General Common Fixed Point Theorems and Applications

The main result is a common fixed point theorem for a pair of multivalued maps on a complete metric space extending a recent result of Đorić and Lazović (2011) for a multivalued map on a metric space satisfying Ćirić-Suzuki-type-generalized contraction. Further, as a special case, we obtain a generalization of an important common fixed point theorem of Ćirić (1974). Existence of a common solution for a class of functional equations arising in dynamic programming is also discussed

Common Fixed Point for Self-Mappings Satisfying an Implicit Lipschitz-Type Condition in Kaleva-Seikkala's Type Fuzzy Metric Spaces

We first introduce the new real function class ℱ satisfying an implicit Lipschitz-type condition. Then, by using ℱ-type real functions, some common fixed point theorems for a pair of self-mappings satisfying an implicit Lipschitz-type condition in fuzzy metric spaces (in the sense of Kaleva and Seikkala) are established. As applications, we obtain the corresponding common fixed point theorems in metric spaces. Also, some examples are given, which show that there exist mappings which satisfy the conditions in this paper but cannot satisfy the general contractive type conditions

Remarks on fixed point assertions in digital topology, 4

[EN] We continue the work of [4, 2, 3], in which we discuss published assertions that are incorrect or incorrectly proven; that are severely limited or reduce to triviality; or that we improve upon.Boxer, L. (2020). Remarks on fixed point assertions in digital topology, 4. Applied General Topology. 21(2):265-284. https://doi.org/10.4995/agt.2020.13075OJS265284212L. Boxer, A classical construction for the digital fundamental group, Journal of Mathematical Imaging and Vision 10 (1999), 51-62. https://doi.org/10.1023/A:1008370600456L. Boxer, Remarks on fixed point assertions in digital topology, 2, Applied General Topology 20, no. 1 (2019), 155-175. https://doi.org/10.4995/agt.2019.10667L. Boxer, Remarks on fixed point assertions in digital topology, 3, Applied General Topology 20, no. 2 (2019), 349-361. https://doi.org/10.4995/agt.2019.11117L. Boxer and P. C. Staecker, Remarks on fixed point assertions in digital topology, Applied General Topology 20, no. 1 (2019), 135-153. https://doi.org/10.4995/agt.2019.10474S. Dalal, Common fixed point results for weakly compatible map in digital metric spaces, Scholars Journal of Physics, Mathematics and Statistics 4, no. 4 (2017), 196-201.S. Dalal, I. A. Masmali, and G. Y. Alhamzi, Common fixed point results for compatible map in digital metric space, Advances in Pure Mathematics 8 (2018), 362-371. https://doi.org/10.4236/apm.2018.83019O. Ege, D. Jain, S. Kumar, C. Park and D. Y. Shin, Commuting and compatible mappings in digital metric spaces, Journal of Fixed Point Theory and Applications 22, no. 5 (2020). https://doi.org/10.1007/s11784-019-0744-5O. Ege and I. Karaca, Digital homotopy fixed point theory, Comptes Rendus Mathematique 353, no. 11 (2015), 1029-1033. https://doi.org/10.1016/j.crma.2015.07.006S.-E. Han, Banach fixed point theorem from the viewpoint of digital topology, Journal of Nonlinear Science and Applications 9 (2016), 895-905. https://doi.org/10.22436/jnsa.009.03.19K. Jyoti and A. Rani, Fixed point theorems for $beta$ - $psi$ - $phi$-expansive type mappings in digital metric spaces, Asian Journal of Mathematics and Computer Research 24, no. 2 (2018), 56-66.K. Jyoti, A. Rani, and A. Rani, Common fixed point theorems for compatible and weakly compatible maps satisfying E.A. and CLR($T$) property in digital metric space, IJAMAA 13, no. 1 (2017), 117-128. https://doi.org/10.9734/JAMCS/2017/34278H. K. Pathak and M. S. Khan, Compatible mappings of type (B) and common fixed point theorems of Gregus type, Czechoslovak Math. J. 45 (1995), 685-698. https://doi.org/10.21136/CMJ.1995.128555A. Rani, K. Jyoti, and A. Rani, Common fixed point theorems in digital metric spaces, International Journal of Scientific & Engineering Research 7, no. 12 (2016), 1704-1716.A. Rosenfeld, 'Continuous' functions on digital pictures, Pattern Recognition Letters 4 (1986), 177-184. 1986. https://doi.org/10.1016/0167-8655(86)90017-

