A fixed point theorem and Ulam stability of a general linear functional equation in random normed spaces

We prove a very general fixed point theorem in the space of functions taking values in a random normed space (RN-space). Next, we show several of its consequences and, among others, we present applications of it in proving Ulam stability results for the general inhomogeneous linear functional equation with several variables in the class of functions f mapping a vector space X into an RN-space. Particular cases of the equation are for instance the functional equations of Cauchy, Jensen, Jordan–von Neumann, Drygas, Fréchet, Popoviciu, the polynomials, the monomials, the p-Wright affine functions, and several others. We also show how to use the theorem to study the approximate eigenvalues and eigenvectors of some linear operators.


Introduction
In this paper, we prove a fixed point theorem for classes of functions taking values in a random normed space (RN-space) and show some applications of it to several issues connected with Ulam-type stability.
The study on such stability was initiated by a question of Ulam from 1940 (cf., e.g., [48,96]) asking if an "approximate" solution of the functional equation of group homomorphisms must be "close" to an exact solution of the equation. The first answer was provided by Hyers [48], who considered the question for the Cauchy functional equation in Banach spaces and used the method that subsequently was called the direct method. He defined the equation solution explicitly as a pointwise limit of a sequence of mappings constructed from the given approximate solution.
functional equation, and Pinelas et al. [76] used the direct and the fixed point method to show stability of a new type of the n-dimensional cubic functional equation. We also refer to the book of Cho et al. [28] for more details on that type of stability in random normed spaces.
In this paper, we will first show a general fixed point theorem for classes of functions taking values in a random normed space. This is the random normed space version of the fixed point theorems in [20,22] (see also [24]), which turned out to be very useful in investigations of the stability of various functional equations. Next, we show how to use the theorem to study the Ulam stability of various functional equations in a single variable and investigate the approximate eigenvalues and eigenvectors in the spaces of function taking values in RN-spaces.
Finally, using this fixed point theorem, we prove the very general results on the stability of the functional equation for functions mapping a linear space X into a random normed space Y , with a given function D : X n → Y . As special cases of this result, we can obtain the stability criteria for numerous functional equations in several variables, in the framework of random normed spaces.

Preliminaries
In the sequel, we use the definitions and properties of the random normed space (RN-space) as in [7,28,44,45,64,68,87,[89][90][91]. However, for the convenience of the reader, we remind some of them. Since T is commutative and associative, it is easy to show by induction that T m+n+l i=m a i = T T m+n i=m a i , T m+n+l i=m+n+1 a i , m, n, l ∈ N 0 , l > 0. (2.1) Note yet that, by (c), the sequence (T m+n i=m a i ) n∈N is non-increasing for every m ∈ N and therefore always convergent. So, for each m ∈ N, we may introduce the following notation: A t-norm T can be extended in a unique way to an n-ary operation taking: T (a 1 , . . . , a n ) := T n i=1 a i . To shorten some long formulas, we will write T (a) := T (a, a), a ∈ [0, 1].
It is easy to show by induction on k (using the associativity and commutativity of T ) that T k j=1 T T m+n i=m a ij = T T m+n i=m T k j=1 a ij (2.2) for every k, n, m ∈ N 0 , k ≥ 1, and a ij ∈ [0, 1] with j = 1, . . . , k and i = m, . . . , m + n. We need that property a bit later.
Definition 2.5. Let Y be a real vector space, F : x → F x a mapping from Y into D + , and T a continuous t-norm. We say that (Y, F, T ) is a random normed space (briefly RN-space) if the following conditions are satisfied: for all x ∈ Y, t > 0 and α = 0; For more information on the RN-spaces, we refer to [41,45,65,87,89].
The same remains true if Definition 2.6. (Cf., e.g.,, [41,65]) Let (Y, F, T ) be an RN-space. (1) A sequence (x n ) n∈N in Y is said to converge (or to be convergent) to x ∈ Y (which we denote by: i.e., for each > 0 and each t > 0, there exists an i.e., for each > 0, and each t > 0, there exists i.e., for every > 0, k ∈ N and t > 0, there exists an Remark 2.7. Since every M -Cauchy sequence is also G-Cauchy, it is easily seen that each G-complete RN-space is M -complete.

A general fixed point theorem in RN-spaces
Our first main result is a very general RN-space version of a fixed point theorem in [20]; actually, we follow the approach from [22] (see also [24]). We provide some applications of it in the next sections.
