# On Approximate Solutions of Functional Equations in Vector Lattices

Authors: Bogdan Batko
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

We provide a method of approximation of approximate solutions of functional equations in the class of functions acting into a Riesz space (algebra). The main aim of the paper is to provide a general theorem that can act as a tool applicable to a possibly wide class of functional equations. The idea is based on the use of the Spectral Representation Theory for Riesz spaces. The main result will be applied to prove the stability of an alternative Cauchy functional equationF(x+y)+F(x)+F(y)≠0⇒F(x+y)=F(x)+F(y) in Riesz spaces, the Cauchy equation with squares F(x+y)2=(F(x)+F(y))2 in f-algebras, and the quadratic functional equation F(x+y)+F(x-y)=2F(x)+2F(y) in Riesz spaces.


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## Body

## 1. 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 [1–7].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 spaceL by extended (admitting infinite values) real continuous functions on a certain topological space X which are finite on a dense subset of X (C∞(X)). The above means that a given Riesz space L (under some additional assumptions) is Riesz isomorphic with a Riesz subspace of C∞(X). Unfortunately, it appears that, in general, the whole of C∞(X) 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 C∞(X) 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(1)F(x+y)+F(x)+F(y)≠0⟹F(x+y)=F(x)+F(y)
is stable in Riesz spaces (see Section 4). In Section 5 we prove that the Cauchy equation with squares
(2)F(x+y)2=(F(x)+F(y))2
is stable in f-algebras; however unlike in the case of real-valued functions it is not superstable. The third one is the quadratic functional equation
(3)F(x+y)+F(x-y)=2F(x)+2F(y).
We prove its stability in Riesz spaces in Section 6.


## 2. Preliminaries

Throughout the paperℕ, 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].We say that a partially ordered real linear spaceL (we denote the order in L by ≤) is aRiesz space (vector lattice) if sup⁡{x,y} exists for all x,y∈L (cf. [8, Definition 11.1]). We define the absolute value of x∈L by the formula |x|∶=sup⁡{x,-x}≥0. A Riesz space L is calledArchimedean if, for each x∈L, the inequality x≤0 holds whenever the set {nx:n∈ℕ} is bounded from above (cf. [8, Definition 22.1]). We say that L is aRiesz algebra if L is a Riesz space equipped with the common algebra multiplication satisfying xy≥0 whenever x,y≥0. A Riesz algebra L is termed an f-algebra, whenever inf⁡{x,y}=0 implies inf⁡{xz,y}=inf⁡{zx,y}=0 for every z≥0.A Riesz spaceL is said to beDedekind complete (σ-complete) if any nonempty (at most countable) subset of L which is bounded from above has a supremum (cf. [8, Definition 1.1]).In the following the notion of therelatively uniform convergence will be used (cf. [8, Definition 39.1]). Let L be a Riesz space and let u∈L+∶={u∈L:u≥0}. A sequence (fn)n∈ℕ in L is said toconvergeu-uniformly to an element f∈L whenever, for every ɛ>0, there exists a positive integer n0 such that |f-fn|≤ɛu holds for all n≥n0. We say that (fn)n∈ℕ isrelatively uniformly convergent if (fn)n∈ℕ is u-uniformly convergent with some u∈L+. A sequence (fn)n∈ℕ in L is called u-uniform Cauchy sequence whenever, for every ɛ>0, there exists a positive integer n1 such that |fm-fn|≤ɛu holds for all m,n≥n1.In general theu-uniform limit of a sequence may depend on the choice of u∈L+ and does not have to be unique. However, if L is Archimedean, the u-uniform limit, if it exists, is unique. In this case the fact that (fn)n∈ℕ converges u-uniformly to f will