Using such a clever idea, they could present a short and simple proof for the stability of the cauchy functional equation. In the last section of this paper, we treat a fixed point approach to the stability of the cauchyjensen functional equation. Pdf we prove a general ulamhyers stability theorem for a nonlinear equation in probabilistic metric spaces, which is then used to obtain stability. Fixed points and the stability of jensens functional equation emis. Apr 17, 2012 we established hyersulamrassias stability of a cubic functional equation in fuzzy normed spaces by using fixed point alternative theorem. Vaezpour3 1 department of mathematics, science and research branch, islamic azad university iau, tehran, iran. Pdf fixed points and stability for functional equations in. A fixed point approach to the stability of a mean value type functional equation soonmo jung 1, id and yanghi lee 2 1 mathematics section, hongik university, sejong 30016, korea 2 department of mathematics education, gongju national university of education, gongju 32553, korea. Similarly, smalls book 38 is a very enjoyable, well written book and focuses on the most essential aspects of functional equations. We say w is a fixed point of a function f if and only if w is in the domain of f and f w w.
Fixed point, hyersulamrassias stability,quadratic functional equation. Some fixed point theorems of functional analysis by f. Fixedpoint iteration convergence criteria sample problem functional fixed point iteration now that we have established a condition for which gx has a unique. A fixed point approach to the stability of pexider quadratic. Seong sik kim1, john michael rassias2 and soo hwan kim1.
Khodaei department of mathematics, semnan university, p. The fixed point alternative and the stability of functional equations viorel radu west university of timi. In this paper, we introduce a setvalued cubic functional equation and a setvalued quartic functional equation and prove the hyersulam stability of the setvalued cubic functional equation and the setvalued quartic functional equation by using the fixed point method. A fixed point approach to the stability of pexider.
A fixed point approach to stability of functional equations request. Dynamic optimization we have the following problem max. Fixed point theorems fixed point theorems concern maps f of a set x into itself that, under certain conditions, admit a. A fixed point approach to the stability of quadratic. The banach fixed point theorem gives a general criterion. A fixed point approach to the stability of a mean value type. Solving equations using fixed point iterations instructor. Fixed point equations and nonlinear eigenvalue problems in. Having information on the fixed points of functions often help to solve. However many necessary andor sufficient conditions for the existence of such points involve a mixture of algebraic order theoretic or topological properties of. A fixed point approach of quadratic functional equations. Applications of fixed point theorems to the hyersulam stability. Baker 2 used the banach fixed point theorem to give hyersulam stability results for a nonlinear functional. In a recent paper of this journal, a fuzzy version of a generalized hyersulam stability for the jensen functional equation, in the fuzzy normed linear space setting is presented.
Fixed points for functions of several variables previously, we have learned how to use xedpoint iteration to solve a single nonlinear equation of the form fx 0. A fixed point approach to the stability of a nonic. A fixed point approach to the stability of a functional equation of the spiral of theodorus. Fixed point method for setvalued functional equations. Fixed points and stability of the cauchyjensen functional. Market equilibrium steady state nash equilibrium solution of bellman equations.
Functional fixed point iteration fixedpoint algorithm to. Moreover, we prove the stability of the quintic and sextic functional equations in quasinormed spaces via fixed point method, and also using gajdas example to give two counterexamples for a singular case. Econ 2010 fall 20 fixed point theory serves as an essential tool for various branches of mathematical analysis and. The fixed point method for fuzzy approximation of a. Cadariu and radu applied the fixed point method to the investigation of cauchy and jensen functional equations. Jun 10, 2008 cadariu and radu applied the fixed point method to the investigation of cauchy and jensen functional equations. By means of iterative techniques and by using topological tools, fixed point theorems for completely continuous maps in ordered banach spaces are deduced, and particular attention is paid to the derivation of multiplicity results. Using the fixed point method, we prove the hyersulam stability of a cauchyjensen type additive setvalued functional equation, a jensen type additivequadratic setvalued functional equation, a generalized quadratic setvalued functional equation and a jensen type cubic setvalued functional equation. Results of this kind are amongst the most generally useful in mathematics. We build an iterative method, using a sequence wich converges to a fixed point of g, this fixed point is the exact solution of fx0. This paper gives a survey over some of the most important methods and results of nonlinear functional analysis in ordered banach spaces. Introduction a classical question in the theory of functional equations is the following.
Applications of fixed point theorems to the hyersulam. Lectures on some fixed point theorems of functional analysis. In mathematics, a fixed point theorem is a result saying that a function f will have at least one fixed point a point x for which f x x, under some conditions on f that can be stated in general terms. Fixed point representations require the programmer to create a virtual decimal place in between two bit locations for a given length of data variable type. A fixed point approach to the stability of a functional equation of. The fixed point method for fuzzy stability of the jensen. Cadariu and radu applied the fixed point theorem to prove the stability theorem of cauchy and jensen functional equations. We may also be able to make a clever choice for xor yto make the functional equation become pleasant. Box 35195363, semnan, iran correspondence should be addressed to m. Pdf on may 27, 2015, muhmmad saeed ahmad and others published new fixed point iterative method for solving nonlinear functional equations find, read and cite all the research you need on. Department of mathematics, faculty of science, king abdulaziz university, p. Pdf a fixed point approach of quadratic functional equations. Mathematics free fulltext a fixed point approach to the. A fixed point approach to the stability of a mean value.
