Best Proximity Pairs for Upper Semicontinuous Set-Valued Maps in Hyperconvex Metric Spaces

Best Proximity Pairs for Upper Semicontinuous Set-Valued Maps in Hyperconvex Metric Spaces Best Proximity Pairs for Upper Semicontinuous Set-Valued Maps in Hyperconvex Metric Spaces //// Hindawi Publishing Corporation Home Journals About Us About this Journal Submit a Manuscript Table of Contents Journal Menu Abstracting and Indexing Aims and Scope Article Processing Charges Articles in Press Author Guidelines Bibliographic Information Contact Information Editorial Board Editorial Workflow Free eTOC Alerts Reviewers Acknowledgment Subscription Information Open Special Issues Published Special Issues Special Issue Guidelines Abstract Full-Text PDF Full-Text HTML Linked References How to Cite this Article Fixed Point Theory and Applications Volume 2008 (2008), Article ID 648985, 5 pages doi:10.1155/2008/648985 Research Article <h2>Best Proximity Pairs for Upper Semicontinuous Set-Valued Maps in Hyperconvex Metric Spaces</h2> A. Amini-Harandi , 1 A. P. Farajzadeh , 2 D. O'Regan , 3 and R. P. Agarwal 4 1 Department of Mathematics, Faculty of Basic Sciences, University of Shahrekord, Shahrekord 88186-34141, Iran 2 Department of Mathematics, School of Science, Razi University, Kermanshah 67149, Iran 3 Department of Mathematics, College of Arts, Social Sciences and Celtic Studies, National University of Ireland, Galway, Ireland 4 Department of Mathematical Sciences, College of Science, Florida Institute of Technology, Melbourne, FL 32901, USA Received 14 July 2008; Accepted 27 October 2008 Academic Editor: Nan-jing Huang Copyright © 2008 A. Amini-Harandi et al. This is an open access article distributed under the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. <h4>Abstract</h4> A best proximity pair for a set-valued map F : A ⊸ B with respect to a map g : A → A is defined, and new existence theorems of best proximity pairs for upper semicontinuous set-valued maps with respect to a homeomorphism are proved in hyperconvex metric spaces. 1. Introduction and Preliminaries Let be a metric space and let and be nonempty subsets of . Let and let be a set-valued map. Now, is called a best proximity pair for with respect to if where Best proximity pair theorems establish conditions under which the problem of minimizing the real-valued function has a solution. In the setting of normed linear spaces, the best proximity pair problem has been studied by many authors for , see [ 1 – 5 ]. Very recently, Al-Thagafi and Shahzad [ 1 ] proved some existence theorems for a finite family of Kakutani set-valued maps in a normed space setting. In the present paper, our aim is to prove new results in hyperconvex metric spaces. In the rest of this section, we recall some definitions and theorems which are used in Section 2 . Let and be topological spaces with and . Let be a set-valued map with nonempty values. The image of under is the set and the inverse image of under is . Now, is said to be upper semicontinuous, if for each closed set , is closed in . A topological space is said to be contractible if the identity map of is homotopic to a constant map and acyclic if all of its reduced ech homology groups over the rationals vanish. Note that a contractible space is acyclic. For topological spaces and we define (1.1) We denote by the set of all finite composites of maps in . Let be a metric space and let denote the closed ball with center and radius . Let (1.2) If , we say that is admissible subset of . Note that is admissible and the intersection of any family of admissible subsets of is admissible. The following definition of a hyperconvex metric space is due to Aronszajn and Pantichpakdi [ 6 ]. Definition 1.1. A metric space is said to be a hyperconvex metric space if for any collection of points of and any collection of nonnegative real numbers with , one has (1.3) The simplest examples of hyperconvex spaces are finite dimensional real Banach spaces endowed with the maximum norm. For other examples of hyperconvex metric spaces which are not linear spaces, see [ 7 ]. Note that an admissible subset of a hyperconvex metric space is hyperconvex and contractible [ 8 ]. Let be a subset of . The -parallel of is defined as (1.4) The following result is due to Sine [ 9 ]. Lemma 1.2. The -parallel sets of an admissible subset of a hyperconvex metric space are also admissible sets. For , the set is called the set of best approximations in to . The map is called the metric projection on . The following lemma is well known. We give its proof for completeness. Lemma 1.3. Let be a nonempty, admissible, and compact subset of a hyperconvex metric space . Then . Proof. Since is compact, then is nonempty. We now show that is contractible and so is acyclic. To see this, notice that (1.5) Then is admissible (note that is admissible) and therefore is contractible. Now, we show that is upper semicontinuous. Let be a closed subset of , , and . Then there exists a sequence such that . Since is compact and , without loss of generality, we may assume that . Thus, (1.6) Therefore, and the set is closed. To prove our main result, we need the following fixed point theorem, which is particular form of Theorem 4 in [ 10 ]. Theorem 1.4. Let be a nonempty compact admissible subset of a hyperconvex metric space and . Then has a fixed point. Corollary 1.5. Let be a nonempty compact admissible subset of a hyperconvex metric space , a homeomorphism, and . Then there exists an such that . Proof. Since is a homeomorphism, then . Hence, by Theorem 1.4 , has a fixed point, say . Therefore, . 