A Study of Banach Fixed Point Theorem and It’s Applications

This paper aims at treating a study of Banach fixed point theorem for mapping results that introduced in the setting of normed space. The classical Banach fixed point theorem is a generalization of this work. A fixed point theory is a beautiful mixture of Mathematical analysis to explain some conditions in which maps give excellent solutions. Here later many mathematicians used this fixed point theory to establish their results, see for instance, Picard-Lindel of Theorem, The Picard theorem, Implicit function theorem etc. Also, we developed ideas that many of known fixed point theorems can easily be derived from the Banach theorem. It extends some recent works on the extension of Banach contraction principle to metric space with norm spaces.

ON THE EXISTENCE OF ZERO POINTS OF A CONTINUOUS FUNCTION

A sufficient condition is given to assert that a continuous mapping between Rm and Rn has a zero. The constructive proof of the result is based upon continuation methods and supplies the existence of a path leading to the zero point.

INITIAL BOUNDARY VALUE PROBLEM FOR THE 3D MAGNETIC-CURVATURE-DRIVEN RAYLEIGH-TAYLOR MODEL

This article studies the initial-boundary value problem for a three dimensional magnetic-curvature-driven Rayleigh-Taylor model.We first obtain the global existence of weak solutions for the full model equation by employing the Galerkin’s approximation method.Secondly,for a slightly simplified model,we show the existence and uniqueness of global strong solutions via the Banach’s fixed point theorem and vanishing viscosity method.

Re ned rigorous perturbation bounds for the SR decomposition

In this article,some new rigorous perturbation bounds for the SR decomposition un-der normwise or componentwise perturbations for a given matrix are derived.Also,the explicit expressions for the mixed and componentwise condition numbers are presented by utilizing the block matrix-vector equation approach.Hypothetical and trial results demonstrate that these new bounds are constantly more tightly than the comparing ones in the literature.

Existence and Controllability Result for Nonlinear Neutral Evolution Integrodifferential Systems

In this paper, we establish sufficient conditions for existence and controllability of nonlinear neutral evolution integroditferential systems in Banach spaces. The result is obtained by using the resolvent operators and fixed point analysis approach.

EXISTENCE OF STABLE EQUILIBRIUM FOR A CLASS OF MULTI-SPECIES ECOSYSTEM

In this paper, we discuss the existence of the stable equilibrium in multispecies Scheoner ecosystem. By means of Banach fixed point theorem, we obtain a sufficent condition for the system to have a unique positive equilibrium. We also discuss the global stability of the equilibrium.

EXISTENCE OF MILD SOLUTIONS TO SEMILINEAR FRACTIONAL ORDER FUNCTIONAL DIFFERENTIAL EQUATIONS

In this paper,we use the analytic semigroup theory of linear operators and fixed point method to prove the existence of mild solutions to a semilinear fractional order functional differential equations in a Banach space.

Analytic Solutions of an Iterative Differential Equation under Brjuno Condition

In this paper, the differential equation involving iterates of the unknown function,x＇（z）=[a^2-x^2（z）]x^[m]（z）with a complex parameter a, is investigated in the complex field C for the existence of analytic solutions. First of all, we discuss the existence and the continuous dependence on the parameter a of analytic solution for the above equation, by making use of Banach fixed point theorem. Then, as well as in many previous works, we reduce the equation with the SchrSder transformation x（z） = y（αy^-1（z）） to the following another functional differential equation without iteration of the unknown functionαy＇（αz）=[a^2-y^2（αz）]y＇（z）y（α^mz）,which is called an auxiliary equation. By constructing local invertible analytic solutions of the auxiliary equation, analytic solutions of the form y（αy^-1 （z）） for the original iterative differential equation are obtained. We discuss not only these α given in SchrSder transformation in the hyperbolic case 0 〈 ｜α｜ 〈 1 and resonance, i.e., at a root of the unity, but also those α near resonance （i.e., near a root of the unity） under Brjuno condition. Finally, we introduce explicit analytic solutions for the original iterative differential equation by means of a recurrent formula, and give some particular solutions in the form of power functions when a = 0.

Existence and controllability results to second order neutral differential equation with non-instantaneous impulses

In this paper,existence and uniqueness of the solutions of the second order neutral differential equation with non-instantaneous impulses in a Hilbert space X are studied.This is followed by a brief discussion on the exact and trajectory controllability.Further,the strongly continuous cosine family of linear operators and Banach fixed point theorem are used to establish the results.Finally,an example is given to illustrate the analytical findings.

On controllability for a class of second order semilinear stochastic systems

Many dynamical systems are modeled as second order differential systems.In this paper we address the problem of controllability for second order semilinear stochastic systems in finite dimensional spaces.Sufficient criteria for the complete controllability are formulated under some appropriate assumptions.The results are obtained by using the Banach fixed point theorem.Finally an illustrative example is provided.