ANTIPODAL FIXED POINT THEORY FOR VOLTERRA MAPS
|
|
- Ferdinand Parker
- 6 years ago
- Views:
Transcription
1 Dynamic Systems and Applications 17 (2008) ANTIPODAL FIXED POINT THEORY FOR VOLTERRA MAPS YEOL JE CHO, DONAL O REGAN, AND SVATOSLAV STANEK Department of Mathematics Education and the RINS, College of Education, Gyeongsang National University, Chinju , Korea Department of Mathematics, National University of Ireland, Galway, Ireland Department of Mathematical Analysis, Faculty of Science, Palacky University, Tomkova 40, Olomouc, Czech Republic ABSTRACT. New antipodal fixed point theorems for compact Kakutani maps between Fréchet spaces are presented. Keywords. Fixed point theory, projective limits. AMS (MOS) Subject Classification. 47H INTRODUCTION This paper presents new antipodal fixed point theorem for compact Kakutani maps between Fréchet spaces. The proofs rely on an antipodal fixed point theorem for compact Kakutani maps due to O Regan and Peran [4] and viewing a Fréchet space as the projective limit of a sequence of Banach spaces. In the literature [1, 2] one usually assumes the map F is defined on a subset X of a Fréchet space E and its restriction (again called F ) is well defined on X n (see Section 2). In general of course for Volterra operators the restriction is always defined on X n and in most applications it is in fact defined on X n and usually even on E n (see Section 2). In this paper we make use of the fact that the restriction is well defined on X n and we only assume it admits an extension (satisfying certain properties) on X n. We also show in Section 2 how easily one can extend antipodal fixed point theory in Banach spaces to fixed point theory in Fréchet spaces. Existence in Section 2 will be based on an antipodal fixed point theorem for compact Kakutani maps due to O Regan and Peran [4]. We state a particular case of it here for the convenience of the reader. Let X and Y be topological vector spaces. We say F : X CK(Y ) is a Kakutani map if F is upper semicontinuous; here CK(Y ) denotes the family of nonempty compact convex subsets of Y. We write F K(X, Y ) if F is a compact Kakutani map. Received September 14, $15.00 c Dynamic Publishers, Inc.
2 326 Y. J. CHO, D. O REGAN, AND S. STANEK Theorem 1.1. Let E be a Banach space and M a closed, bounded, symmetric subset of E with 0 M and let F K(M, E) with F (x) ( F ( x)) for all x M. Then F has a fixed point in M. Now let I be a directed set with order and let E α } α I be a family of locally convex spaces. For each α I, β I for which α β let π α,β : E β E α continuous map. Then the set } x = (x α ) α I E α : x α = π α,β (x β ) α, β I, α β be a is a closed subset of α I E α and is called the projective limit of E α } α I and is denoted by lim E α (or lim E α, π α,β } or the generalized intersection [3, pp. 439] α I E α.) 2. FIXED POINT THEORY IN FRÉCHET SPACES Let E = (E, n } n N ) be a Fréchet space with the topology generated by a family of seminorms n : n N}; here N = 1, 2,... }. We assume that the family of seminorms satisfies (2.1) x 1 x 2 x 3 for every x E. A subset X of E is bounded if for every n N there exists r n > 0 such that x n r n for all x X. For r > 0 and x E we denote B(x, r) = y E : x y n r n N}. To E we associate a sequence of Banach spaces (E n, n )} described as follows. For every n N we consider the equivalence relation n defined by (2.2) x n y iff x y n = 0. We denote by E n = (E / n, n ) the quotient space, and by (E n, n ) the completion of E n with respect to n (the norm on E n induced by n and its extension to E n are still denoted by n ). This construction defines a continuous map µ n : E E n. Now since (2.1) is satisfied the seminorm n induces a seminorm on E m for every m n (again this seminorm is denoted by n ). Also (2.2) defines an equivalence relation on E m from which we obtain a continuous map µ n,m : E m E n since E m / n can be regarded as a subset of E n. Now µ n,m µ m,k = µ n,k if n m k and µ n = µ n,m µ m if n m. We now assume the following condition holds: for each n N, there exists a Banach space (E n, n ) (2.3) and an isomorphism (between normed spaces) j n : E n E n. Remark 2.1. (i). For convenience the norm on E n is denoted by n. (ii). In our applications E n = E n for each n N.
3 ANTIPODAL FIXED POINT THEORY FOR VOLTERRA MAPS 327 (iii). Note if x E n (or E n ) then x E. However if x E n then x is not necessaily in E and in fact E n is easier to use in applications (even though E n is isomorphic to E n ). For example if E = C[0, ), then E n E which coincide on the interval [0, n] and E n = C[0, n]. Finally we assume E 1 E 2 and for each n N, (2.4) j n µ n,n+1 jn+1 1 x n x n+1 x E n+1 consists of the class of functions in (here we use the notation from [3] i.e. decreasing in the generalized sense). lim E n (or 1 E n where 1 is the generalized intersection [3]) denote the projective limit of E n } n N (note π n,m = j n µ n,m jm 1 : E m E n for m n) and note lim E n = E, so for convenience we write E = lim E n. For each X E and each n N we set X n = j n µ n (X), and we let X n, int X n and X n denote respectively the closure, the interior and the boundary of X n with respect to n in E n. Also the pseudo-interior of X is defined by Let pseudo int (X) = x X : j n µ n (x) X n \ X n for every n N}. The set X is pseudo-open if X = pseudo int (X). For r > 0 and x E n we denote B n (x, r) = y E n : x y n r}. Let M E and consider the map F : M 2 E. Assume for each n N and x M that j n µ n F (x) is closed. Let n N and M n = j n µ n (M). Since we only consider Volterra type operators we assume (2.5) if x, y E with x y n = 0 then H n (F (x), F (y)) = 0; here H n denotes the appropriate generalized Hausdorff distance (alternatively we could assume n N, x, y M if j n µ n (x) = j n µ n (y) then j n µ n F (x) = j n µ n F (y) and of course here we do not need to assume that j n µ n F (x) is closed for each n N and x M). Now (2.5) guarantees that we can define (a well defined) F n on M n as follows: For y M n there exists a x M with y = j n µ n (x) and we let F n (y) = j n µ n F (x) (we could of course call it F y since it is clear in the situation we use it); note F n : M n C(E n ) and note if there exists a z M with y = j n µ n (z) then j n µ n F (x) = j n µ n F (z) from (2.5) (here C(E n ) denotes the family of nonempty closed subsets of E n ). In this paper we assume F n will be defined on M n i.e. we assume the F n described above admits an extension (again we call it F n ) F n : M n 2 En (we will assume certain properties on the extension). We now show how easily one can extend fixed point theory in Banach spaces to applicable fixed point theory in Fréchet spaces.
