Abstract
Since the year 1922, Banach’s contraction principle, due to its simplicity and usability, has become a popular tool in modern analytics, particularly in nonlinear analysis, including the use of equations, differential equations, variance, equilibrium problems, and much more (see, e.g., [1,2,3,4,5,6,7,8,9,10]).
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References
Z. Golubović, Z. Kadelburg, S. Radenović, Coupled coincidence points of mappings in ordered partial metric spaces. Abstr. Appl. Anal. 2012, Article ID 192581, 18 p (2012)
M.A. Khamsi, A. Latif, H. Al-Sulami, KKM and Ky Fan theorems in modular function spaces. Fixed Point Theory Appl. 2011, 57 (2011)
M.A. Khamsi, Quasicontraction mappings in modular spaces without \(\Delta _{2}\)-condition. Fixed Point Theory Appl. 2008, Article ID 916187, 6 p (2008)
K. Kuaket, P. Kumam, Fixed points of asymptotic pointwise contractions in modular spaces. Appl. Math. Lett. 24, 1795–1798 (2011)
A. Amini-harandini, Fixed point theory for generalized quasicontraction maps in vector modular spaces. Comp. Math. Appl. 61, 1891–1897 (2011)
H. Aydi, H.K. Nashine, B. Samet, H. Yazidi, Coincidence and common fixed point results in partially ordered cone metric spaces and applications to integral equations. Nonlinear Anal. 74(17), 6814–6825 (2011)
H. Aydi, Common fixed point results for mappings satisfying (\(\psi \),\(\phi \))-weak contractions in ordered partial metric spaces. Int. J. Math. Stat. 12(2), 53–64 (2012)
W. Sintunavarat, P. Kumam, Weak condition for generalized multi-valued \((f,\alpha, \beta )\)-weak contraction mappings. Appl. Math. Lett. 24, 460–465 (2011)
W. Sintunavarat, Y.J. Cho, P. Kumam, Common fixed point theorems for \(c\)-distance in ordered cone metric spaces. Comput. Math. Appl. 62, 1969–1978 (2011)
C. Mongkolkeha, P. Kumam, Fixed point theorems for generalized asymptotic pointwise \(\rho \)-contraction mappings involving orbits in modular function spaces. Appl. Math. Lett. https://doi.org/10.1016/j.aml.2011.11.027 (2012) (in press)
M. Geraghty, On contractive mappings. Proc. Am. Math. Soc. 40, 604–608 (1973)
A. Amini-Harandi, H. Emani, A fixed point theorem for contraction type maps in partially ordered metric spaces and application to ordinary differential equations. Nonlinear Anal. 72, 2238–2242 (2010)
M. Eshaghi Gordji, M. Ramezani, Y.J. Cho, S. Pirbavafa, A generalization of Geraghty’s theorem in partially ordered metric spaces and application to ordinary differential equations. Fixed point Theory Appl. 2012, 74 (2012)
V.V. Chistyakov, Modular metric spaces, I: basic concepts. Nonlinear Anal. 72, 1–14 (2010)
H. Nakano, in Modulared Semi-Ordered Linear Spaces, vol. 1 (Tokyo Math. Book Ser.) (Maruzen Co., Tokyo, 1950)
H. Nakano, in Topology and Linear Topological Spaces, vol. 3 (Tokyo Math. Book Ser.) (Maruzen Co., Tokyo, 1951)
P. Kumam, Some geometric properties and fixed point theorem in modular spaces, in Fixed Point Theorem and Its Applications, ed. by J. Garcia Falset, L. Fuster, B. Sims (Yokohama Publishers, Yokohama, 2004), pp. 173–188
P. Kumam, Fixed point theorems for nonexpansive mappings in modular spaces. Archivum Mathematicum (BRONO) 40(4), 345–353 (2004)
