Generalized Cayley operator with applications to Cayley inclusions in uniformly smooth Banach spaces

Tirth Ram; Mohd Iqbal

doi:10.3934/mfc.2022043

Article Contents

2024, Volume 7, Issue 1: 20-37. Doi: 10.3934/mfc.2022043

This issue Previous Article Stancu variant of Jakimovski-Leviatan-Durrmeyer operators involving Brenke type polynomials Next Article Self–normalized large deviations for sums indexed by the generations of a Galton–Watson process

Generalized Cayley operator with applications to Cayley inclusions in uniformly smooth Banach spaces

Tirth Ram^, and
Mohd Iqbal

Department of Mathematics, University of Jammu, Jammu 180006, India

^*Corresponding author: Tirth Ram
^*Corresponding author: Tirth Ram

Received: June 2022

Revised: October 2022

Early access: October 2022

Published: February 2024

Abstract / Introduction Full Text(HTML) Figure(2) / Table(2) Related Papers Cited by

Abstract

In this paper, we present a generalized Cayley operator and a generalized Cayley inclusion problem $ (GCIP) $. A fixed point formulation of $ (GCIP) $ is established and using this an iterative algorithm is developed to show the existence and convergence of the solutions of $ (GCIP) $. We also establish the equivalence of the $ (GCIP) $ and generalized resolvent equation problem $ (GREP) $, and develop an iterative algorithm and some of its equivalent forms to approximate the solution of $ (GREP) $. To support our results, we construct a numerical example and convergence graphs using MATLAB programming.

Keywords:

Mathematics Subject Classification: Primary: 49J40; Secondary: 47H09.

Citation:

Full Text(HTML)

Figure 1. The convergence of $ \left\{u_n\right\} $ with initial values $ u_0 = 1,\; u_0 = 2 $ and $ u_0 = 4. $

Download: Full-size image PowerPoint slide

Figure 2. The convergence of $ \left\{u_n\right\} $ and $ \left\{s_n\right\} $ with initial values $ s_0 = 1 $ and $ s_0 = 4. $

Download: Full-size image PowerPoint slide

Table 1. Computational results for different intial values of $ u_0 $

No. of iterations	$ u_0 = 1.0 $ $ u_n $	No. of iterations	$ u_0 = 2 $ $ u_n $	No. of iterations	$ u_0 = 4 $ $ u_n $
1	1.0000	1	2.0000	1	4.0000
2	0.6938	2	1.3876	2	2.7752
3	0.4813	3	0.9627	3	1.9258
4	0.3339	4	0.6679	4	1.3361
5	0.2316	5	1.4633	5	0.9269
10	0.0371	10	0.0743	10	0.1489
15	0.0058	15	0.0118	15	0.0238
20	0.0008	20	0.0016	20	0.0037
25	0.0000	25	0.0001	25	0.0004
26	0.0000	26	0.0000	26	0.0002
27	0.0000	27	0.0000	27	0.0001
28	0.0000	28	0.0000	28	0.0000
29	0.0000	29	0.0000	29	0.0000
30	0.0000	30	0.0000	30	0.0000

| Show Table

DownLoad: CSV

Table 2. Computational results for different intial values of $ s_0 $

	$ s_0 = 1.0 $			$ s_0 = 4 $
No. of iterations	$ u_n $	$ s_n $	No. of iterations	$ u_n $	$ s_n $
1	1.0000	1.0000	1	4.0000	4.0000
2	0.7142	0.6938	2	2.8568	2.7752
3	0.5101	0.4813	3	2.0403	1.9258
4	0.3643	0.3339	4	1.4571	1.3361
5	0.2601	0.2316	5	1.0406	0.9269
10	0.0482	0.0371	10	0.1932	0.1489
15	0.0088	0.0058	15	0.0357	0.0238
20	0.0015	0.0008	20	0.0065	0.0037
25	0.0002	0.0000	25	0.0010	0.0004
30	0.0000	0.0000	30	0.0001	0.0000
31	0.0000	0.0000	31	0.0000	0.0000
32	0.0000	0.0000	32	0.0000	0.0000

