When M-convexity of a Function Implies its N-convexity for Some Means M and N
|
|
- Merilyn Lynch
- 5 years ago
- Views:
Transcription
1 International Mathematical Forum,, 007, no. 47, When M-convexity of a Function Implies its N-convexity for Some Means M and N Janusz Matkowski Institute of Mathematics, University of Zielona Góra PL Zielona Góra, Poland and Institute of Mathematics, Silesian University PL Katowice, Poland J.Matkowski@im.uz.zgora.pl Devoted to Professor Boris Paneah on the occasion of his 70th birthday Abstract. Let M be an arbitrary strict mean in an interval J and M p [ϕ] be a weighted quasi-arithmetic mean of the weight p (0; 1) and a generator ϕ : J R. We prove that, for all intervals I J and for all continuous functions f : I J, the condition f is M p [ϕ] -affine implies f is M-convex is satisfied iff M is a quasi-arithmetic mean. Some variants of this result are proved and an open problem is posed. Mathematics Subject Classification: Primary 6B5, 6E60; Secondary 39B Keywords: mean, quasi-arithmetic mean, M convex function (convex function with respect to a mean M), M-affine function 1. Introduction Let J R be an interval. A function M : J R is called a mean in J if min(x, y) M(x, y) max(x, y), x,y J. If, for all x, y J, x y, these inequalities are strict, M is called strict; and symmetric, if M(x, y) =M(y, x). Every mean M in J is reflexive, i.e. M(x, x) =x, x J; consequently, M(I )=I for every subinterval I J and M is a mean in I. Obviously, every reflexive function M : J R which is increasing in each variable is a mean.
2 3 J. Matkowski Let I J be an interval. A function f : I J is said to be M-convex, M-concave, M-affine if, respectively, (cf. J. Aczél [1], G. Aumann [3], and J. Matkowski & J. Rätz [8]), f (M(x, y)) M(f(x),f(y)), x,y I; f (M(x, y)) M(f(x),f(y)), x,y I; f (M(x, y)) = M(f(x),f(y)), x,y I. If M = A where A(x, y) := x+y, then A-convexity coincides with the Jensenconvexity. Let us mention that the convexity of a function with respect to a non arithmetic mean appears in some characterization of L p -norm [7] and the Gamma function [5]. Let N and M be some means in an interval J. In the present paper we show that: if N is a weighted quasi-arithmetic mean and for any subinterval I J and any continuous function f : I J, f is N-affine implies that f is M-convex, then M must be quasi-arithmetic too. If moreover both means are symmetric, then M = N. We also show that assuming either some special condition on the generator of quasi-arithmetic mean N or the continuity of M, one can substantially weaken the assumptions in this result. A simple example shows that the quasi-arithmeticity of the means is meaningful. An open problem is posed.. Auxiliary results Recall that for every continuous and strictly monotonic function ϕ : J R and p [0, 1], the function M p [ϕ] : J J, M p [ϕ] (x, y) :=ϕ 1 (pϕ(x)+(1 p)ϕ(y)), x,y J, is a mean. M p [ϕ] is called a weighted quasi-arithmetic mean; the function ϕ is referred to as its generator, and the numbers p and 1 p its weights. Forp = 1 this mean is denoted by M [ϕ] and is called quasi-arithmetic. Remark 1. The mean M p [ϕ] is strict iff p (0, 1). Moreover we have M [ϕ] 0 (x, y) = y and M [ϕ] 1 (x, y) =x for all x, y I, i.e. M [ϕ] 0 and M [ϕ] 1 are the projective means. Remark. Suppose that ϕ, φ : J R are continuous and strictly monotonic, and p, q (0, 1). Then M p [ϕ] = M q [φ] if, and only if, q = p and there are a, b R, a 0, such that ϕ(x) =aφ(x)+b, x J,
3 When M-convexity implies N-convexity 33 (cf. [], Corollary 5, p. 46 where the case p = q = 1 is considered). This remark allows to assume, without any loss of generality, that the generator ϕ of the mean M p [ϕ] is increasing and, if it is convenient, that 0 ϕ(j). Remark 3. Let M be a mean in an interval J. Considering M-convex (Maffine) functions we can assume, without any loss of generality, that 0 int J (or 0 J). To show this take an arbitrary x 0 J, put J x 0 := {x x 0 : x J}, and define N :(J x 0 ) R by N(u, v) :=M(u + x 0,v+ x 0 ) x 0, u,v J x 0. It is easy to verify that N is a mean in J x 0, and if M = M [ϕ] p then N = M [φ] p where φ :(J x 0 ) R is given by φ(u) :=ϕ(u + x 0 ). Moreover, if f : I J is M-convex, then g :(I x 0 ) J x 0 defined by g(u) :=f(u + x 0 ) x 0, u J x 0, is N-convex. Indeed, for all u, v J x 0 we have g(n(u, v)) = f (N(u, v)+x 0 ) x 0 = f(m(u + x 0,v+ x 0 )) x 0 M(f(u + x 0 ),f(v + x 0 )) x 0 = M([f(u + x 0 ) x 0 ]+x 0, [f(v + x 0 )) x 0 ]+x 0 ) x 0 = M(g(u)+x 0,g(v)+x 0 ) x 0 = N(g(u),g(v)). If f : I J is M-affine, then a similar reasoning shows that g is N-affine. Let J R be an interval and p (0, 1). In the sequel we say that a function g : J R is p-convex (resp., p-concave, p-affine) ifg is convex (resp. concave, affine) with respect to the weighted arithmetic mean A p (x, y) := px+(1 p)y. In particular, 1-convexity ( 1-concavity, 1 -affinity) coincides with Jensen convexity (Jensen cocavity, Jensen affinity, respectively). Remark 4. It follows from the Daróczy-Páles identity ( x + y = p (1 p)x + p x + y ) ( +(1 p) (1 p) x + y ) + py that every p-convex ( p-concave, p-affine) function is Jensen convex (resp., Jensen concave, Jensen affine). Lemma 1. Let J R be an interval, ϕ : J R continuous and strictly increasing and p (0, 1). Then 1. f : I J is M p [ϕ] -convex (M p [ϕ] ϕ f ϕ 1 is p-convex (p-concave, p-affine) in ϕ(i); -concave, M p [ϕ] -affine) iff the function. if f : I J is M p [ϕ] -affine and continuous at least at one point, then there are a, b R such that ϕ f ϕ 1 (u) =au + b for all u ϕ(i).