Remarks on fixed point assertions in digital topology, 2

[EN] Several recent papers in digital topology have sought to obtain fixed point results by mimicking the use of tools from classical topology, such as complete metric spaces. We show that in many cases, researchers using these tools have derived conclusions that are incorrect, trivial, or limited.Boxer, L. (2019). Remarks on fixed point assertions in digital topology, 2. Applied General Topology. 20(1):155-175. https://doi.org/10.4995/agt.2019.10667SWORD155175201L. Boxer, A classical construction for the digital fundamental group, Journal of Mathematical Imaging and Vision 10 (1999), 51-62. https://doi.org/10.1023/A:1008370600456L. Boxer, Generalized normal product adjacency in digital topology, Applied General Topology 18, no. 2 (2017), 401-427. https://doi.org/10.4995/agt.2017.7798L. Boxer, Alternate product adjacencies in digital topology, Applied General Topology 19, no. 1 (2018), 21-53. https://doi.org/10.4995/agt.2018.7146L. Boxer, O. Ege, I. Karaca, J. Lopez and J. Louwsma, Digital fixed points, approximate fixed points, and universal functions, Applied General Topology 17, no. 2 (2016), 159-172. https://doi.org/10.4995/agt.2016.4704L. Boxer and P. C. Staecker, Remarks on fixed point assertions in digital topology, Applied General Topology, Applied General Topology 20, no. 1 (2019), https://doi.org/10.4995/agt.2019.10474S. Dalal, I. A. Masmali and G. Y. Alhamzi, Common fixed point results for compatible map in digital metric space, Advances in Pure Mathematics 8 (2018), 362-371. https://doi.org/10.4236/apm.2018.83019U. P. Dolhare and V. V. Nalawade, Fixed point theorems in digital images and applications to fractal image compression, Asian Journal of Mathematics and Computer Research 25, no. 1 (2018), 18-37.O. Ege and I. Karaca, Banach fixed point theorem for digital images, Journal of Nonlinear Sciences and Applications 8 (2015), 237-245. https://doi.org/10.22436/jnsa.008.03.08G. Herman, Oriented surfaces in digital spaces, CVGIP: Graphical Models and Image Processing 55 (1993), 381-396. https://doi.org/10.1006/cgip.1993.1029S.-E. Han, Banach fixed point theorem from the viewpoint of digital topology, Journal of Nonlinear Science and Applications 9 (2016), 895-905. https://doi.org/10.22436/jnsa.009.03.19A. Hossain, R. Ferdausi, S. Mondal and H. Rashid, Banach and Edelstein fixed point theorems for digital images, Journal of Mathematical Sciences and Applications 5, no. 2 (2017), 36-39. https://doi.org/10.12691/jmsa-5-2-2D. Jain, Common fixed point theorem for intimate mappings in digital metric spaces, International Journal of Mathematics Trends and Technology 56, no. 2 (2018), 91-94. https://doi.org/10.14445/22315373/IJMTT-V56P511K. Jyoti and A. Rani, Fixed point results for expansive mappings in digital metric spaces, International Journal of Mathematical Archive 8, no. 6 (2017), 265-270.K. Jyoti and A. Rani, Digital expansions endowed with fixed point theory, Turkish Journal of Analysis and Number Theory 5, no. 5 (2017), 146-152. https://doi.org/10.12691/tjant-5-5-1K. Jyoti and A. Rani, Fixed point theorems for β−ψ−φ-expansive type mappings in digital metric spaces, Asian Journal of Mathematics and Computer Research 24, no. 2 (2018), 56-66.L. N. Mishra, K. Jyoti, A. Rani and Vandana, Fixed point theorems with digital contractions image processing, Nonlinear Science Letters A 9, no. 2 (2018), 104-115.C. Park, O. Ege, S. Kumar, D. Jain and J. R. Lee, Fixed point theorems for various contraction conditions in digital metric spaces, Journal of Computational Analysis and Applications 26, no. 8 (2019), 1451-1458.A. Rani, K. Jyoti and A. Rani, Common fixed point theorems in digital metric spaces, International Journal of Scientific & Engineering Research 7, no. 12 (2016), 1704-1715.A. Rosenfeld, 'Continuous' functions on digital images, Pattern Recognition Letters 4 (1986), 177-184. https://doi.org/10.1016/0167-8655(86)90017-6B. Samet, C. Vetro, and P. Vetro, Fixed point theorems for contractive mappings, Nonlinear Analysis: Theory, Methods & Applications 75, no. 4 (2012), 2154-2165. https://doi.org/10.1016/j.na.2011.10.014K. Sridevi, M. V. R. Kameswari and D. M. K. Kiran, Fixed point theorems for digital contractive type mappings in digital metric spaces, International Journal of Mathematics Trends and Technology 48, no. 3 (2017), 159-167. https://doi.org/10.14445/22315373/IJMTT-V48P522K. Sridevi, M. V. R. Kameswari and D. M. K. Kiran, Common fixed points for commuting and weakly compatible self-maps on digital metric spaces, International Advanced Research Journal in Science, Engineering and Technology 4, no. 9 (2017), 21-27