be denoted by lim⁡n→∞ufn=f.A Riesz spaceL is called u-uniformly complete (with a given u∈L+) whenever every u-uniform Cauchy sequence has a u-uniform limit. We say that L isuniformly complete if it is u-uniformly complete with every u∈L+ (cf. [8, Definition 39.3]).There is a large class of spaces satisfying the above conditions. In particular every Dedekindσ-complete space is Archimedean and uniformly complete.The elemente∈L+ is called astrong unit if for every l∈L there exists α∈ℝ such that |l|≤αe.The elemente∈L+ is called aweak unit if the band generated by e is the whole of L (cf. [8, Definition 21.4]). Recall that a Riesz subspace M of L is anideal if it issolid, that is, whenever it follows from f∈M, g∈L, and |g|≤|f| that g∈M. An ideal M is termed aband in L, whenever a subset of M has a supremum in L, that supremum is an element of M (cf. [8, Definition 17.1]). If L is Archimedean then e∈L+ is a weak unit if and only if {e}⊥={0}, where {e}⊥ stands for the disjoint complement of e (cf. [9, 353L]).A linear mappingπ:L→M between Riesz spaces L and M is called aRiesz homomorphism if
(4)π(sup⁡{v,w})=sup⁡{πv,πw}forv,w∈L.Now we define the familyC∞(X) of extended (admitting infinite values) real continuous functions on a given topological space X that are finite-valued on a dense subset of X and discuss their elementary properties.Given a topological spaceX, any continuous mapping f of X into R∞∶=ℝ∪{-∞}∪{+∞} with the usual topology, such that the set
(5)R(f)={x:|f(x)|<∞}
is dense in X, is called anextended(real-valued)continuous function on X. The set of all extended (real-valued) continuous functions on X will be denoted by C∞(X). We consider the pointwise order in C∞(X) and the pointwise multiplication by scalars, where it is understood that 0·∞=0. For any f∈C∞(X), the set R(f) is open and dense. If f,g,h∈C∞(X) and h(x)=f(x)+g(x) holds for all x∈R(f)∩R(g), then (by definition) h is calledthe sum offandg (notation h=f+g). Since the set R(f)∩R(g) is dense, the function h=f+g is uniquely defined if it exists. It occurs that C∞(X) is not necessarily closed with respect to the operation of addition (cf. [8, p. 295]). Any subset of C∞(X) closed under the operation of addition, multiplication by scalars and the taking of finite infima and suprema, is obviously a Riesz space with respect to pointwise ordering. Accordingly, any subset of this kind is called aRiesz space of extended real continuous functions on X.The difficulties with the addition disappear ifX isextremally disconnected; that is, the closure of every open set is open. Then C∞(X) is a Riesz space (cf. [8, Theorem 47.2]) and even an f-algebra with the multiplicative identity under the appropriate definition of the multiplication. Given f,g∈C∞(X), the set R(f)∩R(g) is open and dense. Thus the function which is equal to f(x)g(x) for every x∈R(f)∩R(g) is finite-valued and continuous on R(f)∩R(g). Hence, this function can uniquely be extended to an extended continuous function on X (cf. [8, Theorem 47.1]), and let, by definition, this extended function be fg.Proposition 1 (cf. [3, Proposition 1]).
IfX is an extremally disconnected topological space then C∞(X) is an f-algebra with a multiplicative identity.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.LetX and Y be Banach spaces and let ɛ>0. Then for every g:X→Y with sup⁡x,y∈X∥g(x+y)-g(x)-g(y)∥≤ɛ there is a unique f:X→Y such that sup⁡x∈X∥g(x)-f(x)∥≤ɛ and
(6)f(x+y)=f(x)+f(y)forx,y∈X.
To describe this result we used to say that the Cauchy functional equation (6) isHyers-Ulam stable in the class of functionsYX. 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].


