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On the strong convergence of the proximal point algorithm with an application to Hammerstein euations

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Abstract

Let E be a real normed space. A new notion of quasi-boundedness for operators \(A:E\rightarrow 2^E\) is introduced and the following general important result for accretive operators is proved: an accretive operator with zero in the interior of its domain is quasi-bounded. Using this result, a new strong convergence theorem for approximating a zero of an m-accretive operator is proved in a uniformly smooth real Banach space. This result complements the celebrated proximal point algorithm for approximating solutions of \(0\in Au\) in a real Hilbert space where A is a maximal monotone operator. Furthermore, as an application of our theorem, a new strong convergence theorem for approximating a solution of a Hammerstein equation is proved. Finally, several numerical experiments are presented to illustrate the strong convergence of the sequence generated by our algorithm and the results obtained are compared with those obtained using some recent important algorithms.

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Acknowledgements

The authors would like to thank the referees for their comments and suggestions which helped in the to improve the presentation of this paper.

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Authors and Affiliations

  1. African University of Science and Technology, Abuja, Nigeria

    C. E. Chidume, A. Adamu & U. V. Nnyaba

  2. Bayero University, Kano, Kano, Nigeria

    M. S. Minjibir

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  1. C. E. Chidume
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  2. A. Adamu
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  3. M. S. Minjibir
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  4. U. V. Nnyaba
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Correspondence to C. E. Chidume.

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Chidume, C.E., Adamu, A., Minjibir, M.S. et al. On the strong convergence of the proximal point algorithm with an application to Hammerstein euations. J. Fixed Point Theory Appl. 22, 61 (2020). https://doi.org/10.1007/s11784-020-00793-6

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