35-959, Rzeszów, Poland b Institute of Computer Science, Jagiellonian University,

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1 This article was downloaded by: [Polska Akademia Nauk Instytut Matematyczny] On: 07 March 03, At: 03:7 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 0795 Registered office: Mortimer House, 37- Mortimer Street, London WT 3JH, UK Journal of Difference Equations and Applications Publication details, including instructions for authors and subscription information: Applications of de Rham theorem in approximate midconvexity Anna Mureńko a, Jacek Tabor b & Józef Tabor a a Institute of Mathematics, University of Rzeszów, Rejtana 6A, , Rzeszów, Poland b Institute of Computer Science, Jagiellonian University, Łojasiewicza 6, 30-38, Kraków, Poland Version of record first published: Jan 0. To cite this article: Anna Mureńko, Jacek Tabor & Józef Tabor (0): Applications of de Rham theorem in approximate midconvexity, Journal of Difference Equations and Applications, 8:3, To link to this article: PLEASE SCROLL DOWN FOR ARTICLE Full terms and conditions of use: This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae, and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand, or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material.

2 Journal of Difference Equations and Applications Vol. 8, No. 3, March 0, Applications of de Rham theorem in approximate midconvexity Anna Mureńko a *, Jacek Tabor b and Józef Tabor a Downloaded by [Polska Akademia Nauk Instytut Matematyczny] at 03:7 07 March 03 a Institute of Mathematics, University of Rzeszów, Rejtana 6A, Rzeszów, Poland; b Institute of Computer Science, Jagiellonian University, Łojasiewicza 6, Kraków, Poland (Received 5 December 008; final version received 9 June 009) Let be a normed space, V be an open convex subset of and let Q : ½0; Þ! ½0; Þ be a given function. A function f : V! R is called Q-midconvex if x þ y f ðxþþfðyþ f # þ Qðkx ykþ for x; y [ V: By the result of Tabor and Tabor, we know that under respective conditions on Q,iff is Q-midconvex and locally bounded above at a point then there exists a continuous function w : ½0; Š! R such that f ðtx þðtþyþ # tf ðxþþðtþfðyþþwðtþqðkx yjkþ for x; y [ V; t [ ½0; Š: In this paper we determine the smallest function w satisfying the above inequality. The required conditions on Q are such that the functions wðtþ ¼t p, p [ ½; Š satisfy them. As the main tool we use de Rham theorem. Keywords: approximately midconvex function; Takagi function; Jensen difference Mathematics Subject Classification (00): Primary 6B5. Introduction Let be a normed space and V a convex subset of. A function f : V! R is called convex if f ðtx þð tþyþ # tf ðxþþð tþf ðyþ for x; y [ V; t [ ½0; Š: If the above inequality holds for t ¼ =, that is x þ y f ðxþþfðyþ f # for x; y [ V; then f is called midconvex (Jensen convex). The notion of approximate convexity was introduced by Hyers and Ulam [5]. Let d $ 0. A function f : V! R is called d-convex if f ðtx þð tþyþ # tf ðxþþð tþf ðyþþd for x; y [ V; t [ ½0; Š; *Corresponding author. aniam@univ.rzeszow.pl ISSN print/issn online q 0 Taylor & Francis

