Journal articles: 'Hadamard spaces'

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Moslemipour, Ali, Mehdi Roohi, Mohammad Mardanbeigi, and Mahdi Azhini. "Monotone relations in Hadamard spaces." Filomat 33, no. 19 (2019): 6347–58. http://dx.doi.org/10.2298/fil1919347m.

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In this paper, the notion of W-property for subsets of X x X? is introduced and investigated, where X is an Hadamard space and X? is its linear dual space. It is shown that an Hadamard space X is flat if and only if X x X? has W-property. Moreover, the notion of monotone relation from an Hadamard space to its linear dual space is introduced. A characterization result for monotone relations with W-property (and hence in flat Hadamard spaces) is given. Finally, a type of Debrunner-Flor Lemma concerning extension of monotone relations in Hadamard spaces is proved.

2

LANG, U., and V. SCHROEDER. "Quasiflats in Hadamard spaces." Annales Scientifiques de l’École Normale Supérieure 30, no. 3 (1997): 339–52. http://dx.doi.org/10.1016/s0012-9593(97)89923-5.

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Hirayama, Michihiro. "Entropy of Hadamard spaces." Topology and its Applications 158, no. 4 (March 2011): 627–35. http://dx.doi.org/10.1016/j.topol.2010.12.011.

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Yakhshiboyev, M. U. "On Boundedness of Fractional Hadamard Integration and Hadamard-Type Integration in LebesgueSpaces with Mixed Norm." Contemporary Mathematics. Fundamental Directions 68, no. 1 (April 20, 2022): 178–89. http://dx.doi.org/10.22363/2413-3639-2022-68-1-178-189.

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In this paper, we consider the boundedness of integrals of fractional Hadamard integration and Hadamard-type integration (mixed and directional) in Lebesgue spaces with mixed norm. We prove Sobolev-type theorems of boundedness of one-dimensional and multidimensional Hadamard-type fractional integration in weighted Lebesgue spaces with mixed norm.

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Yakhshiboyev, M. U. "On Boundedness of Fractional Hadamard Integration and Hadamard-Type Integration in LebesgueSpaces with Mixed Norm." Contemporary Mathematics. Fundamental Directions 68, no. 1 (April 20, 2022): 178–89. http://dx.doi.org/10.22363/2413-3639-2022-68-1-178-189.

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In this paper, we consider the boundedness of integrals of fractional Hadamard integration and Hadamard-type integration (mixed and directional) in Lebesgue spaces with mixed norm. We prove Sobolev-type theorems of boundedness of one-dimensional and multidimensional Hadamard-type fractional integration in weighted Lebesgue spaces with mixed norm.

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Berdellima, Arian. "On a weak topology for Hadamard spaces." Sbornik: Mathematics 214, no. 10 (2023): 1373–89. http://dx.doi.org/10.4213/sm9808e.

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We investigate whether the existing notion of weak sequential convergence in Hadamard spaces can be induced by a topology. We provide an affirmative answer in what we call weakly proper Hadamard spaces. Several results from functional analysis are extended to the setting of Hadamard spaces. Our weak topology coincides with the usual one in the case of a Hilbert space. Finally, we compare our topology with other existing notions of weak topologies. Bibliography: 24 titles.

7

Wulan, Hasi, and Yanhua Zhang. "Hadamard products and QK spaces." Journal of Mathematical Analysis and Applications 337, no. 2 (January 2008): 1142–50. http://dx.doi.org/10.1016/j.jmaa.2007.04.020.

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Bocci, Cristiano, Enrico Carlini, and Joe Kileel. "Hadamard products of linear spaces." Journal of Algebra 448 (February 2016): 595–617. http://dx.doi.org/10.1016/j.jalgebra.2015.10.008.

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Pavlović, Miroslav. "Hadamard product in Qp spaces." Journal of Mathematical Analysis and Applications 305, no. 2 (May 2005): 589–98. http://dx.doi.org/10.1016/j.jmaa.2004.12.040.

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Alexander, Stephanie B., and Richard L. Bishop. "Warped products of Hadamard spaces." manuscripta mathematica 96, no. 4 (August 1, 1998): 487–505. http://dx.doi.org/10.1007/s002290050078.

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BERTRAND, JÉRÔME, and BENOÎT KLOECKNER. "A GEOMETRIC STUDY OF WASSERSTEIN SPACES: HADAMARD SPACES." Journal of Topology and Analysis 04, no. 04 (December 2012): 515–42. http://dx.doi.org/10.1142/s1793525312500227.