New common fixed point theorems for multivalued maps

[EN] Common fixed point theorems for a new class of multivalued maps are obtained, which generalize and extend classical fixed point theorems of Nadler and Reich and some recent Suzuki type fixed point theorems.Singh, SL.; Kamal, R.; Chugh, R.; Mishra, SN. (2014). New common fixed point theorems for multivalued maps. Applied General Topology. 15(2):111-119. doi:http://dx.doi.org/10.4995/agt.2014.2815.SWORD111119152Assad, N., & Kirk, W. (1972). Fixed point theorems for set-valued mappings of contractive type. Pacific Journal of Mathematics, 43(3), 553-562. doi:10.2140/pjm.1972.43.553Lj. B. Ciric, Fixed points for generalized multivalued contractions, Mat. Vesnik 9, no. 24 (1972), 265-272.Ciric, L. B. (1974). A Generalization of Banach’s Contraction Principle. Proceedings of the American Mathematical Society, 45(2), 267. doi:10.2307/2040075Damjanovic, B., & Djoric, D. (2011). Multivalued generalizations of the Kannan fixed point theorem. Filomat, 25(1), 125-131. doi:10.2298/fil1101125dKikkawa, M., & Suzuki, T. (2008). Three fixed point theorems for generalized contractions with constants in complete metric spaces. Nonlinear Analysis: Theory, Methods & Applications, 69(9), 2942-2949. doi:10.1016/j.na.2007.08.064M. Kikkawa and T. Suzuki, Some notes on fixed point theorems with constants, Bull. Kyushu Inst. Technol. Pure Appl. Math. 56 (2009), 11-18.Moţ, G., & Petruşel, A. (2009). Fixed point theory for a new type of contractive multivalued operators. Nonlinear Analysis: Theory, Methods & Applications, 70(9), 3371-3377. doi:10.1016/j.na.2008.05.005Nadler, S. (1969). Multi-valued contraction mappings. Pacific Journal of Mathematics, 30(2), 475-488. doi:10.2140/pjm.1969.30.475S. B. Nadler, Hyperspaces of Sets, Marcel Dekker, New York, 1978.Popescu, O. (2009). Two fixed point theorems for generalized contractions with constants in complete metric space. Central European Journal of Mathematics, 7(3), 529-538. doi:10.2478/s11533-009-0019-2S. Reich, Fixed points of multi-valued functions. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 51, no. 8 (1971), 32-35.Rhoades, B. E. (1977). A comparison of various definitions of contractive mappings. Transactions of the American Mathematical Society, 226, 257-257. doi:10.1090/s0002-9947-1977-0433430-4I. A. Rus, Fixed point theorems for multivalued mappings in complete metric spaces, Math. Japon. 20 (1975), 21-24.I. A. Rus, Generalized Contractions And Applications, Cluj-Napoca, 2001.K. P. R. Sastry and S. V. R. Naidu, Fixed point theorems for generalized contraction mappings, Yokohama Math. J. 25 (1980), 15-29.Singh, S. L., & Mishra, S. N. (2011). Fixed point theorems for single-valued and multi-valued maps. Nonlinear Analysis: Theory, Methods & Applications, 74(6), 2243-2248. doi:10.1016/j.na.2010.11.029Suzuki, T. (2007). A generalized Banach contraction principle that characterizes metric completeness. Proceedings of the American Mathematical Society, 136(05), 1861-1870. doi:10.1090/s0002-9939-07-09055-7Suzuki, T. (2009). A new type of fixed point theorem in metric spaces. Nonlinear Analysis: Theory, Methods & Applications, 71(11), 5313-5317. doi:10.1016/j.na.2009.04.01

