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Hyers-Ulam stability of a nonlinear partial integro-differential equation of order three

  • Daniela Marian EMAIL logo , Sorina Anamaria Ciplea and Nicolaie Lungu
Published/Copyright: June 13, 2024
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Open Mathematics
From the journal Open Mathematics Volume 22 Issue 1

Abstract

In this article, we study the Hyers-Ulam stability of a nonlinear partial integro-differential equation of order three, of hyperbolic type, using Bielecki norm. Sufficient conditions are established to ensure Hyers-Ulam and Hyers-Ulam-Rassias stability for these equations. These types of equations appear in various applications in engineering, biology, chemistry, economics (price fluctuation – Black-Scholes equation), etc.

Keywords: integro-differential equation of hyperbolic type; Hyers-Ulam stability; Bielecki norm
MSC 2010: 35B35; 35B99; 45G10

1 Introduction

Hyers-Ulam stability and its generalizations are the concern of many mathematicians in recent years. The problem was posed by Ulam in 1940 [1]. The first result in this direction was established by Hyers [2], in 1941, for additive Cauchy equation in Banach spaces. Obloza [3] and Alsina and Ger [4] established the first results in this direction, regarding the Hyers-Ulam stability of differential equations. The stability of first-order linear differential equations and linear differential equations of higher order was also studied in [5–7]. Next, integral equations and integro-differential equations began to be studied. Jung [8] investigated the Hyers-Ulam-Rassias stability of a Volterra integral equation, with the fixed point method, using the idea of Cadariu and Radu from [9]. Castro and Simões [10,11] and Simões et al. [12] also investigated integro-differential equations using the generalized Bielecki metric. Other integral equations were studied in [11–22]. Partial differential equations were studied for the first time by Prastaro and Rassias in [23] and then in [24–30]. Brzdek et al. [31] and Tripathy [32] presented a collection of results related to the Hyers-Ulam stability. Recent contributions to the Hyers-Ulam stability and shadowing for nonautonomous dynamics are given in [33,34].

Semi-Hyers-Ulam-Rassias stability was introduced in [10], as in-between the Hyers-Ulam and Hyers-Ulam-Rassias stabilities, where the following equation was studied:

y ′ = f x , y ( x ) , ∫ a x k ( x , t , y ( t ) , y ( α ( t ) ) ) d t , y ( a ) = c ∈ R ,

with y ∈ C 1 ( [ a , b ] ) , f : [ a , b ] × C × C ⟶ C , and k : [ a , b ] × [ a , b ] × C × C ⟶ C are the continuous functions, and α : [ a , b ] → [ a , b ] is a continuous delay function that fulfills α ( t ) ≤ t for all t ∈ [ a , b ] .

In [11], the following equation was studied:

y ( n ) ( x ) = f x , y ( x ) , ∫ a x k ( x , t , y ( t ) , y ( α ( t ) ) ) d t , y ( j ) ( a ) = 0 , j ∈ { 0 , 1 , … , n − 1 } ,

with y ∈ C n ( [ a , b ] ) , f : [ a , b ] × C × C ⟶ C , and k : [ a , b ] × [ a , b ] × C × C ⟶ C are the continuous functions, and α : [ a , b ] → [ a , b ] is a continuous delay function that fulfills α ( t ) ≤ t for all t ∈ [ a , b ] .

In [12] was examined the equation:

y ( n ) ( x ) = f x , y ( x ) , ∫ a x k ( x , t , y ( t ) , y ′ ( t ) , … , y ( n − 1 ) ( t ) ) d t , y ( j ) ( a ) = 0 , j ∈ { 0 , 1 , … , n − 1 } ,

with y ∈ C n ( [ a , b ] ) , f : [ a , b ] × C × C ⟶ C , and k : [ a , b ] × [ a , b ] × C n ⟶ C are the continuous functions.

Inspired by these types of equations, in this work, we study the Hyers-Ulam stability and the Hyers-Ulam-Rassias stability for a nonlinear hyperbolic partial differential equation of order three, using Bielecki metric.

In the whole work, we consider a , b , c ∈ ( 0 , ∞ ) , D = [ 0 , a ] × [ 0 , b ] × [ 0 , c ] , u ∈ C 3 ( D , R ) , E = { ( x , y , z , r , s , t ) ∣ 0 ≤ r ≤ x ≤ a , 0 ≤ s ≤ y ≤ b , 0 ≤ t ≤ z ≤ c } , f ∈ C ( D × R , R ) , g ∈ C ( D × R × R , R ) , k ∈ C ( E × R , R ) , χ 1 ∈ C 1 ( [ 0 , a ] × [ 0 , b ] , R ) , χ 2 ∈ C 1 ( [ 0 , a ] × [ 0 , c ] , R ) , χ 3 ∈ C 1 ( [ 0 , b ] × [ 0 , c ] , R ) , w 1 ∈ C ( [ 0 , a ] , R ) , w 2 ∈ C ( [ 0 , b ] , R ) , w 3 ∈ C ( [ 0 , c ] , R ) and w 0 ∈ R .