In what follows, X is a non-empty set, (Y, F, T ) is an RN-space, N 0 := N∪{0} and R + := [0, +∞) (the set of non-negative real numbers). If U and V are nonempty sets, then as usual U V denotes the family of all mappings from V to U . If F ∈ U U , then F n stands for the n-th iterate of F , i.e., F 0 (x) = x and F n+1 (x) = F (F n (x)) for x ∈ U and n ∈ N 0 . The space Y X is endowed with the coordinatewise operations, so that it is a linear space.
To simplify some expressions, for given φ ∈ D X + and x ∈ X, we write For every ϕ, ψ ∈ D + the inequality ϕ ≤ ψ means that ϕ(t) ≤ ψ(t) for each t > 0. We use this abbreviation to simplify formulas whenever the variable t is not necessary to express them precisely.
Definition 3.1. Let Λ : D X + → D X + and J : Y X → Y X be given. We say that the operator J is Λ-contractive if, for every ξ, η ∈ Y X and every φ ∈ D X + , The convergence in D + will mean the pointwise convergence. Therefore, we say that a sequence (ψ n ) n∈N in D + converges to some ψ ∈ D + if lim n→∞ ψ n (t) = ψ(t), t > 0.
We need yet the following hypothesis on Λ : D X Then (C 0 ) actually means the continuity of Λ at the point χ 0 (with respect to the pointwise convergence topologies in D X + and D + ) and the property: Λχ 0 = χ 0 . Let ν ∈ N, ξ 1 , . . . , ξ ν : X → X, and L 1 , . . . , L ν : X → (0, ∞) be fixed. A natural example of operator Λ fulfilling hypothesis (C 0 ) can be defined by We refer to Remark 3.10 for further comments on this situation.
In what follows, Ω stands for the family of all real sequences (ω n ) n∈N0 with ω n ∈ (0, 1) for each n ∈ N 0 and ∞ i=0 ω i = 1.
Let us first state the following lemma, which will be used in the sequel. Lemma 3.3. Let Λ : D X + → D X + and : X → D + be arbitrary. Then, for every x ∈ X, k ∈ N 0 , ω ∈ Ω, and t > 0, the limits Proof. Fix k ∈ N 0 , x ∈ X and t > 0 and write Vol. 25 (2023) A fixed point theorem and Ulam stability Page 7 of 38 33 Since (Λ i ) x ∈ D + , it is a non-decreasing function for each i ∈ N. Hence, Consequently, whence the sequence (τ m (x, t, k)) m∈N is non-increasing and, therefore, for every k ∈ N 0 , x ∈ X and t > 0, the following limit exists Next, fix ω ∈ Ω, k ∈ N 0 , x ∈ X and t > 0, and write Then, . This means that the sequence (ρ m (x, t, k)) m∈N is non-increasing. Therefore, there exists the limit Analogous equalities are valid for ω σ k x with any ω ∈ Ω.
In the sequel, given Λ : D X + → D X + and : for every x ∈ X, k ∈ N 0 and t > 0, where σ k x (t) and ω σ k x (t) are defined by (3.2) and (3.3). (3.8) and one of the following three conditions holds.
Then, for every x ∈ X, the limit exists in Y and ψ ∈ Y X thus defined is a fixed point of J with Moreover, in case (i) or (ii) holds, ψ is the unique fixed point of J such that there exists α ∈ (0, 1) with Proof. First we show by induction that, for every n ∈ N 0 , Thus, we have proved that (3.15) holds for every n ∈ N 0 . Consequently, for every n ∈ N 0 , m ∈ N, x ∈ X and t > 0 we have (3.16) and analogously, as (3.17) Now, we show that the limit (3.12) exists in Y for every x ∈ X. First consider the case of (i). Then, by (3.16), for all k, m ∈ N, n ∈ N 0 , x ∈ X and t > 0, Consequently, by (c), Hence, (3.9), (b) and the continuity of T at (1, 1) yield Thus we have proved that, for every x ∈ X, (J n f (x)) n∈N is an M-Cauchy sequence and, as (Y, F, T ) is M-complete, the limit (3.12) exists.
