{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,5,31]],"date-time":"2025-05-31T04:08:21Z","timestamp":1748664501303,"version":"3.41.0"},"reference-count":9,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2025,5,20]],"date-time":"2025-05-20T00:00:00Z","timestamp":1747699200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,5,20]],"date-time":"2025-05-20T00:00:00Z","timestamp":1747699200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/100015528","name":"Universidad de la Laguna","doi-asserted-by":"crossref","id":[{"id":"10.13039\/100015528","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Optim Theory Appl"],"published-print":{"date-parts":[[2025,7]]},"abstract":"<jats:title>Abstract<\/jats:title>\n          <jats:p>Assume <jats:italic>X<\/jats:italic> is a rotund Banach space with Eisenfeld-Lakshmikantham measure of nonconvexity <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$\\nu $$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u03bd<\/mml:mi>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula>. Let <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$Y\\subset X$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>Y<\/mml:mi>\n                    <mml:mo>\u2282<\/mml:mo>\n                    <mml:mi>X<\/mml:mi>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> be nonvoid and bounded, although not necessarily convex. Then, every isometric self-map <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$f:Y\\rightarrow Y$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>f<\/mml:mi>\n                    <mml:mo>:<\/mml:mo>\n                    <mml:mi>Y<\/mml:mi>\n                    <mml:mo>\u2192<\/mml:mo>\n                    <mml:mi>Y<\/mml:mi>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> for which <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$\\lim _{n\\rightarrow \\infty }\\nu (f^n(Y))=0$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:msub>\n                      <mml:mo>lim<\/mml:mo>\n                      <mml:mrow>\n                        <mml:mi>n<\/mml:mi>\n                        <mml:mo>\u2192<\/mml:mo>\n                        <mml:mi>\u221e<\/mml:mi>\n                      <\/mml:mrow>\n                    <\/mml:msub>\n                    <mml:mi>\u03bd<\/mml:mi>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:msup>\n                        <mml:mi>f<\/mml:mi>\n                        <mml:mi>n<\/mml:mi>\n                      <\/mml:msup>\n                      <mml:mrow>\n                        <mml:mo>(<\/mml:mo>\n                        <mml:mi>Y<\/mml:mi>\n                        <mml:mo>)<\/mml:mo>\n                      <\/mml:mrow>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                    <mml:mo>=<\/mml:mo>\n                    <mml:mn>0<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> has a fixed point, under either one of two additional requirements: (<jats:italic>a<\/jats:italic>) <jats:italic>Y<\/jats:italic> is weakly compact; (<jats:italic>b<\/jats:italic>) <jats:italic>X<\/jats:italic> is reflexive, <jats:italic>Y<\/jats:italic> is closed and <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$\\lim _{n\\rightarrow \\infty }\\nu (\\widehat{Y}_n)=0$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:msub>\n                      <mml:mo>lim<\/mml:mo>\n                      <mml:mrow>\n                        <mml:mi>n<\/mml:mi>\n                        <mml:mo>\u2192<\/mml:mo>\n                        <mml:mi>\u221e<\/mml:mi>\n                      <\/mml:mrow>\n                    <\/mml:msub>\n                    <mml:mi>\u03bd<\/mml:mi>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:msub>\n                        <mml:mover>\n                          <mml:mi>Y<\/mml:mi>\n                          <mml:mo>^<\/mml:mo>\n                        <\/mml:mover>\n                        <mml:mi>n<\/mml:mi>\n                      <\/mml:msub>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                    <mml:mo>=<\/mml:mo>\n                    <mml:mn>0<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula>, where, for each <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$n\\in \\mathbb {N}$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>n<\/mml:mi>\n                    <mml:mo>\u2208<\/mml:mo>\n                    <mml:mi>N<\/mml:mi>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula>, <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$\\widehat{Y}_n$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msub>\n                    <mml:mover>\n                      <mml:mi>Y<\/mml:mi>\n                      <mml:mo>^<\/mml:mo>\n                    <\/mml:mover>\n                    <mml:mi>n<\/mml:mi>\n                  <\/mml:msub>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> consists of all those <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$z\\in Y$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>z<\/mml:mi>\n                    <mml:mo>\u2208<\/mml:mo>\n                    <mml:mi>Y<\/mml:mi>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> satisfying <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$\\rho (Y)\\le \\rho (z)\\le \\rho (Y)+n^{-1}$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>\u03c1<\/mml:mi>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:mi>Y<\/mml:mi>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                    <mml:mo>\u2264<\/mml:mo>\n                    <mml:mi>\u03c1<\/mml:mi>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:mi>z<\/mml:mi>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                    <mml:mo>\u2264<\/mml:mo>\n                    <mml:mi>\u03c1<\/mml:mi>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:mi>Y<\/mml:mi>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                    <mml:mo>+<\/mml:mo>\n                    <mml:msup>\n                      <mml:mi>n<\/mml:mi>\n                      <mml:mrow>\n                        <mml:mo>-<\/mml:mo>\n                        <mml:mn>1<\/mml:mn>\n                      <\/mml:mrow>\n                    <\/mml:msup>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> and <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$\\rho (Y)$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>\u03c1<\/mml:mi>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:mi>Y<\/mml:mi>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> denotes the Chebyshev radius of <jats:italic>Y<\/jats:italic>. More precisely, if (<jats:italic>b<\/jats:italic>) holds then <jats:italic>Y<\/jats:italic> has a unique Chebyshev center <jats:italic>c<\/jats:italic>, which is fixed by any such isometry <jats:italic>f<\/jats:italic>. Thus, previous results of Lim <jats:italic>et al.<\/jats:italic> (2003) and of Gordon (2020) are generalized by weakening the hypotheses on <jats:italic>X<\/jats:italic> (just rotundity and reflexivity instead of uniform convexity) and\/or dropping the condition that <jats:italic>Y<\/jats:italic> be convex.<\/jats:p>","DOI":"10.1007\/s10957-025-02703-7","type":"journal-article","created":{"date-parts":[[2025,5,20]],"date-time":"2025-05-20T15:28:30Z","timestamp":1747754910000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Fixed Points for Isometries on Rotund Banach Spaces Without Convexity"],"prefix":"10.1007","volume":"206","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9421-9198","authenticated-orcid":false,"given":"Isabel","family":"Marrero","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,5,20]]},"reference":[{"key":"2703_CR1","doi-asserted-by":"publisher","first-page":"655","DOI":"10.1080\/00029890.1971.11992823","volume":"78","author":"J Baker","year":"1971","unstructured":"Baker, J.: Isometries in normed spaces. 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