P. Kumam, On uniform opial condition and uniform Kadec-Klee property in modular spaces. J. Interdisc. Math. 8(3), 377–385 (2005)
P. Kumam, On nonsquare and Jordan-Von Neumann constants of modular spaces. SE Asian Bull. Math. 30(1), 67–77 (2006)
C. Mongkolkeha, P. Kumam, Common fixed point for generalized weak contraction mapping in Modular spaces. Scientiae Mathematicae Japonicae, online, e-2012, 117–127
Y.J. Cho, R. Saadati, G. Sadeghi, Quasi-contractive mappings in modular metric spaces. J. Appl. Math. 2012, Article ID 907951, 5 p (2012)
P. Chaipunya, C. Mongkolkeha, W. Sintunavarat, P. Kumam, Fixed-point theorems for multivalued mappings in modular metric spaces. Abstr. Appl. Anal. 2012, Article ID 503504, 14 p (2012)
C. Mongkolkeha, W. Sintunavarat, P. Kumam, Fixed point theorems for contraction mappings in modular metric spaces. Fixed Point Theory Appl. 2011, 93 (2011)
P. Chaipunya, J.Y. Cho, P. Kumam, Geraghty-type theorems in modular metric spaces with an application to partial differential equation. Adv. Differ. Equ. 2012, Article ID 83 (2012)
M.R. Taskovic, A generalization of Banach’s contraction principle. Nouvell serie tome. 23(37), 179–191 (1978)
L.J. Ćirić, A generalization of Banachs contraction principle. Proc. Am. Math. Soc. 45, 267273 (1974)
S. Czerwik, Contraction mappings in \(b\)-metric spaces. Acta Math. Inform. Univ. Ostrav. 1, 5–11 (1993)
T. Suzuki, Generalized distance and existence theorems in complete metric spaces. J. Math. Anal. App. 253(2), 440458 (2001)
H.L. Guang, Z. Xian, Cone metric spaces and fixed point theorems of contractive mappings. J. Math. Anal. Appl 332, 14681476 (2007)
B. Samet, C. Vetro, P. Vetro, Fixed point theorems for \(\alpha \)-\(\psi \)-contractive type mappings. Nonlinear Anal. 75, 2154–2165 (2012)
D. Gopal, C. Vetro, M. Abbas, D.K. Patel, Some coincidence and periodic points results in a metric space endowed with a graph and applications. Banach J. Math. Anal. 9(3), 128–139 (2015)
D. Gopal, M. Abbas, D.K. Patel, C. Vetro, Fixed points of \(\alpha \)-type \(F\)-contractive mappings with anapplication to nonlinear fractional differential equation. Acta Math. Scientia 36B(3), 1–14 (2016)
N. Hussain, A. Latif, I. Iqbal, Fixed point results for generalized \(F\)-contractions in modular metric and fuzzy metric spaces. Fixed Point Theory Appl. 2015, 158 (2015)
A. Padcharoen, D. Gopal, P. Chaipunya, P. Kumam, Fixed point and periodic point results \(\alpha \)-type \(F\)-contractions in modular metric spaces. Fixed Point Theory Appl. 2016 12 p (2016)
D. Wardowski, Fixed points of new type of contractive mappings in complete metric space. Fixed Point Theory Appl. (2012). https://doi.org/10.1186/1687-1812-2012-94
A.N.N. Abdou, M.A. Khamsi, Fixed point results of pointwise contractions in modular metric spaces. Fixed Point Theory Appl. (1), 163 (2013)
J. Jachymski, The contraction principle for mappings on a metric space with a graph. Proc. Am. Math. Soc. 1(136), 1359–1373 (2008)
A.C. Ran, M.C.B. Reurings, A fixed point theorem in partially ordered sets and some applications to matrix equations. Proc. Am. Math. Soc. 132(5), 1435–1443 (2004)
G. Chartrand, L. Lesniak, P. Zhang, Graphs & Digraphs (CRC Press, New York, 2011)
