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Pythagorean Property and Best-Proximity Point Theorems

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Abstract

In this paper, a notion called proximally complete pair of subsets of a metric space is introduced, which weakens earlier notions in the theory of best-proximity points. By means of this notion, existence and convergence results of best-proximity points are proven for cyclic contraction mappings, which extent other recent results. By observing geometrical properties of Hilbert spaces, the so-called Pythagorean property is introduced. This property is employed to provide sufficient conditions for a cyclic map to be a cyclic contraction.

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Acknowledgments

The authors would like to thank the anonymous referee for the attentive reading of the submitted manuscript. Rafael Espínola was supported by DGES, Grant MTM2012-34847C02-01 and Junta de Andalucía, Grant FQM-127. G. Sankara Raju Kosuru would like to thank Council of Scientific and Industrial Research (CSIR), National Board of Higher Mathematics (NBHM) and Department of Atomic Energy (DAE), India for financial support.

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Authors and Affiliations

  1. Departamento de Análisis Matemático, Facultad de Matemáticas, IMUS, Universidad de Sevilla, 41010 , Sevilla, Spain

    Rafael Espínola

  2. Mathematics and Statistics Unit, Indian Statistical Institute Bangalore, R. V. College post, Bangalore, 560 059, India

    G. Sankara Raju Kosuru

  3. Department of Mathematics, Indian Institute of Technology Madras, Chennai, 600 036, India

    P. Veeramani

Authors
  1. Rafael Espínola
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  2. G. Sankara Raju Kosuru
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  3. P. Veeramani
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Correspondence to Rafael Espínola.

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Communicated by Stefan Rolewicz.

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Espínola, R., Kosuru, G.S.R. & Veeramani, P. Pythagorean Property and Best-Proximity Point Theorems. J Optim Theory Appl 164, 534–550 (2015). https://doi.org/10.1007/s10957-014-0583-x

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  • DOI: https://doi.org/10.1007/s10957-014-0583-x

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