On Enriched Weakly Contractive Maps in Hilbert Spaces
Keywords:
Enriched weakly contractive mapping, Strong and weak convergence, Stability, Hilbert spaces
Abstract
This paper introduces a class of enriched weakly contractive mappings for approximating a non-Picard average operator on a closed convex subset of Hilbert spaces. By imposing the enriched weakly mapping on the average operator, we establish and prove some results concerning convergence theorems (strong and weak), stability, and convergent rate. The validity and generality of the new class of enriched weakly mappings are examined with the aid of practical examples. The results harmonize and improve some recent results on enriched contractive-type mappings.
References
Banach S. (1922), Sur les opérations dans les ensembles abstraits et leurs applications aux équations intégrales, Fund Math., 3, pp. 133-181.
Rakotch E. (1962), A note on contractive mappings, Proc. Amer. Math. Soc., 13, pp. 459-465.
Geraghty M. (1973), On contractive mappings, Proc. Am. Math. Soc., 40, pp. 604-608.
Reich S. (1971), Some remarks concerning contraction mappings, Oanad. Math. Bull., 14, pp. 121-124.
Boyd D. W. and Wong J. S. W. (1969), On nonlinear contractions, Proc. Amer. Math. Soc., 20, pp. 458-464.
Alber Ya. I. and Guerre-Delabriere S. (1997), Principles of weakly contractive maps in Hilbert spaces, in: I. Gohberg, Yu. Lyubich (Eds.), New Results in Operator Theory, in: Advances and Appl., 98, Birkhäuser, Basel, pp. 7-22.
Rhoades B. E. (2001), Some theorems on weakly contractive maps, Nonlinear Anal., 47, pp. 2683-2693.
Dutta P. N. and Choudhury B. S. (2008), A generalisation of contraction principle in metric spaces, Fixed Point Theory and Applications, 2008, pp. 1-8.
Doric D. (2009), Common fixed point for generalized
(
ψ
,
ϕ
)
(ψ,ϕ)-weak contractions, Applied Mathematics Letters, 22, pp. 1896-1900.
Zhang Q. and Yisheng S. (2009), Fixed point theory for generalized
ϕ
ϕ-weak contractions, Applied Mathematics Letters, 22(1), pp. 75-78.
Nashine H. K. and Samet B. (2011), Fixed point results for mappings satisfying
(
ψ
,
ϕ
)
(ψ,ϕ)-weakly contractive condition in partially ordered metric spaces, Nonlinear Analysis: Theory, Methods & Applications, 74(1), pp. 2201-2209.
Tijani K. R., Wahab O. T., Usamot I. F. and Alata S. M., Common Coupled Fixed Point Theorems without Compatibility in Partially Ordered Metric Spaces, Int. J. Math. Sci. Optim.: Theory and Appl., 8(1), pp. 88-100.
Radenović S., Kadelburg Z., Jandrić D. and Jandrić A. (2012), Some results on weakly contractive maps, Bulletin of the Iranian Mathematical Society, 38(3), pp. 625-645.
Erhan I. M., Karapinar E. and Sekulić T. (2012), Fixed points of
(
ψ
,
ϕ
)
(ψ,ϕ) contractions on rectangular metric spaces, Fixed Point Theory and Applications, 2012(1), pp. 1-12.
Omidire O. J., Ansari A. H., Ariyo R. D. and Aduragbemi M. (2025), Approximating fixed point of generalized C-class contractivity conditions, Int. J. Math. Sci. Optim.: Theory and Appl., 11(1), 1-10.
Singh S. L., Kamal R., Se La San M. and Chugh R. (2015), A fixed point theorem for generalized weak contractions, Filomat, 29(7), pp. 1481-1490.
Wahab O. T. and Musa S. A. (2021), On general class of nonlinear contractive maps and their performance estimates, Aust. J. Math. Anal. Appl., 18(2), 17.
Berinde V. and Pacurar M. (2020), Approximating fixed points of enriched contractions in Banach spaces, J. Fixed Point Theory Appl., 22(10), 38.
