Solvability of a Class of Generalized System of Variational Inclusion Problems Involving \(\oplus\) Operation
DOI:
https://doi.org/10.26713/cma.v14i5.2099Keywords:
Variational inclusion, ⊕ operation, Resolvent operator, Algorithm, ConvergenceAbstract
In this paper, a new type of operator known as \((\alpha,\rho)\)-XOR-NODSM operator and its associated resolvent operator is introduced. Further, some important properties of the resolvent operator associated with the \((\alpha,\rho)\)-XOR-NODSM operator, supported by a well constructed example, have been given. As an application, we have considered a generalized system of variational inclusion problems involving XOR operator in the setting of real ordered positive Hilbert space. Using the resolvent operator technique, we have proved the existence of solution for the system considered. Furthermore, the approximation solvability of the generalized system of variational inclusion problems involving the XOR operator has been studied. The results presented in this paper can be treated as the refinement and generalization of many known results present in the literature in this direction.
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R. Ahmad, I. Ahmad, I. Ali, S. Alhomidan and Y. H. Wang, H(·,·)-ordered-compression mapping for solving XOR-variational inclusion problem, Journal of Nonlinear and Convex Analysis 19(2) (2018), 2189 – 2201, URL: http://www.yokohamapublishers.jp/online2/opjnca/vol19/p2189.html.
I. Ahmad, R. Ahmad and J. Iqbal, A resolvent approach for solving a set-valued variational inclusion problem using weak-RRD set-valued mapping, Korean Journal of Mathematics 24(2) (2016), 199 – 213, DOI: 10.11568/kjm.2016.24.2.199.
R. Ahmad, I. Ali, S. Husain, A. Latif and C. F. Wen, Generalized implicit set-valued variational inclusion problem with ⊕ operation, Mathematics 7(5) (2019), 421, DOI: 10.3390/math7050421.
X. P. Ding and H. R. Feng, The p-step iterative algorithms for a system of generalized mixed quasi-variational inclusions with (A,η)-accretive operators in q-uniformly smooth Banach spaces, Journal of Computational and Applied Mathematics 220(1-2) (2008), 163 – 174, DOI: 10.1016/j.cam.2007.08.003.
X. P. Ding and C. L. Lou, Perturbed proximal point algorithms for general quasi-variationallike inclusions, Journal of Computational and Applied Mathematics 113(1-2) (2000), 153 – 165, DOI: 10.1016/S0377-0427(99)00250-2.
Y.-P. Fang and N.-P. Huang, H-Monotone operator and resolvent operator technique for variational inclusions, Applied Mathematics and Computation 145(2-3) (2003), 795 – 803, DOI: 10.1016/S0096-3003(03)00275-3.
Y.-P. Fang, N.-J. Huang and H. B. Thompson, A new system of variational inclusions with (H,η)-monotone operators in Hilbert spaces, Computers & Mathematics with Applications 49(2-3) (2005), 365 – 374, DOI: 10.1016/j.camwa.2004.04.037.
X.-F. He, J. Lou and Z. He, Iterative methods for solving variational inclusions in Banach spaces, Journal of Computational and Applied Mathematics 203(1) (2007), 80 – 86, DOI: 10.1016/j.cam.2006.03.011.
N.-J. Huang and Y.-P. Fang, A new class of general variational inclusions involving maximal η-monotone mappings, Publicationes Mathematicae Debrecen 62(1-2) (2003), 83 – 98, DOI: 10.5486/PMD.2003.2629.
K. R. Kazmi, M. I. Bhat and N. Ahmad, An iterative algorithm based on M-proximal mappings for a system of generalized implicit variational inclusions in Banach spaces, Journal of Computational and Applied Mathematics 233(2) (2009), 361 – 371, DOI: 10.1016/j.cam.2009.07.028.
K. R. Kazmi, F. A. Khan and M. Shahzad, A system of generalized variational inclusions involving generalized H(·,·)-accretive mapping in real q-uniformly smooth Banach spaces, Applied Mathematics and Computation 217(23) (2011), 9679 – 9688, DOI: 10.1016/j.amc.2011.04.052.
H.-G. Li, Approximation solution for generalized nonlinear ordered variational inequality and ordered equation in ordered Banach space, Nonlinear Analysis Forum 13(2) (2008), 205 – 214, URL: http://prof.ks.ac.kr/bslee/naf/table/vol-1302/NAF130209.pdf.
H.-G. Li, A nonlinear inclusion problem involving (α,λ)-NODM set-valued mappings in ordered Hilbert space, Applied Mathematics Letters 25(10) (2012), 1384 – 1388, DOI: 10.1016/j.aml.2011.12.007.
M. A. Malik, M. I. Bhat and H. G. Hyun, Graph convergence and generalized cayley operator with an application to a system of cayley inclusions in semi-inner product spaces, Nonlinear Functional Analysis and Applications 28(1) (2023), 265 – 286, DOI: 10.22771/nfaa.2023.28.01.14.
M. A. Malik, M. I. Bhat and B. Zahoor, Solvability of a class of set-valued implicit quasi-variational inequalities: A Wiener–Hopf equation method, Results in Control and Optimization 9 (2022), 100169, DOI: 10.1016/j.rico.2022.100169.
M. Sarfaraz, M. K. Kaleem and A. Kilicman, Approximation solution for system of generalized ordered variational inclusions with ⊕ operator in ordered Banach space, Journal of Inequalities and Applications 2017 (2017), Article number: 81, DOI: 10.1186/s13660-017-1351-x.
S.-Q. Shan, Y.-B. Xiao and N.-J. Huang, A new system of generalized implicit set-valued variational inclusions in Banach spaces, Nonlinear Functional Analysis and Applications 22(5) (2017), 1091 – 1105, URL: http://nfaa.kyungnam.ac.kr/journal-nfaa/index.php/NFAA/article/view/1031.
G. Stampacchia, Formes bilineaires coercitives sur les ensembles convexes, Académie des Sciences de Paris 258 (1964), 4413 – 4416
L.-C. Zeng, S.-M. Guu and J.-C. Yao, Characterization of H-monotone operators with applications to variational inclusions, Computers & Mathematics with Applications 50(3-4) (2005), 329 – 337, DOI: 10.1016/j.camwa.2005.06.001.
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