Approximation properties of nonlinear Durrmeyer type operators

This thesis deals with the approximation properties of the nonlinear Durrmeyer type operators in the approximation theory of mathematical analysis. The thesis comprises six chapters in total. The first chapter presents the historical development of the solution, beginning with Bernstein polynomials and addressing the problem posed by the German mathematician Karl Weierstrass. In the second chapter, fundamental definitions essential for the thesis are provided, along with auxiliary results necessary for the fundamental theorems concerning the approximation problems addressed in the thesis. Moving to the third chapter, crucial definitions for constructing nonlinear Durrmeyer-type operators are introduced, along with accompanying auxiliary theorems derived from these definitions. Furthermore, the chapter includes the statement and proof of the existence theorem for these operators. In the fourth chapter, where the convergence results of the thesis are given, the convergence of the operators is discussed at the Lebesgue point, then the convergence properties are analyzed at the points of discontiunity of the functions ψ∘|f| and f respectively. Finally, the convergence results for these operators are given when the functions f and ψ∘|f| are derivatives of bounded variation. In the fifth chapter of the thesis, a conclusion about the thesis and ideas for future work are given. The final chapter of the thesis, Chapter six, includes the references that were utilized throughout the study.