scholarly journals Analytical Methods for Transport Equations in Similarity Form

2007 ◽  
Author(s):  
Abhishek Tiwari ◽  
Kaveh A. Tagavi ◽  
J. M. McDonough
Keyword(s):  
Fixed Point ◽  
Analytical Solutions ◽  
Contraction Mapping ◽  
Contraction Mapping Principle ◽  
Transport Equations ◽  
Fixed Point Iteration ◽  
Novel Approach ◽  
Derivative Term ◽  
Range Of Values ◽  
Limit Of Solution

We present a novel approach for deriving analytical solutions to transport equations expressed in similarity variables. We apply a fixed-point iteration procedure to these transformed equations by formally solving for the highest derivative term and, from this (via requirements for convergence given by the contraction mapping principle), deduce a range of values for the outer limit of solution domain, for which the fixed-point iteration gives a converged solution.

scholarly journals Periodicity and positivity in nonlinear neutral integro-dynamic equations with variable delay

Mathematica ◽  
2020 ◽  
Vol 62 (85) (2) ◽  
pp. 117-132
Author(s):  
Malik Belaid ◽  
Abdelouaheb Ardjouni ◽  
Ahcene Djoudi
Keyword(s):  
Fixed Point ◽  
Contraction Mapping ◽  
Contraction Mapping Principle ◽  
Dynamic Equations ◽  
Variable Delay ◽  
Positive Periodic Solutions ◽  
Krasnoselskii’S Fixed Point Theorem ◽  
Completely Continuous ◽  
Continuous Map ◽  
Uniqueness Results

Let T be a periodic time scale. We use Krasnoselskii's fixed point theorem for a sum of two operators to show new results on the existence of periodic and positive periodic solutions of a nonlinear neutral integro-dynamic equation with variable delay. We invert this equation to construct a sum of a contraction and a completely continuous map which is suitable for applying Krasnoselskii's theorem. The uniqueness results of this equation are studied by the contraction mapping principle.

scholarly journals On a functional equation arising in mathematical biology and theory of learning

2015 ◽  
Vol 24 (1) ◽  
pp. 9-16
Author(s):  
VASILE BERINDE ◽  
◽  
ABDUL RAHIM KHAN ◽  
◽  
Keyword(s):  
Functional Equation ◽  
Mathematical Biology ◽  
Fixed Point ◽  
Contraction Mapping ◽  
Contraction Mapping Principle ◽  
Approximation Result ◽  
Banach Contraction ◽  
The Right ◽  
Banach Contraction Mapping Principle ◽  
Banach Contraction Mapping

V. Istrat¸escu [Istr ˘ at¸escu, V. I., ˘ On a functional equation, J. Math. Anal. Appl., 56 (1976), No. 1, 133–136] used the Banach contraction mapping principle to establish an existence and approximation result for the solution of the functional equation ϕ(x) = xϕ((1 − α)x + α) + (1 − x)ϕ((1 − β)x), x ∈ [0, 1], (0 < α ≤ β < 1), which is important for some mathematical models arising in biology and theory of learning. This equation has been studied by Lyubich and Shapiro [A. P. Lyubich, Yu. I. and Shapiro, A. P., On a functional equation (Russian), Teor. Funkts., Funkts. Anal. Prilozh. 17 (1973), 81–84] and subsequently, by Dmitriev and Shapiro [Dmitriev, A. A. and Shapiro, A. P., On a certain functional equation of the theory of learning (Russian), Usp. Mat. Nauk 37 (1982), No. 4 (226), 155–156]. The main aim of this note is to solve this functional equation with more general arguments for ϕ on the right hand side, by using appropriate fixed point tools.

scholarly journals Maia type fixed point theorems for Ćirić-Prešić operators

2018 ◽  
Vol 10 (1) ◽  
pp. 18-31
Author(s):  
Margareta-Eliza Balazs
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Contraction Mapping ◽  
Fixed Point Theorems ◽  
Acta Math ◽  
Contraction Mapping Principle ◽  
Contraction Condition ◽  
Banach Contraction ◽  
Banach Contraction Mapping Principle ◽  
Banach Contraction Mapping

Abstract The main aim of this paper is to obtain Maia type fixed point results for Ćirić-Prešić contraction condition, following Ćirić L. B. and Prešić S. B. result proved in [Ćirić L. B.; Prešić S. B. On Prešić type generalization of the Banach contraction mapping principle, Acta Math. Univ. Comenian. (N.S.), 2007, v 76, no. 2, 143–147] and Luong N. V. and Thuan N. X. result in [Luong, N. V., Thuan, N. X., Some fixed point theorems of Prešić-Ćirić type, Acta Univ. Apulensis Math. Inform., No. 30, (2012), 237–249]. We unified these theorems with Maia’s fixed point theorem proved in [Maia, Maria Grazia. Un’osservazione sulle contrazioni metriche. (Italian) Rend. Sem. Mat. Univ. Padova 40 1968 139–143] and the obtained results are proved is the present paper. An example is also provided.

scholarly journals A Fixed Point Result with a Contractive Iterate at a Point

Mathematics ◽  
2019 ◽  
Vol 7 (7) ◽  
pp. 606 ◽  
Author(s):  
Badr Alqahtani ◽  
Andreea Fulga ◽  
Erdal Karapınar
Keyword(s):  
Fixed Point ◽  
Contraction Mapping ◽  
Contraction Mapping Principle ◽  
Continuity Condition ◽  
Ulam Stability ◽  
Banach Contraction ◽  
Iteration Number ◽  
The Given ◽  
Banach Contraction Mapping Principle ◽  
Banach Contraction Mapping

