| Citation: |
Lili Chen, Yunyi Jiang, Yanfeng Zhao. ITERATIVE ALGORITHMS AND FIXED POINT THEOREMS FOR SET-VALUED G-CONTRACTIONS IN GRAPHICAL CONVEX METRIC SPACES[J]. Journal of Applied Analysis & Computation, 2024, 14(6): 3558-3580. doi: 10.11948/20240087
|
ITERATIVE ALGORITHMS AND FIXED POINT THEOREMS FOR SET-VALUED G-CONTRACTIONS IN GRAPHICAL CONVEX METRIC SPACES
-
Abstract
In this article, we present a series of fixed point results of the Ishikawa iterative algorithm and the SP iterative algorithm in graphical convex metric spaces. First, we introduce the Ishikawa sequence and the SP sequence in the above space. Furthermore, we study the existence and uniqueness of fixed points for set-valued $ G $-contractions in graphical convex metric spaces. Finally, by providing an example, we demonstrate the hypotheses of the existence theorem of fixed points for set-valued $ G $-contractions in $ G $-complete graphical convex metric spaces are sufficient but not necessary.
-
-
References
[1] R. P. Agarwal, D. O'Regan and D. R. Sahu, Iterative construction of fixed points of nearly asymptotically nonexpansive mappings, J. Nonlinear Convex A., 2007, 8, 61–79. [2] H. Ahmad, M. Younis and A. A. N. Abdou, Bipolar $b$-metric spaces in graph setting and related fixed points, Symmetry, 2023, 15(6), 1227. doi: 10.3390/sym15061227 CrossRef $b$-metric spaces in graph setting and related fixed points" target="_blank">Google Scholar
[3] R. Behl and E. Martínez, A new high-order and efficient family of iterative techniques for nonlinear models, Complexity, 2020, 2020, 1–11. [4] M. F. Bota, L. Guran and G. Petrusel, Fixed points and coupled fixed points in $b$-metric spaces via graphical contractions, Carpathian J. Math., 2023, 39(1), 85–94. $b$-metric spaces via graphical contractions" target="_blank">Google Scholar
[5] L. L. Chen, C. B. Li, R. Kaczmarek and Y. F. Zhao, Several fixed point theorems in convex $b$-metric spaces and applications, Mathematics, 2020, 8(2), 242. doi: 10.3390/math8020242 CrossRef $b$-metric spaces and applications" target="_blank">Google Scholar
[6] L. L. Chen, L. Gao and D. Y. Chen, Fixed point theorems of mean nonexpansive set-valued mappings in Banach spaces, J. Fixed Point Theory Appl., 2017, 19, 2129–2143. doi: 10.1007/s11784-017-0401-9 [7] L. B. Ćirić, A generalization of Banach's contraction principle, Proc. Am. Math. Soc., 1974, 45, 267–273. [8] L. L. Chen, N. Yang, Y. F. Zhao and Z. H. Ma, Fixed points theorems for set-valued $G$-contractions in a graphical convex metric space with applications, J. Fixed Point Theory Appl., 2020, 22, 88. doi: 10.1007/s11784-020-00828-y CrossRef $G$-contractions in a graphical convex metric space with applications" target="_blank">Google Scholar
[9] H. Covitz and S. B. Nadler, Multi-valued contraction mappings in generalized metric spaces, Isr. J. Math., 1970, 8, 5–11. doi: 10.1007/BF02771543 [10] S. S. Chauhan, N. Kumar, M. Imdad and M. Asim, New fixed point iteration and its rate of convergence, Optimization, 2022, 72(9), 2415–2432. [11] T. Dinevari and M. Frigon, Fixed point results for multivalued contractions on ametric space with a graph, J. Math. Anal. Appl., 2013, 405, 507–517. doi: 10.1016/j.jmaa.2013.04.014 [12] X. P. Ding, Iteration processes for nonlinear mappings in convex metric spaces, J. Math. Anal. Appl., 1988, 132, 114–122. doi: 10.1016/0022-247X(88)90047-9 [13] C. Garodia and I. Uddin, A new fixed point algorithm for finding the solution of a delay differential equation, Mathematics, 2020, 5(4), 3182–3200. doi: 10.3934/math.2020205 [14] K. Goebel and W. A. Kirk, Iteration processes for nonexpansive mappings, Contemp. Math., 1983, 21, 115–123. [15] A. Hanjing and S. Suantai, Concidence point and fixed point theorems for a new type of $G$-contraction multivalued mappings on a metric space endowed with a graph, J. Fixed Point Theory Appl., 2015, 2015, 171. doi: 10.1186/s13663-015-0420-4 CrossRef $G$-contraction multivalued mappings on a metric space endowed with a graph" target="_blank">Google Scholar
