Solutions and stability of generalized Kannappan's and Van Vleck's functional equations

We study the solutions of the integral Kannappan's and Van Vleck's functional equations $$\int_{S}f(xyt)d\mu(t)+\int_{S}f(x\sigma(y)t)d\mu(t) = 2f(x)f(y), \;x,y\in S;$$ $$\int_{S}f( x\sigma(y)t)d\mu(t)-\int_{S}f(xyt)d\mu(t) = 2f(x)f(y), \;x,y\in S,$$ where $S$ is a semigroup, $\sigma$ is an involutive automorphism of $S$ and $\mu$ is a linear combination of Dirac measures $(\delta_{z_{i}})_{i\in I}$, such that for all $i\in I$, $z_{i}$ is contained in the center of $S$. We show that the solutions of these equations are closely related to the solutions of the d'Alembert's classic functional equation with an involutive automorphism. Furthermore, we obtain the superstability theorems that these functional equations are superstable in the general case, where $\sigma$ is an involutive morphism.

Stetkaer [33,Exercise 9.18] found the complex-valued solutions of the functional equation (1.4) f (xy −1 z 0 ) − f (xyz 0 ) = 2f (x)f (y), x, y ∈ G on groups G, where z 0 is a fixed element in the center of G. Perkins and Sahoo [27] replaced the group inversion by an involutive anti-automorphism σ: G −→ G and they obtained the abelian, complex-valued solutions of the functional equation Stetkaer [31] extends the results of Perkins and Sahoo [27] about equation (1.5) to the more general case where G is a semigroup and the solutions are not assumed to be abelian and z 0 is a fixed element in the center of G.
Recently, Bouikhalene and Elqorachi [5] obtained the solutions of an extension of Van Vleck's functional equation (1.6) χ(y)f (xσ(y)z 0 ) − f (xyz 0 ) = 2f (x)f (y), x, y ∈ S on semigroup S, and where χ is a multiplicative function such that χ(xσ(x)) = 1 for all x ∈ S. There has been quite a development of the theory of d'Alembert's functional equation (1.7) g(xy) + g(xσ(y)) = 2g(x)g(y), x, y ∈ G, during the last ten years on non abelian groups. The non-zero solutions of d'Alembert's functional equation (1.7) for general groups, even monoids are the normalized traces of certain representations of the group G on C 2 [10,11]. Stetkaer [30] expressed the complex-valued solutions of Kannappan's functional equation (1.8) f (xyz 0 ) + f (xσ(y)z 0 ) = 2f (x)f (y), x, y ∈ S on semigroups in terms of solutions of d'Alembert's functional equation (1.7). Elqorachi [13] extended the results of Stetkaer [30,31] to the following generalizations of Kannappan's functional equation and Van Vleck functional equation where µ is a linear combination of Dirac measures (δ zi ) i∈I , with z i contained in the center of the semigroup S, for all i ∈ I and where σ is an involutive antiautomorphism of S.
Recently, Zeglami and Fadli [37] obtained the continuous and central solutions of (1.9) and (1.10) on locally compact groups. Related studies of functional equations like (1.9) can be found in [1,15,16,17]. The stability of functional equations highlighted a phenomenon which is usually called superstability. Consider the functional equation E(f ) = 0 and assume we are in a framework where the notion of boundedness of f and of E(f ) makes sense.
We say that the equation E(f ) = 0 is superstable if the boundedness of E(f ) implies that either f is bounded or f is a solution of E(f ) = 0. This property was first observed when the following theorem was proved by Baker, Lawrence, and Zorzitto [3]: Let V ba a vector space. If a function f : The result was generalized by Baker [2], by replacing V by a semigroup and R by a normed algebra E, in which the norm is multiplicative, by Ger andŠemrl [20], where E is an arbitrary commutative complex semisimple Banach algebra and by Lawrence [26] in the case where E is the algebra of all n × n matrices. Different generalization of the result of Baker, Lawrence and Zorzitto have been obtained.
