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International Journal of Enhanced Research Publications, ISSN: XXXX-XXXX Vol. 2 Issue 4, April-2013, pp: (1-4), Available online at: www.erpublications.com On Non Standard Description of Solutions of Functional Equations Tahir H. Ismail1, Maha F. KHalf2 1,2 Department of Mathematics, College of Computer Sciences and Mathematics, University of Mosul, Mosul 41002, Iraq Abstract: In this paper, we use some concepts of nonstandard analysis to establish whether all solutions of the functional equation π(π + π) = π(π) + π(π) can be described in terms of a functional equation of the form π(π) = ππ , where π is unlimited or limited. Keywords: Non Standard analysis, infinitely near, functional equation. 1. Introduction It is well known that all the continuous solutions of the functional equation π(π₯ + π¦) = π(π₯) + π(π¦) (1) Where πis a real valued function given by π(π₯) = ππ₯. Many papers on the subject have dealt mainly with the problem of finding the conditions on an additive function that ensure its continuity, one such condition is that the function be bounded on some interval. (see [1], [10], [7], [2]). The dichotomy between the continuous and discontinuous solution of (1) is striking particularly in view of the simplicity of the equation. The problem of finding solutions to (1) is nearly related to the problem of finding all the characters of an additive subgroup π of πΌπ , that is , all the complex valued functions π on π such that |π(π₯)| = 1 for all π₯ β π , and π(π₯ + π¦) = π(π₯) + π(π¦) (2) The solutions of this character problem are of two kinds continuous and discontinuous, all of the forms having the form π(π₯) = π πππ₯ (π β β). The following theorem gives important information about the discontinuous solution of (2). Theorem 1.1[8] If S is a subgroup of β and X is character of S. Then for every π > 0 and every finite subset {π‘1 , π‘2 , β¦ β¦ , π‘π } of S, there exists a continuous character π0 (π₯) = π πππ₯ such that |π(π₯π ) β π0 (π₯π )| < π , π = 1,2, β¦ β¦ , π It follows from the approximation theorem that for each characterπ of a subgroup π of β relationπ1 (π₯, π, π) defined by the inequality |π(π₯) β π πππ₯ | < π is concurrent in sense of Robinson A. [9]. Hence there exists an element π of an enlargement β β of β such that π(π₯) = π πππ₯ for all β π, thus every character of π, continuous or not, is of the form π(π₯) β π βπππ₯ for some π β ββ. To deal with additive functions, we introduce another form of the approximation theorem. Through this paper we need the following definitions and notations. Definition 1.2 (see [6], [3], [4]) A real number π is called unlimited if its absolute value is larger than any standard in tagger, so a nonstandard π is also unlimited real number. Definition 1.3 (see [5], [8], [11]) 1 A real number is called infinitesimal if its absolute is smaller than for any standard number, of course 0 is infinitesimal. π Definition 1.4 [6] A real number is called appreciable if it is neither unlimited nor infinitesimal. Page | 1 International Journal of Enhanced Research Publications, ISSN: XXXX-XXXX Vol. 2 Issue 4, April-2013, pp: (1-4), Available online at: www.erpublications.com Definition 1.5 [4] Two real number π₯ and π¦ are called infinitely near, denoted by π₯ β π¦ , if π₯ β π¦ is infinitesimal. 2. The main results Theorem 2.1 If π is a real valued function of a subgroup πof β, such that π(π₯ + π¦) = π(π₯) + π(π¦)(πππ π) (3) for some positive number π. Then for each π > 0 and every finite subset {π‘1 , π‘2 , β¦ β¦ , π‘π } of S, there exists a number π such that |π(π₯π ) β ππ₯π | < π (πππ π), π = 1,2, β¦ β¦ , π In other words such that |π(π₯π ) β ππ₯π | is always within, of some integral multiple of π. Proof: Each character in β is of the form π(π₯)=exp(ππππ(π₯)/π) Where π satisfies (3). By the character approximation theorem, there exists for each πΏ > 0 a number π such that |π(π₯π ) β exp( 2ππππ₯π π )| < πΏ π = 1,2, β¦ β¦ β¦ , π Since the Logarithmic function is continuous, it follows that to each π > 0 there corresponds a number π such that | 2ππ(π₯π ) π β 2πππ₯π /π| < 2ππ π (πππ 2π) , π = 1,2, β¦ β¦ , π In other words, such that |π(π₯π ) β ππ₯π | < π (πππ π), π = 1,2 β¦ β¦ , π For unlimited values of π the linear function (π π₯) is clearly unlimited for all nonzero standardπ₯ . Thus to get a standard additive function, we must reduce the values of the function ππ₯ to finite values. Let πΎ be the additive subgroup of ββ generated by the set of unlimited power of 2, that is let πΎ = {π§2π : π§ πππ π πππ πππ‘πππππ , π > 0 ππ π’ππππππ‘ππ} π β© If πΎπ is the additive subgroup of πΌπ generated by ( π is standard positive integer), then πΎ can also be written πΎ = πΎ . 