4 328 Y. J. CHO, D. O REGAN, AND S. STANEK Theorem 2.1. Let E and E n be as described in the beginning of Section 2, M a bounded symmetric subset of E, F : Y 2 E with Y E and with j n µ n (0) M n and M n Y n for each n N. Also assume for each n N and x Y that j n µ n F (x) is closed and in addition for each n N that F n : M n 2 En is as described above. Suppose the following conditions are satisfied: (2.6) for each n N, F n K(M n, E n ) (2.7) for each n N, we have F n (x) ( F n ( x)) for x M n and (2.8) for each n 2, 3,... } if y M n solves y F n y in E n then j k µ k,n jn 1 (y) M k for k 1,..., n 1}. Then F has a fixed point in E. Remark 2.2. Note in Theorem 2.1 if x M n then x Y n so there exists a y Y with x = j n µ n (y) and so F n (x) = j n µ n F (y). PROOF: Fix n N. We would like to apply Theorem 1.1. To do so we need to show (2.9) M n is a bounded symmetric subset of E n. To see that M n is symmetric let ˆx µ n (M). Then for every x µ 1 n (ˆx) we have x M since M is symmetric and of course ˆx = µ n (x). Now its easy to see µ n (x) = µ n ( x) so ˆx = µ n (x) = µ n ( x) µ n (M). As a result µ n (M) is symmetric and since j n is linear (in particular j n ( y) = j n (y) for y E n ) we have that M n = j n µ n (M) is symmetric and so M n is symmetric. Also M n is bounded (so M n is bounded) since M is bounded (note if y M n then there exists x M with y = j n µ n (x)). Thus (2.9) holds. For each n N (see Theorem 1.1) there exists y n M n with y n F n (y n ). Lets look at y n } n N. Notice y 1 M 1 and j 1 µ 1,k j 1 k (y k ) M 1 for k N\1} from (2.8). Note j 1 µ 1,n jn n) F 1 (j 1 µ 1,n jn n)) in E 1 ; to see note for n N fixed there exists a x E with y n = j n µ n (x) so j n µ n (x) F n (y n ) = j n µ n F (x) on E n so on E 1 we have j 1 µ 1,n jn 1 (y n) = j 1 µ 1,n jn 1 j n µ n (x) j 1 µ 1,n jn 1 j n µ n F (x) = j 1 µ 1,n µ n F (x) = j 1 µ 1 F (x) = F 1 (j 1 µ 1 (x)) = F 1 (j 1 µ 1,n j 1 n j n µ n (x)) = F 1 (j 1 µ 1,n jn 1 (y n)). As a result j 1 µ 1,n jn n) F 1 (j 1 µ 1,n jn n)) in E 1, j 1 µ 1,n jn 1 (y n) M 1 for n N, together with (2.6) implies there is a subsequence N1 of N and a z 1 M 1 with j 1 µ 1,n jn 1 (y n) z 1 in E 1 as n in N1 and z 1 F 1 (z 1 ) since F 1 is upper
5 ANTIPODAL FIXED POINT THEORY FOR VOLTERRA MAPS 329 semicontinuous. Let N 1 = N 1 \ 1}. Now j 2 µ 2,n j 1 n (y n) M 2 for n N 1 together with (2.6) guarantees that there exists a subsequence N 2 of N 1 and a z 2 M 2 with j 2 µ 2,n jn 1 (y n) z 2 in E 2 as n in N2 and z 2 F 2 (z 2 ). Note from (2.4) and the uniqueness of limits that j 1 µ 1,2 j2 1 z 2 = z 1 in E 1 since N2 N 1 (note j 1 µ 1,n jn 1 (y n ) = j 1 µ 1,2 j2 1 j 2 µ 2,n jn 1 (y n ) for n N2 ). Let N 2 = N2 \ 2}. Proceed inductively to obtain subsequences of integers N 1 N 2, N k k, k + 1,... } and z k M k with j k µ k,n jn 1 (y n ) z k in E k as n in Nk and z k F k (z k ). Note j k µ k,k+1 j 1 k+1 z k+1 = z k in E k for k 1, 2,... }. Also let N k = Nk \ k}. Fix k N. Now z k F k (z k ) in E k. Note as well that z k = j k µ k,k+1 j 1 k+1 z k+1 = j k µ k,k+1 j 1 k+1 j k+1 µ k+1,k+2 j 1 k+2 z k+2 = j k µ k,k+2 j 1 k+2 z k+2 = = j k µ k,m jm 1 z m = π k,m z m for every m k. We can do this for each k N. As a result y = (z k ) lim E n = E and also note z k M k Y k for each k N. Thus for each k N we have so y F (y) in E. j k µ k (y) = z k F k (z k ) = j k µ k F (y) in E k In Theorem 2.1 it is possible to replace M n Y n with M n a subset of the closure of Y n in E n provided Y is a closed subset of E so in this case we could have Y = M if M was closed. To see this note from y = (z k ) lim E n = E and π k,m (y m ) z k in E k as m we can conclude that y Y = Y (note q Y iff for every k N there exists (x k,m ) Y, x k,m = π k,n (x n,m ) for n k with x k,m j k µ k (q) in E k as m ). Thus z k = j k µ k (y) Y k and so j k µ k (y) j k µ k F (y) in E k as before. For completeness we state these results. Theorem 2.2. Let E and E n be as described in the beginning of Section 2, M a bounded symmetric subset of E, F : Y 2 E with Y a closed subset of E and with j n µ n (0) M n and M n a subset of the closure of Y n in E n for each n N. Also assume for each n N and x Y each n N that F n : M n 2 En (2.8) hold. Then F has a fixed point in E. that j n µ n F (x) is closed and in addition for is as described above. Suppose (2.6), (2.7) and Corollary 2.1. Let E and E n be as described in the beginning of Section 2, M a closed bounded symmetric subset of E, F : M 2 E with j n µ n (0) M n for each n N. Also assume for each n N and x M that j n µ n F (x) is closed and in addition for each n N that F n : M n 2 En is as described above. Suppose (2.6), (2.7) and (2.8) hold. Then F has a fixed point in E.
