The fixed point theory is an important tool not only of nonlinear analysis, but also for many other branches of modern mathematics. It has numerous applications within mathematics and has been applied in diverse fields such as medical sciences, chemistry, economics, management, engineering, game theory, and physics.

Historically the beginning of metric fixed point theory goes back two centuries, but its name was coined only in 1922 after the pioneer work of Polish mathematician Stefan Banach in his Ph.D. dissertation. Many remarkable results of fixed point theory have been obtained during nineteen sixtees and nineteen seventies such as Caristi's fixed point theorem, Nadlar's fixed point theorem, etc. A large number of research papers have already appeared in the literature on extensions and generalizations of the Banach contraction principle.

On the other hand, Ivar Ekeland established a result on the existence of an approximate minimizer of a bounded below and lower semicontinuous function in 1972. Such a result is now known as Ekeland's variational principle. It is one of the most elegant and applicable results that appeared in the area of nonlinear analysis with diverse applications in fixed point theory, optimization, optimal control theory, game theory, nonlinear equations, dynamical systems, etc. Later, it was found that several well-known results, namely, the Caristi–Kirk fixed point theorem, Takahashi's minimization theorem, the Petal theorem, and the Daneš drop theorem, from nonlinear analysis are equivalent to Ekeland's variational principle in the sense that one can be derived by using the other results.

The set-valued maps, also called multivalued maps or point-to-set maps or multifunctions, are first considered in the famous book on topology by Kuratowski. Other eminent mathematicians, namely, Painlevé, Hausdorff, and Bouligand, have also visualized the vital role of set-valued maps as one often encounters such objects in concrete and real-life problems.

During the last decade of the last century, the theory on equilibrium problems emerged as one of the popular and hot topics in nonlinear analysis, optimization, optimal control, game theory, mathematical economics, etc. The equilibrium problem is a unified model of several fundamental mathematical problems, namely, optimization problems, saddle point problems, fixed point problems, minimax inequality problems, Nash equilibrium problem, complementarity problems, variational inequality problems, etc.