 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.


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## Body

## 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: (1.1)∂2u∂x∂y(x,y)=f(x,y,u(x,y),∂u∂x(x,y),∂u∂y(x,y)),0≤x<a,0≤y<b,
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.Leta,b∈(0,∞], ɛ>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:(2.1)∂2u∂x∂y(x,y)=f(x,y,u(x,y),∂u∂x(x,y),∂u∂y(x,y)),0≤x<a,0≤y<b,
where f∈C([0,a)×[0,b)×𝔹3,𝔹).We also consider the following inequalities:(2.2)|∂2v∂x∂y(x,y)-f(x,y,v(x,y),∂v∂x(x,y),∂v∂y(x,y))|≤ɛ,x∈[0,a),y∈[0,b),(2.3)|∂2v∂x∂y(x,y)-f(x,y,v(x,y),∂v∂x(x,y),∂v∂y(x,y))|≤φ(x,y),x∈[0,a),y∈[0,b),(2.4)|∂2v∂x∂y(x,y)-f(x,y,v(x,y),∂v∂x(x,y),∂v∂y(x,y))|≤ɛφ(x,y),x∈[0,a),y∈[0,b).Definition 2.1.
A functionu is a solution to (2.1) if
(2.5)u∈C1([0,a)×[0,b),B),∂2u∂x∂y∈C([0,a)×[0,b),B)
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 Cf1, Cf2, and Cf3>0 such that for any ɛ>0 and for any solution v to the inequality (2.2) there exists a solution u to (2.1) with
(2.6)|v(x,y)-u(x,y)|≤Cf1ɛ,∀x∈[0,a),∀y∈[0,b),|∂v∂x(x,y)-∂u∂x(x,y)|≤Cf2ɛ,∀x∈[0,a),∀y∈[0,b),|∂v∂y(x,y)-∂u∂y(x,y)|≤Cf3ɛ,∀x∈[0,a),∀y∈[0,b).Definition 2.3.
Equation (2.1) is generalised Ulam-Hyers-Rassias stable if there exist the real numbers Cf,φ1, Cf,φ2, and Cf,φ3>0 such that for any ɛ>0 and for any solution v to the inequality (2.3) there exists a solution u to (2.1) with
(2.7)|v(x,y)-u(x,y)|≤Cf,φ1φ(x,y),∀x∈[0,a),∀y∈[0,b),|∂v∂x(x,y)-∂u∂x(x,y)|≤Cf,φ2φ(x,y),∀x∈[0,a),∀y∈[0,b),|∂v∂y(x,y)-∂u∂y(x,y)|≤Cf,φ3φ(x,y),∀x∈[0,a),∀y∈[0,b).Remark 2.4.
A functionv 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 allɛ>0,|g(x,y)|≤ɛ, for all x∈[0,a), for all y∈[0,b);(ii)
For allx∈[0,a), for all y∈[0,b):(2.8)∂2v∂x∂y(x,y)=f(x,y,v(x,y),∂v∂x(x,y),∂v∂y(x,y))+g(x,y).Remark 2.5.
A functionv 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 allx∈[0,a), for all y∈[0,b):(2.9)∂2v∂x∂y(x,y)=f(x,y,v(x,y),∂v∂x(x,y),∂v∂y(x,y))+g(x,y).Remark 2.6.
A functionv 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 allɛ>0,|g(x,y)|≤ɛφ(x,y), for all x∈[0,a), for all y∈[0,b);(ii)
For allx∈[0,a), for all y∈[0,b):(2.10)∂2v∂x∂y(x,y)=f(x,y,v(x,y),∂v∂x(x,y),∂v∂y(x,y))+g(x,y).Throughout this paper we denote(2.11)u1(x,y)=∂u∂x(x,y),u2(x,y)=∂u∂y(x,y),v1(x,y)=∂v∂x(x,y),v2(x,y)=∂v∂y(x,y).Theorem 2.7.
Ifv is a solution to the inequality (2.2), then (v,v1,v2) satisfies the following system of integral inequalities:
(2.12)|v(x,y)-v(x,0)-v(0,y)+v(0,0)-∫0x∫0yf(s,t,v(s,t),v1(s,t),v2(s,t))dsdt|≤ɛxy,|v1(x,y)-v1(x,0)-∫0yf(x,t,v(x,t),v1(x,t),v2(x,t))dt|≤ɛy,|v2(x,y)-v2(0,y)-∫0xf(s,y,v(s,y),v1(s,y),v2(s,y))ds|≤ɛx,
for all x∈[0,a) and y∈[0,b).Proof.
