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Approximation of common solution of finite family of monotone inclusion and fixed point problems for demicontractive multivalued mappings in CAT(0) spaces

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Abstract

In this paper, we use the gate condition on two multivalued k-demicontractive mappings to approximate a common solution of a finite family of monotone inclusion problem and fixed point problem in CAT(0) space. Furthermore, we propose a Halpern-type proximal point algorithm and prove its strong convergence to a common solution of a finite family of monotone inclusion problems and fixed point problem for two multivalued k-demicontractive mappings in a complete CAT(0) space. We also applied our result to the problem of finding a common solution of a finite family of minimization problem and fixed point problem in CAT(0) space. Finally, numerical experiments of our result are presented to further show its applicability.

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References

  1. Aremu, K.O., Izuchukwu, C., Ugwunnadi, G.C., Mewomo, O.T.: On the proximal point algorithm and demimetric mappings in CAT(0) spaces. Demonstr. Math. 51, 277–294 (2018)

    MathSciNet  MATH  Google Scholar 

  2. Abass, H., Okeke, C.C., Mewomo, O.T.: On split equality mixed equilibrium and fixed point problems of countable families of generalized \(k\)-strictly pseudocontractive mappings. Dym. Contin. Discrete Impul. Syst. Ser B. Appl. Algorithms 25, 369–395 (2018)

    MATH  Google Scholar 

  3. Bacak, M.: The proximal point algorithm in metric spaces. Israel J. Math. 194, 689–701 (2013)

    MathSciNet  MATH  Google Scholar 

  4. Bartolini, I., Ciaccia, P., and Pattela, M.: String Matching with Trees Using an Approximate Distance. SPIR Lecture Notes in Computer Science, vol. 2476. Spring, Berlin (1999)

  5. Berg, I.D., Nikolaev, I.G.: Quasilinearization and curvature of Alexandrov spaces. Geom. Dedicata 133, 195–218 (2008)

    MathSciNet  MATH  Google Scholar 

  6. Bestvina, M.: \(\mathbb{R}-\)trees in topology, geometry and group theory. In: Sher, R.B., Daverman, R.J. (eds.) Handbook of Geometric Topology North-Holland, pp. 55–91. Elsevier, Amsterdam (2002)

    Google Scholar 

  7. Bridson, M.R., Haeiger, A.: Metric Spaces of Non-Positive Curvature, Fundamental Principle of Mathematical Sciences, vol. 319. Springer, Berlin, Germany (1999)

    Google Scholar 

  8. Brown, K.S.: Buildings. Springer, New York, NY (1989)

    MATH  Google Scholar 

  9. Bruhat, F., and Tits, J.: Groupes réductits sur un cor local, I. Donneés Radicielles Valueés, Institut. des Hautes Études Scientifiques, 41, 5–251 (1972)

  10. Chabrowski, J.H.: On the existence of a solution to a class of variational inequalities. Ricerche Mat. 60(2), 333–350 (2011)

    MathSciNet  MATH  Google Scholar 

  11. Chidume, C. E., Bello, A .U., and Ndambomve, P: Strong and \(\Delta \) -convergence theorems for common fixed points of a finite family of multivalued demicontractive mappings in CAT(0) spaces. Abstr. Appl. Anal. 2014, 805168 (2014)

  12. Combettes, P.L.: Monotone operator theory in convex optimization. Math. Program. Ser. B 170, 177–206 (2018)

    MathSciNet  MATH  Google Scholar 

  13. Dehghan, H., Izuchukwu, C., Mewomo, O.T., Taba, D.A., Ugwunnadi, G.C.: Iterative algorithm for a family of monotone inclusion problems in CAT(0) spaces. Quaest. Math. (2019). https://doi.org/10.2989/16073606.20

  14. Dehghan, H., and Rooin, J: Metric projection and convergence theorems for nonexpansive mapping in Hadamard spaces. arXiv:1410.1137VI [math.FA], 5 Oct. (2014)

  15. Dhompongsa, S., Kirk, W.A., Panyanak, B.: Nonexpansive set-valued mappings in metric and Banach spaces. J. Nonlinear Convex Anal. 8, 35–45 (2007)

    MathSciNet  MATH  Google Scholar 

  16. Dhompongsa, S., Panyanak, B.: On \(\Delta \)-convergence theorems in CAT(0) spaces. Comp. Math. Appl. 56, 2572–2579 (2008)

    MathSciNet  MATH  Google Scholar 

  17. Espinola, R., Kirk, W.A.: Fixed point theorems in \(\mathbb{R}-\)trees with applications to graph theory. Topol. Appl. 153(7), 1046–1055 (2006)

    MathSciNet  MATH  Google Scholar 

  18. Ghoussoub, N.: Self-Dual Partial Differential Systems and Their Variational Principles. Springer, New York (2009)

    MATH  Google Scholar 

  19. Goebel, K., Reich, S.: Uniform Convexity, Hyperbolic Geometry and Nonexpansive Mappings. Marcel Dekker, New York (1984)