Fixed points of α-Θ-Geraghty type and Θ-Geraghty graphic type contractions

[EN] In this paper, by using the concept of the α-Garaghty contraction, we introduce the new notion of the α-Θ-Garaghty type contraction and prove some fixed point results for this contraction in partial metric spaces. Also, we give some examples and applications to illustrate the main results.The first author would like to thank the Research Professional Development Project Under the Science Achievement Scholarship of Thailand (SAST) for the Master’s degree Program at KMUTT. This project was supported by the Theoretical and Computational Science (TaCS) Center under Computational and Applied Science for Smart Innovation Research Cluster (CLASSIC), Faculty of Science, KMUTT.Onsod, W.; Kumam, P.; Cho, YJ. (2017). Fixed points of α-Θ-Geraghty type and Θ-Geraghty graphic type contractions. Applied General Topology. 18(1):153-171. https://doi.org/10.4995/agt.2017.6694SWORD153171181T. Abdeljawad, Meir-Keeler α-contractive fixed and common fixed point theorems, Fixed Point Theory Appl. 19 (2013). https://doi.org/10.1186/1687-1812-2013-19T. Abdeljawad and D. Gopal, Erratum to Meir-Keeler $alpha$-contractive fixed and common fixed point theorems, Fixed Point Theory Appl. 110 (2013). H. Alikhani, D. Gopal, M. A. Miandaragh, Sh. Rezapour and N. Shahzad, Some endpoint results for β-generalized weak contractive multifunctions, The Scientific World Journal (2013), Article ID 948472.Beg, I., Butt, A. R., & Radojević, S. (2010). The contraction principle for set valued mappings on a metric space with a graph. Computers & Mathematics with Applications, 60(5), 1214-1219. doi:10.1016/j.camwa.2010.06.003A. Branciari, A fixed point theorem of Banach-Caccioppoli type on a class of generalized metric spaces, Publ. Math. (Debr.) 57 (2000), 31-37.Cho, S.-H., Bae, J.-S., & Karapınar, E. (2013). Fixed point theorems for α-Geraghty contraction type maps in metric spaces. Fixed Point Theory and Applications, 2013(1), 329. doi:10.1186/1687-1812-2013-329Chandok, S. (2015). Some fixed point theorems for (α, β)-admissible Geraghty type contractive mappings and related results. Mathematical Sciences, 9(3), 127-135. doi:10.1007/s40096-015-0159-4Geraghty, M. A. (1973). On contractive mappings. Proceedings of the American Mathematical Society, 40(2), 604-604. doi:10.1090/s0002-9939-1973-0334176-5GOPAL, D., ABBAS, M., PATEL, D. K., & VETRO, C. (2016). Fixed points of α -type F -contractive mappings with an application to nonlinear fractional differential equation. Acta Mathematica Scientia, 36(3), 957-970. doi:10.1016/s0252-9602(16)30052-2Gordji, M., Ramezani, M., Cho, Y., & Pirbavafa, S. (2012). A generalization of Geraghty’s theorem in partially ordered metric spaces and applications to ordinary differential equations. Fixed Point Theory and Applications, 2012(1), 74. doi:10.1186/1687-1812-2012-74Hussain, N., Karapinar, E., Salimi, P., & Akbar, F. (2013). alpha-Admissible mappings and related Fixed point Theorems. Journal of Inequalities and Applications, 2013(1), 114. doi:10.1186/1029-242x-2013-114Jachymski, J. (2007). The contraction principle for mappings on a metric space with a graph. Proceedings of the American Mathematical Society, 136(04), 1359-1373. doi:10.1090/s0002-9939-07-09110-1X. D. Liu, S. S. Chang, Y. Xiao and L. C. Zhao, Existence of fixed points for Θ-type contraction and Θ-type Suzuki contraction in complete metric spaces, Fixed Point Theory Appl. 8 (2016). J. Martinez-Moreno, W. Sintunavarat and Y. J. Cho, Common fixed point theorems for Geraghty's type contraction mappings using the monotone property with two metrics, Fixed Point Theory Appl. 174 (2015).S. G. Mathews, Partial metric topology, in Proceedings of the 11th Summer Conference on General Topology and Applications 728 (1995), 183-197, The New York Academy of Sci. C. Mongkolkehai, Y. J. Cho and P. Kumam, Best proximity points for Geraghty's proximal contraction mappings, Fixed Point Theory Appl. 180 (2013).W. Onsod and P. Kumam, Common fixed point results for φ-ψ-weak contraction mappings via f-α-admissible Mappings in intuitionistic fuzzy metric spaces, Communications in Mathematics and Applications 7 (2016), 167-178.V. L. Rosa and P. Vetro, Fixed point for Geraghty-contractions in partial metric spaces, J. Nonlinear Sci. Appl. 7 (2014), 1-10.Samet, B., Vetro, C., & Vetro, P. (2012). Fixed point theorems for -contractive type mappings. Nonlinear Analysis: Theory, Methods & Applications, 75(4), 2154-2165. doi:10.1016/j.na.2011.10.01