## 3. Main Results

From now on let(G,+) be a groupoid and L a Riesz space (f-algebra) and let ℰ:LG→LGk; k∈ℕ. We will say that a function F:G→L is asolution of equation
(7)ℰF=0
if ℰF(x¯)=0 for x¯∈Gk. Given u∈L+, any F:G→L with
(8)|ℰF(x¯)|≤uforx¯∈Gk
will be called a u-solution of (7). F will be termed anapproximate solution of (7) if it is a u-solution of (7) with some u∈L+. Finally, we will say that (7) isstable (orHyers-Ulam stable) if for any u∈L+ there is Δ(u)∈L+ such that for each u-solution F:G→L of (7) there exists a solution A:G→L of (7) with
(9)|F(x)-A(x)|≤Δ(u)forx∈G.We will focus on a class of functional equations that possess the following property.Definition 2.
We will say that (7) has theuniformℝ-approximation property (URAP for short) if there exist ℋn:LG→LG, (n∈ℕ), δ:ℝ→ℝ, and real sequences αn→0, βn→0, γn→γ∈ℝ, and ρn→0 such that if we take ℝ, with the ordinary order, as a realisation of L then for any ɛ>0 and any ɛ-solution f:G→ℝ of (7) the following conditions hold:(P1)
|ℋnf(x)-ℋmf(x)|≤(αn+βm)δ(ɛ),(P2)
|ℋnf(x)-f(x)|≤γnδ(ɛ),(P3)
|ℰℋnf(x¯)|≤ρnδ(ɛ),for x∈G,  x¯∈Gk, and m,n∈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 ℋn, (n∈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 f. 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(G,+) be a groupoid. Assume that (7) is Hyers-Ulam stable in the class of real-valued functions defined on G and that there exist μn,en→∞ and ηn:G→G, (n∈ℕ), such that for any solution a:G→ℝ of (7)
(10)a∘ηn=μnaforn∈ℕ
and for any ɛ-solution f:G→ℝ of (7)
(11)1enℰf=ℰ1μnfforn∈ℕ.
Then (7) possesses the URAP.Proof.
Letf:G→ℝ be an ɛ-solution of (7). We define ℋnf(x)∶=f(ηn(x))/μn for x∈G, n∈ℕ. By the Hyers-Ulam stability of (7) there exist a solution a:G→ℝ of (7) and δ(ɛ)∈ℝ with
(12)|f(x)-a(x)|≤δ(ɛ)forx∈G.
Applying (12) for ηn(x) in place of x and taking into account (10) we obtain
(13)|f(ηn(x))-μna(x)|≤δ(ɛ)forx∈G,n∈ℕ.
Dividing the above inequality by μn, side by side, we have
(14)|ℋnf(x)-a(x)|≤1μnδ(ɛ)forx∈G,n∈ℕ
on account of the definition of ℋn. By (14) and (12) we have (P2) with γn∶=1+(1/μn).
If we rewrite (14) for given n∈ℕ and m∈ℕ and then add the resulting inequalities, side by side, we obtain (P1) with αn=1/μn and βm=1/μm.
Sincef is an ɛ-solution of (7) then using (8) for ηn(x¯)∶=(ηn(x1),…,ηn(xk)) in place of x¯ we have
(15)|ℰf(ηn(x¯))|≤ɛforx¯∈Gk,n∈ℕ.
Dividing the above inequality by en and taking into account (11), in view of the definition of ℋn, we arrive at
(16)|ℰℋnf(x¯)|≤1enɛforx¯∈Gk,n∈ℕ,
which means that (P3) holds with ρn=ɛ/enδ(ɛ) provided that δ(ɛ)≠0. The case δ(ɛ)=0 means that f satisfies (7) and, therefore, (P3) holds with any nonnegative ρn.Remark 4.
Let us observe that all the assumptions of Lemma3 are fulfilled if we assume that (7) is Hyers-Ulam stable in the class of real-valued functions and that there exists λn→∞ such that any solution a:G→ℝ of (7) is (λn,α)-homogeneous with some α>0; that is,
(17)a(λnx)=λnαa(x)forx∈G,n∈ℕ,
and ℰ is (λn-α,β)-homogeneous with some β>0; that is,
(18)ℰ(λn-αf)=λn-αβℰfforn∈ℕ,f∈ℝG.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.Corollary 5.
Letf:G→ℝ map a groupoid (G,+) into ℝ. Assume that there exist an additive function a:G→ℝ and a real number α≥0 such that
(19)|f(x)-a(x)|≤αforx∈G.
Let ℋnf(x)∶=(1/n)f(nx) for x∈G and n∈ℕ. Then (P1)–(P3) hold with αn=βn=1/n, γn=1+1/n, ρn=3/n, for n∈ℕ and δ=α.Proof.
Routine.Let〈x1,x2,…,xk〉 stand for the substructure of G generated by x1,…,xk∈G and let Δ∈L. We will consider the following hypotheses:(H1)
there exist a topological spaceX and a Riesz (f-algebra) isomorphism π:L→L^⊂C∞(X),(H2)
π and ℰ commute; that is, π(ℰF(x¯))=ℰ(πF)(x¯) for any F∈LG and x¯∈Gk,(H3)
π and ℋn commute; that is, π(ℋnF(x))=(ℋnπF)(x) for any F∈LG, x∈G, n∈ℕ,(H4)
for eachx¯=(x1,x2,…xk)∈Gk there exists an open and dense subset wx¯ of X such that
(20)wx¯⊂R(πF(z))forz∈〈x1,x2,…,xk〉(H5)
for eachx¯∈Gk, if lim⁡n→∞ΔℋnF(x¯) exists, then
(21)lim⁡n→∞ΔℰℋnF(x¯)=ℰ(lim⁡n→∞ΔℋnF)(x¯).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 spaceX as well as a space of representatives L^ and a Riesz isomorphism π, depending on the properties of a given Riesz space L (cf., e.g., [8, Ch.7]).It is easy to see that ifℰ and ℋn 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 functionF:G→L that yield (H4) (for k=2). Assume that (H1) holds and that we are given mappings B:L→L, C:L×L→L, D:L×L×L→L and open and dense subsets RC, RD of X such that
(22)R(πB(v))⊂R(πv)R(πv)∩R(πw)∩RC⊂R(πC(v,w))R(πv)∩R(πw)∩R(πt)∩RD⊂R(πD(v,w,t))forv,w,t∈L.
For fixed x∈G we consider the following hypotheses. (L1)
(G,+) is an Abelian group and
(23)B(F(y))≤C(F(y+z),F(z))fory,z∈〈x〉,(L2)
(G,+) is an Abelian semigroup and
(24)B(F(y+z))≤C(F(y),F(z))fory,z∈〈x〉,(L3)
(G,+) is an Abelian group and
(25)B(F(y+z))≤D(F(y),F(z),F(y-z))fory,z∈〈x〉.Lemma 6.
LetF:G→L map a groupoid (G,+) into a Riesz space L and let (H1) hold. If for any x∈G at least one of the hypotheses (L1), (L2), and (L3) holds then (H4) (for k=2) is fulfilled.Proof.
Assume, at first, that (L1) holds. For fixedx∈G we define
(26)wx∶=R(πF(x))∩R(πF(0))∩RC.
By (23) and (H1), for every y,z∈〈x〉, we have
(27)πB(F(y))≤πC(F(y+z),F(z)),
which means that
(28)R(πC(F(y+z),F(z)))⊂R(πB(F(y))).
Therefore, by (22), we see that
(29)R(πF(y+z))∩R(πF(z))∩RC⊂R(πF(y))fory,z∈〈x〉.
Replacing y by -x and z by x in (29) we observe that wx⊂R(πF(-x)). Now, suppose that wx⊂R(πF(nx)) for given n∈ℕ and apply (29) with y and z replaced by (n+1)x and -x, respectively, in order to obtain wx⊂R(πF((n+1)x)). By induction we arrive at
(30)wx⊂R(πF(nx))forn∈ℕ.
On the other hand, using (29) with -nx in place of y and nx in place of z, we receive wx∩R(πF(nx))⊂R(πF(-nx)). This along with (30) yields
(31)wx⊂R(πF(z))forz∈〈x〉.
For arbitrary x1,x2∈G we define wx1,x2∶=wx1∩wx2, where wx1 and wx2 are given by (26). By (31) wx1⊂R(πF(y)) for y∈〈x1〉 and wx2⊂R(πF(y)) for y∈〈x2〉. Let us consider y∈〈x1〉 and z∈〈x2〉. Using (29) with y and z replaced by y+z and -z, respectively, we observe that wx1,x2⊂R(πF(y+z)), which completes the proof of (20).
Now we assume that (L2) holds. Letx∈G and let
(32)wx∶=R(πF(x))∩RC.
Similarly as in case (L1) one can observe that (H1), (24), and (22) yield
(33)R(πF(y))∩R(πF(z))∩RC⊂R(πF(y+z))fory,z∈〈x〉.
By the definition wx⊂R(πF(x)). Suppose that wx⊂R(πF(nx)) for given n∈ℕ and apply (33) with y and z replaced by nx and x, respectively, in order to obtain wx⊂R(πF((n+1)x)). By induction we receive
(34)wx⊂R(πF(nx))forn∈ℕ;
thus (31) holds.
Fix arbitraryx1,x2∈G and define wx1,x2:=wx1∩wx2, where wx1 and wx2 are given by (32). By (31) wx1⊂R(πF(y)) for y∈〈x1〉 and wx2⊂R(πF(y)) for y∈〈x2〉. Let us consider y∈〈x1〉, z∈〈x2〉. Using (33) for y and z, we observe that wx1,x2⊂R(πF(y+z)). This completes the proof of (20).
Finally, assume that (L3) holds. Letx∈G and let
(35)wx∶=R(πF(x))∩R(πF(-x))∩R(πF(0))∩RD.
Observe that due to (H1), (25), and (22) we have
(36)R(πF(y))∩R(πF(z))∩R(πF(y-z))∩RD⊂R(πF(y+z))fory,z∈〈x〉.
By the definition wx⊂R(πF(x))∩R(πF(0)). Let n∈ℕ, suppose that wx⊂R(πF(kx)) for k∈{0,…,n}, and apply (36) with y and z replaced by nx and x, respectively, in order to obtain wx⊂R(πF((n+1)x)). By induction we obtain
(37)wx⊂R(πF(nx))forn∈ℕ.
Exchanging x with -x and repeating the lines of the proof of (37) we conclude that wx⊂R(πF(-nx)) for n∈ℕ, which together with (37) results with (31).
