Approximating fixed points of enriched contractions in Banach spaces. (English) Zbl 1460.47039
The authors introduce a class of mappings called enriched contractions. This class of mappings includes the contractive type mappings, the Picard-Banach contractions, some nonexpansive mappings, and other important nonlinear mappings. They establish the existence and uniqueness of fixed points for this class of mappings and construct a Kransnoselskij iterative process to approximate the fixed points of these mappings. The authors show that the sequence generated by their method converges strongly to the fixed point of an enriched contraction (Theorem 2.4). The authors further extend their main results to the case of asymptotic enriched Banach contractions and consider a local variant of the Picard-Banach fixed point theorem (Section 3). Some examples are presented to illustrate the generality of their results.
Reviewer: Mewomo Oluwatosin Temitope (Durban)
MSC:
| 47J26 | Fixed-point iterations |
| 47H05 | Monotone operators and generalizations |
| 47H09 | Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc. |
Keywords:
Banach space; enriched contraction; fixed point; Krasnoselskij iteration; strong convergence; local enriched contraction; Maia-type enriched contractionReferences:
| [1] | Alghamdi, M., Berinde, V., Shahzad, N.: Fixed point of multivalued nonself almost contractions. J. Appl. Math. 2013, Art. ID 621614 (2013) · Zbl 1271.54071 |
| [2] | Baillon, JB; Bruck, RE; Reich, S., On the asymptotic behavior of nonexpansive mappings and semigroups in Banach spaces, Houst. J. Math., 4, 1, 1-9 (1978) · Zbl 0431.47034 |
| [3] | Banach, S., Sur les opérations dans les ensembles abstraits et leurs applications aux équations intégrales, Fund. Math., 3, 133-181 (1922) · JFM 48.0201.01 · doi:10.4064/fm-3-1-133-181 |
| [4] | Berinde, V., Approximating fixed points of weak contractions using the Picard iteration, Nonlinear Anal. Forum, 9, 1, 43-53 (2004) · Zbl 1078.47042 |
| [5] | Berinde, V., Iterative Approximation of Fixed Points. Lecture Notes in Mathematics (2007), Berlin: Springer, Berlin · Zbl 1165.47047 |
| [6] | Berinde, V., Approximating fixed points of implicit almost contractions, Hacet. J. Math. Stat., 41, 1, 93-102 (2012) · Zbl 1279.47084 |
| [7] | Berinde, V., Approximating fixed points of enriched nonexpansive mappings by Krasnoselskij iteration in Hilbert spaces, Carpathian J. Math., 35, 3, 293-304 (2019) · Zbl 1463.47155 |
| [8] | Berinde, V.: Weak and strong convergence theorems for the Krasnoselskij iterative algorithm in the class of enriched strictly pseudocontractive operators. An. Univ. Vest Timiş. Ser. Mat.-Inform. 56(2), 13-27 (2018). 10.2478/awutm-2018-0013 · Zbl 1513.47101 |
| [9] | Berinde, V.; Petric, M., Fixed point theorems for cyclic non-self single-valued almost contractions, Carpathian J. Math., 31, 3, 289-296 (2015) · Zbl 1389.47130 |
| [10] | Berinde, V., Păcurar, M.: Iterative approximation of fixed points of single-valued almost contractions. In: Fixed Point Theory and Graph Theory. Elsevier, Amsterdam, pp. 29-97 (2016) · Zbl 1353.54002 |
| [11] | Berinde, V., Păcurar, M.: Fixed point theorems for Kannan type mappings with applications to split feasibility and variational inequality problems (submitted) · Zbl 1309.47071 |
| [12] | Berinde, V., Păcurar, M.: Fixed point theorems for Chatterjea type mappings in Banach spaces (submitted) · Zbl 1288.54033 |
| [13] | Borwein, J.; Reich, S.; Shafrir, I., Krasnoselski-Mann iterations in normed spaces, Can. Math. Bull., 35, 1, 21-28 (1992) · Zbl 0712.47050 · doi:10.4153/CMB-1992-003-0 |
| [14] | Caccioppoli, R., Un teorema generale sull’esistenza di elementi uniti in una transformazione funzionale, Rend. Accad. Lincei, 11, 794-799 (1930) · JFM 56.0359.01 |
| [15] | Goebel, K., Reich, S.: Uniform convexity, hyperbolic geometry, and nonexpansive mappings. In: Monographs and Textbooks in Pure and Applied Mathematics, vol. 83. New York: Marcel Dekker, Inc. (1984) · Zbl 0537.46001 |
| [16] | Granas, A.; Dugundji, J., Fixed Point Theory (2003), New York: Springer, New York · Zbl 1025.47002 |
| [17] | Maia, M., Un’osservazione sulle contrazioni metriche, Rend. Semin. Mat. Univ. Padova, 40, 139-143 (1968) · Zbl 0188.45603 |
| [18] | Păcurar, M., Iterative Methods for Fixed Point Approximation (2009), Cluj-Napoca: Editura Risoprint, Cluj-Napoca · Zbl 1213.54002 |
| [19] | Rhoades, BE, A comparison of various definitions of contractive mappings, Trans. Am. Math. Soc., 226, 257-290 (1977) · Zbl 0365.54023 · doi:10.1090/S0002-9947-1977-0433430-4 |
| [20] | Rus, IA, Generalized Contractions and Applications (2001), Cluj-Napoca: Cluj-University Press, Cluj-Napoca · Zbl 0968.54029 |
| [21] | Rus, IA, Weakly Picard operators and applications, Semin. Fixed Point Theory Cluj-Napoca, 2, 41-57 (2001) · Zbl 1035.47044 |
| [22] | Rus, IA; Petruşel, A.; Petruşel, G., Fixed Point Theory (2008), Cluj-Napoca: Cluj University Press, Cluj-Napoca · Zbl 1171.54034 |
| [23] | Shioji, N.; Suzuki, T.; Takahashi, W., Contractive mappings, Kannan mappings and metric completeness, Proc. Am. Math. Soc., 126, 10, 3117-3124 (1998) · Zbl 0955.54009 · doi:10.1090/S0002-9939-98-04605-X |
| [24] | Subrahmanyam, PV, Remarks on some fixed-point theorems related to Banach’s contraction principle, J. Math. Phys. Sci., 8, 445-457 (1974) · Zbl 0294.54033 |
| [25] | Suzuki, T., Contractive mappings are Kannan mappings, and Kannan mappings are contractive mappings in some sense, Comment. Math. (Prace Mat.), 45, 1, 45-58 (2005) · Zbl 1098.54024 |
| [26] | Zeidler, E., Nonlinear Functional Analysis and its Applications. III. Variational Methods and Optimization (1985), New York: Springer, New York · Zbl 0583.47051 |
| [27] | Zeidler, E., Nonlinear Functional Analysis and its Applications. I. Fixed-Point Theorems (1986), New York: Springer, New York · Zbl 0583.47050 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
