On Approximate Solutions of Functional Equations in Vector Lattices

and Applied Analysis 3 To describe this result we used to say that the Cauchy functional equation (6) is Hyers-Ulam stable in the class of functions Y. It is worth to mention here that, probably, the first known result in this direction is due to Pólya and Szegö (cf. [12]). Next, the stability of functional equations has beenwidely investigated and generalized in various directions by many authors. For the extensive discussion concerning possible definitions of the stability of functional equations and differences between them we refer the interested reader to [13]. Examples of various recent results concerning the subject as well as a list of numerous references connected with it can be found in the survey paper [14].


Introduction
In this paper we deal with a method of treating approximate solutions of functional equations in a class of functions taking values in Riesz spaces (algebras).Some recent results concerning stability of functional equations in ordered spaces can be found in [1][2][3][4][5][6][7].
In view of the fact that the idea of applying the Spectral Representation Theory (SRT for short) for Riesz spaces to investigate approximate solutions of functional equations in vector lattices has appeared fruitful for various functional equations (cf.[2,3,5]) it seems to be valuable to formulate a general theorem that could play a role of a tool hopefully applicable to a wide class of functional equations.The main purpose of this paper is to provide such a result (see Section 3).
As it has already been mentioned, in the following we are going to make use of the SRT for Riesz spaces that provides a representation of vectors of a given Riesz space  by extended (admitting infinite values) real continuous functions on a certain topological space  which are finite on a dense subset of  ( ∞ ()).The above means that a given Riesz space  (under some additional assumptions) is Riesz isomorphic with a Riesz subspace of  ∞ ().Unfortunately, it appears that, in general, the whole of  ∞ () is not necessarily a Riesz space and that causes some difficulties.The second inconvenience we have to defeat stems from the fact that functions from  ∞ () may attain infinite values.
Once the main results of the paper are achieved, we show their benefits.We apply them to investigate approximate solutions of three selected functional equations.The first two of them have the common origin, but they exhibit different stability behaviours (at least in the class of real-valued functions).We show that an alternative Cauchy functional equation  ( + ) +  () +  () ̸ = 0 ⇒  ( + ) =  () +  () is stable in Riesz spaces (see Section 4).In Section 5 we prove that the Cauchy equation with squares ( + ) 2 = ( () +  ())

Preliminaries
Throughout the paper N, Z, R, and R + are used to denote the sets of all positive integers, integers, real numbers, and nonnegative real numbers, respectively.For the reader's convenience we quote basic definitions and properties concerning Riesz spaces following [8].
A Riesz space  is said to be Dedekind complete (complete) if any nonempty (at most countable) subset of  which is bounded from above has a supremum (cf.[8,Definition 1.1]).
In the following the notion of the relatively uniform convergence will be used (cf.[8,Definition 39.1]).Let  be a Riesz space and let  ∈  + := { ∈  :  ≥ 0}.A sequence (  ) ∈N in  is said to converge -uniformly to an element  ∈  whenever, for every  > 0, there exists a positive integer  0 such that | −   | ≤  holds for all  ≥  0 .We say that (  ) ∈N is relatively uniformly convergent if (  ) ∈N is uniformly convergent with some  ∈  + .A sequence (  ) ∈N in  is called -uniform Cauchy sequence whenever, for every  > 0, there exists a positive integer  1 such that |  −  | ≤  holds for all ,  ≥  1 .
In general the -uniform limit of a sequence may depend on the choice of  ∈  + and does not have to be unique.However, if  is Archimedean, the -uniform limit, if it exists, is unique.In this case the fact that (  ) ∈N converges uniformly to  will be denoted by lim   → ∞   = .A Riesz space  is called -uniformly complete (with a given  ∈  + ) whenever every -uniform Cauchy sequence has a -uniform limit.We say that  is uniformly complete if it is -uniformly complete with every  ∈  + (cf.[8, Definition

39.3]).