In what follows, we consider the equation

(1.1) ∂ 3 u ( x , y , z ) ∂ x ∂ y ∂ z = f ( x , y , z , u ( x , y , z ) ) + g x , y , z , u ( x , y , z ) , ∫ 0 x ∫ 0 y ∫ 0 z k ( x , y , z , r , s , t , u ( r , s , t ) ) d r d s d t ,

and the following conditions, analogously as in [29,30,35]:

(1.2) u ( x , y , 0 ) = χ 1 ( x , y ) , ( x , y ) ∈ [ 0 , a ] × [ 0 , b ] , u ( x , 0 , z ) = χ 2 ( z , x ) , ( x , z ) ∈ [ 0 , a ] × [ 0 , c ] , u ( 0 , y , z ) = χ 3 ( y , z ) , ( y , z ) ∈ [ 0 , b ] × [ 0 , c ] ,

where χ 1 , χ 2 , and χ 3 satisfy

(1.3) χ 1 ( x , 0 ) = χ 2 ( 0 , x ) = w 1 ( x ) , x ∈ [ 0 , a ] , χ 3 ( y , 0 ) = χ 1 ( 0 , y ) = w 2 ( y ) , y ∈ [ 0 , b ] , χ 2 ( z , 0 ) = χ 3 ( 0 , z ) = w 3 ( z ) , z ∈ [ 0 , c ] , w 1 ( 0 ) = w 2 ( 0 ) = w 3 ( 0 ) = w 0 .

We also consider a real number τ > 0 and the set C ( D , R ) endowed with the Bielecki’s norm

∥ u ∥ τ ≔ sup ( x , y , z ) ∈ D ( ∣ u ( x , y , z ) ∣ e − τ ( x + y + z ) ) .

It is clear that ( C ( D , R ) , ∥ ⋅ ∥ τ ) is a Banach space.

The importance of studying type (1.1) equations lies, among others, in the multiple applications in various fields of activity, among which we mention the field of economy (Black-Scholes equation [36], price fluctuation [37]), engineering (pseudoparabolic equation [38]), chemistry, biology, etc.

The outline of this article is the following: in Section 2, the stability notions are presented (Definitions 2.1 and 2.2), and Corollaries 2.1 and 2.2 are proven. The first main result (Theorem 3.1) is given in Section 3, and it concerns the Hyers-Ulam stability of equation (1.1) + (1.2) + (1.3). Example 3.1 is also presented. The second main result (Theorem 4.1) is given in Section 4 and refers to the Hyers-Ulam-Rassias stability of equation (1.1) + (1.2) + (1.3). Example 4.1 is also established. In the last section, some conclusions and future concerns are presented.

2 Preliminary notions and results

We continue to give some definitions and useful results. Let ε > 0 , arbitrary, φ ∈ C ( D , R + ) , fixed and the inequalities

(2.1) ∂ 3 v ( x , y , z ) ∂ x ∂ y ∂ z − f ( x , y , z , v ( x , y , z ) ) − g x , y , z , v ( x , y , z ) , ∫ 0 x ∫ 0 y ∫ 0 z k ( x , y , z , r , s , t , v ( r , s , t ) ) d r d s d t ≤ ε ,

(2.2) ∂ 3 v ( x , y , z ) ∂ x ∂ y ∂ z − f ( x , y , z , v ( x , y , z ) ) − g x , y , z , v ( x , y , z ) , ∫ 0 x ∫ 0 y ∫ 0 z k ( x , y , z , r , s , t , v ( r , s , t ) ) d r d s d t ≤ ε φ ( x , y , z ) , ( x , y , z , r , s , t ) ∈ E .

Definition 2.1

Equation (1.1) + (1.2) + (1.3) is called Hyers-Ulam stable if there exists a real number C 1 > 0 such that, for every ε > 0 and for each solution v of inequality (2.1), satisfying (1.2) + (1.3), there exists a solution u of equation (1.1), satisfying (1.2) + (1.3), with

(2.3) ∣ v ( x , y , z ) − u ( x , y , z ) ∣ ≤ C 1 ⋅ ε , ∀ ( x , y , z ) ∈ D .

The number C 1 from the aforementioned definition is called an Ulam constant for equation 1.1. Denote by C 1 the infimum of all Ulam constants of equation (1.1). In general, C 1 is not necessarily an Ulam constant for the equation. If it is also an Ulam constant of the equation, then is called the best Ulam constant.

Definition 2.2

Equation (1.1) + (1.2) + (1.3) is called Hyers-Ulam-Rassias stable if there exists a real number C 1 > 0 such that, for every ε > 0 and for each solution v of inequality (2.2), satisfying (1.2) + (1.3), there exists a solution u of equation (1.1), satisfying (1.2) + (1.3), with

(2.4) ∣ v ( x , y , z ) − u ( x , y , z ) ∣ ≤ C 1 ⋅ ε ⋅ φ ( x , y , z ) , ∀ ( x , y , z ) ∈ D .

Remark 2.1

A function v is a solution of (2.1) if and only if there exists a function p ∈ C ( D , R ) such that

  1. ∣ p ( x , y , z ) ∣ ≤ ε , ∀ ( x , y , z ) ∈ D ,

  2. ∂ 3 v ( x , y , z ) ∂ x ∂ y ∂ z = f ( x , y , z , v ( x , y , z ) ) + g x , y , z , v ( x , y , z ) , ∫ 0 x ∫ 0 y ∫ 0 z k ( x , y , z , r , s , t , v ( r , s , t ) ) d r d s d t + p ( x , y , z ) , ∀ ( x , y , z ) ∈ D .

