SOLUTION OF A CLASS OF COMPLEX VECTOR LINEAR FUNCTIONAL EQUATIONS

Ice B. Risteski

Abstract. In this paper a class of complex vector linear functional equations is solved in the general case, so that the results obtained generalize the results given in [1].

Introduction. First we introduce the following notations.

Let V be a finite dimensional complex vector space and let there exist mappings f_{i}: V^{p} → V (1 ≤ i ≤ n). Throughout this paper Z_{i} (1 ≤ i ≤ n) are vectors in V and C_{i} are constant vectors in the same space. We may assume that Z_{i} = (z_{i1}(t), …, z_{in}(t))^{T}, where the components z_{ij}(t) (1 ≤ i ≤ p; 1 ≤ j ≤ n) are complex functions and O = (0, 0, …, 0)^{T} is the zero vector in V.

In the present paper we will solve the following complex vector functional equation

∑_{i=1}^{n} f_{i}(Z_{i}, Z_{i+1}, …, Z_{i+p−1}) = O (Z_{n+i} ≡ Z_{i}).  (1)

A special case of the above functional equation if p < n < 2p − 1 is solved in [1]. Some other particular cases of the functional equation (1) are considered in [2,3] under the hypothesis that the functions and the independent variables are real.

If we put f_{i} = a_{i}f (1 ≤ i ≤ n), where a_{i} are complex constants, into (1), then we obtain the following functional equation

∑_{i=1}^{n} a_{i}f(Z_{i}, Z_{i+1}, …, Z_{i+p−1}) = O (Z_{n+i} ≡ Z_{i})  (2)

which is a special case of (1). Functional equation (2) will be solved here.

1. Main Results. We will prove the following results.

\underline{Theorem 1}. The general solution of complex vector functional equation (1) is given by the following formulas.

f_{i}(Z_{i}, Z_{i+1}, …, Z_{i+p−1})  (3)

= ∑_{j=1}^{i−1}(−1)^{i+1}F_{ji}({Z_{i}, Z_{i+1}, …, Z_{i+p−1}} ∩ {Z_{j}, Z_{j+1}, …, Z_{j+p−1}})

+ ∑_{j=i+1}^{n}(−1)^{j}F_{ij}({Z_{i}, Z_{i+1}, …, Z_{i+p−1}} ∩ {Z_{j}, Z_{j+1}, …, Z_{j+p−1}}) (1 ≤ i ≤ n),

where F_{ij} (1 ≤ i ≤ n − 1, i + 1 ≤ j ≤ n) are arbitrary complex vector functions from V such that

F_{ij}({Z_{i}, Z_{i+1}, …, Z_{i+p−1}} ∩ {Z_{j}, Z_{j+1}, …, Z_{j+p−1}}) = A_{ij}

if

{Z_{i}, Z_{i+1}, …, Z_{i+p−1}} ∩ {Z_{j}, Z_{j+1}, …, Z_{j+p−1}} = ∅,

where A_{ij} are constant complex vectors from V and ∑_{a}^{s} = O (a > s).

\underline{Proof}. We will prove the statement of the theorem by mathematical induction. For n = 2, equation (1) becomes

f_{1}(Z_{1}, Z_{2}, …, Z_{p}) + f_{2}(Z_{2}, Z_{3}, …, Z_{p+1}) = O.  (4)

Putting Z_{1} = C_{1} into equation (4), we get

f_{2}(Z_{2}, Z_{3}, …, Z_{p+1}) = −F_{12}(Z_{2}, Z_{3}, …, Z_{p}),  (5)

where the following notation is introduced

F_{12}(Z_{2}, Z_{3}, …, Z_{p}) = f_{1}(C_{1}, Z_{2}, …, Z_{p}).

If we put (5) into (4), we get

f_{1}(Z_{1}, Z_{2}, …, Z_{p}) = F_{12}(Z_{2}, Z_{3}, …, Z_{p}).  (6)

Therefore, for i = 1, 2 from (3) we obtain (6) and (5), respectively, which means that the theorem holds for n = 2.

For some fixed n we suppose that the general solution of functional equation (1) is given by (3).

Now, consider the functional equation

∑_{i=1}^{n+1} g_{i}(Z_{i}, Z_{i+1}, …, Z_{i+p−1}) = O.  (7)

If we put Z_{i} = C_{i} (1 ≤ i ≤ n) into (7), we find that function g_{n+1} has the following form

g_{n+1}(Z_{n+1}, Z_{n+2}, …, Z_{n+p})  (8)

= ∑_{j=1}^{n}(−1)^{n}F_{j,n+1}({Z_{n+1}, Z_{n+2}, …, Z_{n+p}} ∩ {Z_{j}, Z_{j+1}, …, Z_{j+p−1}}).