3 336 and it is called d-midconvex if x þ y f # A. Mureńko et al. f ðxþþfðyþ þ d for x; y [ V: Downloaded by [Polska Akademia Nauk Instytut Matematyczny] at 03:7 07 March 03 The notion of approximate midconvexity has been recently successively generalized. It has been done by replacing d in the previous inequality by respective function of kx yk. The first step on this way was done by Házy and Páles []. They replaced d by kx ykþd with $ 0, d $ 0. Házy [3] replaced d by kx yk p þ d with p $ 0. A function f : V! R is called ð; d; pþ-midconvex if x þ y f # f ðxþþfðyþ þ kx yk p þ d for x; y [ V: Tabor and Tabor considered in [0] the following generalization of the previous definitions. Definition.. Let Q : ½0; Þ! ½0; Þ be a given function. A function f : V! R is Q-midconvex if x þ y f # f ðxþþfðyþ þ Qðkx ykþ for x; y [ V: In all the above mentioned papers, the relation between approximate midconvexity and approximate convexity was investigated. The versions of Bernstain Doetsch theorem have been proved there. Assume additionally that V is open and let f : V! R be locally bounded above at a point. Ng and Nikodem [6] proved that if f is d-midconvex, then it is d convex. They also showed that the constant d cannot be improved. Házy and Páles [] proved that if f is ð; dþ-midconvex, i.e. then x þ y f # f ðxþþfðyþ þ kx ykþd for x; y [ V; f ðtx þð tþyþ # tf ðxþþð tþf ðyþþwðtþkx ykþd for x; y [ V; t [ ½0; Š; where w : ½0; Š! R is the Takagi function wðtþ U dist ð k t; ZÞ k for t [ ½0; Š: Házy showed in [3] that if f : V! R is ð; d; pþ-midconvex with p [ ð0; Þ, then f ðtx þð tþyþ # tf ðxþþð tþf ðyþþw p ðtþkx yk p þ d for x; y [ V; t [ ½0; Š;

4 Journal of Difference Equations and Applications 337 where w p : ½0; Š! R can be defined as follows w p ðtþ U ðdist ð k t; ZÞÞ p k for t [ ½0; Š: Tabor and Tabor proved in [0] that if f : V! R is Q-midconvex, then there exists a real function w depending on kx yk and t such that f ðtx þð tþyþ # tf ðxþþð tþf ðyþþwðt; kx ykþ for x; y [ V; t [ ½0; Š: Downloaded by [Polska Akademia Nauk Instytut Matematyczny] at 03:7 07 March 03 Three such different functions w are presented there. Some of them appear under certain additional assumptions on Q. The above-mentioned results yield a very natural question whether occurring in the right-hand sides convexity deviations are sharp. In some particular cases, this problem has been already solved. As we have mentioned for d-midconvexity, it was positively answered by Ng and Nikodem [6]. For ð; 0Þ-midconvexity also a positive answer was given by Boros []. It happens that in the case of ð; 0; pþ-midconvexity with p. the smallest function w which can be used is w ; 0. Rolewicz proved in [8] that if f : V! R is ð; 0; pþ-midconvex with p. and locally bounded above at a point, then f is convex. Numerical simulations confirm the hypothesis that for ð; 0; pþ-midconvexity sharp functions w p in the case p [ ð0; Þ and p [ ð; Þ are of different shapes. In [9] it has been determined sharp convexity deviations ð; 0; pþ-midconvex function in case p [ ½; Š. In this paper we generalize the above mentioned result. We will need the function d : R! R defined as follows dðtþ U distðt; ZÞ for t [ R: Tabor and Tabor proved in [0] the following theorem. Theorem.. Let Q : ½0; Þ! ½0; Þ be a non-decreasing function satisfying the condition Q k, : ðþ Let V be an open convex subset of and let f : V! R be a Q-midconvex function which is locally bounded above at a point of V. Then f is continuous and f ðtxþðtþyþ # tf ðxþþðtþf ðyþþ Q kxyk k dð k tþ for x;y [ V; t [ ½0;Š: ðþ The question whether the convexity deviation P Qðkx yk=k Þdð k tþ occurring in () is sharp at present seems to be too complicated to deal with in general. It is a function of two variables: kx yk and t. Therefore, we will assume a certain condition on Q to simplify this function. The main idea consists of replacing this function of two variables by the product of two functions of one variable. As a direct corollary from Theorem. we obtain the following result.