4 34 J. Matkowski Proof. Suppose that f is M p [ϕ] -convex in convex in I. Then, for all x, y I. f ( ϕ 1 (pϕ(x)+(1 p)ϕ(y)) ) ϕ ( 1 ϕ 1 (pϕ(f(x)) + (1 p)ϕ(f(y))) ). For arbitrary u, v ϕ(i), taking here x := ϕ 1 (u), y:= ϕ 1 (v) and making use of the increasing monotonicity of ϕ, we hence get ϕ f ϕ 1 (pu +(1 p)v) pϕ f ϕ 1 (u)+(1 p)ϕ f ϕ 1 (u), which proves that ϕ f ϕ 1 is p-convex. In the same way we can show the remaining assertions of part 1. Now the second part of the lemma is a consequence of the Daróczy-Páles identity lemma and the classical theory of Jensen convex (or affine) functions (cf. M. Kuczma p. [6]). 3. Some results Clearly, every M-affine function is M-convex. We begin with the following Theorem 1. Let ϕ : J R be continuous and strictly monotonic in an open interval J R such that ϕ(j) =R, and p (0, 1) a fixed number. Suppose that M : J J is a mean. If, for all continuous functions f : J J, f is M p [ϕ] -affine = f is M-convex, then M = M q [ϕ] for some q [0, 1]. If, moreover, M is a strict mean, then q (0, 1). Furthermore, if p = 1 and M is strict and symmetric, then M = M [ϕ]. Proof. By Lemma 1, taking into account that ϕ(j) =R, we infer that, for all a, b R, the function f : J J given by f(x) =ϕ 1 (aϕ(x)+b), x J, is M p [ϕ] -affine. From the assumed implication we infer that this function f is M-convex, that is ϕ 1 (aϕ(m(x, y)) + b) M ( ϕ 1 (aϕ(x)+b),ϕ 1 (aϕ(y)+b) ), x,y J, for all a, b R. By Remark, without any loss of generality, we can assume that ϕ is strictly increasing. Therefore, repleacing x by ϕ 1 (u) and y by ϕ 1 (v), we hence get aϕ(m(ϕ 1 (u),ϕ 1 (v))) + b ϕ ( M ( ϕ 1 (au + b),ϕ 1 (av + b) )), u,v R, for all a, b R, that is (1) am (u, v)+b M (au + b, av + b), a,b,u,v R, where M : R R R is defined by M (u, v) :=ϕ(m(ϕ 1 (u),ϕ 1 (v))). For b = 0 we hence get am (u, v) M (au, av), a,u,v R,
5 which, of course, implies that () that is M is homogeneous. Taking a = 1 in (1) we get When M-convexity implies N-convexity 35 am (u, v) =M (au, av), a,u,v R, M (u, v)+b M (u + b, v + b), b,u,v R. Repleacing here u by u b and v by v b we obtain whence Consequently, (3) M (u b, v b) M (u, v) b, b, u, v R, M (u + b, v + b) M (u, v)+b, b, u, v R. M (u + b, v + b) =M (u, v)+b, b, u, v R. Applying in turn (3) and () we obtain, for all u.v R, where M (u, v) =M ((u v)+v, 0+v) =M (u v, 0) + v =(u v)m (1, 0) + v = qu +(1 q)v, Hence, by the definition of M, q := M (1, 0). M(x, y) =ϕ 1 (qϕ(x)+(1 q)ϕ(y)), x,y J. Since M is a mean, we have q [0, 1]. This completes the proof. Denote by Q the set of rational numbers. The continuity of the mean M in Theorem 1 (as well as in Corollary 1) allows to weaken the basic assumption significally. Theorem. Let ϕ : J R be continuous and strictly monotonic in an open interval J R, ϕ(j) = R, and p (0, 1) a fixed number. Suppose that M : J J is a continuous mean. log b If there are a, b, c, d R\{0}, 0 < a < 1 < b, / Q such that the log a functions and the function ϕ 1 (aϕ), ϕ 1 (bϕ), ϕ 1 (cϕ + d) are M-convex ϕ 1 ( ϕ) is M-affine, then M = M [ϕ] q for some q [0; 1]. If moreover M is strict, then q (0; 1).
6 36 J. Matkowski Proof. We can assume that ϕ is strictly increasing. From the M-convexity of the function ϕ 1 (aϕ) we have ϕ 1 (aϕ(m(x, y))) M(ϕ 1 (aϕ(x)),ϕ 1 (aϕ(y))), x,y J. Repleacing x by ϕ 1 (u) and y by ϕ 1 (v), we hence get aϕ(m(ϕ 1 (u),ϕ 1 (v))) ϕ(m ( ϕ 1 (au),ϕ 1 (av) ) ), u,v R, whence, by induction, a n ϕ(m(ϕ 1 (u),ϕ 1 (v))) ϕ(m ( ϕ 1 (a n u),ϕ 1 (a n v) ) ), u,v R, n N. From the M-convexity of the function ϕ 1 (bϕ), in the same way, we get b m ϕ(m(ϕ 1 (u),ϕ 1 (v))) ϕ(m ( ϕ 1 (b m u),ϕ 1 (b m v) ) ), u,v R, n N. Making use both these inequalities we obtain a n b m ϕ(m(ϕ 1 (u),ϕ 1 (v))) ϕ(m ( ϕ 1 (a n b m u),ϕ 1 (a n b m v) ) ) for all u, v R, n N. Since, by assumption, 0 <a<1 <b, log g / Q, the set log a {a n b m : n, m N} is dense in (0, ). The last inequality and the continuity of M imply that tϕ(m(ϕ 1 (u),ϕ 1 (v))) ϕ(m ( ϕ 1 (tu),ϕ 1 (tv) ) ), u,v R, t>0. Replacing u and v by u/t and v/t, respectively, we obtain the reversed inequality, whence tϕ(m(ϕ 1 (u),ϕ 1 (v))) = ϕ(m ( ϕ 1 (tu),ϕ 1 (tv) ) ), u,v R, t>0. This proves that the mean M : R R defined by (4) M (u, v) :=ϕ(m(ϕ 1 (u),ϕ 1 (v))), u,v R, is positively homogeneous. Since, by assumption, ϕ 1 ( ϕ) ism-affine, we have ϕ 1 ( ϕ(m(x, y))) = M(ϕ 1 ( ϕ(x)),ϕ 1 ( ϕ(y))), x,y J, which can be written in the form ϕ(m(ϕ 1 (u),ϕ 1 (v))) = ϕ(m ( ϕ 1 ( u),ϕ 1 ( v) ) ), u,v R, whence, by the definition of M, (5) M ( u, v) := M (u, v), u,v R. This relation and the positively homogeneity of M imply that M is homogeneous, i.e. (6) M (tu, tv) :=tm (u, v), t,u,v R. By assumption, there are real nonzero c, d such that the function ϕ 1 (cϕ+d) is M-convex. Consequently, ϕ 1 (cϕ(m(x, y)) + d) M ( ϕ 1 (cϕ(x)+d),ϕ 1 (cϕ(y)+d) ), x,y J,
7 When M-convexity implies N-convexity 37 which can be written in the form cϕ(m(ϕ 1 (u),ϕ 1 (v))) + d ϕ(m ( ϕ 1 (cu + d),ϕ 1 (cv + d) ) ), u,v R, whence cm (u, v)+d M (cu + d, cv + d), u,v R. Hence, applying (6), the homogeneity of M, we get M (cu, cv)+d M (cu + d, cv + d), u,v R, and, consequently, (7) M (u, v)+d M (u + d, v + d), u,v R. Replacing here u by u d and v by v d we get M (u d, v d) M (u, v) d, u, v R, whence, by replacing u by u and v by v, M ( u d, v d) M ( u, v) d, u, v R. Making use of (5), we obtain M (u + d, v + d) M (u, v) d, u, v R, that is (8) M (u + d, v + d) M (u, v)+d, u, v R, The inequalities (7) and (8) imply that M (u + d, v + d) =M (u, v)+d, u, v R. Hence, by the homogeneity of M, M (tu + td, tv + td) =M (tu, tv)+td, t, u, v R, whence M (u + td, v + td) =M (u, v)+td, t, u, v R. Since t R is arbitrary, we conclude that (9) M (u + w, v + w) =M (u, v)+w, u, v, w R. Now, similarly as in the proof of Theorem 1, applying (6) and (9) we obtain M (u, v) =M ((u v)+v, 0+v) =M (u v, 0) + v =(u v)m (1, 0) + v) =qu +(1 q)v for all u, v R, and the result is a consequence of (4). In the case ϕ(j) Rthe following result holds true:
8 38 J. Matkowski Theorem 3. Let ϕ : J R be continuous and strictly monotonic in an open interval J R and let p (0, 1) be a fixed number. Suppose that M : J J is a mean. If for all compact intervals I J and for all continuous functions f : I J, f is M p [ϕ] -affine = f is M-convex, then M = M q [ϕ] for some q [0; 1]. If moreover M is a strict mean, then q (0; 1). Furthermore, if M is symmetric, then M = M [ϕ]. Proof. By Remark we may assume that 0 int ϕ(j). By Remark 3 we can also assume that 0 int J. Take a compact subinterval I J such that 0 int(i) and a continuous function f : I J. By Lemma 1, if f is M p [ϕ] -affine, then there are a, b R such that (4) Conversely, for all a, b R such that f(x) =ϕ 1 (aϕ(x)+b), x I. (aϕ(i)+b) ϕ(j), the function f given by this formula is M p [ϕ] -affine. Since I J is compact, there are α = α(i) > 1 and β = β(i) > 0 such that this inclusion holds true for all a, b R such that 0 a<αand b <β.suppose that condition 1 is satisfied. Then, for all a, b R, such that 0 a<αand b <β,we have ϕ 1 (aϕ(m(x, y)) + b) M ( ϕ 1 (aϕ(x)+b),ϕ 1 (aϕ(y)+b) ), x,y I, or equivalently, for all a, b R, such that 0 a<αand b <β, aϕ(m(ϕ 1 (u),ϕ 1 (v))) + b ϕ ( M ( ϕ 1 (au + b),ϕ 1 (av + b) )), (5) Thus, for all a, b R, such that 0 a<αand b <β, am 1 (u, v)+b M 1 (au + b, av + b), where M 1 : ϕ(j) ϕ(j) ϕ(j) is defined by M 1 (u, v) :=ϕ(m(ϕ 1 (u),ϕ 1 (v))), Taking b = 0 we hence get am 1 (u, v) M 1 (au, av), for all a R such that 0 a<α,which implies that (6) am 1 (u, v) =M 1 (au, av), u,v ϕ(i), u,v ϕ(j). u,v ϕ(i), u,v ϕ(i), for all a R such that 0 a<α. For an arbitrary (u, v) R R take t>0 such that u t, v t M (u, v) :=tm 1 ( u t, v t ). u,v ϕ(i). ϕ(i) and put
9 When M-convexity implies N-convexity 39 To show that M (u, v) does not depend on the choice of t, take an s>0 such that u, v ϕ(i). Without any loss of generality we can assume that s>t. s s Since 0 < s < 1, from (6) we have t tm 1 ( u t, v t )=s t s M 1( u t, v t )=sm 1( t u s t, t v s t )=sm 1( u s, v s ). This proves that the function M : R R R is correctly defined. Clearly, M is positively homogeneous, that is M (tu, tv) =tm (u, v), u,v R, t 0. Taking a = 1 in (1) we get, for all b R, b <β, M 1 (u, v)+b M 1 (u + b, v + b), u,v ϕ(i). Repleacing here u by u b and v by v b we obtain M 1 (u b, v b) M 1 (u, v) b, u, v ϕ(i), for all b R, b <β.therefore M 1 (u + b, v + b) =M 1 (u, v)+b, b <β; u, v ϕ(i). Take arbitrary u, v R, b R and choose a t>0 such that u, v ϕ(i) and u t b t <β.then, by the definition of M and its homogeneity, we have ( u M (u + b, v + b) =tm 1 t + b t, v t + b ) [ ( u = t M 1 t t, v ) + b ] t t ( u = tm 1 t, v ) + b = M (u, v)+b. t Now the same resoning as in the proof of Theorem 1 shows that there is a q [0, 1] such that M (u, v) =qu +(1 q)v, u, v R. Since the function M is uniquelly determined and the definition does not depend on the the choice of the compact interval I, we infer that ϕ(m(ϕ 1 (u),ϕ 1 (v))) = M (u, v), u,v ϕ(j). Consequently, M(x, y) =ϕ 1 (qϕ(u)+(1 q)ϕ(y)), x,y J. Since remaining statements are obvious, the proof is complete.
10 330 J. Matkowski 4. A remark and open problem Remark 5. In the basic supposition of the above results that N-affinity of some f implies its M-convexity, we assume that N is a quasi-arithmetic mean. This assumption is essential because there are different non-quasiarithemetic means with the same classes of affine functions. For instance, the logarithmic mean L :(0, ) (0, ), { x y L(x, y) =, x y log x log y x, x = y and the mean M :(0, ) (0, ), M(x, y) := x + xy + y, 3 x + y are symmetric and have the same classes continuous L-affine functions and M-affine functions coincide (cf. [9], [10]). We end up this paper with the following open Problem 1. Let M and N be strict, symmetric and continuous means in an interval J. Suppose that for all intervals I J and for all continuous functions f : I J, the implication f is N-convex = f is M-convex holds true. Is then M = N? Acknowledgement 1. The research was supported by the Silesian University Mathematical Department (Iterative Functional Equations and Real Analysis program). 5. References [1] J. Aczél, A generalization of the notion of convex functions, Norske Vid. Selsk. Forhdl. 19, Nr 4 (1947), [] J. Aczél, J. Dhombres, Functional equations in several variables, Encyclopedia of Mathematics and its applications 31, Cambridge University Press, Cambridge, New York, New Rochelle, Melbourne, Sydney, [3] G. Aumann, Konvexe Funktionen und Induktion bei Ungleichungen zwischen Mittelverten S.-B. math.-naturw. Abt. Bayer. Akad. Wiss. München (1933), [4] Z. Daróczy, Zs. Páles, Convexity with given infinite weight sequences, Stochastica II(1987), 5-1. [5] D. Gronau, J. Matkowski, Geometrical convexity and generalization of the Bohr-Mollerup theorem on the Gamma function, Math. Pannonica 4/(1993), [6] M. Kuczma, An Introduction to the Theory of Functional Equations and Inequalities, Cauchy s Equation and Jensen s inequality, Prace Nauk. Uniw. Ślask. 489, Polish Scientific Publishers, Warszawa - Kraków - Katowice, 1985.
