Complex Vector Functional Equations

Ice Risteski Valery Covachev

World Scientific

(complex Vector Functional Equations

Complex Vector Functional Equations

Ice Risteski Valery Covachev Bulgarian Academy of Sciences

V^fe World Scientific « •

New Jersey • London »Sint • Singapore • Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

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COMPLEX VECTOR FUNCTIONAL EQUATIONS Copyright © 2001 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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ISBN 981-02-4683-8

Printed in Singapore by Uto-Print

To the authors' sons Branko Risteski and Svetoslav Covachev

Preface

Complex vector functional equations are a new field in the theory of functional equations. Their development up to now has not been particularly dynamical. As every new mathematical discipline, complex vector functional equations call for their more detailed investigation. From this standpoint, in the last few years we dedicated a series of papers to these equations. Our multi-annual experience and the results obtained by the research in this new area show us the necessity to write a strictly scientific systematized monograph. The object of this monograph is to give an overview to the recently obtained results of the authors from all aspects of utilization, which is obviously not possible with partial results in papers published in distinct mathematical journals. We have divided our exposition on the field of complex vector functional equations in two parts which are devoted respectively to linear and nonlinear equations. Part One consists of five chapters. In the first chapter the general classes of cyclic functional equations in a complex vector form are considered. Among these are: the basic cyclic functional equation, the derived cyclic functional equation, the semicyclic functional equation, the special cyclic functional equation and the condensed cyclic functional equation. In the second chapter the complex vector functional equations with operations between arguments are considered. First an operator functional equation is solved. Also, in this chapter the generalized equation of operator vii

Vlll

Preface

type is solved. Next, some simple functional equations and functional equations with distinct functions are solved. Of this type of functional equations we can mention Frechet's functional equations, to which an adequate place in this chapter is given. Finally, some complex vector functional equations with two operations between arguments are also solved. T h e functional equations with constant parameters are considered in the third chapter. T h e equations of this type which are solved here are: the general parametric functional equation, the special parametric functional equation, the expanded parametric functional equation and the general expanded parametric functional equation. T h e linear complex vector functional equations with constant coefficients are considered in the fourth chapter by a new m a t r i x approach. In particular, we pay attention b o t h to homogeneous and nonhomogeneous functional equations with constant coefficients. Finally, a paracyclic functional equation with complex constant coefficients is solved. In the last fifth chapter of P a r t One two types of systems of functional equations are studied. First some systems in which each equation contains all unknown functions are solved, and then some systems in which not all equations contain all unknown functions. P a r t Two is also divided in five chapters which are described below quite briefly. C h a p t e r s 6, 7 and 8 are devoted respectively to quadratic, modified quadratic and expanded quadratic equations. Higher order complex vector functional equations are considered in the ninth chapter. Finally, in C h a p ter 10 systems of nonlinear (quadratic and higher order) complex vector functional equations are studied. T h e present monograph is intended for mathematicians, physicists and engineers who use functional equations in their investigations. T h e authors are well aware t h a t all aspects of the complex vector functional equations cannot be included in an adequate way in a newly written book, a n d they will be grateful for all remarks and suggestions which will be valuable for the correction and completion of a subsequent edition. T h e second author acknowledges partial support by G r a n t M M - 7 0 6 with the Bulgarian Science Fund. A part of the book was written during the work of the second author at Fatih University, Istanbul, Turkey.

Contents

Preface PART 1

vii Linear Complex Vector Functional Equations

Chapter 1 General Classes of Cyclic Functional Equations 1 Basic Cyclic Functional Equation 2 Derived Cyclic Functional Equation 3 Paracyclic Functional Equation 4 Semicyclic Functional Equation 5 Special Cyclic Functional Equation 6 Condensed Cyclic Functional Equation

1 3 3 6 29 65 69 72

Chapter 2 Functional Equations with Operations between Arguments 75 7 Operator Functional Equation 75 8 Generalized Functional Equation 81 9 Simple Functional Equations 94 10 Functional Equations with Several Unknown Functions 99 11 Frechet's Functional Equations 106 12 Functional Equations with Two Operations Ill Chapter 3 Functional Equations with Constant Parameters 13 General Parametric Functional Equation 14 Special Parametric Functional Equation 15 Expanded Parametric Functional Equation 16 General Expanded Parametric Functional Equation ix

121 121 135 142 161

x

Contents

Chapter 4 Functional Equations with Constant Coefficients 173 17 Matrix Equations 173 18 Homogeneous Functional Equations with Constant Coefficients 175 19 Nonhomogeneous Functional Equations with Constant Coefficients 190 20 Paracyclic Functional Equations with Constant Coefficients . . 197 Chapter 5 Systems of Linear Functional Equations 21 Systems in Which Each Equation Contains All Unknown Functions 22 Systems in Which Not All Equations Contain All Unknown Functions

205

PART 2

217

Nonlinear Complex Vector Functional Equations

205 214

Chapter 6 Quadratic Functional Equations 23 Simple Quadratic Functional Equation 24 Special Quadratic Functional Equation

219 219 227

Chapter 7 Modified Quadratic Functional Equations 25 Basic Quadratic Functional Equation 26 Permuted Quadratic Functional Equation 27 First Modified Quadratic Functional Equation 28 Second Modified Quadratic Functional Equation 29 Third Modified Quadratic Functional Equation

231 231 234 238 241 244

Chapter 8 Expanded Quadratic Functional Equations 247 30 Expanded Quadratic Functional Equations with Functional Arguments 247 31 Expanded Quadratic Functional Equations with the Same Signs between the Functions 251 32 Expanded Quadratic Functional Equation with Alternating Signs between the Functions 259 33 Generalized Quadratic Functional Equation 262 Chapter 9 Higher Order Functional Equations 34 Higher Order Functional Equation without Parameters 35 Higher Order Functional Equation with Complex Parameters 36 Nonlinear Operator Functional Equation

265 265 267 273

Contents

xi

Chapter 10 Systems of Nonlinear Functional Equations 37 Systems of Quadratic Functional Equations 38 Systems of Higher Order Functional Equations

285 285 311

Bibliography

319

Index

323

PART 1

Linear Complex Vector Functional Equations

Chapter 1

General Classes of Cyclic Functional Equations

In this chapter six general classes of cyclic partial linear complex vector functional equations are solved, namely the basic cyclic functional equation, the derived cyclic functional equation, the paracyclic functional equation, the semicyclic functional equation, the special cyclic functional equation and the condensed cyclic functional equation. The results given here were obtained in [I. B. Risteski (to appear A); I. B. Risteski and V. C. Covachev (2000); I. B. Risteski et al. (2000A)].

1

Basic Cyclic Functional Equation

First we will introduce the following notations. Let V be a finite dimensional complex vector space and let there exist a mapping / : Vn i-> V, Zj (1 < i < n) are vectors in V and C is a constant complex vector in the same space. We assume that Zj = (za(t),--- ,Zin(t))T, where zy(t) (1 < i < n) are complex functions and O = (0,0, • • • , 0) T is the zero vector in V. Now we will give the following results which were obtained in [I. B. Risteski et al. (2000A)]. Theorem 1.1 The general solution of the basic cyclic complex vector functional equation n

2__/f(Zi,Zi+i,---,Zi+n-i)

= o

4=1

3

( z n + j = Zj)

(i.i)

4

General Classes of Cyclic Functional

Equations

is given by / ( Z 1 , Z 2 , - - . , Z „ ) = F ( Z i , Z 2 ) - " ,Zn)-F(Z2,Z3,---,Zn,Z1) where F is an arbitrary Proof.

complex vector function

(1.2)

with values in V.

Prom Eq. (1.1) it immediately follows t h a t / ( Z i , Z 2 , • • • ,Z„) = —/(Z2,Z3,- • • ,Z„,Zi) —

/(Z3,Z4)- • • ,Zi,Z2) — • • • — /(Zra,Zi, • • • ,Z„_i)

and n

f(Zi,

Z 2 , • • • , Z „ ) = / ( Z i , Z 2 , • • • , Z „ _ ! , Zn) — / ( Z 2 , Z 3 , • • • , Zn, Z i ) +/(Zi,Z2, • • • ,Z„_i,Zn) - /(Z3,Z4,- • • ,Zi,Z2) + • • • +/(Zi,Z2, •• • ,Zn_i,Zn) - /(Z„,Zi, • •• ,Zn_2,Z„_i) = /(Zi,Z2,--- ,Z„_i,Z„) - /(Z2,Z3,--- ,Z„,Zi) +/(Zi,Z2,--- ,Zn_i,Z„) — /(Z2,Z3, • •• ,Z„,Zi) + / ( Z 2 , Z 3 , • • • , Z „ , Zj) - / ( Z 3 , Z 4 , • • • , Z i , Z 2 ) + / ( Z i , Z 2 , • • • , Z „ _ i , Z n ) — / ( Z 2 , Z 3 , • • • , Zn,

Z{)

+ / ( Z 2 , Z 3 , • • • , Z „ , Z i ) - / ( Z 3 , Z4, • • • , Z i , Z 2 ) +/(Z3,Z4,-- • , Zi,Z2) - /(Z4,Z5,-- • ,Z2,Z3) H +/(Zi,Z2, • •• ,Zn_i,Z„) — /(Z2,Z3, • •• ,Z„,Zi)

+/(z 2 ,z 3 , • • • ,z„,Zi) — /(z 3 ,z 4 , • • • ,Zi,z 2 ) + • • • + / ( Z „ - i , Z n , • • • , Z n _ 3 , Z n _ 2 ) — f(Zn,

Zi, • • • , Z„_2, Z„_i).

