Fixed points and the stability of jensens functional equation. Two interesting and important works were investigated by rassias and rus 14, in which they dealt with the stability of a linear mappings in the banach space and the ulam sta bility of ordinary differential equations. S s be a contraction operator with constant k1 f f lf f, snn n10 converges to the unique fixed point. We consider a paper of banas and rzepka which deals with existence and asymptotic stability of an integral equation by means of fixedpoint theory and. In the case of partial differential equations, the dimension of the problem is reduced in this process. Any inve stigation of the stability of an equation, or system of equations, using l iapunovs direct method. Stability behavior of eulers method we consider the socalled linear test equation y. We know the eigenvalues of this matrix will be the roots to the following equation its form also allows us to reduce the equation from a quartic equation to a quadratic equation and. In this paper, we investigate the existence and stability results for a class of nonlinear caputo nabla fractional difference equation. Stability by fixed point theory for functional differential. The stability of the lagrange points in the circular restricted three body problem is derived. Chapter 7 integral equations theorem banach fixed point theorem, 1922 let s be a nonempty closed subset of banach spacem, sm. As applications of the generalized diazmargoliss fixed point theorem, we present some existence theorems of the hyersulam stability for a general class of the nonlinear volterra integral equations in. Fixed points and stability in neutral nonlinear differential.
For an n nmatrix a, kakis the largest entry of ain absolute value. Stability of a nonlinear volterra integrodifferential equation via a. Method of successive approximations for fredholm ie s e i. An asymptotic mean square stability theorem with a necessary and sufficient condition is proved, which improves and generalizes some results due to burton, zhang and luo. Equilibrium points for nonlinear differential equations youtube. Stability by fixed point methods for highly nonlinear delay equations, fixed point theory 52004, 320. Theory, methods and applications 1, gordon and breach publ. Introduction recently, bana and rzepka 1 studied an integral equation by means of a modification of a fixed point theorem of darbo using measures of noncompactness.
To obtain the existence and stability results, we used schauders fixed point theorem, banach contraction principle and krasnoselskiis fixed point theorem. Fixed points and stability in neutral stochastic differential. Positive kernels, fixed points, and integral equations. In this case the solution is exponentially decaying. Jung 12 carried out a work involving fixed point for stability of integral equations of volterra. Read the stability of neutral stochastic delay differential equations with poisson jumps by fixed points, journal of computational and applied mathematics on deepdyve, the largest online rental service for scholarly research with thousands of academic publications available at your fingertips. Stability analysis by fixed point theorems for a class of. The levels of population at which this equilibrium is achieved depend on the chosen values of the parameters. Jun 16, 2008 we consider the mean square asymptotic stability of a generalized linear neutral stochastic differential equation with variable delays by using the fixed point theory. Cadariu and radu applied the fixed point method to the investigation of the cauchy additive functional equation. Other results on fixed points and stability properties in equations with variable delays can be found in. Pdf on nov 1, 2019, s c sihombing and others published fixed. A stability theorem with a necessary and sufficient condition is given. Fixed points and stability of a class of integrodifferential.
Two xed point theorems are proved enabling us to solve the integral equation. The question of the stability or instability of the solution. By means of krasnoselskiis fixed point theorem we obtain boundedness and stability results of a neutral nonlinear differential equation with variable delays. Zhang nonlinear analysis 63 2005 e233e242 delay differential equations of the type considered here arise in a variety of applications including control systems, electrodynamics, mixing liquids, neutron transportation. A fixed point approach to the stability of a volterra integral equation article pdf available in fixed point theory and applications 2007 june 2007 with 187 reads how we measure reads. Keywords fixed points, stability, integral equations, nonuniqueness. Let be a finite interval, and let and be integral equations in which is a nonlinear integral map.