2. Best Proximity Theorems Let and be nonempty subsets of . Define (2.1) Notice that is nonempty if and only if is nonempty. Theorem 2.1. Let and be nonempty subsets of . Then the following statement holds. (i) If and are admissible, then is admissible. (ii) If and are compact, then is compact. Proof. To prove (a), notice that (2.2) Since is admissible, then by Lemma 1.2 , is also admissible. Thus, is admissible (note, that is admissible). (b) Let be a sequence in such that . Then there exists sequence in such that . Since is compact, we may assume that . Then (2.3) Thus, . Therefore, is closed and so compact. Theorem 2.2. Let be a hyperconvex metric space, and are admissible. Let be a homeomorphism, and let be an upper semicontinuous set-valued map with admissible values. Assume that is compact and admissible and is nonempty, for each . Then there exists such that . Proof. We use some ideas from [ 1 , Theorem 3.2]. From [ 11 , Proposition 2.8], is nonempty. Since and are admissible, it follows from Theorem 2.1 (a) that is admissible. From Lemma 1.3 , (note, that is nonempty, admissible, and compact and is hyperconvex since is an admissible subset of ). Define by . Since is upper semicontinuous with nonempty admissible values and is admissible, then is upper semicontinuous with admissible (in particular acyclic) values. From Lemma 1.3 (see proof), is upper semicontinuous with admissible (in particular acyclic) values. Since , it follows from Corollary 1.5 that there exists such that . Thus, there exists such that . Hence, and . Since , there exists such that and hence (2.4) Then (2.5) If we take , we get the following corollary. Corollary 2.3. Let be a hyperconvex metric space, and are admissible. Let be an upper semicontinuous set-valued map with admissible values. Assume that is nonempty, for each and is compact and admissible. Then there exists such that . Corollary 2.4 . (see [ 8 , Corollary 5.6]). Let be a hyperconvex metric space and nonempty, compact, and admissible. Let be an upper semicontinuous set-valued map with admissible values. Assume that is nonempty, for each . Then has a fixed point. <h4>References</h4> M. A. Al-Thagafi and N. Shahzad, “ Best proximity pairs and equilibrium pairs for Kakutani multimaps ,” Nonlinear Analysis: Theory, Methods & Applications , vol. 70, no. 3, pp. 1209–1216, 2009. W. K. Kim and K. H. Lee, “ Corrigendum to: “Existence of best proximity pairs and equilibrium pairs” ,” Journal of Mathematical Analysis and Applications , vol. 329, no. 2, pp. 1482–1483, 2007. W. K. Kim and K. H. Lee, “ Existence of best proximity pairs and equilibrium pairs ,” Journal of Mathematical Analysis and Applications , vol. 316, no. 2, pp. 433–446, 2006. S. Sadiq Basha and P. Veeramani, “ Best proximity pair theorems for multifunctions with open fibres ,” Journal of Approximation Theory , vol. 103, no. 1, pp. 119–129, 2000. P. S. Srinivasan and P. Veeramani, “ On best proximity pair theorems and fixed-point theorems ,” Abstract and Applied Analysis , vol. 2003, no. 1, pp. 33–47, 2003. N. Aronszajn and P. Panitchpakdi, “Extension of uniformly continuous transformations and hyperconvex metric spaces,” Pacific Journal of Mathematics , vol. 6, pp. 405–439, 1956. M. Borkowski, D. Bugajewski, and H. Przybycień, “Hyperconvex spaces revisited,” Bulletin of the Australian Mathematical Society , vol. 68, no. 2, pp. 191–203, 2003. J.-H. Kim and S. Park, “ Comments on some fixed point theorems in hyperconvex metric spaces ,” Journal of Mathematical Analysis and Applications , vol. 291, no. 1, pp. 154–164, 2004. R. Sine, “ Hyperconvexity and approximate fixed points ,” Nonlinear Analysis: Theory, Methods & Applications , vol. 13, no. 7, pp. 863–869, 1989. S. Park and H. Kim, “Coincidences of composites of u.s.c. maps on H -spaces and applications,” Journal of the Korean Mathematical Society , vol. 32, no. 2, pp. 251–264, 1995. W. A. Kirk, S. Reich, and P. Veeramani, “ Proximinal retracts and best proximity pair theorems ,” Numerical Functional Analysis and Optimization , vol. 24, no. 7-8, pp. 851–862, 2003. // http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Fixed Point Theory and Applications Hindawi Publishing Corporation

Best Proximity Pairs for Upper Semicontinuous Set-Valued Maps in Hyperconvex Metric Spaces

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Best Proximity Pairs for Upper Semicontinuous Set-Valued Maps in Hyperconvex Metric Spaces //// Hindawi Publishing Corporation Home Journals About Us About this Journal Submit a Manuscript Table of Contents Journal Menu Abstracting and Indexing Aims and Scope Article Processing Charges Articles in Press Author Guidelines Bibliographic Information Contact Information Editorial Board Editorial Workflow Free eTOC Alerts Reviewers Acknowledgment Subscription Information Open Special Issues Published Special Issues Special Issue Guidelines Abstract Full-Text PDF Full-Text HTML Linked References How to Cite this Article Fixed Point Theory and Applications Volume 2008 (2008), Article ID 648985, 5 pages doi:10.1155/2008/648985 Research Article <h2>Best Proximity Pairs for Upper Semicontinuous Set-Valued Maps in Hyperconvex Metric Spaces</h2> A. Amini-Harandi , 1 A. P. Farajzadeh , 2 D. O'Regan , 3 and R. P. Agarwal 4 1 Department of Mathematics, Faculty of Basic Sciences, University of Shahrekord, Shahrekord 88186-34141, Iran 2 Department of Mathematics, School of Science, Razi University, Kermanshah 67149, Iran 3 Department of Mathematics, College of Arts, Social Sciences and Celtic Studies, National University of Ireland, Galway, Ireland 4 Department of Mathematical Sciences, College of Science, Florida Institute of Technology, Melbourne, FL 32901, USA Received 14 July 2008; Accepted 27 October 2008 Academic Editor: Nan-jing Huang Copyright © 2008 A. Amini-Harandi et al. This is an open access article distributed under the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. <h4>Abstract</h4> A best proximity pair for a set-valued map F : A ⊸ B with respect to a map g : A → A is defined, and new existence theorems of best proximity pairs for upper semicontinuous set-valued maps with respect to a homeomorphism are proved in hyperconvex metric spaces. 1. Introduction and Preliminaries Let be a metric space and let and be nonempty subsets of . Let and let be a set-valued map. Now, is called a best proximity pair for with respect to if where Best proximity pair theorems establish conditions under which the problem of minimizing the real-valued function has a solution. In the setting of normed linear spaces, the best proximity pair problem has been studied by many authors for , see [ 1 – 5 ]. Very recently, Al-Thagafi and Shahzad [ 1 ] proved some existence theorems for a finite family of Kakutani set-valued maps in a normed space setting. In the present paper, our aim is to prove new results in hyperconvex metric spaces. In the rest of this section, we recall some definitions and theorems which are used in Section 2 . Let and be topological spaces with and . Let be a set-valued map with nonempty values. The image of under is the set and the inverse image of under is . Now, is said to be upper semicontinuous, if for each closed set , is closed in . A topological space is said to be contractible if the identity map of is homotopic to a constant map and acyclic if all of its reduced ech homology groups over the rationals vanish. Note that a contractible space is acyclic. For topological spaces and we define (1.1) We denote by the set of all finite composites of maps in . Let be a metric space and let denote the closed ball with center and radius . Let (1.2) If , we say that is admissible subset of . Note that is admissible and the intersection of any family of admissible subsets of is admissible. The following definition of a hyperconvex metric space is due to Aronszajn and Pantichpakdi [ 6 ]. Definition 1.1. A metric space is said to be a hyperconvex metric space if for any collection of points of and any collection of nonnegative real numbers with , one has (1.3) The simplest examples of hyperconvex spaces are finite dimensional real Banach spaces endowed with the maximum norm. For other examples of hyperconvex metric spaces which are not linear spaces, see [ 7 ]. Note that an admissible subset of a hyperconvex metric space is hyperconvex and contractible [ 8 ]. Let be a subset of . The -parallel of is defined as (1.4) The following result is due to Sine [ 9 ]. Lemma 1.2. The -parallel sets of an admissible subset of a hyperconvex metric space are also admissible sets. For , the set is called the set of best approximations in to . The map is called the metric projection on . The following lemma is well known. We give its proof for completeness. Lemma 1.3. Let be a nonempty, admissible, and compact subset of a hyperconvex metric space . Then . Proof. Since is compact, then is nonempty. We now show that is contractible and so is acyclic. To see this, notice that (1.5) Then is admissible (note that is admissible) and