In the case of (iii), in view of (3.16), whence (3.11) and the continuity of T at (1, 1) imply that Thus, for every To this end, observe that (3.16) implies for every m ∈ N. Hence, by (3.12) and the continuity of T at the point 1, σ n x (αt) , by letting m → +∞, we obtain (3.18). Next, we show that So, fix ω ∈ Ω and note that (3.17) implies for every m ∈ N. Hence, by (3.12) and the continuity of T at the point 1, ω σ n x (αt) , by letting m → +∞, we obtain Furthermore, by the Λ-contractivity of J, Since (3.12) means that Whence, on account of (3.23), and consequently Thus, we have shown that ψ is a fixed point of J. It remains to prove the statements on the uniqueness of ψ. So, assume that (i) or (ii) holds and ψ 1 , ψ 2 ∈ Y X are two fixed points of J such that Note yet that, in view of (3.4) and (3.5), each of the conditions (3.9) and (3.10) implies Hence, by letting k → ∞ in (3.24), by the continuity of T at the point (1, 1), we finally obtain that which means that ψ 1 = ψ 2 .
Remark 3.6. If, for a given k ∈ N 0 and x ∈ X, the function σ k x is left continuous (which is not necessarily the case, because this depends on the forms of and T ), then it is easily seen that (3.13) can be replaced by Otherwise, for every fixed x ∈ X and k ∈ N 0 , the inequality in (3.13) can of course be replaced by Remark 3.7. The assumptions (i) and (ii) in Theorem 3.5 look nearly the same and (i) is a bit simpler than (ii). However, as we will see below, in some situations (3.10) (with some sequence (ω k n ) n∈N0 ∈ Ω) and (3.11) are fulfilled, while (3.9) is not.
Namely, let T = T M and Λ have the following simple form: x ∈ X, t > 0, δ ∈ D X + , with some a, b ∈ (0, ∞) (cf. the proof of Corollary 6.4). Then, Write e 0 := ba −p . Clearly, for every n ∈ N, x ∈ X and t > 0, and therefore Assume that e 0 > 1. Then, by (3.27), which means that (3.11) holds. Further, for every k ∈ N 0 , (3.28) yields Consequently, for every x ∈ X and t > 0, Hence, (3.9) is not valid and σ k x makes no contribution in estimation (3.13). On the other hand, for every x ∈ X, t > 0 and ω = (ω n ) n∈N0 ∈ Ω, we have we have Therefore, for every x ∈ X and t > 0, This means that (3.10) holds with ω k = ω for k ∈ N 0 and For the situation where (3.10) is valid with sequences ω k ∈ Ω that are not the same for all k ∈ N, we refer to Remark 5.3.
Remark 3.8. Note that in the proof of Theorem 3.5, we have only used continuity of T at the points of the form (1, ξ) for ξ ∈ (0, 1]. Actually, even that assumption can be weakened. Namely, it is enough to assume that T is continuous only at the point (1, 1), but then we have to modify inequality in (3.13) basing it only on (3.19) and (3.21) without taking the limits.

Remark 3.9.
Observe that the properties of the t-norm yield and not necessarily conversely, assumption (iii) is not weaker than (i).
with a function H : X × Y ν → Y satisfying the following Lipschitz-type condition: , t > 0, (3.31) for all x ∈ X and y 1 , . . . , y ν , z 1 , . . . , z ν ∈ Y , then such J is Λ-contractive with Λ defined by (3.1) and such Λ fulfills hypothesis (C 0 ) (see Remark 3.2). Clearly, (3.31) holds if H has the following simple form: In particular, such J satisfies the following Lipschitz-type condition: x ∈ X, t > 0. If we want to admit functions L i taking values in R (i.e., in particular taking the value zero), then we can rewrite that condition in the subsequent form: Note that if, in such a situation, L i (x) = 0 for some i ∈ {1, . . . , ν} and some x ∈ X, then In view of Remark 3.10, for operators J : Y X → Y X fulfilling condition (3.34), we have the following particular case of Theorem 3.5, with a stronger statement on the uniqueness of fixed point (because under the weaker assumption that (3.14) holds only for k = 0). (3.34), and f : X → Y fulfil (3.8). Assume that one of the conditions (i)-(iii) of Theorem 3.5 holds. Then, for every x ∈ X, the limit (3.12) exists and the function ψ ∈ Y X , defined in this way, is a fixed point of J satisfying (3.14).
