A self adaptive inertial algorithm for solving split variational inclusion and fixed point problems with applications

Timilehin Opeyemi Alakoya; Lateef Olakunle Jolaoso; Oluwatosin Temitope Mewomo

doi:10.3934/jimo.2020152

Article Contents

2022, Volume 18, Issue 1: 239-265. Doi: 10.3934/jimo.2020152

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A self adaptive inertial algorithm for solving split variational inclusion and fixed point problems with applications

School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban, South Africa

^* Corresponding author: Oluwatosin Temitope Mewomo
^* Corresponding author: Oluwatosin Temitope Mewomo

Received: October 31, 2019

Revised: February 29, 2020

Early access: October 2020

Published: January 2022

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Abstract

We propose a general iterative scheme with inertial term and self-adaptive stepsize for approximating a common solution of Split Variational Inclusion Problem (SVIP) and Fixed Point Problem (FPP) for a quasi-nonexpansive mapping in real Hilbert spaces. We prove that our iterative scheme converges strongly to a common solution of SVIP and FPP for a quasi-nonexpansive mapping, which is also a solution of a certain optimization problem related to a strongly positive bounded linear operator. We apply our proposed algorithm to the problem of finding an equilibrium point with minimal cost of production for a model in industrial electricity production. Numerical results are presented to demonstrate the efficiency of our algorithm in comparison with some other existing algorithms in the literature.

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Mathematics Subject Classification: Primary: 47H10, 47J25; Secondary: 65J15.

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Figure 1. Example 5.1, Top Left: $ N = 50 $; Top Left: $ N = 100 $; Bottom Left: $ N = 500 $; Bottom Right: $ N = 1000 $

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Figure 2. Example 5.2, Top Left: Case I; Top Left: Case II; Bottom Left: Case III; Bottom Right: Case IV

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Figure 3. Example 5.3, Top Left: Choice (i); Top Left: Choice (ii); Bottom Left: Choice (iii); Bottom Right: Choice (iv)

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Table 1. Numerical results for Example 5.1

	No of Iteration	CPU time (sec)
$ N= 50 $	19	0.0289
$ N=100 $	19	0.0386
$ N=500 $	41	0.1386
$ N=1000 $	138	0.3523

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Table 2. Numerical results for Example 5.2

		Algorithm 3.1	Algorithm 1.1	Algorithm 1.2
Case I	CPU time (sec)	0.0021	0.0071	0.0036
$ x_0 = 1, x_1 = 0.5 $	No of Iter.	8	22	16
Case II	CPU time (sec)	0.0021	0.0041	0.0047
$ x_0 = -0.5, x_1 = 2 $	No. of Iter.	9	25	17
Case III	CPU time (sec)	0.0044	0.0532	0.0095
$ x_0 = 5, x_1 = 10 $	No of Iter.	10	28	19
Case IV	CPU time (sec)	0.0062	0.0589	0.0071
$ x_0 = -5, x_1 = 2 $	No of Iter.	10	27	19

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Table 3. Numerical results for Example 5.3

		Algorithm 3.1	Algorithm 1.1
Choice (i)	CPU time (sec)	1.7859	5.1231
	No of Iter.	11	23
Choice (ii)	CPU time (sec)	1.4997	13.3981
	No. of Iter.	13	27
Choice (iii)	CPU time (sec)	2.6789	9.1093
	No of Iter.	7	12
Choice (iv)	CPU time (sec)	6.3222	24.5622
	No of Iter.	11	24

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