Some Coupled Fixed Point Results on Partial Metric Spaces

In this paper we give some coupled fixed point results for mappings satisfying different contractive conditions on complete partial metric spaces.Comment: To appear in International Journal of Mathematics and Mathematical Science

Coupled fixed point results on metric spaces defined by binary operations

In parallel with the various generalizations of the Banach fixed point theorem in metric spaces, this theory is also transported to some different types of spaces including ultra metric spaces, fuzzy metric spaces, uniform spaces, partial metric spaces, $b$-metric spaces etc. In this context, first we define a binary normed operation on nonnegative real numbers and give some examples. Then we recall the concept of $T$-metric space and some important and fundamental properties of it. A $T$-metric space is a $3$-tuple $(X, T, \diamond)$, where $X$ is a nonempty set, $\diamond$ is a binary normed operation and $T$ is a $T$-metric on $X$. Since the triangular inequality of $T$-metric depends on a binary operation, which includes the sum as a special case, a $T$-metric space is a real generalization of ordinary metric space. As main results, we present three coupled fixed point theorems for bivariate mappings satisfying some certain contractive inequalities on a complete $T$-metric space. It is easily seen that not only existence but also uniqueness of coupled fixed point guaranteed in these theorems. Also, we provide some suitable examples that illustrate our results

On the construction of metrics from fuzzy metrics and its application to the fixed point theory of multivalued mappings

[EN] We present a procedure to construct a compatible metric from a given fuzzy metric space. We use this approach to obtain a characterization of a large class of complete fuzzy metric spaces by means of a fuzzy version of Caristi’s fixed point theorem, obtaining, in this way, partial solutions to a recent question posed in the literature. Some illustrative examples are also given.The authors thank the referees for several useful suggestions. Salvador Romaguera and Pedro Tirado acknowledge the support of the Ministry of Economy and Competitiveness of Spain, grant MTM2012-37894-C02-01.Castro Company, F.; Romaguera Bonilla, S.; Tirado Peláez, P. (2015). On the construction of metrics from fuzzy metrics and its application to the fixed point theory of multivalued mappings. Fixed Point Theory and Applications. 2015:226. https://doi.org/10.1186/s13663-015-0476-1S2015:226Kelley, JL: General Topology. Springer, New York (1955)Schweizer, B, Sklar, A: Statistical metric spaces. Pac. J. Math. 10, 314-334 (1960)Klement, E, Mesiar, R, Pap, E: Triangular Norms. Kluwer Academic, Dordrecht (2000)Hamacher, H: Über logische Verknüpfungen unscharfer Aussagen und deren zugehörige Bewertungsfunktionen. In: Progress in Cybernetics and Systems Research, pp. 276-287. Hemisphere, New York (1975)Kramosil, I, Michalek, J: Fuzzy metrics and statistical metric spaces. Kybernetika 11, 326-334 (1975)George, A, Veeramani, P: On some results in fuzzy metric spaces. Fuzzy Sets Syst. 64, 395-399 (1994)Gregori, V, Romaguera, S: Some properties of fuzzy metric spaces. Fuzzy Sets Syst. 115, 485-489 (2000)Radu, V: On the triangle inequality in PM-spaces. STPA, West University of Timişoara 39 (1978)Abbas, M, Ali, B, Romaguera, S: Multivalued Caristi’s type mappings in fuzzy metric spaces and a characterization of fuzzy metric completeness. Filomat 29(6), 1217-1222 (2015)Cho, YJ, Grabiec, M, Radu, V: On Nonsymmetric Topological and Probabilistic Structures. Nova Science Publishers, New York (2006)Hadžić, O, Pap, E: Fixed Point Theory in Probabilistic Metric Spaces. Kluwer Academic, Dordrecht (2001)Mihet, D: A note on Hicks type contractions on generalized Menger spaces. STPA, West University of Timişoara 133 (2002)Mihet, D: A Banach contraction theorem in fuzzy metric spaces. Fuzzy Sets Syst. 144, 431-439 (2004)Radu, V: Some fixed point theorems in PM spaces. In: Stability Problems for Stochastic Models. Lecture Notes in Mathematics, vol. 1233, pp. 125-133. Springer, Berlin (1985)Radu, V: Some remarks on the probabilistic contractions on fuzzy Menger spaces (The Eighth Intern. Conf. on Applied Mathematics and Computer Science, Cluj-Napoca, 2001). Autom. Comput. Appl. Math. 11(1), 125-131 (2002)Chauhan, S, Shatanawi, W, Kumar, S, Radenović, S: Existence and uniqueness of fixed points in modified intuitionistic fuzzy metric spaces. J. Nonlinear Sci. Appl. 7, 28-41 (2014)Hussain, N, Salimi, P, Parvaneh, V: Fixed point results for various contractions in parametric and fuzzy b-metric spaces. J. Nonlinear Sci. Appl. 8, 719-739 (2015)Mihet, D: Common coupled fixed point theorems for contractive mappings in fuzzy metric spaces. J. Nonlinear Sci. Appl. 6, 35-40 (2013)Hicks, TL: Fixed point theory in probabilistic metric spaces. Zb. Rad. Prir.-Mat. Fak. (Novi Sad) 13, 63-72 (1983)Radu, V: Some suitable metrics on fuzzy metric spaces. Fixed Point Theory 5, 323-347 (2004)O’Regan, D, Saadati, R: Nonlinear contraction theorems in probabilistic spaces. Appl. Math. Comput. 195, 86-93 (2008)Caristi, J: Fixed point theorems for mappings satisfying inwardness conditions. Trans. Am. Math. Soc. 215, 241-251 (1976)Kirk, WA: Caristi’s fixed-point theorem and metric convexity. Colloq. Math. 36, 81-86 (1976)Ansari, QH: Metric Spaces: Including Fixed Point Theory and Set-Valued Maps. Alpha Science, Oxford (2010

