Abstract
We introduce an iterative method to approximate a common solution of split variational inclusion problem and fixed point problem for nonexpansive semigroups with a way of selecting the stepsizes which does not need any prior information about the operator norms in Hilbert spaces. We prove that the sequences generated by the proposed algorithm converge strongly to a common element of the set of solutions of a split variational inclusion and the set of common fixed points of one-parameter nonexpansive semigroups. Moreover, numerical results demonstrate the performance and convergence of our result, which may be viewed as a refinement and improvement of the previously known results announced by many other researchers.
Highlights
We introduce an iterative method to approximate a common solution of split variational inclusion problem and fixed point problem for nonexpansive semigroups with a way of selecting the stepsizes which does not need any prior information about the operator norms in Hilbert spaces
Moudafi [1] shows that split monotone variational inclusion problem (SMVIP) (1) includes, as special cases, the split variational inequality problem, the split common fixed point problem, split zero problem, and split feasibility problem [1,2,3,4,5,6,7] which have already been studied and used in practice as a model in intensity-modulated radiation therapy treatment planning
We know that (2) is the variational inclusion problem and denote its solution set by SOLVIP(B1)
Summary
Moudafi [1] proposed the following split monotone variational inclusion problem (SMVIP): find a point x∗ ∈ H1 such that 0 ∈ f1 (x∗) + B1 (x∗) , (1). Xn+1 = anf (xn) + (I − an) Sun, where λ > 0, A∗ is the adjoint of A, L is the spectral radius of the operator ‖A∗A‖, and γ ∈ (0, 1/L) They proved the sequence {xn} generated by (5) strongly converges to the fixed point of nonexpansive mapping S and the solution set Γ of SVIP ((2)-(3)). Following the work of Moudafi [1], Kazmi and Rizvi [10], and Byrne et al [4], we introduce and study an iterative method to approximate a common solution of split variational inclusion problem and fixed point problem for nonexpansive semigroups with a way of selecting the stepsizes which does not need any prior information about the operator norms in Hilbert spaces. Numerical results are proposed to show that our algorithm is more suitable for SVIP ((2)-(3)) than the proposed algorithms (4) and (6)
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