[16] J. Jachymski, The contraction principle for mappings on a metric space with a garph, Proc. Am. Math. Soc., 2008, 136, 1359–1373. [17] G. Mani, R. Ramaswamy, A. J. Gnanaprakasam, A. Elsonbaty, O. A. A. Abdelnaby and S. Radenović, Application of fixed points in bipolar controlled metric space to solve fractional differential equation, Fractal Fract., 2023, 7(3), 242. doi: 10.3390/fractalfract7030242 [18] A. K. Mirmostafaee, Fixed point theorems for set-valued mappings in $b$-metric spaces, J. Fixed Point Theory Appl., 2017, 18(1), 305–314. doi: 10.24193/fpt-ro.2017.1.24 [19] D. Mihet, A Banach contraction theorem in fuzzy metric spaces, Fuzzy sets syst., 2004, 144(3), 431–439. doi: 10.1016/S0165-0114(03)00305-1 [20] A. K. Mirmostafaee, Coupled fixed points for mappings on a $b$-metric space with a graph, Math. Vesnik, 2017, 69(3), 214–225. $b$-metric space with a graph" target="_blank">Google Scholar
[21] S. B. Nadler, Multi-valued contraction mappings, Pac. J. Math., 1969, 30, 475–488. doi: 10.2140/pjm.1969.30.475 [22] A. Nicolae, D. O'Regan and A. Petrusel, Fixed point theorems for singlevalued and multivalued generalized contractions in metric spaces endowed with a graph, Georgian. Math. J., 2011, 18, 307–327. doi: 10.1515/gmj.2011.0019 [23] O. Popescu and G. Stan, A generalization of Nadlers fixed point theorem, Results Math., 2017, 72, 1525–1534. doi: 10.1007/s00025-017-0694-4 [24] A. Pansuwan, W. Sintunavarat, V. Parvaneh and Y. J. Cho, Some fixed point theorems for $(\alpha, \theta, k)$-contractive multi-valued mappings with some applications, J. Fixed Point Theory Appl., 2015, 2015, 132. doi: 10.1186/s13663-015-0385-3 CrossRef $(\alpha, \theta, k)$-contractive multi-valued mappings with some applications" target="_blank">Google Scholar
[25] O. Popescu, A new type of contractive multivalued operators, Bull. Sci. Math., 2013, 137, 30–44. doi: 10.1016/j.bulsci.2012.07.001 [26] W. Phuengrattana and S. Suantai, On the rate of convergence of Mann, Ishikawa, Noor and SP-iterations for continuous functions on an arbitrary interval, J. Comput. Appl. Math., 2011, 235, 3006–3014. doi: 10.1016/j.cam.2010.12.022 [27] A. C. M. Ran and M. C. B. Reurings, A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. Am. Math. Soc., 2004, 132, 1435–1443. [28] S. Reich, Fixed points of contractive functions, Boll. Un. Mat. Ital., 1972, 5, 26–42. [29] S. Reich and I. Shafrir, Nonexpansive iterations in hyperbolic spaces, Nonlinear Anal., 1990, 19, 537–558. [30] L. Y. Shi, T. Ling, X. L. Tong, Y. Cao and Y. S. Peng, Fixed Point Theory and Applications: Recent Developments, Springer Nature, 2023. [31] D. R. Sahu and A. Petrusel, Strong convergence of iterative methods by strictly pseudocontractive mappings in Banach spaces, Nonlinear Anal-Theor., 2011, 74, 6012–6023. doi: 10.1016/j.na.2011.05.078 [32] S. Shukla, N. Dubey and R. Shukla, Fixed point theorems in graphical cone metric spaces and application to a system of initial value problems, J. Inequal. Appl., 2023, 