By denoting t h e sum of members with positive sign by nF(Zi, we obtain Eq. (1.2).

Z2, • • • , Zn)

Basic Cyclic Functional

Equation

5

On the other hand, every function of the form Eq. (1.2) satisfies the complex vector functional equation (1.1). • Theorem 1.2 The general solution of the basic cyclic complex vector functional equation n

^ / ( Z i . Z i + i , - - - ,Zi+p_i) = 0

(n>2p~l;Zn+i

= Zi)

(1.3)

i=\

is given by / ( Z i , Z 2 , • • • , Z„) = F(Z1,Z2,

••• , Z p _ x ) - F ( Z 2 , Z 3 , • • • , Z p )

(1.4)

tu/tere F is an arbitrary complex vector function with values in V. Proof. A direct calculation shows that every function F of the form Eq. (1.4) satisfies the functional equation (1.3). Now we will prove the converse, i.e. that / has the form Eq. (1.4). For Zj = C (1 < i < n) where C is a constant complex vector from V, the equation (1.3) gives nf(C,C,--,C) = 0,i.e. f{C,C,--- ,C) = 0. By putting Zj = C (p + 1 < i < n) into Eq. (1.3) we obtain / ( Z 1 , Z 2 , - - - , Z P ) + / ( Z 2 , Z 3 ) - - - , Z P , C ) + ---

(1.5)

+/(Zp,C,C,..-,C) + /(C,C,-.-,C,Z1) +f(C,C,---

,C,Z1,Z2)

+ --- + f(C,Zl,Z2,---,Zp-1)

= 0.

If we substitute Zp = C into the above equality, we obtain / ( Z 1 , Z 2 , - - - , Z P _ 1 , C ) + / ( Z 2 , Z 3 , - - - , Z P _ 1 , C , C ) + --+f(Zp-1,C,C,---,C) +f(C,C,---

+

(1.6)

f(C,C,---,C,Z1)

, C , Z i , Z 2 ) + --- + / ( C ) Z 1 , Z 2 > " - ,Z p _i) = 0 .

By subtracting Eq. (1.6) from Eq. (1.5), we get / ( Z i , Z 2 , • • • ,Z P ) = / ( Z i , Z 2 , • • • , Z p _ i , C ) — / ( Z 2 , Z 3 , • • • , Z P , C ) +

/ ( Z 2 , Z 3 , . . . , Z P _ 1 , C , C ) - / ( Z 3 , Z 4 , . . - , Z P , C , C ) + ---

+

/ ( Z P - i , C , C , • • • ,C) — f(Zp,C,C,

• • • ,C).

(1.7)

6

General Classes of Cyclic Functional Equations

By putting F ( Z i , Z 2 , - - - ,Z p _i) = f(Z1,Z2>+

f(Z2,Z3,---

,Zp_1,C,C)

•• , Z p _ i , C )

+ --- + f(Zp_1,C,C,---

,C),

the equality (1.7) takes the form Eq. (1.4).

•

Some particular cases of the functional equation (1.3) are considered in [J. Aczel et al. (1960); M. Ghermanescu (1940); M. Hosszii (1961)] under the hypothesis that the functions and the independent variables are real.

2

Derived Cyclic Functional Equation

Let V be a finite dimensional complex vector space and let there exist mappings ft : Vp >-» V (1 < i < k). Throughout this section Zj (1 < i < n) are vectors in V, and d are constant vectors in the same space. We may assume that Zj = (zji(t),--- ,zin(t))T, where the components Zij(t) (1 < i < n; 1 < j < n) are complex functions, and that O = (0,0, • • • , 0) T is the zero vector in V. Next we will give the following results obtained in [I. B. Risteski and V. C. Covachev (2000)]. T h e o r e m 2.1 The general solution of the derived cyclic complex vector functional equation n

2_jfi\^ii

Z*+i> • • • ; Zj + n _j) = O

(Zn+i = Zi, fn+i =

fa)

(2.1)

i=l

is given by fi(ZuZ2,..-,Zn) =

Fi(Z1,Z2,---

(2.2) ,Zn)-Fi+1(Z2,Z3,---

,Zn,Zi)

(l ^"i+n —1 j

+/j+n-p+l(Zj+n_p+l,Zj+n_p+2,'- • • ,Zj+n_i,Zj) + • • • +fi+n-l(Zi+n-l,

Z j , • • • , Zj+ p _2) = O .

By p u t t i n g Zi+P = Z j + p + 1 = • • • = Z i + n _ i = C in Eq. (2.6), where C is a fixed vector from V, we obtain /j(Zj,Zj+i,-• • ,Zj+p_i) + / j + i ( Z j + i , Z j + 2 , • • • ,Zj+p_i,C) + / t + 2 ( Z j + 2 , Z j + 3 , • • • , Z j + p _ i , C , C) + • • • + / i + p - i ( Z j + p _ i , C, C, • • • , C) 4- fi+p(C,

C, • • • ,C)

(2.7)

Derived Cyclic Functional Equation

+ft+p+i (C> C, • • • , C) + • • • + fi+n-p(C,

9

C, • • • ,C)

+fi+n-p+l (C, C, • • • ,C, Zj) + /j+„_p_|_2 (C, C, • • • , C, Zj,Zj+i) + • • •

+ /i+n-l(C, Zj,Zj + i, • • • ,Zj + p_ 2 ) — O. The substitution Zj+p_i = C in Eq. (2.7) yields i+1 j Z j + 2 , • ' • , Z j + p _ 2 , C,C)

+ -..

+/i+p-2(Zj+ p _2,C,C, • • • , C) + fi+p-i(C, C, • • • ,C) +fi+p(C,C,

• • • ,C) + • • • + fi+n-p(C,C,

iji+n—p+l

(2.8)

• • • ,C)

P+2(C,C, • • • ,C, Zj, Zj + i) + • • •

+fi+n-l{C ,?'i,'Z'i+l, • • • ,Zj + p_ 2 ) = O. By subtracting Eq. (2.8) from Eq. (2.7), we get the formula /j(Zi,Zj + 1 ,- • • ,Zj + p _i)

(2.9)

= /j(Zj,Z; + i, • • • ,Zj + p_2,C) — fi+l(Zj+i, Z i+ 2, • • • ,Z; + p_i,C) + /t+l(Zi+l,Zj+2, • ' • , Z; + p_2,C, C)-/i+2(Zj + 2,Zj + 3,- • • , Z,+ p _i, C, C) + + fi+p-2 (Zi+p-2 ,C,C,- • • , C) — fi+p-1 (Zj+p_i ,C,C,y

+/ i + p_ 1 (C,C,---,C)

• ,C)

(l Z i + f c + 2 - n , • • • i Z j + f c + p - n )

( —1)™ _ = < K

Zj = C; for i £ fc + 1, It + 2 , . . . , k + p (mod n)

-Fi+fc+l,n-i(Zfc+l, Zfc + 2, . . . , Z p+ fc_|_j,

Zi+fc+1,Zi+fc+2,...,Zp+t)

(-l)n-i+1Fi+k+ltn-i(Zk+1,

(i = n - k , n - k

+

Z f c + 2 , . . . ,Zp+k+i)

l,...,p-l),

(i=p,p+l,...,n-l).

Substituting Eq. (2.26) into Eq. (2.20) and introducing new functions by 9i = fi + (-l)1+k-iFi,1+k-i

(i = l,...,fc),

(2.27)

we obtain Eq. (2.19). On the basis of the inductive hypothesis and Eq. (2.27), the general solution of Eq. (2.20) is n—p

/ r ( Z i , Z 2 , . . . ,Zp) = 2 _ , ( - l ) * t=l

^rt(Zj+i,Zj+2, . . . ,Z p )

Derived Cyclic Functional

Equation

15

k—r +

(— 1 ) * _ • P ' r i ( Z t + l , Z i + 2 , . . . , Z p , Z i , Z 2 , • . . , Z p + i )

2^ i=n—p+1

+ ( — 1)

_r

-Fr,fc+l-r(Zfc+2-r, Zfc+3_r, . . . , Z p , Z i , Z2, . . . ,

Zp+k-r+l)

n—p +

(— l ) t _ -Fri(Zi+l,Zj+2, . . . , Z p )

2_j

i=n—r+1 p-1 +

(~l)"-t-Fi+r,n-i(Zl,Z2, . . . ,Zp+i,Zi+i,Zi+2, . . . ,Zp)

2_^

i=max(n—r+l,n—p+1) n-1

+

(-l) n _ i i ; i+r,n-i(Zi,Z2,...,Zp + i )

J2

(r = l , 2 , . . . , p + * - n ) ;

i=max(r»-r+l,p)

(2.28) k-r /r(Zi,Z2,...,Zp) = ^ ( - 1 ) * i=l

+ ( — 1)

r ;l

i

_

Fri(Zi+l,Zi+2,.

r)fc+i_r(Zfc+2-r,

. . , Zp)

Zfc+3_r, . . . , Zp)

n—p +

2 ^ (— 1 ) * ~ i=n—r+1

•Fri(Zi+l,Zj+2,...,Zp)

p-1 +

(-l)"_t-fi+r,n-i(Zi,Z2,. . . ,Zp+j,Zi+i,Zi+2, . . .,Zp)

^ ,

i=max(n—r+l,n—p+1) n-1

+

13

(-l) n ~ i i r i+r,n-i(Z i + 1 ,Z i + 2 ,...,Zp)(r=p+fc-n + l,...,A;).

i=max(n—r+l,p)

On the basis of Eqs. (2.26) and (2.28), the general solution of Eq. (2.20) is determined by Eq. (2.15), where fc must be replaced by k + 1.