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Optimal transport enables one to construct a metric on the set of (sufficiently small at infinity) probability measures on any (not too wild) metric space X, called its Wasserstein space [Formula: see text]. In this paper we investigate the geometry of [Formula: see text] when X is a Hadamard space, by which we mean that X has globally non-positive sectional curvature and is locally compact. Although it is known that, except in the case of the line, [Formula: see text] is not non-positively curved, our results show that [Formula: see text] have large-scale properties reminiscent of that of X. In particular we define a geodesic boundary for [Formula: see text] that enables us to prove a non-embeddablity result: if X has the visibility property, then the Euclidean plane does not admit any isometric embedding in [Formula: see text].

12

Giles, J. R., and Scott Sciffer. "On weak Hadamard differentiability of convex functions on Banach spaces." Bulletin of the Australian Mathematical Society 54, no. 1 (August 1996): 155–66. http://dx.doi.org/10.1017/s000497270001515x.

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We study two variants of weak Hadamard differentiability of continuous convex functions on a Banach space, uniform weak Hadamard differentiability and weak Hadamard directional differentiability, and determine their special properties on Banach spaces which do not contain a subspace topologically isomorphic to l1.

13

Turcanu, Teodor, and Mihai Postolache. "On Enriched Suzuki Mappings in Hadamard Spaces." Mathematics 12, no. 1 (January 3, 2024): 157. http://dx.doi.org/10.3390/math12010157.

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We define and study enriched Suzuki mappings in Hadamard spaces. The results obtained here are extending fundamental findings previously established in related research. The extension is realized with respect to at least two different aspects: the setting and the class of involved operators. More accurately, Hilbert spaces are particular Hadamard spaces, while enriched Suzuki nonexpansive mappings are natural generalizations of enriched nonexpansive mappings. Next, enriched Suzuki nonexpansive mappings naturally contain Suzuki nonexpansive mappings in Hadamard spaces. Besides technical lemmas, the results of this paper deal with (1) the existence of fixed points for enriched Suzuki nonexpansive mappings and (2) Δ and strong (metric) convergence of Picard iterates of the α-averaged mapping, which are exactly Krasnoselskij iterates for the original mapping.

14

Mleczko, Paweł. "Hadamard Multipliers and Abel Dual of Hardy Spaces." Journal of Function Spaces 2016 (2016): 1–7. http://dx.doi.org/10.1155/2016/3262761.

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The paper is devoted to the study of Hadamard multipliers of functions from the abstract Hardy classes generated by rearrangement invariant spaces. In particular the relation between the existence of such multiplier and the boundedness of the appropriate convolution operator on spaces of measurable functions is presented. As an application, the description of Hadamard multipliers intoH∞is given and the Abel type theorem for mentioned Hardy spaces is proved.

15

NAGANO, Koichi. "Asymptotic rigidity of Hadamard 2-spaces." Journal of the Mathematical Society of Japan 52, no. 4 (October 2000): 699–723. http://dx.doi.org/10.2969/jmsj/05240699.

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Wan, Lili. "Composite Minimization Problems in Hadamard Spaces." Journal of Applied Mathematics and Physics 08, no. 04 (2020): 597–608. http://dx.doi.org/10.4236/jamp.2020.84046.

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Banert, Sebastian. "Backward–backward splitting in Hadamard spaces." Journal of Mathematical Analysis and Applications 414, no. 2 (June 2014): 656–65. http://dx.doi.org/10.1016/j.jmaa.2014.01.054.

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Eikrem, Kjersti Solberg. "Hadamard gap series in growth spaces." Collectanea Mathematica 64, no. 1 (June 4, 2012): 1–15. http://dx.doi.org/10.1007/s13348-012-0065-0.

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19

Adams, Scot, and Werner Ballmann. "Amenable isometry groups of Hadamard spaces." Mathematische Annalen 312, no. 1 (September 1, 1998): 183–96. http://dx.doi.org/10.1007/s002080050218.

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Karapetrović, Boban, and Javad Mashreghi. "Hadamard products in weighted Bergman spaces." Journal of Mathematical Analysis and Applications 494, no. 2 (February 2021): 124617. http://dx.doi.org/10.1016/j.jmaa.2020.124617.