R. Johnsonbaugh, Discrete Mathematics (Prentice Hall, New York, 1997)
M.R. Alfuraidan, On monotone Ćirić quasi-contraction principle for mappings with a graph. Fixed Point Theory Appl. 2015, 93 11 p (2015)
A. Padcharoen, P. Kumam, D. Gopal, Coincidence and periodic point results in a modular metric space endowed with a graph and applications. Creat. Math. Inform. 26(1), 95–104 (2017)
G.S. Jeong, B.E. Rhoades, Maps for which \(F(T) = F(T^n)\). Fixed Point Theory Appl. 6, 87–131 (2005)
C. Chifu, G. Petrusel, Generalized contractions in metric space endowed with a graph. Fixed Point Theory Appl. 2012, 161, 9 p (2012)
W. Sintunavarat, P. Kumam, Common fixed point theorems for a pair of weakly compatible mappings in fuzzy metric spaces. J. Appl. Math. 2011, Article ID 637958, 14 p
P. Sumalai, P. Kumam, Y.J. Cho, A. Padcharoen, The \((CLR_g)\)-property for coincidence point theorems and Fredholm integral equations in modular metric spaces. Eur. J. Pure Appl. Math. 10(2), 238–254 (2017)
P. Chaipunya, C. Mongkolkeha, W. Sintunavarat, P. Kumam, Erratum to “fixed-point theorems for multivalued mappings in modular metric spaces”. Abstr. Appl. Anal. 2012, 2 (2012)
P. Chaipunya, P. Kumam, An observation on set-valued contraction mappings in modular metric spaces. Thai J. Math. 13(1), 9–17 (2015)
S.B. Nadler Jr., Multi-valued contraction mappings. Pac. J. Math. 30, 475–488 (1969)
M. Sgroi, C. Vetro, Multi-valued \(F\)-contractions and the solution of certain functional and integral equations. Filomat 27, 1259–1268 (2013)
D. Jain, A. Padcharoen, P. Kumam, D. Gopal, A new approach to study fixed point of multivalued mappings in modular metric spaces and applications. Mathematics 4, 51 (2016). https://doi.org/10.3390/math4030051
S. Moradi, M. Mohammadi Anjedani, E. Analoei, On existence and uniqueness of solutions of a nonlinear Volterra-Fredholm integral equation. Int. J. Nonlinear Anal. Appl. 6, 62–68 (2015)
M. Cosentino, P. Vetro, Fixed point results for \(F\)-contractive mappings of Hardy-Rogers-type. Filomat 28(4), 715–722 (2014)
H. Piri, P. Kumam, Some fixed point theorems concerning \(F\)-contraction in complete metric spaces. Fixed Point Theory Appl. 2014, 210, 11 p (2014)
B.E. Rhoades, Some theorems on weakly contractive maps. Nonlinear Anal. 47, 2683–2693 (2001)
V.V. Chistyakov, Fixed points of modular contractive maps. Doklady Math. 86(1), 515–518 (2012)
V.V. Chistyakov, Modular contractions and their application, in Models, Algorithms, and Technologies for Network Analysis, volume 32 of Springer Proceedings in Mathematics & Statistics, ed. by B. Goldengorin, V.A. Kalyagin, P.M. Pardalos (Springer, New York, 2013), pp. 65–92
O. Acar, G. Durmaz, G. Minak, Generalized multivalued \(F\)-contractions on complete metric spaces. Bull. Iranian Math. Soc. 40, 1469–1478 (2014)
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Padcharoen, A., Chaipunya, P., Kumam, P. (2018). Existence Theory on Modular Metric Spaces. In: Agarwal, P., Dragomir, S., Jleli, M., Samet, B. (eds) Advances in Mathematical Inequalities and Applications. Trends in Mathematics. Birkhäuser, Singapore. https://doi.org/10.1007/978-981-13-3013-1_2
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