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2021•arXiv (Cornell University)Open access

Fixed points of asymptotically nonexpansive mappings with center 0 and\n applications

Abdelkader Dehici, Sami Atailia, Redjel, Najeh

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Abstract

In this paper, we investigate the existence of fixed points for\nasymptotically nonexpansive mappings with center 0 defined on closed convex\nsubsets of various Banach spaces. Three applications are given. Firstly, we\nprove that our results refine those concerning alternate convexically\nnonexpansive (in short; ACN) mappings studied by P. N. Dowling in " On a fixed\npoint result of Amini-Harandi in strictly convex Banach spaces, Acta. Math.\nHungar., 112 (1-2), (2006), 85-88" . Secondly, by using Lau's result in "\nClosed convex invariant subsets of $L_p(G)$, Trans. Amer. Math. Soc., {\\bf\n232}, (1977), 131-142", we give another characterization for the noncompactness\nof locally compact groups $G$. Finally, we discuss the existence of a solution\nfor a nonlinear transport equation without using compactness results.\n

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In this paper, we investigate the existence of fixed points for\nasymptotically nonexpansive mappings with center 0 defined on closed convex\nsubsets of various Banach spaces. Three applications are given. Firstly, we\nprove that our results refine those concerning alternate convexically\nnonexpansive (in short; ACN) mappings studied by P. N. Dowling in " On a fixed\npoint result of Amini-Harandi in strictly convex Banach spaces, Acta. Math.\nHungar., 112 (1-2), (2006), 85-88" . Secondly, by using Lau's result in "\nClosed convex invariant subsets of $L_p(G)$, Trans. Amer. Math. Soc., {\\bf\n232}, (1977), 131-142", we give another characterization for the noncompactness\nof locally compact groups $G$. Finally, we discuss the existence of a solution\nfor a nonlinear transport equation without using compactness results.\n

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Available abstract

In this paper, we investigate the existence of fixed points for\nasymptotically nonexpansive mappings with center 0 defined on closed convex\nsubsets of various Banach spaces. Three applications are given. Firstly, we\nprove that our results refine those concerning alternate convexically\nnonexpansive (in short; ACN) mappings studied by P. N. Dowling in " On a fixed\npoint result of Amini-Harandi in strictly convex Banach spaces, Acta. Math.\nHungar., 112 (1-2), (2006), 85-88" . Secondly, by using Lau's result in "\nClosed convex invariant subsets of $L_p(G)$, Trans. Amer. Math. Soc., {\\bf\n232}, (1977), 131-142", we give another characterization for the noncompactness\nof locally compact groups $G$. Finally, we discuss the existence of a solution\nfor a nonlinear transport equation without using compactness results.\n

Key concepts: Fixed point, Mathematics, Banach space, Regular polygon, Pure mathematics, Compact space, Invariant (physics), Center (category theory)

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