11 When M-convexity implies N-convexity 331 [7] J. Matkowski, L p -like paranorms, in Selected Topics in Functional Equations and Iteration Theory, Proceedings of the Austrian-Polish Seminar, Universität Graz, October 4-6, 1991, edited by D. Gronau and L. Reich, Grazer Math. Ber. 316(199), [8] J. Matkowski, J. Rätz, Convex functions with respect to an arbitrary mean, Internat. Ser. Num. Math. 13(1997), [9] J. Matkowski, Affine functions with respect to the logarithmic mean, Colloq. Math. 95(003), [10] J. Matkowski, Convex and affine functions with respect to a mean and a characterizationn of the quasi-arithmetic means, Real Anal. Exchange 9 (1), 003/004, Received: November 9, 006
CONVEX FUNCTIONS WITH RESPECT TO A MEAN AND A CHARACTERIZATION OF QUASI-ARITHMETIC MEANS
Real Analsis Echange Vol. 29(1), 2003/2004, pp. 229246 Janusz Matkowski, Institute of Mathematics, Universit of Zielona Góra, PL-65-246 Zielona Góra, Poland. email: J.Matkowski@im.uz.zgora.pl CONVEX FUNCTIONS
More informationWHEN LAGRANGEAN AND QUASI-ARITHMETIC MEANS COINCIDE
Volume 8 (007, Issue 3, Article 71, 5 pp. WHEN LAGRANGEAN AND QUASI-ARITHMETIC MEANS COINCIDE JUSTYNA JARCZYK FACULTY OF MATHEMATICS, COMPUTER SCIENCE AND ECONOMETRICS, UNIVERSITY OF ZIELONA GÓRA SZAFRANA
More informationCauchy quotient means and their properties
Cauchy quotient means and their properties 1 Department of Mathematics University of Zielona Góra Joint work with Janusz Matkowski Outline 1 Introduction 2 Means in terms of beta-type functions 3 Properties
More informationOn a Problem of Alsina Again
Journal of Mathematical Analysis and Applications 54, 67 635 00 doi:0.006 jmaa.000.768, available online at http: www.idealibrary.com on On a Problem of Alsina Again Janusz Matkowski Institute of Mathematics,
More informationarxiv: v1 [math.ca] 12 Jul 2018
arxiv:1807.04811v1 [math.ca] 12 Jul 2018 A new invariance identity and means Jimmy Devillet and Janusz Matkowski Abstract. The invariance identity involving three operations D f,g : X X X of the form D
More informationCHINI S EQUATIONS IN ACTUARIAL MATHEMATICS
Annales Univ. Sci. Budapest. Sect. Comp. 41 (2013) 213 225 CHINI S EQUATIONS IN ACTUARIAL MATHEMATICS Agata Nowak and Maciej Sablik (Katowice Poland) Dedicated to Professors Zoltán Daróczy and Imre Kátai
More informationDirectional Monotonicity of Fuzzy Implications
Acta Polytechnica Hungarica Vol. 14, No. 5, 2017 Directional Monotonicity of Fuzzy Implications Katarzyna Miś Institute of Mathematics, University of Silesia in Katowice Bankowa 14, 40-007 Katowice, Poland,
More informationON CONTINUITY OF MEASURABLE COCYCLES
Journal of Applied Analysis Vol. 6, No. 2 (2000), pp. 295 302 ON CONTINUITY OF MEASURABLE COCYCLES G. GUZIK Received January 18, 2000 and, in revised form, July 27, 2000 Abstract. It is proved that every
More informationSome results on stability and on characterization of K-convexity of set-valued functions
ANNALES POLONICI MATHEMATICI LVIII. (1993) Some results on stability and on characterization of K-convexity of set-valued functions by Tiziana Cardinali (Perugia), Kazimierz Nikodem (Bielsko-Bia la) and
More informationOn (M, N)-convex functions
Notes on Number Theory and Discrete Mathematics ISSN 1310 513 Vol. 1, 015, No. 4, 40 47 On (M, N)-convex functions József Sándor 1 and Edith Egri 1 Babeş Bolyai University, Department of Mathematics, Cluj-Napoca,
More informationCharacterization of quadratic mappings through a functional inequality
J. Math. Anal. Appl. 32 (2006) 52 59 www.elsevier.com/locate/jmaa Characterization of quadratic mappings through a functional inequality Włodzimierz Fechner Institute of Mathematics, Silesian University,
More informationRELATIONSHIPS BETWEEN HOMOGENEITY, SUBADDITIVITY AND CONVEXITY PROPERTIES
Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. 16 (2005), 77 87. Available electronically at http: //pefmath.etf.bg.ac.yu RELATIONSHIPS BETWEEN HOMOGENEITY, SUBADDITIVITY AND CONVEXITY PROPERTIES Pál
More informationUNIFORMLY CONTINUOUS COMPOSITION OPERATORS IN THE SPACE OF BOUNDED Φ-VARIATION FUNCTIONS IN THE SCHRAMM SENSE
Opuscula Mathematica Vol. 3 No. 01 http://dx.doi.org/10.7494/opmath.01.3..39 UNIFORMLY CONTINUOUS COMPOSITION OPERATORS IN THE SPACE OF BOUNDED Φ-VARIATION FUNCTIONS IN THE SCHRAMM SENSE Tomás Ereú, Nelson
More informationRafał Kapica, Janusz Morawiec CONTINUOUS SOLUTIONS OF ITERATIVE EQUATIONS OF INFINITE ORDER
Opuscula Mathematica Vol. 29 No. 2 2009 Rafał Kapica, Janusz Morawiec CONTINUOUS SOLUTIONS OF ITERATIVE EQUATIONS OF INFINITE ORDER Abstract. Given a probability space (, A, P ) and a complete separable
More informationA Functional Equation with Conjugate Means Derived from a Weighted Arithmetic Mean ABSTRACT 1. INTRODUCTION
Malaysian Journal of Mathematical Sciences 9(1): 21-31 (2015) MALAYSIAN JOURNAL OF MATHEMATICAL SCIENCES Journal homepage: http://einspem.upm.edu.my/journal A Functional Equation with Conjugate Means Derived
More informationSEQUENCES OF ITERATES OF RANDOM-VALUED VECTOR FUNCTIONS AND CONTINUOUS SOLUTIONS OF RELATED EQUATIONS. Rafa l Kapica Silesian University, Poland
GLASNIK MATEMATIČKI Vol. 42(62)(2007), 389 399 SEQUENCES OF ITERATES OF RANDOM-VALUED VECTOR FUNCTIONS AND CONTINUOUS SOLUTIONS OF RELATED EQUATIONS Rafa l Kapica Silesian University, Poland Abstract.
