Abstract
A technical extension is given for the fixed point result in Liu et al. [J. Appl. Math., Volume 2012, Article ID: 786061].
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References
M. Altman, An integral test for series and generalized contractions. Am. Math. Mon. 82, 827–829 (1975)
S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fundam. Math. 3, 133–181 (1922)
V. Berinde, M. Păcurar, The role of the Pompeiu-Hausdorff metric in fixed point theory. Creat. Math. Inform. 22, 35–42 (2013)
P. Bernays, A system of axiomatic set theory: Part III. Infinity and enumerability analysis. J. Symb. Log. 7, 65–89 (1942)
D.W. Boyd, J.S.W. Wong, On nonlinear contractions. Proc. Am. Math. Soc. 20, 458–464 (1969)
B.S. Choudhury, N. Metiya, Fixed point theorems for almost contractions in partially ordered metric spaces. Ann. Univ. Ferrara 58, 21–36 (2012)
L.B. Cirić, Fixed point theorems for multi-valued contractions in complete metric spaces. J. Math. Anal. Appl. 348, 499–507 (2008)
L.B. Cirić, Multi-valued nonlinear contraction mappings. Nonlinear Anal. 71, 2716–2723 (2009)
P.J. Cohen, Set Theory and the Continuum Hypothesis (Benjamin, New York, 1966)
P.Z. Daffer, H. Kaneko, Fixed points of generalized contractive multi-valued mappings. J. Math. Anal. Appl. 192, 655–666 (1995)
W.-S. Du, F. Khojasteh, Y.-N. Chiu, Some generalizations of Mizoguchi-Takahashi’s fixed point theorem with new local constraints. Fixed Point Theory Appl. 2014, 31 (2014)
Y. Feng, S. Liu, Fixed point theorems for multi-valued contractive mappings and multi-valued Caristi type mappings. J. Math. Anal. Appl. 317, 103–112 (2006)
P.R. Halmos, Naive Set Theory (Van Nostrand Reinhold Co., New York, 1960)
P. Hitzler, Generalized metrics and topology in logic programming semantics. PhD Thesis, Natl. Univ. Ireland, Univ. College Cork, 2001
J. Jachymski, Common fixed point theorems for some families of mappings. Indian J. Pure Appl. Math. 25, 925–937 (1994)
J. Jachymski, The contraction principle for mappings on a metric space with a graph. Proc. Am. Math. Soc. 136, 1359–1373 (2008)
M. Javahernia, A. Razani, F. Khojasteh, Fixed point of multi-valued contractions via manageable functions and Liu’s generalization. Cogent Math. 3, 1276818 (2016)
C.F.K. Jung, On generalized complete metric spaces. Bull. Amer. Math. Soc. 75, 113–116 (1969)
T. Kamran, Mizoguchi-Takahashi’s type fixed point theorem. Comput. Math. Appl. 57, 507–511 (2009)
S. Kasahara, On some generalizations of the Banach contraction theorem. Publ. Res. Inst. Math. Sci. Kyoto Univ. 12, 427–437 (1976)
M.S. Khan, M. Swaleh, S. Sessa, Fixed point theorems by altering distances between the points. Bull. Aust. Math. Soc. 30, 1–9 (1984)
F. Khojasteh, V. Rakočević, Some new common fixed point results for generalized contractive multi-valued nonself-mappings. Appl. Math. Lett. 25, 287–293 (2012)
D. