Some fixed point theorems on non-convex sets

Abstract

<span style="color: #000000;">In this paper, we prove that if </span><span style="color: #008000;">$K$</span><span style="color: #000000;"> is a </span><span style="text-decoration: underline; color: #000000;">nonempty</span><span style="color: #000000;"> weakly compact set in a </span><span style="text-decoration: underline; color: #000000;">Banach</span><span style="color: #000000;"> space </span><span style="color: #008000;">$X$</span><span style="color: #000000;">, </span><span style="color: #008000;">$T:K\to K$</span><span style="color: #000000;"> is a </span><span style="text-decoration: underline; color: #000000;">nonexpansive</span><span style="color: #000000;"> map satisfying </span><span style="color: #008000;">$\frac{x+Tx}{2}\in K$</span><span style="color: #000000;"> for all </span><span style="color: #008000;">$x\in K$</span><span style="color: #000000;"> and if </span><span style="color: #008000;">$X$</span><span style="color: #000000;"> is </span><span style="color: #008000;">$3-$</span><span style="color: #000000;">uniformly convex or </span><span style="color: #008000;">$X$</span><span style="color: #000000;"> has the </span><span style="text-decoration: underline; color: #000000;">Opial</span><span style="color: #000000;"> property, then </span><span style="color: #008000;">$T$</span><span style="color: #000000;"> has a fixed point in </span><span style="color: #008000;">$K.$ <br /></span>