From this remark, it follows:

Corollary 2.1

If v is a solution of (2.1), then v is a solution of the integral inequality

(2.5) v ( x , y , z ) − χ 1 ( x , y ) − χ 2 ( z , x ) − χ 3 ( y , z ) + w 1 ( x ) + w 2 ( y ) + w 3 ( z ) − w 0 − ∫ 0 x ∫ 0 y ∫ 0 z f ( r , s , t , v ( r , s , t ) ) d r d s d t − ∫ 0 x ∫ 0 y ∫ 0 z g r , s , t , v ( r , s , t ) , ∫ 0 r ∫ 0 s ∫ 0 t k ( r , s , t , r 1 , s 1 , t 1 , v ( r 1 , s 1 , t 1 ) ) d r 1 d s 1 d t 1 d r d s d t ≤ ε x y z , ∀ ( x , y , z ) ∈ D .

Proof

Indeed, from (1.2), (1.3), and Remark 2.1, we have

v ( x , y , z ) = χ 1 ( x , y ) + χ 2 ( z , x ) + χ 3 ( y , z ) − w 1 ( x ) − w 2 ( y ) − w 3 ( z ) + w 0 + ∫ 0 x ∫ 0 y ∫ 0 z f ( r , s , t , v ( r , s , t ) ) d r d s d t + ∫ 0 x ∫ 0 y ∫ 0 z g r , s , t , v ( r , s , t ) , ∫ 0 r ∫ 0 s ∫ 0 t k ( r , s , t , r 1 , s 1 , t 1 , v ( r 1 , s 1 , t 1 ) ) d r 1 d s 1 d t 1 d r d s d t + ∫ 0 x ∫ 0 y ∫ 0 z p ( r , s , t ) d r d s d t ,

and from here, we obtain

v ( x , y , z ) − χ 1 ( x , y ) − χ 2 ( z , x ) − χ 3 ( y , z ) + w 1 ( x ) + w 2 ( y ) + w 3 ( z ) − w 0 − ∫ 0 x ∫ 0 y ∫ 0 z f ( r , s , t , v ( r , s , t ) ) d r d s d t − ∫ 0 x ∫ 0 y ∫ 0 z g r , s , t , v ( r , s , t ) , ∫ 0 r ∫ 0 s ∫ 0 t k ( r , s , t , r 1 , s 1 , t 1 , v ( r 1 , s 1 , t 1 ) ) d r 1 d s 1 d t 1 d r d s d t ≤ ∫ 0 x ∫ 0 y ∫ 0 z p ( r , s , t ) d r d s d t ≤ ∫ 0 x ∫ 0 y ∫ 0 z ∣ p ( r , s , t ) ∣ d r d s d t ≤ ε x y z .

Hence, inequality (2.5) is satisfied.□

Analogously, we have the following remark and consequence.

Remark 2.2

A function v is a solution of (2.2) if and only if there exists a function p ∈ C ( D , R ) such that

  1. ∣ p ( x , y , z ) ∣ ≤ ε φ ( x , y , z ) , ∀ ( x , y , z ) ∈ D ,

  2. ∂ 3 v ( x , y , z ) ∂ x ∂ y ∂ z = f ( x , y , z , v ( x , y , z ) ) + g x , y , z , v ( x , y , z ) , ∫ 0 x ∫ 0 y ∫ 0 z k ( x , y , z , r , s , t , v ( r , s , t ) ) d r d s d t + p ( x , y , z ) , ∀ ( x , y , z ) ∈ D .

Corollary 2.2

If v is a solution of (2.2), then v is a solution of the integral inequality

(2.6) v ( x , y , z ) − χ 1 ( x , y ) − χ 2 ( z , x ) − χ 3 ( y , z ) + w 1 ( x ) + w 2 ( y ) + w 3 ( z ) − w 0 − ∫ 0 x ∫ 0 y ∫ 0 z f ( r , s , t , v ( r , s , t ) ) d r d s d t − ∫ 0 x ∫ 0 y ∫ 0 z g r , s , t , v ( r , s , t ) , ∫ 0 r ∫ 0 s ∫ 0 t k ( r , s , t , r 1 , s 1 , t 1 , v ( r 1 , s 1 , t 1 ) ) d r 1 d s 1 d t 1 d r d s d t ≤ ε ∫ 0 x ∫ 0 y ∫ 0 z φ ( r , s , t ) d r d s d t , ∀ ( x , y , z ) ∈ D .

3 Hyers-Ulam stability

The following result refers to the Hyers-Ulam stability of equation (1.1) + (1.2) + (1.3).

Theorem 3.1

If:

  1. there exists l f > 0 such that

    ∣ f ( x , y , z , u 1 ) − f ( x , y , z , u 2 ) ∣ ≤ l f ∣ u 1 − u 2 ∣ , ∀ ( x , y , z ) ∈ D , u 1 , u 2 ∈ R ;

  2. there exists l k > 0 such that

    ∣ k ( x , y , z , r , s , t , u 1 ) − k ( x , y , z , r , s , t , u 2 ) ∣ ≤ l k ∣ u 1 − u 2 ∣ , ∀ ( x , y , z , r , s , t ) ∈ E , u 1 , u 2 ∈ R ;

  3. there exists l g > 0 such that

    ∣ g ( x , y , z , u 1 , v 1 ) − g ( x , y , z , u 2 , v 2 ) ∣ ≤ l g ( ∣ u 1 − u 2 ∣ + ∣ v 1 − v 2 ∣ ) , ∀ ( x , y , z ) ∈ D , u 1 , u 2 , v 1 , v 2 ∈ R ;

  4. τ > 0 is such that ( l f + l g ) τ 3 + l g l k τ 6 < 1 ,

then:

  1. problem (1.1) + (1.2) + (1.3) has a unique solution;

  2. equation (1.1) + (1.2) + (1.3) is Hyers-Ulam stable.

