Fixed point properties for Lorentz sequence spaces $$\ell _{\rho ,\infty }^0$$ and $$\ell _{\rho ,1}$$

Abstract

In this paper, we work on Lorentz sequence spaces and explore the fixed point property for $$\ell _{\rho ,\infty }^0$$ and $$\ell _{\rho ,1}$$ space. We prove that both spaces enjoy the weak fixed point property for nonexpansive mappings. To prove the weak fixed point property for $$\ell _{\rho ,1}$$, we provide alternative methods such that one gives exact value of its Riesz angle. We show that both spaces fail the fixed point property for nonexpansive mappings since $$\ell _{\rho ,1}$$ contains an asymptotically isometric copy of $$\ell ^1$$ and $$\ell _{\rho ,\infty }^0$$ contains an asymptotically isometric copy of $$c_0$$. We also show that there exists a non-empty, closed, bounded and convex subset C of $$\ell _{\rho ,1}$$ and a fixed point-free affine, nonexpansive mapping $$T:C\longrightarrow C$$ and so $$\ell _{\rho ,1}$$ fails to have the fixed point property for affine nonexpansive mappings. Contrary to this, in the final section, we get a Goebel and Kuczumow analogy for $$\ell _{\rho ,1}$$ by proving that there exists a large class of non-weak* compact, closed, bounded and convex subsets of $$\ell _{\rho ,1}$$ with the fixed point property for affine nonexpansive mappings.