In this manuscript, we define generalized Kincses-Totik type contractions within the context of metric space and consider the existence of a fixed point for such operators. Kincses-Totik type contractions extends the renowned Banach contraction mapping principle in different aspects. First, the continuity condition for the considered mapping is not required. Second, the contraction inequality contains all possible geometrical distances. Third, the contraction inequality is formulated for some iteration of the considered operator, instead of the dealing with the given operator. Fourth and last, the iteration number may vary for each point in the domain of the operator for which we look for a fixed point. Consequently, the proved results generalize the acknowledged results in the field, including the well-known theorems of Seghal, Kincses-Totik, and Banach-Caccioppoli. We present two illustrative examples to support our results. As an application, we consider an Ulam-stability of one of our results.

scholarly journals Generalizations of the Converse of the Contraction Mapping Principle

1966 ◽  
Vol 18 ◽  
pp. 1095-1104 ◽  
Author(s):  
James S. W. Wong
Keyword(s):  
Fixed Point ◽  
Metric Space ◽  
Contraction Mapping ◽  
Unique Fixed Point ◽  
Complete Metric Space ◽  
Contraction Mapping Principle ◽  
Converse Statement ◽  
Natural Formulation

This paper is an outgrowth of studies related to the converse of the contraction mapping principle. A natural formulation of the converse statement may be stated as follows: “Let X be a complete metric space, and T be a mapping of X into itself such that for each x ∈ X, the sequence of iterates ﹛Tnx﹜ converges to a unique fixed point ω ∈ X. Then there exists a complete metric in X in which T is a contraction.” This is in fact true, even in a stronger sense, as may be seen from the following result of Bessaga (1).

scholarly journals Existence and Uniqueness of Positive Solutions for a Fractional Switched System

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Zhi-Wei Lv ◽  
Bao-Feng Chen
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Positive Solutions ◽  
Existence And Uniqueness ◽  
Contraction Mapping ◽  
Contraction Mapping Principle ◽  
Constant Function ◽  
Switched System ◽  
Piecewise Constant Function ◽  
Piecewise Constant

We discuss the existence and uniqueness of positive solutions for the following fractional switched system: (Dc0+αu(t)+fσ(t)(t,u(t))+gσ(t)(t,u(t))=0,t∈J=[0,1]);(u(0)=u′′(0)=0,u(1)=∫01u(s) ds), whereDc0+αis the Caputo fractional derivative with2<α≤3,σ(t):J→{1,2,…,N}is a piecewise constant function depending ont, andℝ+=[0,+∞),  fi,gi∈C[J×ℝ+,ℝ+],i=1,2,…,N. Our results are based on a fixed point theorem of a sum operator and contraction mapping principle. Furthermore, two examples are also given to illustrate the results.

scholarly journals Fixed Points and Exponential Stability for Impulsive Time-Delays BAM Neural Networks via LMI Approach and Contraction Mapping Principle

2016 ◽  
Vol 2016 ◽  
pp. 1-8 ◽  
Author(s):  
Ruofeng Rao ◽  
Zhilin Pu ◽  
Shouming Zhong ◽  
Xinggui Li
Keyword(s):  
Neural Networks ◽  
Fixed Point ◽  
Exponential Stability ◽  
Time Delays ◽  
Large Scale ◽  
Contraction Mapping ◽  
Contraction Mapping Principle ◽  
Bam Neural Networks ◽  
Lmi Approach ◽  
The Stability

The fixed point technique has been employed in the stability analysis of time-delays bidirectional associative memory (BAM) neural networks with impulse. By formulating a contraction mapping in a product space, a new LMI-based exponential stability criterion was derived. Lately, fixed point methods have educed various good results inspiring this work, but those criteria cannot be programmed by a computer. In this paper, LMI conditions of the obtained result can be applicable to computer Matlab LMI toolbox which meets the need of the large-scale calculation in real engineering. Moreover, a numerical example and a comparable table are presented to illustrate the effectiveness of the proposed methods.

scholarly journals Revisiting of some outstanding metric fixed point theorems via E-contraction

2018 ◽  
Vol 26 (3) ◽  
pp. 73-98
Author(s):  
Andreea Fulga ◽  
Erdal Karapınar
Keyword(s):  
Fixed Point ◽  
Existence And Uniqueness ◽  
Contraction Mapping ◽  
Fixed Point Theorems ◽  
Contractive Mapping ◽  
Contraction Mapping Principle ◽  
Contraction Mappings ◽  
Banach Contraction ◽  
Banach Contraction Mapping Principle ◽  
Banach Contraction Mapping

AbstractIn this paper, we introduce the notion of α-ψ-contractive mapping of type E, to remedy of the weakness of the existing contraction mappings. We investigate the existence and uniqueness of a fixed point of such mappings. We also list some examples to illustrate our results that unify and generalize the several well-known results including the famous Banach contraction mapping principle.

scholarly journals Existence of Solutions for Integrodifferential Equations of Fractional Order with Antiperiodic Boundary Conditions

2009 ◽  
Vol 2009 ◽  
pp. 1-9 ◽  
Author(s):  
Ahmed Alsaedi
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Fractional Order ◽  
Contraction Mapping ◽  
Existence Of Solutions ◽  
Contraction Mapping Principle ◽  
Integrodifferential Equations ◽  
Krasnoselskii’S Fixed Point Theorem

We discuss the existence of solutions for a nonlinear antiperiodic boundary value problem of integrodifferential equations of fractional orderq∈(1,2]. The contraction mapping principle and Krasnoselskii's fixed point theorem are applied to establish the results.

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