Proof

(a) We consider the operator A : C ( D , R + ) → C ( D , R + ) ,

(3.1) A ( u ) ( x , y , z ) = χ 1 ( x , y ) + χ 2 ( z , x ) + χ 3 ( y , z ) − w 1 ( x ) − w 2 ( y ) − w 3 ( z ) + w 0 + ∫ 0 x ∫ 0 y ∫ 0 z f ( r , s , t , u ( r , s , t ) ) d r d s d t + ∫ 0 x ∫ 0 y ∫ 0 z g r , s , t , u ( r , s , t ) , ∫ 0 r ∫ 0 s ∫ 0 t k ( r , s , t , r 1 , s 1 , t 1 , u ( r 1 , s 1 , t 1 ) ) d r 1 d s 1 d t 1 d r d s d t .

We prove that A is a contraction. We have:

∣ A ( u 1 ) ( x , y , z ) − A ( u 2 ) ( x , y , z ) ∣ ≤ l f ∫ 0 x ∫ 0 y ∫ 0 z ∣ u 1 ( r , s , t ) − u 2 ( r , s , t ) ∣ d r d s d t + l g ∫ 0 x ∫ 0 y ∫ 0 z ∣ u 1 ( r , s , t ) − u 2 ( r , s , t ) ∣ d r d s d t + l g ∫ 0 x ∫ 0 y ∫ 0 z ∫ 0 r ∫ 0 s ∫ 0 t ∣ k ( r , s , t , r 1 , s 1 , t 1 , u 1 ( r 1 , s 1 , t 1 ) ) − k ( r , s , t , r 1 , s 1 , t 1 , u 2 ( r 1 , s 1 , t 1 ) ) ∣ d r 1 d s 1 d t 1 d r d s d t ≤ l f ∫ 0 x ∫ 0 y ∫ 0 z ∥ u 1 − u 2 ∥ τ e τ ( r + s + t ) d r d s d t + l g ∫ 0 x ∫ 0 y ∫ 0 z ∥ u 1 − u 2 ∥ τ e τ ( r + s + t ) d r d s d t + l g l k ∫ 0 x ∫ 0 y ∫ 0 z ∫ 0 r ∫ 0 s ∫ 0 t ∥ u 1 − u 2 ∥ τ e τ ( r 1 + s 1 + t 1 ) d r 1 d s 1 d t 1 d r d s d t ≤ l f τ 3 + l g τ 3 + l g l k τ 6 ∥ u 1 − u 2 ∥ τ e τ ( x + y + z ) .

Hence,

∥ A ( u 1 ) − A ( u 2 ) ∥ τ ≤ ( l f + l g ) τ 3 + l g l k τ 6 ∥ u 1 − u 2 ∥ τ .

Using condition (iv), we obtain that A is a contraction with respect to the Bielecki norm. Hence, problem (1.1) + (1.2) + (1.3) has a unique solution.

(b) Let v be a solution of (2.1) satisfying (1.2) + (1.3) and let u be the unique solution of problem (1.1) + (1.2) + (1.3). Inequality (2.5) and conditions (i)–(iv) imply

∣ v ( x , y , z ) − u ( x , y , z ) ∣ ≤ ∣ v ( x , y , z ) − χ 1 ( x , y ) − χ 2 ( z , x ) − χ 3 ( y , z ) + w 1 ( x ) + w 2 ( y ) + w 3 ( z ) − w 0 − ∫ 0 x ∫ 0 y ∫ 0 z f ( r , s , t , v ( r , s , t ) ) d r d s d t − ∫ 0 x ∫ 0 y ∫ 0 z g r , s , t , v ( r , s , t ) , ∫ 0 r ∫ 0 s ∫ 0 t k ( r , s , t , r 1 , s 1 , t 1 , v ( r 1 , s 1 , t 1 ) ) d r 1 d s 1 d t 1 d r d s d t + ∫ 0 x ∫ 0 y ∫ 0 z ∣ f ( r , s , t , v ( r , s , t ) ) − f ( r , s , t , u ( r , s , t ) ) ∣ d r d s d t + ∫ 0 x ∫ 0 y ∫ 0 z g r , s , t , v ( r , s , t ) , ∫ 0 r ∫ 0 s ∫ 0 t k ( r , s , t , r 1 , s 1 , t 1 , v ( r 1 , s 1 , t 1 ) ) d r 1 d s 1 d t 1 − g r , s , t , u ( r , s , t ) , ∫ 0 r ∫ 0 s ∫ 0 t k ( r , s , t , r 1 , s 1 , t 1 , u ( r 1 , s 1 , t 1 ) ) d r 1 d s 1 d t 1 d r d s d t ≤ ε x y z + l f ∫ 0 x ∫ 0 y ∫ 0 z ∣ v ( r , s , t ) − u ( r , s , t ) ∣ d r d s d t + l g ∫ 0 x ∫ 0 y ∫ 0 z ∣ v ( r , s , t ) − u ( r , s , t ) ∣ d r d s d t + l g l k ∫ 0 x ∫ 0 y ∫ 0 z ∫ 0 r ∫ 0 s ∫ 0 t ∣ v ( r , s , t ) − u ( r , s , t ) ∣ d r 1 d s 1 d t 1 d r d s d t , ∀ ( x , y , z , r , s , t ) ∈ E .

From here, we obtain

∣ v ( x , y , z ) − u ( x , y , z ) ∣ ≤ ε x y z + ( l f + l g ) τ 3 + l g l k τ 6 ∥ v − u ∥ τ e τ ( x + y + z ) .

