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This book started its life as a series of lectures given by the second author from the 1970’s onwards to students in their third and fourth years in the Department of Mechanics and Mathematics at Rostov State University. For these lectures there was also an audience of engineers and applied mechanicists who wished to understand the functional analysis used in contemporary research in their fields. These people were not so much interested in functional analysis itself as in its applications; they did not want to be told about functional analysis in its most abstract form, but wanted a guided tour through those parts of the analysis needed for their applications. The lecture notes evolved over the years as the first author started to make more formal typewritten versions incorporating new material. About 1990 the first author prepared an English version and submitted it to Kluwer Academic Publishers for inclusion in the series Solid Mechanics and its Applications. At that state the notes were divided into three long chapters covering linear and nonlinear analysis. As Series Editor, the third author started to edit them. The requirements of lecture notes and books are vastly different. A book has to be complete (in some sense), self contained, and able to be read without the help of an instructor.

The Traditional Chinese Medicine is a national treasure of China, and an important part of the treasure of the entire world as well. The Traditional Chinese Medicine has a long history for thousands of years since the beginning of the written literature on medicine in China. Even in ancient time with very low developed culture, science, and technology, our ancestry had made a gigantic contribution to the healthy multiplying of the Chinese people of all nationalities through the application of acupuncture techniques and medicinal herbs as well as massage and Qigong. After the People's Republic of China was founded, the administration has been standing for combining of Traditional Chinese Medicine with Western Medicine in order to use reference to each other, which has gained gratifying achievements in the treatment for a lot of ordi­ nary illnesses and hypochondria. With the reformation and opening of the People's Republic of China, the Traditional Chinese Medicine has been recommended to the of returning to nature is rising western countries. While the voice higher and higher, more and more friends over the world have ac­ cepted the Traditional Chinese Medicine, and many of them hope to deeply understand the Traditional Chinese Medicine, study its the­ ory and learn how to apply medical herbs and techniques of acupunc­ ture, massage and Qigong.

This Special Issue provides a platform to showcase the latest achievements in many branches of theoretical and practical mathematical studies. These relate to Riemannian theories, generalized Riemannian spaces and their mappings. The scope of this Special Issue also includes Finsler geometry, Kenmotsu manifolds, Kaehler manifolds, manifolds with non-symmetric linear connections, cosymplectic manifolds, contact manifolds, statistical manifolds, Minkowski spaces, geodesic mappings, almost-geodesic mappings, holomorphically projective mappings, warped products of manifolds, complex space forms, quaternionic space forms, golden manifolds, inequalities, invariants, immersions, etc.

The aim of this reprint is to highlight the significance of difference equations and their applications in applied mathematics. Difference equations provide a robust and widely recognized framework for modeling complex dynamical systems. In recent years, non-integer order derivative operators have gained attention, particularly for understanding anomalous behaviors in social and physical sciences. The following Special Issue introduces several important developments, including new partial fractional derivatives, a novel nonlinear delayed integral inequality for studying integro-differential equations, and a neural network approach for solving nonlinear partial differential equations. The existence and uniqueness of solutions for nonlinear differential equations with impulses at variable times are discussed, as well as the oscillatory behavior of nonlinear second-order differential equations. Ultimately, this reprint bridges theory and application, offering novel insights with a focus on differential and difference equations.