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Moslemipour, A., M. Roohi, M. R. Mardanbeigi, and M. Azhini. "Fitzpatrick transform of monotone relations in Hadamard spaces." Analele Universitatii "Ovidius" Constanta - Seria Matematica 28, no. 2 (July 1, 2020): 173–93. http://dx.doi.org/10.2478/auom-2020-0026.

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AbstractIn the present paper, monotone relations and maximal monotone relations from an Hadamard space to its linear dual space are investigated. Fitzpatrick transform of monotone relations in Hadamard spaces is introduced. It is shown that Fitzpatrick transform of a special class of monotone relations is proper, convex and lower semi-continuous. Finally, a representation result for monotone relations is given.

22

Ogwo, Grace N., Chinedu Izuchukwu, Kazeem O. Aremu, and Oluwatosin T. Mewomo. "On θ-generalized demimetric mappings and monotone operators in Hadamard spaces." Demonstratio Mathematica 53, no. 1 (July 3, 2020): 95–111. http://dx.doi.org/10.1515/dema-2020-0006.

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AbstractOur main interest in this article is to introduce and study the class of θ-generalized demimetric mappings in Hadamard spaces. Also, a Halpern-type proximal point algorithm comprising this class of mappings and resolvents of monotone operators is proposed, and we prove that it converges strongly to a fixed point of a θ-generalized demimetric mapping and a common zero of a finite family of monotone operators in a Hadamard space. Furthermore, we apply the obtained results to solve a finite family of convex minimization problems, variational inequality problems and convex feasibility problems in Hadamard spaces.

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Moslemipour, Ali, and Mehdi Roohi. "Monotonicity of sets in Hadamard spaces from polarity point of view." Filomat 36, no. 13 (2022): 4459–70. http://dx.doi.org/10.2298/fil2213459m.

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This paper is devoted to introduce and investigate the notion of monotone sets in Hadamard spaces. First, flat Hadamard spaces are introduced and investigated. It is shown that an Hadamard space X is flat if and only if X ? X? has Fl-property, where X? is the linear dual of X. Moreover, monotone and maximal monotone sets are introduced and also monotonicity from polarity point of view is considered. Some characterizations of (maximal) monotone sets, specially based on polarity, are given. Finally, it is proved that any maximal monotone set is sequentially bw?????-closed in X ? X?.

24

Ceng, Lu-Chuan, Yeong-Cheng Liou, Ching-Feng Wen, Hui-Ying Hu, Long He, and Yun-Ling Cui. "On the Hadamard Well-Posedness of Generalized Mixed Variational Inequalities in Banach Spaces." Journal of Mathematics 2021 (November 18, 2021): 1–9. http://dx.doi.org/10.1155/2021/9527107.

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We introduce a new concept of Hadamard well-posedness of a generalized mixed variational inequality in a Banach space. The relations between the Levitin–Polyak well-posedness and Hadamard well-posedness for a generalized mixed variational inequality are studied. The characterizations of Hadamard well-posedness for a generalized mixed variational inequality are established.

25

Ruiz-Garzón, Gabriel, Rafaela Osuna-Gómez, and Jaime Ruiz-Zapatero. "Necessary and Sufficient Optimality Conditions for Vector Equilibrium Problems on Hadamard Manifolds." Symmetry 11, no. 8 (August 12, 2019): 1037. http://dx.doi.org/10.3390/sym11081037.

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The aim of this paper is to show the existence and attainability of Karush–Kuhn–Tucker optimality conditions for weakly efficient Pareto points for vector equilibrium problems with the addition of constraints in the novel context of Hadamard manifolds, as opposed to the classical examples of Banach, normed or Hausdorff spaces. More specifically, classical necessary and sufficient conditions for weakly efficient Pareto points to the constrained vector optimization problem are presented. The results described in this article generalize results obtained by Gong (2008) and Wei and Gong (2010) and Feng and Qiu (2014) from Hausdorff topological vector spaces, real normed spaces, and real Banach spaces to Hadamard manifolds, respectively. This is done using a notion of Riemannian symmetric spaces of a noncompact type as special Hadarmard manifolds.

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HARUNA, LAWAL YUSUF, GODWIN CHIDI UGWUNNADI, and BASHIR ALI. "On the Proximal Point Algorithm of Hybrid-Type in Flat Hadamard Spaces with Applications." Kragujevac Journal of Mathematics 48, no. 6 (2024): 845–57. http://dx.doi.org/10.46793/kgjmat2406.845h.