We obtain

∥ v − u ∥ τ ≤ ε x y z e − τ ( x + y + z ) + ( l f + l g ) τ 3 + l g l k τ 6 ∥ v − u ∥ τ .

So

∥ v − u ∥ τ ≤ ε τ 6 τ 6 − ( l f + l g ) τ 3 − l g l k x y z e − τ ( x + y + z ) ,

and

∣ v ( x , y , z ) − u ( x , y , z ) ∣ ≤ ε τ 6 a b c τ 6 − ( l f + l g ) τ 3 − l g l k = ε C 1 ,

where C 1 = τ 6 a b c τ 6 − ( l f + l g ) τ 3 − l g l k ; hence, equation (1.1) + (1.2) + (1.3) is Hyers-Ulam stable.□

Remark 3.1

The inequality ( l f + l g ) τ 3 + l g l k τ 6 < 1 is satisfied for every τ > τ 1 , where τ 1 = l f + l g + ( l f + l g ) 2 + 4 l g ⋅ l k 2 3 .

Remark 3.2

The function f : ( τ 1 , ∞ ) , f ( τ ) = τ 6 τ 6 − ( l f + l g ) τ 3 − l f ⋅ l g is strictly decreasing and lim τ ⟶ ∞ f ( τ ) = 1 . Hence,

∣ v ( x , y , z ) − u ( x , y , z ) ∣ ≤ ε a b c = ε C , ∀ ( x , y , z ) ∈ D ,

where C = a b c is the infimum of the constants C 1 from above, for τ > τ 1 . This C is the best possible in our approach but is not the best Ulam constant. This is difficult to calculate.

Remark 3.3

If a = b = c = ∞ , then equation (1.1) is not always Ulam-Hyers stable. Let χ 1 ∈ C 1 ( [ 0 , ∞ ) × [ 0 , ∞ ) , R ) , χ 1 ( x , y ) = x + y , χ 2 ∈ C 1 ( [ 0 , ∞ ) × [ 0 , ∞ ) , R ) , χ 2 ( x , z ) = x + z , χ 3 ∈ C 1 ( [ 0 , ∞ ) × [ 0 , ∞ ) , R ) , χ 3 ( y , z ) = y + z , w 1 ∈ C ( [ 0 , ∞ ) , R ) , w 1 ( x ) = x , w 2 ∈ C ( [ 0 , ∞ ) , R ) , w 2 ( y ) = y , w 3 ∈ C ( [ 0 , ∞ ) , R ) , w 3 ( z ) = z , and w 0 = 0 . We consider the homogeneous equation ∂ 3 u ( x , y , z ) ∂ x ∂ y ∂ z = 0 . In this case, u ( x , y , z ) = x + y + z is a solution of equation (1.1) satisfying

(3.2) u ( x , y , 0 ) = χ 1 ( x , y ) = x + y , ( x , y ) ∈ [ 0 , ∞ ) × [ 0 , ∞ ) , u ( x , 0 , z ) = χ 2 ( z , x ) = x + z , ( x , z ) ∈ [ 0 , ∞ ) × [ 0 , ∞ ) , u ( 0 , y , z ) = χ 3 ( y , z ) = y + z , ( y , z ) ∈ [ 0 , ∞ ) × [ 0 , ∞ ) ,

and

(3.3) χ 1 ( x , 0 ) = χ 2 ( 0 , x ) = w 1 ( x ) = x , x ∈ [ 0 , ∞ ) , χ 3 ( y , 0 ) = χ 1 ( 0 , y ) = w 2 ( y ) = y , y ∈ [ 0 , ∞ ) , χ 2 ( z , 0 ) = χ 3 ( 0 , z ) = w 3 ( z ) = z , z ∈ [ 0 , ∞ ) , w 1 ( 0 ) = w 2 ( 0 ) = w 3 ( 0 ) = w 0 = 0 .

Consider the solution v ( x , y , z ) = ε x y z + x + y + z of inequality (2.1) satisfying (3.2) + (3.3). In this case, ∣ v ( x , y , z ) − u ( x , y , z ) ∣ = ∣ ε x y z ∣ is unbounded when x , y , and z tend to ∞ .

Example 3.1

Let D = [ 0 , 1 ] × [ 0 , 1 ] × [ 0 , 1 ] , u ∈ C 3 ( D , R ) , f ∈ C ( D × R , R ) , g ∈ C ( D × R × R , R ) , k ∈ C ( E × R , R ) . Let χ 1 ∈ C 1 ( [ 0 , 1 ] × [ 0 , 1 ] , R ) , χ 2 ∈ C 1 ( [ 0 , 1 ] × [ 0 , 1 ] , R ) , χ 3 ∈ C 1 ( [ 0 , 1 ] × [ 0 , 1 ] , R ) , w 1 ∈ C ( [ 0 , 1 ] , R ) , w 2 ∈ C ( [ 0 , 1 ] , R ) , and w 3 ∈ C ( [ 0 , 1 ] , R ) , w 0 ∈ R and such that conditions (1.2) + (1.3) are satisfied.

We consider the equation

(3.4) ∂ 3 u ( x , y , z ) ∂ x ∂ y ∂ z = x y z u ( x , y , z ) + e x + y + z − 3 2 ∫ 0 x ∫ 0 y ∫ 0 z sin ( x + y + x − r − s − t ) ⋅ u ( r , s , t ) d r d s d t .

