#### Abstract

Let S be a nonunital commutative semigroup, an involution, and the set of complex numbers. In this paper, first we determine the general solutions of Wilson’s generalizations of d’Alembert’s functional equations   and on nonunital commutative semigroups, and then using the solutions of these equations we solve a number of other functional equations on more general domains.

#### 1. Introduction

The functional equation is known as the d’Alembert’s functional equation. It has a long history going back to d’Alembert [1]. As the name suggests this functional equation was introduced by d’Alembert in connection with the composition of forces and plays a central role in determining the sum of two vectors in Euclidean and non-Euclidean geometries [2].

Several authors have determined the general solution of the functional equation for all , where is an automorphism of order . Stetkær [3] studied (2) when , is an abelian topological group and and are continuous. Sinopoulos [4] determined the general solution of (2) when is a commutative semigroup, is a quadratically closed commutative field of characteristic different from 2, and is an endomorphism of of order 2.

Wilson’s functional equation is a generalization of d’Alembert’s functional equation. Among others Wilson’s functional equation was studied by Wilson [5] and [6], Kaczmarz [7], van der Lyn [8], Fenyö [9], Chung et al. [10], and Aczél [11]. Stetkær (see [12] and [3]) studied Wilson’s equation with an involution; namely, where is a topological monoid (i.e., is a semigroup with an identity element), , are continuous functions defined on and taking values on , and is a continuous automorphism of order 2. He gave several properties of the solution of Wilson’s equation (see [3, Chapter 11, Lemmas 11.3 and 11.4]). In Chapter 11 of [3] (see Corollary 11.7) a complete and satisfactory description of the solutions of Wilson’s equation (3) was given on abelian groups.

The functional equation was introduced in [6] (see also [3] Chapter 11). For more on known results related to Wilson’s functional equations we refer the reader to [1320]. The above functional equation with an involution, namely, was studied by Stetkær (see [21]) on abelian groups.

In this paper, first we present the solutions of Wilson’s functional equations (3) and (4) on nonunital commutative semigroups and then using the general solutions of these functional equations we determine the general solutions of several functional equations in two variables in more general domains. In particular, we determine the general solutions of functional equations; namely, treated by Riedel and Sahoo in [22], on more general domains. We also determine the general solutions of functional equations:

#### 2. Solution of Wilson’s Equations on Commutative Nonunital Semigroups

In this section, we present the solutions of Wilson’s functional equations (3) and (4) on nonunital semigroups. We solve these equations on more general domains than the domains used in [3]. In particular we give the description of the solutions of Wilson’s equations (3) and (4) on nonunital semigroups whereas in [3] the description of the solutions of (3) is given on commutative topological monoids (i.e., semigroups with an identity element). Note that Lemmas 11.3 and 11.4 of Chapter  11 in [3] are not very useful for solving (3) and (4) on nonunital commutative semigroups due to the lack of identity element. Our method of proof is quite elementary and transparent.

Throughout this paper we denote by a -divisible commutative semigroup with no identity element and denote by the set of complex numbers. A function is said to be an involution if , for all , and , for all . For simplicity we write instead of . A function is called an exponential function provided that , for all , and is called an additive function provided that , for all .

As a direct consequence of a theorem of Sinopoulos [4] we have the following lemma.

Lemma 1. Let satisfy for all . Then there exists an exponential function such that for all .

In the following, we exclude the trivial cases when or .

Theorem 2. Let satisfy the functional equation for all . Then either , are of the form for all , where is an exponential function satisfying and , or else where is an exponential function satisfying , and is an additive function satisfying and .

Proof. Choose an such that . Replacing by in (9) and dividing the result by we have for all . Multiplying both sides of (12) by and using (9) we have for all . By Lemma 1, has the form for all . Now, we find . Let and be the even part and the odd part of with respect to ; that is, for all . Then we have for all . Replacing by in (14) we obtain for all . Replacing by in (9) and using (17) we have for all . Adding (18) to (9) and dividing the result by we have for all . Replacing by in (19) and using (16) we have Putting in (19) and (20) such that and equating the right hand sides of the results we have for all . From now on, we determine . Subtracting (18) from (9) and dividing the result by we have for all . Replacing by and using (16) we have for all . Subtracting (23) from (22) and dividing the result by we have for all . We divide into two cases when (i) , for some , and (ii) , for all .
Case 1. First, we consider the case when , for some . We show that if , then is not an exponential function. Indeed, from (10) we have for all , and for all . If is exponential, equating (25) and (26) and multiplying the result by we have for all , which implies , for all , contradicting our assumption that , for some . Thus, is not exponential. Let Then it is easy to see that satisfies the functional equation (22). Since is not exponential, we can choose such that . Let be an arbitrary solution of the functional equation (22). We can choose (not both zero) such that Let for all . Then and satisfies the functional equation (22) and satisfies for all . We claim that . If not, that is, there exists such that ; then putting in (31) and dividing the result by we have for all . Putting in (31) and using (17) and (30) we have for all . Now, from (32) and (33) we have for all , which contradicts the fact that . Thus, we conclude that and it follows from (30) that for all . If , then and hence for all , which contracts the assumption for some . Thus, and from (35) we have for all . From (21), (28), and (37) we have for all , where . Thus, we get (10).
Case 2. Next, we assume that , for all . Then from (14) we have for all . Thus, from (24) and (39) we have for all .
Let . Then is an ideal in and is a subsemigroup of . Dividing (40) by and using (39) we have for all . Let for all . Then we have for all . Replacing by in (43) we have Replacing by in (43) we have and replacing by in (43) and using (44) we have From (45) and (46) we have for all . Replacing by in (47) and using (43) we have for all . From (21), (39), (42), and (48) we have for all . Let . Then we have . Thus, replacing both and by in (40) we have and also by (21) . Hence for any . Thus, from (39), (48), and (49) we get (11). This completes the proof of the theorem.