Berinde V. (2019), Approximating fixed points of enriched nonexpansive mappings by Krasnoselskij iteration in Hilbert spaces, Carpathian J. Math., 35(3), pp. 293-304.
Abbas M., Anjum R. and Berinde V. (2021), Enriched multivalued contractions with applications to differential inclusions and dynamic programming, Symmetry, 13(8), 1350.
Berinde V. (2020), Approximating fixed points of enriched nonexpansive mappings in Banach spaces by using a retraction-displacement condition, Carpathian J. Math., 36(1), pp. 27-34.
Berinde V. and Pacurar M. (2021), Kannan's fixed point approximation for solving split feasibility and variational inequality problems, J. Comput. Appl. Math., 386, 113217.
Berinde V. (2018), Weak and strong convergence theorems for the Krasnoselskij iterative algorithm in the class of enriched strictly pseudocontractive operators, An. Univ. Vest Timis. Ser. Mat. Inform., 56, pp. 13-27.
Berinde V. and Pacurar M. (2021), Existence and approximation of fixed points of enriched contractions and enriched
ϕ
ϕ-contractions, Symmetry, 13, 498.
Berinde V. and Pacurar M. (2021), Fixed points theorems for unsaturated and saturated classes of contractive mappings in Banach spaces, Symmetry, 13, 713.
Berinde V. and Pacurar M. (2021), Fixed point theorems for enriched Ciric-Reich-Rus contractions in Banach spaces and convex metric spaces, Carpathian J. Math., 37, pp. 173-184.
Górnicki J. and Bisht R. K. (2021), Around averaged mappings, J. Fixed Point Theory Appl., 23, 48.
Suantai S., Chumpungan D. and Sarmeta P. (2021), Existence of fixed points of weak enriched nonexpansive mappings in Banach spaces, Carpathian J. Math., 37(2), pp. 287-294.
Krasnoselskij M. A. (1955), Two remarks on the method of successive approximations (in Russian), Uspehi Mat. Nauk., 10(1), pp. 123-127.
Browder F. E. (1965), Fixed point theorems for noncompact mappings in Hilbert space, PYOC. Natl. Acad. Sci. U.S.A., 53, pp. 1272-1276.
Rakotch E. (1962), A note on contractive mappings, Proc. Amer. Math. Soc., 13, pp. 459-465.
Geraghty M. (1973), On contractive mappings, Proc. Am. Math. Soc., 40, pp. 604-608.
Reich S. (1971), Some remarks concerning contraction mappings, Oanad. Math. Bull., 14, pp. 121-124.
Boyd D. W. and Wong J. S. W. (1969), On nonlinear contractions, Proc. Amer. Math. Soc., 20, pp. 458-464.
Alber Ya. I. and Guerre-Delabriere S. (1997), Principles of weakly contractive maps in Hilbert spaces, in: I. Gohberg, Yu. Lyubich (Eds.), New Results in Operator Theory, in: Advances and Appl., 98, Birkhäuser, Basel, pp. 7-22.
Rhoades B. E. (2001), Some theorems on weakly contractive maps, Nonlinear Anal., 47, pp. 2683-2693.
Dutta P. N. and Choudhury B. S. (2008), A generalisation of contraction principle in metric spaces, Fixed Point Theory and Applications, 2008, pp. 1-8.
Doric D. (2009), Common fixed point for generalized
(
ψ
,
ϕ
)
(ψ,ϕ)-weak contractions, Applied Mathematics Letters, 22, pp. 1896-1900.
Zhang Q. and Yisheng S. (2009), Fixed point theory for generalized
ϕ
ϕ-weak contractions, Applied Mathematics Letters, 22(1), pp. 75-78.
Nashine H. K. and Samet B. (2011), Fixed point results for mappings satisfying
(
ψ
,
ϕ
)
(ψ,ϕ)-weakly contractive condition in partially ordered metric spaces, Nonlinear Analysis: Theory, Methods & Applications, 74(1), pp. 2201-2209.