We have f ( x , y , z , u ) = x y z u , and

∣ f ( x , y , z , u 1 ) − f ( x , y , z , u 2 ) ∣ ≤ ∣ u 1 − u 2 ∣ , ∀ ( x , y , z ) ∈ D , u 1 , u 2 ∈ R .

Hence l f = 1 .

We also have k ( x , y , z , r , s , t , u ) = sin ( x + y + x − r − s − t ) ⋅ u , and

∣ k ( x , y , z , r , s , t , u 1 ) − k ( x , y , z , r , s , t , u 2 ) ∣ ≤ ∣ u 1 − u 2 ∣ ,

∀ ( x , y , z , r , s , t ) ∈ E , u 1 , u 2 ∈ R ; hence, l k = 1 .

We have g ( x , y , z , u , v ) = e x + y + z − 3 2 ⋅ v , where

v ( x , y , z , r , s , t , u ( x , y , z ) ) = ∫ 0 x ∫ 0 y ∫ 0 z sin ( x + y + x − r − s − t ) ⋅ u ( r , s , t ) d r d s d t

and

∣ g ( x , y , z , u 1 , v 1 ) − g ( x , y , z , u 2 , v 2 ) ∣ ≤ e x + y + z − 3 2 ∣ v 1 − v 2 ∣ ≤ ∣ v 1 − v 2 ∣ ,

∀ ( x , y , z ) ∈ D , u 1 , u 2 , v 1 , v 2 ∈ R , i.e., l g = 1 . From Remark 3.1, we have that ( l f + l g ) τ 3 + l g l k τ 6 = 2 τ 3 + 1 τ 6 < 1 , for each τ > 1 + 2 3 . We remark that all the conditions from Theorem 3.1 are satisfied; hence, equation (3.4) + (1.2)+ (1.3) is Hyers-Ulam stable. If v is a solution of (2.1) satisfying (1.2) + (1.3) and u is the unique solution of problem (3.4) + (1.2) + (1.3), then

∣ v ( x , y , z ) − u ( x , y , z ) ∣ ≤ ε τ 6 τ 6 − 2 τ 3 − 1 , ∀ τ > 1 + 2 3 , ∀ ( x , y , z ) ∈ D .

Remark 3.4

From Remark 3.2, we obtain

∣ v ( x , y , z ) − u ( x , y , z ) ∣ ≤ ε , ∀ ( x , y , z ) ∈ D .

4 Hyers-Ulam-Rassias stability

We study now, Hyers-Ulam-Rassias stability of equation (1.1) + (1.2) + (1.3).

Theorem 4.1

If:

  1. conditions (i)–(iv) from Theorem  3.1  are satisfied; and

  2. there exists λ φ > 0 such that

    ∫ 0 x ∫ 0 y ∫ 0 z φ ( r , s , t ) d r d s d t ≤ λ φ ⋅ φ ( x , y , z ) , ∀ ( x , y , z ) ∈ D ;

then problem (1.1) + (1.2) + (1.3) is Hyers-Ulam-Rassias stable.

Proof

Let v be a solution of (2.2) satisfying (1.2) + (1.3) and let u be the unique solution of problem (1.1) + (1.2) + (1.3). Using (2.6), we obtain

∣ v ( x , y , z ) − u ( x , y , z ) ∣ ≤ ε ∫ 0 x ∫ 0 y ∫ 0 z φ ( r , s , t ) d r d s d t + ( l f + l g ) τ 3 + l g l k τ 6 ∥ v − u ∥ τ e τ ( x + y + z ) .

Taking account condition (b), we obtain

∥ v − u ∥ τ ≤ ε λ φ φ ( x , y , z ) e − τ ( x + y + z ) + ( l f + l g ) τ 3 + l g l k τ 6 ∥ v − u ∥ τ ,

and from here,

(4.1) ∥ v − u ∥ τ ≤ ε λ φ τ 6 τ 6 − ( l f + l g ) τ 3 − l g l k φ ( x , y , z ) e − τ ( x + y + z ) .

Hence,

∣ v ( x , y , z ) − u ( x , y , z ) ∣ ≤ ε λ φ τ 6 τ 6 − ( l f + l g ) τ 3 − l g l k φ ( x , y , z ) = C 1 ⋅ ε ⋅ φ ( x , y , z ) ,

∀ ( x , y , z ) ∈ D , where C 1 = λ φ τ 6 τ 6 − ( l f + l g ) τ 3 − l g l k , i.e., equation (1.1) + (1.2) + (1.3) is Hyers-Ulam-Rassias stable.□

Remark 4.1

Since the function f : ( τ 1 , ∞ ) , f ( τ ) = τ 6 τ 6 − ( l f + l g ) τ 3 − l f ⋅ l g , τ 1 defined in Remark 3.1, is strictly decreasing and lim τ ⟶ ∞ f ( τ ) = 1 , we obtain

∣ v ( x , y , z ) − u ( x , y , z ) ∣ = C ⋅ ε ⋅ φ ( x , y , z ) , ∀ ( x , y , z ) ∈ D ,

where C = λ φ is the infimum of the constants C 1 from above, for τ > τ 1 .

Example 4.1

Let D = [ 0 , 1 ] × [ 0 , 1 ] × [ 0 , 1 ] , u ∈ C 3 ( D , R ) , f ∈ C ( D × R , R ) , g ∈ C ( D × R × R , R ) , k ∈ C ( E × R , R ) . Let χ 1 ∈ C 1 ( [ 0 , 1 ] × [ 0 , 1 ] , R ) , χ 2 ∈ C 1 ( [ 0 , 1 ] × [ 0 , 1 ] , R ) , χ 3 ∈ C 1 ( [ 0 , 1 ] × [ 0 , 1 ] , R ) , w 1 ∈ C ( [ 0 , 1 ] , R ) , w 2 ∈ C ( [ 0 , 1 ] , R ) , w 3 ∈ C ( [ 0 , 1 ] , R ) , w 0 ∈ R and such that the conditions (1.3) are satisfied.

