On a class of abstract convex cone valued functional equations

We present an approach to solving a number of functional equations for functions with values in abstract convex cones. Such cones seem to be good generalizations of, e.g., families of nonempty compact and convex subsets or nonempty closed, bounded and convex subsets of a normed space. Moreover, we study some related stability problems.


Introduction
In [14] we solve the functional equation (1) for an integer β > 1, where f is a function from a group uniquely divisible by β to a family of nonempty compact and convex subsets of a locally convex linear metric space with an invariant metric. The form of solutions of this single variable equation helped us to find solutions of a number of set-valued functional equations in several variables. The idea for studying (1) and all the computations leading to this equation are described in [11]. There are also several applications concerning stability considerations presented there. The ranges of functions in the mentioned papers were Banach spaces (which in fact could easily be generalized to complete linear metric spaces or sequentially complete linear topological spaces) or hyperspaces of nonempty compact and convex subsets of a locally convex linear metric space.
A natural question, particularly after studying the latter case, is whether similar results can be obtained for functions with values in arbitrary semigroups.
In the present paper we solve (1) for functions with values in abstract convex cones which seem to be a good generalization of a family of nonempty compact and convex subsets of a normed space.
Besides questions concerning solutions of (1), one may state questions connected with the approximation of a function satisfying (1) or its stability. In the paper, we address a question about the stability of (1) and its equivalent form.
We will consider an abstract convex cone with a metric : S × S → [0, ∞). We say further that is translation invariant or positively homogeneous provided that (s + u, t + u) = (s, t) for all s, t, u ∈ S, or (λs, λt) = λ (s, t) for all s, t ∈ S, λ > 0, respectively.
In 1952, Rådström [10] showed that such a cone can be embedded isometrically and isomorphically into a normed space. Examples of abstract convex cones: 1 • an arbitrary normed vector space (X, · ) with the induced metric (s, t) = s − t , s, t ∈ X, and the operations + and · from X. If S ⊂ X is a convex cone (i.e., s + t, λs ∈ S for all s, t ∈ S and λ 0) then S is an abstract convex cone. It is complete whenever X is a Banach space and S is closed in X; 2 • a collection of nonempty, compact and convex subsets of a real normed linear space Y (denoted cc(Y )) with the Hausdorff distance and operations: Vol. 94 (2020) On a class of abstract convex cone valued functional equations 537 (for some details see, e.g., [2]); 3 • a collection of nonempty, closed, bounded and convex subsets of a real normed linear space Y (denoted bcl(Y )) with the Hausdorff distance and operations: Let S be a semigroup with the cancelation law. For elements s, t, u ∈ S we say that u is an abstract S-difference of s and t, i.e., u = s − t if and only if s = u + t. If this difference exists, then it is unique. In the space cc(Y ) (see 2 • above) such a difference is known as the Hukuhara difference (see [6]).
The following lemmas will be useful in the next part of the paper. The proof of the above lemma goes along the same lines as in [9, Lemma 1]. In particular cases of abstract convex cones we do not need to assume the completeness of the space (see [14,Lemma 2.6] for the case of the family of nonempty, compact and convex subsets of a locally convex linear metric space with an invariant metric). Also, if our cone is a closed proper subset of a metric space, we obtain the same assertion as in Lemma 2.1.

Lemma 2.2.
Let (X, ) be a metric space and let S X be an abstract convex cone with the metric inherited from X, which is translation invariant and positively homogeneous in S. Assume that the cone S is closed in X. If for s, t, s n , t n ∈ S, n ∈ N, we have s n → s, t n → t and for every n ∈ N there exists an abstract S-difference s n −t n , then there exists an abstract S-difference s − t and s n − t n → s − t.
Proof. By the assumptions, for every n ∈ N there exists u n := s n − t n which means that s n = u n + t n . We have On the other side, u n + t ∈ S + t for all n ∈ N and s ∈ S + t, since S + t is closed. Consequently, s = u + t for some u ∈ S, there exists the abstract S-difference s−t and, finally, u n → u.
Remark 2.1. It is worth observing that the abstract convex cone cc(Y ), where Y is a normed linear space, is closed in bcl(Y ) or in the space of all nonempty subsets of Y (cf., Lemma 2.6 in [14]).

J. Sikorska AEM
The following easy example shows that we may not omit the assumption about the completeness or closedness of S.

