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Numerical Solution of Systems of Nonlinear Equations
Published in Azmy S. Ackleh, Edward James Allen, Ralph Baker Kearfott, Padmanabhan Seshaiyer, Classical and Modern Numerical Analysis, 2009
Azmy S. Ackleh, Edward James Allen, Ralph Baker Kearfott, Padmanabhan Seshaiyer
The Brouwer fixed point theorem is a partial strengthening of the contraction mapping theorem (Theorem 8.2 on page 442). In particular, one of the assumptions in the contraction mapping theorem is that G map D into itself. That is the only assumption in the Brouwer fixed point theorem. However, the conclusion of the Brouwer fixed point theorem is somewhat weaker, since the Brouwer fixed point theorem doesn’t mention an iteratively defined sequence that converges to the fixed point, nor does it claim that the fixed point is unique.
Measures of Noncompactness
Published in Eberhard Malkowsky, Vladimir Rakočević, Advanced Functional Analysis, 2019
Eberhard Malkowsky, Vladimir Rakočević
([201, Remark 1.5]) In the case of one variable, the Brouwer fixed point theorem is the following:Every continuous function of the interval [−1, 1] onto itself has a fixed point, or equivalently,every continuous function of the interval [−1, 1] onto itself intersects the main diagonal at some point.
Stability on a patch-structured Nicholson's blowflies system incorporating mature delay and feedback delay
Published in International Journal of Control, 2023
For arbitrary , owing to the Brouwer fixed point theorem, there exists obeying that Noting the compactness of D, it is an easy matter to find a subsequence of such that Label Then, for all , and Therefore, and then is also a positive equilibrium point of system (2). This ends the evidence of Lemma 2.5.
Time-dependent pricing strategies for metro lines considering peak avoidance behaviour of commuters
Published in Transportmetrica A: Transport Science, 2022
Ning Huan, Stephane Hess, Enjian Yao, Wenting Liu
Equations (24) and (25) express the conservation conditions of the flows of the mode-related nests and the departure time-related alternatives, respectively. Equation (26) is a non-negative constraint. Equations (27) and (28) illustrate the relationship between the perceived cost of the commuters and the minimum expected cost in the SUE problem. The flow-related constraints of the feasible region ensure a non-empty convex set. In addition, the utility functions in Equations (16) and (17) as well as the functions for calculating the probability of the alternatives being chosen in Equations (22) and (23) are continuous. The Brouwer fixed-point theorem ensures the existence of a fixed point, which also indicates the existence of a solution of the equivalent NL–SUE problem.
Global attractors and exponential stability of partly dissipative reaction diffusion systems with exponential growth nonlinearity
Published in Applicable Analysis, 2021
Existence. For each , let be the projection given in the proof of Theorem 2.1. We will find an approximate stationary solution on in the following systems: for any test function . To do this, we define the following operators determined by for any . Using the Cauchy inequality and the conditions (1.3), (1.5), (1.7) and (1.8), we can derive where κ and ℓ are given in (5.2). So, we deduce that for any satisfying . For each , by a corollary of the Brouwer fixed point theorem (see [15, Chapter 2, Lemma 1.4]), there exists such that and Hence the sequence is uniformly bounded in , and the sequence is uniformly bounded in . By the compactness of the injection , we can extract a subsequence of (relabeled by ) such that Taking in (5.5), we have Using the condition (1.7), the Cauchy inequality and the estimate (5.6), we obtain the uniform boundedness of . By the same argument as in the proof of Theorem 2.1, we conclude that Note that from the conditions (1.7) and (1.8), we have the uniform boundedness of and in . Using the strong convergence of to in and the Lebesgue dominated convergence theorem, we can take the limits in (5.5) to obtain that is a weak stationary solution to the system (1.2). The estimate (5.1) directly follows from the estimate (5.6) as n tends to infinity.