There is a large class of spaces satisfying the above conditions.In particular every Dedekind -complete space is Archimedean and uniformly complete.
The element  ∈  + is called a strong unit if for every  ∈  there exists  ∈ R such that || ≤ .
The element  ∈  + is called a weak unit if the band generated by  is the whole of  (cf.[8,Definition 21.4]).Recall that a Riesz subspace  of  is an ideal if it is solid, that is, whenever it follows from  ∈ ,  ∈ , and || ≤ || that  ∈ .An ideal  is termed a band in , whenever a subset of  has a supremum in , that supremum is an element of  (cf.[8,Definition 17.1]).If  is Archimedean then  ∈  + is a weak unit if and only if {} ⊥ = {0}, where {} ⊥ stands for the disjoint complement of  (cf.[9, 353L]).
A linear mapping  :  →  between Riesz spaces  and  is called a Riesz homomorphism if  (sup {V, }) = sup {V, } for V,  ∈ . (4) Now we define the family  ∞ () of extended (admitting infinite values) real continuous functions on a given topological space  that are finite-valued on a dense subset of  and discuss their elementary properties.
Given a topological space , any continuous mapping  of  into  ∞ := R ∪ {−∞} ∪ {+∞} with the usual topology, such that the set is dense in , is called an extended (real-valued) continuous function on .The set of all extended (real-valued) continuous functions on  will be denoted by  ∞ ().We consider the pointwise order in  ∞ () and the pointwise multiplication by scalars, where it is understood that 0 ⋅ ∞ = 0.For any  ∈  At the end of this section we briefly remind the notion of the Hyers-Ulam stability originated by the well-known problem posed by Ulam (cf.[10]) during his talk at the University of Wisconsin in 1940 and the answer given by Hyers (cf.[11]), which we quote below.
Let  and  be Banach spaces and let  > 0. Then for every  :  →  with sup ,∈ ‖( + ) − () − ()‖ ≤  there is a unique  :  →  such that sup ∈ ‖() − ()‖ ≤  and  ( + ) =  () +  () for ,  ∈ . ( To describe this result we used to say that the Cauchy functional equation ( 6) is Hyers-Ulam stable in the class of functions   .It is worth to mention here that, probably, the first known result in this direction is due to Pólya and Szegö (cf.[12]).
Next, the stability of functional equations has been widely investigated and generalized in various directions by many authors.For the extensive discussion concerning possible definitions of the stability of functional equations and differences between them we refer the interested reader to [13].Examples of various recent results concerning the subject as well as a list of numerous references connected with it can be found in the survey paper [14].

Main Results
From now on let (, +) be a groupoid and  a Riesz space (algebra) and let E :   →    ;  ∈ N. We will say that a function  :  →  is a solution of equation if E() = 0 for  ∈   .Given  ∈  + , any  :  →  with will be called a -solution of ( 7). will be termed an approximate solution of ( 7) if it is a -solution of ( 7) with some  ∈  + .Finally, we will say that ( 7) is stable (or Hyers-Ulam stable) if for any  ∈  + there is Δ() ∈  + such that for each -solution  :  →  of ( 7) there exists a solution  :  →  of (7) with We will focus on a class of functional equations that possess the following property.Definition 2. We will say that (7) has the uniform Rapproximation property (URAP for short) if there exist H  :   →   , ( ∈ N),  : R → R, and real sequences   → 0,   → 0,   →  ∈ R, and   → 0 such that if we take R, with the ordinary order, as a realisation of  then for any  > 0 and any -solution  :  → R of ( 7) the following conditions hold: for  ∈ ,  ∈   , and ,  ∈ N.
