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Open Access Research Article

An approximating solution of the hierarchical fixed-point problem

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pp. 741–753Vol. 28Issue 3-AApril 2025DOI: 10.47974/JIM-1969XML
Received:
03 May 2024
Published Online:
08 May 2025
Article type:
Research Article
Language:
EN
Article no.:
JIM-1969
Pages:
741–753

Abstract

In Hilbert spaces, a problem of variational inequality is termed hierarchical if is treated on the collection of common fixed points of mappings. This article is devoted to studying the situation of this type of inequality by projection method and approximating of contraction algorithm to a fixed point. This is a way for solving some types of nonlinear problems. Here, a new iterative projection sequence is presented, and then the examination of the strong convergence of this sequence to a solution of hierarchical-variational inequality (HFI) is studied under some appropriate conditions. Current iterative sequence constructs for three elements, which are mean non-expansive mapping, non-expansive mapping and projection mapping considered on nonvoid and closed subset of a real Hilbert space. We also established strong convergence results for the iterative projection sequence to obtain an approximation to the solution of HFI by showing that the sequence is an asymptotic regularity and demi-closedness principle, and another strong convergence result for a more general iterative projection sequence for the same three mappings defined above is studied.

Keywords

Subject Classifications

47J2049J40

References

[1] G. Stampacchia, “Formes bilineaires coercitives sur les ensembles convexes,” Comptes Rendus de l’Académie des Sciences, vol. 258, no. 18, pp. 4413–4416 (1964).
[2] S. Hu, Y. Wang, B. Tan, and F. Wang, “Inertial iterative method for solving variational inequality problems of pseudo-monotone operators and fixed point problems of non-expansive mappings in Hilbert spaces,” Journal of Industrial and Management Optimization, vol. 19, no. 4 (2023).
[3] M. A. Noor, K. I. Noor, and M. T. Rassias, “New trends in general variational inequalities,” Acta Applicandae Mathematicae, vol. 170, pp. 981–1064 (2020).
[4] T.-Y. Zhao, D.-Q. Wang, L.-C. Ceng, L. He, C.-Y. Wang, and H.-L. Fan, “Quasi-inertial Tseng’s extragradient algorithms for pseudo-monotone variational inequalities and fixed point problems of quasi-nonexpansive operators,” Numerical Functional Analysis and Optimization, vol. 42, no. 1, pp. 69–90 (2020).
[5] R. P. Agarwal, D. O’Regan, and D. Sahu, Fixed Point Theory for Lipschitzian-Type Mappings with Applications. New York: Springer (2009).
[6] A. M. Hashim and A. T. Hashim, “Some new fixed point theorems in weak partial metric spaces,” Baghdad Science Journal, vol. 20, no. 1, p. 0175 (2023).
[7] Sh. Albundi, “Iterated function system in ϑϑ-metric spaces,” Boletim da Sociedade Paranaense de Matemática, vol. 40, pp. 1–10 (2022).
[8] H. A. Satar and R. K. Naji, “Stability and bifurcation in a prey–predator–scavenger system with Michaelis–Menten type of harvesting function,” Differential Equations and Dynamical Systems, vol. 30, pp. 1–24 (2019).
[9] N. S. Taresh, Sh. Albundi, A. H. S. Fadhil, and A. R. Al-Shaikhli, “Convergence of iterative algorithms in CAT(0) spaces,” Iraqi Journal of Science, vol. 63, no. 1, pp. 233–240 (2022).
[10] M. Sahni, D. Shah, and R. Sahni, “A new modified accelerated iterative scheme using an amalgamation of fixed point and N-R method,” Journal of Interdisciplinary Mathematics, pp. 679–688 (2019).
[11] M. F. Abduljabbar and S. S. Abed, “Equivalence between iterative schemes in modular spaces,” Journal of Interdisciplinary Mathematics, vol. 22, no. 8, pp. 1529–1535 (2020).
[12] L. A. Al-Swidi and F. S. S. Awad, “On soft turning points,” Baghdad Science Journal, vol. 15, no. 3, p. 0352 (2018).
[13] R. I. Sabri and B. A. A. Ahmed, “Fixed point results for almost contraction mappings in fuzzy metric space,” Baghdad Science Journal (2024).
[14] N. S. Taresh, “On stability of Picard-Mann iteration and S-iteration,” Journal of Mustansiriyah for Science and Education, vol. 19, no. 2, pp. 233–244 (2018).
[15] A. N. Abed and S. S. Abed, “Convergence and stability of iterative scheme for a monotone total asymptotically non-expansive mapping,” Iraqi Journal of Science, vol. 63, no. 1, pp. 241–250 (2022).
[16] P. E. Maingé and A. Moudafi, “Strong convergence of an iterative method for hierarchical fixed-point problems,” Pacific Journal of Optimization, vol. 3, no. 3, pp. 529–538 (2007).
[17] A. Moudafi and P. E. Maingé, “Towards viscosity approximations of hierarchical fixed-point problems,” Fixed Point Theory and Applications, pp. 1–10 (2006).
[18] A. Moudafi, “Krasnoselski-Mann iteration for hierarchical fixed-point problems,” Inverse Problems, vol. 23, no. 4, pp. 1635–1640 (2007).
[19] H.-K. Xu, “Viscosity approximation methods for nonexpansive mappings,” Journal of Mathematical Analysis and Applications, vol. 298, no. 1, pp. 279–291 (2004).
[20] Y. Yao, Y. J. Cho, and Y.-C. Liou, “Iterative algorithms for hierarchical fixed-point problems and variational inequalities,” Mathematical and Computer Modelling, vol. 52, no. 9-10, pp. 1697–1705 (2010).
[21] E. Zeidler, Applied Functional Analysis: Applications to Mathematical Physics. New York: Springer (1995).
[22] E. Zeidler, Nonlinear Functional Analysis and Its Applications I: Fixed Point Theorems. New York: Springer-Verlag (1986).
[23] C.-X. Wu and L.-J. Zhang, “Fixed points for mean non-expansive mappings,” Acta Mathematicae Applicatae Sinica, English Series, vol. 23, no. 3, pp. 489–494 (2007).
[24] W. Takahashi, Nonlinear Functional Analysis: Fixed Point Theory and Its Applications. Yokohama (2000).
[25] G. Marino and H.-K. Xu, “A general iterative method for nonexpansive mappings in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 318, no. 1, pp. 43–52 (2006).
[26] P. E. Maingé, “Approximation methods for common fixed points of nonexpansive mappings in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 325, no. 1, pp. 469–479 (2007).

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