    MATH  Google Scholar 

  20. Heydari, M.T., Khadem, A., Ranjbar, S.: Approximating a common zero of finite family of monotone operators in Hadamard spaces. Optimization 66(12), 2233–2244 (2017)

    MathSciNet  MATH  Google Scholar 

  21. Izuchukwu, C., Aremu, K.O., Mebawondu, A.A., Mewomo, O.T.: A viscosity iterative technique for equilibrium and fixed point problems in a Hadamard space. Appl. Gen. Topol. 20(1), 193–210 (2019)

    MathSciNet  MATH  Google Scholar 

  22. Izuchukwu, C., Ugwunnadi, G.C., Mewomo, O.T., Khan, A.R., Abbas, M.: Proximal-type algorithms for split minimization problem in p-uniformly convex metric space. Numer. Algorithms (2018). https://doi.org/10.1007/s11075-018-0633-9

  23. Jolaoso, L.O., Ogbuisi, F.U., Mewomo, O.T.: An iterative method for solving minimization, variational inequality and fixed point problems in reflexive Banach spaces. Adv. Pure Appl. Math. 9(3), 167–184 (2017)

    MathSciNet  MATH  Google Scholar 

  24. Jolaoso, L.O., Oyewole, K.O., Okeke, C.C., Mewomo, O.T.: A unified algorithm for solving split generalized mixed equilibrium problem and fixed point of nonspreading mapping in Hilbert space. Demonstr. Math. 51, 211–232 (2018)

    MathSciNet  MATH  Google Scholar 

  25. Kakavandi, B.A., Amini, M.: Duality and subdifferential for convex functions on complete CAT(0) metric spaces. Nonlinear Anal. 73, 3450–3455 (2010)

    MathSciNet  MATH  Google Scholar 

  26. Khatibzadeh, H., Ranjbar, S.: Monotone operators and the proximal point algorithm in complete CAT(0) metric spaces. J. Aust. Math. Soc. 103, 70–90 (2017)

    MathSciNet  MATH  Google Scholar 

  27. Kirk, W.A.: Fixed point theorems in CAT(0) spaces and \(\mathbb{R}\)-trees. Fixed Point Theory Appl. 2004(4), 309–316 (2004)

    MathSciNet  MATH  Google Scholar 

  28. Kirk, W.A.: Some recent results in metric fixed point theory. Fixed Point Theory Appl. 2, 195–207 (2007)

    MathSciNet  MATH  Google Scholar 

  29. Kōmura, Y.: Nonlinear semi-groups in Hilbert space. J. Math. Soc. Jpn. 19, 493–507 (1967)

    MathSciNet  MATH  Google Scholar 

  30. Kumam, P., and Chaipunya, P.: Equilibrium Problems and Proximal Algorithms in Hadamard Spaces, arXiv: 1807.10900v1 [math.oc]. Accessed 28 Mar 2018

  31. Laowang, W., Panyanak, B.: Strong and \(\Delta -\)convergence theorems for multivalued mappings in CAT(0) spaces. J. Inequal. Appl. 1, 730132 (2009)

    MathSciNet  MATH  Google Scholar 

  32. Leustean, L.: Nonexpansive iterations uniformly cover W-hyperbolic spaces. Nonlinear analysis and optimization 1: nonlinear analysis. Contemp. Math. Am. Math. Soc. 513, 193–209 (2010)

    MATH  Google Scholar 

  33. Maingé, P.E.: Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization. Set-Valued Anal. 16, 899–912 (2008)

    MathSciNet  MATH  Google Scholar 

  34. Markin, J.T.: Fixed points, selections and best approximation for multivalued mappings in \(\mathbb{R}-\)trees. Nonlinear Anal. 67, 2712–2716 (2007)

    MathSciNet  MATH  Google Scholar 

  35. Martinet, B.: Regularisation d’ inequations varaiationnelles par approximations successives. Rev. Fr. Inform. Rec. Oper. 4, 154–158 (1970)

    MATH  Google Scholar 

  36. Mewomo, O.T., Ogbuisi, F.U.: Convergence analysis of iterative method for multiple set split feasibility problems in certain Banach spaces. Quaest. Math. 41(1), 129–148 (2018)

    MathSciNet  MATH  Google Scholar 

  37. Moreau, J.J.: Fonctionnelles sous-différentiables. C.R. Acad. Sci. Paris A257, 4117–4119 (1963)

    MATH  Google Scholar 

  38. Okeke, C.C., Izuchukwu, C.: A strong convergence theorem for monotone inclusion and minimization problems in complete CAT(0) spaces. Optim. Methods Softw. (2018). https://doi.org/10.1080/10556788.2018.1472259