By Remark2.4 we have that
(2.13)v(x,y)=v(x,0)+v(0,y)-v(0,0)+∫0x∫0yf(s,t,v(s,t),v1(s,t),v2(s,t))dsdt+∫0x∫0yg(s,t)dsdtv1(x,y)=v1(x,0)+∫0yf(x,t,v(x,t),v1(x,t),v2(x,t))dt+∫0yg(x,t)dtv2(x,y)=v2(0,y)+∫0xf(s,y,v(s,y),v1(s,y),v2(s,y))ds+∫0xg(s,y)ds,
and we have the following inequalities:
(2.14)|v(x,y)-v(x,0)-v(0,y)+v(0,0)-∫0x∫0yf(s,t,v(s,t),v1(s,t),v2(s,t))dsdt|=|∫0x∫0yg(s,t)dsdt|≤∫0x∫0y|g(s,t)|dsdt≤ɛxy,|v1(x,y)-v1(x,0)-∫0yf(x,t,v(x,t),v1(x,t),v2(x,t))dt|≤∫0y|g(x,t)|dt≤ɛy,|v2(x,y)-v2(0,y)-∫0xf(s,y,v(s,y),v1(s,y),v2(s,y))ds|≤∫0x|g(s,y)|ds≤ɛx.The following two theorems are obtained in a similar fashion.Theorem 2.8.
Ifv is a solution to the inequality (2.3), then (v,v1,v2) satisfies the following system of integral inequalities:
(2.15)|v(x,y)-v(x,0)-v(0,y)+v(0,0)-∫0x∫0yf(s,t,v(s,t),v1(s,t),v2(s,t))dsdt|≤∫0x∫0yφ(s,t)dsdt,|v1(x,y)-v1(x,0)-∫0yf(x,t,v(x,t),v1(x,t),v2(x,t))dt|≤∫0yφ(x,t)dt,|v2(x,y)-v2(0,y)-∫0xf(s,y,v(s,y),v1(s,y),v2(s,y))ds|≤∫0xφ(s,y)ds,
for all x∈[0,a) and y∈[0,b).Theorem 2.9.
Ifv is a solution to the inequality (2.4), then (v,v1,v2) satisfies the following system of integral inequalities:
(2.16)|v(x,y)-v(x,0)-v(0,y)+v(0,0)-∫0x∫0yf(s,t,v(s,t),v1(s,t),v2(s,t))dsdt|≤ɛ∫0x∫0yφ(s,t)dsdt,|v1(x,y)-v1(x,0)-∫0yf(x,t,v(x,t),v1(x,t),v2(x,t))dt|≤ɛ∫0yφ(x,t)dt,|v2(x,y)-v2(0,y)-∫0xf(s,y,v(s,y),v1(s,y),v2(s,y))ds|≤ɛ∫0xφ(s,y)ds,
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 anyt≥t0 one has
(2.17)u(t)≤h(t)+∫t0tv(s)u(s)ds;(iii)
h(t) is positive and nondecreasing.Then,
(2.18)u(t)≤h(t)exp⁡∫stv(r)dr,foranyt≥t0.


## 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 existsLf>0 such that
(3.1)|f(x,y,z1,z2,z3)-f(x,y,t1,t2,t3)|≤Lfmax⁡{|zi-ti|,i=1,2,3}, for all x∈[0,a], y∈[0,b] and z1,z2,z3,t1,t2,t3∈𝔹.
Then,(a)
forϕ∈C1([0,a],𝔹) and ψ∈C1([0,b],𝔹) (2.1) has a unique solution, which satisfies
(3.2)u(x,0)=ϕ(x),∀x∈[0,a],u(0,y)=ψ(y),∀y∈[0,b];(b)
(2.1) is Ulam-Hyers stable.Proof.
(a) This is a known result (see, e.g, [9, 10]).
(b) Letv be a solution to the inequality (2.2) and let u be the unique solution to (2.1), which satisfies the following conditions:
(3.3)u(x,0)=v(x,0),∀x∈[0,a],u(0,y)=v(0,y),∀y∈[0,b].
From Theorem 2.7, the hypothesis (iii), and Gronwall Lemma 2.10, it follows that
(3.4)|v(x,y)-u(x,y)|≤|v(x,y)-v(x,0)-v(0,y)+v(0,0)-∫0x∫0yf(s,t,v(s,t),v1(s,t),v2(s,t))dsdt|+∫0x∫0y|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≤ɛxy+Lf∫0x∫0ymax⁡i∈{1,2,3}|vi(s,t)-ui(s,t)|dsdt≤ɛabexp⁡(Lfab)=Cf1ɛ,whereCf1∶=abexp⁡(Lfab).
Similarly we have
(3.5)|v1(x,y)-u1(x,y)|≤|v1(x,y)-v1(x,0)-∫0yf(x,t,v(x,t),v1(x,t),v2(x,t))dt|+∫0y|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≤ɛy+Lf∫0ymax⁡i∈{1,2,3}|vi(x,t)-ui(x,t)|dt≤ɛbexp⁡(Lfb)=Cf2ɛ,whereCf2∶=bexp⁡(Lfb),|v2(x,y)-u2(x,y)|≤|v2(x,y)-v2(0,y)-∫0xf(s,y,v(s,y),v1(s,y),v2(s,y))ds|+∫0x|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≤ɛx+Lf∫0xmax⁡i∈{1,2,3}|vi(s,y)-ui(s,y)|ds≤ɛaexp⁡(Lfa)=Cf3ɛ,whereCf3∶=aexp⁡(Lfa).Remark 3.2.
