Nonself KKM Maps and Corresponding Theorems in Hadamard Manifolds

[EN] In this paper, we consider the KKM maps defined for a nonself map and the correlated intersection theorems in Hadamard manifolds. We also study some applications of the intersection results. Our outputs improved the results of Raj and Somasundaram [17, V. Sankar Raj and S. Somasundaram, KKM-type theorems for best proximity points, Appl. Math. Lett., 25(3): 496–499, 2012.].The first author is supported by the Thailand Research Fund through the Royal Golden Jubilee Ph.D. Program (Grant No. PHD/0045/2555) and the King Mongkut’s University of Technology Thonburi under the RGJ-Ph.D. scholarship. Moreover, we would like to gratefully thank the anonymous referees for their suggestions, which improve this paper signifi- cantlyChaipunya, P.; Kumam, P. (2015). Nonself KKM Maps and Corresponding Theorems in Hadamard Manifolds. Applied General Topology. 16(1):37-44. https://doi.org/10.4995/agt.2015.2305SWORD3744161Bigi, G., Capătă, A., & Kassay, G. (2012). Existence results for strong vector equilibrium problems and their applications. Optimization, 61(5), 567-583. doi:10.1080/02331934.2010.528761E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems, The Mathematics Student 63, no. 1-4 (1994), 123-145.W. Oettli, A remark on vector-valued equilibria and generalized monotonicity, Acta Math. Vietnam. 22, no. 1 (1997), 213-221.A. Papadopoulos, Metric Spaces, Convexity and Nonpositive Curvature, IRMA lectures in mathematics and theoretical physics, European Mathematical Society, 2005.S. Park, Some coincidence theorems on acyclic multifunctions and applications to kkm theory, Fixed Point Theory and Applications (1992), pp. 248-277.S. Park, Ninety years of the brouwer fixed point theorem, Vietnam J. Math. 27 (1997), 187-222.Sturm, K.-T. (2003). Probability measures on metric spaces of nonpositive curvature. Contemporary Mathematics, 357-390. doi:10.1090/conm/338/0608

Metric fixed point theory on hyperconvex spaces: recent progress

In this survey we present an exposition of the development during the last decade of metric fixed point theory on hyperconvex metric spaces. Therefore we mainly cover results where the conditions on the mappings are metric. We will recall results about proximinal nonexpansive retractions and their impact into the theory of best approximation and best proximity pairs. A central role in this survey will be also played by some recent developments on R-trees. Finally, some considerations and new results on the extension of compact mappings will be shown

Dynamic Processes, Fixed Points, Endpoints, Asymmetric Structures, and Investigations Related to Caristi, Nadler, and Banach in Uniform Spaces

Research ArticleIn uniform spaces (...) with symmetric structures determined by the D-families of pseudometrics which define uniformity in these spaces, the new symmetric and asymmetric structures determined by the J-families of generalized pseudodistances on (...) are constructed; using these structures the set-valued contractions of two kinds of Nadler type are defined and the new and general theorems concerning the existence of fixed points and endpoints for such contractions are proved. Moreover, using these new structures, the single-valued contractions of two kinds of Banach type are defined and the new and general versions of the Banach uniqueness and iterate approximation of fixed point theorem for uniform spaces are established. Contractions defined and studied here are not necessarily continuous. One of the main key ideas in this paper is the application of our fixed point and endpoint version of Caristi type theorem for dissipative set-valued dynamic systems without lower semicontinuous entropies in uniform spaces with structures determined by J-families. Results are new also in locally convex and metric spaces. Examples are provided

On a dual characterisation in best approximation problem

We establish a dual characterization of solutions of Ky Fan best approximation problem and as consequence we obtain an existence criterium under conditions formulated for the weak topology

Best approximation and fixed-point theorems for discontinuous increasing maps in Banach lattices

In this paper, we extend and prove Ky Fan’s Theorem for discontinuous increasing maps f in a Banach lattice X when f has no compact conditions. The main tools of analysis are the variational characterization of the generalized projection operator and order-theoretic fixed-point theory. Moreover, we establish a sequence {xn} which converges strongly to the unique best approximation point. As an application of our best approximation theorems, a fixed-point theorem for non-self maps is established and proved under some conditions. Our results generalize and improve many recent results obtained by many authors

Boyd-Wong contractions in F-metric spaces and applications

[EN] The main aim of this paper is to  study the Boyd-Wong type fixed point result in the  F-metric context and apply it to obtain  some existence and uniqueness criteria of solution(s) to a second order initial value problem and a Caputo fractional differential equation. We substantiate our obtained result  by finding a suitable non-trivial example.The Research is funded by the Ministry of Human Resource and Development, Government of India and by the Council of Scientific and Industrial Research (CSIR), Government of India under the Grant Number: 25(0285)/18/EMR-II. This project is funded by National Research Council of Thailand (NRCT) N41A640092.Bera, A.; Dey, LK.; Som, S.; Garai, H.; Sintunavarat, W. (2022). Boyd-Wong contractions in F-metric spaces and applications. Applied General Topology. 23(1):157-167. https://doi.org/10.4995/agt.2022.1535615716723

EXISTENCE OF BEST PROXIMITY POINTS: GLOBAL OPTIMAL APPROXIMATE SOLUTION

Abstract. Given non-empty subsets A and B of a metric space, let S : A → B and T : A → B be non-self mappings. Taking into account the fact that, given any element x in A, the distance between x and Sx, and the distance between x and T x are at least d(A, B), a common best proximity point theorem affirms global minimum of both functions x → d(x, Sx) and x → d(x, T x) by imposing a common approximate solution of the equations Sx = x and T x = x to satisfy the constraint that d(x, Sx) = d(x, T x) = d(A, B). In this work we introduce a new notion of proximally dominating type mappings and derive a common best proximity point theorem for proximally commuting non-self mappings, thereby producing common optimal approximate solutions of certain simultaneous fixed point equations when there is no common solution. We furnish suitable examples to demonstrate the validity of the hypotheses of our results