General Classes of Cyclic Functional

16

Equations

Therefore, forn— p+1< k < p the theorem holds for k + 1 if it holds for k, i.e. the theorem is true for all such k, and also for k — p. 3° Let p < k < n - 1. For Zt = d when i ^ k+l,k (mod n) Eq. (2.20) becomes

+ 2,

...,k+p

n—p

/*+i(Z*+i, Zk+2, • •., Zfc+p) = 2_^(—l)1

Fk+iti(Zi+i, Z j + 2 , . . . , Z p ) (2.29)

i=n—k p-1 + 2 ^ (_l)n_lfi+fc+l,n-i(Zfc+l,Zfc+2, . . .,Zp+fc+i,Zj+fc+1,Zj+fc+2, . . i=n—p+1

.,Zp+k)

n-1 + / ^,( — 1 ) " *-Pi+fc+l,n-t(ZAi + l , Z f c + 2 , . . . , Zp+fc+j), i=p

where we have introduced the notation fi+k+l-nC^i+k+l-n,

Zj+fc+2-rai • • • > Z j + f c + p _ n )

Z; = C, for t ^ fc + 1, fc + 2,. .. , k + p (mod n)

' (-iyFk+iti(Zi+i,Zi+2,...,Zp) =
1 + f c _.

(i = l,...,k + l-p), (i = k + 2-p,...,k).

{2 di})

-

If we substitute the function fk+x determined by Eq. (2.29) into Eq. (2.20), in view of Eq. (2.30) we obtain the functional equation (2.19). On the basis of Eq. (2.30) and the general solution of the functional equation (2.19), we obtain that the general solution of the functional equation (2.20) is determined by n—p

fr{Z\, Z2, . . . , Z p ) = 2_j(~iy j=l

-Pr«(Zj+l, Zj+2, . . . , Z p )

Derived Cyclic Functional Equation

17

p-l

+ J2 (-l) i_1 F ri (Z i+i,Zi_|_2)- •• , Z p , Z i , Z 2 , . . .

,Zp+i)

&—r + ^^(~"^-)n

l

^ + r , n - i ( Z i , Z 2 , . . . , Zp-j-j)

i=p

+ (— l ) n

r

-ffc,n-fc+l-r(Zi, Z2, • • • , Z p + & _ r )

n—p +

^

("I)*

-Pri(Zi+i,Zi+2, • •• , Zp)

i=n—r+1 p-i +

(— 1 ) "

2 ^

!

-Pi+r,n-i(Zl, Z2, . . . , Z p + j , Z j + i , Zj+2j • • • , Z p )

i=max(n—r+l,n—p+1) n-1

+

(-^n~iFi+r,n-i(Zi,Z2,...,Zp+i)(r

J2

=

l,2,...,n-k-p+l);

i=max(n—r+l,p)

(2.31) n—p / r ( Z l , Z 2 , . . . , Zp) = 2 ^ ( - l )

l _

Frt(Zi+l,Zj+2, •••, Zp)

i=l fc-r +

2^,

("•'•)'

•Pri(Zi+l,Zj+2, . . . , Z p , Z i , Z 2 , . . . , Z p + j )

i=n—p+1

r

+ ( — 1)

F,.,fc_,.+i(Zk_r_|_2, Zfc_r+3, . . . , Z p , Z i , Z 2 , . . . , Z p + ^ _

r +

i)

n—p +

^

(— I ) '

-Fri ( Z j + 1 , Z j + 2 , . . . , Z p )

i = n —r+1 p-l +

2-^i

(—-U

i=max(n—r+l,n—p+1)

-fi+r,n-j(Zl, Z2, ..., Zp+j, Z j

+ 1

, Zj+2, ..., Zp)

18

General Classes of Cyclic Functional

Equations

n-1

+

(-l)n~iFi+r,n-i(Z1,Z2,...,Zp+i)(r

Yl

=

n-k-p+2,...,p-l);

i=raax(n- r+l,p) k—r / r ( Z i , Z 2 , . . . , Z p ) = 2_^(~^~ i=l + ( — 1) ~TFrtk+l-r(Zk+2-r,

Fri(Zi+i,Zi+2,

Zk+3-r,

•••

,Zp)

• • • : Zp)

n—p

+

2_^ (—l) t_ FTi(Zi+i,Zi+2,...

,Z P )

i=n—r+1 P-l + ^L/ ( _ 1)™ i=max(n—r+l,n—p+1)

l

-Fi+r,n-t(Zl,Z2, . . . ,Zp+i,Zj_|_i,Zi+2,. . . ,Zp)

n-1

+

X)

(-l)n_ii?i+r,n-i(Zi,Z2,...,Zp+i)

(r = p , p + l , . . . , A ) .

i=max(n—r+1 ,p)

On the basis of Eqs. (2.29) and (2.31) we obtain that the general solution of the functional equation (2.20) in the case p < f c < n - l i s determined by Eq. (2.15), where k must be replaced by k + 1. • Therefore, the solution of the research problem given in [D. S. Mitrinovic and D. Z. Djokovic (1963)] is presented by this theorem. As particular cases see the results given in [M. Hosszii (1961); D. S. Mitrinovic (1963B)]. Example 2.1 For n = 8, p = 5 and k = 6 the complex vector functional equation Eq. (2.14) becomes / l ( Z l , Z 2 , Z 3 , Z 4 , Z 5 ) + /2(Z2,Z3,Z 4 ,Z5,Z 6 ) + /3(Z3,Z4,Z 5 ,Z 6 ,Z 7 ) +fi{Zi,

Z5, ZQ, Z7, Zg)+/5(Z5, Z6, Z7, Zg, Zi) + /e(Z6, Z7, Zg, Zi, Z2) = O,

whose general solution is given by /l(Zl,Z2,Z3,Z4,Z5) = Fn(Z2,Z3,Z4,Z5) - .^12^3, Z4, Z5) +Fi 3 (Z4,Z 5 ) — Fi4(Z5,Zi) - F 6 3(Zi,Z 2 ),

Derived Cyclic Functional Equation

19

/ 2 ( Z i , Z 2 , Z 3 , Z 4 , Z 5 ) = i*2i(Z2,Z3,Z4,Z 5 ) - F 2 2 ( Z 3 , Z 4 , Z 5 )

+F 2 3 (Z 4 , Z5) - F 2 4 (Z 5 , Zi) - F n ( Z i , Z 2 , Z 3 , Z 4 ), /3(Zl,Z 2 ,Z3,Z4,Z 5 ) = i 7 3 1 (Z 2 ,Z3,Z4,Z 5 ) — F 32 (Z3,Z4, Z 5 ) +F 3 3 (Z4,Z 5 ) + F 1 2 (Zi,Z2,Z 3 ) - i r 2i(Zi,Z2,Z3,Z 4 ),

/4(Z 1 ,Z2,Z3,Z4,Z 5 ) = F4i(Z 2 ,Z3,Z4,Z 5 ) - F4 2 (Z 3 , Z4, Z 5 ) - F i 3 ( Z 1 , Z 2 ) +F22(Zi,Z2,Z 3 ) - F3i(Zi,Z 2 ,Z3,Z4),

/5(Zl,Z2,Z 3 ,Z4,Z5) = F5i(Z2,Z3,Z4,Z 5 ) + Fi4(Zi,Zs) — F 2 3(Zi,Z 2 ) + F32(Zi,Z 2 ,Z3) — F4i(Z 1 ) Z2,Z3,Z 4 ), /6(Z 1 ,Z 2 ,Z3,Z4,Z 5 ) = F 6 3(Z 4 ,Z 5 ) + F24(Zi,Z 5 ) — F33(Zi,Z 2 ) + -F 4 2(Zi,Z2,Z3) — F5i(Zi,Z 2 ,Z3,Z4), where Fij are arbitrary complex vector functions from V. Now we will give two particular cases of the above theorem. Theorem 2.4 The general solution of the derived cyclic complex vector functional equation n

£)/i(ZilZj+1>...,Zi+p_1) = 0

( p < n < 2 p - l ; Z n + j = ZO

(2.32)

is given by n—p

/ r ( Z i , Z 2 > . . . , Z„) = ^ ( - l J ' - ^ Z i + i , Z i + 2 , . . . , Z p ) »=i

(2.33)

General Classes of Cyclic Functional

20

Equations

min (n—r,p—1)

+

2^i

(~^y

^ r i ( Z j + l , Z j + 2 , . . .,Z p ,Zi,Z 2 , . . .,Zp+i)

i=n—p+l p-1

+

(-l) n _ 1 -fi+r,n-i(Zl,Z 2 , . . .,Z p+ j,Zj + i,Zj_|_2, . . .,Z p )

2_j

i=max (n—r+l,n—p+1)

+ X>l) n ~ i i r «+r,n-*(Zi>Z2,...,Z p+i )