In general, ifa=∞ 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 existslf∈C1([0,∞)×[0,∞),ℝ+) such that
(4.1)|f(x,y,z1,z2,z3)-f(x,y,t1,t2,t3)|≤lf(x,y)max⁡{|zi-ti|,i=1,2,3},
for all x,y∈[0,∞);(iii)
there existλφ1,λφ2,λφ3>0 such that
(4.2)∫0x∫0yφ(s,t)dsdt≤λφ1φ(x,y),∀x,y∈[0,∞),∫0yφ(x,t)dt≤λφ2φ(x,y),∀x,y∈[0,∞),∫0xφ(s,y)ds≤λφ3φ(x,y),∀x,y∈[0,∞);(iv)
φ:ℝ+×ℝ+→ℝ+ is increasing.Then (2.1) (a=∞ and b=∞) is generalised Ulam-Hyers-Rassias stable.Proof.
Letv be a solution to the inequality (2.3). Denote by u the unique solution to the Darboux problem:
(4.3)∂2u∂x∂y(x,y)=f(x,y,u(x,y),u1(x,y),u2(x,y)),∀x,y∈[0,∞),u(x,0)=v(x,0),∀x∈[0,∞),u(0,y)=v(0,y),∀y∈[0,∞).
If u is a solution to the Darboux problem (4.3), then (u,u1,u2) is a solution to the following system:
(4.4)u(x,y)=v(x,0)+v(0,y)-v(0,0)+∫0x∫0yf(s,t,u(s,t),u1(s,t),u2(s,t))dsdt,u1(x,y)=v1(x,0)+∫0yf(x,t,u(x,t),u1(x,t),u2(x,t))dt,u2(x,y)=v2(0,y)+∫0xf(s,y,u(s,y),u1(s,y),u2(s,y))ds.
From Theorem 2.8 and the hypothesis (iii), it follows that
(4.5)|v(x,y)-v(x,0)-v(0,y)+v(0,0)-∫0x∫0yf(s,t,v(s,t),v1(s,t),v2(s,t))dsdt|≤∫0x∫0yφ(s,t)dsdt≤λφ1φ(x,y),∀x,y∈[0,∞),|v1(x,y)-v1(x,0)-∫0yf(x,t,v(x,t),v1(x,t),v2(x,t))dt|≤∫0yφ(x,t)dt≤λφ2φ(x,y),∀x,y∈[0,∞),|v2(x,y)-v2(0,y)-∫0xf(s,y,v(s,y),v1(s,y),v2(s,y))ds|≤∫0xφ(s,y)ds≤λφ3φ(x,y),∀x,y∈[0,∞).
Using (4.5) gives us
(4.6)|v(x,y)-u(x,y)|≤|v(x,y)-v(x,0)-v(0,y)+v(0,0)-∫0x∫0yf(s,t,v(s,t),v1(s,t),v2(s,t))dsdt|+∫0x∫0y|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≤λφ1φ(x,y)+∫0x∫0ylf(s,t)max⁡i∈{1,2,3}(|vi(s,t)-ui(s,t)|)dsdt.
From Lemma 2.10 it follows that
(4.7)|v(x,y)-u(x,y)|≤λφ1exp⁡(∫0∞∫0∞lf(s,t)dsdt)φ(x,y),∀x,y∈[0,∞)or|v(x,y)-u(x,y)|≤Cf,φ1φ(x,y),∀x,y∈[0,∞),
where Cf,φ1∶=λφ1exp∫0∞∫0∞lf(s,t)dsdt.
Similarly, we have(4.8)|v1(x,y)-u1(x,y)|≤λφ2φ(x,y)+∫0ylf(x,t)max⁡i∈{1,2,3}(|vi(x,t)-ui(x,t)|)dt,
and from Lemma 2.10 we get
(4.9)|v1(x,y)-u1(x,y)|≤λφ2exp⁡(∫0∞lf(x,t)dt)φ(x,y),∀x,y∈[0,∞)or|v1(x,y)-u1(x,y)|≤Cf,φ2φ(x,y),∀x,y∈[0,∞),
where Cf,φ2∶=λφ2exp∫0∞lf(x,t)dt.
Also,(4.10)|v2(x,y)-u2(x,y)|≤λφ3φ(x,y)+∫0xlf(s,y)max⁡i∈{1,2,3}(|vi(s,y)-ui(s,y)|)ds.
By using Lemma 2.10 we obtain
(4.11)|v2(x,y)-u2(x,y)|≤λφ3exp⁡(∫0∞lf(s,y)ds)φ(x,y),∀x,y∈[0,∞)or|v2(x,y)-u2(x,y)|≤Cf,φ3φ(x,y),∀x,y∈[0,∞),
where Cf,φ3∶=λφ3exp∫0∞lf(s,y)ds.
So, (2.1) is generalised Ulam-Hyers-Rassias stable.



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