The Meir-Keeler Fixed Point Theorem for Quasi-Metric Spaces and Some Consequences

[EN] We obtain quasi-metric versions of the famous Meir¿Keeler fixed point theorem from which we deduce quasi-metric generalizations of Boyd¿Wong¿s fixed point theorem. In fact, one of these generalizations provides a solution for a question recently raised in the paper ¿On the fixed point theory in bicomplete quasi-metric spaces¿, J. Nonlinear Sci. Appl. 2016, 9, 5245¿5251. We also give an application to the study of existence of solution for a type of recurrence equations associated to certain nonlinear difference equationsPedro Tirado acknowledges the support of the Ministerio de Ciencia, Innovación y Universidades, under grant PGC2018-095709-B-C21Romaguera Bonilla, S.; Tirado Peláez, P. (2019). The Meir-Keeler Fixed Point Theorem for Quasi-Metric Spaces and Some Consequences. Symmetry (Basel). 11(6):1-10. https://doi.org/10.3390/sym11060741S110116Alegre, C., Dağ, H., Romaguera, S., & Tirado, P. (2016). On the fixed point theory in bicomplete quasi-metric spaces. Journal of Nonlinear Sciences and Applications, 09(08), 5245-5251. doi:10.22436/jnsa.009.08.10Boyd, D. W., & Wong, J. S. W. (1969). On nonlinear contractions. Proceedings of the American Mathematical Society, 20(2), 458-458. doi:10.1090/s0002-9939-1969-0239559-9Meir, A., & Keeler, E. (1969). A theorem on contraction mappings. Journal of Mathematical Analysis and Applications, 28(2), 326-329. doi:10.1016/0022-247x(69)90031-6Aydi, H., & Karapinar, E. (2012). A Meir-Keeler common type fixed point theorem on partial metric spaces. Fixed Point Theory and Applications, 2012(1). doi:10.1186/1687-1812-2012-26Chen, C.-M. (2012). Fixed point theory for the cyclic weaker Meir-Keeler function in complete metric spaces. Fixed Point Theory and Applications, 2012(1). doi:10.1186/1687-1812-2012-17Chen, C.-M. (2012). Fixed point theorems for cyclic Meir-Keeler type mappings in complete metric spaces. Fixed Point Theory and Applications, 2012(1). doi:10.1186/1687-1812-2012-41Chen, C.-M., & Karapınar, E. (2013). Fixed point results for the α-Meir-Keeler contraction on partial Hausdorff metric spaces. Journal of Inequalities and Applications, 2013(1). doi:10.1186/1029-242x-2013-410Choban, M. M., & Berinde, V. (2017). Multiple fixed point theorems for contractive and Meir-Keeler type mappings defined on partially ordered spaces with a distance. Applied General Topology, 18(2), 317. doi:10.4995/agt.2017.7067Di Bari, C., Suzuki, T., & Vetro, C. (2008). Best proximity points for cyclic Meir–Keeler contractions. Nonlinear Analysis: Theory, Methods & Applications, 69(11), 3790-3794. doi:10.1016/j.na.2007.10.014Jachymski, J. (1995). Equivalent Conditions and the Meir-Keeler Type Theorems. Journal of Mathematical Analysis and Applications, 194(1), 293-303. doi:10.1006/jmaa.1995.1299Karapinar, E., Czerwik, S., & Aydi, H. (2018). (α,ψ)-Meir-Keeler Contraction Mappings in Generalized b-Metric Spaces. Journal of Function Spaces, 2018, 1-4. doi:10.1155/2018/3264620Mustafa, Z., Aydi, H., & Karapınar, E. (2013). Generalized Meir–Keeler type contractions on G-metric spaces. Applied Mathematics and Computation, 219(21), 10441-10447. doi:10.1016/j.amc.2013.04.032Nashine, H. K., & Romaguera, S. (2013). Fixed point theorems for cyclic self-maps involving weaker Meir-Keeler functions in complete metric spaces and applications. Fixed Point Theory and Applications, 2013(1). doi:10.1186/1687-1812-2013-224Park, S., & Bae, J. S. (1981). Extensions of a fixed point theorem of Meir and Keeler. Arkiv för Matematik, 19(1-2), 223-228. doi:10.1007/bf02384479Piątek, B. (2011). On cyclic Meir–Keeler contractions in metric spaces. Nonlinear Analysis: Theory, Methods & Applications, 74(1), 35-40. doi:10.1016/j.na.2010.08.010Rhoades, B. ., Park, S., & Moon, K. B. (1990). On generalizations of the Meir-Keeler type contraction maps. 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Applications of the complexity space to the General Probabilistic Divide and Conquer Algorithms. Journal of Mathematical Analysis and Applications, 348(1), 346-355. doi:10.1016/j.jmaa.2008.07.026Mohammadi, Z., & Valero, O. (2016). A new contribution to the fixed point theory in partial quasi-metric spaces and its applications to asymptotic complexity analysis of algorithms. Topology and its Applications, 203, 42-56. doi:10.1016/j.topol.2015.12.074Romaguera, S., & Tirado, P. (2011). The complexity probabilistic quasi-metric space. Journal of Mathematical Analysis and Applications, 376(2), 732-740. doi:10.1016/j.jmaa.2010.11.056Romaguera, S., & Tirado, P. (2015). A characterization of Smyth complete quasi-metric spaces via Caristi’s fixed point theorem. Fixed Point Theory and Applications, 2015(1). doi:10.1186/s13663-015-0431-1Stevo, S. (2002). The recursive sequence xn+1 = g(xn, xn−1)/(A + xn). Applied Mathematics Letters, 15(3), 305-308. doi:10.1016/s0893-9659(01)00135-

