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If general relativity is correct, then the origin of the universe is a simple mathematical problem. The Friedmann equation in cosmology is a well-structured ordinary differential equation, and the global properties of its solutions can be qualitatively analyzed by the phase-trajectory method. In this paper we show that the total energy density of matter in the universe is positive, and the total pressure near the Big Bang is negative. By analyzing the global properties of the solutions to the Friedmann equation according to these two conditions of state functions, we find that the Big Bang is impossible, and the space must be a closed 3-dimensional sphere, the cosmological constant is likely to be zero, and the evolution of the universe should be cyclic. The analysis and the proof are simple and straight forward, therefore these conclusions should be reliable.
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On “A new three-step fixed point iteration scheme with strong convergence and applications”
Article ID: 2918
Vol 3, Issue 1, 2025
DOI: https://doi.org/10.54517/mss2918
Vol 3, Issue 1, 2025
Received: 31 August 2024; Accepted: 12 November 2024; Available online: 19 November 2024; Issue release: 31 March 2025
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Abstract
This note delves into the convergence analysis of several iterative methods and elucidates their behaviors. Furthermore, we demonstrate that the findings presented in “A new three-step fixed point iteration scheme with strong convergence and applications” are not entirely novel. In particular, some of the results either overlap with or restate previously established methods without introducing significant innovations.
Keywords
algorithms; iterative methods; convergence order
References
Traub JF. Iterative methods for the solution of equations. Prentice-Hall Englewood Cliffs; 1964.
Kang SM, Rafiq A, Kwun YC. A new second-order iteration method for solving nonlinear equations. Abst. Appl. Anal. 2013; 1.
Sharma P, Ramos H, Behl R, Kanwar V. A new three-step fixed point iteration scheme with strong convergence and applications. J. Comput. Appl. Math. 2023; 430: 115–242.
Mann WR. Mean value methods in iteration. Proc. Amer. Math. Soc. 1953; 4: 506–610.
Ishikawa S. Fixed point by a new iteration method. Proc. Amer. Math. Soc. 1974; 4(1): 147–150.
Noor MA. Some approximation schemes for general variational inequalities. J. Math. Anal. Appl. 2000; 251: 217–229.
Kirk WA. On successive approximations for nonexpansive mappings in Banach spaces. Glasg. Math. J. 1971; 12: 6–9.
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Cairo University, Egypt
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