We consider the equation

(4.2) ∂ 3 u ( x , y , z ) ∂ x ∂ y ∂ z = x 2 y 2 z 2 u ( x , y , z ) + 1 x + y + z + 1 ∫ 0 x ∫ 0 y ∫ 0 z 2 π arctan ( x + y + x − r − s − t ) ⋅ u ( r , s , t ) d r d s d t .

We also consider φ ( x , y , z ) = 1 3 e x + y + z . We remark that

∫ 0 x ∫ 0 y ∫ 0 z φ ( r , s , t ) d r d s d t = ∫ 0 x ∫ 0 y ∫ 0 z 1 3 e r + s + t d r d s d t ≤ λ φ φ ( x , y , z ) ,

∀ x , y , z ∈ [ 0 , 1 ] , with λ φ = 1 3 ; we have f ( x , y , z , u ) = x 2 y 2 z 2 u , and

∣ f ( x , y , z , u 1 ) − f ( x , y , z , u 2 ) ∣ ≤ ∣ u 1 − u 2 ∣ , ∀ ( x , y , z ) ∈ D , u 1 , u 2 ∈ R .

Hence, l f = 1 .

We also have k ( x , y , z , r , s , t , u ) = 2 π arctan ( x + y + x − r − s − t ) ⋅ u , and

∣ k ( x , y , z , r , s , t , u 1 ) − k ( x , y , z , r , s , t , u 2 ) ∣ ≤ ∣ u 1 − u 2 ∣ ,

∀ ( x , y , z , r , s , t ) ∈ E , u 1 , u 2 ∈ R . Hence, l k = 1 .

We have g ( x , y , z , u , v ) = 1 x + y + z + 1 ⋅ v , where

v ( x , y , z , r , s , t , u ( x , y , z ) ) = ∫ 0 x ∫ 0 y ∫ 0 z 2 π arctan ( x + y + x − r − s − t ) ⋅ u ( r , s , t ) d r d s d t

and

∣ g ( x , y , z , u 1 , v 1 ) − g ( x , y , z , u 2 , v 2 ) ∣ ≤ 1 x + y + z + 1 ∣ v 1 − v 2 ∣ ≤ ∣ v 1 − v 2 ∣ ,

∀ ( x , y , z ) ∈ D , u 1 , u 2 , v 1 , v 2 ∈ R , i.e., l g = 1 . We remark that for each τ > 1 + 2 3 , we have ( l f + l g ) τ 3 + l g l k τ 6 < 1 (Remark 3.1). So all the conditions from Theorem 4.1 are satisfied; hence, the equation is Hyers-Ulam-Rassias stable. If v is a solution of (2.2) and u is the unique solution of problem (4.2) + (1.2) + (1.3), using (4.1), we obtain

∣ v ( x , y , z ) − u ( x , y , z ) ∣ ≤ ε λ φ τ 6 τ 6 − 2 τ 3 − 1 φ ( x , y , z ) = C 1 ⋅ ε ⋅ φ ( x , y , z ) ,

∀ ( x , y , z ) ∈ D , where C 1 = τ 6 3 [ τ 6 − ( l f + l g ) τ 3 − l g l k ] .

5 Conclusions

We have studied, in this article, the Hyers-Ulam stability of a perturbed partial integro-differential equation of order three, of hyperbolic type, (1.1) + (1.2) + (1.3), using the Bielecki metric. The Ulam stability of these general types of partial differential equations has not yet been studied. We intend to study the Ulam stability of other general kinds of partial differential equations. We also want to deal with the best Ulam constant.

Acknowledgement

The authors thank the referee for his valuable comments and remarks.

  1. Funding information: The authors state no funding involved.

  2. Author contributions: All authors have accepted responsibility for the entire content of this manuscript and consented to its submission to the journal, reviewed all the results, and approved the final version of the manuscript. DM, SAC, and NL conceptualized this article. DM and SAC developed the methodology and investigated the results. NL supervised the results. DM prepared and edited the manuscript with contributions from all co-authors.

  3. Conflict of interest: The authors state no conflict of interest.

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Received: 2023-11-08
Revised: 2024-03-18
Accepted: 2024-04-23
Published Online: 2024-06-13

