Split monotone variational inclusion problem involving Cayley operators
-
Mijanur Rahaman
,
Iqbal Ahmad
Abstract
In this article, we introduce a new kind of split monotone variational inclusion problem involving Cayley operator in the setting of infinite-dimensional Hilbert spaces. We develop a general iterative method to approximate the solution of the split monotone variational inclusion problem involving Cayley operator. Under some suitable conditions, a convergence theorem for the sequences generated by the proposed iterative scheme is established, which also solves certain variational inequality problems related to strongly positive linear operators. Finally, a numerical example is presented to study the efficiency of the proposed algorithm through MATLAB programming.
References
[1] I. Ahmad, V. N. Mishra, R. Ahmad and M. Rahaman, An iterative algorithm for a system of generalized implicit variational inclusions, SpringerPlus 5 (2016), Article ID 1283. 10.1186/s40064-016-2916-8Search in Google Scholar PubMed PubMed Central
[2] C. Byrne, Iterative oblique projection onto convex sets and the split feasibility problem, Inverse Problems 18 (2002), no. 2, 441–453. 10.1088/0266-5611/18/2/310Search in Google Scholar
[3] C. Byrne, Y. Censor, A. Gibali and S. Reich, The split common null point problem, J. Nonlinear Convex Anal. 13 (2012), no. 4, 759–775. Search in Google Scholar
[4] A. Cegielski, Iterative Methods for Fixed Point Problems in Hilbert Spaces, Lecture Notes in Math. 2057, Springer, Heidelberg, 2012. 10.1007/978-3-642-30901-4Search in Google Scholar
[5] Y. Censor, T. Bortfeld, N. Martin and A. Trofimov, A unified approach for inversion problem in intensity-modulated radiation therapy, Phys. Med. Biol. 51 (2006), no. 10, 2353–2365. 10.1088/0031-9155/51/10/001Search in Google Scholar PubMed
[6] Y. Censor and T. Elfving, A multiprojection algorithm using Bregman projections in a product space, Numer. Algorithms 8 (1994), no. 2–4, 221–239. 10.1007/BF02142692Search in Google Scholar
[7] Y. Censor, T. Elfving, N. Kopf and T. Bortfeld, The multiple-sets split feasibility problem and its applications for inverse problems, Inverse Problems 21 (2005), no. 6, 2071–2084. 10.1088/0266-5611/21/6/017Search in Google Scholar
[8] Y. Censor, A. Gibali and S. Reich, Extensions of Korpelevich’s extragradient method for the variational inequality problem in Euclidean space, Optimization 61 (2012), no. 9, 1119–1132. 10.1080/02331934.2010.539689Search in Google Scholar
[9] S. S. Chang, Set-valued variational inclusions in Banach spaces, J. Math. Anal. Appl. 248 (2000), no. 2, 438–454. 10.1006/jmaa.2000.6919Search in Google Scholar
[10] S.-S. Chang, J.-C. Yao, J. K. Kim and L. Yang, Iterative approximation to convex feasibility problems in Banach space, Fixed Point Theory Appl. 2007 (2007), Article ID 46797. 10.1155/2007/46797Search in Google Scholar
[11] J. Douglas, Jr. and H. H. Rachford, Jr., On the numerical solution of heat conduction problems in two and three space variables, Trans. Amer. Math. Soc. 82 (1956), 421–439. 10.1090/S0002-9947-1956-0084194-4Search in Google Scholar
[12] P.-E. Maingé, Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization, Set-Valued Anal. 16 (2008), no. 7–8, 899–912. 10.1007/s11228-008-0102-zSearch in Google Scholar
[13] G. Marino and H.-K. Xu, A general iterative method for nonexpansive mappings in Hilbert spaces, J. Math. Anal. Appl. 318 (2006), no. 1, 43–52. 10.1016/j.jmaa.2005.05.028Search in Google Scholar
[14] A. Moudafi, Split monotone variational inclusions, J. Optim. Theory Appl. 150 (2011), no. 2, 275–283. 10.1007/s10957-011-9814-6Search in Google Scholar
[15] Z. A. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc. 73 (1967), 591–597. 10.1090/S0002-9904-1967-11761-0Search in Google Scholar
[16] R. J. Palta and T. R. Mackie, Intensity-Modulated Radiation Therapy: The State of Art, Medical Phys. Monogr., Medical Physics, Madison, 2003. 10.1118/1.1628279Search in Google Scholar
[17] M. Rahaman, M. Ishtyak, R. Ahmad and I. Ali, The Yosida approximation iterative technique for split monotone Yosida variational inclusions, Numer. Algorithms 82 (2019), no. 1, 349–369. 10.1007/s11075-018-0607-ySearch in Google Scholar
[18] S. Takahashi, W. Takahashi and M. Toyoda, Strong convergence theorems for maximal monotone operators with nonlinear mappings in Hilbert spaces, J. Optim. Theory Appl. 147 (2010), no. 1, 27–41. 10.1007/s10957-010-9713-2Search in Google Scholar
[19] R. Wangkeeree, K. Rattanaseeha and R. Wangkeeree, The general iterative methods for split variational inclusion problem and fixed point problem in Hilbert spaces, J. Comput. Anal. Appl. 25 (2018), no. 1, 19–31. Search in Google Scholar
[20] H. K. Xu, An iterative approach to quadratic optimization, J. Optim. Theory Appl. 116 (2003), no. 3, 659–678. 10.1023/A:1023073621589Search in Google Scholar
© 2022 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- On the Hölder continuity of ring Q-homeomorphisms
- New regularity results for the heat equation and application to non-homogeneous Burgers equation
- Maximum inequalities in rearrangements of orthogonal series
- A parallel type decomposition scheme for quasi-linear abstract hyperbolic equation
- Reversible ring property via idempotent elements
- Infinitely many solutions for perturbed Λγ-Laplace equations
- On an alternative approach for mixed boundary value problems for the Laplace equation
- Split monotone variational inclusion problem involving Cayley operators
- Duals and approximate duals of von Neumann–Schatten p-frames
- Properties of general Fourier coefficients of differentiable functions
- On linear spline algorithms of computerized tomography in the space of n-orbits
- Barbashin type characterizations for the uniform polynomial stability and instability of evolution families
Articles in the same Issue
- Frontmatter
- On the Hölder continuity of ring Q-homeomorphisms
- New regularity results for the heat equation and application to non-homogeneous Burgers equation
- Maximum inequalities in rearrangements of orthogonal series
- A parallel type decomposition scheme for quasi-linear abstract hyperbolic equation
- Reversible ring property via idempotent elements
- Infinitely many solutions for perturbed Λγ-Laplace equations
- On an alternative approach for mixed boundary value problems for the Laplace equation
- Split monotone variational inclusion problem involving Cayley operators
- Duals and approximate duals of von Neumann–Schatten p-frames
- Properties of general Fourier coefficients of differentiable functions
- On linear spline algorithms of computerized tomography in the space of n-orbits
- Barbashin type characterizations for the uniform polynomial stability and instability of evolution families
