chapter  2
32 Pages

## Fixed Points of Nonlinear Semigroups in Modular Function Spaces

Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia

M. A. Khamsi

Department of Mathematical Sciences, University of Texas at El Paso, El Paso, USA, and Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.2 Basic Definitions and Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 2.3 Some Geometric Properties of Modular Function Spaces . . . . . . . 53 2.4 Some Fixed-Point Theorems in Modular Spaces . . . . . . . . . . . . . . . . 59 2.5 Semigroups in Modular Function Spaces . . . . . . . . . . . . . . . . . . . . . . . . 61 2.6 Fixed Points of Semigroup of Mappings . . . . . . . . . . . . . . . . . . . . . . . . . 64

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

Nonlinear semigroup theory is not only of intrinsic interest, but is also important in the study of evolution problems. In the last forty years, the general theory of flows of holomorphic mappings has been of great interest in the theory of Markov stochastic branching processes, the theory of composition operators, control theory, and optimization. It transpires that the asymptotic behavior of solutions to evolution equations is applicable to the study of the geometry of certain domains in complex spaces. In this chapter, we will discuss the existence of common fixed points of nonlinear semigroups acting in modular function spaces. For the theory of a common fixed point of mappings, we refer to the excellent papers [5, 6, 10, 12, 41].