Linear stability of fixed points for the case of linear systems, stability of xed points can readily be determined from the fundamental matrix. Our aim is to establish some existence and stability results for integral solutions to the problem at. Pdf a fixed point approach to the stability of a volterra integral. Recently, zhao and yuan investigate the stability of a generalized volterralevin equation. Rao, thoery of integrodifferential equations, stability and control. For example, when ut is a single valued mapping, it is easy to see that hukuhara derivative and the integral reduce to the ordinary vector derivative and the integral, and therefore, the results obtained. A fixed point approach to the stability of a volterra integral equation soonmo jung 1 fixed point theory and applications volume 2007, article number. Research article, report by international journal of analysis. By choosing a different fixedpoint theorem, we show that the measures of noncompactness can be avoided and the existence and stability can be proved under weaker conditions.
Stability analysis by fixed point theorems for a class of non. Space stations at lagrange points are briefly mentioned. Pdf a fixed point approach to the stability of a volterra. It is worth noting that integral equations often do not have an analytical solution, and must be solved numerically. They give new conditions on the stability of zero solution of this equation. By choosing a different fixedpoint theorem, we show that the measures of. Fixed points and stability of the cauchy functional equation in algebras fixed points and. By choosing a different fixedpoint theorem, we show that the measures of noncompactness can be avoided and the existence and stability can be proved under weaker.
Burton northwest research institute, 732 caroline street port angeles, wa 98362, u. Jung, a fixed point approach to the stability of a volterra integral equation, fixed point theory and applications, vol 2007, article id 57064, 9 pages, 2007. In this paper, we study the mean square asymptotic stability of a class of generalized nonlinear neutral stochastic differential equations with variable time delays by using fixed point theory. Pdf a fixed point approach to the mittaglefflerhyers. Two examples are also given to illustrate our results. Burton tetsuo furumochi northwest research institute department of mathematics 732 caroline street shimane university port angeles, wa 98362 matsue, japan 6908504 taburton at furumochiatmath. The results follow from the darbo fixed point theorem. Hyersulam stability of nonlinear integral equation hyersulam stability of nonlinear integral equation. In this paper, we have presented and studied two types of the mittaglefflerhyersulam stability of a fractional integral equation. We will apply the fixed point method for proving the hyersulamrassias stability of a volterra integral equation of the second kind. In this paper we begin a study of stability theory for ordinary. We consider the mean square asymptotic stability of a generalized linear neutral stochastic differential equation with variable delays by using the fixed point theory. Using such a clever idea, they could present another proof for the hyersulam stability of that equation 1419. In this paper, we deal with a class of fractional integrodifferential equations involving impulsive effects and nonlocal conditions, whose principal part is of diffusionwave type.
Jung 9 applied the fixed point method to the investigation of the volterra integral equation adhering to the notion of cadariu and radu 2. Fixed points and stability of an integral equation. The reason for doing this is that it may make solution of the problem easier or, sometimes, enable us to prove fundamental results on the existence and uniqueness of the solution. A fixed point approach to the mittaglefflerhyersulam stability of a. Solution of differential and integral equations using. In the circular restricted three body problem, there are a set of 5 points that if we place our spacecraft there, itll never move relative to the two bodies. Stability of fixed points for a differential equation.
Burton1 and tetsuo furumochi2 1northwest research institute 732 caroline st. Additionally, as an application, we apply the obtained fixed point theorems to study the nonlinear functional integral equations. Jim lambers mat 605 fall semester 201516 lecture 6 notes these notes correspond to sections 2. Hyersulam stability of differential equation pdf paperity. We prove a fixed point theorem and show its applications in investigations of the hyersulam type stability of some functional equations in single and many variables in riesz spaces. The analysis of the theoretical results depends on the structure of nabla discrete mittagleffler.