therefore is contractible. Now, we show that is upper semicontinuous. Let be a closed subset of , , and . Then there exists a sequence such that . Since is compact and , without loss of generality, we may assume that . Thus, (1.6) Therefore, and the set is closed. To prove our main result, we need the following fixed point theorem, which is particular form of Theorem 4 in [ 10 ]. Theorem 1.4. Let be a nonempty compact admissible subset of a hyperconvex metric space and . Then has a fixed point. Corollary 1.5. Let be a nonempty compact admissible subset of a hyperconvex metric space , a homeomorphism, and . Then there exists an such that . Proof. Since is a homeomorphism, then . Hence, by Theorem 1.4 , has a fixed point, say . Therefore, . 2. Best Proximity Theorems Let and be nonempty subsets of . Define (2.1) Notice that is nonempty if and only if is nonempty. Theorem 2.1. Let and be nonempty subsets of . Then the following statement holds. (i) If and are admissible, then is admissible. (ii) If and are compact, then is compact. Proof. To prove (a), notice that (2.2) Since is admissible, then by Lemma 1.2 , is also admissible. Thus, is admissible (note, that is admissible). (b) Let be a sequence in such that . Then there exists sequence in such that . Since is compact, we may assume that . Then (2.3) Thus, . Therefore, is closed and so compact. Theorem 2.2. Let be a hyperconvex metric space, and are admissible. Let be a homeomorphism, and let be an upper semicontinuous set-valued map with admissible values. Assume that is compact and admissible and is nonempty, for each . Then there exists such that . Proof. We use some ideas from [ 1 , Theorem 3.2]. From [ 11 , Proposition 2.8], is nonempty. Since and are admissible, it follows from Theorem 2.1 (a) that is admissible. From Lemma 1.3 , (note, that is nonempty, admissible, and compact and is hyperconvex since is an admissible subset of ). Define by . Since is upper semicontinuous with nonempty admissible values and is admissible, then is upper semicontinuous with admissible (in particular acyclic) values. From Lemma 1.3 (see proof), is upper semicontinuous with admissible (in particular acyclic) values. Since , it follows from Corollary 1.5 that there exists such that . Thus, there exists such that . Hence, and . Since , there exists such that and hence (2.4) Then (2.5) If we take , we get the following corollary. Corollary 2.3. Let be a hyperconvex metric space, and are admissible. Let be an upper semicontinuous set-valued map with admissible values. Assume that is nonempty, for each and is compact and admissible. Then there exists such that . Corollary 2.4 . (see [ 8 , Corollary 5.6]). Let be a hyperconvex metric space and nonempty, compact, and admissible. Let be an upper semicontinuous set-valued map with admissible values. Assume that is nonempty, for each . Then has a fixed point. <h4>References</h4> M. A. Al-Thagafi and N. Shahzad, “ Best proximity pairs and equilibrium pairs for Kakutani multimaps ,” Nonlinear Analysis: Theory, Methods & Applications , vol. 70, no. 3, pp. 1209–1216, 2009. W. K. Kim and K. H. Lee, “ Corrigendum to: “Existence of best proximity pairs and equilibrium pairs” ,” Journal of Mathematical Analysis and Applications , vol. 329, no. 2, pp. 1482–1483, 2007. W. K. Kim and K. H. Lee, “ Existence of best proximity pairs and equilibrium pairs ,” Journal of Mathematical Analysis and Applications , vol. 316, no. 2, pp. 433–446, 2006. S. Sadiq Basha and P. Veeramani, “ Best proximity pair theorems for multifunctions with open fibres ,” Journal of Approximation Theory , vol. 103, no. 1, pp. 119–129, 2000. P. S. Srinivasan and P. Veeramani, “ On best proximity pair theorems and fixed-point theorems ,” Abstract and Applied Analysis , vol. 2003, no. 1, pp. 33–47, 2003. N. Aronszajn and P. Panitchpakdi, “Extension of uniformly continuous transformations and hyperconvex metric spaces,” Pacific Journal of Mathematics , vol. 6, pp. 405–439, 1956. M. Borkowski, D. Bugajewski, and H. Przybycień, “Hyperconvex spaces revisited,” Bulletin of the Australian Mathematical Society , vol. 68, no. 2, pp. 191–203, 2003. J.-H. Kim and S. Park, “ Comments on some fixed point theorems in hyperconvex metric spaces ,” Journal of Mathematical Analysis and Applications , vol. 291, no. 1, pp. 154–164, 2004. R. Sine, “ Hyperconvexity and approximate fixed points ,” Nonlinear Analysis: Theory, Methods & Applications , vol. 13, no. 7, pp. 863–869, 1989. S. Park and H. Kim, “Coincidences of composites of u.s.c. maps on H -spaces and applications,” Journal of the Korean Mathematical Society , vol. 32, no. 2, pp. 251–264, 1995. W. A. Kirk, S. Reich, and P. Veeramani, “ Proximinal retracts and best proximity pair theorems ,” Numerical Functional Analysis and Optimization , vol. 24, no. 7-8, pp. 851–862, 2003. //

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