Abstract
A fixed point theorem for nonlinear contraction in the modular space is proved. Moreover, a fixed point theorem for asymptotic contraction in this space is studied.
Highlights
The theory of modular space was initiated by Nakano [1] in connection with the theory of order spaces and was redefined and generalized by Musielak and Orlicz [2]
Even though a metric is not defined, many problems in fixed point theory for nonexpansive mappings can be reformulated in modular spaces
A fixed point theorem for nonlinear contraction in the modular space is proved
Summary
The theory of modular space was initiated by Nakano [1] in connection with the theory of order spaces and was redefined and generalized by Musielak and Orlicz [2]. Particular Banach spaces of functions can be considered. Metric fixed theory for these Banach spaces of functions has been widely studied (see [3]). Another direction is based on considering and abstractly given functional which control the growth of the functions. Even though a metric is not defined, many problems in fixed point theory for nonexpansive mappings can be reformulated in modular spaces. A fixed point theorem for nonlinear contraction in the modular space is proved. Kirk’s fixed point theorem for asymptotic contraction is presented in this space.
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