Fixed point theorem in a uniformly convex paranormed space and its application (Q1939212)

Let \((X,p)\) be a complete uniformly convex paranormed space with a continuous modulus of convexity and \(C\subset X\) be a nonempty bounded closed convex subset. The main results in the paper are as follows. Theorem 1. Let \(T:C\to C\) be nonexpansive (i.e., \(p(Tx-Ty)\leq p(x-y)\) for all \(x,y\in C\)). Then \(T\) has a fixed point. An application of this result is given to the functional equation \(x(\tau)=h(\tau,x(f(\tau)))\) over the measure subspace \(S^\varphi(\Omega,\Sigma,\mu)\). Further, as a by-product of Theorem 1, the following fixed point result is given. Theorem 2. Let \((X,p)\) and \(C\) be as before and \(T:C\to C\) be radially continuous at all points, except for at most two. If there exist two sequences of positive numbers \((c_n)\) and \((t_n)\) with \(\liminf_n c_n=1\), \(\lim_n t_n=0\) such that for all \(n\in \mathbb{N}\) and for all \(x,y\in C\), \(p(x-y)=t_n\) implies \(p(Tx-Ty)\leq c_nt_n\), then \(T\) has a fixed point. In particular, when \(X\) is a uniformly convex Banach space, one obtains the usual Browder-Göhde-Kirk fixed point theorem.