C1 ^ r ^ n )>

i=p

Wiere F^ are arbitrary complex vector functions from V. Proof. The proof of this theorem immediately follows from the previous theorem for k = n. • This theorem generalizes the results given in [D. Z. Djokovic (1964)]. Theorem 2.5 The general solution of the basic cyclic complex vector functional equation n

53/(Zi>Zj+i,...,Zj+p_i) = 0

(p < n< 2p - 1; Zn+i = Zt)

(2.34)

is given by /(ZllZ2,...,Zp)=Fo(Zl!Z2,...,Zp_1)-Fo(Z2jZ3j...,Zp)

(2.35)

p-[n/2]

+ 2_^

[-fi(Zi,z2,...,Zj,zn_p+j+i,...,zp)

— F!i(Z p _j + i,..., Z p , Zi, Z 2 , . . . ,Z 2 p _ n _j)J, where Fi (0 < i < p— [n/2]) are arbitrary complex vector functions from V. Proof. By summing up the functions fi (1 < i < n) determined by Eq. (2.33) and putting / i = f2 = ••• = fn = f, we obtain Eq. (2.35), where

Derived

Cyclic

Functional

Equation

21

we have introduced the following notations 1

F 0 ( Z i , Z 2 , . . . ,Z p _i)

=

G>(Zi,Z 2 , . . . , Z p _ r )

_

n—p n-p

rr

—y n

^ Gr(Zt,Zi+i,... ,Zp_r_i+i),

r=l

i=l

/

y(-l)

r r

i ir(Zi,Z2,. • . ,Zp_r),

i=l

•Fi(Zl,Z2,. . . , Z j , Z n _ p + i + i , . . . , Z p ) n—p+» ( ~ l ) i + 1 " """ / y •Pr,p-t(Zl,Z2,. . . , Z j , Z n _ p + j + i , . . . ,Zp) n p—i

— ^•FV,n-p+i(Z n -p+i+l,. . .,Zp,Zi,Z2,. . . ,Zj) r=l

(1 "p—mi

— Frm(Zm+i,...

^ m + l i • • • j ^p)

, Z p , Z i , . . . , Z p _ m ) (1 < r < m). D

We have noticed that in [P. M. Vasic and R. Z. Djordjevic (1965)] special generalized cases of Eqs. (2.14), (2.32) and (2.34) are considered. They are solved in a complicated manner using a cyclic operator. At the end of the present section we give two more general theorems obtained in [I. B. Risteski (to appear A)]. Theorem 2.6 The general solution of the derived cyclic complex vector functional equation n

/

y/i(Zi,Zt+1,...,Zj+p_1)

= O

( Z n + i = Zj)

(2.36)

22

General Classes of Cyclic Functional

Equations

is given by / r ( Z r , Z r + i , . . . ,Z r + p _i)

(2.37)

r-l

— 2_/_1)r

^ J > ( { Z r , Z r + i , . . . , Z r + p _ i } n {Zj,Zj+i,.

..

,Zj+p-i})

i=i n

+

( - l ^ ^ ^ j Z r j Z r + i , . . . , Z r + p _ i } ft { Z j , Z j + i , . . . , Z j + p _ i } )

^ j=r+l

(1 < r < n), luftere F r j (1 < r < n - 1, r + 1 < j < n) are arbitrary complex vector functions from V such that

}n{z j( Z j + i » •

• • > Zj+p—i)) — A r j

{z r ,z r + i,... ,z r+p _i} n {Zj,Zj+i,... ,Zj+p_i} = 0, V

where Arj are constant vectors from V and ^ = O for a > v. a

Proof. We will prove the assertion of the theorem by mathematical induction. For n — 2, the equation (2.36) becomes /1(Z1,Z2,---,Zp) + /2(Z2)Z3,---)Zp+1) = 0 .

(2.38)

Putting Zi = Ci into the equation (2.38), we get /2(Z 2 ,Z3, • • • ,Z P + 1 ) = - F i 2 ( Z 2 , Z 3 , • • • ,Z P )

(2.39)

where the following notation is introduced Fi 2 (Z 2 ,Z3, • • • ,Z p ) = / i ( C i , Z 2 , • • • ,Z P ). If we put Eq. (2.39) into Eq. (2.38), we get / i ( Z i , Z 2 , • • • , Zp) = F 1 2 (Z 2 , Z 3 , • • • , Z p ).

(2.40)

Therefore, for i = 1,2 from Eq. (2.37) we obtain Eqs. (2.40) and (2.39), respectively, which means that the theorem holds for n = 2.

Derived Cyclic Functional

23

Equation

For some fixed n let us suppose that the general solution of the functional equation (2.36) is given by Eq. (2.37). Now, we will consider the functional equation n+l

^fli(Zi,Zi+1,---,Zi+p_1) = 0.

(2.41)

t=i

If we put Zj = Ci (1 < i < n) into Eq. (2.41), we obtain that the function gn+i has the following form • • • , Z n + p )

(2.42)

n

=

/ ,(-l)"-Fj,n+l({Z n +l|Z n + 2,- •• ,Zn+p}n{Zj,Zj+i,---

,Zj+p_i}).

3=1

Substituting Eq. (2.42) into Eq. (2.41) and introducing the new notations /i(Zj,Zj+i, • • • , Z; + p _i) = z j + i > ' " > z i+P-i}) j=i+l — (-l)nFitTl+i({Zn+i,Zn+2,-

• • ,Zn+p} (1

X p , Y i , Y 2 , . . . , Y j + 9 ) ft-r 1

Pi+r,n—i(Xi,X2,..

. , Xj+p, Y i , Y 2 , . . . , Yj+9)

n-1 +

2^i ("I)" *-f1»+r,n-t(Xi,X2,...,Xj+p, Yi, Y2,. . . , Yi+9) t=max (p,n—r+1)

(1 < r < jfe),

Paracyclic Functional Equation where F^ are arbitrary complex vector functions T h e o r e m 3.4 general solution

Ifq

^ri(Xj+i,Xj+2,- • • ,X p )

i=max (q,n—r+1) p-1 +

(-l)7l~*-Ft+r,n-i(Xi,X2,. . . ,Xi+p,Xi+i,Xi+2, ••• ,Xp)

2^

i=max (n—p+l,n—r+1) n—q

+

2_^i

(~1)" '-PVfr.n-i(Xi,X2,. .. ,Xj+ p )

i=max (p,n—r+1) n-1 + 2^ ( _ 1)" *-Fi+r,n-i(Xi,X2, . • . , X i + p , Y i , Y2, . . . , Yj+9) i=max (n—g+l,n—r+1)

+ ( — 1)

r r

i rfc+i_r(Xfc_r+2,Xfc_r+3) • • • >Xp, Yfc_r+2, Yft_r+3, . . . ,

Yq)

k+l-r —

(— I ) '

2L,

^rri(Xj+i,Xj+2,. . . , X p , Y j + 1 , Yj+2, . . . ,

Yq)

t=l

q-1 +

2^

(— 1 ) * ~ • f r t ( X j + i , X i + 2 , . . . , X p , Y i + i , Y j + 2 , - " )

Y9)

i=n—r+1 n—p +

2^ ( —1)*~ - F r t ( X j + i , X j + 2 , . . . , X p ) i=max (q,n—r+1)

p-1 + 2 ^ (—l)n_t-Pi+r,n-i(Xi,X2, . . . ,Xj+p,Xj+1,Xt+2) ••• ,Xp) i=max (n—p+l,n—r+1)

38

General Classes of Cyclic Functional Equations

• P i + r , n - i ( X i , X 2 , . . . , Xj+p) i=max (p,n—r+1) n-1

+

( _ 1)™ l -Fi+r,n-i(X 1 ,X 2 ) . . . , X i + p , Y i , Y 2 , . . . , Y i + 9 )

2^

i=max (n—X p , Y j + i , Yj+2, . . . , Yq) i=n—r+1 n—p

+

2__,

( - I ) * " •fri(Xi + i,Xj + 2, . . . ,X p )

i=max (q,n—r+1) p-1 + 2^, ( —l)n_t-Fi+r,n-i(Xi,X2,. . . ,X,+p,Xj+i,Xt+2, . . • ,Xp) i=max (n—p+1,n—r+1)

+

2^,

(~1)" l -Fi+r,n-i(Xi,X2, . . . ,Xj+p)

i=max (p,n—r+1)

42

General Classes of Cyclic Functional

Equations

n-1 +

2^i (— 1 ) " ' • P « + r , n - i ( X i , X 2 , . . . , X j + p , Y i , Y 2 , . . . , Y j + g ) i=m ax (n—g+1, n—r+1)

(1 < r < fc+p-n); /r(Xi,X2, . . . ,X p , Yi, Y2, . . . , Y,) g-i =

E^-^)'

^ri(Xj+i,Xj+2, . • . ,X p , Yj+i, Yj+2,. . . , Yq)

t=l

A:—r +

^^(

—

1)'

^ri(Xj+i,Xj+2,...,Xp)

9-1

+

(-l)

E

l _

-Fri(Xi+i,Xi+2,...,Xp,Yi+i,Yi+2,..., Y,)

i=n—r+1 n—p + 2__, { — 1)* i=max (q,n—r+1)

-Fr«(X{+i,X;+2, • • • , X p )

p-1 +

^_,

( — 1)™ ' • F i + r , n - i ( X i , X 2 , . . . , X i + p , X j + i , X i + 2 , . . . , X p )

i=max (n—p+l,n—r+1) n—g + E (*"•'•)" i=max (p,n—r+1)

l

-^i+r,n-i(Xi,X2,. .. ,Xj+p)

n-1 i

Fi+r,n-i(X\,

X 2 , . . . ,Xj_|_p, Y i , Y 2 , . . . , Y j + g )

i=max (n—g+l,?!—r+1)

+ ( — 1)

_r

-Fr,A+l-r(Xj;+2_r, Xfc+3_r, ..., X p )

(r = k + p - n + 1,... ,k - q + 1);

Paracyclic Functional Equation

43

/ r ( X 1 , X 2 , . . . ,Xp, Y i , Y2, . . . , Yg) k—r ^ ( — I )

=

1

•PVt(Xi+i,Xi+2,... , X P , Yj+i, Y j + 2 , . . . , Yg)

9-1

+

2^i (~^Y i=n—r+1

-Pri ( X j + i , X j + 2 , . . - , X p , Y j + i , Y j + 2 , . . . ,

Yq)

n—p +

(—1)'~ •fri(Xj+i,Xj+2, . . . , X p )

2_^ i=max (g,n—r+1) p-1

+

^ ,

(~1)

- P i + r . n - i ( X i , X 2 , ••• , X j + p ,

i=max (n—p+l,n—r+1) X i + i , X j + 2 , . . ., X p ) n—q

+

2~2

(-1)"" 1

Fi+r,n-i(X-i,

X2, . . . , Xj+p)

i^max (p,n—r+1) n-1 2_^i ( — 1)™ ' • f i + r , n - i ( X i , X 2 , . • . , X j + p , Y i , Y 2 , . . . , Y j + g ) i=max(n—q+l,n—r+1) + ( — 1) r F r ] f c + i _ r ( X f c + 2 - r , X f c + 3 _ r , . . . , X p , Y f c + 2 _ r , Y f c + 3 _ r , . . . , Yq) +

(r = k-q

+

2,...,k).

We can write the above equalities in a general form / r ( X i , X 2 , . . . , Xp, Y i , Y2,. . . , Yq)

(3.19)

min (q—l,fc+l—r) =

2_^

(— I ) *

•Pri(Xi+i,Xi_(-2, . . . , X p , Y j + i , Yj+2) • • • ; Y g )

i=l min (n—p,fc+l — r) +

^

+

fc+l-r 2 ^ (— ^ ) * i=n—p+1

(_1)J

^ri(Xi+i,Xi+2,...,Xp)

-fri(Xi+i,Xi+2,. . . , X p , X i , X 2 , • . . ,Xj+p)

44

General Classes of Cyclic Functional Equations

9-1 +

( _ I)'"

2^i

-^rt(Xj + i,Xj+2,- . . , X p , Y j + i , Y j + 2 , . . . , Yq)

i—n—r+1 n—p +

(_1)'

2^,

i=max

_

^r»(Xi+i,Xj+2, •.. ,Xp)

(q,n—r+1)

p-1 +

^ ^

(""I)"

' • P ' » + r , n - i ( X i , X 2 , . . . ,Xj_|_p,

t=max (n—p+l,n—r+1) Xj+i, Xj+2 > • • •j Xp)

n—q +

2 ^

(—1)"

*-Fi+r,n-i(Xi,X2,. . . ,Xj+p)

i = m a x (p,n—r+1) n-1 * - f i + r , n — t ( X i , X 2 , • • • , X i + p , Y i , Y 2 , . . . , Y^_j_g) i = m ax ( n—^4-1, n-^-l)

(1 < r < k). On the basis of the equalities (3.18) and (3.19), the theorem holds for this case as well, and also for k = p. 4° Let p < k < n - q + 1. Putting Eq. (3.12) into Eq. (3.11), we obtain /fc+i(Xjfc+i,XA;+2,... ,Xp + f c , Yfc+1, Yk+2,...

,Yq+k)

n—p =

/\_^ (~I)*

-pA+l,i(Xj + ft +1 ,Xj + fc + 2, . . . ,Xp+fc)

i=n—k p-1 " J -Fi+*+l,n-i(X Xp) n—9 +

2^

(—1)" '•fi+r,n-t(Xi,X2, ••. ,Xj+p)

i=max (p,n—r+1) n-1 A-i+p, i=max(n—q+l,n—r+1) Yi, Y2,. .. , Yj+,)

54

General Classes of Cyclic Functional

Equations

(r = p - n + k + 1,..., k - q + 1); / r ( X i , X 2 , . . . ,X p , Yl, Y2,. . . , Yq) k—r ~

^(~1)

1

^ r t ( X i + l , X j + 2 , . . • , X p , Y j + i , Y i + 2 , . . . , Yq)

i=l r

+ ( — 1)

Fr^k-r+\ (Xfc+2-D Xft+3_r, . . . , X p ,

Yfc+2-n Yfc+3_r, . . . , Yq)

9-1

+

2^i (~tyl i=n—r+1

-Pri(Xi + i,X i+ 2,. . . ,X p , Y i + 1 , Yj + 2,. . . , Yq)

+ 2^, ( —I)* i=max (?,n—r+1)

•fri(Xi+i,Xj+2I-• • ,Xp)

p-1 + 2_^ \~~ 1)™ i=max (n—p+l,n—r+1)

^i+r,n-i(Xi,X2,. ..,Xj+p,Xi+i,Xj+2i ••• |Xp)

n—q '•Pi+r,n-i(Xi,X2, . . . , Xj+p) i=max (p,n—r+1) n-1 +

2^i

(~^)

l

-^i+r,n-t(Xi,X2, .. . ,Xj+p,

i=max(n—g+l,n—r+1) Y l , Y 2 , . . . , Y'j+g)

(r =fc-g+ 2,...,fc)We can write the previous equalities in the following general form /r(Xi,X 2 ,...,Xp,Yi,Y2,.. -,Y ? )

(3-24)

min(q>—l,fc+l—r) =

2-j i=l

(_1)*~ •fri(Xi+i,Xj+2,...,Xp, Yj + i, Yi+2,..., Y g )

Paracyclic Functional Equation

55

min (fc+1—r,n—p)

+

i-^)1

E

Fri(Xi+ i , X j + 2 ) . .. ,X p )

i=g min (fc+1—r,p—1) +

E i=n—p+1

(~^)*

^ri(Xj+i,Xi+2, . . . ,Xp,Xi,X2, . . . ,Xj+p)

min (fc+1—r,n—q)

+

( _ l ) n _ t - f i + r , n - i ( X i , X 2 , . . . ,X i + p)

E

fc+l-r +

E

( — 1 ) " ' • ^ 1 * + r , n - t ( X i , X 2 , . . . , X i + p , Y i , Y j , . . ., Y j + g )

i=ra—g+1

+

9-1 E ( — •'•)' i=n—r+1

^ ? r t ( X i + i , X i + 2 , . . . ,Xp, Y j + i , Yj+2, . . . ,

Yq)

n—p + E ( —•*•)* i=max (g,n—r+1)

•FVi(Xi+i,Xj+2, . . . , X p )

p-1 +

E ,

(— 1 )

-Pi+r,n-i(Xi,X2, . . . ,Xi+p,Xj+i,Xj-|-2,. . . ,Xp)

t=max (n—p+1,n—r+1) n—q *-fi+r,n-i(Xi, X2, . . . , Xj+p) i=max (p,n—r+1) n-1 •^i+r,n-i ( X i , X 2 , . . . , X j + p , Y j , Y 2 , . . . , Y j + g ) i=max(n^-g+l,n—r+1)

(1 < r < fc). Therefore, the theorem holds for n - q + 1 < k < n.

•

General Classes of Cyclic Functional Equations

56

Now we will solve two particular cases of the equation (3.1). a) By p u t t i n g k = n into equation (3.1), we obtain the functional equation n

5 ^ / j ( X i , X j + i , . • • ) X j + p _ i , Y j , Y j + i , . . . , Yi+q-i)

= O.

(3.25)

Therefore, if we put k = n into Eqs. (3.2), (3.3), (3.4), (3.5) a n d (3.6), t h e n we obtain the general solution of the functional equation (3.25) in the cases considered. For example, if we put k = n into Eq. (3.4), we obtain t h a t t h e general solution of the functional equation (3.25) for q < p < n < 2q — 1 < 2p — 1 is given by the formulae / r ( X i , X 2 , . . . , X P , Y i , Y 2 , . . . , Yq) =

n—p ^(~1)

1 -

Fri(Xi+i,Xi+2,

(3.26)

• • • ,XP, Yj+i, Y i + 2 , . . . , Y,)

t=i n—q

+ J2 (-l)i_1-Fri(X i+ljXi_|_2

3

• - • )Xp,

i=n—p+1

X i , X 2 , • • • ,Xp+», Y i + i , Yj+2, • • • , Yq) min (n—r, g—1)

» + i ) X j + 2 , . . . , X p , X i X 2 , . . . , Xp+j, i=n—q+l Y»+i, Yj+2j • . • , Yq, Y i , Y 2 , . . .

,Yg+i)

9-1

+ 2_j ( — 1) i=max (n—q-\-l,n—r+1)

t

- f i + r , n - i ( X i , X 2 , . . . ,Xp+j,Xi_|_i,Xj_|-2, . . . , X p ,

Y l , Y 2 , . . . , Y g + j , Y j + i , Y j + 2 , . . . , Yq)

P-l l

Fi+r,n-i ( X i , X 2 , . . . , Xp+j,

X j + l , X j + 2 , . . . , X p , Y i , Y2, . . .

,Yq+i)

Paracyclic Functional Equation

%

+ Z~/(~*)"

Fi+r,n-i(X-i,X.2,

57

• • • , X j + p , Y i , Y2, . . . , Yg+j)

i=p

(1 < r < n), where Fy are arbitrary complex vector functions from V. The functional equation (3.1) for q < p < 2q - 1 < 2p - 1 < n had not been previously investigated, but for this case we must additionally determine the general solution of the equation (3.25). Now we will give the following result which treats a more general case than the previous one. Theorem 3.6

The general solution of the functional equation (3.25) for

—-— > max (p, q) is given by the formulae Li

/ r ( X i , X 2 , . . . , X P , Y i , Y 2 , . . . , Yq) =

(3.27)

r

i r(Xi,X2,... ,Xp_i, Yi, Y 2 , . . . , Yg_i)

— ^r+l(X2,X3,. . . , X p , Y 2 , Y 3 , . . ., Yq) (l • • • > * i+q—pi

* n — p + i + l > * n—p+i+21

• • • j I 9J

— Fj(Xp_i+i, Xp_j+2, . . . , X p , X i , X 2 , . . • , X 2 p _ n _ j , * p—i+1;

* p—i+2> • • • ) I g j ' l ) * 2 ) . . . j

1 p + « — n—ijj >

w/iere Fj (i = 0 , l , . . . , p — [n/2]) are arbitrary complex vector functions from V. Theorem 3.10 The general solution of the functional equation (3.36) for q < p t(i), Zt(2), • • •, Zt(fc_i)) = - 5 J JrpCZtih), Z t (j 2 ),..., Z t ( i m ) ) ,

(4.8)

where J r p ( Z t ( i l ) , Z t ( i 2 ) , . . . , Z t ( i m ) ) (m < k - 2) is obtained from G>p(Zrp(1), Z r p ( 2 ), • • •, Z rp ( fc _ : )) by putting Z{ - C for all i but t(l),t(2), ... ,t(k — 1). The sum on the right-hand side of Eq. (4.8) is extended over certain (not all) pairs of indices r and p. Consider a certain summand JroPo on the right-hand side of Eq. (4.8). Let wo 6 S% and VQ € Sj^_1 be such that UQVQ = *• We can construct a sequence of ordered pairs (U0,VQ),

( « 0 , W o ) , ( U l>^l)> ( " l . ^ l ) . •••>

which satisfy the following conditions

(uq,wq)

68

General Classes of Cyclic Functional Equations

2° («„«;,) = (r0,po); 3° Ui_iiOi_i = UjUj (1 < i < q); 4° the sequence UiiUj(l),... ,UiWi(k — 1) (0 < i < g) contains the sequence t(i\),..., t(im) as a subsequence. Let us put

We note that the representation Eq. (4.4) is still valid with G*. tV. and G*. <Wi instead of GUi >Vi and GUi tWi, respectively. We have Hf = Ht + JroPO. On the other hand, if t £ P", i.e. Ht = O, then also Hf = O. If the same procedure is applied to all summands of the right-hand side of Eq. (4.8), we conclude that the new function Ht is identically the zero vector. This contradicts the minimum property of the number s. Hence, s = 0 which proves the theorem. • This theorem generalizes the results given in [D. Z. Djokovic (1965A)]. E x a m p l e 4.1

If n = 5 and k = 4, the equation (4.1) is

/(Zi,Z2,Z3,Z4) + +

0(Zi,Z 2 ,Z3,Z 5 ) + /i(Zi,Z 2 ,Z4,Z 5 ) i(Z1}Z3,Z4,Z5)+j(Z2,Z3,Z4,Z5)

= O.

Its general solution is given by /(Zi,Z2,Z3,Z4)

= +

/i(Zi,Z2,Z3) +/2(Zi,Z2,Z4) /3(Zl,Z 3 ,Z4) + / 4 ( Z 2 , Z 3 , Z 4 ) ,

g{ZuZ2,Z3,Z5)

=

5l(Z1,Z2,Z3)+52(Z1,Z2,Z5)

+

5 3 (Zi,Z 3 ,Z 5 ) + 5 4 ( Z 2 , Z 3 , Z 5 ) ,

=

/ix(Zi,Z 2 ,Z4) + /i 2 (Zi,Z 2 ,Z 5 )

+

/ i 3 ( Z i , Z 4 , Z 5 ) + /i 4 (Z 2 ,Z 4 ,Z 5 ),

=

i!(Zi,Z 3 ,Z4) + i 2 ( Z i , Z 3 , Z 5 )

+

i3(Zi,Z 4 ,Z 5 ) + i 4 ( Z 3 , Z 4 , Z 5 ) ,

=

j1(Z2,Z3,Z4)+j2(Z2,Z3,Z5)

+

j3(Z2,Zi,Z5)

/i(Zi,Z2,Z4,Z5) z

»( ijZ3,Z4,Z5)

j(Z2,Z3,Z4,Z5)

+

u(Z3,Z4,Z5),

Special Cyclic Functional Equation

69

where / i ( Z i , Z 2 , Z 3 ) + 9i (Zi

Z3)

/ 3 ( Z i , Z 3 , Z 4 ) + /n(Zi / 4 (Z 2 ,Z3,Z 4 ) + j i ( Z a

z4) = o, z4) = o, z4) = o,

g2(Z1,Z2,Z5)

ZB)

/3(Zi,Z3,Z4) + n(Zi + /u(Zi

=

=

O,

O,

93(Zi,Z3,Z 5 ) + i 2 (Zi

z5) = o,

94(Z2,Z 3 ) Z 5 ) + J2(Z 2

ZB)

=

O,

/i 3 (Zi,Z 4 ,Z 5 ) + i 3 (Zi

Z5)

=

O,

/i 4 (Z 2 ,Z 4 ,Z 5 ) + j 3 ( Z 2

Z5)

=

O,

j4(Z3,Z4,Z5) + j4(Z3

ZB)

=

O.

Hence we may take / i , / 2 , / 3 , / 4 ,92,93,94 /i 3 , hi, i 4 to be arbitrary complex vector functions from the complex vector space V and 91 = - / l ,

5

/*1 = -J*2, /i2 = - 9 2 ,

»1 = ~ / 3 i ii = -93,

jl = ~/4, h = ~94,

h = -h,3,

J3 = -hi, ji = -u-

Special Cyclic Functional Equation

The notations for the vectors in this section are the same as in Sec. 1. Let V be the vector space and let there exist mappings fi-.Vi+1^V

(lZ2n+2)-

This theorem generalizes the result given in [R. Z. Djordjevic (1965A)].

6

Condensed Cyclic Functional Equation

Consider the condensed cyclic complex vector functional equation h

E ^ Z i ( i ) > Z i ( 2 ) > - - - > Z i ( S i ) ) = °>

(6-1)

j=l

where 1 < st < n, fc : VSi •-> V (1 < i < k), ~i(v) 6 {1,2,...,n} 1ZT2(2)'---'Z12(t12))'

/2(Z2(l)»Z2(2)»---'Z2(s2))

=

•Pl2(Z12(l)'Z12(2)'---'Zl2(ti2))

for an arbitrary function F\i : V*12 •-> V, i.e., for k = 2 the general solution of Eq. (6.1) is given by Eq. (6.2). For some fixed k suppose that the general solution of the functional equation (6.1) is given by the formula Eq. (6.2). Now, let us consider the equation fc+i

E/«( z ;(i)> z ;(2).---> z 7( Si )) =

a

(6-3)

i=l

If we put Zj = d for i & {k+T(l),k~+l{2),.. representation

.,k+l(sk+1)},

we obtain the

/*+i( z FfT(i)> ZFPT(2)> • • •' Zfc+T(sfc+i)) k

=

2 ^ F3,k+l ( Z pTT(l)' Z P+I(2) » • • • ' Z p+I(ty, fc+ i))

where ^\*+l( Z J>+I(l)> ZIfc+T(2)' • • • ' =

~

/i(zJ(i).Zj(2).---'Zi(,3))

z

Z

JJ+l(ti,h+1))

- = c ' for » (2 {* + !(!), * + l(2),...,fc + l ( » i + 1 ) }

(6-4)

74

General Classes of Cyclic Functional

Equations

If we substitute Eq. (6.4) into Eq. (6.3) and denote 5i(Zi(l)>Zi(2)>--->Zi(Si))

=

+

/*(Zi(l)>Zi(2)>---'Z7(Si)) -Fi,fc+i( Z i^+T(i)j Z Mfc+T(2)' • • • ' Z 7 ^ + T ( t i , f e + 1 ) ) '

then equation (6.3) becomes k

X] fl ' i ( Z i(l)' Z i(2)'---' Z i(s0)

=

°"

i=l

By assumption its general solution is i-l Z

Z

Z

5i( i(l) I i(2)'---' i( S i ))

~

-

2^^'*( Z J*(l)'" , ' Z ii(*J«))

22

Fi Z

i( ij(i)>--->Zij(tij))>

j=i+l

and hence i-l /i(z7(i)>Zi(2)>--->Zi(Si))

-

-

2^fi»(Zji(i)>---'Zj7(ij.o) i=i fc+i

J2

F

ij(Zij(l)'--->Zij(Ui))

for 1 < i < k. Together with Eq. (6.4) this yields Eq. (6.2) with instead of k.

k+1 •

The last theorem which is obtained in [I. B. Risteski and V. C. Covachev (2000)] generalized all previous results given in this chapter.