The first purpose of this paper is to extend the results of Stetkaer [31,30] about the Kannappan's functional equation (1.9) and Van Vleck's functional equation (1.10) to the case, where σ is an involutive automorphism of S. By using similar methods and computations to those in [13] we prove that the solutions of (  [29] (general solutions) for a special involutive automorphism σ of the Heisenberg group. We show that any solution of (1.7) is proportional to a solution of (1.7). We prove that all solutions of the integral Van Vleck's functional equation (1.10) are abelian and as an application we obtain some results about abelian solutions of (1.7). In our proofs we do not need the crucial proposition [33, Proposition 8.14] used in the proofs of the main results in [13] and [31,30]. The second purpose of this paper is to prove the superstability of equations (1.9) and (1.10). We show that the superstability of these functional equations is closely related to the superstability of the Wilson's classic functional equation f (xy) + f (xσ(y)) = 2f (x)g(y) x, y ∈ S, and consequently, we obtain the superstability theorems of equations (1.9) and (1.10) on semigroups that are not necessarily abelian and where σ is an involutive morphism.

Integral Kannappan's functional equation on semigroups
In this section we study the complex-valued solutions of the functional equation (1.9), where σ is an involutive automorphism and µ is a linear combination of Dirac measures (δ zi ) i∈I , such that z i is contained in the center of S for all i ∈ I. Throughout this paper we use in (all) proofs without explicit mentioning the assumption that for all i ∈ I; z i is in the center of S and its consequence σ(z i ) is in the center of S. The following lemma has been obtained in [13] for σ an involutive anti-automorphism. It is still true, where σ is an involutive automorphism. In the proof we adapt similar computations used in [13]. f (x) = f (σ(x)), The following notations will be used later. -A consists of the solutions of g : S −→ C of d'Alembert's functional equation (1.7) with S g(t)dµ(t) = 0 and satisfying the condition -To any g ∈ A we associate the function T g = S g(t)dµ(t)g : S −→ C.
(2) Any non-zero solution f : S −→ C of the integral Kannappan's functional equation (1.9) is of the form f = S g(t)dµ(t)g, where g ∈ A. Furthermore, f (x) = S g(xt)dµ(t) = S g(xσ(t))dµ(t) = S g(t)dµ(t)g(x) for all x ∈ S. Proof. Similar computations to those of [13], where σ anti-automorphism involutive, can be adapted to the present situation. The only assertion that need proof is that the function [13] satisfies the condition (2.5). By replacing x by xks and y by r in (1.9) and integrating the result obtained with respect to k, s and r we get By replacing x by xs and y by kr in (1.9) and integrating the result obtained with respect to k, s and r we obtain

From (2.3) and (2.4) we have
In view of (2.6) and (2.7) we deduce that So, by using the expression of g we obtain for all x ∈ S. This completes the proof. Remark 2.3. In Stetkaer's paper [30] about Kannappan's functional equation on semigroups, more precisely in the definition of the set A other assertions which are equivalent to (2.5) are needed to prove the main result in [30]. We notice here that we do not need these statements to show the main result. The same note is also valid for the manuscript [13]. Now, we extend Stetkaer's result [30] from anti-automorphisms to the more general case of morphism as follows.
Remark 2.6. The result of the Corollary 2.5 is also true without the assumption that µ is a linear combination of Dirac measures δ zi with z i contained in the center of S (see [18]).  (1.9), with σ an involutive automorphism of S. In view of [32], there exists a non-zero multiplicative function χ: S −→ C such that This completes the proof.

Superstability of the Intergral Kannappan functional equation (1.9)
In this section we obtain the superstability result of equation (1.9) on semigroups not necessarily abelian. Later, we need the following Lemma.
Lemma 3.1. Let σ be an involutive morphism of S. Let µ be a complex measure that is a linear combination of Dirac measures (δ zi ) i∈I , such that z i is contained in the center of S for all i ∈ I. Let δ > 0 be fixed. If f : S −→ C is an unbounded function which satisfies the inequality The function g defined by is unbounded on S and satisfies the following inequalities.
for all x, y ∈ S. Furthermore, g is a non-zero solution of d'Alembert's functional equation (1.7) and satisfies the condition (2.5).
