The ordered implicit relations and related fixed point problems in the cone $b$-metric spaces

In this paper, we introduce an ordered implicit relation. We present some examples for the illustration of the ordered implicit relation. We investigate conditions for the existence of the fixed points of an implicit contraction. We obtain some fixed point theorems in the cone $b$-metric spaces and hence answer a fixed-point problem. We present several examples and consequences to explain the obtained theorems. We solve an homotopy problem and show existence of solution to a Urysohn Integral Equation as applications of the obtained fixed point theorem.