Substituting (8) into (7) and introducing the new notations

f_{i}(Z_{i}, Z_{i+1}, …, Z_{i+p−1}) = g_{i}(Z_{i}, Z_{i+1}, …, Z_{i+p−1})  (9)

+ (−1)^{n}F_{i,n+1}({Z_{n+1}, Z_{n+2}, …, Z_{n+p}} ∩ {Z_{i}, Z_{i+1}, …, Z_{i+p−1}}) (1 ≤ i ≤ n),

we obtain equation (1). According to the inductive hypothesis, the general solution of this equation is given by the formulas in (3). Therefore, from (3), (8), and (9) we get

g_{i}({Z_{i}, Z_{i+1}, …, Z_{i+p−1}})

= ∑_{j=1}^{i−1}(−1)^{i+1}F_{ji}({Z_{i}, Z_{i+1}, …, Z_{i+p−1}} ∩ {Z_{j}, Z_{j+1}, …, Z_{j+p−1}})

+ ∑_{j=i+1}^{n}(−1)^{j}F_{ij}({Z_{i}, Z_{i+1}, …, Z_{i+p−1}} ∩ {Z_{j}, Z_{j+1}, …, Z_{j+p−1}})

− (−1)^{n}F_{i,n+1}({Z_{n+1}, Z_{n+2}, …, Z_{n+p}} ∩ {Z_{i}, Z_{i+1}, …, Z_{i+p−1}}) (1 ≤ i ≤ n), i.e.

g_{i}(Z_{i}, Z_{i+1}, …, Z_{i+p−1})

= ∑_{j=1}^{i−1}(−1)^{i+1}F_{ji}({Z_{i}, Z_{i+1}, …, Z_{i+p−1}} ∩ {Z_{j}, Z_{j+1}, …, Z_{j+p−1}})

+ ∑_{j=i+1}^{n+1}(−1)^{j}F_{ij}({Z_{i}, Z_{i+1}, …, Z_{i+p−1}} ∩ {Z_{j}, Z_{j+1}, …, Z_{j+p−1}})

(1 ≤ i ≤ n + 1).

The theorem holds for n + 1 if it holds for n.

\underline{Example 1}. The general solution of the complex vector functional equation

f_{1}(Z_{1}, Z_{2}) + f_{2}(Z_{2}, Z_{3}) + f_{3}(Z_{3}, Z_{4}) + f_{4}(Z_{4}, Z_{1}) = O,

which is a particular case for n = 4 and p = 2 of equation (1), is given by

f_{1}(Z_{1}, Z_{2}) = F_{12}(Z_{2}) − A_{13} + F_{14}(Z_{1}),
f_{2}(Z_{2}, Z_{3}) = −F_{12}(Z_{2}) − F_{23}(Z_{3}) + A_{24},
f_{3}(Z_{3}, Z_{4}) = A_{13} + F_{23}(Z_{3}) + F_{34}(Z_{4}),
f_{4}(Z_{4}, Z_{1}) = −F_{14}(Z_{1}) − A_{24} − F_{34}(Z_{4}),

where F_{ij} (1 ≤ i ≤ 3; 2 ≤ j ≤ 4) are arbitrary complex vector functions from V and A_{ij} (i = 1, 2; j = 3, 4) are arbitrary constant complex vectors also from V.

\underline{Example 2}. Consider the functional equation

f_{1}(Z_{1}, Z_{2}, Z_{3}) + f_{2}(Z_{2}, Z_{3}, Z_{4}) + f_{3}(Z_{3}, Z_{4}, Z_{1}) + f_{4}(Z_{4}, Z_{1}, Z_{2}) = O, (10)

where f_{i}: V^{3} → V (1 ≤ i ≤ 4). This equation is a particular case for p = 3 and n = 4 of functional equation (1).