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In this paper, we introduce a hybrid-type proximal point algorithm for approximating zero of monotone operator in Hadamard-type spaces. We then prove that a sequence generated by the algorithm involving Mann-type iteration converges strongly to a zero of the said operator in the setting of ﬂat Hadamard spaces. To the best of our knowledge, this result presents the ﬁrst hybrid-type proximal point algorithm in the space. The result is applied to convex minimization and ﬁxed point problems.

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Bačák, Miroslav. "Computing Medians and Means in Hadamard Spaces." SIAM Journal on Optimization 24, no. 3 (January 2014): 1542–66. http://dx.doi.org/10.1137/140953393.

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Zhou, Jizhen. "Hadamard products inF(p,q,s) spaces." Journal of Function Spaces and Applications 8, no. 3 (2010): 257–70. http://dx.doi.org/10.1155/2010/765341.

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Salem, Hussein A. H. "Hadamard-type fractional calculus in Banach spaces." Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas 113, no. 2 (March 30, 2018): 987–1006. http://dx.doi.org/10.1007/s13398-018-0531-y.

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Lang, U., B. Pavlović, and V. Schroeder. "Extensions of Lipschitz maps into Hadamard spaces." Geometric and Functional Analysis 10, no. 6 (December 2000): 1527–53. http://dx.doi.org/10.1007/pl00001660.

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Ugwunnadi, Godwin Chidi, Chinedu Izuchkwu, and Oluwatosin Temitope Mewomo. "On nonspreading-type mappings in Hadamard spaces." Boletim da Sociedade Paranaense de Matemática 39, no. 5 (January 1, 2021): 175–97. http://dx.doi.org/10.5269/bspm.41768.

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In this paper, we studied a new class of nonspreading-type mappings more general than the class of strictly pseudononspreading and the class of generalized nonspreading mappings. We state and prove some strong convergence theorems of the Mann-type and Ishikawa-type algorithms for approximating fixed points of our class of mappings in Hadamard spaces.

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Hruska, G. Christopher, and Bruce Kleiner. "Erratum to “Hadamard spaces with isolated flats”." Geometry & Topology 13, no. 2 (January 1, 2009): 699–707. http://dx.doi.org/10.2140/gt.2009.13.699.

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Izuchukwu, Chinedu, Kazeem Olalekan Aremu, Olawale Kazeem Oyewole, Oluwatosin Temitope Mewomo, and Safeer Hussain Khan. "On Mixed Equilibrium Problems in Hadamard Spaces." Journal of Mathematics 2019 (October 13, 2019): 1–13. http://dx.doi.org/10.1155/2019/3210649.

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The main purpose of this paper is to study mixed equilibrium problems in Hadamard spaces. First, we establish the existence of solution of the mixed equilibrium problem and the unique existence of the resolvent operator for the problem. We then prove a strong convergence of the resolvent and a Δ-convergence of the proximal point algorithm to a solution of the mixed equilibrium problem under some suitable conditions. Furthermore, we study the asymptotic behavior of the sequence generated by a Halpern-type PPA. Finally, we give a numerical example in a nonlinear space setting to illustrate the applicability of our results. Our results extend and unify some related results in the literature.

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Trybuła, Maria. "Hadamard Multipliers on Spaces of Holomorphic Functions." Integral Equations and Operator Theory 88, no. 2 (April 27, 2017): 249–68. http://dx.doi.org/10.1007/s00020-017-2369-7.

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Бeрделлима, Ариан, and Arian Berdellima. "On a weak topology for Hadamard spaces." Математический сборник 214, no. 10 (2023): 25–43. http://dx.doi.org/10.4213/sm9808.

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Изучается вопрос о топологизуемости секвенциальной слабой сходимости в пространствах Адамара. Положительный ответ на этот вопрос дается для так называемых слабо собственных пространств Адамара. Ряд результатов из функционального анализа переносится на случай пространств Адамара. Показывается, что вводимое определение слабой топологии совпадает с обычным определением слабой топологии для гильбертовых пространств. Дается сравнение вводимой топологии с существующими определениями других слабых топологий. Библиография: 24 названия.

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Bačák, Miroslav. "Old and new challenges in Hadamard spaces." Japanese Journal of Mathematics 18, no. 2 (September 2023): 115–68. http://dx.doi.org/10.1007/s11537-023-1826-0.