Common fixed point theorems for mappings satisfying (E.A)-property via C-class functions in b-metric spaces

[EN] In this paper, we consider and generalize recent b-(E.A)-property results in [11] via the concepts of C-class functions in b- metric spaces. A example is given to support the result.Ozturk, V.; Ansari, AH. (2017). Common fixed point theorems for mappings satisfying (E.A)-property via C-class functions in b-metric spaces. Applied General Topology. 18(1):45-52. doi:10.4995/agt.2017.4573.SWORD4552181Aamri, M., & El Moutawakil, D. (2002). Some new common fixed point theorems under strict contractive conditions. Journal of Mathematical Analysis and Applications, 270(1), 181-188. doi:10.1016/s0022-247x(02)00059-8Jungck, G. (1986). Compatible mappings and common fixed points. International Journal of Mathematics and Mathematical Sciences, 9(4), 771-779. doi:10.1155/s016117128600093

A New General Iterative Method for Solution of a New General System of Variational Inclusions for Nonexpansive Semigroups in Banach Spaces

We introduce a new general system of variational inclusions in Banach spaces and propose a new iterative scheme for finding common element of the set of solutions of the variational inclusion with set-valued maximal monotone mapping and Lipschitzian relaxed cocoercive mapping and the set of fixed point of nonexpansive semigroups in a uniformly convex and 2-uniformly smooth Banach space. Furthermore, strong convergence theorems are established under some certain control conditions. As applications, finding a common solution for a system of variational inequality problems and minimization problems is given

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. Journal of Mathematical Analysis and Applications, 146(2), 482-494. doi:10.1016/0022-247x(90)90318-aSamet, B. (2010). Coupled fixed point theorems for a generalized Meir–Keeler contraction in partially ordered metric spaces. Nonlinear Analysis: Theory, Methods & Applications, 72(12), 4508-4517. doi:10.1016/j.na.2010.02.026Samet, B., Vetro, C., & Yazidi, H. (2013). A fixed point theorem for a Meir-Keeler type contraction through rational expression. Journal of Nonlinear Sciences and Applications, 06(03), 162-169. doi:10.22436/jnsa.006.03.02Schellekens, M. (1995). The Smyth Completion. Electronic Notes in Theoretical Computer Science, 1, 535-556. doi:10.1016/s1571-0661(04)00029-5Romaguera, S., & Schellekens, M. (1999). Quasi-metric properties of complexity spaces. Topology and its Applications, 98(1-3), 311-322. doi:10.1016/s0166-8641(98)00102-3García-Raffi, L. M., Romaguera, S., & Schellekens, M. P. (2008). 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-

Unified common fixed point theorems under weak reciprocal continuity or without continuity