Periodicity and stability in neutral nonlinear differential equations by. Pdf we will apply the fixed point method for proving the hyersulamrassias stability of a volterra integral equation of the second kind. Stability of a class of nonlinear neutral stochastic. Port angeles, wa 98362 2department of mathematics shimane university matsue, japan 6908504 abstract. We prove that the fractional order delay integral equation is mittaglefflerhyersulam stable on a compact interval. The stability of equilibria of a differential equation, analytic approach. Jun 17, 2007 a fixed point approach to the stability of a volterra integral equation soonmo jung 1 fixed point theory and applications volume 2007, article number. A fixed point approach to the stability of a volterra. Stability of a cauchyjensen functional equation in quasibanach spaces stability of a cauchyjensen functional equation in quasibanach spaces. It contains an extensive collection of new and classical examples worked in detail and presented in an elementary manner.
Mt5802 integral equations introduction integral equations occur in a variety of applications, often being obtained from a differential equation. Fractional differential and integral equations can serve as excellent tools for description of mathematical modelling of systems and processes in the fields of. The following nonlinear quadratic integral equation of hammerstein type is studied. Fixed points and stability in neutral nonlinear differential equations with variable delays abdelouaheb ardjouni and ahcene djoudi abstract. To state results concerning stability, we use the following norms. The second solution represents a fixed point at which both populations sustain their current, nonzero numbers, and, in the simplified model, do so indefinitely. A fixed point theorem and the hyersulam stability in. Using fixed point theory existence and uniqueness of solution of differential and integral equation can be verified. Krasnoselskiis fixed point theorem and stability t. Stability of the lagrange points three body problem. Method of successive approximations for fredholm ie s e i r e s n n a m u e n 2.
The hyersulam stability for two functional equations in a single variable mihet, dorel, banach journal of mathematical analysis, 2008. A generalization of diazmargoliss fixed point theorem. In 1940, ulam gave a wide ranging talk before the mathematics club of the university of wisconsin in which he discussed a number of important unsolved problems among those was the question concerning the. A fixed point approach to the stability of an integral equation. Apr 08, 2016 equilibrium points for nonlinear differential equations.
Fixed points and stability in neutral nonlinear differential equations with variable delays. Journal of inequalities in pure and applied mathematics 2003,4. Introduction over the past few decades, fixed point theory of lipschitzian mappings has been. Generalizations of darbos fixed point theorem for new condensing. An example of this is evaluating the electricfield integral equation efie or magneticfield integral equation mfie over an arbitrarily shaped object in an electromagnetic scattering problem.
Stability, fixed points, and inverses of delays leigh c. The reason for doing this is that it may make solution of the problem easier or, sometimes, enable us to prove fundamental results on. Contraction, differential equations, fixed points, integral equations. On a fixed point theorem with application to integral equations. Fixed points and stability in neutral stochastic differential equations with variable delays. Fixed points and stability in differential equations with. A fixed point approach to the stability of an integral. Pdf fixed point theorem on volterra integral equation. Abstractwe consider a paper of banas and rzepka which deals with existence and asymptotic stability of an integral equation by means of fixedpoint theory and measures of noncompactness. Hyersulam stability of nonlinear integral equation. In this paper, we showed that has the hyersulam stability.
We will apply the fixed point method for proving the generalized hyersulam stability of the integral equation which is strongly related to the wave equation. Fixed points and stability of neutral stochastic delay. We consider a paper of banas and rzepka which deals with existence and asymptotic stability of an integral equation by means of fixedpoint theory and measures of noncompactness. Fixed points and stability in the following example, we illustrate how the xed points of a dynamical system can be associated. Mathematics duality theory mathematics fixed point theory integral equations mappings mathematics maps mathematics mathematical research. A fixed point approach to the stability of an equation of the square spiral jung, soonmo, banach journal of. Fixed points and stability of the cauchy functional equation in algebras fixed points and stability of the cauchy functional equation in algebras. Equilibrium points for nonlinear differential equations. This book is the first general introduction to stability of ordinary and functional differential equations by means of fixed point techniques.
In this paper, a generalization of diazmargoliss fixed point theorem is established. Introduction motivated by a problem in implicit function theory, we formulate an existence problem in integral equations. A fixed point approach to the stability of a volterra integral equation. To show the existence and the asymptotic stability of periodic solutions, we transform.
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