Fixed point theory in [alpha]-complete metric spaces with applications

The aim of this paper is to introduce new concepts of [alpha]-[eta]-complete metric space and [alpha]-[eta]-continuous function and establish fixed point results for modified [alpha]-[eta]-[psi]-rational contraction mappings in [alpha]-[eta]-complete metric spaces. As an application, we derive some Suzuki type fixed point theorems and new fixed point theorems for [psi]-graphic- rational contractions. Moreover, some examples and an application to integral equations are given here to illustrate the usability of the obtained results. This paper is dedicated to Professor Miodrag Mateljevic on the occasion of his 65th birthday