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Introduction
Published in Vladimir A. Dobrushkin, Applied Differential Equations, 2018
Corollary 2.2: Suppose the function f(x,y) satisfies the conditions of Picard Theorem 1.3 and that y1(x) and y2(x) are solutions to the equation y'=f(x,y) subject to the initial conditions yx0=y01 and yx0=y02, respectively. Then y1(x)-y2(x)⩽y01-y02eLx-x0
On the second-order asymptotical regularization of linear ill-posed inverse problems
Published in Applicable Analysis, 2020
Denote by , and rewrite (4) as a first-order differential equation where , and I denotes the identity operator in . Since A is a bounded linear operator, both and B are also bounded linear operators. Hence, by the Cauchy–Lipschitz–Picard theorem, the first-order autonomous system (15) has a unique global solution for the given initial data .
The use of phase portraits to visualize and investigate isolated singular points of complex functions
Published in International Journal of Mathematical Education in Science and Technology, 2019
The advantage of visualizing essential singularities is not only that students can observe the behaviour of the function near this kind of singularity, but also it might help them to explore and comprehend more abstract mathematical results such as the Great Picard Theorem, which tells us that any analytic function with an essential singularity at takes on all possible complex values (with at most a single exception) infinitely often in any neighbourhood of (Krantz, 2004, pp. 28–30).
Global existence of weak solutions for the 3D incompressible Keller–Segel–Navier–Stokes equations with partial diffusion
Published in Applicable Analysis, 2023
Jijie Zhao, Xiaoyu Chen, Qian Zhang
First, we use the Picard theorem to obtain a local-in-time solution of problem (13). Assuming with , we obtain the following inequalities and where C is a constant with . We only give the proof of (14), and the other two inequalities can be proved by similar methods.