According to Theorem 1, the general solution of functional equation (10) is

f_{1}(Z_{1}, Z_{2}, Z_{3}) = F_{12}(Z_{2}, Z_{3}) − F_{13}(Z_{1}, Z_{3}) + F_{14}(Z_{1}, Z_{2}),  (11)
f_{2}(Z_{2}, Z_{3}, Z_{4}) = −F_{12}(Z_{2}, Z_{3}) − F_{23}(Z_{3}, Z_{4}) + F_{24}(Z_{2}, Z_{4}),  (12)
f_{3}(Z_{3}, Z_{4}, Z_{1}) = F_{13}(Z_{1}, Z_{3}) + F_{23}(Z_{3}, Z_{4}) + F_{34}(Z_{1}, Z_{4}),  (13)
f_{4}(Z_{4}, Z_{1}, Z_{2}) = −F_{14}(Z_{1}, Z_{2}) − F_{24}(Z_{2}, Z_{4}) − F_{34}(Z_{1}, Z_{4}),  (14)

where F_{12}, F_{13}, F_{14}, F_{23} and F_{34} are arbitrary complex vector functions from V.

\underline{Theorem 2}. The general solution of complex vector functional equation (2) is given by

f(Z_{1}, Z_{2}, …, Z_{p})  (15)

= ∑_{i=1}^{n} α_{i}[∑_{j=1}^{i−1}(−1)^{i+1}C^{n+1−i}G_{ji}({Z_{i}, Z_{i+1}, …, Z_{i+p−1}} ∩ {Z_{j}, Z_{j+1}, …, Z_{j+p−1}})

+ ∑_{j=i+1}^{n}(−1)^{j}C^{n+1−i}G_{ij}({Z_{i}, Z_{i+1}, …, Z_{i+p−1}} ∩ {Z_{j}, Z_{j+1}, …, Z_{j+p−1}})],

where C is a cyclic operator such that CG(Z_{1}, Z_{2}, …, Z_{n}) = G(Z_{2}, Z_{3}, …, Z_{n}, Z_{1}); G_{ij} (1 ≤ i ≤ n − 1; i + 1 ≤ j ≤ n) are arbitrary complex vector functions from V and ∑_{a}^{s} = O (a > s).

\underline{Proof}. If we apply operator C^{n+1−i} to both sides of (3) (with f_{i} replaced by a_{i}f) and then multiply by indefinite complex constants α_{i} (1 ≤ i ≤ n), we have

a_{i}α_{i}C^{n+1−i}f(Z_{i}, Z_{i+1}, …, Z_{i+p−1})  (16)

= α_{i}[∑_{j=1}^{i−1}(−1)^{i+1}C^{n+1−i}F_{ji}({Z_{i}, Z_{i+1}, …, Z_{i+p−1}} ∩ {Z_{j}, Z_{j+1}, …, Z_{j+p−1}})

+ ∑_{j=i+1}^{n}(−1)^{j}C^{n+1−i}F_{ij}(Z_{i}, Z_{i+1}, …, Z_{i+p−1}} ∩ {Z_{j}, Z_{j+1}, …, Z_{j+p−1}})]

(1 ≤ i ≤ n).

By summing up the above functions (16), we obtain

(∑_{i=1}^{n} a_{i}α_{i})f(Z_{1}, Z_{2}, …, Z_{p})  (17)

= ∑_{i=1}^{n} α_{i}[∑_{j=1}^{i−1}(−1)^{i+1}C^{n+1−i}F_{ji}({Z_{i}, Z_{i+1}, …, Z_{i+p−1}} ∩ {Z_{j}, Z_{j+1}, …, Z_{j+p−1}})

+ ∑_{j=i+1}^{n}(−1)^{j}C^{n+1−i}F_{ij}({Z_{i}, Z_{i+1}, …, Z_{i+p−1}} ∩ {Z_{j}, Z_{j+1}, …, Z_{j+p−1}})].

From equation (2) and equality (17) we find

O = ∑_{r=1}^{n} a_{r}(∑_{i=1}^{n} a_{i}α_{i})C^{r−1}f(Z_{1}, Z_{2}, …, Z_{p})

= ∑_{r=1}^{n} a_{r}{ ∑_{i=1}^{n} α_{i}[∑_{j=1}^{i−1}(−1)^{i+1}C^{n+r−i}

F_{ji}({Z_{i}, Z_{i+1} …, Z_{i+p−1}} ∩ {Z_{j}, Z_{j+1}, …, Z_{j+p−1}})

+ ∑_{j=i+1}^{n−1}(−1)^{j}C^{n+r−i}

F_{ij}({Z_{i}, Z_{i+1}, …, Z_{i+p−1}} ∩ {Z_{j}, Z_{j+1}, …, Z_{j+p−1}})]}

= ∑_{j=1}^{n−1} ∑_{i=j+1}^{n}(−1)^{i+1}α_{i} ∑_{r=1}^{n} a_{r}C^{n+r−i}

F_{ji}({Z_{i}, Z_{i+1}, …, Z_{i+p−1}} ∩ {Z_{j}, Z_{j+1}, …, Z_{j+p−1}})

