Common fixed point theorems for cyclic contractive mappings in partial cone b-metric spaces and applications to integral equations

In this paper, we introduce the concept of partial cone b-metric spaces as a generalization of partial metric, cone metric and b-metric spaces and establish some topological properties of partial cone b-metric spaces. Moreover, we also prove some common fixed point theorems for cyclic contractive mappings in such spaces. Our results generalize and extend the main results of Huang and Zhang [Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl., 332:1468–1476, 2007], Stanić et al. [Common fixed point under contractive condition of Ćirić's type on cone metric type spaces, Fixed Point Theory Appl., 2012:35, 2012] and Latif et al. [Fixed point results for generalized (α,ψ)‐Meir–Keeler contractive mappings and applications, J. Inequal. Appl., 2014:68, 2014]. Some examples and an application are given to support the usability of the obtained results

Unique Fixed Point Theorems for Generalized Weakly Contractive Condition in Ordered Partial Metric Spaces

The aim of this paper to prove some fixed point theorems for generalized weakly contractive condition in ordered partial metric spaces. The result extend the main theorems of Nashine and altun[17] on the class of ordered partial metric ones. Keywords: - Partial metric, ordered set, fixed point, common fixed point. AMS subject classification: - 54H25, 47H10, 54E5

Common fixed points of generalized Mizoguchi-Takahashi type contractions in partial metric spaces

We give some common fixed point results for multivalued mappings in the setting of complete partial metric spaces. Our theorems extend and complement analogous results in the existing literature on metric and partial metric spaces. Finally, we provide an example to illustrate the new theory

Fixed point results for generalized cyclic contraction mappings in partial metric spaces