Fix arbitraryx1,x2∈G and define wx1,x2∶=wx1∩wx2∩R(πF(x1-x2)), where wx1 and wx2 are given by (35). By (31) wx1,x2⊂R(πF(y)) for y∈〈x1〉 and wx1,x2⊂R(πF(y)) for y∈〈x2〉. We will prove that wx1,x2⊂R(πF(kx1+lx2)) for k,l∈ℤ. It is evident that wx1,x2⊂R(πF(x1+x2)). Let k∈ℕ and suppose that wx1,x2⊂R(πF(mx1+x2)) for m∈{0,…,k}. Applying (36) with y and z replaced by mx1+x2 and x1, respectively, we observe that wx1,x2⊂R(πF((m+1)x1+x2)). Thus, by induction
(38)wx1,x2⊂R(πF(kx1+x2))fork∈ℕ.
To observe that (38) holds also for negative integers it is enough to apply (36) with x2 and -kx1 in place of y and z, respectively, and take into account (38). Thus,
(39)wx1,x2⊂R(πF(kx1+x2))fork∈ℤ.
Now, let k∈ℤ be fixed and assume that, for given l∈ℕ, it is wx1,x2⊂R(πF(kx1+mx2)) for m∈{0,…,l}. Then by (36) with y and z replaced by kx1+lx2 and x2, respectively, we obtain wx1,x2⊂R(πF(kx1+(l+1)x2)). By induction,
(40)wx1,x2⊂R(πF(kx1+lx2))fork∈ℤ,l∈ℕ.
If l is a negative integer, we use (36) with y and z replaced by kx1 and lx2, 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(G,+) be a groupoid, let L be an Archimedean Riesz space, and let (7) possess the URAP. Assume that, for given u∈L+, F:G→L is a u-solution of (7) and that hypotheses (H1)–(H5) hold. Let Δ(u)∈L be such that πΔ(u)(s)≥δ(πu(s)) for s∈R(πΔ(u)) and assume that L is Δ(u)-uniformly complete. Then there exists a solution A:G→L of (7) such that
(41)|F(x)-A(x)|≤γΔ(u)forx∈G.
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 L.Proof.
The proof runs in three steps.
Step
1. Assume that F:G→L is a u-solution of (7). We will prove that, for every x∈G, the sequence (ℋnF(x))n∈ℕ is Δ(u)-uniformly convergent.
Fixx∈G. By (H4) there exists an open and dense subset wx of X with
(42)wx⊂R(πF(z))forz∈〈x〉.
Let Wx:=wx∩R(πu)∩R(πΔ(u)). By (H1) and (H2) for arbitrarily fixed s∈Wx and x¯∈〈x〉k we have
(43)|ℰ(πF(·)(s))(x¯)|≤πu(s),
where πF(·)(s) stands for the function mapping G into ℝ∞ given by πF(x)(s)∶=π(F(x))(s) for x∈G. This means that πF(·)(s)|〈x〉 is a real-valued πu(s)-solution of (7) on 〈x〉. Since (7) has the URAP, by (P1),
(44)|ℋnπF(·)(s)(x)-ℋmπF(·)(s)(x)|≤(αn+βm)δ(πu(s)),forn,m∈ℕ,
with some αn→0 and βm→0. Similarly, by (P2) we obtain
(45)|ℋnπF(·)(s)(x)-πF(x)(s)|≤γnδ(πu(s)),forn∈ℕ
with some γn→γ. According to (H3) and the definition of Δ the above inequalities imply
(46)|πℋnF(x)(s)-πℋmF(x)(s)|≤(αn+βm)πΔ(u)(s),forn,m∈ℕ,|πℋnF(x)(s)-πF(x)(s)|≤γnπΔ(u)(s),forn∈ℕ,
respectively. Since s∈Wx is arbitrary, Wx is open and dense in X; moreover all the functions in the above inequalities (as functions of variable s) are continuous; we obtain
(47)|πℋnF(x)-πℋmF(x)|≤(αn+βm)πΔ(u),forn,m∈ℕ,|πℋnF(x)-πF(x)|≤γnπΔ(u),forn∈ℕ.
Due to the fact that π is a Riesz homomorphism, the above inequalities result in
(48)|ℋnF(x)-ℋmF(x)|≤(αn+βm)Δ(u),forn,m∈ℕ,(49)|ℋnF(x)-F(x)|≤γnΔ(u),forn∈ℕ,
respectively. Inequality (48) means that (ℋnF(x))n∈ℕ is a Δ(u)-uniform Cauchy sequence in a Δ(u)-uniformly complete Riesz space and, therefore, relatively uniformly convergent. This, due to the fact that x∈G was arbitrarily fixed, proves that A:G→L given by
(50)A(x)∶=lim⁡n→∞Δ(u)ℋnF(x),forx∈G
is well defined.
Lettingn→∞ in (49) we obtain (41) as L is Archimedean.
Step
2. We will prove that A is a solution of (7). Let x¯=(x1,x2,…,xk)∈Gk. By (H4) there exists an open and dense subset wx¯ of X with (20). Let Wx¯:=wx¯∩R(πu)∩R(πΔ(u)). By (H1) and (H2) for arbitrarily fixed s∈Wx and y¯∈〈x1,x2,…,xk〉k we have
(51)|ℰπF(·)(s)(y¯)|≤πu(s)
which means that πF(·)(s)|〈x1,x2,…,xk〉 is a real-valued πu(s)-solution of (7) on 〈x1,x2,…,xk〉. Since (7) has the URAP, by (P3),
(52)|ℰℋnπF(·)(s)(x¯)|≤ρnδ(πu(s))forn∈ℕ
with some ρn→0 and consequently, taking into account the definition of Δ, (H2), and (H3), we have
(53)|πℰℋnF(x¯)(s)|≤ρnπΔ(u)(s)forn∈ℕ.
Since the last inequality is valid for any s from the open and dense subset Wx¯ of X and all the functions in the above inequality are continuous, we obtain
(54)|πℰℋnF(x¯)|≤ρnπΔ(u)forn∈ℕ.
