Study of approximation properties of nonlinear operators via summability process

In this thesis, the nonlinear integral operators of the convolution type are investigated with the help of summability methods. More general versions of the known approximation results are obtained through the use of summability methods. As known, a summability method can make a divergent sequence convergent (in some sense) and may accelerate the rate of convergence of a convergent sequence. For example, although the Fourier series of a continuous and periodic function may be divergent, its arithmetic mean (that is, Cesàro mean) always approaches to the function itself. In another example, a suitable sub-sequence matrix transformation of a convergent sequence accelerates the rate of convergence of the sequence. All these cases reveal the importance of summability theory. The theory of summability has so far been involved in many areas of mathematics, particularly in the approximation theory. For instance, it is used for the principle of analytic continuation in function theory, for the production of iterative methods for the solution of linear equation systems in applied mathematics and for the convergence of Fourier series in the approximation theory as mentioned above. In this thesis, a general summability method defined by Bell in 1971 is used in the convergence of nonlinear operators for the first time. In this way, Angeloni and Vinti's results which are obtained in 2006 are improved. In this study, the family of non-negative regular matrices are considered. These methods contain not only the classical convergence but also the arithmetic mean convergence given by Cesàro, the almost convergence introduced by Lorentz, and the order summability defined by Jurkat and Peyerimhoff. With nonlinear operators, we first approximate to univariate and 2pi-periodic functions and then to multivariate functions. In the approximations, both variation semi-norm and classical supremum norm will be considered. We use functions of bounded variations in the variation seminorm, while uniformly continuous functions in the supremum norm. After that, rates of convergence are calculated with the help of suitable Lipschitz classes. Using the estimation results obtained, some characterizations of absolute continuity and uniform continuity are given. Finally, in order to support our approximation results we construct some sequences of operators and display their graphs and also compute the error estimations numerically. Considering the investigation on nonlinear operators and using summability methods effectively, the results obtained in this thesis offer an original contribution to the literature and we expect that it provides a basis for studies in the future.