2 π β πΌπ π Definition 2.2 π β π (πππ πΎ) means that the distance between π β π and some element of πΎ is infinitesimal. Theorem 2.3 Forβ ββ , let π denote the subgroup of β consisting of all π₯ such that (π π₯) is congruent to some limited number π(π₯) modulo πΎ, define a function π on π by π(π₯) = π§(π₯). Then π is additive on π . Conversely if π is additive on a subgroup πof β. Then there exists a number π β ββ such that for each π₯ β β , ππ₯ is limited modulo, and π(π₯) β ππ₯(πππ πΎ). Proof: Supposeπ β ββ, and if π₯, π¦ β β have the property that ππ₯ and ππ¦ are limited modulo πΎ, then certainly π(π₯ + π¦) = ππ₯ + ππ¦ has the some property. Now by definition of π(π₯) + π(π¦) β π(π₯) + π(π¦) = π(π₯ + π¦) β π(π₯ + π¦)(ππππ’ππ πΎ) Since the values of π are standard, and no two standard numbers can be congruent modulo πΎwithout being equal π(π₯ + π¦) = π(π₯) + π(π¦). Page | 2 International Journal of Enhanced Research Publications, ISSN: XXXX-XXXX Vol. 2 Issue 4, April-2013, pp: (1-4), Available online at: www.erpublications.com Now suppose that π is an additive function on a subgroup πof β, then certainly π(π₯ + π¦) = π(π₯) + π(π¦)(ππππ’ππ 2π )πππ ππ£πππ¦ πππ ππ‘ππ£π πππ‘ππππ π By the second approximation theorem, for any π > 0 , for positive integers π1 , π2 , β¦ β¦ , ππ , and for elements π₯1 , π₯2 , β¦ β¦ , π₯π β π , there exists some π β β such that |π(π₯π ) β ππ₯π | < π (πππ 2ππ ), π = 1,2, β¦ β¦ , π , π = π1 , π2 , β¦ β¦ , ππ Chooseπ = max(2π ) , π = π1 , π2 , β¦ β¦ , ππ . The relation π2 (π, π₯, π, π) defined by the inequality |π(π₯π ) β ππ₯π | < π (πππ 2π ) Is therefore can, hence there exists π β ββ such that, for every π₯ β π and every limited positive integer πΌ π(π₯) β ππ₯(πππ 2πΌ ) This implies that π(π₯) β ππ₯(πππ πΎ). Remark 2.4: Since the elements of πΎ are numerous, in the sense that each interval of unlimited length in ββ contains unlimited many elements of πΎ. It is natural to ask whether, for each unlimited π β ββ, ππ₯ is necessarily finite modulo πΎ for every π₯ β β ? Theorem 2.5: There exists an element π¦ β ββ such that the distance from π¦ to each element of πΎ is unlimited. Proof: Consider the sequence of numbers defined inductively by the conditionπ¦1 = 1, π¦π+1 = 2π β π¦π . Ifβπβπ denotes the distance form π₯to the nearest integral multiple of 2π , then the induction argument shows that βπ¦π βπ = π¦π , π = 1, 2, β¦ β¦ , π For each π , the relation ππ (π, π¦) defined by statement βπ¦π β = π¦π is concurrent therefore there existsπ¦ β ββ such that βπ¦π β = π¦π for every standard positive integer π. Now suppose that π¦ within a limited distance, say 2π , of some element of πΎ . This element is evidently a multiple of 2π+π for each standard π , since the sequence {π¦π } is increasing and not bounded which implies contradiction to inequalities |π¦π | < 2π < π¦π+π Hence π¦ is of unlimited distance from each element of πΎ. References [1]. J. AczπΜ l and J. Dhombres, Functional Equations in SeveralVariables, Cambridge University Press,Cambridge, 1989. [2]. L. BERG, βIterative functional equations,β Rostock. Math. Kolloq., vol. 64, pp. 3-10,2009. [3]. M. DAVIS, Applied Nonstandard Analysis , New York , JohnWiley and Sons, 1977. [4]. F. DIENER and M. DIENER, Nonstandard Analysis In Practice , Springer-verlag, Berlin , Heidelberg, 1995. [5]. LUJZ, and M. Goze, Nonstandard Analysis , Practical Guide with Applications, Lecture Notes in Mathematics No. 881,Springerβ verlag, 1981. [6]. E. NEISON, βInterual Set Theory A new Approach to Nonstandard Analysis,β Bull. Amer. Math. Soc., Vol. 83, pp. 1165-1198, 1977. [7]. N. Nicolae, βAbout Some Classical Functional Equations,β Tr. J. of Mathematics, vol. 22 , pp. 119 β126, 1998. [8]. N. Paisan, βAn Alternative JensenΚΌs Functional Equation On Semigroups,β ScienceAsia, vol. 38, pp. 408 β413, 2012. [9]. A. ROBINSON, Nonstandard Analysis 2ππ. ed. Noth β HolaudPub. Comp. Amster, 1970. [10]. P. K. Sahoo and T. Riedel, Mean Value Theorems andFunctional Equations . World Scientific Pub. Comp.,1998. [11]. K. D. STROYAN and W. A. LUXEMBURG, Introduction toTheory of Infinitesimal, New York Academic Press,1976. Page | 3