Finally, taking into account the fact that π is a Riesz homomorphism, we arrive at
(55)|ℰℋnF(x¯)|≤ρnΔ(u)forn∈ℕ.
Letting n→∞ and taking into account the definition of A, (H5) and the fact that L is Archimedean, we have ℰA(x¯)=0 which proves that A satisfies (7) as x¯∈Gk was chosen arbitrarily.
Step
3. We will prove the uniqueness of A satisfying (41) under the assumption that in the class of real-valued functions a solution of (7) which approximates F is uniquely determined. Contrary, suppose that two solutions A1,A2:G→L of (7) satisfy
(56)|F(x)-Ai(x)|≤γΔ(u)forx∈G,i∈{1,2}.
Fix x∈G. By (H4) there exists an open and dense subset wx of X such that
(57)wx⊂R(πF(z))forz∈〈x〉.
Let Wx:=wx∩R(πu)∩R(πΔ(u)). According to (H1) and (H2), the above means that, for arbitrarily fixed s∈Wx, function πF(·)(s)|〈x〉 is a real-valued πu(s)-solution of (7) on 〈x〉. On the other hand, by (56),
(58)|πF(y)(s)-πAi(y)(s)|≤γπΔ(u)(s)fory∈〈x〉
as π is a Riesz homomorphism. This along with (H2) means that πAi(·)(s)|〈x〉, (i∈{1,2}) are real-valued solutions of (7) that approximate πF(·)(s)|〈x〉 on 〈x〉. Hence πA1(x)(s)=πA2(x)(s). But then πA1(x)=πA2(x) as s∈Wx was chosen arbitrarily, Wx is open and dense in X, and πA1(x) and πA2(x) are continuous. Since π is injective we infer that A1(x)=A2(x). This completes the proof, as x∈G was arbitrarily fixed.Remark 8.
Theorem7 remains valid for more involved functional equations, for instance, alternative (conditional) functional equations
(59)(ℰ1∨ℰ2)F=0
for ℰ1,ℰ2:LG→LGk. By (59) we mean that F:G→L satisfies (59) if
(60)ℰ1F(x¯)=0orℰ2F(x¯)=0forx¯∈Gk.
Given u1,u2∈L+, any F:G→L is a (u1,u2)-near solution of (59) if
(61)|ℰ1F(x¯)|≤u1or|ℰ2F(x¯)|≤u2forx¯∈Gk.
We assume that both operators ℰ1 and ℰ2 satisfy (H2) and (H5). Since, in fact, δ is now two-place function, we assume that Δ(u1,u2) satisfies πΔ(u1,u2)(s)≥δ(πu1(s),πu2(s)) for s∈R(πΔ(u1,u2)). Moreover, defining open and dense subsets of X we replace any occurrence of R(πu) by R(πu1)∩R(πu2).Remark 9.
Similarly, instead of an alternative in (59) one may consider a conjunction, which is useful to investigate systems of functional equations.Remark 10.
Observe that Theorem7 remains valid if one considers a slightly more general definition of the URAP. Namely, one can allow sequences αn, βn, γn, ρn to be dependent on x. Now we assume the convergence of αn(x), βn(x), ρn(x) to 0 and γn(x) to γ(x) at each point x∈G. Moreover, one can consider different deltas on the right-hand sides of (P1)–(P3).Remark 11.
Let us note that one can replace condition (P3) in Definition2 of the URAP with the following one:(P3′) the sequence(62)ξn∶=sup⁡x¯∈Gk|ℰℋnf(x¯)|is convergent to 0.
Then, accordingly, Step 2 of the proof of Theorem7, that is, the proof that A defined by (50) satisfies (7), should be replaced by the following reasoning.
Letx¯=(x1,x2,…,xk)∈Gk. By (H4) there exists an open and dense subset wx¯ of X with (20). Let Wx¯∶=wx¯∩R(πu)∩R(πΔ(u))∩R(πℰA(x¯)). By the definition of A (50) and (H5) we have
(63)lim⁡n→∞ΔℰℋnF(x¯)=ℰA(x¯).
Let s∈Wx¯ be fixed. Then,
(64)lim⁡n→∞πℰℋnF(x¯)(s)=πℰA(x¯)(s).
On the other hand, by (H1) and (H2), for any y¯∈〈x1,x2,…,xk〉k, we have
(65)|ℰπF(·)(s)(y¯)|≤πu(s)
which means that πF(·)(s)|〈x1,x2,…,xk〉 is a real-valued πu(s)-solution of (7) on 〈x1,x2,…,xk〉. Since (7) has the URAP, then (P3′), (H2), and (H3) yield
(66)lim⁡n→∞πℰℋnF(x¯)(s)=lim⁡n→∞ℰℋnπF(·)(s)(x¯)=0.
Taking into account (64), we have
(67)πℰA(x¯)(s)=0.
Since the last equality is valid for any s from the open and dense subset Wx¯ of X and all the functions in the above inequality are continuous, we obtain
(68)πℰA(x¯)=0.
Consequently, we infer that
(69)ℰA(x¯)=0
as π is a Riesz isomorphism. This, due to the fact that x¯∈Gk was chosen arbitrarily, completes the proof that A satisfies (7).


