This book presents extended forms of the Maxwell equations as well as electromagnetic fields, based on a non-zero divergence of the electric field and a non-zero electric conductivity in vacuo. These approaches, which predict new features of the electromagnetic field, such as the existence of both longitudinal and transverse solutions, the existence of space-charge current in vacuo, and steady electromagnetic equilibria, have possible applications to charge and neutral leptons and new photon physics. The present theory can also clear up some unsolved problems, such as the total reflection of light at the interface between a vacuum and a dissipative medium, and the appearance of an angular momentum of the photon, thereby leading to a rest mass and an axial magnetic field component of the photon. This axial magnetic field component may be related to the B(3) field proposed by Evans and Vigier. A new gauge condition has been proposed to maintain consistency of the theory with the non-zero photon mass. Several consequences of the non-zero mass of the photon are also discussed, especially in the astrophysical context.Contents: Historical SurveyBasis of Present ApproachesNew Features Introduced by Present ApproachesGeneral Properties of Steady Axisymmetric EquilibriaElectrically Charged and Neutral Axisymmetric EquilibriaNontransverse Plane Waves Due to Space Charges in VacuoTotal Reflection of Plane WavesAxisymmetric Wave ModesNonzero Conductivity in Vacuo and Nonzero Rest Mass of PhotonAxial Magnetic Field and Longitudinal SolutionsGauge ConditionPossible ImplicationsReadership: Researchers in physics and chemistry.Key Features:Our book gives a complete treatment of the diverse and manifold use of fixed point theorems and their many applications throughout Mathematics and is another example of the unity within MathematicsAs such it serves as a valuable resource for researchers in diverse fields of MathematicsIt also serves as solid introduction for students to several subjects in modern Mathematics such as Functional Analysis, Topology, Differential Geometry, Dynamical Systems and Algebraic Groups