Aequat. Math. 89 (2015), 591 603 c The Author(s) 2013. Ths artcle s publshed wth open access at Sprngerlnk.com 0001-9054/15/030591-13 publshed onlne December 6, 2013 DOI 10.1007/s00010-013-0244-4 Aequatones Mathematcae Some generalzaton of Cauchy s and Wlson s functonal equatons on abelan groups Rados law Lukask Abstract. We fnd the solutons f, g,h: G X, α: G K of the functonal equaton f(x + λy) = K g(x)+ α(x)h(y), x,y G, where (G, +) s an abelan group, K s a fnte, abelan subgroup of the automorphsm group of G, X s a lnear space over the feld K {R, C}. Mathematcs Subject Classfcaton (2010). 39B52. Keywords. Wlson s functonal equaton, Cauchy s functonal equaton. 1. Introducton The followng generalzaton f(x + y)+f(x + σy) =2f(x)+2f(y), x,y G, of the quadratc functonal equaton, where σ s an automorphsm of an abelan group G such that σ σ = d G and f : G C, was nvestgated by Stetkær [9]. In hs other work [10] he solved the functonal equaton N 1 1 f(z + ω n ζ)=f(z)+g(z)h(ζ), z,ζ C, N n=0 where N {2, 3,...},ω s a prmtve N th root of unty, f,g,h: C C are contnuous. Lukask [5, 6] derved explct formulas for the solutons of the functonal equaton f(x + λy) = K α(y)g(x)+ K h(y), x,y G,
592 R. Lukask AEM where (G, +) s an abelan group, K s a fnte abelan subgroup of the automorphsm group of G, X s a lnear space over the feld K {R, C},f,g,h: G X, α: G K. The functonal equaton f(x + λy) = K g(x)h(y), x,y G, where (G, +) s an abelan group, K s a fnte subgroup of the automorphsm group on G, f, g, h: G C, was studed by Förg-Rob and Schwager [3], Gajda [4], Stetkær [7, 8], Badora [2]. Aczél et al. [1] studed the complex-valued solutons of the equaton f(x + y)+f(x y) 2f(x) =g(x)h(y), x,y G, where (G, +) s a group and f,g,h: G C. The purpose of ths paper s to fnd the solutons of the functonal equaton f(x + λy) = K g(x)+α(x)h(y), x,y G, where f,g,h: G X, α: G K, (G, +) s an abelan group, K s a fnte, abelan subgroup of the automorphsm group of G, X s a lnear space over the feld K {R, C}. We fnd these solutons under the assumpton f λ const, α λ const, α(0) 0 and they are some combnatons of multplcatve and mult-addtve functons. Our results generalze all the results mentoned above (except the papers by Lukask). 2. Man result Throughout the present paper, we assume that X s a lnear space over the feld K {R, C},(G, +) s an abelan group, K s a fnte, abelan subgroup of the automorphsm group of G. In ths work we use some theorems. The frst gves us the form of the solutons of a generalzaton of Jensen s functonal equaton. Theorem 1. [5, Theorem5]Let (S, +) be an abelan semgroup, K be a fnte subgroup of the automorphsm group of S, (H, +) be an abelan group unquely dvsble by K!. Then the functon f : S H satsfes the equaton f(x + λy) = K f(x), x,y S (1)
Vol. 89 (2015) Generalzaton of Cauchy s and Wlson s functonal equatons 593 f and only f there exst k-addtve, symmetrc mappngs A k : S k H, k {1,..., K 1} and A 0 H such that f(x) =A 0 + A 1 (x)+ + A K 1 (x,...,x), x S, A k (x,...,x,λy,...,λy)=0, x,y S, 1 k K 1. The second theorem shows all solutons of a generalzaton of Wlson s functonal equaton. Theorem 2. [6, Theorem 4,5] Let f : G X,f 0,ϕ: G K. They satsfy the equaton f(x + λy) = K ϕ(y)f(x), x,y G, (2) f and only f there exsts a homomorphsm m: G C, such that ϕ(x) = 1 m(λx), x G, K and () f X s complex, then there exst A λ 0 X, k-addtve, symmetrc mappngs A λ k : Gk X,k {1,..., K 0 1},λ K 1 such that f(x) = K m(λx) A λ 0 + A λ (x,...,x), x G, 1 A λ k(x,...,x,μy,...,μy)=0, x,y G, 1, 1 k K 0 1, 0 () f X s real, then there exst A λ 0 X,B0 λ X, k-addtve, symmetrc mappngs A λ k,bλ k : Gk X,k {1,..., K 0 1},λ K 1 such that f(x) = K Re (m(λx)) A λ 0 + A λ (x,...,x) 1 Im (m(λx)) B 0 λ + K B λ (x,...,x), x G, A λ k(x,...,x,μy,...,μy)=0, x,y G, λ K 1, 1 k K 0 1, 0 Bk λ (x,...,x,μy,...,μy)=0, x,y G, 1, 1 k K 0 1, 0 where K 0 := {λ K : m λ = m},k 1 s the set of representatves of cosets of the quotent group K/K 0.