The URAP is closely related to the Hyers-Ulam stability of (7) in the class of real-valued functions, where the role of the operators H  , ( ∈ N), is played by the so called Hyers operators (cf.[10,11]).The term uniform in the name of the property refers to the fact that the right-hand sides of (P1)-(P3) do not depend on .It is evident that URAP implies the Hyers-Ulam stability.As it will be shown below, in many cases the converse is also true.Lemma 3. Let (, +) be a groupoid.Assume that (7) is Hyers-Ulam stable in the class of real-valued functions defined on  and that there exist   ,   → ∞ and   :  → , ( ∈ N), such that for any solution  :  → R of (7) and for any -solution  :  → R of (7) Then (7) possesses the URAP.
If we rewrite (14) for given  ∈ N and  ∈ N and then add the resulting inequalities, side by side, we obtain (P1) with   = 1/  and   = 1/  .
Since  is an -solution of (7) then using (8) for   () := (  ( 1 ), . . .,   (  )) in place of  we have Dividing the above inequality by   and taking into account (11), in view of the definition of H  , we arrive at which means that (P3) holds with   = /  () provided that () ̸ = 0.The case () = 0 means that  satisfies (7) and, therefore, (P3) holds with any nonnegative   .Remark 4. Let us observe that all the assumptions of Lemma 3 are fulfilled if we assume that (7) is Hyers-Ulam stable in the class of real-valued functions and that there exists   → ∞ such that any solution  :  → R of ( 7) is (  , )homogeneous with some  > 0; that is, and E is ( −  , )-homogeneous with some  > 0; that is, In particular, any functional equation which is Hyers-Ulam stable in the class of real-valued functions, whose solution form additive functions, has the URAP.
Let us note that the SRT for Riesz spaces provides results which guarantee (H1).Various classical spectral representation theorems offer effective constructions of a topological space  as well as a space of representatives L and a Riesz isomorphism , depending on the properties of a given Riesz space  (cf., e.g., [8,Ch.7]).
It is easy to see that if E and H  are defined with the use of the ordinary Riesz space (algebra) operations, that is, linear operations or lattice operations, then (H2), (H3), and (H5) are automatically satisfied.Now we are going to prove a lemma that provides some properties of a function  :  →  that yield (H4) (for  = 2).Assume that (H1) holds and that we are given mappings  :  → ,  :  ×  → ,  :  ×  ×  →  and open and dense subsets   ,   of  such that For fixed  ∈  we consider the following hypotheses.Proof.Assume, at first, that (L1) holds.For fixed  ∈  we define By ( 23) and (H1), for every ,  ∈ ⟨⟩, we have which means that  ( ( ( + ) ,  ())) ⊂  ( ( ())) .
Finally, assume that (L3) holds.Let  ∈  and let Observe that due to (H1), (25), and ( 22) we have If  is a negative integer, we use (36) with  and  replaced by  1 and  2 , respectively, together with (40) to complete the proof of ( 20).Now we are in a position to formulate and prove the main theorem of the paper.Theorem 7. Let (, +) be a groupoid, let  be an Archimedean Riesz space, and let (7) possess the URAP.Assume that, for given  ∈  + ,  :  →  is a -solution of (7) and that hypotheses (H1)-(H5) hold.Let Δ() ∈  be such that Δ()() ≥ (()) for  ∈ (Δ()) and assume that  is Δ()-uniformly complete.Then there exists a solution  :  →  of (7) such that | () −  ()| ≤ Δ () for  ∈ . (41) Moreover, the solution of (7) satisfying (41) is unique provided that it is unique in the case where we consider R as a realisation of .
Proof.The proof runs in three steps.
Remark 8. Theorem 7 remains valid for more involved functional equations, for instance, alternative (conditional) functional equations for E 1 , E 2 :   →    .By (59) we mean that  :  →  satisfies (59) if We assume that both operators E 1 and E 2 satisfy (H2) and (H5).Since, in fact,  is now two-place function, we assume that Δ( is convergent to 0. Then, accordingly, Step 2 of the proof of Theorem 7, that is, the proof that  defined by (50) satisfies (7), should be replaced by the following reasoning.
