| Summary: | This thesis discusses the Holder functions, Holder space and their properties.
Furthermore, we will introduce the superposition operator in general and the
properties of the superposition operator in Holder space. The properties discussed
are the following: the acting condition, the boundedness condition and the
continuity condition. The superposition operator which discussed a nonlinear
operator so that the boundedness and the continuity is not equivalent. The
superposition operator F generated by f maps holder room into relation room does not imply its
boundedness, except for the autonomous case. However if F is bounded, then F
is not continuous even though for the autonomous case, and only in case
superposition operator F generated by f maps holder room into sub relation and holder room
.
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