594 R. Lukask AEM Frst we start wth a corollary of Theorem 2. Corollary 1. A nonzero functon α: G K satsfes the equaton α(λy), x,y G, (3) α(x + λy) =α(x) f and only f there exsts a homomorphsm m: G C and () f K = C, then there exst a λ 0 C, k-addtve, symmetrc mappngs a λ k : Gk C,k {1,..., K 0 1},λ K 1 such that α(x) = K m(λx) a λ 0 + a λ (x,...,x), x G, 1 1, 1 k< K 0, 0 1 () f K = R, then there exst a λ 0,b λ 0 R, k-addtve, symmetrc mappngs a λ k,bλ k : Gk R,k {1,..., K 0 1},λ K 1 such that α(x) = K Re (m(λx)) a λ 0 + a λ (x,...,x) 1 Im (m(λx)) b λ 0 + K b λ (x,...,x), x G, 1, 1 k< K 0, 0 b λ k(x,...,x,μy,...,μy)=0, x,y G, λ K 1, 1 k< K 0, 0 1 where K 0 := {λ K : m λ = m},k 1 s the set of representatves of cosets of the quotent group K/K 0. Moreover, f α has the above form, then m(λx), x G. α(λx) = Proof. Assume that α satsfes (3). Let ϕ: G K be gven by the formula ϕ = 1 α λ. K
Vol. 89 (2015) Generalzaton of Cauchy s and Wlson s functonal equatons 595 Then α and ϕ satsfy the equaton 1 α(x + λy) =ϕ(y)α(x), x,y G. K In vew of Theorem 2 we have the form of α such as n the statement of ths corollary and the equalty ϕ(x) = 1 m(λx), x G. K We observe that () f K = C, then K ϕ(x) = α(μx) = K m(λμx) a λ 0 + a λ (μx,...,μx) 1 = K m(λμx) K 0 a λ 0 + a λ (σμx,...,σμx) 1 1 σ K 0 = m(λμx) K 0 a λ 0 = m(μx) a λ 0, x,y G, 1 1 1 () f K = R, then K ϕ(x) = α(μx) = K 0 1 Re (m(λμx)) a λ 0 + a λ (μx,...,μx) 1 K 0 1 Im (m(λμx)) b λ 0 + b λ (μx,...,μx) 1 = K 0 1 Re (m(λμx)) K 0 a λ 0 + a λ (σμx,...,σμx) 1 1 σ K 0 K 0 1 Im (m(λμx)) K 0 b λ 0 + b λ (σμx,...,σμx) 1 1 σ K 0 = Re (m(λμx)) K 0 a λ 0 Im (m(λμx)) K 0 b λ 0 1 1 1 1 = Re (m(μx)) a λ 0 Im (m(μx)) b λ 0 1 1 = m(μx) a λ 0, x,y G. 1
596 R. Lukask AEM Hence 1 a λ 0 = 1 and on the other hand a functon α, whch has the form such as n the statement of ths corollary, satsfes Eq. (3). Theorem 3. Let functons f : G X,α: G K be such that f λ 0, α 0, α λ K. They satsfy the equaton f(x + λy) = K f(x)+α(x) f(λy), x,y G, (4) f and only f there exst a homomorphsm m: G C,A 0 X, k-addtve, symmetrc mappngs A k : G k X,k {1,...,L 1} such that K 1 f(x) =A 0 + A (x,...,x) α(x)a 0, x G, (5) A k (x,...,x,μy,...,μy)=0, x,y G, 1 k< K, (6) and () f K = C, then there exst a λ 0 C, k-addtve, symmetrc mappngs a λ k : Gk C,k {1,..., K 0 1},λ K 1 such that α(x) = K m(λx) a λ 0 + a λ (x,...,x), x G, (7) 1 1, 1 k< K 0, 0 1 () f K = R, then there exst a λ 0,b λ 0 R, k-addtve, symmetrc mappngs a λ k,bλ k : Gk R,k {1,..., K 0 1},λ K 1 such that α(x) = K Re (m(λx)) a λ 0 + a λ (x,...,x)] (10) 1 Im (m(λx)) b λ 0 + K (8) (9) b λ (x,...,x), x G, (11) 1, 1 k< K 0, 0 (12)
Vol. 89 (2015) Generalzaton of Cauchy s and Wlson s functonal equatons 597 b λ k(x,...,x,μy,...,μy)=0, x,y G, λ K 1, 1 k< K 0, 0 (13) (14) 1 where K 0 := {λ K : m λ = m},k 1 s the set of representatves of cosets of the quotent group K/K 0. Moreover ( f(λx) = K ) α(λx) A 0, x G. (15) Proof. Let f and α satsfy (4). We observe that f(x + λy + μz) = K f(x + λy) + α(x + λy) f(μz) = K 2 f(x)+ K α(x) f(λy) + α(x + λy) f(μz), x,y,z G, and f(x + λ(y + μz)) f(x + λy + μz) = = K 2 f(x)+α(x) f(λy + μz) = K 2 f(x)+ K α(x) f(λy)+α(x) α(λy) f(μz), x,y,z G. Hence we have ( ) α(λy) f(μz) =0, x,y,z G, α(x + λy) α(x) and we obtan that α satsfes Eq. (3). In vew of Corollary 1 we get the form of α. Now we notce that K f(λx)+ α(λx) f(μy) = f(λx + μy) = f(μy + λx) = K f(μy)+ α(μy) f(λx), x,y G.
598 R. Lukask AEM Hence ( K ) α(λx) f(μy) = K α(μy) f(λx), x,y G. whch gves us Eq. (15) for some A 0 X. Now, we can wrte Eq. (4) nthe form ( f(x + λy) = K f(x)+α(x) K ) α(λy) A 0, x,y G. Let q : G X be gven by the formula q(x) =f(x)+α(x)a 0, x G. Then from equaltes (3), (4), (15) wehave α(x + λy)a 0 q(x + λy) = f(x + λy)+ = K f(x)+α(x) f(λy)+α(x) α(λy)a 0 = K f(x)+ K α(x)a 0 = K q(x), x,y G. In vew of Theorem 1 there exst c X, k-addtve, symmetrc mappngs A k : G k X,k {1,..., K 1} such that K 1 q(x) =c + A (x,...,x), x G, A k (x,...,x,μy,...,μy)=0, x,y G, 1 k< K. Snce c = q(0) = f(0) + α(0)a 0 = A 0, we have K 1 f(x) =A 0 + A (x,...,x) α(x)a 0, x G. Now we assume that f satsfes condtons (5) (6) andα satsfes condtons (7) (9) n the complex case or (10) (14) n the real case. In vew of Theorem 1 a functon f + αa 0 satsfes Eq. (1) and n vew of Corollary 1 α satsfes Eq. (3). We have α(x + λy)a 0 f(x + λy) = (f(x + λy)+α(x + λy)a 0 ) = K f(x)+ K α(x)a 0 α(x) α(λy)a 0, x,y G.