Solutions of the equation
We start this section with presenting the main result concerning solutions of the equation in a single variable introduced at the beginning of the paper. Moreover, Conversely, each function f of the form a + q, where a, q : G → S are such that (2) and (3) hold, satisfies (1).
Proof. By induction, we get from (1) From Lemma 2.1 we derive that for every x ∈ G there exists the abstract for all x ∈ G then f (x) = a(x) + q(x), x ∈ G, and q is even.
Remark 3.1. Instead of the completeness of S in Theorem 3.1 (and in the following Theorem 3.2), on account of Lemma 2.2 we may assume that S is a proper closed subset of a metric space.
Theorem 3.1 brings a number of immediate applications in solving equations in several variables for abstract convex cone-valued functions. We present here a few of them (cf., e.g., [4,7,8,12,14,15]): (a) the Drygas equation: In all the above examples we obtain solutions of the form f = a + q (in (d)-(f) up to a constant) with a and q having properties as in the assertion of the mentioned theorem. In the last three cases we assume additionally that for every x ∈ G there exists the abstract S-difference f (x) − f (0). The forms of equations (a)-(f) imply additionally that a is additive (i.e., a(x + y) = a(x) + a(y) for all x, y ∈ G) and q is quadratic (i.e., q(x+y)+q(x−y) = 2q(x)+2q(y) for all x, y ∈ G).
As an example we show such an application for the Drygas equation.
Then there exist an additive function a : G → S and a quadratic function q : G → S such that f = a + q and such a representation is unique. Moreover, each function of the form a + q, where a : G → S is additive and q : G → S is quadratic, is a solution of (4).
Proof. Setting y = x in (4) we get Small rearrangements lead to which means that (1) is satisfied with β = 2. By Theorem 3.1, there exist functions a, q : G → S such that f = a + q and Using the form of f in (4), writing it also for −x instead of x and −y instead of y, and finally adding both equations side by side, we conclude that q is quadratic, and consequently, a satisfies Since a(0) = 0, it follows from the above that a is additive. The rest is proved in a standard way.

Stability considerations
In what follows we answer the question whether for a function f : G → S satisfying inequality with some control function ϕ : G → [0, ∞), there exists a function g : G → S which satisfies (1) and which is close (with respect to the metric ) to f . In case of a positive answer we say that (1) is stable (the reader interested in learning more on stability is referred, e.g., to [1,5]). We will show, however, that without any additional assumption imposed on the class of functions for which (5) is considered, the equation is not stable (cf., [13]).
To see this, consider S = cc(R), the family of nonempty, closed and bounded intervals. It is easy to see that for for some functions p, q : G → R, p q, then the condition with some ε 0 (i.e., we have (5) in the simplest case where ϕ(x) ≡ ε), is equivalent to the system of conditions Example 4.1. Fix ε > 0 and define functions p, q : R → R, Then p < q and functions p and q satisfy system (8). Consider f : R → cc(R) of the form (6). As a consequence, f satisfies (7). However, there does not exist a function g : R → cc(R) satisfying (1) and such that with some constant M depending on ε. Indeed, suppose to the contrary that such g does exist. Then for all x ∈ R and b : R → cc(R) is even. It follows that a is single-valued and odd.
It follows from (8) and [11,Corollary 2.2] that there exist uniquely determined functions p, q : R → R satisfying (1) and such that for all x ∈ R. It is easy to verify that in our case p(x) = 1 2 ε x 2 + x and q(x) = ε x 2 + x for all x ∈ R.
Since the function is bounded, by the uniqueness of p and q, it follows that This is impossible since the odd parts of p and q equal to 1 2 εx and εx, respectively, are different. β n ϕ(β n x) < ∞ (with β > 1), are not stable either.
As we have just seen, (1) is in general not stable. However, considering again S = cc(R) and making some additional assumptions on p and q, we obtain some positive stability results. Theorem 4.1. Let β be a positive integer different from 1, (G, +) be a group uniquely divisible by β, and let f : G → cc(R) of the form (6) satisfy (7) with some ε > 0. If q(x)−p(x) > 2βε β−1 for all x ∈ G\{0}, then there exists a unique function g : G → cc(R) satisfying (1) and such that Proof. By the assumption, functions p and q satisfy (8) and from [11,Corollary 2.2] it follows that there exist functions p, q : G → R which satisfy (1) and (9) for all x ∈ G. As a result, Since Surely, the function g : G → cc(R) defined by satisfies (1) and, moreover, To prove the uniqueness, let g be another function satisfying (1) and such that the (Hausdorff) distance between f and g is bounded, that is, f (x), g(x) M for all x ∈ G and some M > 0. Hence, By Theorem 3.1, g = a 1 + b 1 and g = a 2 + b 2 with some functions a 1 , a 2 : G → cc(R) such that a i (βx) = βa i (x) for i ∈ {1, 2}, and b 1 , b 2 : G → cc(R) such that b i (βx) = β 2 b i (x) for i ∈ {1, 2}. Therefore, and Dividing by β 2n and letting n tend to infinity we obtain b 1 = b 2 . Now, it is easy to see that a 1 = a 2 , and the proof is complete.  (1) and such that Proof. On account of (8), Moreover, In fact, function d o is given by (see [  For the uniqueness of g, we use the same argument as in the proof of the previous theorem. Assume (G, +) is a group uniquely divisible by β. Substitute x β and − x β in the place of x in (1). After small rearrangements we get Open Access. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
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