+ ∑_{i=1}^{n−1} ∑_{j=i+1}^{n}(−1)^{j}α_{i} ∑_{r=1}^{n} a_{r}C^{n+r−i}

F_{ij}({Z_{i}, Z_{i+1}, …, Z_{i+p−1}} ∩ {Z_{j}, Z_{j+1}, …, Z_{j+p−1}})

= ∑_{i=1}^{n−1} ∑_{j=i+1}^{n}(−1)^{j}{ ∑_{r=1}^{n}[α_{i}C^{n+r−i}

F_{ij}({Z_{i}, Z_{i+1}, …, Z_{i+p−1}} ∩ {Z_{j}, Z_{j+1}, …, Z_{j+p−1}}) − α_{j}C^{n+r−j}F_{ij}({Z_{i}, Z_{i+1}, …, Z_{i+p−1}} ∩ {Z_{j}, Z_{j+1}, …, Z_{j+p−1}})]}

= ∑_{i=1}^{n−1} ∑_{j=i+1}^{n}(−1)^{j}C^{n+1−i}{ ∑_{r=1}^{n} a_{r}[α_{i}C^{r−1}

F_{ij}({Z_{i}, Z_{i+1}, …, Z_{i+p−1}} ∩ {Z_{j}, Z_{j+1}, …, Z_{j+p−1}}) − α_{j}C^{r−1}F_{ij}({Z_{i}, Z_{i+1}, …, Z_{i+p−1}} ∩ {Z_{2i−j}, Z_{2i−j+1}, …, Z_{2i−j+p−1}})]}.

Now, we may determine the constants α_{i} (1 ≤ i ≤ n) from the identities

∑_{r=1}^{n} a_{r}[α_{i}C^{r−1}F_{ij}({Z_{i}, Z_{i+1}, …, Z_{i+p−1}} ∩ {Z_{j}, Z_{j+1}, …, Z_{j+p−1}})  (18)

− α_{j}C^{r−1}F_{ij}({Z_{i}, Z_{i+1}, …, Z_{i+p−1}} ∩ {Z_{2i−j}, Z_{2i−j+1}, …, Z_{2i−j+p−1}})] = O (1 ≤ i ≤ n − 1; i + 1 ≤ j ≤ n).

Equation (17) may be rewritten in the following form

(∑_{i=1}^{n} a_{i}α_{i})T = S,  (19)

and because it has a unique solution T, by introducing the new functions

\frac{F_{ij}}{∑_{i=1}^{n} a_{i}α_{i}} = G_{ij} (1 ≤ i ≤ n − 1; i + 1 ≤ j ≤ n)  (20)

(15) follows.

\underline{Example 3}. We consider the functional equation

f(Z_{1}, Z_{2}, Z_{3}) − f(Z_{2}, Z_{3}, Z_{4}) + f(Z_{3}, Z_{4}, Z_{1}) − f(Z_{4}, Z_{1}, Z_{2}) = O.  (21)

According to (11), (12), (13) and (14) we obtain

f(Z_{1}, Z_{2}, Z_{3}) = F_{12}(Z_{2}, Z_{3}) − F_{13}(Z_{1}, Z_{3}) + F_{14}(Z_{1}, Z_{2}),
− f(Z_{1}, Z_{2}, Z_{3}) = −F_{12}(Z_{1}, Z_{2}) − F_{23}(Z_{2}, Z_{3}) + F_{24}(Z_{1}, Z_{3}),
f(Z_{1}, Z_{2}, Z_{3}) = F_{13}(Z_{3}, Z_{1}) + F_{23}(Z_{1}, Z_{2}) + F_{34}(Z_{3}, Z_{2}),
− f(Z_{1}, Z_{2}, Z_{3}) = −F_{14}(Z_{2}, Z_{3}) − F_{24}(Z_{3}, Z_{1}) − F_{34}(Z_{2}, Z_{1}).

