Analysis of the solvability and stability of the operator-valued Fredholm integral equation in Hölder space

In this paper, the solvability of an operator-valued integral equation in Hölder spaces, i.e.,

$$w({\zeta }_1)=y({\zeta }_1)+w({\zeta }_1){\int }_{\mathbf{J}}\kappa ({\zeta }_1,\phi )(T_{1}w)(\phi )d\phi +z({\zeta }_1){\int }_{\mathbf{J}}h(\phi ,(T_{2}w)(\phi ))d\phi ,$$

for ${\zeta }_1\in \mathbf{J}=[0,1],$ is studied by using Darbo's fixed point theorem (FPT). The process of the measure of noncompactness of the operators which constitute an intermediary of contraction and compact mappings can be explained with the help of Darbo's FPT. The greater effectiveness of Darbo's FPT due to its non-involvement of the compactness property gives a better scope when dealing with the Schauder FPT, where compactness is an essential property. To obtain a unique solution, we apply the Banach fixed point theorem and discuss the Hyers-Ulam stability of the integral equation. We also give some important examples to illustrate the existence and uniqueness of the results.