Rus (Approx. Convexity 3:171–178, 2005) introduced the concept of cyclic contraction mapping. P˘acurar and Rus (Nonlinear Anal. 72:1181–1187, 2010) proved some fixed point results for cyclic φ-contraction mappings on a metric space. Karapinar (Appl. Math. Lett. 24:822–825, 2011) obtained a unique fixed point of cyclic weak φ- contraction mappings and studied well-posedness problem for such mappings. On the other hand, Matthews (Ann. New York Acad. Sci. 728:183–197, 1994) introduced the concept of a partial metric as a part of the study of denotational semantics of dataflow networks. He gave a modified version of the Banach contraction principle, more suitable in this context. In this paper, we initiate the study of fixed points of generalized cyclic contraction in the framework of partial metric spaces. We also present some examples to validate our results.S. Romaguera acknowledges the support of the Ministry of Science and Innovation of Spain, grant MTM2009-12872-C02-01.Abbas, M.; Nazir, T.; Romaguera Bonilla, S. (2012). Fixed point results for generalized cyclic contraction mappings in partial metric spaces. Revista- Real Academia de Ciencias Exactas Fisicas Y Naturales Serie a Matematicas. 106(2):287-297. https://doi.org/10.1007/s13398-011-0051-5S2872971062Abdeljawad T., Karapinar E., Tas K.: Existence and uniqueness of a common fixed point on partial metric spaces. Appl. Math. Lett. 24(11), 1894–1899 (2011). doi: 10.1016/j.aml.2011.5.014Altun, I., Erduran A.: Fixed point theorems for monotone mappings on partial metric spaces. Fixed Point Theory Appl. article ID 508730 (2011). doi: 10.1155/2011/508730Altun I., Sadarangani K.: Corrigendum to “Generalized contractions on partial metric spaces” [Topology Appl. 157 (2010), 2778–2785]. Topol. Appl. 158, 1738–1740 (2011)Altun I., Simsek H.: Some fixed point theorems on dualistic partial metric spaces. J. Adv. Math. Stud. 1, 1–8 (2008)Altun I., Sola F., Simsek H.: Generalized contractions on partial metric spaces. Topol. Appl. 157, 2778–2785 (2010)Aydi, H.: Some fixed point results in ordered partial metric spaces. arxiv:1103.3680v1 [math.GN](2011)Boyd D.W., Wong J.S.W.: On nonlinear contractions. Proc. Am. Math. Soc. 20, 458–464 (1969)Bukatin M., Kopperman R., Matthews S., Pajoohesh H.: Partial metric spaces. Am. Math. Monthly 116, 708–718 (2009)Bukatin M.A., Shorina S.Yu. et al.: Partial metrics and co-continuous valuations. In: Nivat, M. (eds) Foundations of software science and computation structure Lecture notes in computer science vol 1378., pp. 125–139. Springer, Berlin (1998)Derafshpour M., Rezapour S., Shahzad N.: On the existence of best proximity points of cyclic contractions. Adv. Dyn. Syst. Appl. 6, 33–40 (2011)Heckmann R.: Approximation of metric spaces by partial metric spaces. Appl. Cat. Struct. 7, 71–83 (1999)Karapinar E.: Fixed point theory for cyclic weak $${\phi}$$ -contraction. App. Math. Lett. 24, 822–825 (2011)Karapinar, E.: Generalizations of Caristi Kirk’s theorem on partial metric spaces. Fixed Point Theory Appl. 2011,4 (2011). doi: 10.1186/1687-1812-2011-4Karapinar E.: Weak $${\varphi}$$ -contraction on partial metric spaces and existence of fixed points in partially ordered sets. Math. Aeterna. 1(4), 237–244 (2011)Karapinar E., Erhan I.M.: Fixed point theorems for operators on partial metric spaces. Appl. Math. Lett. 24, 1894–1899 (2011)Karpagam S., Agrawal S.: Best proximity point theorems for cyclic orbital Meir–Keeler contraction maps. Nonlinear Anal. 74, 1040–1046 (2011)Kirk W.A., Srinavasan P.S., Veeramani P.: Fixed points for mapping satisfying cylical contractive conditions. Fixed Point Theory. 4, 79–89 (2003)Kosuru, G.S.R., Veeramani, P.: Cyclic contractions and best proximity pair theorems). arXiv:1012.1434v2 [math.FA] 29 May (2011)Matthews S.G.: Partial metric topology. in: Proc. 8th Summer Conference on General Topology and Applications. Ann. New York Acad. Sci. 728, 183–197 (1994)Neammanee K., Kaewkhao A.: Fixed points and best proximity points for multi-valued mapping satisfying cyclical condition. Int. J. Math. Sci. Appl. 1, 9 (2011)Oltra S., Valero O.: Banach’s fixed theorem for partial metric spaces. Rend. Istit. Mat. Univ. Trieste. 36, 17–26 (2004)Păcurar M., Rus I.A.: Fixed point theory for cyclic $${\phi}$$ -contractions. Nonlinear Anal. 72, 1181–1187 (2010)Petric M.A.: Best proximity point theorems for weak cyclic Kannan contractions. Filomat. 25, 145–154 (2011)Romaguera, S.: A Kirk type characterization of completeness for partial metric spaces. Fixed Point Theory Appl. (2010, article ID 493298, 6 pages).Romaguera, S.: Fixed point theorems for generalized contractions on partial metric spaces. Topol. Appl. (2011). doi: 10.1016/j.topol.2011.08.026Romaguera S., Valero O.: A quantitative computational model for complete partial metric spaces via formal balls. Math. Struct. Comput. Sci. 19, 541–563 (2009)Rus, I.A.: Cyclic representations and fixed points. Annals of the Tiberiu Popoviciu Seminar of Functional equations. Approx. Convexity 3, 171–178 (2005), ISSN 1584-4536Schellekens M.P.: The correspondence between partial metrics and semivaluations. Theoret. Comput. Sci. 315, 135–149 (2004)Valero O.: On Banach fixed point theorems for partial metric spaces. Appl. Gen. Top. 6, 229–240 (2005)Waszkiewicz P.: Quantitative continuous domains. Appl. Cat. Struct. 11, 41–67 (2003

Some coupled coincidence and common fixed point results for a hybrid pair of mappings in 0-complete partial metric spaces

In this paper we extend some coupled coincidence and common fixed point theorems for a hybrid pair of mappings obtained by Abbas et al. (Fixed Point Theory Appl. 2012:4, 2012, doi:10.1186/1687-1812-2012-4) from the complete metric space to 0-complete partial metric spaces. An example showing that this extension is proper is given

SOME COMMON FIXED POINT THEOREMS IN COMPLETE WEAK PARTIAL METRIC SPACES INVOLVING AUXILIARY FUNCTIONS

In this paper, we establish some common fixed point theorems and a coincidence point theorem on complete weak partial metric spaces using auxiliary functions. We also give examples in support of the result. The results proved in this paper extend and generalize several results from the existing literature

ON SOME COMMON FIXED POINT THEOREMS FOR GENERALIZED INTEGRAL TYPE FF-CONTRACTIONS IN PARTIAL METRIC SPACES

In this article, we prove some common fixed point theorems for generalized integral type $F$-contractions in the setting of complete partial metric spaces and give some consequences of the main result. Also we give an example in support of the result. Our result extends and generalizes several results from the existing literature

Interpolative Reich-Rus-Ciric and Hardy-Rogers Contraction on Quasi-Partial b-Metric Space and Related Fixed Point Results