Moreover, if (i) or (ii) holds, then ψ is the unique fixed point of J such that there is α ∈ (0, 1) with Then, by (3.34), Hence, J is Λ-contractive. Moreover, as we have noticed in Remark 3.10, Λ satisfies hypothesis (C 0 ). Hence, by Theorem 3.5, limit (3.12) exists for every x ∈ X and so defined function ψ is a fixed point of J satisfying (3.14). It remains to show the statement on uniqueness of ψ. So, let τ ∈ Y X be a fixed point of J such that, for some α ∈ (0, 1), Vol. 25 (2023) A fixed point theorem and Ulam stability Page 15 of 38 33 Fix x ∈ X, ω = (ω n ) n∈N0 ∈ Ω and t > 0. We show that, for every n ∈ N 0 , we have This is the case for n = 0, because by the continuity of T , for every x ∈ X and t > 0, we have Analogously, Since T is continuous, we finally get Now assume that (3.38) is valid for some n ∈ N 0 . Then, by (2.2) and the continuity of T , for every x ∈ X and t > 0, Next, assume that (3.39) is valid for some n ∈ N 0 . Then in the same way, by (2.2) and the continuity of T , for every x ∈ X, t > 0, and ω ∈ Ω, Thus, we have proved (3.38) and (3.39) for every n ∈ N, x ∈ X, ω ∈ Ω, and t > 0. Whence Now, if (3.9) holds, then by letting n → +∞ in (3.40), by the continuity of T , we get F (τ −ψ)(x) (t) = 1 for every x ∈ X and t > 0, which means that τ = ψ. Similarly, if (3.10) holds, then we argue analogously by letting n → +∞ in (3.41).
As for the uniqueness of the fixed points of J in Theorem 3.5, we also have the following proposition.
Then, for every f : X → Y, J has at most one fixed point ψ 0 with Proof. Fix f : X → Y and assume that ψ 1 , ψ 2 ∈ Y X are fixed points of J satisfying Then, by the Λ-contractivity of J, and consequently, for every m ∈ N 0 , x ∈ X and t > 0. Hence, by letting m tend to ∞, by (3.42) and the continuity of T at the point (1, 1), If X has only one element, then Y X can actually be identified with Y and Theorem 3.5 becomes an analog of the classical Banach Contraction Principle (somewhat generalized), given in Corollary 3.14 below. To present it, we need the following hypothesis, concerning mappings λ : D + → D + , which is a special case of hypothesis (C 0 ).
(C) The sequence λ(F zn ) n∈N converges pointwise to H 0 for each sequence (z n ) n∈N in Y , which converges to 0.
To avoid any ambiguity, let us give one more definition, which is a special case of an earlier definition, namely: Definition 3.1.
Definition 3.13. Let λ : D + → D + be given. We say that a mapping h : for every z, w ∈ Y and φ ∈ D + with F z−w ≥ φ. Corollary 3.14. Let λ : D + → D + satisfy hypothesis (C) and h : Y → Y be λ-contractive. Let ∈ D + be such that

43)
and assume that one of the following three conditions holds. Then, for every ω = (ω n ) n∈N0 ∈ Ω and t > 0, the limits Y and R, respectively) and z 0 is a fixed point of h such that

Approximate eigenvalues
In this section, we show an application of Theorem 3.5 in investigation of the approximate eigenvalues and eigenvectors, which corresponds to the results in [36,47]. It is well known that Y X is a real linear space with the operations defined pointwise in the usual way: The next corollary is an example of a result concerning approximate eigenvalues of some linear operators on Y X . Actually, the assumption of linearity of the operators is not necessary in the proof, but the notion of eigenvalue might be ambiguous without it (see, e.g., [86]) and therefore we confine only to the linear case. (4.1)

If one of the conditions (i), (ii) and (iii) of Theorem 3.5 is valid with
then γ is an eigenvalue of J 0 , the limits exist for every x ∈ X, ω = (ω n ) n∈N0 ∈ Ω and t > 0, and the function ψ 0 ∈ Y X , given by is an eigenvector of J 0 , with the eigenvalue γ, such that Proof. Let ϕ := γh and J : Y X → Y X be given by: Then, in view of the Λ 0 -contractivity and linearity of J 0 , for every μ, ξ ∈ Y X and δ ∈ D X which means that J is Λ -contractive. Next, we can write (4.1) in the form: for every x ∈ X, ω = (ω n ) n∈N0 ∈ Ω and t > 0. Moreover, the function ψ : X → Y , defined by (3.12), is a fixed point of J with Write ψ 0 := γ −1 ψ. Now, it is easily seen that J 0 ψ 0 = Jψ = ψ = γψ 0 , (4.6) is equivalent to (4.8), and (3.12) yields (4.3).
