In this study, we present a link between approximation theory and summability methods by constructing bivariate Bernstein-Kantorovich type operators on an extended domain with reparametrized knots. We use a statistical convergence type and power series method to obtain certain Korovkin type theorems, and we study certain rates of convergences related to these summability methods. Furthermore, we numerically analyze the theoretical results and provide some computer graphics to emphasize the importance of this study.

Integral equations, which are defined as “the equation containing an unknown function under the integral sign”, have many applications of real-world problems. The second type of Fredholm integral equations is generally used in radiation transfer theory, kinetic theory of gases, and neutron transfer theory. A special case of these equations, known as the quadratic Chandrasekhar integral equation, given by $x\left(s\right)=1+\lambda x\left(s\right){\int }_0^1\frac{s}{t+s}x\left(t\right)dt,$ can be very often encountered in many applications, where x is the function to be determined, $\lambda$ is a parameter, and $t,s\in [0,1]$. In this paper, using a fixed-point theorem, the existence conditions for the solution of Fredholm integral equations of the form $\chi \left(l\right)=\varrho \left(l\right)+\chi \left(l\right){\int }_p^{q}k(l,z)\left(V\chi \right)\left(z\right)dz$ are investigated in the space $C_{\omega }\left[p,q\right],$ where $\chi$ is the unknown function to be determined, V is a given operator, and $\varrho ,k$ are two given functions. Moreover, certain important applications demonstrating the applicability of the existence theorem presented in this paper are provided.

This study investigates the shape-preserving characteristics of 𝜆-Schurer operators, a class of operators derived from a modified version of the classical Schurer bases by incorporating a shape parameter 𝜆. The primary focus is on understanding how these operators maintain the geometric features of the functions they approximate, which is crucial in fields like computer graphics and geometric modelling. By examining the fundamental properties and the divided differences associated with 𝜆-Schurer bases, we derive vital results that confirm the operators’ capability to preserve essential shape attributes under various conditions. The findings have significant implications for the application of these operators in computational analysis and other related areas, providing a solid foundation for future research.