More informationGeneralized Convexity and Separation Theorems
Journal of Convex Analysis Volume 14 (2007), No. 2, 239 247 Generalized Convexity and Separation Theorems Kazimierz Nikodem Department of Mathematics, University of Bielsko-Biała, ul. Willowa 2, 43-309
More informationA Conditional Gofcjb-Schinzel Equation
Anzeiger Abt. II (2000) 137: 11-15 A n z e i g e r M athem atisch -n a tu rw isse n schaftliche Klasse Abt. II M athem a tisc h e, P h ysikalische u n d Technische W issenscha ften Ö sterreichische A k
More informationSOME REMARKS ON THE SPACE OF DIFFERENCES OF SUBLINEAR FUNCTIONS
APPLICATIONES MATHEMATICAE 22,3 (1994), pp. 419 426 S. G. BARTELS and D. PALLASCHKE (Karlsruhe) SOME REMARKS ON THE SPACE OF DIFFERENCES OF SUBLINEAR FUNCTIONS Abstract. Two properties concerning the space
More informationOn a Cauchy equation in norm
http://www.mathematik.uni-karlsruhe.de/ semlv Seminar LV, No. 27, 11 pp., 02.10.2006 On a Cauchy equation in norm by R O M A N G E R (Katowice) and P E T E R V O L K M A N N (Karlsruhe) Abstract. We deal
More informationImprovements of the Giaccardi and the Petrović inequality and related Stolarsky type means
Annals of the University of Craiova, Mathematics and Computer Science Series Volume 391, 01, Pages 65 75 ISSN: 13-6934 Improvements of the Giaccardi and the Petrović inequality and related Stolarsky type
More informationOptimal lower and upper bounds for the geometric convex combination of the error function
Li et al. Journal of Inequalities Applications (215) 215:382 DOI 1.1186/s1366-15-96-y R E S E A R C H Open Access Optimal lower upper bounds for the geometric convex combination of the error function Yong-Min
More informationOn intermediate value theorem in ordered Banach spaces for noncompact and discontinuous mappings
Int. J. Nonlinear Anal. Appl. 7 (2016) No. 1, 295-300 ISSN: 2008-6822 (electronic) http://dx.doi.org/10.22075/ijnaa.2015.341 On intermediate value theorem in ordered Banach spaces for noncompact and discontinuous
More informationJournal of Inequalities in Pure and Applied Mathematics
Journal of Inequalities in Pure and Applied Mathematics CONTINUITY PROPERTIES OF CONVEX-TYPE SET-VALUED MAPS KAZIMIERZ NIKODEM Department of Mathematics, University of Bielsko Biała Willowa 2, PL-43-309
More informationChapter 37 On Means Which are Quasi-Arithmetic and of the Beckenbach Gini Type
Chapter 37 On Means Which are Quasi-Arithmetic and of the Beckenbach Gini Type Janusz Matkowski Dedicated to the memory of S.M. Ulam on the 100th anniversary of his birth Abstract The class of quasi-arithmetic
More informationThe aim of this paper is to obtain a theorem on the existence and uniqueness of increasing and convex solutions ϕ of the Schröder equation
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 125, Number 1, January 1997, Pages 153 158 S 0002-9939(97)03640-X CONVEX SOLUTIONS OF THE SCHRÖDER EQUATION IN BANACH SPACES JANUSZ WALORSKI (Communicated
More informationJournal of Inequalities in Pure and Applied Mathematics
Journal of Inequalities in Pure and Applied Mathematics THE EXTENSION OF MAJORIZATION INEQUALITIES WITHIN THE FRAMEWORK OF RELATIVE CONVEXITY CONSTANTIN P. NICULESCU AND FLORIN POPOVICI University of Craiova
More informationON APPROXIMATE HERMITE HADAMARD TYPE INEQUALITIES
ON APPROXIMATE HERMITE HADAMARD TYPE INEQUALITIES JUDIT MAKÓ AND ATTILA HÁZY Abstract. The main results of this paper offer sufficient conditions in order that an approximate lower Hermite Hadamard type
More informationChapter 1 Preliminaries
Chapter 1 Preliminaries 1.1 Conventions and Notations Throughout the book we use the following notations for standard sets of numbers: N the set {1, 2,...} of natural numbers Z the set of integers Q the
More informationIf g is also continuous and strictly increasing on J, we may apply the strictly increasing inverse function g 1 to this inequality to get
18:2 1/24/2 TOPIC. Inequalities; measures of spread. This lecture explores the implications of Jensen s inequality for g-means in general, and for harmonic, geometric, arithmetic, and related means in
More informationEQUICONTINUITY AND SINGULARITIES OF FAMILIES OF MONOMIAL MAPPINGS
STUDIA UNIV. BABEŞ BOLYAI, MATHEMATICA, Volume LI, Number 3, September 2006 EQUICONTINUITY AND SINGULARITIES OF FAMILIES OF MONOMIAL MAPPINGS WOLFGANG W. BRECKNER and TIBERIU TRIF Dedicated to Professor
More informationJournal of Inequalities in Pure and Applied Mathematics
Journal of Inequalities in Pure and Applied Mathematics ON SOME APPROXIMATE FUNCTIONAL RELATIONS STEMMING FROM ORTHOGONALITY PRESERVING PROPERTY JACEK CHMIELIŃSKI Instytut Matematyki, Akademia Pedagogiczna
More informationInequalities among quasi-arithmetic means for continuous field of operators
Filomat 26:5 202, 977 99 DOI 0.2298/FIL205977M Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Inequalities among quasi-arithmetic
More informationAvailable online at ISSN (Print): , ISSN (Online): , ISSN (CD-ROM):
American International Journal of Research in Formal, Applied & Natural Sciences Available online at http://www.iasir.net ISSN (Print): 2328-3777, ISSN (Online): 2328-3785, ISSN (CD-ROM): 2328-3793 AIJRFANS
More informationConvex Functions and Optimization
Chapter 5 Convex Functions and Optimization 5.1 Convex Functions Our next topic is that of convex functions. Again, we will concentrate on the context of a map f : R n R although the situation can be generalized
More informationPM functions, their characteristic intervals and iterative roots
ANNALES POLONICI MATHEMATICI LXV.2(1997) PM functions, their characteristic intervals and iterative roots by Weinian Zhang (Chengdu) Abstract. The concept of characteristic interval for piecewise monotone
More informationFUNCTION BASES FOR TOPOLOGICAL VECTOR SPACES. Yılmaz Yılmaz
Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 33, 2009, 335 353 FUNCTION BASES FOR TOPOLOGICAL VECTOR SPACES Yılmaz Yılmaz Abstract. Our main interest in this
More informationDedicated to Prof. Jaroslav Kurzweil on the occasion of his 80th birthday
131 (2006) MATHEMATICA BOHEMICA No. 4, 365 378 ON MCSHANE-TYPE INTEGRALS WITH RESPECT TO SOME DERIVATION BASES Valentin A. Skvortsov, Piotr Sworowski, Bydgoszcz (Received November 11, 2005) Dedicated to
More informationStrongly concave set-valued maps
Mathematica Aeterna, Vol. 4, 2014, no. 5, 477-487 Strongly concave set-valued maps Odalis Mejía Universidad Central de Venezuela Escuela de Matemáticas Paseo Los Ilustres, VE-1020 A Caracas, Venezuela
More informationActa Academiae Paedagogicae Agriensis, Sectio Mathematicae 30 (2003) A NOTE ON NON-NEGATIVE INFORMATION FUNCTIONS