6 330 Y. J. CHO, D. O REGAN, AND S. STANEK REFERENCES [1] R.P. Agarwal, M. Frigon and D. O Regan, A survey of recent fixed point theory in Fréchet spaces, Nonlinear analysis and applications: to V. Lakshmikantham on his 80th birthday, Vol 1, 75 88, Kluwer Acad. Publ., Dordrecht, [2] R.P. Agarwal and D. O Regan, A Lefschetz fixed point theorems for admissible maps in Fréchet spaces, Dynamic Systems and Applications, 16(2007), [3] L.V. Kantorovich and G.P. Akilov, Functional analysis in normed spaces, Pergamon Press, Oxford, [4] D. O Regan and J. Peran, Fixed points of compact Kakutani maps with antipodal boundary conditions, Proc. American Math. Soc., 134(2005),
A LEFSCHETZ FIXED POINT THEOREM FOR ADMISSIBLE MAPS IN FRÉCHET SPACES
Dynamic Systems and Applications 16 (2007) 1-12 A LEFSCHETZ FIXED POINT THEOREM FOR ADMISSIBLE MAPS IN FRÉCHET SPACES RAVI P. AGARWAL AND DONAL O REGAN Department of Mathematical Sciences, Florida Institute
More informationShih-sen Chang, Yeol Je Cho, and Haiyun Zhou
J. Korean Math. Soc. 38 (2001), No. 6, pp. 1245 1260 DEMI-CLOSED PRINCIPLE AND WEAK CONVERGENCE PROBLEMS FOR ASYMPTOTICALLY NONEXPANSIVE MAPPINGS Shih-sen Chang, Yeol Je Cho, and Haiyun Zhou Abstract.
More informationYuqing Chen, Yeol Je Cho, and Li Yang
Bull. Korean Math. Soc. 39 (2002), No. 4, pp. 535 541 NOTE ON THE RESULTS WITH LOWER SEMI-CONTINUITY Yuqing Chen, Yeol Je Cho, and Li Yang Abstract. In this paper, we introduce the concept of lower semicontinuous
More informationHAIYUN ZHOU, RAVI P. AGARWAL, YEOL JE CHO, AND YONG SOO KIM
Georgian Mathematical Journal Volume 9 (2002), Number 3, 591 600 NONEXPANSIVE MAPPINGS AND ITERATIVE METHODS IN UNIFORMLY CONVEX BANACH SPACES HAIYUN ZHOU, RAVI P. AGARWAL, YEOL JE CHO, AND YONG SOO KIM
More informationAN ITERATIVE METHOD FOR COMPUTING ZEROS OF OPERATORS SATISFYING AUTONOMOUS DIFFERENTIAL EQUATIONS
Electronic Journal: Southwest Journal of Pure Applied Mathematics Internet: http://rattler.cameron.edu/swjpam.html ISBN 1083-0464 Issue 1 July 2004, pp. 48 53 Submitted: August 21, 2003. Published: July
More informationCONVERGENCE OF HYBRID FIXED POINT FOR A PAIR OF NONLINEAR MAPPINGS IN BANACH SPACES
International Journal of Analysis and Applications ISSN 2291-8639 Volume 8, Number 1 2015), 69-78 http://www.etamaths.com CONVERGENCE OF HYBRID FIXED POINT FOR A PAIR OF NONLINEAR MAPPINGS IN BANACH SPACES
More informationCOMMON FIXED POINT THEOREMS FOR A PAIR OF COUNTABLY CONDENSING MAPPINGS IN ORDERED BANACH SPACES
Journal of Applied Mathematics and Stochastic Analysis, 16:3 (2003), 243-248. Printed in the USA c 2003 by North Atlantic Science Publishing Company COMMON FIXED POINT THEOREMS FOR A PAIR OF COUNTABLY
More informationFixed Point Theorems for Condensing Maps
Int. Journal of Math. Analysis, Vol. 2, 2008, no. 21, 1031-1044 Fixed Point Theorems for Condensing Maps in S-KKM Class Young-Ye Huang Center for General Education Southern Taiwan University 1 Nan-Tai
More informationThe antipodal mapping theorem and difference equations in Banach spaces
Journal of Difference Equations and Applications Vol., No.,, 1 13 The antipodal mapping theorem and difference equations in Banach spaces Daniel Franco, Donal O Regan and Juan Peran * Departamento de Matemática
More informationThe Journal of Nonlinear Sciences and Applications
J. Nonlinear Sci. Appl. 2 (2009), no. 3, 195 203 The Journal of Nonlinear Sciences Applications http://www.tjnsa.com ON MULTIPOINT ITERATIVE PROCESSES OF EFFICIENCY INDEX HIGHER THAN NEWTON S METHOD IOANNIS
More informationSome topological properties of fuzzy cone metric spaces
Available online at www.tjnsa.com J. Nonlinear Sci. Appl. 9 2016, 799 805 Research Article Some topological properties of fuzzy cone metric spaces Tarkan Öner Department of Mathematics, Faculty of Sciences,
More informationY. J. Cho, M. Matić and J. Pečarić
Commun. Korean Math. Soc. 17 (2002), No. 4, pp. 583 592 INEQUALITIES OF HLAWKA S TYPE IN n-inner PRODUCT SPACES Y. J. Cho, M. Matić and J. Pečarić Abstract. In this paper, we give Hlawka s type inequalities
More informationA PROOF OF A CONVEX-VALUED SELECTION THEOREM WITH THE CODOMAIN OF A FRÉCHET SPACE. Myung-Hyun Cho and Jun-Hui Kim. 1. Introduction
