The utilization of Bernstein polynomials in constructing and numerically integrating bivariate fractal interpolation functions is a prominent research focus. It involves the construction and numerical integration of these functions over various domains. Furthermore, a significant aspect of this field pertains to the fixed-point results derived for the sum of two operators and their application in solving differential equations. In a related context, the existence and uniqueness of a coupled best attractor for proximal Iterated Function Systems (IFS) are explored. This concept plays a fundamental role in understanding the behavior of fractal interpolations. Additionally, the investigation into the existence of common best proximity points within a topological space is a crucial topic. This endeavor aims to establish foundational principles and theorems in mathematical analysis and functional analysis, enhancing our understanding of these intricate mathematical concepts.

Vision

• Construction and Numerical Integration of Bivariate Fractal Interpolation Functions on various domains.
• The fixed-point result for the sum of two operators and its application to differential equations.

Mission

• Bernstein Polynomials in the Construction and Numerical Integration of Bivariate Fractal Interpolation Functions.
• Existence and uniqueness of a coupled best attractor for proximal IFS.
• Existence common best proximity point in topological space.

Proposed initiative

• Bernstein Polynomials in the Construction and Numerical Integration of Bivariate Fractal Interpolation Functions.
• To find the fixed-point result for the sum of two operators and its application to differential equations.
• To extend our best proximity point result “Poonguzali, G., Golubović, Z., Radenović, S. Existence of best proximity points for the sum of two operators (2023) Filomat, 37 (23), pp. 7831-7837” to a larger group of maps.
• We are trying to work on the existence and uniqueness of fixed points for some special type of contractive map in different types of metric spaces.
• Existence of fractal via best proximity point.
• Existence and uniqueness of a coupled best attractor for proximal IFS.
• Existence common best proximity point in topological space.