Vol. 89 (2015) Generalzaton of Cauchy s and Wlson s functonal equatons 599 Hence we obtan f(λy) = K f(0) + K α(0)a 0 α(0) = K A 0 α(λy)a 0, x,y G α(λy)a 0 and f(x + λy) = K f(x)+ K α(x)a 0 α(x) α(λy)a 0 = K f(x)+α(x) f(λy), x,y G, whch ends the proof. Remark 1. Let f : G X, α: G K,α =0or f λ = 0. Then they satsfy Eq. (15) f and only f f satsfes Eq. (1). Hence, n vew of Theorem 1, we know the form of f. Remark 2. Let f : G X, α: G K, α λ = K. If they satsfy Eq. (15) then α satsfes Eq. (1) and we know ts form. At the present moment we don t know the form of f. Now we can prove the man theorem of ths paper whch s a pexderzed verson of Theorem 3. Theorem 4. Let functons f,g,h: G X,α: G K be such that f λ const, α λ const, α(0) 0. They satsfy the equaton f(x + λy) = K g(x)+α(x)h(y), x,y G, (16) f and only f there exst a homomorphsm m: G C,A,B,A 0 X, k- addtve, symmetrc mappngs A k : G k X,k {1,..., K 1} such that K 1 f(x) =A + A 0 + A (x,...,x) α(x) α(0) A 0, x G, (17) and K 1 g(x) =A + A 0 + A (x,...,x) α(x) α(0) [A + A 0 B], x G, (18) [( h(x) = 1 K ) ] α(λx) A 0 + K (A B), x G, (19) α(0) α(0) A k (x,...,x,μy,...,μy)=0, x,y G, 1 k< K, (20)
600 R. Lukask AEM () f K = C, then there exst a λ 0 C, k-addtve, symmetrc mappngs a λ k : G k C, k {1,..., K 0 1},λ K 1 such that α(x) =α(0) K m(λx) a λ 0 + a λ (x,...,x), x G, (21) 1 1, 1 k< K 0, 0 1 (22) (23) () f K = R, then there exst a λ 0,b λ 0 R, k-addtve, symmetrc mappngs a λ k,bλ k : Gk R,k {1,..., K 0 1},λ K 1 such that α(x) =α(0) 1 Im (m(λx)) b λ 0 + Re (m(λx)) a λ 0 + K K a λ (x,...,x) (24) b λ (x,...,x), x G, (25) 1, 1 k< K 0, 0 (26) b λ k(x,...,x,μy,...,μy)=0, x,y G, λ K 1, 1 k< K 0, 0 1 (27) (28) where K 0 := {λ K : m λ = m},k 1 s the set of representatves of cosets of the quotent group K/K 0. Proof. Puttng x = 0n(16) wehave f(λy) = K g(0) + α(0)h(y), y G. Puttng y =0n(16) weget K f(x) = K g(x)+α(x)h(0), x G.