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37

Dragomir, S. S. "Inequalities of Hermite-Hadamard Type." Moroccan Journal of Pure and Applied Analysis 1, no. 1 (June 1, 2015): 1–21. http://dx.doi.org/10.7603/s40956-015-0001-x.

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AbstractSome inequalities of Hermite-Hadamard type for λ-convex functions defined on convex subsets in real or complex linear spaces are given. Applications for norm inequalities are provided as well.

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WOLTER, THOMAS H. "GEOMETRY OF HOMOGENEOUS HADAMARD MANIFOLDS." International Journal of Mathematics 02, no. 02 (April 1991): 223–34. http://dx.doi.org/10.1142/s0129167x91000430.

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Using result of R. Azencott and E. Wilson on the algebraic structure of homogeneous manifolds of nonpositive sectional curvature, we discuss the geometry of these manifolds and give several conditions how to distinguish the symmetric spaces among them.

39

Stojiljković, Vuk, Rajagopalan Ramaswamy, Ola A. Ashour Abdelnaby, and Stojan Radenović. "Some Refinements of the Tensorial Inequalities in Hilbert Spaces." Symmetry 15, no. 4 (April 16, 2023): 925. http://dx.doi.org/10.3390/sym15040925.

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Hermite–Hadamard inequalities and their refinements have been investigated for a long period of time. In this paper, we obtained refinements of the Hermite–Hadamard inequality of tensorial type for the convex functions of self-adjoint operators in Hilbert spaces. The obtained inequalities generalize the previously obtained inequalities by Dragomir. We also provide useful Lemmas which enabled us to obtain the results. The examples of the obtained inequalities for specific convex functions have been given in the example and consequences section. Symmetry in the upper and lower bounds can be seen in the last Theorem of the paper given, as the upper and lower bounds differ by a constant.

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Liu, Weiwei, and Lishan Liu. "Properties of Hadamard Fractional Integral and Its Application." Fractal and Fractional 6, no. 11 (November 13, 2022): 670. http://dx.doi.org/10.3390/fractalfract6110670.

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We begin by introducing some function spaces Lcp(R+),Xcp(J) made up of integrable functions with exponent or power weights defined on infinite intervals, and then we investigate the properties of Mellin convolution operators mapping on these spaces, next, we derive some new boundedness and continuity properties of Hadamard integral operators mapping on Xcp(J) and Xp(J). Based on this, we investigate a class of boundary value problems for Hadamard fractional differential equations with the integral boundary conditions and the disturbance parameters, and obtain uniqueness results for positive solutions to the boundary value problem under some weaker conditions.

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SALISU, SANI, POOM KUMAM, and SONGPON SRIWONGSA. "On Fixed Points of Enriched Contractions and Enriched Nonexpansive Mappings." Carpathian Journal of Mathematics 39, no. 1 (July 30, 2022): 237–54. http://dx.doi.org/10.37193/cjm.2023.01.16.

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We apply the concept of quasilinearization to introduce some enriched classes of Banach contraction mappings and analyse the fixed points of such mappings in the setting of Hadamard spaces. We establish existence and uniqueness of the fixed point of such mappings. To approximate the fixed points, we use an appropriate Krasnoselskij-type scheme for which we establish $\Delta$ and strong convergence theorems. Furthermore, we discuss the fixed points of local enriched contractions and Maia-type enriched contractions in Hadamard spaces setting. In addition, we establish demiclosedness-type property of enriched nonexpansive mappings. Finally, we present some special cases and corresponding fixed point theorems.

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Eftekharinasab, Kaveh. "A Version of the Hadamard–Lévy Theorem for Fréchet Spaces." Proceedings of the Bulgarian Academy of Sciences 75, no. 8 (August 31, 2022): 1099–104. http://dx.doi.org/10.7546/crabs.2022.08.01.

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Borwein, J. M., and M. Fabian. "On Convex Functions Having Points of Gateaux Differentiability Which are Not Points of Fréchet Differentiability." Canadian Journal of Mathematics 45, no. 6 (December 1, 1993): 1121–34. http://dx.doi.org/10.4153/cjm-1993-062-8.

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AbstractWe study the relationships between Gateaux, Fréchet and weak Hadamard differentiability of convex functions and of equivalent norms. As a consequence we provide related characterizations of infinite dimensional Banach spaces and of Banach spaces containing ł1. Explicit examples are given. Some renormings of WCG Asplund spaces are made in this vein.