Remark 3. The solution in (11) can be written in the form where is an exponential function and is an additive function. Also, let us assume that has the property: for any , there exists a positive integer and such that which is satisfied by every group and also most of well known semigroups such as . Then, if for some , then for any we can choose a positive integer and such that . Thus, we have which implies for all . Thus, and in (11) are given by for all .

In view of the proof of Theorem 2, we can see that -divisibility of is necessary only in showing that , for all . Thus, if is a commutative group possibly without -divisibility, we have the following.

Corollary 4. Let be a commutative group and satisfy the functional equation for all . Then either , are of the form for all , where is an exponential function satisfying and , or else for all , where is an exponential function satisfying , is an additive function satisfying , and .

Remark 5. Note that in general -divisible commutative semigroup, the solution can not be written in the simple form as above (e.g., : see Theorem 18).

Example 6. Let in Corollary 4 and let with and let Then is an involution on and we can consider the functional equation for all . As a direct consequence of Corollary 4 we can exhibit all regular solutions of (58) in a transparent form. By Corollary 4, has the form for all , where is an exponential function. In view of the proof in [4], is given by , for all and for some . Thus, if is Lebesgue measurable, so is and has the form for some . Thus, has the form for all . Also, by Corollary 4, if , that is, , then has the form for all and for some , and if , then has the form for all and for some .

In the following theorem, we present the general solution of the second Wilson’s functional equation.

Theorem 7. Let satisfy for all . Then there exists an exponential function and with such that for all .

Proof. Replacing by in (63) we see that for all . Replacing by in (63) and using (65) we have for all . Choosing such that , putting in (63) and (66), and equating the right hand sides of the results we have for all . Replacing by in (63) and multiplying the result by we have for all . By Lemma 1 and (67) we get (64). This completes the proof.

#### 3. Applications of Wilson’s Equations in Solving Several Other Equations

Let be a commutative semigroup, and . As an application of Theorems 2 and 7, we determine all general solutions of the functional equations for all . For the case , (69) was treated by Riedel and Sahoo [22]. In the following we exclude the trivial cases when or .

Theorem 8 (c.f. [22]). Let satisfy the functional equation (69) for all . Then either , are of the form for all , where are exponential functions such that and , or else where , is an additive function, is an exponential function, and .

Proof. Let for all . Then by Theorem 2 for the case when , all general solutions of (69) are given by for all , where is an exponential function and . Since can be written in the form , for some exponential functions , from (73) we get (71). Assume that ; that is, , for all . Then we have for all . Let . If there exists , putting in (74) we have , for all , and hence . Since we exclude the case or , we have . Similarly, we have and . From (74), we have for all , which implies for all and for some . Thus, we have for all , which implies and hence . Thus, from (73) and (74) we have for all . Since if and only if or , by Remark 3 we have for all , where is an additive function and . Thus, we get (72). This completes the proof.

Remark 9. Since has the property stated in Remark 3, we have . Thus, Theorem 8 includes the result of Riedel and Sahoo [22].

Using Theorem 7, we have the following.

Theorem 10. Let satisfy the functional equation (70) for all . Then , are of the form for all , where are exponential functions and with .

Using polar form of complex numbers, we have the following two lemmas. In the following we define for .

Lemma 11. Let satisfy the functional equation for all . Then has the form for all , where is a multiplicative function, that is, , for all , and is an exponential function such that for all .

Lemma 12. Let satisfy the functional equation for all . Then has the form for all , where is a multiplicative function and is an additive function such that for all .

Using Theorem 2 and Lemma 11 we have the following.

Theorem 13. Let satisfy the functional equation for all . Then , are of the form for all , or for all , where is a multiplicative function, is an exponential function such that , for all , and is an additive function such that , for all , and .

Remark 14. If some regularities on are assumed in Theorem 13, it can be verified that and . Thus, all regular solutions (e.g., continuous solutions) of (85) are given by for all , where .

Using Theorem 7 and Lemma 11 we have the following.

Theorem 15. Let satisfy the functional equation for all . Then , are of the form for all , where is a multiplicative function and .

As consequence of Theorems 13 and 15, we also find general solutions of the following functional equations: for all , where .

Theorem 16. Let satisfy the functional equation (91). Then , are of the form for all , or for all , where is a multiplicative function, is an exponential function such that , for all , and is an additive function such that , for all , and .

Proof. Define by for all . Then, (91) is reduced to for all . Using Theorem 13 we get the result.

Similarly, using Theorem 15 we have the following.

Theorem 17. Let satisfy the functional equation (92). Then , are of the form for all , where is a multiplicative function, is an exponential function such that , for all , and .

Theorem 18. Let satisfy the functional equation (93). Then either , are of the form for all , where are multiplicative functions such that and , or where is a logarithmic function, that is, , is a multiplicative function, and , or for all .

Proof. Define by for all . Then, (93) is reduced to for all . Applying Theorem 8 with for , we get (100). It is easy to see that or in the case when of Theorem 8. If , from (71) we get (101) with . If , then and it follows that . In this case, is a logarithmic function on ; that is, satisfies for all . Putting in (105) we have . Thus, from (71) we get (102). This completes the proof.

Define by for all . Then, (94) is reduced to for all . Using Theorem 10 we have the following.

Theorem 19. Let satisfy the functional equation (94). Then either or has the form for all , where are multiplicative functions and with .

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

The authors are very thankful to the referee for valuable suggestions that improved the presentation of the paper. This work was done while the first author visited the University of Louisville from Kunsan National University during 2012-13. This work was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (MEST) (no. 2012008507).