    Article  MATH  Google Scholar 

  39. Phuengrattana, W.: Approximation of common fixed points of two strictly pseudononspreading multivalued mappings in \(\mathbb{R}-\)trees. Kyungpook Math. J. 55, 378–382 (2015)

    MathSciNet  MATH  Google Scholar 

  40. Puttansontiphot, T.: Mann and Ishikawa iteration schemes for multivalued mappings in CAT(0) spaces. Appl. Math. Sci. 4(61), 3005–3018 (2010)

    MathSciNet  MATH  Google Scholar 

  41. Ranjbar, S., Khatibzadeh, H.: Strong and \(\Delta -\)convergence to a zero of a monotone operator in CAT(0) spaces. Mediterr. J. Math. (2017). https://doi.org/10.1007/s00009-017-0885-y

  42. Reich, S., Shafrir, I.: Nonexpansive iterations in hyperbolic spaces. Nonlinear Anal. 15, 537–558 (1990)

    MathSciNet  MATH  Google Scholar 

  43. Rockafellar, R.T.: Monotone Operators and the proximal point algorithm. SIAM J. Control Optim. 14, 877–898 (1976)

    MathSciNet  MATH  Google Scholar 

  44. Rockafellar, R.T.: Characterization of the subdifferentials of convex function. Pac. J. Math. 17, 497–510 (1966)

    MathSciNet  MATH  Google Scholar 

  45. Samanmit, K.: A convergence theorem for finite family of multivalued a \(k-\)strictly pseudononspreading mappings in \(\mathbb{R}-\)trees. Thai J. Math. 13(3), 581–591 (2015)

    MathSciNet  MATH  Google Scholar 

  46. Samanmit, K., Panyanak, B.: On multivalued nonexpansive mappings in \(\mathbb{R}\)trees. J. Appl. Math. 38(6), 13 (2012)

    MATH  Google Scholar 

  47. Semple, C., Steel, M.: Phylogenetics. Oxford Lecture Series in Mathematics and Its Applications, vol. 24. Oxford University Press, Oxford (2003)

    MATH  Google Scholar 

  48. Senakka, P., Cholamjiak, P.: Approximation method for solving fixed point problem of Bregman strongly nonexpansive mappings in reflexive Banach spaces. Ricerche Mat. 65(1), 206–220 (2016)

    MathSciNet  MATH  Google Scholar 

  49. Shahzad, N.: Fixed points for multimaps in CAT(0) spaces. Topol. Appl. 156(5), 997–1001 (2009)

    MathSciNet  MATH  Google Scholar 

  50. Shehu, Y., Mewomo, O.T.: Further investigation into split common fixed point problem for demicontractive operators. Acta Math. Sin. (Engl. Ser.) 32(11), 1357–1376 (2016)

    MathSciNet  MATH  Google Scholar 

  51. Tufa, A.R., Zegeye, H.: Krasnoselskii–Mann method for multivalued non self mappings in CAT(0) spaces. Filomat 31(14), 4629–4640 (2017)

    MathSciNet  Google Scholar 

  52. Tufa, A.R., Zegeye, H., Thuto, M.: Convergence theorems for non self mappings in CAT(0) spaces. Numer. Funct. Anal. Optim. 38(6), 705–722 (2017)

    MathSciNet  MATH  Google Scholar 

  53. Ugwunnadi, G.C., Izuchukwu, C., Mewomo, O.T.: Proximal point algorithm involving fixed point of nonexpansive mapping in p-uniformly convex metric space. Adv. Pure Appl. Math. (2018). https://doi.org/10.1515/apam-2018-0026

  54. Ugwunnadi, G.C., Izuchukwu, C., Mewomo, O.T.: Strong convergence theorem for monotone inclusion problem in CAT(0) Spaces. Afr. Mat. 31(1–2), 151–169 (2019)

    MathSciNet  MATH  Google Scholar 

  55. Vaĭnberg, M.M.: Variational Method and Method of Monotone Operators in the Theory of Nonlinear (Equations English translation). Wiley, New York (1973)

    MATH  Google Scholar 

  56. Xu, H.K.: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. 66, 240–256 (2002)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors thank the anonymous referee for valuable and useful suggestions and comments which led to the great improvement of the paper.

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  1. School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban, South Africa

    K. O. Aremu, L. O. Jolaoso, C. Izuchukwu & O. T. Mewomo

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  1. K. O. Aremu
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  2. L. O. Jolaoso
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  3. C. Izuchukwu
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  4. O. T. Mewomo
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Correspondence to O. T. Mewomo.

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Aremu, K.O., Jolaoso, L.O., Izuchukwu, C. et al. Approximation of common solution of finite family of monotone inclusion and fixed point problems for demicontractive multivalued mappings in CAT(0) spaces. Ricerche mat 69, 13–34 (2020). https://doi.org/10.1007/s11587-019-00446-y

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