Comm. Korean Math. Soc. 16 (2001), No. 2, pp. 277 285 A PROOF OF A CONVEX-VALUED SELECTION THEOREM WITH THE CODOMAIN OF A FRÉCHET SPACE Myung-Hyun Cho and Jun-Hui Kim Abstract. The purpose of this paper
More informationNew w-convergence Conditions for the Newton-Kantorovich Method
Punjab University Journal of Mathematics (ISSN 116-2526) Vol. 46(1) (214) pp. 77-86 New w-convergence Conditions for the Newton-Kantorovich Method Ioannis K. Argyros Department of Mathematicsal Sciences,
More informationStrong convergence theorems for asymptotically nonexpansive nonself-mappings with applications
Guo et al. Fixed Point Theory and Applications (2015) 2015:212 DOI 10.1186/s13663-015-0463-6 R E S E A R C H Open Access Strong convergence theorems for asymptotically nonexpansive nonself-mappings with
More informationSolvability of Discrete Neumann Boundary Value Problems
Solvability of Discrete Neumann Boundary Value Problems D. R. Anderson, I. Rachůnková and C. C. Tisdell September 6, 2006 1 Department of Mathematics and Computer Science Concordia College, 901 8th Street,
More informationAn improved convergence theorem for the Newton method under relaxed continuity assumptions
An improved convergence theorem for the Newton method under relaxed continuity assumptions Andrei Dubin ITEP, 117218, BCheremushinsaya 25, Moscow, Russia Abstract In the framewor of the majorization technique,
More informationFUNCTIONAL COMPRESSION-EXPANSION FIXED POINT THEOREM
Electronic Journal of Differential Equations, Vol. 28(28), No. 22, pp. 1 12. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) FUNCTIONAL
More informationDIFFERENCE EQUATIONS IN ABSTRACT SPACES
J. Austral. Math. Soc. Series A) 64 199), 277 24 DIFFERENCE EQUATIONS IN ABSTRACT SPACES RAVI P. AGARWAL and DONAL O REGAN Received 10 January 1997; revised 14 October 1997) Communicated by J. R. J. Groves
More informationFréchet algebras of finite type
Fréchet algebras of finite type MK Kopp Abstract The main objects of study in this paper are Fréchet algebras having an Arens Michael representation in which every Banach algebra is finite dimensional.
More informationConvergence to Common Fixed Point for Two Asymptotically Quasi-nonexpansive Mappings in the Intermediate Sense in Banach Spaces
Mathematica Moravica Vol. 19-1 2015, 33 48 Convergence to Common Fixed Point for Two Asymptotically Quasi-nonexpansive Mappings in the Intermediate Sense in Banach Spaces Gurucharan Singh Saluja Abstract.
More informationLocally convex spaces, the hyperplane separation theorem, and the Krein-Milman theorem
56 Chapter 7 Locally convex spaces, the hyperplane separation theorem, and the Krein-Milman theorem Recall that C(X) is not a normed linear space when X is not compact. On the other hand we could use semi
More informationProblem Set 2: Solutions Math 201A: Fall 2016
Problem Set 2: s Math 201A: Fall 2016 Problem 1. (a) Prove that a closed subset of a complete metric space is complete. (b) Prove that a closed subset of a compact metric space is compact. (c) Prove that
More informationOn the effect of α-admissibility and θ-contractivity to the existence of fixed points of multivalued mappings
Nonlinear Analysis: Modelling and Control, Vol. 21, No. 5, 673 686 ISSN 1392-5113 http://dx.doi.org/10.15388/na.2016.5.7 On the effect of α-admissibility and θ-contractivity to the existence of fixed points
More informationi=1 β i,i.e. = β 1 x β x β 1 1 xβ d
66 2. Every family of seminorms on a vector space containing a norm induces ahausdorff locally convex topology. 3. Given an open subset Ω of R d with the euclidean topology, the space C(Ω) of real valued
More informationWeak and strong convergence theorems of modified SP-iterations for generalized asymptotically quasi-nonexpansive mappings
Mathematica Moravica Vol. 20:1 (2016), 125 144 Weak and strong convergence theorems of modified SP-iterations for generalized asymptotically quasi-nonexpansive mappings G.S. Saluja Abstract. The aim of
More informationON WEAK AND STRONG CONVERGENCE THEOREMS FOR TWO NONEXPANSIVE MAPPINGS IN BANACH SPACES. Pankaj Kumar Jhade and A. S. Saluja
MATEMATIQKI VESNIK 66, 1 (2014), 1 8 March 2014 originalni nauqni rad research paper ON WEAK AND STRONG CONVERGENCE THEOREMS FOR TWO NONEXPANSIVE MAPPINGS IN BANACH SPACES Pankaj Kumar Jhade and A. S.