Pdf on a fixed point theorem with application to functional. The stability of a general sextic functional equation by. We find out the general solution of a generalized cauchyjensen functional equation and prove its stability. In this paper, we show that the theorems of hyers, rassias and gajda concerning the stability of the cauchys functional equation in banach spaces, are direct consequences. Request pdf a fixed point approach to stability of functional equations we prove a simple fixed point theorem for some not necessarily linear operators and. Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in related fields.
The stability of a quadratic functional equation with the fixed point alternative park, choonkil and kim, jihye, abstract and applied analysis, 2009 solution and hyersulamrassias stability of generalized mixed type additivequadratic functional equations in fuzzy banach spaces eshaghi gordji, m. Functional equation and fixed points mathematics stack exchange. The purpose of this paper is to study behavior of a rational type contraction introduced in a fixed point theorem for contractions of rational type in partially ordered metric spaces, ann. For the purposes of this paper the notion of a q point for a fixed point number is introduced. Fixed point theory orders of convergence mthbd 423 1. The fixed point method for fuzzy approximation of a functional equation associated with inner product spaces m. The center of a linear homogeneous differential equation of the second order is an example of a neutrally stable fixed point. Using fixed point method, we prove the hyersulam stability of the cauchyjensen functional equation in fuzzy banach algebras. Vedak no part of this book may be reproduced in any form by print, micro. Stability, jensens functional equation, fixed point, modular space. Radu, we will discuss the stability of the solutions for this functional equation. A fixed point approach to the stability of a nonic functional. Fixedpoint theory a solution to the equation x gx is called a.
A fixed point approach to the stability of a functional. Park, fixed points and hyersulamr assias stability of cauchyjensen functional equations in banach algebr as, fixed point. The banach fixedpoint theorem gives a general criterion guaranteeing that, if it is satisfied, the procedure of iterating a function yields a fixed point by contrast, the brouwer fixedpoint theorem is a nonconstructive result. In this note we give an alternative proof of this result, using the fixed point method. Pdf a fixed point approach to the stability of a functional. Fuzzy stability of a cubic functional equation via fixed. Research open access fuzzy stability of a cubic functional. A fixed point approach to the stability of pexider quadratic functional equation with involution m. On the stability of setvalued functional equations with the.
A fixed point approach to the stability of an additive. In this paper, we prove the stabilities and nonstabilities of a new mixed type cubic and quartic functional equation in 2banach space using fixed point method in the sense of hyersulam. Generally g is chosen from f in such a way that fr0 when r gr. Mathematics free fulltext a fixed point approach to. The hyersulam stability for two functional equations in a single variable mihet, dorel, banach journal of mathematical analysis, 2008. A number is a fixed point for a given function if root finding 0 is related to fixedpoint iteration given a rootfinding problem 0, there are many with fixed points at. In this paper, we prove the generalized hyersulam stability via the fixed point method and investigate new theorems via direct method concerning the stability of a general quadratic functional equation. View fulltext download pdf cite this paper abstract we prove the generalized hyersulam stability of a mean value type functional equation f x.
A fixed point approach to stability of functional equations. Stability of quadratic functional equations via the fixed. A fixed point approach to stability of functional equations in. Mar 18, 2017 in this paper, we introduce a setvalued cubic functional equation and a setvalued quartic functional equation and prove the hyersulam stability of the setvalued cubic functional equation and the setvalued quartic functional equation by using the fixed point method. If the functional equation is true for any two variables xand y, then we may try to let x yand get a functional equation which is true for all x. Fixed points and the stability of jensens functional equation. Yunpeng li, mark cowlishaw, nathanael fillmore our problem, to recall, is solving equations in one variable. A fixed point approach to the stability of an equation of the square spiral jung, soonmo, banach journal of. On the stability of setvalued functional equations with. Graphically, these are exactly those points where the graph of f, whose equation is y fx, crosses the diagonal, whose equation is y x. We established hyersulamrassias stability of a cubic functional equation in fuzzy normed spaces by using fixed point alternative theorem. Fixed point method allows us to solve non linear equations.
We will present a fixed point method for the stability theorems of functional equations of jensen type as given by s. A fixed point approach to the stability of a cauchyjensen. A fixed point approach park, choonkil, taiwanese journal of mathematics, 2010. Multiple attractive points can be collected in an attractive fixed set. In this paper, we will adopt the idea of cadariu and radu to prove the hyersulamrassias stability of the quadratic functional equation with involution. The stability of a quadratic functional equation with the fixed point alternative park, choonkil and kim, jihye, abstract and applied analysis, 2009 ulam stability of a quartic functional equation bodaghi, abasalt, alias, idham arif, and ghahramani, mohammad hosein, abstract and applied analysis, 2012. Fixed points is a fixed points of function f if example. Hyersulamrassias stability, cubic functional equation, fuzzy normed space, fixed point 1 introduction, definitions and notations fuzzy set theory is a powerful hand set for modeling uncertainty and vagueness in var. Park, a fixed point approach to the stability of an additivequadraticcubicquartic functional equation, fixed point theory and applications, vol. In fact, we investigate the existence of a cauchyjensen mapping related to the generalized cauchyjensen functional equation and prove its uniqueness. Jensen functional equation, fixed point, stability.
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