On the basis of the identities in (18), we determine the constants α_{i} (i = 1, 2, 3, 4) as follows

α_{1}F_{12}(Z_{2}, Z_{3}) − α_{2}F_{12}(Z_{1}, Z_{2}) − α_{1}F_{12}(Z_{3}, Z_{4}) + α_{2}F_{12}(Z_{2}, Z_{3}) + α_{1}F_{12}(Z_{4}, Z_{1}) − α_{2}F_{12}(Z_{3}, Z_{4}) − α_{1}F_{12}(Z_{1}, Z_{2}) + α_{2}F_{12}(Z_{3}, Z_{1}) = O,

− α_{1}F_{13}(Z_{1}, Z_{3}) + α_{3}F_{13}(Z_{3}, Z_{1}) + α_{1}F_{13}(Z_{2}, Z_{4}) − α_{3}F_{13}(Z_{4}, Z_{2}) − α_{1}F_{13}(Z_{3}, Z_{1}) + α_{3}F_{13}(Z_{1}, Z_{3}) + α_{1}F_{13}(Z_{4}, Z_{2}) − α_{3}F_{13}(Z_{2}, Z_{4}) = O,

α_{1}F_{14}(Z_{1}, Z_{2}) − α_{4}F_{14}(Z_{2}, Z_{3}) − α_{1}F_{14}(Z_{2}, Z_{3}) + α_{4}F_{14}(Z_{3}, Z_{4}) + α_{1}F_{14}(Z_{3}, Z_{4}) − α_{4}F_{14}(Z_{4}, Z_{1}) − α_{1}F_{14}(Z_{4}, Z_{1}) + α_{4}F_{14}(Z_{1}, Z_{2}) = O,

− α_{2}F_{23}(Z_{2}, Z_{3}) + α_{3}F_{23}(Z_{1}, Z_{2}) + α_{2}F_{23}(Z_{3}, Z_{4}) − α_{3}F_{23}(Z_{2}, Z_{3})
− α_{2}F_{23}(Z_{4}, Z_{1}) + α_{3}F_{23}(Z_{3}, Z_{4}) + α_{2}F_{23}(Z_{1}, Z_{2}) − α_{3}F_{23}(Z_{4}, Z_{1}) = O,
α_{2}F_{24}(Z_{1}, Z_{3}) − α_{4}F_{24}(Z_{3}, Z_{1}) − α_{2}F_{24}(Z_{2}, Z_{4}) + α_{4}F_{24}(Z_{4}, Z_{2})
+ α_{2}F_{24}(Z_{3}, Z_{1}) − α_{4}F_{24}(Z_{1}, Z_{3}) − α_{2}F_{24}(Z_{4}, Z_{2}) + α_{4}F_{24}(Z_{1}, Z_{4}) = O,
α_{3}F_{34}(Z_{3}, Z_{2}) − α_{4}F_{34}(Z_{2}, Z_{1}) − α_{3}F_{34}(Z_{4}, Z_{3}) + α_{4}F_{34}(Z_{3}, Z_{2})
+ α_{3}F_{34}(Z_{1}, Z_{4}) − α_{4}F_{34}(Z_{4}, Z_{3}) − α_{3}F_{34}(Z_{2}, Z_{1}) + α_{4}F_{34}(Z_{1}, Z_{4}) = O.

Thus, we obtain

α_{1} = −α_{2}, α_{1} = α_{3}, α_{1} = −α_{4}, α_{2} = −α_{3}, α_{2} = α_{4}, α_{3} = −α_{4},

which means that α_{1} = −α_{2} = α_{3} = −α_{4} = 1.

The general solution of functional equation (21) is

f(Z_{1}, Z_{2}, Z_{3}) = F (Z_{1}, Z_{2}) + F (Z_{2}, Z_{3}) + G(Z_{1}, Z_{3}) − G(Z_{3}, Z_{1}),

where

F (Z_{1}, Z_{2}) = F_{14}(Z_{1}, Z_{2}) + F_{12}(Z_{1}, Z_{2}) + F_{23}(Z_{1}, Z_{2}) + F_{34}(Z_{2}, Z_{1}), G(Z_{1}, Z_{3}) = −F_{13}(Z_{1}, Z_{3}) − F_{24}(Z_{1}, Z_{3}).

\underline{Acknowledgement}. The author expresses his gratitude to Prof. V. C. Covachev from the Bulgarian Academy of Sciences for helpful discussion on the results of the present paper.

\underline{References}

1. I. B. Risteski and V. C. Covachev, “On Some General Classes of Partial Linear Complex Vector Functional Equations,” Sci. Univ. Tokyo J. Math., 36 (2000), 105–146.

2. P. M. Vasić, “Sur Une Class d’Équations Fonctionnelle Linéaires,” C. R. Acad. Sci. Paris, 260 (1965), 5666–5667.

3. P. M. Vasić, “Résolution d’Une Classe d’Équations Fonctionnelle Linéaires,” Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz., No. 181 – No. 196 (1967), 53–60.

Ice B. Risteski 2 Milepost Place # 606 Toronto, Ontario M4H 1C7, Canada email: iceristeski@hotmail.com