5 338 A. Mureńko et al. Proposition.. Let Q : ½0; Þ! ½0; Þ be a non-decreasing function satisfying the following condition Q t # QðtÞ for t $ 0; ð3þ p with certain p [ ð0; Š. Let V be an open convex subset of and let f : V! R be a Q-midconvex function which is bounded above at a point of V. Then f is continuous and Downloaded by [Polska Akademia Nauk Instytut Matematyczny] at 03:7 07 March 03 f ðtx þð tþyþ # tf ðxþþð tþf ðyþþv p ðtþqðkx ykþ for x; y [ V; t [ ½0; Š; ðþ where v p ðtþ U kp dðk tþ for t [ R: Furthermore v p is continuous. Obviously, a constant p satisfying (3) is not determined uniquely. We are interested in the greatest p with this property. Let p Q U sup{p [ ð0; Š such that Q t # QðtÞ for every t $ 0}: p Clearly, if Q satisfies (3) with some p [ ð0; Š, then Q satisfies (3) with p Q. In the particular case when QðtÞ ¼t p, p. 0 we have p Q ¼ p. Now we are ready to present precisely the main question we will deal with. Problem.. Let Q : ½0; Þ! ½0; Þ be a non-decreasing function satisfying condition (3). Denote by F a family of all functions w : ½0; Š! ½0; Þ such that for each normed space, each open convex subset V of, each Q-midconvex function f : V! R locally bounded above at a point the following inequality holds true f ðtx þð tþyþ # tf ðxþþð tþf ðyþþwðtþqðkx ykþ for x; y [ V; t [ ½0; Š: ð5þ Does there exist the smallest element in F? If so find it. The next result shows one of the ways to solve the above problem. Theorem.. Let Q : ½0; Þ! ½0; Þ be a non-decreasing function satisfying condition (3), and let QðÞ ¼. Ifv pq is Q-midconvex, then v pq is the smallest element in F. Proof. From Proposition. we obtain that v pq [ F. Let us consider an arbitrary w [ F. Let ¼ R, let V be an open, convex subset of R such that ½0; Š, V, and let f U v pq j V. Inserting into (5) x ¼, y ¼ 0, we obtain v pq ðtþ # wðtþqðþ ¼wðtÞ for t [ ½0; Š; i.e. v pq j ½0;Š # w. A

6 Journal of Difference Equations and Applications 339 The above result in the case QðtÞ U t p, p [ ½0; Š was presented by Páles at the st International Symposium on Functional Equations [7]. It is worthy of mentioning that the condition QðÞ ¼ in determining the smallest element of F is only convenient, not essential one. Suppose that QðÞ 0 and consider an arbitrary Q-midconvex function f : V! R. Put ~f U QðÞ f ; ~ Q U QðÞ Q: Downloaded by [Polska Akademia Nauk Instytut Matematyczny] at 03:7 07 March 03 Then ~f is ~ Q-midconvex and ~ QðÞ ¼. We can now apply Theorem... Approximate midconvexity of the function v p In this section we will prove that, under respective additional assumptions on Q, the function v pq is Q-midconvex. In view of Theorem. it will mean that v pq is the smallest element of the family F. As a consequence we obtain sharp estimations of the convexity deviations for QðxÞ U x p, p [ ½; Š. We would like to sketch the main idea of our considerations. We begin with the particular case QðxÞ ¼x and then we reduce the general case to this particular one. Let f : V! R be any function. By Jf we denote the Jensen difference of f, i.e. Jf ðx; yþ U f x þ y ðf ðxþþfðyþþ for x; y [ V: By BðR; RÞ we denote the Banach space of all bounded functions from R into R with the supremum norm. In the paper we apply the fixed point method (for related approach see []). A crucial role in our considerations will play the following simple lemma. Lemma.(modified version of de Rham theorem). Let a; b [ R,a[ ½0; Þ and let a bounded function f : R! R be given. Let T : BðR; RÞ! BðR; RÞ be an operator defined as follows ðtgþðxþ U f ðxþþagðbxþ for g [ BðR; RÞ; x [ R: Then T is a contraction and consequently it has a unique fixed point w [ BðR; RÞ. If f is continuous, then so is w. Moreover, if g [ BðR; RÞ is such a function that Tg # g, then w # g. Proof. The proof is obvious and therefore it is omitted. A Lemma.. We have for p. 0 v p ðxþ ¼v ðxþþ p i¼ v ð i xþ ip for x [ R:

7 30 A. Mureńko et al. Proof. Sketch of the proof of this Lemma can be found in [9, Proposition.]. For the convenience of the reader, we present the complete reasoning below. Fix arbitrarily x [ R. Then we have v ðxþþ p i¼ v ð i xþ ip Downloaded by [Polska Akademia Nauk Instytut Matematyczny] at 03:7 07 March 03 ¼ ¼ ¼ ¼ ¼ ¼ ¼ l dðl xþþ p i¼ l dðl xþþ p i¼ ip ip l dðl xþþ p i¼ l¼i l dðl xþþ p i;l[n;i#l l dðl xþþ p l l¼ i¼ l þ p l i¼ lp dðl xþ¼v p ðxþ: ip li l¼i dð iþk xþ k dð l xþ li dð l xþ ip li dð l xþ ip li dð l xþ ip li! dð l xþ Theorem.. We assume that Q : ½0; Þ! ½0; Þ is a non-decreasing function satisfying the following conditions: (i) there exists a p [ ð0; Š such that Q t ¼ QðtÞ for t [ ½0; Š; p A (ii) Then Qð tþ QðtÞ #tþ for t [ ; : J v p ðx; yþ # Q d x y for x; y [ R:

8 Journal of Difference Equations and Applications 3 Proof. Notice first that the case Q ; 0 is trivial. Assume now that Q ò 0. Then p is determined uniquely and consequently p Q ¼ p. One can check by direct calculation, see also [9, Theorem.], that J v ðx; yþ # d x y By the above formula and Lemma. we have for x; y [ R: Downloaded by [Polska Akademia Nauk Instytut Matematyczny] at 03:7 07 March 03 J v p ðx; yþ ¼J v ðx; yþþ p i¼ ¼ J v ðx; yþþ p i¼0 # d x y þ p Therefore, it is sufficient to prove that c p ðrþ U ðdðrþþ þ p i¼0 ip J v ð i x; i yþ ip J v ð iþ x; iþ yþ i¼0 It is easy to check by direct calculation that ip d iþ x y for x; y [ R: ip ðdðiþ rþþ # QðdðrÞÞ for r [ R: ð6þ c p ðrþ ¼ p c pðrþþðdðrþþ ðdðrþþ for r [ R: ð7þ We define an operator T p : BðR; RÞ! BðR; RÞ by ðt p cþðrþ U p cðrþþðdðrþþ ðdðrþþ for c [ BðR; RÞ; r [ R: It follows from (7) that c p is a fixed point of T p. To be able to apply Lemma. we will prove that ðt p ðq + dþþðrþ # ðq + dþðrþ for r [ R: ð8þ The function d is -periodic and symmetric with respect to /. Therefore, it is sufficient to prove (8) for r [ ½0; =Š. Let r [ ½0; =Š. Making use of the definition of T p and (i) we obtain ðt p ðq + dþþðrþ ¼ p ðq + dþðrþþðdðrþþ ðdðrþþ ¼ p Qð rþþðrþ ð rþ ¼ QðrÞ ¼ðQ + dþðrþ:

9 3 A. Mureńko et al. Now we assume that r [ ½=; =Š. Thenr [ ½=; Š. Applying (i) and (ii) we get ðt p ðq + dþþðrþ ¼ p Qðð rþþ þ ðrþ ðð rþþ ¼ Qðð rþþ þ r # QðrÞ ¼ðQ + dþðrþ: p We have proved (8). Since c p is a fixed point of T p, by Lemma. we have c p # Q + d, which means that (6) is valid. A Downloaded by [Polska Akademia Nauk Instytut Matematyczny] at 03:7 07 March 03 Lemma.3. Let p [ ½; Š. Then the function QðxÞ ¼x p for x [ ½0; Þ is non-decreasing and satisfies conditions (i) and (ii). Proof. Only condition (ii) needs verification. We should prove that gðxþ U ð xþ p x p þ x # 0 for x [ ; : Obviously, g is continuous and g 00 ðxþ ¼pðp Þðð xþ p x p Þ $ 0 for x [ ; : Hence g is convex. Furthermore gð=þ ¼ 0 ¼ gðþ. Thus gðxþ # 0 for x [ ; : From Theorem. and Lemma.3 we directly obtain the following result. Corollary.. Let p [ ½; Š. Then J v p ðx; yþ # d p x y # jx yj p for x; y [ R: ð9þ Corollary. yields immediately the question if inequality (9) is valid for 0, p, or p.. It happens that it is not the case. We have 3 J vp ; 0 3 ¼ v p 8 v p 3 ¼ 3 þ p þ p p ¼ þ p : A We are going to show that þ p. 3 p for p [ ð0; Þ < ð; Þ:

10 The above inequality is equivalent to the following one Let Journal of Difference Equations and Applications 33 p þ. 3 p for p [ ð0; Þ < ð; Þ: ð0þ gðxþ U x þ 3 x for x [ R: Downloaded by [Polska Akademia Nauk Instytut Matematyczny] at 03:7 07 March 03 Obviously, g is continuous and gðþ ¼0 ¼ gðþ. One can check easily that g 0 ðxþ, 0 for x [ ð0; Þ and g 0 ðxþ. 0 for x [ ð; Þ. It means that gðxþ. 0 for x [ ð0; Þ < ð; Þ, which proves (0). As we have shown condition (9) does not hold for p [ ð0; Þ < ð; Þ. Consequently, in this case another way of finding the best convexity deviation is needed. The case p [ ð; Þ has been in fact already solved by the result of Rolewicz [8] stating that if the real function f defined in a real interval J satisfies, for some p., the inequality f ðtx þð tþyþ # tf ðxþþð tþf ðyþþcjx yj p for x; y [ J; t [ ½0; Š; then f is convex. This result can be easily generalized by replacing interval J by a convex subset of a normed space. It follows from the generalized version of Rolewicz s result that the zero function is the smallest convexity deviation for QðtÞ ¼t p, p.. It happens also that in the case p [ ð0; Þ the function v p is not optimal. The optimal function in this case will be determined in a separate paper. We know that for p [ ½; Š the function QðxÞ U x p for x [ ½0; Þ; is non-decreasing, QðÞ ¼ and that it satisfies conditions (i) and (ii). Another family of functions satisfying these conditions is given below. Example.. Let p [ ½; Š. We define Q in the following way: Qð0Þ ¼0; Qð n Þ¼ pn for n [ Z; and Q is piecewise linear and continuous. It is obvious that Q is non-decreasing, QðÞ ¼ and that condition (i) is satisfied. It is also clear that condition (ii) is satisfied for t ¼ ð= n Þ, n [ N. Since Q is piecewise linear and continuous it proves (ii). It would be interesting to characterize non-decreasing functions Q : ½0; Þ! ½0; Þ satisfying conditions (i), (ii) and such that QðÞ ¼. Notes. tabor@ii.uj.edu.pl. tabor@univ.rzeszow.pl

11 3 A. Mureńko et al. Downloaded by [Polska Akademia Nauk Instytut Matematyczny] at 03:7 07 March 03 References [] Z. Boros, An inequality for the Takagi function, Math. Inequal. Appl. / (008), pp [] L. Cadariu and V. Radu, Fixed point methods for the generalized stability of functional equations in a single variable, Fixed Point Theory Appl. (008), Art. ID 7939, 5 pp, doi:0.55/008/7939. [3] A. Házy, On approximate t-convexity, Math. Inequal. Appl. 8/3 (005), pp [] A. Házy and Zs. Páles, On approximately midconvex functions, Bull. London Math. Soc. 36(3) (00), pp [5] D.H. Hyers and S. Ulam, Approximately convex functions, Proc. Amer. Math. Soc. 3 (95), pp [6] C.T. Ng and K. Nikodem, On approximately convex functions, Proc. Amer. Math. Soc. 8 (993), pp [7] Zs. Páles, Problem in Report of Meeting, the Forty-first International Symposium on Functional Equations, Aequationes Math. 67 (00), p [8] S. Rolewicz, On g-paraconvex multifunctions, Math. Japonica /3 (979), pp [9] J. Tabor and J. Tabor, Takagi functions and approximate midconvexity, J. Math. Anal. Appl. 356 (009), pp [0] J. Tabor and J. Tabor, Generalized approximate midconvexity, Control and Cybernetics 38(3) (009), pp