New discontinuity and fixed disc results via meir-keeler and caristi techniques on metric spaces

Abstract

Metric fixed-point theory has been extensively studied with effective approaches. There are some open problems about fixed-point theory. One of them is the Rhoades’ discontinuity problem and another is the fixed-circle (or fixed-figure) problem. In this paper, we focus on these two open problems on metric spaces. To give some solutions, we combine Caristi and Meir-Keeler techniques. So, we present new answers to the stated problems.