More informationA fixed point theorem for a Ćirić-Berinde type mapping in orbitally complete metric spaces
CARPATHIAN J. MATH. 30 (014), No. 1, 63-70 Online version available at http://carpathian.ubm.ro Print Edition: ISSN 1584-851 Online Edition: ISSN 1843-4401 A fixed point theorem for a Ćirić-Berinde type
More informationWEAK CONVERGENCE OF RESOLVENTS OF MAXIMAL MONOTONE OPERATORS AND MOSCO CONVERGENCE
Fixed Point Theory, Volume 6, No. 1, 2005, 59-69 http://www.math.ubbcluj.ro/ nodeacj/sfptcj.htm WEAK CONVERGENCE OF RESOLVENTS OF MAXIMAL MONOTONE OPERATORS AND MOSCO CONVERGENCE YASUNORI KIMURA Department
More informationA note on the σ-algebra of cylinder sets and all that
A note on the σ-algebra of cylinder sets and all that José Luis Silva CCM, Univ. da Madeira, P-9000 Funchal Madeira BiBoS, Univ. of Bielefeld, Germany (luis@dragoeiro.uma.pt) September 1999 Abstract In
More informationFixed points of mappings in Klee admissible spaces
Control and Cybernetics vol. 36 (2007) No. 3 Fixed points of mappings in Klee admissible spaces by Lech Górniewicz 1 and Mirosław Ślosarski 2 1 Schauder Center for Nonlinear Studies, Nicolaus Copernicus
More informationOn Positive Solutions of Boundary Value Problems on the Half-Line
Journal of Mathematical Analysis and Applications 259, 127 136 (21) doi:1.16/jmaa.2.7399, available online at http://www.idealibrary.com on On Positive Solutions of Boundary Value Problems on the Half-Line
More informationON GENERAL BEST PROXIMITY PAIRS AND EQUILIBRIUM PAIRS IN FREE GENERALIZED GAMES 1. Won Kyu Kim* 1. Introduction
JOURNAL OF THE CHUNGCHEONG MATHEMATICAL SOCIETY Volume 19, No.1, March 2006 ON GENERAL BEST PROXIMITY PAIRS AND EQUILIBRIUM PAIRS IN FREE GENERALIZED GAMES 1 Won Kyu Kim* Abstract. In this paper, using
More informationOn Common Fixed Point and Approximation Results of Gregus Type
International Mathematical Forum, 2, 2007, no. 37, 1839-1847 On Common Fixed Point and Approximation Results of Gregus Type S. Al-Mezel and N. Hussain Department of Mathematics King Abdul Aziz University
More informationApproximating Fixed Points of Asymptotically Quasi-Nonexpansive Mappings by the Iterative Sequences with Errors
5 10 July 2004, Antalya, Turkey Dynamical Systems and Applications, Proceedings, pp. 262 272 Approximating Fixed Points of Asymptotically Quasi-Nonexpansive Mappings by the Iterative Sequences with Errors
More informationSome Fixed Point Theorems for G-Nonexpansive Mappings on Ultrametric Spaces and Non-Archimedean Normed Spaces with a Graph
J o u r n a l of Mathematics and Applications JMA No 39, pp 81-90 (2016) Some Fixed Point Theorems for G-Nonexpansive Mappings on Ultrametric Spaces and Non-Archimedean Normed Spaces with a Graph Hamid
More informationApplied Mathematics Letters. Functional inequalities in non-archimedean Banach spaces
Applied Mathematics Letters 23 (2010) 1238 1242 Contents lists available at ScienceDirect Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml Functional inequalities in non-archimedean
More informationMÖNCH TYPE RESULTS FOR MAPS WITH WEAKLY SEQUENTIALLY CLOSED GRAPHS
Dynamic Systems and Applications 24 (2015) 129--134 MÖNCH TYPE RESULTS FOR MAPS WITH WEAKLY SEQUENTIALLY CLOSED GRAPHS DONAL O REGAN School of Mathematics, Statistics and Applied Mathematics National University
More informationCONVERGENCE THEOREMS FOR STRICTLY ASYMPTOTICALLY PSEUDOCONTRACTIVE MAPPINGS IN HILBERT SPACES. Gurucharan Singh Saluja
Opuscula Mathematica Vol 30 No 4 2010 http://dxdoiorg/107494/opmath2010304485 CONVERGENCE THEOREMS FOR STRICTLY ASYMPTOTICALLY PSEUDOCONTRACTIVE MAPPINGS IN HILBERT SPACES Gurucharan Singh Saluja Abstract
More informationASYMPTOTICALLY NONEXPANSIVE MAPPINGS IN MODULAR FUNCTION SPACES ABSTRACT
ASYMPTOTICALLY NONEXPANSIVE MAPPINGS IN MODULAR FUNCTION SPACES T. DOMINGUEZ-BENAVIDES, M.A. KHAMSI AND S. SAMADI ABSTRACT In this paper, we prove that if ρ is a convex, σ-finite modular function satisfying
More informationThe weak topology of locally convex spaces and the weak-* topology of their duals
The weak topology of locally convex spaces and the weak-* topology of their duals Jordan Bell jordan.bell@gmail.com Department of Mathematics, University of Toronto April 3, 2014 1 Introduction These notes
More informationApproximate additive and quadratic mappings in 2-Banach spaces and related topics
Int. J. Nonlinear Anal. Appl. 3 (0) No., 75-8 ISSN: 008-68 (electronic) http://www.ijnaa.semnan.ac.ir Approximate additive and quadratic mappings in -Banach spaces and related topics Y. J. Cho a, C. Park
More information(convex combination!). Use convexity of f and multiply by the common denominator to get. Interchanging the role of x and y, we obtain that f is ( 2M ε
1. Continuity of convex functions in normed spaces In this chapter, we consider continuity properties of real-valued convex functions defined on open convex sets in normed spaces. Recall that every infinitedimensional
More informationB. Appendix B. Topological vector spaces
B.1 B. Appendix B. Topological vector spaces B.1. Fréchet spaces. In this appendix we go through the definition of Fréchet spaces and their inductive limits, such as they are used for definitions of function
More informationCommon fixed points of two generalized asymptotically quasi-nonexpansive mappings
An. Ştiinţ. Univ. Al. I. Cuza Iaşi. Mat. (N.S.) Tomul LXIII, 2017, f. 2 Common fixed points of two generalized asymptotically quasi-nonexpansive mappings Safeer Hussain Khan Isa Yildirim Received: 5.VIII.2013
More informationInternational Journal of Scientific & Engineering Research, Volume 7, Issue 12, December-2016 ISSN
1750 Approximation of Fixed Points of Multivalued Demicontractive and Multivalued Hemicontractive Mappings in Hilbert Spaces B. G. Akuchu Department of Mathematics University of Nigeria Nsukka e-mail:
More informationYour first day at work MATH 806 (Fall 2015)
Your first day at work MATH 806 (Fall 2015) 1. Let X be a set (with no particular algebraic structure). A function d : X X R is called a metric on X (and then X is called a metric space) when d satisfies
More informationREMARKS ON THE KKM PROPERTY FOR OPEN-VALUED MULTIMAPS ON GENERALIZED CONVEX SPACES
J. Korean Math. Soc. 42 (2005), No. 1, pp. 101 110 REMARKS ON THE KKM PROPERTY FOR OPEN-VALUED MULTIMAPS ON GENERALIZED CONVEX SPACES Hoonjoo Kim and Sehie Park Abstract. Let (X, D; ) be a G-convex space
More informationThe Proper Generalized Decomposition: A Functional Analysis Approach
The Proper Generalized Decomposition: A Functional Analysis Approach Méthodes de réduction de modèle dans le calcul scientifique Main Goal Given a functional equation Au = f It is possible to construct
More informationMethods of constructing topological vector spaces
CHAPTER 2 Methods of constructing topological vector spaces In this chapter we consider projective limits (in particular, products) of families of topological vector spaces, inductive limits (in particular,
More informationPROPERTIES OF FIXED POINT SET OF A MULTIVALUED MAP
PROPERTIES OF FIXED POINT SET OF A MULTIVALUED MAP ABDUL RAHIM KHAN Received 15 September 2004 and in revised form 21 November 2004 Properties of the set of fixed points of some discontinuous multivalued
More information1 Lecture 4: Set topology on metric spaces, 8/17/2012
Summer Jump-Start Program for Analysis, 01 Song-Ying Li 1 Lecture : Set topology on metric spaces, 8/17/01 Definition 1.1. Let (X, d) be a metric space; E is a subset of X. Then: (i) x E is an interior
More informationConstrained controllability of semilinear systems with delayed controls
BULLETIN OF THE POLISH ACADEMY OF SCIENCES TECHNICAL SCIENCES Vol. 56, No. 4, 28 Constrained controllability of semilinear systems with delayed controls J. KLAMKA Institute of Control Engineering, Silesian
More informationPositive Periodic Solutions of Systems of Second Order Ordinary Differential Equations
Positivity 1 (26), 285 298 26 Birkhäuser Verlag Basel/Switzerland 1385-1292/2285-14, published online April 26, 26 DOI 1.17/s11117-5-21-2 Positivity Positive Periodic Solutions of Systems of Second Order
More informationFixed point of ϕ-contraction in metric spaces endowed with a graph
Annals of the University of Craiova, Mathematics and Computer Science Series Volume 374, 2010, Pages 85 92 ISSN: 1223-6934 Fixed point of ϕ-contraction in metric spaces endowed with a graph Florin Bojor
More informationRESEARCH ANNOUNCEMENTS OPERATORS ON FUNCTION SPACES
BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY Volume 78, Number 5, September 1972 RESEARCH ANNOUNCEMENTS The purpose of this department is to provide early announcement of significant new results, with
More informationReal Analysis, 2nd Edition, G.B.Folland Elements of Functional Analysis
Real Analysis, 2nd Edition, G.B.Folland Chapter 5 Elements of Functional Analysis Yung-Hsiang Huang 5.1 Normed Vector Spaces 1. Note for any x, y X and a, b K, x+y x + y and by ax b y x + b a x. 2. It
More informationStability of Adjointable Mappings in Hilbert
Stability of Adjointable Mappings in Hilbert arxiv:math/0501139v2 [math.fa] 1 Aug 2005 C -Modules M. S. Moslehian Abstract The generalized Hyers Ulam Rassias stability of adjointable mappings on Hilbert
More informationarxiv:math/ v1 [math.fa] 26 Oct 1993
arxiv:math/9310217v1 [math.fa] 26 Oct 1993 ON COMPLEMENTED SUBSPACES OF SUMS AND PRODUCTS OF BANACH SPACES M.I.Ostrovskii Abstract. It is proved that there exist complemented subspaces of countable topological
More informationSOME GENERALIZATION OF MINTY S LEMMA. Doo-Young Jung
J. Korea Soc. Math. Educ. Ser. B: Pure Appl. Math. 6(1999), no 1. 33 37 SOME GENERALIZATION OF MINTY S LEMMA Doo-Young Jung Abstract. We obtain a generalization of Behera and Panda s result on nonlinear
More informationSingularities and Laplacians in Boundary Value Problems for Nonlinear Ordinary Differential Equations
Singularities and Laplacians in Boundary Value Problems for Nonlinear Ordinary Differential Equations Irena Rachůnková, Svatoslav Staněk, Department of Mathematics, Palacký University, 779 OLOMOUC, Tomkova
More informationSTRONG CONVERGENCE OF AN IMPLICIT ITERATION PROCESS FOR ASYMPTOTICALLY NONEXPANSIVE IN THE INTERMEDIATE SENSE MAPPINGS IN BANACH SPACES
Kragujevac Journal of Mathematics Volume 36 Number 2 (2012), Pages 237 249. STRONG CONVERGENCE OF AN IMPLICIT ITERATION PROCESS FOR ASYMPTOTICALLY NONEXPANSIVE IN THE INTERMEDIATE SENSE MAPPINGS IN BANACH
More informationStrong Convergence Theorems for Nonself I-Asymptotically Quasi-Nonexpansive Mappings 1
Applied Mathematical Sciences, Vol. 2, 2008, no. 19, 919-928 Strong Convergence Theorems for Nonself I-Asymptotically Quasi-Nonexpansive Mappings 1 Si-Sheng Yao Department of Mathematics, Kunming Teachers
More informationAlfred O. Bosede NOOR ITERATIONS ASSOCIATED WITH ZAMFIRESCU MAPPINGS IN UNIFORMLY CONVEX BANACH SPACES
F A S C I C U L I M A T H E M A T I C I Nr 42 2009 Alfred O. Bosede NOOR ITERATIONS ASSOCIATED WITH ZAMFIRESCU MAPPINGS IN UNIFORMLY CONVEX BANACH SPACES Abstract. In this paper, we establish some fixed
More informationRelationships between upper exhausters and the basic subdifferential in variational analysis
J. Math. Anal. Appl. 334 (2007) 261 272 www.elsevier.com/locate/jmaa Relationships between upper exhausters and the basic subdifferential in variational analysis Vera Roshchina City University of Hong
More informationMaximilian GANSTER. appeared in: Soochow J. Math. 15 (1) (1989),
A NOTE ON STRONGLY LINDELÖF SPACES Maximilian GANSTER appeared in: Soochow J. Math. 15 (1) (1989), 99 104. Abstract Recently a new class of topological spaces, called strongly Lindelöf spaces, has been
More informationON PEXIDER DIFFERENCE FOR A PEXIDER CUBIC FUNCTIONAL EQUATION
ON PEXIDER DIFFERENCE FOR A PEXIDER CUBIC FUNCTIONAL EQUATION S. OSTADBASHI and M. SOLEIMANINIA Communicated by Mihai Putinar Let G be an abelian group and let X be a sequentially complete Hausdorff topological
More information("-1/' .. f/ L) I LOCAL BOUNDEDNESS OF NONLINEAR, MONOTONE OPERA TORS. R. T. Rockafellar. MICHIGAN MATHEMATICAL vol. 16 (1969) pp.
I l ("-1/'.. f/ L) I LOCAL BOUNDEDNESS OF NONLINEAR, MONOTONE OPERA TORS R. T. Rockafellar from the MICHIGAN MATHEMATICAL vol. 16 (1969) pp. 397-407 JOURNAL LOCAL BOUNDEDNESS OF NONLINEAR, MONOTONE OPERATORS
More informationExtension of continuous convex functions II
Interactions between Logic, Topological structures and Banach spaces theory May 19-24, 2013 Eilat Joint work with Libor Veselý Notations and preliminaries X real topological vector space. X topological
More informationFixed Points for Multivalued Mappings in b-metric Spaces
Applied Mathematical Sciences, Vol. 10, 2016, no. 59, 2927-2944 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2016.68225 Fixed Points for Multivalued Mappings in b-metric Spaces Seong-Hoon
More informationKantorovich s Majorants Principle for Newton s Method
Kantorovich s Majorants Principle for Newton s Method O. P. Ferreira B. F. Svaiter January 17, 2006 Abstract We prove Kantorovich s theorem on Newton s method using a convergence analysis which makes clear,
More informationON COMPLEMENTED SUBSPACES OF SUMS AND PRODUCTS OF BANACH SPACES
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 124, Number 7, July 1996 ON COMPLEMENTED SUBSPACES OF SUMS AND PRODUCTS OF BANACH SPACES M. I. OSTROVSKII (Communicated by Dale Alspach) Abstract.
More informationMAT 578 FUNCTIONAL ANALYSIS EXERCISES
MAT 578 FUNCTIONAL ANALYSIS EXERCISES JOHN QUIGG Exercise 1. Prove that if A is bounded in a topological vector space, then for every neighborhood V of 0 there exists c > 0 such that tv A for all t > c.
More informationTHE KNASTER KURATOWSKI MAZURKIEWICZ THEOREM AND ALMOST FIXED POINTS. Sehie Park. 1. Introduction
Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 16, 2000, 195 200 THE KNASTER KURATOWSKI MAZURKIEWICZ THEOREM AND ALMOST FIXED POINTS Sehie Park Abstract. From the
More informationFIXED POINT THEOREMS AND CHARACTERIZATIONS OF METRIC COMPLETENESS. Tomonari Suzuki Wataru Takahashi. 1. Introduction
Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 8, 1996, 371 382 FIXED POINT THEOREMS AND CHARACTERIZATIONS OF METRIC COMPLETENESS Tomonari Suzuki Wataru Takahashi
More informationViscosity approximation methods for nonexpansive nonself-mappings
J. Math. Anal. Appl. 321 (2006) 316 326 www.elsevier.com/locate/jmaa Viscosity approximation methods for nonexpansive nonself-mappings Yisheng Song, Rudong Chen Department of Mathematics, Tianjin Polytechnic
More informationCOUPLED FIXED POINT THEOREMS FOR MONOTONE MAPPINGS IN PARTIALLY ORDERED METRIC SPACES
Kragujevac Journal of Mathematics Volume 382 2014, Pages 249 257. COUPLED FIXED POINT THEOREMS FOR MONOTONE MAPPINGS IN PARTIALLY ORDERED METRIC SPACES STOJAN RADENOVIĆ Abstract. In this paper, by reducing
More informationExistence and Approximation of Fixed Points of. Bregman Nonexpansive Operators. Banach Spaces
Existence and Approximation of Fixed Points of in Reflexive Banach Spaces Department of Mathematics The Technion Israel Institute of Technology Haifa 22.07.2010 Joint work with Prof. Simeon Reich General
More informationON COUNTABLE FAMILIES OF TOPOLOGIES ON A SET
Novi Sad J. Math. Vol. 40, No. 2, 2010, 7-16 ON COUNTABLE FAMILIES OF TOPOLOGIES ON A SET M.K. Bose 1, Ajoy Mukharjee 2 Abstract Considering a countable number of topologies on a set X, we introduce the
More informationAnother Low-Technology Estimate in Convex Geometry
Convex Geometric Analysis MSRI Publications Volume 34, 1998 Another Low-Technology Estimate in Convex Geometry GREG KUPERBERG Abstract. We give a short argument that for some C > 0, every n- dimensional
More informationIntrinsic Definition of Strong Shape for Compact Metric Spaces
Volume 39, 0 Pages 7 39 http://topology.auburn.edu/tp/ Intrinsic Definition of Strong Shape for Compact Metric Spaces by Nikita Shekutkovski Electronically published on April 4, 0 Topology Proceedings
More informationThe tensor algebra of power series spaces
The tensor algebra of power series spaces Dietmar Vogt Abstract The linear isomorphism type of the tensor algebra T (E) of Fréchet spaces and, in particular, of power series spaces is studied. While for
More informationMircea Balaj. Comment.Math.Univ.Carolinae 42,4 (2001)
Comment.Math.Univ.Carolinae 42,4 (2001)753 762 753 Admissible maps, intersection results, coincidence theorems Mircea Balaj Abstract. We obtain generalizations of the Fan s matching theorem for an open
More informationNotes for Functional Analysis
Notes for Functional Analysis Wang Zuoqin (typed by Xiyu Zhai) November 6, 2015 1 Lecture 18 1.1 The convex hull Let X be any vector space, and E X a subset. Definition 1.1. The convex hull of E is the
More informationCOUNTEREXAMPLES IN ROTUND AND LOCALLY UNIFORMLY ROTUND NORM
Journal of Mathematical Analysis ISSN: 2217-3412, URL: http://www.ilirias.com Volume 5 Issue 1(2014), Pages 11-15. COUNTEREXAMPLES IN ROTUND AND LOCALLY UNIFORMLY ROTUND NORM F. HEYDARI, D. BEHMARDI 1
More informationSIMULTANEOUS SOLUTIONS OF THE WEAK DIRICHLET PROBLEM Jan Kolar and Jaroslav Lukes Abstract. The main aim of this paper is a geometrical approach to si
SIMULTANEOUS SOLUTIONS OF THE WEAK DIRICHLET PROBLEM Jan Kolar and Jaroslav Lukes Abstract. The main aim of this paper is a geometrical approach to simultaneous solutions of the abstract weak Dirichlet
More informationSang-baek Lee*, Jae-hyeong Bae**, and Won-gil Park***
JOURNAL OF THE CHUNGCHEONG MATHEMATICAL SOCIETY Volume 6, No. 4, November 013 http://d.doi.org/10.14403/jcms.013.6.4.671 ON THE HYERS-ULAM STABILITY OF AN ADDITIVE FUNCTIONAL INEQUALITY Sang-baek Lee*,
More informationFRAGMENTABILITY OF ROTUND BANACH SPACES. Scott Sciffer. lim <Jl(x+A.y)- $(x) A-tO A
222 FRAGMENTABILITY OF ROTUND BANACH SPACES Scott Sciffer Introduction A topological. space X is said to be fragmented by a metric p if for every e > 0 and every subset Y of X there exists a nonempty relatively
More informationLocally uniformly rotund renormings of the spaces of continuous functions on Fedorchuk compacts
Locally uniformly rotund renormings of the spaces of continuous functions on Fedorchuk compacts S.TROYANSKI, Institute of Mathematics, Bulgarian Academy of Science joint work with S. P. GUL KO, Faculty
More informationON THE CONVERGENCE OF MODIFIED NOOR ITERATION METHOD FOR NEARLY LIPSCHITZIAN MAPPINGS IN ARBITRARY REAL BANACH SPACES
TJMM 6 (2014), No. 1, 45-51 ON THE CONVERGENCE OF MODIFIED NOOR ITERATION METHOD FOR NEARLY LIPSCHITZIAN MAPPINGS IN ARBITRARY REAL BANACH SPACES ADESANMI ALAO MOGBADEMU Abstract. In this present paper,
More informationON JOINT CONVEXITY AND CONCAVITY OF SOME KNOWN TRACE FUNCTIONS
ON JOINT CONVEXITY ND CONCVITY OF SOME KNOWN TRCE FUNCTIONS MOHMMD GHER GHEMI, NHID GHRKHNLU and YOEL JE CHO Communicated by Dan Timotin In this aer, we rovide a new and simle roof for joint convexity
More informationON A CERTAIN GENERALIZATION OF THE KRASNOSEL SKII THEOREM
Journal of Applied Analysis Vol. 9, No. 1 23, pp. 139 147 ON A CERTAIN GENERALIZATION OF THE KRASNOSEL SKII THEOREM M. GALEWSKI Received July 3, 21 and, in revised form, March 26, 22 Abstract. We provide
More informationTopological properties
CHAPTER 4 Topological properties 1. Connectedness Definitions and examples Basic properties Connected components Connected versus path connected, again 2. Compactness Definition and first examples Topological
More information1 Introduction The study of the existence of solutions of Variational Inequalities on unbounded domains usually involves the same sufficient assumptio
Coercivity Conditions and Variational Inequalities Aris Daniilidis Λ and Nicolas Hadjisavvas y Abstract Various coercivity conditions appear in the literature in order to guarantee solutions for the Variational
More informationAn Alternative Proof of Primitivity of Indecomposable Nonnegative Matrices with a Positive Trace
An Alternative Proof of Primitivity of Indecomposable Nonnegative Matrices with a Positive Trace Takao Fujimoto Abstract. This research memorandum is aimed at presenting an alternative proof to a well
More informationQUOTIENTS OF F-SPACES
QUOTIENTS OF F-SPACES by N. J. KALTON (Received 6 October, 1976) Let X be a non-locally convex F-space (complete metric linear space) whose dual X' separates the points of X. Then it is known that X possesses
More informationSOME REMARKS ON SUBDIFFERENTIABILITY OF CONVEX FUNCTIONS
Applied Mathematics E-Notes, 5(2005), 150-156 c ISSN 1607-2510 Available free at mirror sites of http://www.math.nthu.edu.tw/ amen/ SOME REMARKS ON SUBDIFFERENTIABILITY OF CONVEX FUNCTIONS Mohamed Laghdir
More informationCountably condensing multimaps and fixed points
Electronic Journal of Qualitative Theory of Differential Equations 2012, No. 83, 1-9; http://www.math.u-szeged.hu/ejqtde/ Countably condensing multimaps and fixed points Tiziana Cardinali - Paola Rubbioni
More informationGeneralized Monotonicities and Its Applications to the System of General Variational Inequalities
Generalized Monotonicities and Its Applications to the System of General Variational Inequalities Khushbu 1, Zubair Khan 2 Research Scholar, Department of Mathematics, Integral University, Lucknow, Uttar
More informationResearch Article Coupled Fixed Point Theorems for Weak Contraction Mappings under F-Invariant Set
Abstract and Applied Analysis Volume 2012, Article ID 324874, 15 pages doi:10.1155/2012/324874 Research Article Coupled Fixed Point Theorems for Weak Contraction Mappings under F-Invariant Set Wutiphol
More information