2023, 91. doi: 10.1186/s13660-023-03002-3 [33] S. Shukla, S. Radenović and C. Vetro, Graphical metric space: A generalized setting in fixed point theory, RACSAM, 2017, 111(3), 641–655. doi: 10.1007/s13398-016-0316-0 [34] A. Sultana and V. Vetrivel, Fixed points of Mizoguchi-Takahashi contraction on a metric space with a graph and applications, J. Math. Anal. Appl., 2014, 417(1), 336–344. doi: 10.1016/j.jmaa.2014.03.015 [35] D. C. Sun, Hausdorff metric space, Journal of South China Normal Universuty (Natural Science Edition), 2002, 2. [36] W. Takahashi, A convexity in metric space and nonexpansive mappings, I. Kodai Math. Semin. Rep., 1970, 22, 142–149. [37] Y. H. Wang, B. Huang, B. N. Jiang, T. T. Xu and K. Wang, A general hybrid relaxed CQ algorithm for solving the fixed-point problem and split-feasibility problem, Mathematics, 2023, 8(10), 24310–24330. doi: 10.3934/math.20231239 [38] M. Younis, H. Ahmad, L. L. Chen and M. A. Han, Computation and convergence of fixed points in graphical spaces with an application to elastic beam deformations, J. Geom. Phys., 2023, 192, 104955. doi: 10.1016/j.geomphys.2023.104955 [39] H. Yingtaweesitikul, Suzuki type fixed point theorems for generalized multi-valued mappings in $b$-metric spaces, J. Fixed Point Theory Appl., 2013, 2013, 215. doi: 10.1186/1687-1812-2013-215 [40] M. Younis and D. Bahuguna, A unique approach to graph-based metric spaces with an application to rocket ascension, Comput. Appl. Math., 2023, 42, 44. doi: 10.1007/s40314-023-02193-1 [41] M. Younis, D. Singh, I. Altun and V. Chauhan, Graphical structure of extended $b$-metric spaces: An application to the transverse oscillations of a homogeneous bar, International Journal of Nonlinear Sciences and Numerical Simulation, 2022, 23, 1239–1252. doi: 10.1515/ijnsns-2020-0126 CrossRef $b$-metric spaces: An application to the transverse oscillations of a homogeneous bar" target="_blank">Google Scholar
[42] M. Younis, D. Singh and A. Goyal, A novel approach of graphical rectangular $b$-metric spaces with an application to the vibrations of a vertical heavy hanging cable, J. Fixed Point Theory Appl., 2019, 21, 1–33. doi: 10.1007/s11784-018-0638-y CrossRef $b$-metric spaces with an application to the vibrations of a vertical heavy hanging cable" target="_blank">Google Scholar
[43] M. Younis, D. Singh, L. L. Chen and M. Metwali, A study on the solutions of notable engineering models, Math. Model. Anal., 2022, 27(3), 492–509. doi: 10.3846/mma.2022.15276 [44] M. Younis, D. Singh and A. A. N. Abdou, A fixed point approach for tuning circuit problem in dislocated $b$-metric spaces, Math. Meth. Appl. Sci., 2022, 45(4), 2234–2253. doi: 10.1002/mma.7922 [45] M. Younis, D. Singh, M. Asadi and V. Joshi, Results on contractions of Reich type in graphical $b$-metric spaces with applications, Filomat, 2019, 33(17), 5723–5735. doi: 10.2298/FIL1917723Y CrossRef $b$-metric spaces with applications" target="_blank">Google Scholar
[46] K. Zoto, I. Vardhami, D. Bajović, Z. D. Mitrović and S. Radenović, On Some Novel Fixed Point Results for Generalized $F$-Contractions in $b$-Metric-Like Spaces with Application, CMES-Computer Modeling in Engineering & Sciences., 2023, 135(1), 673–686. $F$-Contractions in
$b$ -
-
Export File
Citation
Lili Chen, Yunyi Jiang, Yanfeng Zhao. ITERATIVE ALGORITHMS AND FIXED POINT THEOREMS FOR SET-VALUED G-CONTRACTIONS IN GRAPHICAL CONVEX METRIC SPACES[J]. Journal of Applied Analysis & Computation, 2024, 14(6): 3558-3580. doi: 10.11948/20240087
DownLoad: