Many researchers have been interested in investigating the solutions of systems of nonlinear differential equations where those equations can be seen in many scientific phenomena from physics and natural science. To study nonlinear differential equations, various analytical, approximate-analytical, and numerical methods have been proposed to solve those equations that have been formulated in either integer-order derivatives or fractional-order derivatives such as the method of fictitious domains and homotopy in [I. P. Gavrilyuk and V. L. Makarov, Ukr. Math. J. 72, No. 2, 211–231 (2020; Zbl 1453.65434); translation from Ukr. Mat. Zh. 72, No. 2, 191–208 (2020)], the method of semigroups in [N. Cusimano et al., ESAIM, Math. Model. Numer. Anal. 54, No. 3, 751–774 (2020; Zbl 1452.35237)], the differential transform method in [the reviewer, “Novel methods for solving the conformable wave equation”, J. New Theory 2020, No. 31, 56–85 (2020)], and the double Laplace transform in [the reviewer et al., “New approximate-analytical solutions for the nonlinear fractional Schrödinger equation with second-order spatio-temporal dispersion via double Laplace transform method”, Preprint, arXiv:2010.10977]. We recommend the reader to refer to other related recent research studies such as the numerical study on a proposed mathematical model based on the human eye in [S. Barbeiro and P. Serranho, SEMA SIMAI Springer Ser. 23, 87–101 (2020; Zbl 1454.65182)] and the most recent research works on nonlocal modelling, analysis, and computation in [Q. Du, Nonlocal modeling, analysis, and computation. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM) (2019; Zbl 1423.00007)] (see also [L. C. F. Ferreira et al., Bull. Sci. Math. 153, 86–117 (2019; Zbl 1433.35185); K. Hidano and C. Wang, Sel. Math., New Ser. 25, No. 1, Paper No. 2, 28 p. (2019; Zbl 1428.35662); F. Camilli and A. Goffi, NoDEA, Nonlinear Differ. Equ. Appl. 27, No. 2, Paper No. 22, 37 p. (2020; Zbl 1452.35234); A. Ghanmi and Z. Zhang, Bull. Korean Math. Soc. 56, No. 5, 1297–1314 (2019; Zbl 1432.34012); B. Zhu and B. Han, Mediterr. J. Math. 17, No. 4, Paper No. 113, 12 p. (2020; Zbl 1452.35248); H. Dong and D. Kim, J. Funct. Anal. 278, No. 3, Article ID 108338, 66 p. (2020; Zbl 1427.35316); K. Ryszewska, J. Math. Anal. Appl. 483, No. 2, Article ID 123654, 17 p. (2020; Zbl 1436.35323); S. D. Taliaferro, J. Math. Pures Appl. (9) 133, 287–328 (2020; Zbl 1437.35697); M. Musso et al., Math. Ann. 375, No. 1–2, 361–424 (2019; Zbl 1452.35244)]).
One of the most interesting optimization problems is a complementarity problem that has attracted the interests of many researchers in the fields of topology, fixed point theory, nonlinear analysis, mathematical optimization, variational inequality theory, and equilibrium problems. Therefore, this research paper investigates the dynamic nonlinear complementarity problem by discussing the convergence analysis of the newly studied iterative method, Gauss-Seidel type method.
Given the intial value \(x(0)=x_{0}\), \(x(t)\in\mathbb{R}^{m}\), \(y(t)\in\mathbb{R}_{+}^{n}\), \(F: \mathbb{R}_{+}\times\mathbb{R}^{m}\), and \(G: \mathbb{R}_{+}\times\mathbb{R}^{m}\times\mathbb{R}_{+}^{n}\rightarrow\mathbb{R}^{n}\), then the dynamic nonlinear complementarity problem is as follows: \[ \dot{x}(t)=F(t,x(t),y(t)),\quad 0\leq y(t) \perp G(t,x(t),y(t))\geq0,\text{ where } t\in(0,T). \] From the above problem, it has two parts: a nonlinear system and a complementarity system which are very important in investigating dynamic problems. Due to the high computational cost of solving the nonlinear system, the authors have successfully overcome this challenging issue by solving the above problem iteratively using a Gauss-Seidel-type method which is considered an efficient method for solving the nonlinear system where the authors have proven two various properties of convergence, and the convergence theorems of this method have been studied depending on the one-sided Lipschitz condition for nonlinear system and on the traditional Lipschitz condition for the complementarity system.
One of the interesting results in this research work is that the studied method has been proved to converge superlinearly for a fixed length of time interval with a rate independent of the step-size \(h\), while for a smaller step-size \(h\), the method has been proved to converge faster with a rate \(\mathcal{O}(h)\). In addition, to support and validate all obtained results, the authors have provided two numerical experiments on the 4-diode wave rectifier and projected dynamic systems: the spatial price equilibrium. Finally, this research study is very interesting to every applied mathematician and researcher who is interesting in this topic of research. Therefore, further research work is needed on this research topic.