Tijani K. R., Wahab O. T., Usamot I. F. and Alata S. M., Common Coupled Fixed Point Theorems without Compatibility in Partially Ordered Metric Spaces, Int. J. Math. Sci. Optim.: Theory and Appl., 8(1), pp. 88-100.
Radenović S., Kadelburg Z., Jandrić D. and Jandrić A. (2012), Some results on weakly contractive maps, Bulletin of the Iranian Mathematical Society, 38(3), pp. 625-645.
Erhan I. M., Karapinar E. and Sekulić T. (2012), Fixed points of
(
ψ
,
ϕ
)
(ψ,ϕ) contractions on rectangular metric spaces, Fixed Point Theory and Applications, 2012(1), pp. 1-12.
Omidire O. J., Ansari A. H., Ariyo R. D. and Aduragbemi M. (2025), Approximating fixed point of generalized C-class contractivity conditions, Int. J. Math. Sci. Optim.: Theory and Appl., 11(1), 1-10.
Singh S. L., Kamal R., Se La San M. and Chugh R. (2015), A fixed point theorem for generalized weak contractions, Filomat, 29(7), pp. 1481-1490.
Wahab O. T. and Musa S. A. (2021), On general class of nonlinear contractive maps and their performance estimates, Aust. J. Math. Anal. Appl., 18(2), 17.
Berinde V. and Pacurar M. (2020), Approximating fixed points of enriched contractions in Banach spaces, J. Fixed Point Theory Appl., 22(10), 38.
Berinde V. (2019), Approximating fixed points of enriched nonexpansive mappings by Krasnoselskij iteration in Hilbert spaces, Carpathian J. Math., 35(3), pp. 293-304.
Abbas M., Anjum R. and Berinde V. (2021), Enriched multivalued contractions with applications to differential inclusions and dynamic programming, Symmetry, 13(8), 1350.
Berinde V. (2020), Approximating fixed points of enriched nonexpansive mappings in Banach spaces by using a retraction-displacement condition, Carpathian J. Math., 36(1), pp. 27-34.
Berinde V. and Pacurar M. (2021), Kannan's fixed point approximation for solving split feasibility and variational inequality problems, J. Comput. Appl. Math., 386, 113217.
Berinde V. (2018), Weak and strong convergence theorems for the Krasnoselskij iterative algorithm in the class of enriched strictly pseudocontractive operators, An. Univ. Vest Timis. Ser. Mat. Inform., 56, pp. 13-27.
Berinde V. and Pacurar M. (2021), Existence and approximation of fixed points of enriched contractions and enriched
ϕ
ϕ-contractions, Symmetry, 13, 498.
Berinde V. and Pacurar M. (2021), Fixed points theorems for unsaturated and saturated classes of contractive mappings in Banach spaces, Symmetry, 13, 713.
Berinde V. and Pacurar M. (2021), Fixed point theorems for enriched Ciric-Reich-Rus contractions in Banach spaces and convex metric spaces, Carpathian J. Math., 37, pp. 173-184.
Górnicki J. and Bisht R. K. (2021), Around averaged mappings, J. Fixed Point Theory Appl., 23, 48.
Suantai S., Chumpungan D. and Sarmeta P. (2021), Existence of fixed points of weak enriched nonexpansive mappings in Banach spaces, Carpathian J. Math., 37(2), pp. 287-294.
Krasnoselskij M. A. (1955), Two remarks on the method of successive approximations (in Russian), Uspehi Mat. Nauk., 10(1), pp. 123-127.
Browder F. E. (1965), Fixed point theorems for noncompact mappings in Hilbert space, PYOC. Natl. Acad. Sci. U.S.A., 53, pp. 1272-1276.
Published
2025-09-30
How to Cite
Wahab , O. T., Musa, S. A., & Usman, A. A. (2025). On Enriched Weakly Contractive Maps in Hilbert Spaces. International Journal of Mathematical Sciences and Optimization: Theory and Applications, 11(3), 68 - 79. Retrieved from https://ijmso.unilag.edu.ng/article/view/2903
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