| Show Table

DownLoad: CSV

Related Papers

Cited by

References

[1]	R. P. Agarwal, N.-J. Huang and M.-Y. Tan, Sensitivity analysis for a new system of generalized nonlinear mixed quasi variational inclusions, Appl. Math. Lett., 17 (2004), 345-352. doi: 10.1016/S0893-9659(04)90073-0.
[2]	R. P. Agarwal and R. U. Verma, Generalized system of $(A, \eta)$-maximal relaxed monotone variational inclusion problems based on generalized hybrid algorithms, Commun. Nonlinear Sci. Numer. Simul., 15 (2010), 238-251. doi: 10.1016/j.cnsns.2009.03.037.
[3]	R. Ahmad, I. Ali, M. Rahaman, M. Ishtyak and J. C. Yao, Cayley inclusion problem with its corresponding generalized resolvent equation problem in uniformly smooth Banach spaces, Appl. Anal., 101 (2022), 1354-1368. doi: 10.1080/00036811.2020.1781822.
[4]	R. Ahmad and Q. H. Ansari, An iterative algorithm for generalized nonlinear variational inclusions, Appl. Math. Lett., 13 (2002), 23-26. doi: 10.1016/S0893-9659(00)00028-8.
[5]	R. Ahmad, M. Ishtyak, M. Rahaman and I. Ahmad, Graph convergence and generalized Yosida approximation operator with an application, Math. Sci., 11 (2017), 155-163. doi: 10.1007/s40096-017-0221-5.
[6]	F. E. Browder, Nonlinear mappings of nonexpansive and accretive type in Banach space, Bull. Am. Math. Soc., 73 (1967), 875-882. doi: 10.1090/S0002-9904-1967-11823-8.
[7]	H.-W. Cao, Yosida approximation equations technique for system of generalized set-valued variational inclusions, J. Inequal. Appl., 2013 (2013), Article ID 455. doi: 10.1186/1029-242X-2013-455.
[8]	J. Deepho, P. Thounthong, P. Kumam and S. Phiangsungnoen, A new general iterative scheme for split variational inclusions and fixed point problems of $k$-strict pseudo-contraction mappings with convergence analysis, J. Comput. Appl. Maths., 318 (2017), 293-306. doi: 10.1016/j.cam.2016.09.009.
[9]	Y.-P. Fang and N.-J. Huang, $H$-accretive operator and resolvent operator technique for variational inclusions in Banach spaces, Appl. Math. Lett., 17 (2004), 647-653. doi: 10.1016/S0893-9659(04)90099-7.
[10]	A. Hassouni and A. Moudafi, A perturbed algorithm for variational inclusions, J. Math. Anal. Appl., 183 (1994), 706-712. doi: 10.1006/jmaa.1994.1277.
[11]	N. J. Huang and Y. P. Fang, Generalized m-accretive mappings in Banach spaces, J. Sichuan Univ., 38 (2001), 591-592.
[12]	J. Iqbal, A. K. Rajpoot, M. Islam, R. Ahmad and Y. Wang, System of generalized variational inclusions involving cayley operators and XOR-operation in q-uniformly smooth Banach spaces, Mathematics, 10 (2022), 2837. doi: 10.3390/math10162837.
[13]	T. Kato, Nonlinear semi-groups and evolution equations, J. Math. Soc. Jpn., 19 (1967), 508-520. doi: 10.2969/jmsj/01940508.
[14]	H.-Y. Lan, Generalized Yosida approximation based on relatively $A$-maximal $m$-relaxed monotonicity frameworks, Abs. Appl. Anal., 2013 (2013), Art. ID 157190, 8 pp. doi: 10.1155/2013/157190.
[15]	H.-Y. Lan, Y. J. Cho and R. U. Verma, Nonlinear relaxed cocoercive variational inclusions involving $(A, \eta)$-accretive mappings in Banach spaces, Comput. Math. Appl., 51 (2006), 1529-1538. doi: 10.1016/j.camwa.2005.11.036.
[16]	H.-Y. Lan, J. H. Kim and Y. J. Cho, On a new system of nonlinear $A$-monotone multivalued variational inclusions, J. Math. Anal. Appl., 327 (2007), 481-493. doi: 10.1016/j.jmaa.2005.11.067.
[17]	A. Moudafi, A duality algorithm for solving general variational inclusions, Adv. Model. Optim., 13 (2011), 213-220.
[18]	W. V. Petershyn, A characterization of strictly convexity of Banach spaces and other uses of duality mappings, J. Funct. Anal., 6 (1970), 282-291. doi: 10.1016/0022-1236(70)90061-3.
[19]	Z. A. Rather, R. Ahmad and C.-F. Wen, Variational-like inequality problem involving generalized cayley operator, Axioms, 10 (2021), 133. doi: 10.3390/axioms10030133.
[20]	C. Zhang and Y. Wang, Proximal algorithm for solving monotone variational inclusions, Optimization, 67 (2018), 1197-1209. doi: 10.1080/02331934.2018.1455832.
[21]	Q.-B. Zhang, X.-P. Ding and C.-Z. Chang, Resolvent operator technique for generalized implicit variational-like inclusion in Banach spaces, J. Math. Anal. Appl., 361 (2010), 283-292. doi: 10.1016/j.jmaa.2006.01.090.
[22]	X. Zhao, D. R. Sahu and C.-F. Wen, Iterative methods for system of variational inclusions involving accretive operators and applications, Fixed Point Theory, 19 (2018), 801-821. doi: 10.24193/fpt-ro.2018.2.59.
[23]	Y.-Z. Zou and N.-J. Huang, $H(., .)$-accretive operator with an application for solving variational inclusions in Bananch spaces, Appl. Math. Comput., 204 (2008), 809-816. doi: 10.1016/j.amc.2008.07.024.