## 4. Approximate Solutions of an Alternative Cauchy Equation

In this section we deal with approximate solutions of an alternative Cauchy functional equation(70)F(x+y)+F(x)+F(y)≠0⟹F(x+y)=F(x)+F(y).
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.Theorem 12 (cf. [16, Theorem 1]).
Let(G,+) be an Abelian semigroup and let (E,∥·∥) be a Banach space. If, for some ε1,ε2≥0 and all x,y∈G, a function f:G→E satisfies
(71)∥f(x+y)+f(x)+f(y)∥>ε1⟹∥f(x+y)-f(x)-f(y)∥≤ε2,
then there exists a unique additive function a:G→E such that
(72)∥f(x)-a(x)∥≤max⁡{ε1,ε2}
for all x∈G.The natural question arises if a similar result holds true in ordered spaces. One can rewrite all the sentences of Theorem12 for functions mapping an Abelian semigroup G into a Riesz space L, replacing the norm by the absolute value in L. 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]).
LetL be an Archimedean Riesz space. There exists a Riesz space L^ of extended real continuous functions and a Riesz isomorphism of L onto L^.The main theorem of this section reads as follows.Theorem 14.
Let(G,+) be an Abelian semigroup and let L be an Archimedean Riesz space. Assume that, for given u1,u2∈L+, L is sup⁡{u1,u2}-uniformly complete. If, for every x,y∈G, a function F:G→L satisfies
(73)|F(x+y)+F(x)+F(y)|≤u1or|F(x+y)-F(x)-F(y)|≤u2,
then there exists a unique additive function A:G→L such that
(74)|F(x)-A(x)|≤sup⁡{u1,u2}forx∈G.Proof.
By the Johnson-Kist Spectral Representation Theorem there exist a topological spaceX and a Riesz isomorphism π:L→L^⊂C∞(X); hence (H1) is satisfied. (H2′) and (H5) hold because (70) has the form (59) with ℰ1F(x,y)∶=F(x+y)+F(x)+F(y) and ℰ2F(x,y)∶=F(x+y)-F(x)-F(y), for x,y∈G. If F satisfies (73) then
(75)|F(x+y)|≤|F(x)+F(y)|+sup⁡{u1,u2}forx,y∈G
and, therefore, (L2) holds with B(v)∶=|v|, C(v,w)∶=|v+w|+sup⁡{u1,u2} for v,w∈L and RC∶=R(πsup⁡{u1,u2}). Thus, by Lemma 6, we have (H4). It is clear that ℋn:LG→LG given by ℋnF(x)∶=(1/n)F(nx), for x∈G, n∈ℕ, satisfies (H3). Moreover, by Theorem 12 applied for E=ℝ and Lemma 3, one can easily verify that (59) possesses URAP with αn=βn=1/n, γn=1+1/n, ρn=3/n for n∈ℕ and δ(ε1,ε2)=sup⁡{ε1,ε2}. Putting Δ(u1,u2)=sup⁡{u1,u2} we have πΔ(u1,u2)=sup⁡{πu1,πu2} which means that πΔ(u1,u2)(s)=sup⁡{πu1(s),πu2(s)}=δ(πu1(s),πu2(s)) for s∈R(πΔ(u1,u2)).
Now all the assertions of Theorem14 follow directly from Theorem 7.It is easy to observe that the constant of approximation in Theorem14 is the best possible one.In view of Theorem12 and the meaning of the approximate solution of an alternative functional equation (70), that is condition (71) one may expect the following implication:
(76)|F(x+y)+F(x)+F(y)|>u1⟹|F(x+y)-F(x)-F(y)|≤u2,
for x,y∈G, in Theorem 14 instead of condition (73), with the common meaning of a<b as a≤b and a≠b. Of course, if the order in a Riesz space L 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.
LetL be the Archimedean Riesz space of all real functions of real variable with the pointwise order and let u1∈L be given by
(77)u1(s)∶={1,ifs=00,ifs≠0fors∈ℝ.
Then L is u1-uniformly complete. We define F:ℝ→L by
(78)F(x)(s)∶={x,ifs=x0,ifs≠xforx,s∈ℝ.
Then F is not additive and satisfies (76) with u1 defined above and u2≡0. On the other hand, F cannot be approximated by any additive function. Suppose, for contradiction, that there exists an additive mapping A:ℝ→L satisfying (74). Let us fix x∈ℝ. For s≠0 inequality (74) results with F(x)(s)=A(x)(s) according to the definition of u1. Directly from the definition of F we have F(x)(0)=0. Then by (74) and the additivity of A we obtain A(x)(0)=0. Eventually, we infer that F and A coincide and, therefore, F is additive. We have obtained a contradiction.Let us point out that the assumption that the Riesz spaceL is Archimedean is necessary in order to have the uniqueness of an existing additive function A in Theorem 14, which can be observed in the following simple example.Example 16.
Let us consider the lexicographically ordered planeL=ℝ2. L is then (1,0)-uniformly complete Riesz space. Moreover function F:ℝ→ℝ2 given by F(x)=(x,0) satisfies inequality (73) with u1=(1,0) and u2=(0,0). On the other hand inequality (74) holds true with any additive mapping Aα:ℝ→ℝ2, (α∈ℝ), of the form Aα(x)=(x,αx).