On some fixed point theorems under (α,ψ,ϕ) -contractivity conditions in metric spaces endowed with transitive binary relations

After the appearance of Nieto and Rodríguez-López’s theorem, the branch of fixed point theory devoted to the setting of partially ordered metric spaces have attracted much attention in the last years, especially when coupled, tripled, quadrupled and, in general, multidimensional fixed points are studied. Almost all papers in this direction have been forced to present two results assuming two different hypotheses: the involved mapping should be continuous or the metric framework should be regular. Both conditions seem to be different in nature because one of them refers to the mapping and the other one is assumed on the ambient space. In this paper, we unify such different conditions in a unique one. By introducing the notion of continuity of a mapping from a metric space into itself depending on a function α, which is the case that covers the partially ordered setting, we extend some very recent theorems involving control functions that only must be lower/upper semi-continuous from the right. Finally, we use metric spaces endowed with transitive binary relations rather than partial orders.This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. N Shahzad acknowledges with thanks DSR for financial support. A-F Roldán-López-de-Hierro is grateful to the Department of Quantitative Methods for Economics and Business of the University of Granada. The same author has been partially supported by Junta de Andalucía by project FQM-268 of the Andalusian CICYE

Unified multi-tupled fixed point theorems involving mixed monotone property in ordered metric spaces

In the present article, we introduce a unified notion of multi-tupled fixed points and utilize the same to prove some existence and uniqueness unified multi-tupled fixed point theorems for Boyd-Wong type nonlinear contractions satisfying generalized mixed monotone property in ordered metric spaces. Our results unify several classical and well-known n-tupled (including coupled, tripled and quadrupled ones) fixed point results existing in the literature.Comment: arXiv admin note: substantial text overlap with arXiv: 1601.0251