Proof. Equation (3.2): Replacing y by σ(y) in (3.1) and subtracting resulting inequalities we find after using the triangle inequality that |f (x)(f (y)−f (σ(y)))| ≤ 2δ. Since f is assumed to be unbounded then f (σ(y)) = f (y) for all y ∈ S. Equation (3.2): By replacing x by σ(s) in (3.1) and integrating the result obtained with respect to s we get Equation (3.4): By setting y = s in (3.1) and integrating the result obtained with respect to s we get According to (3.3) and the triangle inequality we deduce (3.4). Equation (3.5): Assume that f is an unbounded function which satisfies the inequality (3.1) and that S f (t)dµ(t) = 0. Replacing x by xs, y by yk in (3.1) and integrating the result obtained with respect to s and k we get In view of (3.3) and (3.4) we have Since S f (t)dµ(t) = 0, then we get From (3.9) we conclude that the function h(x) = S f (xs)dµ(s) is a bounded function on S, in particular the functions (x, y) −→ S f (xys)dµ(s); (x, y) −→ S f (xσ(y)s)dµ(s) are bounded on S × S. So, from (3.1) we deduce that f is a bounded function, which contradict the assumption that f is an unbounded function on S and this proves (3.5). Equation Which gives (3.7). Equation (3.8): For all x ∈ S, we have Replacing x by xsk and y by r in (3.1) and integrating the result obtained with respect to s, k and r we get (3.10) By replacing x by xs and y by kr in (3.1) and integrating the result obtained with respect to s, k and r we get (3.11) From inequalities (3.1), (3.2), (3.3) and the above relations we get Which implies that and this proves (3.8). Now, since g is unbounded and satisfies the inequality (3.7) so, from [6], we deduce that g satisfies the d'Alembert's functional equation (1.7). We will show that S g(xt)dµ(t) = g(x) S g(t)dµ(t) for all x ∈ S.
Since g is an unbounded function on S then we get S g(xt)dµ(t) = g(x) S g(t)dµ(t) for all x ∈ S. This completes the proof. Now, we are ready to prove the main result of the present section. We notice here that same result has been obtained in [6] with other assumptions on µ. for all x ∈ S or f is a solution of the integral Kannappan's functional equation (1.9).
Proof. Assume that f is an unbounded solution of (3.12). Replacing y by s in (3.12) and integrating the result obtained with respect to s we get (3.13) (3.4) and the triangle inequality we get for all x, y ∈ S. Since from (3.5) we have S f (s)dµ(s) = 0. Then the inequality (3.14) can be written as follows (3.15) |f (xy) + f (xσ(y)) − 2f (x)g(y)| ≤ 3δ µ | S f (s)dµ(s)| for all x, y ∈ S and where g is the function defined in Lemma 3.1. Now, by using same computation used in [6, Theorem 2.2(iii)] we conclude that f, g are solutions of Wilson's functional equation (3.16) f (xy) + f (xσ(y)) = 2f (x)g(y) for all x, y ∈ S. By replacing x by t in (5.10) and integrating the result obtained with respect to t we get S f (ty)dµ(t)+ S f (tσ(y))dµ(t) = 2g(y) S f (t)dµ(t). Since .
Since f is assumed to be unbounded then we deduce that β = 1 and then from (3.17) we deduce that f is a solution of (1.9). This completes the proof.
for all x, y ∈ M . Then either f is bounded or f is a solution of the integral Kannappan's functional equation (1.9).
Proof. Let f be an unbouded contious function which satisfies (3.18). Taking y = e in (3.18) we get for all x ∈ M . Since f is unbounded then f (e) = 0, because if f (e) = 0 the functions (x, y) −→ M f (xyt)dµ(t); (x, y) −→ M f (xσ(y)t)dµ(t) are bounded and from (3.18) and the triangle inequality we get f a bounded function on S. This contradict the assumption that f is unbounded. Now, From (3.18), (3.19) and the triangle inequality we obtain (3.20) |f (e)f (xy) Inequality which can be written as follows From [6, Theorem 2.2(iii)] we deduce that f, g are solutions of Wilson's functional equation for all x, y ∈ M , then from [33] f is central. So, Substituting this into (3.18) after computation we get |f (y) for all x ∈ S. Thus, for all x, y ∈ M we get That is f satisfies the integral Kannappan's functional equation (1.9). This completes the proof.