## 5. Approximate Solutions of the Cauchy Equation with Squares

Equation (70) has stemmed from
(79)f(x+y)2=(f(x)+f(y))2,
with a real function f, and next has been investigated in the form
(80)|f(x+y)|=|f(x)+f(y)|,
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:
(81)∥f(x+y)∥=∥f(x)+f(y)∥forx,y∈G.
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 ℝ2 with the Euclidean norm as a target space of f, (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.Theorem 17 (cf. [16, Theorem 2]).
Let(G,+) be an Abelian semigroup. If for a given ɛ≥0 a function f:G→ℂ satisfies
(82)|f(x+y)2-(f(x)+f(y))2|≤ɛforx,y∈G,
then there exists a unique additive function a:G→ℂ such that
(83)|f(x)-a(x)|≤ɛforx∈G.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].The main aim of this section is to apply Theorem7 in order to prove that in the class of functions taking values in an f-algebra equation (79) is stable; however it is not superstable. Our main stability result reads as follows.Theorem 19.
Let(G,+) be an Abelian semigroup and let L be an Archimedean f-algebra with a multiplicative identity e∈L+. Assume that L is u-uniformly complete for given u∈L+. If a function F:G→L satisfies
(84)|F(x+y)2-(F(x)+F(y))2|≤u2forx,y∈G,
then there exists a unique additive function A:G→L such that
(85)|F(x)-A(x)|≤uforx∈G.In the proof of Theorem19 we are going to use Proposition 2 from [3], which is quoted below. This proposition is due to the Ogasawara-Maeda Spectral Representation Theorem for Archimedean Riesz spaces with a weak unit (cf. [8, Theorem 50.1]). The multiplicative identity is a weak order unit (cf. [9, Proposition 353P]) and the topological space—the domain of the Ogasawara-Maeda representatives—appears to be extremally disconnected; hence, according to Proposition 1, C∞(X) is an f-algebra.Proposition 20 (cf. [3, Proposition 2]).
LetL be an Archimedean f-algebra with a multiplicative identity e∈L+. Then there exist a topological space X and an f-subalgebra L^ of the f-algebra C∞(X) and an f-algebra isomorphism of L onto L^.Proof of Theorem19.
By Proposition20 there exist a topological space X and an f-algebra isomorphism π:L→L^⊂C∞(X); hence (H1) is satisfied. Equation (79) has the form (7) with ℰF(x,y)∶=F(x+y)2-(F(x)+F(y))2, for x,y∈G; thus (H2) and (H5) hold. By (84) we have
(86)|F(x+y)2|≤|(F(x)+F(y))2|+u2forx,y∈G
and, therefore, (L2) holds with B(v)=|v2|, C(v,w)∶=|(v+w)2|+u2 for v,w∈L and RC∶=R(πu2). Thus, by Lemma 6 we have (H4). It is evident that ℋn:LG→LG given by ℋnF(x)∶=(1/n)F(nx) for x∈G, n∈ℕ satisfies (H3). Moreover, by Theorem 17 and Lemma 3 (79) possesses the URAP with αn=βn=1/n, γn=1+1/n, and ρn=3/n for n∈ℕ and δ(ɛ2)=ɛ. For Δ(u2)=u we have πΔ(u2)=πu and consequently πΔ(u2)(s)=πu(s)=δ((πu(s))2) for s∈R(πΔ(u2)).
Having applied Theorem7 we finish the proof.Theorem19 states that in f-algebras the Cauchy equation with squares (79) is stable in the Hyers-Ulam sense. According to Remark 18 one can ask if (79) is superstable in the sense of Baker. But it appears that this is not the case.Example 21.
LetB[-1,1] be the Archimedean f-algebra of all bounded real functions on the interval [-1,1] with a multiplicative identity e≡1, the pointwise order, pointwise addition, and multiplication. Let u∈B[-1,1] be given by
(87)u(s)∶={3|s|,ifs∈[-1,0]0,ifs∈(0,1].
Then B[-1,1] is u-uniformly complete and F:ℝ→B[-1,1] given by
(88)F(x)(s)∶={s,ifs∈[-1,0]sx,ifs∈(0,1]forx∈ℝ
is, clearly, neither bounded nor additive. Moreover, one can easily check that F and u satisfy (84).


## 6. Approximate Solutions of the Quadratic Functional Equation

In this section we deal with approximate solutions of the quadratic functional equation(89)F(x+y)+F(x-y)=2F(x)+2F(y).
Stability of this equation, in the class of functions mapping an Abelian group into a Banach space, has been investigated in [23] (cf., e.g., [24–26]).Theorem 22 (cf. [23]).
Let(G,+) be an Abelian group and let (E,∥·∥) be a Banach space. If a function f:G→E satisfies the inequality
(90)∥f(x+y)+f(x-y)-2f(x)-2f(y)∥<ɛ
for some ɛ≥0 and all x,y∈G then there exists a unique quadratic function q:G→E such that
(91)∥f(x)-q(x)∥≤12ɛ
for all x∈G.The main aim of this section is to show that a similar result holds true in the class of Riesz space-valued mappings.Theorem 23.
Let(G,+) be an Abelian group and let L be an Archimedean u-uniformly complete Riesz space, with some u∈L+. If, for every x,y∈G, a function F:G→L satisfies
(92)|F(x+y)+F(x-y)-2F(x)-2F(y)|≤u,
then there exists a unique quadratic function Q:G→L such that
(93)|F(x)-Q(x)|≤12uforx∈G.Proof.
By the Johnson-Kist Spectral Representation Theorem there exist a topological spaceX and a Riesz isomorphism π:L→L^⊂C∞(X); hence (H1) is satisfied. Equation (89) has the form (7) with ℰF(x,y)∶=F(x+y)+F(x-y)-2F(x)-2F(y); hence (H2) and (H5) hold. If F satisfies (92) then
(94)|F(x+y)|≤|F(x-y)-2F(x)-2F(y)|+uforx,y∈G
and therefore (L3) holds with B(v)∶=|v|, D(t,v,w)∶=|t-2v-2w|+u for t,v,w∈L and RD∶=R(πu). Consequently Lemma 6 guaranties (H4). It is clear that ℋn:LG→LG given by ℋnF(x)∶=F(2nx)/4n for x∈G, n∈ℕ satisfies (H3). Moreover, by Theorem 22 applied for E=ℝ one can easily verify that (89) possesses URAP with (ℋnF(x))n∈ℕ defined above and αn=βn=1/4n, γn=1+1/4n, δn=3/4n for n∈ℕ, δ(ɛ)=ɛ/2. Putting Δ(u)=u/2 we have πΔ(u)(s)=(πu(s))/2=δ(πu(s)) for s∈R(πΔ(u)).
Applying Theorem7 we finish the proof.



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