[EN] The aim of this paper was to obtain common fixed point results by using an interpolative contraction condition given by Karapinar in the setting of complete metric space. Here in this paper, we have redefined the Reich-Rus-Ciric type contraction and Hardy-Rogers type contraction in the framework of quasi-partial b-metric space and proved the corresponding common fixed point theorem by adopting the notion of interpolation. The results are further validated with the application based on them.Mishra, VN.; Sánchez Ruiz, LM.; Gautam, P.; Verma, S. (2020). Interpolative Reich-Rus-Ciric and Hardy-Rogers Contraction on Quasi-Partial b-Metric Space and Related Fixed Point Results. Mathematics. 8(9):1-11. https://doi.org/10.3390/math8091598S11189Banach, S. (1922). Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fundamenta Mathematicae, 3, 133-181. doi:10.4064/fm-3-1-133-181Shukla, S. (2013). Partial b-Metric Spaces and Fixed Point Theorems. Mediterranean Journal of Mathematics, 11(2), 703-711. doi:10.1007/s00009-013-0327-4MATTHEWS, S. G. (1994). Partial Metric Topology. Annals of the New York Academy of Sciences, 728(1 General Topol), 183-197. doi:10.1111/j.1749-6632.1994.tb44144.xKARAPINAR, E. (2018). Revisiting the Kannan Type Contractions via Interpolation. Advances in the Theory of Nonlinear Analysis and its Application, 85-87. doi:10.31197/atnaa.431135Reich, S. (1971). Some Remarks Concerning Contraction Mappings. Canadian Mathematical Bulletin, 14(1), 121-124. doi:10.4153/cmb-1971-024-9Hardy, G. E., & Rogers, T. D. (1973). A Generalization of a Fixed Point Theorem of Reich. Canadian Mathematical Bulletin, 16(2), 201-206. doi:10.4153/cmb-1973-036-0Karapinar, E., Agarwal, R., & Aydi, H. (2018). Interpolative Reich–Rus–Ćirić Type Contractions on Partial Metric Spaces. Mathematics, 6(11), 256. doi:10.3390/math6110256Karapınar, E., Alqahtani, O., & Aydi, H. (2018). On Interpolative Hardy-Rogers Type Contractions. Symmetry, 11(1), 8. doi:10.3390/sym11010008Aydi, H., Karapinar, E., & Roldán López de Hierro, A. (2019). ω-Interpolative Ćirić-Reich-Rus-Type Contractions. Mathematics, 7(1), 57. doi:10.3390/math7010057Debnath, P., & de La Sen, M. de L. (2019). Set-Valued Interpolative Hardy–Rogers and Set-Valued Reich–Rus–Ćirić-Type Contractions in b-Metric Spaces. Mathematics, 7(9), 849. doi:10.3390/math7090849Alqahtani, B., Fulga, A., & Karapınar, E. (2018). Fixed Point Results on Δ-Symmetric Quasi-Metric Space via Simulation Function with an Application to Ulam Stability. Mathematics, 6(10), 208. doi:10.3390/math6100208Aydi, H., Chen, C.-M., & Karapınar, E. (2019). Interpolative Ćirić-Reich-Rus Type Contractions via the Branciari Distance. Mathematics, 7(1), 84. doi:10.3390/math7010084Aydi, H., & Karapinar, E. (2012). A Meir-Keeler common type fixed point theorem on partial metric spaces. Fixed Point Theory and Applications, 2012(1). doi:10.1186/1687-1812-2012-26Ćirić, L., Samet, B., Aydi, H., & Vetro, C. (2011). Common fixed points of generalized contractions on partial metric spaces and an application. Applied Mathematics and Computation, 218(6), 2398-2406. doi:10.1016/j.amc.2011.07.005Karapınar, E., Chi, K. P., & Thanh, T. D. (2012). A Generalization of Ćirić Quasicontractions. Abstract and Applied Analysis, 2012, 1-9. doi:10.1155/2012/518734Mlaiki, N., Abodayeh, K., Aydi, H., Abdeljawad, T., & Abuloha, M. (2018). Rectangular Metric-Like Type Spaces and Related Fixed Points. Journal of Mathematics, 2018, 1-7. doi:10.1155/2018/3581768Gupta, A., & Gautam, P. (2016). Topological Structure of Quasi-Partial b-Metric Spaces. International Journal of Pure Mathematical Sciences, 17, 8-18. doi:10.18052/www.scipress.com/ijpms.17.8Gupta, A., & Gautam, P. (2015). Quasi-partial b-metric spaces and some related fixed point theorems. Fixed Point Theory and Applications, 2015(1). doi:10.1186/s13663-015-0260-