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This monograph provides a concise introduction to the main results and methods of the fixed point theory in modular function spaces. Modular function spaces are natural generalizations of both function and sequence variants of many important spaces like Lebesgue, Orlicz, Musielak-Orlicz, Lorentz, Orlicz-Lorentz, Calderon-Lozanovskii spaces, and others. In most cases, particularly in applications to integral operators, approximation and fixed point results, modular type conditions are much more natural and can be more easily verified than their metric or norm counterparts. There are also important results that can be proved only using the apparatus of modular function spaces. The material is presented in a systematic and rigorous manner that allows readers to grasp the key ideas and to gain a working knowledge of the theory. Despite the fact that the work is largely self-contained, extensive bibliographic references are included, and open problems and further development directions are suggested when applicable. The monograph is targeted mainly at the mathematical research community but it is also accessible to graduate students interested in functional analysis and its applications. It could also serve as a text for an advanced course in fixed point theory of mappings acting in modular function spaces.

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Introduction.- Fixed Point Theory in Metric Spaces: An Introduction.- Modular Function Spaces.- Geometry of Modular Function Spaces.- Fixed Point Existence Theorems in Modular Function Spaces.- Fixed Point Construction Processes.- Semigroups of Nonlinear Mappings in Modular Function Spaces.- Modular Metric Spaces.

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"The book is essentially self-contained and it provides an outstanding resource for those interested in this line of research." (María Angeles Japón Pineda, Mathematical Reviews, March, 2016)

"The book is devoted to a comprehensive treatment of what is currently known about the fixed point theory in modular function spaces. ... the book will be useful for all mathematicians whose interests lie in nonlinear analysis, in particular, in the theory of function spaces and fixed point theory." (Peter P. Zabre ko, zbMATH 1318.47002, 2015)

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Mohamed Amine Khamsi, Ph.D. is a Professor in the Department of Mathematical Sciences at the University of Texas at El Paso, Texas, USA. His research interests include functional analysis, fixed point theory, discrete dynamical systems, and logic programming. Dr. Khamsi received his Ph.D. at the University Paris VI in 1987.

Wojciech M. (Walter) Kozlowski, Ph.D. is a professor in the School of Mathematics and Statistics at the University of New South Wales in Sydney, Australia. His research interests include functional analysis, function spaces, fixed point theory, approximation theory and applications. He received his doctorate at the Jagiellonian University in Krakow in 1981. Dr. Kozlowski, a Fulbright Scholar at the California University of Technology in Pasadena in years 1986 - 1988, works also in a capacity of the business consultant for the telecommunications industry.

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