Solutions of the functional equation (1.10)
The solutions of the functional equation (1.10) with σ an involutive anti-automorphism are explicitly obtained by Elqorachi [13] on semigroups not necessarily abelian in terms of multiplicative functions. In this section we express the solutions of (1.10) where σ is an involutive automorphism in terms of multiplicative functions. The following lemma is obtained in [13] for the case where σ is an involutive antiautomorphism. It still holds for the case where σ is an involutive automorphism.
The function defined by Jf Proof. From Lemma 4.1 the formula (4.7) makes sense, and we have g := Jf ∈ B for any f ∈ V.
-Injection: Let f 1 and f 2 be two non-zero solutions of (1.10). If Jf 1 = Jf 2 then we get for all x ∈ S. Since f 1 and f 2 are solutions of (1.10) then we have and (4.10) By multiplying (4.9) by S f 2 (t)dµ(t) and using (4.8) we get By replacing y by s in (4.11) and integrating the result obtained with respect to s we get 2f First we notice that since g is a solution of (1.7) and S g(s)dµ(s) = 0 then if we let y = s in (1.7) and integrating the result obtained with respect to s we deduce that S g(xσ(s))dµ(s) = − S g(xs)dµ(s). We may define f : S −→ C by For all x, y ∈ S we have Furthermore, Thus, we get f = 0. On the other hand for all x ∈ S we have = S S g(xtσ(s))dµ(t)dµ(s) S S g(tσ(s))dµ(t)dµ(s) S S g(xtσ(s))dµ(t)dµ(s) + S S g(xtσ(s))dµ(t)dµ(s) 2 S S g(tσ(s))dµ(t)dµ(s) = 2g(x) S S g(tσ(s))dµ(t)dµ(s) 2 S S g(tσ(s))dµ(t)dµ(s) = g(x).
This completes the proof.
In [13] we use [33,Proposition 8.14] to derive the form of the solutions of (1.10) where σ is an involutive anti-automorphism of S. This reasoning no longer works for the present situation. We will use an elementary approach which works for both situations.
where χ : S −→ C is a multiplicative function such that S χ(t)dµ(t) = 0 and Proof. Let f be a non-zero solution of (1.10). Replacing x by xs in (1.10) and integrating the result obtained with respect to s we get (4.12) By using (4.5) equation (4.12) can be written as follows where g is the function defined in Lemma 4.1. If we replace y by ys in (1.10) and integrate the result obtained with respect to s we get (4.14) By using (4.5) we obtain that (4.15) f (xσ(y)) + f (xy) = 2f (x)g(y), x, y ∈ S . By adding (4.15) and (4.13) we get that the pair f, g satisfies the sine addition law f (xy) = f (x)g(y) + f (y)g(x) for all x, y ∈ S. Now, in view of [12,Lemma 3.4], [33,Theorem 4.1] g is abelian. Since g is a non-zero solution of d'Alembert's functional equation (1.7) there exists a non-zero multiplicative function χ: S −→ C such that g = χ+χ•σ 2 . The rest of the proof is similar to the one used in [13]. This completes the proof. In the present section we prove the superstability theorem of the integral Van Vleck's functional equation (1.10) on semigroups. First, we prove he following useful lemma. The function g defined by is unbounded on S and satisfies the following inequality (5.9) |g(xy) + g(xσ(y)) − 2g(x)g(y)| ≤ 3δ µ 2 (| S f (s)dµ(s)) 2 | for all x, y ∈ S. for all x, y ∈ S. By adding the result of (5.1) and (5.11) and using the triangle inequality we obtain |2f (x)(f (y) + f (σ(y)))| ≤ 2δ for all x ∈ S. Since f is assumed to be unbounded then we get (5.2). Equation That is f is a solution of Van Vleck's functional equation (1.10). This completes the proof.