Clearly, under suitable additional assumptions in Corollary 4.1, we can deduce from Theorem 3.5 some statements on the uniqueness of ψ, and consequently on the uniqueness of ψ 0 .
Given ε ∈ D X + , let us introduce the following definition: γ ∈ R\{0} is an ε-eigenvalue of a linear operator J 0 : It is easily seen that Corollary 4.1 yields the following simple result.

Ulam stability of functional equations in a single variable
In this section, as before, X is a nonempty set and (Y, F, T ) is an RN-space.
As we have mentioned in the Introduction, the main issue of Ulam stability can be very briefly expressed in the following way: when must a function satisfying an equation approximately (in some sense) be near an exact solution to the equation?
Definition 5.1. Let E and C be nonempty subsets of D X + with E ⊂ C. Let T be an operator mapping C into D X + , G be an operator mapping a nonempty set K ⊂ Y X into Y X , and χ 0 ∈ Y X . We say that the equation is (E, T ) -stable provided for any ε ∈ E and φ 0 ∈ K with there exists a solution φ ∈ K of (5.1) such that Roughly speaking, (E, T )-stability of (5.1) means that every approximate (in the sense of (5.2)) solution φ 0 ∈ K of (5.1) is always close (in the sense of (5.3)) to an exact solution φ ∈ K of (5.1). Now, we present a simple Ulam stability outcome that can be derived from the results of the previous sections. To this end, we need the following hypothesis.
Vol. 25 (2023) A fixed point theorem and Ulam stability Page 21 of 38 33 The subsequent corollary can be easily deduced from Theorem 3.11.
The stability of functional equations of form (1.2) (or related to it) has already been studied by several authors. For further information, we refer to [4,23,25]. A very particular case of (5.9), with H given by (3.32), is the linear functional equation of the form: with fixed functions h ∈ Y X and L 1 , . . . , L ν ∈ R X . That equation is called a linear equation of higher order when ξ i = ξ i for i = 1, . . . , ν, with some ξ ∈ X X , i.e., when (5.12) has the form: Some recent results concerning the stability of less general cases of it can be found in [25,26,51,52,70,97]. The simplest case of Eq. (5.13), when ν = 1 and 0 ∈ L 1 (X), can be rewritten in the form: which is also called the linear equation. Special cases of (5.14) are the gamma functional equation with fixed s ∈ R\{0}, and the Abel functional equation For more details on Eq. (5.14) and its various particular versions, we refer to [58,60]. Clearly, Eq. (5.16) is (5.9) with ν = 1, ξ 1 = ξ and H(x, y) = 1 s y for x ∈ X and y ∈ Y . So we have the case as in Corollary 5.2 with Λ : D X + → D X + given by x ∈ X, δ ∈ D X + , t > 0. Further, let E be a normed space, X := E\{0}, p ∈ R, L ∈ (0, ∞) and Assume that ξ n (x) p ≤ a n x p for x ∈ X and n ∈ N 0 , with some sequence (a n ) n∈N0 of positive reals such that lim n→∞ a −1 n |s| n = ∞. Write e n := a −1 n |s| n . Clearly, for every n ∈ N, x ∈ X and t > 0, Further, assume additionally that and write Now, using (5.17), we get Consequently, for every x ∈ X and t > 0, which means that (3.10) holds with ω k := (ω k n ) n∈N0 ∈ Ω. Moreover, for ω := ω 0 we have x and σ 0 x have the same meaning as in Corollary 5.2.

Stability of Eq. (1.2)
In this section, we are concerned with the stability of the functional equation (1.2) for m > 1. So we assume that X is a linear space over a field K ∈ {R, C}, A i , a ij ∈ K for i = 1, . . . , m and j = 1, . . . , n, and that D : X n → Y is a fixed function.
