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doi: 10.3934/jimo.2021210
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## A novel algorithm for approximating common solution of a system of monotone inclusion problems and common fixed point problem

 1 Department of Mathematics, University of Science and Technology of Mazandaran, Behshahr, Iran 2 School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehran, Iran

Received  July 2021 Revised  October 2021 Early access December 2021

Fund Project: This research was in part supported by a grant from IPM (No.1400470032)

In this paper, we study the problem of finding a common element of the set of solutions of a system of monotone inclusion problems and the set of common fixed points of a finite family of generalized demimetric mappings in Hilbert spaces. We propose a new and efficient algorithm for solving this problem. Our method relies on the inertial algorithm, Tseng's splitting algorithm and the viscosity algorithm. Strong convergence analysis of the proposed method is established under standard and mild conditions. As applications we use our algorithm for finding the common solutions to variational inequality problems, the constrained multiple-set split convex feasibility problem, the convex minimization problem and the common minimizer problem. Finally, we give some numerical results to show that our proposed algorithm is efficient and implementable from the numerical point of view.

Citation: Mohammad Eslamian, Ahmad Kamandi. A novel algorithm for approximating common solution of a system of monotone inclusion problems and common fixed point problem. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021210
##### References:

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##### References:
The graph of the error $\|x_n-x_{n-1}\|_2$
The graph of $x_n$
The graph of the value of $\frac{1}{2} \|Tx_n-b\|^2$ for the algorithms
Comparison of the new algorithm and the Algorithm 1 to recovery of a sparse signal with 5% nonzero elements
The results of the new algorithm for Example 5.1
 Starting points CPU time No. iterations $x_0=t^2$ $x_1=sin(2t)$ 0.62 s 1 $x_0=t$ $x_1=3e^{-2t}$ 1.42 s 2 $x_0=10\;\; t\;\; sin(5t)$ $x_1=10\;\; t\;\; sin(5t)$ 0.61 s 1
 Starting points CPU time No. iterations $x_0=t^2$ $x_1=sin(2t)$ 0.62 s 1 $x_0=t$ $x_1=3e^{-2t}$ 1.42 s 2 $x_0=10\;\; t\;\; sin(5t)$ $x_1=10\;\; t\;\; sin(5t)$ 0.61 s 1
Numerical results of comparison of the new algorithm and the Algorithm 1 for Example 5.2
 Starting point(s) Algorithm 1 The new algorithm No. iterations CPU time No. iterations CPU time $[-5,4,-3,2,-1]$ 63 0.0133 31 0.0025 $[-50,40,-30,20,-1]$ 72 0.0263 35 0.0093 $[5,4,3,2,1]$ 64 0.0146 31 0.0120 $[50,40,30,20,10]$ 72 0.0150 29 0.0095
 Starting point(s) Algorithm 1 The new algorithm No. iterations CPU time No. iterations CPU time $[-5,4,-3,2,-1]$ 63 0.0133 31 0.0025 $[-50,40,-30,20,-1]$ 72 0.0263 35 0.0093 $[5,4,3,2,1]$ 64 0.0146 31 0.0120 $[50,40,30,20,10]$ 72 0.0150 29 0.0095
The details of iterations of the new algorithm for Example 5.2
 n $x_n$ $\|x_n-x_{n-1}\|_2$ 0 [-5.000, 4.0000, -3.0000, 2.0000, -1.0000] — 1 [-5.000, 4.0000, -3.0000, 2.0000, -1.0000] — 2 [-6.3896, 5.0620, -4.2310, 2.4069, -1.5759] 2.2520e+00 8 [-0.0295, 0.0190, -0.0547, -0.0021, -0.0336] 1.8203e-01 14 [ 0.0019, -0.0014, 0.0024, -0.0003, 0.0013] 2.6378e-03 20 [ 0.0000, -0.0000, 0.0000, -0.0000, 0.0000] 5.5509e-05 22 [ 0.0000, -0.0000, 0.0000, -0.0000, 0.0000] 8.9023e-06
 n $x_n$ $\|x_n-x_{n-1}\|_2$ 0 [-5.000, 4.0000, -3.0000, 2.0000, -1.0000] — 1 [-5.000, 4.0000, -3.0000, 2.0000, -1.0000] — 2 [-6.3896, 5.0620, -4.2310, 2.4069, -1.5759] 2.2520e+00 8 [-0.0295, 0.0190, -0.0547, -0.0021, -0.0336] 1.8203e-01 14 [ 0.0019, -0.0014, 0.0024, -0.0003, 0.0013] 2.6378e-03 20 [ 0.0000, -0.0000, 0.0000, -0.0000, 0.0000] 5.5509e-05 22 [ 0.0000, -0.0000, 0.0000, -0.0000, 0.0000] 8.9023e-06
Numerical results of comparison of the new algorithm and The Algorithm 1 to recovery of a sparse signal with $p$ % nonzero elements
 p Algorithm 1 New Algorithm CPU time No. iterations CPU time No. iterations 2 % 1.8809 s 606 0.3698 s 129 5 % 2.2180 s 790 0.4061 s 142 10 % 1.1052 s 374 0.5425 s 174
 p Algorithm 1 New Algorithm CPU time No. iterations CPU time No. iterations 2 % 1.8809 s 606 0.3698 s 129 5 % 2.2180 s 790 0.4061 s 142 10 % 1.1052 s 374 0.5425 s 174
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