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Fagbemigun, B. O., A. A. Mogbademu, and J. O. Olaleru. "Double Integral Inequalities of Hermite-Hadamard-Type for ɸh-Convex Functions on Linear Spaces." Journal of Nepal Mathematical Society 2, no. 2 (December 2, 2019): 11–18. http://dx.doi.org/10.3126/jnms.v2i2.33006.

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The concept of Φh-convexity is extended for functions defined on closed Φh-convex subsets of linear spaces. Consequently, some double integral inequalities of Hermite-Hadamard type defined on time-scaled linear spaces are established for Φh-convex functions.

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Abbas, Saïd, Mouffak Benchohra, Naima Hamidi, and Johnny Henderson. "Caputo-Hadamard fractional differential equations in banach spaces." Fractional Calculus and Applied Analysis 21, no. 4 (August 28, 2018): 1027–45. http://dx.doi.org/10.1515/fca-2018-0056.

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Abstract This article deals with some existence results for a class of Caputo–Hadamard fractional differential equations. The results are based on the Mönch’s fixed point theorem associated with the technique of measure of noncompactness. Two illustrative examples are presented.

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UGWUNNADI, G. C., C. C. OKEKE, A. R. KHAN, and L. O. JOLAOSO. "Strong Convergence Results for Variational Inequality and Equilibrium Problem in Hadamard Spaces." Kragujevac Journal of Mathematics 47, no. 6 (December 2023): 825–45. http://dx.doi.org/10.46793/kgjmat2306.825u.

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The main purpose of this paper is to introduce and study a viscosity type algorithm in a Hadamard space which comprises of a demimetric mapping, a ﬁnite family of inverse strongly monotone mappings and an equilibrium problem for a bifunction. Strong convergence of the proposed algorithm to a common solution of variational inequality problem, ﬁxed point problem and equilibrium problem is established in Hadamard spaces. Nontrivial Applications and numerical examples were given. Our results compliment some results in the literature.

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Yee, Tat-Leung, and Kwok-Pun Ho. "Hardy’s inequalities and integral operators on Herz-Morrey spaces." Open Mathematics 18, no. 1 (March 10, 2020): 106–21. http://dx.doi.org/10.1515/math-2020-0008.

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Abstract We obtain some estimates for the operator norms of the dilation operators on Herz-Morrey spaces. These results give us the Hardy’s inequalities and the mapping properties of the integral operators on Herz-Morrey spaces. As applications of this general result, we have the boundedness of the Hadamard fractional integrals on Herz-Morrey spaces. We also obtain the Hilbert inequality on Herz-Morrey spaces.

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Dehghani, Mahdi. "Characterization of inner product spaces by unitary Carlsson type orthogonalities." Filomat 36, no. 14 (2022): 4655–63. http://dx.doi.org/10.2298/fil2214655d.

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In this study, we consider the Hermite-Hadamard type of unitary Carlsson?s orthogonality (UHH-C-orthogonality) to characterize real inner product spaces. We give a necessary and sufficient condition weaker than the homogeneity of symmetric HH-C-orthogonalities which characterizes inner product spaces among normed linear spaces of dimension at least three. In conclusion, some more characterizations of real inner product spaces are provided.

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Render, Hermann, and Andreas Sauer. "Multipliers on vector spaces of holomorphic functions." Nagoya Mathematical Journal 159 (2000): 167–78. http://dx.doi.org/10.1017/s0027763000007467.

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Let G be a domain in the complex plane containing zero and H(G) be the set of all holomorphic functions on G. In this paper the algebra M(H(G)) of all coefficient multipliers with respect to the Hadamard product is studied. Central for the investigation is the domain introduced by Arakelyan which is by definition the union of all sets with w ∈ Gc. The main result is the description of all isomorphisms between these multipliers algebras. As a consequence one obtains: If two multiplier algebras M(H(G1)) and M(H(G2)) are isomorphic then is equal to Two algebras H(G1) and H(G2) are isomorphic with respect to the Hadamard product if and only if G1 is equal to G2. Further the following uniqueness theorem is proved: If G1 is a domain containing 0 and if M(H(G)) is isomorphic to H(G1) then G1 is equal to .

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Vedel, Ya I., and V. V. Semenov. "Adaptive algorithms for equilibrium problems in Hadamard spaces." Reports of the National Academy of Sciences of Ukraine, no. 8 (August 2020): 26–34. http://dx.doi.org/10.15407/dopovidi2020.08.026.

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Journal articles on the topic 'Hadamard spaces'

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