Date on Master's Thesis/Doctoral Dissertation

8-2014

Document Type

Doctoral Dissertation

Degree Name

Ph. D.

Department

Mathematics

Committee Chair

Sahoo, Prasanna K.

Committee Member

Brown, David

Committee Member

Gill, Ryan

Committee Member

Powers, Robert

Committee Member

Riedel, Thomas

Subject

Functional equations

Abstract

Let G be an abelian group, C be the _eld of complex numbers, _ 2 G be any _xed, nonzero element and _ : G ! G be an involution. In Chapter 2, we determine the general solution f; g : G ! C of the functional equation f(x + _y + _) + g(x + y + _) = 2f(x)f(y) for all x; y 2 G. Let G be an arbitrary group, z0 be any _xed, nonzero element in the center Z(G) of the group G, and _ : G ! G be an involution. The main goals of Chapter 3 are to study the functional equations f(x_yz0) ?? f(xyz0) = 2f(x)f(y) and f(x_yz0) + f(xyz0) = 2f(x)f(y) for all x; y 2 G and some _xed element z0 in the center Z(G) of the group G. In Chapter 4, we consider some properties of the general solution to f(xy)f(x_y) = f(x)2 ?? f(y)2. We also _nd the solution to this equation when G is a 2-divisible, perfect group. We end the chapter by discussing the periodicity of the solutions to both the sine functional equation and the sine inequality.

Included in

Mathematics Commons

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