Fixed Point Results and the Ekeland Variational Principle in Vector B-Metric Spaces

1 month ago

(preprint), Fixed point theory, Nonlinear analysis, paper

Abstract

In this paper, we extend the concept of b-metric spaces to the vectorial case, where the distance is vector valued, and the constant in the triangle in equality axiom is replaced by a matrix. For such spaces, we establish results analogous to those in the b-metric setting: fixed-point theorems, stability results, and a variant of Ekeland’s variational principle. As a consequence, we also derive a variant of Caristi’s fixed-point theorem.

Authors

Radu Precup
Faculty of Mathematics and Computer Science and Institute of Advanced Studies in Science and Technology, Babes-Bolyai, University, Cluj-Napoca, Romania
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, Romania

Andrei Stan
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, Romania
Department of Mathematics, Babes-Bolyai University, Cluj-Napoca, Romania

Keywords

tiangle inequality axiom; b-metric space; variational principle; fixed point

Paper coordinates

R. Precup, A. Stan, Fixed Point Results and the Ekeland Variational Principle in Vector B-Metric Spaces, Preprints.org., 10.20944/preprints202502.0815.v1

PDF

https://www.preprints.org/manuscript/202502.0815/v1

About this paper

Journal

Publisher Name

DOI

http://10.20944/preprints202502.0815.v1

Print ISSN

Online ISSN

google scholar link

[1] Wilson, W. A. On Quasi-Metric Spaces. Amer. J. Math. 1931, 675–684.
[2] Bourbaki, N. Topologie Generale; Herman: Paris, France, 1974.
[3] Bakhtin, I.A. Contracting mapping principle in an almost metric space. Funktsionalnyi Analiz 1989, 30, 26–37.
[4] Czerwik, S. Contraction mappings in b-metric spaces. Acta Math. Inform. Univ. Ostrav. 1993, 1, 5–11.
[5] Coifman, R.R.; de Guzman, M. Singular integrals and multipliers on homogeneous spaces. Rev. Un. Mat. Argentina 1970/71, 25, 137–143.
[6] Hyers, D.H. A note on linear topological spaces. Bull. Amer. Math. Soc. 1938, 44, 76–80.
[7] Bourgin, D.G. Linear topological spaces. Amer. J. Math. 1943, 65, 637–659.
[8] Berinde, V.; P˘acurar, M. The early developments in fixed point theory on b-metric spaces: A brief survey and some important related aspects. Carpathian J. Math. 2022, 38, 523–538.
[9] An, T.V.; Van Dung, N.; Kadelburg, Z.; Radenovi´c, S. Various generalizations of metric spaces and fixed point theorems. RACSAM 2015, 109(1), 175–198. Generalized distances and their associate metrics: Impact on fixed point theory. Creat. Math. Inform. 2013, 22(1), 23–32.
[10] Mitrovi´c, Z.D. Fixed point results in b-metric spaces. Fixed Point Theory 2019, 20, 559–566, https://doi.org/10.24193/fpt-ro.2019.2.36.
[11] Boriceanu, M.; Petru¸sel, A.; Rus, I.A. Fixed point theorems for some multivalued generalized contractions in b-metric spaces. Int. J. Math. Stat. 2010, 6, 65–76.
[12] Aydi, H.; Czerwik, S. Modern Discrete Mathematics and Analysis. Springer: Cham, Switzerland, 2018.
[13] Kirk, W.; Shahzad, N. Fixed Point Theory in Distance Spaces. Springer: Cham, Switzerland, 2014.
[14] Reich, S.; Zaslavski, A.J. Well-posedness of fixed point problems. Far East J. Math. Sci. 2001, Special Volume (Functional Analysis and its Applications), Part III, 393–401.
[15] Berinde, V. Generalized contractions in quasimetric spaces. Seminar on Fixed Point Theory, Preprint no. 3, 1993, 3–9.
[16] Miculescu, R.; Mihail, A. New fixed point theorems for set-valued contractions in b-metric spaces. J. Fixed Point Theory Appl. 2017, 19, 2153–2163.
[17] Suzuki, T. Basic inequality on a b-metric space and its applications. J. Inequal. Appl. 2017, 2017, 256.
[18] Bota, M.-F.; Micula, S. Ulam–Hyers stability via fixed point results for special contractions in b-metric spaces. Symmetry 2022, 14, 2461.
[19] Petrus, el, A.; Petrus, el, G. Graphical contractions and common fixed points in b-metric spaces. Arab. J. Math. 2023, 12, 423–430. https://doi.org/10.1007/s40065-022-00396-8.
[20] Bota, M.; Molnar, A.; Varga, C. On Ekeland’s variational principle in b-metric spaces. Fixed Point Theory 2011, 12, 21–28.
[21] Farkas, C; Molnár, A.; Nagy, S. A generalized variational principle in b-metric spaces. Le Matematiche 2014, 69(2), 205–221.
[22] Boriceanu, M. Fixed point theory on spaces with vector-valued b-metrics. Demonstr. Math. 2009, 42, 831–841.
[23] Precup, R. The role of matrices that are convergent to zero in the study of semilinear operator systems. Math. Comput. Model. 2009, 49(3), 703–708.
[24] Berman, A.; Plemmons, R.J. Nonnegative matrices in the mathematical sciences. Academic Press: New York, USA, 1997.
[25] Collatz, L. Aufgaben monotoner Art. Arch. Math. (Basel) 1952, 3, 366–376.
[26] Cobzas, S,.; Czerwik, S. The completion of generalized b-metric spaces and fixed points. Fixed Point Theory 2020, 21(1), 133–150.
[27] Perov, A.I. On the Cauchy problem for a system of ordinary differential equations (Russian). Priblizhen. Metody Reshen. Differ. Uravn. 1964, 2, 115–134.
[28] Perov, A.I. Generalized principle of contraction mappings (Russian). Vestn. Voronezh. Gos. Univ., Ser. Fiz. Mat. 2005, 1, 196–207.
[29] Ortega, J.M.; Rheinboldt, W.C. Iterative Solutions of Nonlinear Equations in Several Variables. Academic Press: New York, USA, 1970.
[30] Avramescu, C. Asupra unei teoreme de punct fix. St. Cerc. Mat. 1970, 22, 215–221.
[31] Ekeland, I. On the variational principle. J. Math. Anal. Appl. 1974, 47(2), 324–353.
[32] Mawhin, J.; Willem, M. Critical Point Theory And Hamiltonian Systems. Applied Mathematical Sciences: Springer, New York, USA, 1989.
[33] De Figueiredo, D.G. Lectures on the Ekeland Variational Principle with Applications and Detours. Tata Institute of Fundamental Research: Bombay, India, 1989.
[34] Meghea, I. Ekeland Variational Principle with Generalizations and Variants. Old City Publishing: Philadelphia, USA, 2009.
[35] Caristi, J. Fixed point theorems for mappings satisfying inwardness conditions. Trans. Amer. Math. Soc. 1976, 215, 241–251.