The origin of fixed-point theory lies in the strategy of progressive approximation utilized to demonstrate the existence of solutions of differential equations first presented in the 19th century. However, classical fixed-point theory was established as an important part of mathematical analysis in the early 20th century, by mathematicians including Luitzen Egbertus Jan Brouwer, Stephan Banach, and Juliusz Schauder. In 1922, Stephan Banach gave the condition under which the presence, as well as the uniqueness, of a fixed point is ensured.

Different outcomes relating to fixed points, fixed circles, fixed discs, fixed ellipses, and fixed elliptic discs for single-valued and multi-valued mappings have been explored for mappings that satisfy distinctive contractive conditions in various settings, and this practice is ongoing. The celebrated Banach contraction principle has several applications in solving nonlinear equations. Yet, one disadvantage is that contraction mapping forces involved mapping to be continuous throughout the metric space. It is interesting to mention here that discontinuous mapping is of great interest, as most phenomena in the real world are discontinuous in nature. In a variety of scientific issues, beginning with different branches of mathematics, the existence of a solution is comparable to the existence of a unique fixed point for a suitable mapping. Fixed-point theorems give adequate conditions under which there exists a unique fixed point for a given function, enabling us to ensure the existence of a solution to the original problem. One of the significant outcomes of fixed-point theory is when mapping under minimal suitable conditions has a unique fixed point. However, in some situations, mapping may not have a unique fixed point. Non-unique fixed points play a crucial role and it is essential to discover the necessary conditions under which mappings have non-unique fixed points and collections of non-unique fixed points include some geometric shape. In particular, non-unique fixed points of discontinuous self-mapping perform an essential role, as if the fixed point is not unique, then the set of non-unique fixed points may form a circle, disc, or ellipse, which has great potential for applications in a variety of fields.

The aim of this Special Issue is to gather research into unique and non-unique fixed points, as well as papers investigating their applications. We welcome both original research and review articles.

Potential topics include but are not limited to the following:

• The existence of discontinuity at the fixed figure and its applications
• Discontinuity, fixed points, and their applications
• Generalizations of a Banach contraction and their applications
• Geometric properties of non-unique fixed points in different spaces and related applications
• Fixed point theorems for multi-valued mappings in different spaces and applications
• Common fixed-point theorems in different spaces and their applications
• Non-unique fixed-point theorems satisfying distinctive contractive conditions and their applications
• Fixed point to fixed disc, fixed circle, fixed ellipse, or elliptic disc and related applications