Chapter 2

Functional Equations with Operations between Arguments

In this chapter some classes of complex vector functional equations are solved, namely such equations in which operations between arguments appear. The results presented in this chapter are given in [I. B. Risteski et al. (1999); I. B. Risteski et al. (2000B); I. B. Risteski et al. (to appear A); K. G. Trencevski et al. (1999)].

7

Operator Functional Equation

Here a new linear operator \t is defined such that $ o $ = O. The general analytic solution of the vector functional equation * / = O is given. The results presented here are obtained in [I. B. Risteski et al. (1999)]. Definition 7.1. Let V and V be complex vector spaces. For an arbitrary mapping / : V n _ 1 i-» V (n > 1) we define a mapping \ t / : Vn •-*• V by (*/)(Z1>Z2>...,Zn_1,Z„) =

(7.1)

1

( - l ) " - / ^ ! , Z 2 , . . . , Z„_i) - / ( Z 2 , Z 3 , . . . , Z„) n-1

+

2 ^ ( - i ) 4 + 1 / ( Z i , z 2 , . . . , Z j + Zj+i,.. . , z n _ i , z n ) . i=l

If n = 1, we define * / = O. Remark 7.1. The definition of the operator * is a variation of the formula giving the differential of the bar construction [S. MacLane (1963)]. 75

76

Functional Equations with Operations between

Arguments

For an arbitrary mapping f : V n _ 1 i-> V it holds

Theorem 7.1

(tf o * ) / ( Z i , Z 2 , . . . , Zn> Z n + 1 ) = O. Proof. obtain

(7.2)

Applying the operator \P to the mapping \£/ : V" H V , we

(*o*)/(Zi,Z2,...,Zn,Zn+1) =

( - 1 ) " ( * / ) ( Z ! , Z 2 , . . . , Z„_i, Z„) - ( * / ) ( Z 2 ) Z 3 ) . . . , Z„, Z n + 1 )

+

^ ( - l ) i + 1 ( * / ) ( Z ! , Z 2 , ...,Zi

n

+ Zi+U...,

Z n> Z n + i ) .

i=l

Using the definition of the operator * inside the sum and replacing i by i + 1 in some of the terms, we can write down the previous equality as (*o*)/(Zi,Z2>...,

(7.3)

= ( - l ) " ( * / ) ( Z i , Z 2 , . . . , Z n _!, Z n ) - (*/)(Z 2 , Z 3 , . . . , Z„, Z„ + 1 )

+ < ^ ( _ l ) » + i [ ( _ i ) « - i / ( Z l , Z 2 , . . . , Zi + Zi+1,...,

Z B _!, Z n )

i=l

+ / ( Z 2 , Z 3 , . . . , Zj + i + Z j + 2 , . . . , Z n , Z „ + i ) j + / ( Z i , Z 2 ) . . . , Z„_i) — / ( Z 3 , . . . , Z„, Z n + i ) n—1 i—1

Z j + l , . . ., Zj+i + Z;+ 2 , . . . , Z n , Z„+i) i=2 J=l

n—2

+ /

y

n

/

y

(-1) 4

i=l j=i+2

,?

/ ( Z i , Z 2 , . . . , Z i + Z i + i , . . .,Zj + Z j - + i , . . . , Z n , Z n + 1 )

Operator Functional Equation

77

In the parentheses {• • • } the terms for j = i, i + 1 are omitted because they cancel each other. Further we will consider the double sum n—1 i—1

/] ^ ( - l ) t + J / ( Z i , Z 2 , . . . , Z j + Z j + i , . . . , Z i + 1 + Z j + 2 , . . . , Z n , Z n + i ) i=2 j=l

n

i-1

— 2 _ , Z ^ ( - ^ ) t + J + / ( Z i , Z 2 , . . . , Zj + Z j + i , . . . , Z t + Z { + i , . . . , z n , z n + i ) i=3 j = l ra—2

= /

n

/ ! • (—-0*

y

3

/ ( Z i , Z 2 , . . .,Zj + Z j + i , . . .,Zj + Z j + i , . . . , Z n , Z n + i )

j = l i=j+2 n—2

= -2_j

n

\~i ( _ i ) , + : ' / ( Z i , Z 2 , . . . , z , + Z j + i , . . . , Z j + Z j + i , . . . , z „ , z n + i ) .

t=l j=i+2

Since the double sums in Eq. (7.3) cancel each other, we obtain Z n - i ) Z n , Zn_|-i) = ( - l ) n ( * / ) ( Z ! , Z 2 , . . . , Z n _ ! , Z n ) - ( * / ) ( Z 2 , Z 3 , •.., Z„, Z n + 1 ) n-1

+ Z(-i) i + 1 [(-i)n_1/(Zi, z2,..., ii + zi+1,..., zn_!, zn) i=l

+ / ( Z 2 , Z 3 , . . . , Z i + 1 + Zj+2,..., Z„, Z„+i) I + / ( Z i , Z2,..., Z n _i) — / ( Z 3 , . . . , Z n , Z n + i )

= (-irw)(Zi, z2>..., zn_i, z„) - (*/)(z2, z3,..., z„, z n+ o + ( - l ) » - i [ ( - l ) » - 1 / ( Z 1 , Z 2 ) . . . , Z n _ 0 - / ( Z 2 , Z 3 , . . . , Zn)

78

Functional Equations with Operations between

Arguments

n-1

+ £ ( - i ) i + 7 ( Z i , z 2l ..., Zi + zi+1,..., zn_u z n )] + [(-i) B - 1 /(z 3 , z 3> ..., z„) - /(z 3 ,..., zn> z n+1 ) n-1

+ ^ (

—

I)*

/(Z2, Z3, . . ., Z; + i + Z j + 2 , . . ., Z„, Z n + i ) I

1=1

=

(-l)n(*/)(Z1>Z2,..,Zn_i,Zn)-(*/)(Z2>Z3j...)Zn,ZB+i)

+(-l)"-1(*/)(Z1,Z2,...,Zn_1,Zn) + (*/)(Z2,Z3,...,Z„,Zn+1) = 0 , which yields the validity of (7.2). • This formula shows that the kernel of the operator \P contains all mappings of the form \ t / . The next theorem provides a complete description of this kernel. T h e o r e m 7.2

The general solution of the operator equation */(Zi,Z2,...,

)= O

(7.4)

in the set of analytic functions f : Vn i-¥ V (n > 1) is given by f(Zu Z 2 ) . . . , Z„) = ( * F ) ( Z i , Z 2 , . . . , Z„_i, Z n ) + L(Zi, Z 2 , . . . , Z n ), (7.5) where F : V " - 1 i-> V' is an arbitrary analytic function andL is an arbitrary mapping: Vn i-+V'(n> 1) linear with respect to each argument. Proof. First note that if n = 1, the equation (*/)(Z1,Z2) = 0 is the Cauchy functional equation /(Z1+Z2)-/(Zi)-/(Za) = 0. The general analytic solution of this equation is / ( Z ) = AZ, where A is an (s x r)-matrix with arbitrary complex constant entries (r = dim V and s = dim V ) . About the solution of the Cauchy matrix functional equation see [0. E. Gheorghiu (1963)] and [A. Kuwagaki (1962)].

Operator Functional

Equation

79

Now let n > 2. The operator equation (7.4) is equivalent to (-l)"/(Z1,Z2,...,Zn)-/(Z2!Z3,...,Zn+1)

(7.6)

n

+

^ ( - l ) i + 1 / ( Z i , Z 2 , . . . , Zi + Z i + 1 , . . . , Z„, Z n + i ) = O.

Note that it is sufficient to prove the theorem if dim V = 1 and the general case is just a consequence. So let us assume that dim V = 1. Note also that / given by Eq. (7.5) is a solution of Eq. (7.4), but we want to prove that each solution is included in Eq. (7.5). Let dim V = r and let Zj = (z»i,--- ,Zir)T (1 < i < n + 1). By differentiating the equation (7.6) partially with respect to z„+i,„ (1 < v < r) at Z„ + i = O, we obtain the following system of r equations " 5 — / ( Z i , Z 2 , . . . , Z n ) = — p„(Z 2 l Z 3 , . . . , Z n ) oznv n-\

^ ( - l ) i + V ( Z i , Z 2 , . . . , Z i + Zi+1,...,Zn_1)Zn)

+

(l Z 2 + Z3,Z4) + / ( Z i , Z 2 , Z 3 + Z 4 ) = 0 . The general analytic solution of this equation is given by /(Z1,Z2)Z3) = F ( Z 1 + Z 2 , Z 3 ) + F(Z1)Z2) -F(ZUZ2 E x a m p l e 7.3

+ Z3) - F ( Z 2 , Z 3 ) +

L(Zi,Z2,Z3).

