Some fixed point theorems in logarithmic convex structures

Abstract

In this paper, we introduce the concept of a logarithmic convex structure. Let X be a set and D: X x X -> [1, infinity) a function satisfying the following conditions: (i) For all x,y is an element of X, D(x,y) >= 1 and D(x,y)= 1 if and only if x = y. (ii) For all x,y is an element of X, D(x,y)= D(y,x). (iii) For all x,y,z is an element of X, D(x,y) D(x,z) <= (z,y). (iv) For all x,y,z is an element of X, z not equal x,y and lambda is an element of(0, 1), D(z,W (x,y, lambda)) <= D-lambda (x, z)D1-lambda(y, z), D(x,y)= D(x,W(x,y,lambda))D(y,W(x,y, lambda)), where W: X x X x [0, 1] -> X is a continuous mapping. We name this the logarithmic convex structure. In this work we prove some fixed point theorems in the logarithmic convex structure.