SOME RESULTS ON A NEW MODEL OF PHASE RELAXATION

2002 ◽  
Vol 12 (03) ◽  
pp. 431-444 ◽  
Author(s):  
VINCENZO RECUPERO
Keyword(s):  
Fixed Point ◽  
Stefan Problem ◽  
Relaxation Parameter ◽  
Well Posedness ◽  
Phase Variable ◽  
Relaxation Dynamics ◽  
Multivalued Maps ◽  
Phase Relaxation ◽  
New Model ◽  
Fixed Point Technique

This paper deals with the analysis of the relaxed Stefan problem with the relaxation dynamics for the phase variable χ [Formula: see text] where θ stands for the temperature. We prove the well-posedness of the problem by means of a fixed point-technique for multivalued maps and show that its solution converges to the solution of the Stefan problem as the relaxation parameter ε tends to zero.

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.

Hyperbolic phase change problems in heat conduction with memory

1993 ◽  
Vol 123 (3) ◽  
pp. 571-592 ◽  
Author(s):  
Pierluigi Colli ◽  
Maurizio Grasselli
Keyword(s):  
Stefan Problem ◽  
Equilibrium Condition ◽  
Past History ◽  
Study Phase ◽  
Phase Variable ◽  
Relaxation Dynamics ◽  
Materials With Memory ◽  
Uniqueness Of The Solution ◽  
Heat Conduction With Memory ◽  
With Memory

SynopsisThe aim of this paper is to formulate and study phase transition problems in materials with memory, based on the Gurtin–Pipkin constitutive assumption on the heat flux. As different phases are involved, the internal energy is allowed to depend on the phase variable (besides the temperature) and to take its past history into account. By considering the standard equilibrium condition at the interface between two phases, we deal with a hyperbolic Stefan problem reckoning with memory effects. Then, substituting this equilibrium condition with a relaxation dynamics, we represent some dissipation phenomena including supercooling or superheating. The application of a fixed point argument helps us to show the existence and uniqueness of the solution to the latter problem (still of hyperbolic type). Hence, by introducing a suitable regularisation and taking the limit as a kinetic parameter goes to zero, we prove an existence result for the former Stefan problem. Moreover, its uniqueness is deduced by contradiction.

scholarly journals Unified multivalued interpolative Reich–Rus–Ćirić-type contractions

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Monairah Alansari ◽  
Muhammad Usman Ali
Keyword(s):  
Fixed Point ◽  
Type Condition ◽  
Multivalued Maps ◽  
Contraction Type ◽  
Type Contraction

AbstractThis article examines new multivalued interpolative Reich–Rus–Ćirić-type contraction conditions and fixed point results for multivalued maps that fulfill these conditions. Earlier defined interpolative contraction type conditions cannot be particularized to any contraction type condition. This slackness of the interpolative contraction type condition is addressed through new multivalued interpolative Reich–Rus–Ćirić-type contraction conditions.

On soft set-valued maps and integral inclusions

2021 ◽  
pp. 1-15
Author(s):  
Monairah Alansari ◽  
Shehu Shagari Mohammed ◽  
Akbar Azam
Keyword(s):  
Fixed Point ◽  
Set Theory ◽  
Fuzzy Set Theory ◽  
Fixed Point Theorems ◽  
Soft Set ◽  
Mathematical Tool ◽  
Multivalued Maps ◽  
Soft Sets ◽  
Special Cases ◽  
New Development

As an improvement of fuzzy set theory, the notion of soft set was initiated as a general mathematical tool for handling phenomena with nonstatistical uncertainties. Recently, a novel idea of set-valued maps whose range set lies in a family of soft sets was inaugurated as a significant refinement of fuzzy mappings and classical multifunctions as well as their corresponding fixed point theorems. Following this new development, in this paper, the concepts of e-continuity and E-continuity of soft set-valued maps and αe-admissibility for a pair of such maps are introduced. Thereafter, we present some generalized quasi-contractions and prove the existence of e-soft fixed points of a pair of the newly defined non-crisp multivalued maps. The hypotheses and usability of these results are supported by nontrivial examples and applications to a system of integral inclusions. The established concepts herein complement several fixed point theorems in the framework of point-to-set-valued maps in the comparable literature. A few of these special cases of our results are highlighted and discussed.

scholarly journals Common fixed points of single-valued and multivalued maps

2005 ◽  
Vol 2005 (19) ◽  
pp. 3045-3055 ◽  
Author(s):  
Yicheng Liu ◽  
Jun Wu ◽  
Zhixiang Li
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Common Fixed Point ◽  
Fixed Point Theorems ◽  
Common Fixed Points ◽  
Partial Answer ◽  
Multivalued Maps ◽  
Hybrid Pair ◽  
Common Fixed Point Theorems

We define a new property which contains the property (EA) for a hybrid pair of single- and multivalued maps and give some new common fixed point theorems under hybrid contractive conditions. Our results extend previous ones. As an application, we give a partial answer to the problem raised by Singh and Mishra.

Sign in / Sign up

Export Citation Format

Share Document