If n = 4, the functional equation (7.6) takes on the form

/ ( Z i , Z 2 , Z 3 , Z 4 ) — /(Z 2 ,Z 3 ,Z4,Z5) + / ( Z i + Z 2 , Z 3 , Z 4 , Z 5 ) — / ( Z i , Z 2 + Z 3 , Z 4 , Z 5 ) + / ( Z i , Z 2 , Z 3 + Z 4 , Z 5 ) - / ( Z i , Z 2 , Z 3 , Z 4 + Z 5 ) = O. According to Eq. (7.9), the general analytic solution of this functional equation is given by / ( Z i , Z 2 , Z 3 , Z 4 ) = F(Zi + Z 2 , Z 3 , Z 4 ) + F ( Z i , Z 2 , Z 3 + Z 4 ) - F ( Z 2 , Z 3 , Z 4 ) - F ( Z i , Z 2 + Z 3 , Z 4 ) - F ( Z i , Z 2 ) Z 3 ) + L(ZU Z 2 , Z 3 , Z 4 ), where F is an arbitrary analytic function, and L is an arbitrary linear in each argument mapping. This method for solving functional equations does not appear in the other references [J. Aczel (1966); M. Ghermanescu (1960); M. Kuczma (1968); G. Valiron (1945)]. 8

Generalized Functional E q u a t i o n

In this section one generalized complex vector functional equation is solved. Further some particular cases are given. Let V, V be arbitrary complex finite dimensional vector spaces. Now we will prove the following result obtained in [I. B. Risteski et al. (2000B)]. T h e o r e m 8.1

The general analytic solution of the functional equation

( - l ) n / n + i ( Z i , Z 2 , • • • ,Zn) — /n+2(Z2,Z3,- • • ,Zn+1) n

+

Z ) ( _ 1 ) i + 1 ^ ( Z l ' Z 2 ' ' • ' ' Zi + Zk+1(Zi,

• • , Zn+1)

• • • , Zfc + Zfc + i,Z fc+2 + Zjfc+3, • • • , Z n + i )

+ (—l)™Ffc,„_i(Zi,- • • , Zfc + Zfc + i,Z fc+2 , • • • , Z n + Z n + i ) + ( ~ l ) n + ^*n(Zl,-"" >Zfc + Zfc+i,--- ,Z n )] —Hi(Z3, Z 4 , • • • , Z„ + i) + i? 2 (Z 2 + Z 3 , Z 4 , • • • , Z n + i ) 1- (—1) Hk(Z2,-

•• ,Zk + Zk+ir • • , Z „ + i )

+ ( - l ) n / n + i ( Z i , Z 2 , - - • ,Z„) - / „ + 2 ( Z 2 , Z 3 , - •• , Z n + i ) n

+ E (-ir +1 / i (Zi,---,z i +z i+1 ,---,z n+1 ) = o. Indeed, from Eqs. (8.7), (8.9) and (8.10) it follows that the statement is true for A; = 2. We will suppose that it holds for k (2 < k < n — 1). If we

Generalized Functional Equation

87

put Z f c + 1 = O and replace Z i + i by Z, (A; + 1 < i < n) in Eq. (8.12), we obtain / f c + i ( Z i , Z 2 , - - - , Z n ) = Fik(Zi

+ Z2,Z3,--- ,Zn)

(8.13)

—Fik ( Z i , Z 2 + Z 3 , • • • , Z n ) + • • • +(-1)

+1

-Ffefc(Zi,--- ,Zfc_i,Zfc + Zfc+i,-- • , Z n )

+[/fc+2(Zi, • • • ,Zfc,0,Zfc + i + Zfc+2,- • • , Z n ) —-Fi,fc+i(Zi + Z 2 , Z 3 , • • • ,Zfc,0,Zfc+i + Zft+2, • • • , Z n ) +-p2,fc+i(Zi,Z 2 + Z 3 , Z 4 , • • • , Zfc,0, Zfc+i + Z A + 2 , • • • , Z n ) - • • • +(—1) ~~ Fk-\,k+i{Zi,--

-,Zk-2,Zk-i

+ Z f c , 0 , Zk+i + Z f c + 2 , Z j ; + 3 , • • - , Z n )

+ ( - 1 ) Ffc,fc + i(Zi, • • • , Z fc , Zfc+i + Zfc+2, • • • , Z n ) ]

-[A+3(Zi,- • •, Zfc,o, Zfc+i,Zfe+2 + Zfc +3) - • • , z „ ) -•f1i,fc+2(Zi + Z 2 , Z 3 , • • • ,Zjt,0,Zfc + i,Zfc + 2 + Z f c + 3 ,- • • , Z n ) + - F 2 , f e + 2 ( Z l , Z 2 + Z 3 , Z 4 , - • • , Z f c , 0 , Z f c + i , Z j f e + 2 + Zfc + 3, • • • , Z „ ) — • • •

+ ( - 1 ) ~~ i 7 fc-i,fc+2(Zi, • • •,Zfc_2,Zfc_ 1 + Z j ; , 0 , Z f c + i , Z f c + 2 + Zk+3,- • - , Z n ) + ( - 1 ) Fk,k+2(Zi,--H

,Zfc + i,Zjt + 2 + Zfc + 3 ,-- • , Z n ) ]

h (—1)"~ ~~ [ / n ( Z i , - • • , Z j t , 0 , Z f c + i , - • • , Z „ _ 2 , Z „ _ i + Z n ) —Fi,n-i{Zi

+ Z 2 , Z 3 , • • • , Z/t, O , Zfc + i, • • • , Z n _ 2 , Z n _ ! + Z n )

+ - f 2 , n - i ( Z i , Z 2 + Z 3 , Z 4 , - • • , Z f c , 0 , Z f c + i , - - • , Z „ _ 2 , Z „ _ i + Zn) — • • • + ( - 1 ) ~ Ffe_i i n-i(Zi,---,Zfc_2,Zfc_ 1 + Z f e , 0 , Z f c + 1 , - - - , Z „ _ 2 , Z n _ i + Z n ) +(—1) F j t , „ _ i ( Z i , • • • , Z n _ 2 , Z n _ i + Z n ) ]

88

Functional Equations with Operations between Arguments + (-l)"

[/n+l(Zi,--- ,Zfc,0,Zfc+i,--- , Z n _ i )

—Fin(Zi

+ 7i2,2i3,- • • ,Zjfe,0,Z&+i, • • • , Z n _ i )

+-p2n(Zl,Z2 + Z 3 , Z 4 , - • • , Z ^ , 0 , Z f c + i , • • • , Z „ _ i ) — • • • + ( — l ) ~ -F*:-i,7i(Zi, • • • , Zfc-2, Zfc_i + Zfc, O , Zfc +1 , • • • , Z n _ i ) +(-l)*Ffcn(ZllZ2,.-->ZB_i)] + (-1)

[/n+2(Z2,-" ) Z f t , 0 , Z * + i , - - - , Z n )

+ i ? i ( Z 3 , - - - , Zfc,0, Zjfc+i,- • • , Z „ ) - i f 2 ( Z 2 + Z 3 , Z 4 , • • • ,Zk,0,Zk+i, H

_

+ (—1)

Hk-i(Z1}-

• • • ,Zn)

• • ,Zfc_2,Zfc_i + Zk,0,Zk+i,

• • • ,Zn)

+(-l)fc-1iJA(Z2,Z3,---,Zn)]. Now we define (-l)*+2*i+i,*+-> V and g : V >->• V . continuous solution of the functional equation

(

n

\

Zi,£z;

n—1

+$>

I

n+i—1

Then the general

\

Zi,z i+1 , Yl Zi

=0

(10.24)

j=2 J i=1 \ j=i+2 J where n > 4 and Z n + ; = Zj for i = 1, • • • , n — 2, is determined by /(Z1,Z2)=JF(Z1+Z2), (10.25) g(Zu Z 2 , Z3) =

^ F ( Z ! + Z 2 + Z 3 ), n —1

where F is an arbitrary continuous complex vector function V \-¥ V. Proof.

If we put /1 = / , f2 = / 3 = • • • = / n - i = 5 into Eq. (10.18), we

obtain Eq. (10.24). Now by the substitution I

1

from Eq. (10.24) we get

(

z

n

\

i-EZi J=2

n—1

/

+ X> /

t=3

n+z—1

Zi.Zi+i, £ \

j=i+2

+ glZn^Zi+J^Zj]

\

Z,-

(10.26) j

=0.

Prom Eqs. (10.25) and (10.26) it follows that the function g satisfies the

Functional Equations with Several Unknown Functions

105

equation glZ2,Z3,Z1+^Zj j -5IZ„,Z2,Z1 + ^ Z

j

j =0

and, more generally,

(

t-l

n

Z i > Z i + 1 , £ z > + X) j=l

\

Z

/

i

j=i+2

=5

n-l

Z n ,Z 2 ) Z 1 + ^ Z

J

\

j=3

\

(10.27)

j

J

(2 < i < n - 1). On the basis of the relation Eq. (10.27), the functional equation Eq. (10.24) becomes flzu^Zj)

+ ( n - 2 ) J z n , Z 2 , Z 1 + ] T z i ) = O.

(10.28)

ra-1

If we put Z2 = Z n = O and 5Z Z_, = S, we obtain 3=3

F{Zi, S) + (n - 2)ff(O, O, Zj + S) = O. Thus /(Z1)S)=F(Z1+S),

(10.29)

where F(Zi) = - ( n - 2 ) f f ( 0 , 0 , Z i ) . On the basis of Eqs. (10.29) and (10.28) it follows that F

( ] C Z ' ) +(n-2)