Acta Academiae Paedagogicae Agriensis, Sectio Mathematicae 30 (2003 3 36 A NOTE ON NON-NEGATIVE INFORMATION FUNCTIONS Béla Brindza and Gyula Maksa (Debrecen, Hungary Dedicated to the memory of Professor
More informationCONVERGENCE PROPERTIES OF COMBINED RELAXATION METHODS
CONVERGENCE PROPERTIES OF COMBINED RELAXATION METHODS Igor V. Konnov Department of Applied Mathematics, Kazan University Kazan 420008, Russia Preprint, March 2002 ISBN 951-42-6687-0 AMS classification:
More informationA Note of the Strong Convergence of the Mann Iteration for Demicontractive Mappings
Applied Mathematical Sciences, Vol. 10, 2016, no. 6, 255-261 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2016.511700 A Note of the Strong Convergence of the Mann Iteration for Demicontractive
More informationConvergence Rates in Regularization for Nonlinear Ill-Posed Equations Involving m-accretive Mappings in Banach Spaces
Applied Mathematical Sciences, Vol. 6, 212, no. 63, 319-3117 Convergence Rates in Regularization for Nonlinear Ill-Posed Equations Involving m-accretive Mappings in Banach Spaces Nguyen Buong Vietnamese
More informationJournal of Inequalities in Pure and Applied Mathematics
Journal of Inequalities in Pure and Applied Mathematics INEQUALITIES FOR GENERAL INTEGRAL MEANS GHEORGHE TOADER AND JOZSEF SÁNDOR Department of Mathematics Technical University Cluj-Napoca, Romania. EMail:
More informationContents: 1. Minimization. 2. The theorem of Lions-Stampacchia for variational inequalities. 3. Γ -Convergence. 4. Duality mapping.
Minimization Contents: 1. Minimization. 2. The theorem of Lions-Stampacchia for variational inequalities. 3. Γ -Convergence. 4. Duality mapping. 1 Minimization A Topological Result. Let S be a topological
More informationInvariant measures and the compactness of the domain
ANNALES POLONICI MATHEMATICI LXIX.(998) Invariant measures and the compactness of the domain by Marian Jab loński (Kraków) and Pawe l Góra (Montreal) Abstract. We consider piecewise monotonic and expanding
More informationSecond Order Elliptic PDE
Second Order Elliptic PDE T. Muthukumar tmk@iitk.ac.in December 16, 2014 Contents 1 A Quick Introduction to PDE 1 2 Classification of Second Order PDE 3 3 Linear Second Order Elliptic Operators 4 4 Periodic
More informationSPECTRAL PROPERTIES OF THE LAPLACIAN ON BOUNDED DOMAINS
SPECTRAL PROPERTIES OF THE LAPLACIAN ON BOUNDED DOMAINS TSOGTGEREL GANTUMUR Abstract. After establishing discrete spectra for a large class of elliptic operators, we present some fundamental spectral properties
More informationON THE CAUCHY EQUATION ON SPHERES
Annales Mathematicae Silesianae 11. Katowice 1997, 89-99 Prace Naukowe Uniwersytetu Śląskiego nr 1665 ON THE CAUCHY EQUATION ON SPHERES ROMAN GER AND JUSTYNA SIKORSKA Abstract. We deal with a conditional
More informationHölder, Chebyshev and Minkowski Type Inequalities for Stolarsky Means
Int. Journal of Math. Analysis, Vol. 4, 200, no. 34, 687-696 Hölder, Chebyshev and Minkowski Type Inequalities for Stolarsky Means Zhen-Hang Yang System Division, Zhejiang Province Electric Power Test
More informationPreservation of graded properties of fuzzy relations by aggregation functions
Preservation of graded properties of fuzzy relations by aggregation functions Urszula Dudziak Institute of Mathematics, University of Rzeszów, 35-310 Rzeszów, ul. Rejtana 16a, Poland. e-mail: ududziak@univ.rzeszow.pl
More informationTHE NEARLY ADDITIVE MAPS
Bull. Korean Math. Soc. 46 (009), No., pp. 199 07 DOI 10.4134/BKMS.009.46..199 THE NEARLY ADDITIVE MAPS Esmaeeil Ansari-Piri and Nasrin Eghbali Abstract. This note is a verification on the relations between
More informationA New Elliptic Mean. Peter Kahlig. (Vorgelegt in der Sitzung der math.-nat. Klasse am 21. März 2002 durch das w. M. Ludwig Reich)
Sitzungsber. Abt. II (00) 11: 137 14 A New Elliptic Mean By Peter Kahlig (Vorgelegt in der Sitzung der math.-nat. Klasse am 1. März 00 durch das w. M. Ludwig Reich) Dedicated to Professor Janusz Matkowski
More informationCONTROLLABILITY OF NONLINEAR DISCRETE SYSTEMS
Int. J. Appl. Math. Comput. Sci., 2002, Vol.2, No.2, 73 80 CONTROLLABILITY OF NONLINEAR DISCRETE SYSTEMS JERZY KLAMKA Institute of Automatic Control, Silesian University of Technology ul. Akademicka 6,
More informationFunctions of several variables of finite variation and their differentiability
ANNALES POLONICI MATHEMATICI LX.1 (1994) Functions of several variables of finite variation and their differentiability by Dariusz Idczak ( Lódź) Abstract. Some differentiability properties of functions
More informationADDITIVE SELECTIONS AND THE STABILITY OF THE CAUCHY FUNCTIONAL EQUATION
ANZIAM J. 44(2003), 323 337 ADDITIVE SELECTIONS AND THE STABILITY OF THE CAUCHY FUNCTIONAL EQUATION ROMAN BADORA 1, ROMAN GER 2 and ZSOLT PÁLES 3 (Received 28 April, 2000; revised 9 March, 2001) Abstract
More informationOn the simplest expression of the perturbed Moore Penrose metric generalized inverse
Annals of the University of Bucharest (mathematical series) 4 (LXII) (2013), 433 446 On the simplest expression of the perturbed Moore Penrose metric generalized inverse Jianbing Cao and Yifeng Xue Communicated
More informationThis is a submission to one of journals of TMRG: BJMA/AFA EXTENSION OF THE REFINED JENSEN S OPERATOR INEQUALITY WITH CONDITION ON SPECTRA
This is a submission to one of journals of TMRG: BJMA/AFA EXTENSION OF THE REFINED JENSEN S OPERATOR INEQUALITY WITH CONDITION ON SPECTRA JADRANKA MIĆIĆ, JOSIP PEČARIĆ AND JURICA PERIĆ3 Abstract. We give
More informationL p Spaces and Convexity
L p Spaces and Convexity These notes largely follow the treatments in Royden, Real Analysis, and Rudin, Real & Complex Analysis. 1. Convex functions Let I R be an interval. For I open, we say a function
More informationarxiv: v1 [math.ap] 16 Jan 2015
Three positive solutions of a nonlinear Dirichlet problem with competing power nonlinearities Vladimir Lubyshev January 19, 2015 arxiv:1501.03870v1 [math.ap] 16 Jan 2015 Abstract This paper studies a nonlinear
More informationBanach Journal of Mathematical Analysis ISSN: (electronic)
Banach J. Math. Anal. 1 (2007), no. 1, 56 65 Banach Journal of Mathematical Analysis ISSN: 1735-8787 (electronic) http://www.math-analysis.org SOME REMARKS ON STABILITY AND SOLVABILITY OF LINEAR FUNCTIONAL
More informationCourse 311: Michaelmas Term 2005 Part III: Topics in Commutative Algebra
Course 311: Michaelmas Term 2005 Part III: Topics in Commutative Algebra D. R. Wilkins Contents 3 Topics in Commutative Algebra 2 3.1 Rings and Fields......................... 2 3.2 Ideals...............................