© 2024 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/math-2024-0017
Keywords for this article
integro-differential equation of hyperbolic type; Hyers-Ulam stability; Bielecki norm
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  51. The fibering method approach for a Schrödinger-Poisson system with p-Laplacian in bounded domains
  52. On discrete inequalities for some classes of sequences
  53. Boundary value problems for integro-differential and singular higher-order differential equations
  54. Existence and properties of soliton solution for the quasilinear Schrödinger system
  55. Hermite-Hadamard-type inequalities for generalized trigonometrically and hyperbolic ρ-convex functions in two dimension
  56. Endpoint boundedness of toroidal pseudo-differential operators
  57. Matrix stretching
  58. A singular perturbation result for a class of periodic-parabolic BVPs
  59. On Laguerre-Sobolev matrix orthogonal polynomials
  60. Pullback attractors for fractional lattice systems with delays in weighted space
  61. Singularities of spherical surface in R4
  62. Variational approach to Kirchhoff-type second-order impulsive differential systems
  63. Convergence rate of the truncated Euler-Maruyama method for highly nonlinear neutral stochastic differential equations with time-dependent delay
  64. On the energy decay of a coupled nonlinear suspension bridge problem with nonlinear feedback
  65. The limit theorems on extreme order statistics and partial sums of i.i.d. random variables
  66. Hardy-type inequalities for a class of iterated operators and their application to Morrey-type spaces
  67. Solving multi-point problem for Volterra-Fredholm integro-differential equations using Dzhumabaev parameterization method
  68. Finite groups with gcd(χ(1), χc (1)) a prime
  69. Small values and functional laws of the iterated logarithm for operator fractional Brownian motion
  70. The hull-kernel topology on prime ideals in ordered semigroups
  71. ℐ-sn-metrizable spaces and the images of semi-metric spaces
  72. Strong laws for weighted sums of widely orthant dependent random variables and applications
  73. An extension of Schweitzer's inequality to Riemann-Liouville fractional integral
  74. Construction of a class of half-discrete Hilbert-type inequalities in the whole plane
  75. Analysis of two-grid method for second-order hyperbolic equation by expanded mixed finite element methods
  76. Note on stability estimation of stochastic difference equations
  77. Trigonometric integrals evaluated in terms of Riemann zeta and Dirichlet beta functions
  78. Purity and hybridness of two tensors on a real hypersurface in complex projective space
  79. Classification of positive solutions for a weighted integral system on the half-space
  80. A quasi-reversibility method for solving nonhomogeneous sideways heat equation
  81. Higher-order nonlocal multipoint q-integral boundary value problems for fractional q-difference equations with dual hybrid terms
  82. Noetherian rings of composite generalized power series
  83. On generalized shifts of the Mellin transform of the Riemann zeta-function
  84. Further results on enumeration of perfect matchings of Cartesian product graphs
  85. A new extended Mulholland's inequality involving one partial sum
  86. Power vector inequalities for operator pairs in Hilbert spaces and their applications
  87. On the common zeros of quasi-modular forms for Γ+0(N) of level N = 1, 2, 3
  88. One special kind of Kloosterman sum and its fourth-power mean
  89. The stability of high ring homomorphisms and derivations on fuzzy Banach algebras
  90. Integral mean estimates of Turán-type inequalities for the polar derivative of a polynomial with restricted zeros
  91. Commutators of multilinear fractional maximal operators with Lipschitz functions on Morrey spaces
  92. Vector optimization problems with weakened convex and weakened affine constraints in linear topological spaces
  93. The curvature entropy inequalities of convex bodies
  94. Brouwer's conjecture for the sum of the k largest Laplacian eigenvalues of some graphs
  95. High-order finite-difference ghost-point methods for elliptic problems in domains with curved boundaries
  96. Riemannian invariants for warped product submanifolds in Q ε m × R and their applications
  97. Generalized quadratic Gauss sums and their 2mth power mean
  98. Euler-α equations in a three-dimensional bounded domain with Dirichlet boundary conditions
  99. Enochs conjecture for cotorsion pairs over recollements of exact categories
  100. Zeros distribution and interlacing property for certain polynomial sequences
  101. Random attractors of Kirchhoff-type reaction–diffusion equations without uniqueness driven by nonlinear colored noise
  102. Study on solutions of the systems of complex product-type PDEs with more general forms in ℂ2
  103. Dynamics in a predator-prey model with predation-driven Allee effect and memory effect
  104. A note on orthogonal decomposition of 𝔰𝔩n over commutative rings
  105. On the δ-chromatic numbers of the Cartesian products of graphs
  106. Binomial convolution sum of divisor functions associated with Dirichlet character modulo 8
  107. Commutator of fractional integral with Lipschitz functions related to Schrödinger operator on local generalized mixed Morrey spaces
  108. System of degenerate parabolic p-Laplacian
  109. Stochastic stability and instability of rumor model
  110. Certain properties and characterizations of a novel family of bivariate 2D-q Hermite polynomials
  111. Stability of an additive-quadratic functional equation in modular spaces
  112. Monotonicity, convexity, and Maclaurin series expansion of Qi's normalized remainder of Maclaurin series expansion with relation to cosine
  113. On k-prime graphs
  114. On the existence of tripartite graphs and n-partite graphs
  115. Classifying pentavalent symmetric graphs of order 12pq
  116. Almost periodic functions on time scales and their properties
  117. Some results on uniqueness and higher order difference equations
  118. Coding of hypersurfaces in Euclidean spaces by a constant vector
  119. Cycle integrals and rational period functions for Γ0+(2) and Γ0+(3)
  120. Degrees of (L, M)-fuzzy bornologies
  121. A matrix approach to determine optimal predictors in a constrained linear mixed model
  122. On ideals of affine semigroups and affine semigroups with maximal embedding dimension
  123. Solutions of linear control systems on Lie groups
  124. A uniqueness result for the fractional Schrödinger-Poisson system with strong singularity
  125. On prime spaces of neutrosophic extended triplet groups
  126. On a generalized Krasnoselskii fixed point theorem
  127. On the relation between one-sided duoness and commutators
  128. Non-homogeneous BVPs for second-order symmetric Hamiltonian systems
  129. Erratum
  130. Erratum to “Infinitesimals via Cauchy sequences: Refining the classical equivalence”
  131. Corrigendum
  132. Corrigendum to “Matrix stretching”
  133. Corrigendum to “A comprehensive review of the recent numerical methods for solving FPDEs”
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