Asymptotic behaviour of the solutions of the supercooled Stefan problem

1997 ◽  
Vol 127 (1) ◽  
pp. 181-189 ◽  
Author(s):  
Fahuai Yi
Keyword(s):  
Surface Tension ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Asymptotic Behaviour ◽  
Specific Heat ◽  
Stefan Problem ◽  
Limit Case ◽  
Uniform Estimates ◽  
Kinetic Condition ◽  
Two Phases

In this paper, we prove that the Hele–Shaw problem with kinetic condition and surface tension is the limit case of the supercooled Stefan problem in the classical sense when specific heat ε goes to zero. The method is the use of a fixed-point theorem; the key is to construct a workable function space. The main feature is to obtain the existence and the uniform estimates with respect to ε > 0 at the same time for the solutions of the supercooled Stefan problem. For the sake of simplicity, we only consider the case of one phase, although the method used here is also applicable in the case of two phases.

On solutions of a stochastic integral equation of the volterra type with applications for chemotherapy

1988 ◽  
Vol 25 (02) ◽  
pp. 257-267 ◽  
Author(s):  
D. Szynal ◽  
S. Wedrychowicz
Keyword(s):  
Banach Space ◽  
Integral Equation ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Stochastic Model ◽  
Asymptotic Behaviour ◽  
Existence Of Solutions ◽  
Stochastic Integral ◽  
Volterra Type ◽  
Stochastic Integral Equation

This paper deals with the existence of solutions of a stochastic integral equation of the Volterra type and their asymptotic behaviour. Investigations of this paper use the concept of a measure of non-compactness in Banach space and fixed-point theorem of Darbo type. An application to a stochastic model for chemotherapy is also presented.

scholarly journals Regional topology and approximative solutions of difference and differential equations

2015 ◽  
Vol 63 (1) ◽  
pp. 183-203 ◽  
Author(s):  
Janusz Migda
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Asymptotic Behaviour ◽  
Point Theorem ◽  
Metric Space ◽  
Schauder’S Fixed Point Theorem ◽  
Locally Compact ◽  
Compact Metric Space ◽  
Real Functions

Abstract We introduce a topology, which we call the regional topology, on the space of all real functions on a given locally compact metric space. Next we obtain new versions of Schauder’s fixed point theorem and Ascoli’s theorem. We use these theorems and the properties of the iterated remainder operator to establish conditions under which there exist solutions, with prescribed asymptotic behaviour, of some difference and differential equations.

scholarly journals FIXED POINT THEOREM FOR AN INFINITE TOEPLITZ MATRIX

2020 ◽  
pp. 1-10
Author(s):  
VYACHESLAV M. ABRAMOV
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Asymptotic Behaviour ◽  
Stochastic Processes ◽  
Positive Solution ◽  
Point Theorem ◽  
Toeplitz Matrix ◽  
Recurrence Relations ◽  
Convolution Type

Abstract For an infinite Toeplitz matrix T with nonnegative real entries we find the conditions under which the equation $\boldsymbol {x}=T\boldsymbol {x}$ , where $\boldsymbol {x}$ is an infinite vector column, has a nontrivial bounded positive solution. The problem studied in this paper is associated with the asymptotic behaviour of convolution-type recurrence relations and can be applied to problems arising in the theory of stochastic processes and other areas.

scholarly journals A Stefan problem on an evolving surface

2015 ◽  
Vol 373 (2050) ◽  
pp. 20140279 ◽  
Author(s):  
Amal Alphonse ◽  
Charles M. Elliott
Keyword(s):  
Fixed Point ◽  
Stefan Problem ◽  
Continuous Dependence ◽  
Existence Of Solutions ◽  
Well Posedness ◽  
Duality Method ◽  
Uniform Estimates ◽  
Natural Treatment ◽  
Evolving Surface ◽  
Evolving Surfaces

We formulate a Stefan problem on an evolving hypersurface and study the well posedness of weak solutions given L 1 data. To do this, we first develop function spaces and results to handle equations on evolving surfaces in order to give a natural treatment of the problem. Then, we consider the existence of solutions for data; this is done by regularization of the nonlinearity. The regularized problem is solved by a fixed point theorem and then uniform estimates are obtained in order to pass to the limit. By using a duality method, we show continuous dependence, which allows us to extend the results to L 1 data.

On solutions of a stochastic integral equation of the volterra type with applications for chemotherapy

1988 ◽  
Vol 25 (2) ◽  
pp. 257-267 ◽  
Author(s):  
D. Szynal ◽  
S. Wedrychowicz
Keyword(s):  
Banach Space ◽  
Integral Equation ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Stochastic Model ◽  
Asymptotic Behaviour ◽  
Existence Of Solutions ◽  
Stochastic Integral ◽  
Volterra Type ◽  
Stochastic Integral Equation

This paper deals with the existence of solutions of a stochastic integral equation of the Volterra type and their asymptotic behaviour. Investigations of this paper use the concept of a measure of non-compactness in Banach space and fixed-point theorem of Darbo type. An application to a stochastic model for chemotherapy is also presented.

ON THE HYPERSTABILITY OF THE DRYGAS FUNCTIONAL EQUATION ON A RESTRICTED DOMAIN

2019 ◽  
Vol 102 (1) ◽  
pp. 126-137
Author(s):  
JEDSADA SENASUKH ◽  
SATIT SAEJUNG
Keyword(s):  
Functional Equation ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Asymptotic Behaviour ◽  
Point Theorem ◽  
Normed Space ◽  
Restricted Domain ◽  
The Fixed Point Theorem ◽  
Misleading Statement

We prove hyperstability results for the Drygas functional equation on a restricted domain (a certain subset of a normed space). Our results are more general than the ones proposed by Aiemsomboon and Sintunavarat [‘Two new generalised hyperstability results for the Drygas functional equation’, Bull. Aust. Math. Soc.95 (2017), 269–280] and our proof does not rely on the fixed point theorem of Brzdęk as was the case there. A characterisation of the Drygas functional equation in terms of its asymptotic behaviour is given. Several examples are given to illustrate our generalisations. Finally, we point out a misleading statement in the proof of the second result in the paper by Aiemsomboon and Sintunavarat and propose its correction.

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