It is easily seen that particular cases of the homogeneous version of (1.2), namely of the equation are the Cauchy functional equation the Jensen functional equation the Drygas equation 4) and the Fréchet functional equation Various information on the Cauchy, Jensen and linear equations can be found in [2,3,59]. Equation (6.3) (the parallelogram law) was used by Jordan and von Neumann [50] in a characterization of the inner product spaces and Eqs. (6.4) and (6.5) were applied for the analogous purposes (cf. [8,38,55]); we refer to [12,15,35,49,53,54,71,72,[77][78][79][80]85] for further related information and stability results for those equations.
It is easily seen that the equations are particular cases of (6.1). Functions f : X → Y satisfying (6.6) and (6.7) are called polynomial functions of order n−1 and monomial functions of order n, respectively (see, e.g., [42,49,59,61,93] for information on their solutions and stability).
Let us mention yet that (6.5) can be written as where and i.e., C 2 f is the Cauchy difference of f of the second order. Recurrently, for x 1 , . . . , x n , u, w ∈ X, and n ∈ N. It is easily seen that the equation also is of the form (6.1) for every n ∈ N.
The functional equation (M, N, m, n being non-zero integers) is another particular case of (6.1). It has been studied in [29][30][31][32]. The Eq. (6.8) with M = m = 3 and N = n = 2 was considered for the first time by Popoviciu [81] in connection with some inequalities for convex functions; for results on solutions and stability of it, we refer to [92,94]. Solutions and stability of (6.8) with M = m = 3 and N = n = 2 have been investigated by Lee [62]. The more general case N = n 2 and M = m 2 of (6.8) has been studied in [63]. For results on a generalization of (6.8) we refer to [95]. Finally, let us recall here the equation of p-Wright affine functions (called also the p-Wright functional equation) where p ∈ R is fixed, which also is of form (6.1). For more information on (6.9) and recent results on its stability we refer to [11,18]. Our main theorem in this section concerns the Ulam-type stability of Eq. (1.2) in RN-spaces. The following two hypotheses are needed to formulate it. (M) There exist μ ∈ {1, · · · , m − 1} and c 1 , . . . , c n ∈ K such that where β i is defined as in hypothesis (M) and d(x) = D(c 1 x, . . . , c n x) for x ∈ X. The next two remarks provide some comments on those hypotheses. For instance, for the Cauchy equation (6.2) and its inhomogeneous form D(x, y), (6.11) we can consider the following two situations (we refer to Corollary 6.4 and its proof for consequences in both of them).
(b3) More generally, if h 1 , . . . , h n : X → Y are solutions to equation (6.1), then the function D : X n → Y , given by fulfills hypothesis (D). In fact, fix x 1 , . . . , x n ∈ X. Then, according to the definition of d, whence we get (6.10).
where T : D X n + → D X n + is given by χ ∈ D X n + , t > 0, x 1 , . . . , x n ∈ X. (6.14) Further, assume that one of the conditions (i)-(iii) of Theorem 3.5 holds with ∈ D X + and Λ : D X + → D X + defined by  1) σ 0 x (αt), t > 0, x ∈ X, (6.18) with σ 0 x defined by (3.7) (see also (3.2) and (3.3)). Moreover, in case where (i) or (ii) holds, there is exactly one solution ψ ∈ Y X of (1.2) such that there exists α ∈ (0, 1) with Proof. Write |α| = 1/A 0 and fix x ∈ X. Putting x j := c j x for j ∈ {1, . . . , n} in (6.17), we get Therefore, θ(c 1 x, . . . , c n x)(A 0 t), t > 0, (6.20) with the operator J : Y X → Y X defined by Note that the assumptions of Theorem 3.11 are satisfied for such J, because, for every ξ, η ∈ Y X and x ∈ X, This means that the condition (3.34) is fulfilled with ξ i (x) = β i x and L i (x) = |A i |/A 0 . Consequently, by Theorem 3.11, for every k ∈ N 0 , x ∈ X and t > 0, the limit (3.12) exists and the function ψ ∈ Y X is a fixed point of J fulfilling (6.18).
Consequently, by (6.22), a ij x j = D(x 1 , . . . , x n ), x 1 , . . . , x n ∈ X. (6.26) To complete the proof, observe that every solution of (1.2) is a fixed point of J and therefore the statement on uniqueness follows directly from the uniqueness property of ψ as a fixed point of J satisfying (6.21).
Using Theorem 6.3, we can obtain various stability results for numerous equations. For instance, for the Cauchy inhomogeneous equation (6.11) we can argue as in the following corollary.