Let  = ( 1 ,  2 , . . .,   ) ∈   .By (H4) there exists an open and dense subset   of  with (20).Let   :=   ∩ () ∩ (Δ()) ∩ (E()).By the definition of  (50) and (H5) we have lim Let  ∈   be fixed.Then, lim On the other hand, by (H1) and (H2), for any  ∈ ⟨ 1 , Taking into account (64), we have Since the last equality is valid for any  from the open and dense subset   of  and all the functions in the above inequality are continuous, we obtain Consequently, we infer that as  is a Riesz isomorphism.This, due to the fact that  ∈   was chosen arbitrarily, completes the proof that  satisfies (7).
This equation belongs to the class of conditional Cauchy equations with the condition dependent on the unknown function.The general solution of (70) is described in [15,Theorem 8].Stability of this equation, in the class of functions mapping an Abelian semigroup into a Banach space, has been investigated in [16] and in a more general setting in [17].For the readers convenience we quote the main result of [16] as it will be used in the sequel.for all  ∈ .
The natural question arises if a similar result holds true in ordered spaces.One can rewrite all the sentences of Theorem 12 for functions mapping an Abelian semigroup  into a Riesz space , replacing the norm by the absolute value in .The main goal of this section is to apply Theorem 7 with the purpose to give an affirmative answer to this question.
We will use one of the most general spectral representation theorems, namely, the Johnson-Kist Spectral Representation Theorem which we quote here.
Theorem 13 (Johnson-Kist representation theorem) (cf.[8,Theorem 44.4]).Let  be an Archimedean Riesz space.There exists a Riesz space L of extended real continuous functions and a Riesz isomorphism of  onto L.
The main theorem of this section reads as follows.
for ,  ∈ , in Theorem 14 instead of condition (73), with the common meaning of  <  as  ≤  and  ̸ = .Of course, if the order in a Riesz space  is linear then conditions (73) and (76) coincide.However, as it will be shown in the example below, in general, assumption (73) cannot be replaced by (76).
Example 15.Let  be the Archimedean Riesz space of all real functions of real variable with the pointwise order and let  1 ∈  be given by Then  is  1 -uniformly complete.We define  : R →  by Then  is not additive and satisfies (76) with  1 defined above and  2 ≡ 0. On the other hand,  cannot be approximated by any additive function.Suppose, for contradiction, that there exists an additive mapping  : R →  satisfying (74).Let us fix  ∈ R. For  ̸ = 0 inequality (74) results with ()() = ()() according to the definition of  1 .Directly from the definition of  we have ()(0) = 0. Then by (74) and the additivity of  we obtain ()(0) = 0. Eventually, we infer that  and  coincide and, therefore,  is additive.We have obtained a contradiction.
Let us point out that the assumption that the Riesz space  is Archimedean is necessary in order to have the uniqueness of an existing additive function  in Theorem 14, which can be observed in the following simple example.

Approximate Solutions of the Cauchy Equation with Squares
which admits further generalisations from the real case to more general structures.Affirmative results concerning stability of (80) are contained in [18] for real-valued functions and, for the class of functions taking values in Riesz spaces, in [5].There are also known results concerning the stability of the generalized equation (80) for functions acting into a normed space, called Fischer-Muszély functional equation: ( + )     =      () +  ()     for ,  ∈ .
It occurs that, despite the fact that on the assumption that the norm is strictly convex (81) is equivalent to the Cauchy functional equation (cf.[19]), even if we consider R 2 with the Euclidean norm as a target space of , (81) fails to be stable in the Hyers-Ulam sense (cf.[18]).However, if we consider the stability of (81) in the class of surjective functions, then the answer is positive (cf.[20]).Finally, concerning (79) in the class of complex functions we have the following stability result.Remark 18.In fact, for complex functions, (79) occurs to be superstable in the sense of Baker which was proved in [21] and, with the use of Theorem 5, in [22].