Solvability of some integral equations in Hölder spaces

In the first part of this thesis, which consists of four chapters, general information about the historical development and usage areas of integral equations and some definitions about these equations were given, and the place and importance of the thesis study in the literature was emphasized. In the second part of the thesis, some definitions and some properties were given about both the Hα[a,b] Hölder space and the Cω[a,b] spaces formed by the modulus of continuity ω, which are related to the (3.0.1) and (4.0.1) equations. Some basic definitions and theorems were given to help understand the other chapters. In addition, some theorems were given with their proofs. The third and fourth chapters constitute the original part of the study. In the third part of the thesis, by researching the recent studies and developments; An existence theorem was given for the solution of the (3.0.1) Fredholm type quadratic integral equation, which is more general than the previously studied (1.1.15)-(1.1.17) equations in the Hölder spaces, and two examples of the applicability of this theorem was presented. In the fourth part of the thesis, the solution of the (4.0.1) Fredholm type quadratic integral equation, which is partly more general of the equation we discussed in the third part of the thesis, was investigated in Cω[a,b] space, which is different from the Hα[a,b] Hölder space in the third part of the thesis, and an existence theorem was obtained. Relative compactness in Cω[a,b] space and Schauder fixed point theorem were used to reach this conclusion. Afterwards, an example of the applicability of this theorem was presented.