More informationSTATISTICAL LIMIT POINTS
proceedings of the american mathematical society Volume 118, Number 4, August 1993 STATISTICAL LIMIT POINTS J. A. FRIDY (Communicated by Andrew M. Bruckner) Abstract. Following the concept of a statistically
More informationMath 5210, Definitions and Theorems on Metric Spaces
Math 5210, Definitions and Theorems on Metric Spaces Let (X, d) be a metric space. We will use the following definitions (see Rudin, chap 2, particularly 2.18) 1. Let p X and r R, r > 0, The ball of radius
More information2 Statement of the problem and assumptions
Mathematical Notes, 25, vol. 78, no. 4, pp. 466 48. Existence Theorem for Optimal Control Problems on an Infinite Time Interval A.V. Dmitruk and N.V. Kuz kina We consider an optimal control problem on
More informationEntropy and Ergodic Theory Lecture 4: Conditional entropy and mutual information
Entropy and Ergodic Theory Lecture 4: Conditional entropy and mutual information 1 Conditional entropy Let (Ω, F, P) be a probability space, let X be a RV taking values in some finite set A. In this lecture
More informationA Note on Nonconvex Minimax Theorem with Separable Homogeneous Polynomials
A Note on Nonconvex Minimax Theorem with Separable Homogeneous Polynomials G. Y. Li Communicated by Harold P. Benson Abstract The minimax theorem for a convex-concave bifunction is a fundamental theorem
More informationOn a functional equation connected with bi-linear mappings and its Hyers-Ulam stability
Available online at www.isr-publications.com/jnsa J. Nonlinear Sci. Appl., 10 017, 5914 591 Research Article Journal Homepage: www.tjnsa.com - www.isr-publications.com/jnsa On a functional equation connected
More informationHyers-Ulam Rassias stability of Pexiderized Cauchy functional equation in 2-Banach spaces
Available online at www.tjnsa.com J. Nonlinear Sci. Appl. 5 (202), 459 465 Research Article Hyers-Ulam Rassias stability of Pexiderized Cauchy functional equation in 2-Banach spaces G. Zamani Eskandani
More informationDivision of the Humanities and Social Sciences. Supergradients. KC Border Fall 2001 v ::15.45
Division of the Humanities and Social Sciences Supergradients KC Border Fall 2001 1 The supergradient of a concave function There is a useful way to characterize the concavity of differentiable functions.
More informationConvex Functions. Daniel P. Palomar. Hong Kong University of Science and Technology (HKUST)
Convex Functions Daniel P. Palomar Hong Kong University of Science and Technology (HKUST) ELEC5470 - Convex Optimization Fall 2017-18, HKUST, Hong Kong Outline of Lecture Definition convex function Examples
More informationAnn. Funct. Anal. 5 (2014), no. 2, A nnals of F unctional A nalysis ISSN: (electronic) URL:www.emis.
Ann. Funct. Anal. 5 (2014), no. 2, 147 157 A nnals of F unctional A nalysis ISSN: 2008-8752 (electronic) URL:www.emis.de/journals/AFA/ ON f-connections OF POSITIVE DEFINITE MATRICES MAREK NIEZGODA This
More informationPOWERS AND LOGARITHMS. Danuta Przeworska-Rolewicz. Abstract
POWERS AND LOGARITHMS Danuta Przeworska-Rolewicz Dedicated to Professor Ivan H. Dimovski on the occasion of his 70th birthday Abstract There are applied power mappings in algebras with logarithms induced
More informationConvergence rate estimates for the gradient differential inclusion
Convergence rate estimates for the gradient differential inclusion Osman Güler November 23 Abstract Let f : H R { } be a proper, lower semi continuous, convex function in a Hilbert space H. The gradient
More informationOn the stability of the functional equation f(x+ y + xy) = f(x)+ f(y)+ xf (y) + yf (x)
J. Math. Anal. Appl. 274 (2002) 659 666 www.academicpress.com On the stability of the functional equation f(x+ y + xy) = f(x)+ f(y)+ xf (y) + yf (x) Yong-Soo Jung a, and Kyoo-Hong Park b a Department of
More informationSubdifferential representation of convex functions: refinements and applications
Subdifferential representation of convex functions: refinements and applications Joël Benoist & Aris Daniilidis Abstract Every lower semicontinuous convex function can be represented through its subdifferential
More informationTopology, Math 581, Fall 2017 last updated: November 24, Topology 1, Math 581, Fall 2017: Notes and homework Krzysztof Chris Ciesielski
Topology, Math 581, Fall 2017 last updated: November 24, 2017 1 Topology 1, Math 581, Fall 2017: Notes and homework Krzysztof Chris Ciesielski Class of August 17: Course and syllabus overview. Topology
More information3. Convex functions. basic properties and examples. operations that preserve convexity. the conjugate function. quasiconvex functions
3. Convex functions Convex Optimization Boyd & Vandenberghe basic properties and examples operations that preserve convexity the conjugate function quasiconvex functions log-concave and log-convex functions
More information1. Subspaces A subset M of Hilbert space H is a subspace of it is closed under the operation of forming linear combinations;i.e.,
Abstract Hilbert Space Results We have learned a little about the Hilbert spaces L U and and we have at least defined H 1 U and the scale of Hilbert spaces H p U. Now we are going to develop additional
More informationOn non-expansivity of topical functions by a new pseudo-metric
https://doi.org/10.1007/s40065-018-0222-8 Arabian Journal of Mathematics H. Barsam H. Mohebi On non-expansivity of topical functions by a new pseudo-metric Received: 9 March 2018 / Accepted: 10 September
More informationErdinç Dündar, Celal Çakan