[EN] The purpose of this paper is two fold. Firstly, using the notion of weak reciprocal continuity due to Pant et al. Weak reciprocal continuity and fixed point theorems, Ann. Univ. Ferrara Sez. VII Sci. Mat. 57(1), 181-190 (2011)], we prove unified common fixed point theorems for various variants of compatible and $R$-weakly commuting mappings in complete metric spaces employing an implicit relation which covers a multitude of contraction conditions yielding thereby known as well as unknown results as corollaries. Secondly, we point out that more natural results can be proved under relatively tighter conditions if we replace the completeness of the space by completeness of suitable subspaces. The realized improvements in our results are also substantiated using appropriate examples.Kadelburg, Z.; Imdad, M.; Chauhan, S. (2014). Unified common fixed point theorems under weak reciprocal continuity or without continuity. Applied General Topology. 15(1):65-84. doi:10.4995/agt.2014.1823SWORD6584151J. Ali and M. Imdad, An implicit function implies several contraction conditions, Sarajevo J. Math. 4, no. 2 (2008), 269-285.S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fund. Math. 3 (1922), 133-181.B. C. Dhage, On common fixed points of coincidentally commuting mappings in $D$-metric spaces, Indian J. Pure Appl. Math. 30, no. 4 (1999), 395-406.Husain, S. A., & Sehgal, V. M. (1975). On common fixed points for a family of mappings. Bulletin of the Australian Mathematical Society, 13(2), 261-267. doi:10.1017/s000497270002445xM. Imdad and J. Ali, Reciprocal continuity and common fixed points of nonself mappings, Taiwanese J. Math. 13, no. 5 (2009), 1457-1473.Imdad, M., Ali, J., & Tanveer, M. (2009). Coincidence and common fixed point theorems for nonlinear contractions in Menger PM spaces. Chaos, Solitons & Fractals, 42(5), 3121-3129. doi:10.1016/j.chaos.2009.04.017M. Imdad and Q. H. Khan, Six mappings satisfying a rational inequality, Rad. Mat. 9, no. 2 (1999), 251-260.Imdad, M., Khan, M. S., & Sessa, S. (1988). On some weak conditions of commutativity in common fixed point theorems. International Journal of Mathematics and Mathematical Sciences, 11(2), 289-296. doi:10.1155/s0161171288000353M. Imdad, S. Kumar and M. S. Khan, Remarks on some fixed point theorems satisfying implicit relations. Rad. Mat. 11, no. 1 (2002), 135-143.Jungck, G. (1976). Commuting Mappings and Fixed Points. The American Mathematical Monthly, 83(4), 261. doi:10.2307/2318216Jungck, G. (1986). Compatible mappings and common fixed points. International Journal of Mathematics and Mathematical Sciences, 9(4), 771-779. doi:10.1155/s0161171286000935G. Jungck and B. E. Rhoades, Fixed points for set valued functions without continuity, Indian J. Pure Appl. Math. 29, no. 3 (1998), 227-238.M. S. Khan and M. Imdad, A common fixed point theorem for a class of mappings, Indian J. Pure Appl. Math. 14 (1983), 1220-1227.S. Kumar and R. Chugh, Common fixed points theorem using minimal commutativity and reciprocal continuity conditions in metric space, Sci. Math. Japon. 56, no. 2 (2002), 269-275.Murthy, P. P. (2001). Important tools and possible applications of metric fixed point theory. Nonlinear Analysis: Theory, Methods & Applications, 47(5), 3479-3490. doi:10.1016/s0362-546x(01)00465-5Pant, R. P. (1994). Common Fixed Points of Noncommuting Mappings. Journal of Mathematical Analysis and Applications, 188(2), 436-440. doi:10.1006/jmaa.1994.1437R. P. Pant, Common fixed points of four mappings, Bull. Cal. Math. Soc. 90 (1998), 281-286.R. P. Pant, Noncompatible mappings and common fixed points, Soochow J. Math. 26 (2000), 29-35.Pant, R. P., Bisht, R. K., & Arora, D. (2011). Weak reciprocal continuity and fixed point theorems. ANNALI DELL’UNIVERSITA’ DI FERRARA, 57(1), 181-190. doi:10.1007/s11565-011-0119-3H. K. Pathak, Y. J. Cho and S. M. Kang, Remarks on $R$-weakly commuting mappings and common fixed point theorems, Bull. Korean Math. Soc. 34, no. 2 (1997), 247-257.H. K. Pathak and M. S. Khan, A comparison of various types of compatible maps and common fixed points, Indian J. Pure Appl. Math. 28, no. 4 (1997), 477-485.V. Popa, Some fixed point theorems for compatible mappings satisfying an implicit relation, Demonstratio Math. 32, no. 1 (1999), 157-163.V. Popa, M. Imdad and J. Ali, Fixed point theorems for a class of mappings governed by strictly contractive implicit function, Southeast Asian Bulletin of Math. 34, no. 5 (2010), 941-952.S. L. Singh and A. Tomar, Weaker forms of commuting maps and existence of fixed points, J. Korea Soc. Math. Educ. Ser. B Pure Appl. Math. 10, no. 3 (2003), 145-161