Vol. 89 (2015) Generalzaton of Cauchy s and Wlson s functonal equatons 601 Hence we get g(x) =f(x) α(x) α(x) h(0) = f(x) [f(0) g(0)], x G, (29) K α(0) [ ] h(y) = 1 f(λy) K g(0), y G. (30) α(0) Let f 0 = f f(0),α 0 = α α(0). From the above equaltes we obtan f 0 (x + λy) = f(x + λy) K f(0) = K g(x)+α(x)h(y) K f(0) [ ] = K f 0 (x) α 0 (x) K [f(0) g(0)] + α 0 (x) f(λy) K g(0) = K f 0 (x)+α 0 (x) f 0 (λy), x,y G. In vew of Theorem 3 there exst a homomorphsm m: G C,A 0 X, k- addtve, symmetrc mappngs A k : G k X,k {1,..., K 1} such that and K 1 f 0 (x) =A 0 + A (x,...,x) α 0 (x)a 0, x G, A k (x,...,x,μy,...,μy)=0, x,y G, 1 k< K, () f K = C, then there exst a λ 0 C, k-addtve, symmetrc mappngs a λ k : G k C, k {1,..., K 0 1},λ K 1 such that α 0 (x) = K m(λx) a λ 0 + a λ (x,...,x), x G, 1 1, 1 k< K 0, 0 1
602 R. Lukask AEM () f K = R, then there exst a λ 0,b λ 0 R, k-addtve, symmetrc mappngs a λ k,bλ k : Gk R,k {1,..., K 0 1},λ K 1 such that α 0 (x) = K Re (m(λx)) a λ 0 + a λ (x,...,x) 1 Im (m(λx)) b λ 0 + K b λ (x,...,x), x G, 1, 1 k< K 0, 0 b λ k(x,...,x,μy,...,μy)=0, x,y G, λ K 1, 1 k< K 0, 0 a λ 0 =1. 1 Moreover ( f 0 (λx) = K ) α 0 (λx) A 0, x G. Hence, puttng A := f(0),b := g(0) and usng equaltes (29), (30), we obtan the form of f,g,h and α. Now we assume that f,g,h satsfy condtons (17) (20) and α satsfes condtons (21) (23) n the complex case and (24) (28) n the real case. In vew of Theorem 1 a functon f + α 0 A 0 satsfes Eq. (1) and n vew of Corollary 1 α 0 satsfes Eq. (3). We have α 0 (x + λy)a 0 f(x + λy) = (f(x + λy)+α 0 (x + λy)a 0 ) = K f(x)+ K α 0 (x)a 0 α 0 (x) α 0 (λy)a 0 = K f(x) [ K α 0 (x)[a B]+α 0 (x) K (A B)+ K A 0 ] α 0 (λy)a 0 = K g(x)+α(x)h(y), x,y G, whch ends the proof. Remark 3. Let f,g,h: G X, α: G K satsfy Eq. (16). () If α(0) = 0, then f λ =const. () If f λ = const, then α λ = const or h = const (n ths case Eq. (16) becomes Eq. (1) and we know ts form). We don t know the form of the solutons n the case when α λ =const.
Vol. 89 (2015) Generalzaton of Cauchy s and Wlson s functonal equatons 603 Open Access. Ths artcle s dstrbuted under the terms of the Creatve Commons Attrbuton Lcense whch permts any use, dstrbuton, and reproducton n any medum, provded the orgnal author(s) and the source are credted. References [1] Acél, J., Chung, J.K., Ng, C.T.: Symmetrc Second Dfferences n Product form on Groups. Topcs n Mathematcal Analyss, 1 22, Seres n Pure Mathematcs, vol. 11. World Scentfc Publshng, Teaneck (1989) [2] Badora, R.: On a generalzed Wlson functonal equaton. Georgan Math. J. 12(4), 595 606 (2005) [3] Förg-Rob, W., Schwager, J.: A generalzaton of the cosne equaton to n summands. Grazer Math. Ber. 316, 219 226 (1992) [4] Gajda, Z.: A remark on the talk of W. Förg-Rob. Grazer Math. Ber. 316, 234 237 (1992) [5] Lukask, R.: Some generalzaton of Cauchy s and the quadratc functonal equatons. Aequ. Math. 83, 75 86 (2012) [6] Lukask, R.: Some generalzaton of the quadratc and Wlson s functonal equaton. Aequ. Math. do:10.1007/s00010-013-0185-y [7] Stetkær, H.: On a sgned cosne equaton of N summands. Aequ. Math. 51(3), 294 302 (1996) [8] Stetkær, H.: Wlson s functonal equaton on C. Aequ. Math. 53(1-2), 91 107 (1997) [9] Stetkær, H.: Functonal equaton on abelan groups wth nvoluton. Aequ. Math. 54(1 2), 144 172 (1997) [10] Stetkær, H.: Functonal equatons nvolvng means of functons on the complex plane. Aequ. Math. 56, 47 62 (1998) Rados law Lukask Insttute of Mathematcs Unversty of Slesa ul. Bankowa 14 40-007 Katowce Poland e-mal: rlukask@math.us.edu.pl Receved: July 30, 2013 Revsed: November 4, 2013