DEMONSTRATIO MATHEMATICA Vol. XLVII No 3 2014 Erdinç Dündar, Celal Çakan ROUGH I-CONVERGENCE Abstract. In this work, using the concept of I-convergence and using the concept of rough convergence, we introduced
More informationEXISTENCE RESULTS FOR QUASILINEAR HEMIVARIATIONAL INEQUALITIES AT RESONANCE. Leszek Gasiński
DISCRETE AND CONTINUOUS Website: www.aimsciences.org DYNAMICAL SYSTEMS SUPPLEMENT 2007 pp. 409 418 EXISTENCE RESULTS FOR QUASILINEAR HEMIVARIATIONAL INEQUALITIES AT RESONANCE Leszek Gasiński Jagiellonian
More informationHalf convex functions
Pavić Journal of Inequalities and Applications 2014, 2014:13 R E S E A R C H Open Access Half convex functions Zlatko Pavić * * Correspondence: Zlatko.Pavic@sfsb.hr Mechanical Engineering Faculty in Slavonski
More informationON GENERALIZED-CONVEX CONSTRAINED MULTI-OBJECTIVE OPTIMIZATION
ON GENERALIZED-CONVEX CONSTRAINED MULTI-OBJECTIVE OPTIMIZATION CHRISTIAN GÜNTHER AND CHRISTIANE TAMMER Abstract. In this paper, we consider multi-objective optimization problems involving not necessarily
More informationExistence result for the coupling problem of two scalar conservation laws with Riemann initial data
Existence result for the coupling problem of two scalar conservation laws with Riemann initial data Benjamin Boutin, Christophe Chalons, Pierre-Arnaud Raviart To cite this version: Benjamin Boutin, Christophe
More informationYET MORE ON THE DIFFERENTIABILITY OF CONVEX FUNCTIONS
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 103, Number 3, July 1988 YET MORE ON THE DIFFERENTIABILITY OF CONVEX FUNCTIONS JOHN RAINWATER (Communicated by William J. Davis) ABSTRACT. Generic
More informationUSING FUNCTIONAL ANALYSIS AND SOBOLEV SPACES TO SOLVE POISSON S EQUATION
USING FUNCTIONAL ANALYSIS AND SOBOLEV SPACES TO SOLVE POISSON S EQUATION YI WANG Abstract. We study Banach and Hilbert spaces with an eye towards defining weak solutions to elliptic PDE. Using Lax-Milgram
More informationA SHORT INTRODUCTION TO BANACH LATTICES AND
CHAPTER A SHORT INTRODUCTION TO BANACH LATTICES AND POSITIVE OPERATORS In tis capter we give a brief introduction to Banac lattices and positive operators. Most results of tis capter can be found, e.g.,
More informationAveraging Operators on the Unit Interval
Averaging Operators on the Unit Interval Mai Gehrke Carol Walker Elbert Walker New Mexico State University Las Cruces, New Mexico Abstract In working with negations and t-norms, it is not uncommon to call
More informationA GENERALIZATION OF JENSEN EQUATION FOR SET-VALUED MAPS
Seminar on Fixed Point Theory Cluj-Naoca, Volume 3, 2002, 317-322 htt://www.math.ubbcluj.ro/ nodeacj/journal.htm A GENERALIZATION OF JENSEN EQUATION FOR SET-VALUED MAPS DORIAN POPA Technical University
More informationON UNIFORM TAIL EXPANSIONS OF BIVARIATE COPULAS
APPLICATIONES MATHEMATICAE 31,4 2004), pp. 397 415 Piotr Jaworski Warszawa) ON UNIFORM TAIL EXPANSIONS OF BIVARIATE COPULAS Abstract. The theory of copulas provides a useful tool for modelling dependence
More informationCharacterizations of Solution Sets of Fréchet Differentiable Problems with Quasiconvex Objective Function
Characterizations of Solution Sets of Fréchet Differentiable Problems with Quasiconvex Objective Function arxiv:1805.03847v1 [math.oc] 10 May 2018 Vsevolod I. Ivanov Department of Mathematics, Technical
More informationA REVIEW OF OPTIMIZATION
1 OPTIMAL DESIGN OF STRUCTURES (MAP 562) G. ALLAIRE December 17th, 2014 Department of Applied Mathematics, Ecole Polytechnique CHAPTER III A REVIEW OF OPTIMIZATION 2 DEFINITIONS Let V be a Banach space,
More informationConvex Analysis and Economic Theory AY Elementary properties of convex functions
Division of the Humanities and Social Sciences Ec 181 KC Border Convex Analysis and Economic Theory AY 2018 2019 Topic 6: Convex functions I 6.1 Elementary properties of convex functions We may occasionally
More informationGEOMETRIC APPROACH TO CONVEX SUBDIFFERENTIAL CALCULUS October 10, Dedicated to Franco Giannessi and Diethard Pallaschke with great respect
GEOMETRIC APPROACH TO CONVEX SUBDIFFERENTIAL CALCULUS October 10, 2018 BORIS S. MORDUKHOVICH 1 and NGUYEN MAU NAM 2 Dedicated to Franco Giannessi and Diethard Pallaschke with great respect Abstract. In
More information2 Sequences, Continuity, and Limits
2 Sequences, Continuity, and Limits In this chapter, we introduce the fundamental notions of continuity and limit of a real-valued function of two variables. As in ACICARA, the definitions as well as proofs
More informationASSOCIATIVE n DIMENSIONAL COPULAS
K Y BERNETIKA VOLUM E 47 ( 2011), NUMBER 1, P AGES 93 99 ASSOCIATIVE n DIMENSIONAL COPULAS Andrea Stupňanová and Anna Kolesárová The associativity of n-dimensional copulas in the sense of Post is studied.
More informationRNDr. Petr Tomiczek CSc.
HABILITAČNÍ PRÁCE RNDr. Petr Tomiczek CSc. Plzeň 26 Nonlinear differential equation of second order RNDr. Petr Tomiczek CSc. Department of Mathematics, University of West Bohemia 5 Contents 1 Introduction
More informationKybernetika. Margarita Mas; Miquel Monserrat; Joan Torrens QL-implications versus D-implications. Terms of use:
Kybernetika Margarita Mas; Miquel Monserrat; Joan Torrens QL-implications versus D-implications Kybernetika, Vol. 42 (2006), No. 3, 35--366 Persistent URL: http://dml.cz/dmlcz/3579 Terms of use: Institute
More informationAvailable online at Advances in Fixed Point Theory, 2 (2012), No. 1, ISSN:
Available online at http://scik.org Advances in Fixed Point Theory, 2 (2012), No. 1, 29-46 ISSN: 1927-6303 COUPLED FIXED POINT RESULTS UNDER TVS-CONE METRIC AND W-CONE-DISTANCE ZORAN KADELBURG 1,, STOJAN
More information