On a generalization of the Cauchy functional equation |
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Authors: | Janusz Brzd?k |
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Institution: | (1) Department of Mathematics, Pedagogical University, Rejtana 16 A, PL-35-310 Rzeszów, Poland |
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Abstract: | Summary While looking for solutions of some functional equations and systems of functional equations introduced by S. Midura and their generalizations, we came across the problem of solving the equationg(ax + by) = Ag(x) + Bg(y) + L(x, y) (1) in the class of functions mapping a non-empty subsetP of a linear spaceX over a commutative fieldK, satisfying the conditionaP + bP P, into a linear spaceY over a commutative fieldF, whereL: X × X Y is biadditive,a, b K\{0}, andA, B F\{0}.
Theorem.Suppose that K is either R or C, F is of characteristic zero, there exist A
1,A
2,B
1,B
2, F\ {0}with L(ax, y) = A
1
L(x, y), L(x, ay) = A
2
L(x, y), L(bx, y) = B
1
L(x, y), and L(x, by) = B
2
L(x, y) for x, y X, and P has a non-empty convex and algebraically open subset. Then the functional equation (1)has a solution in the class of functions g: P Y iff the following two conditions hold: L(x, y) = L(y, x) for x, y X, (2)if L 0, then A
1 =A
2,B
1 =B
2,A = A
1
2
,and B = B
1
2
. (3)
Furthermore, if conditions (2)and (3)are valid, then a function g: P Y satisfies the equation (1)iff there exist a y
0
Y and an additive function h: X Y such that if A + B 1, then y
0 = 0;h(ax) = Ah(x), h(bx) =Bh(x) for x X; g(x) = h(x) + y
0 + 1/2A
1
-1
B
1
-1
L(x, x)for x P. |
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Keywords: | Primary 39B20 Secondary 39B30 |
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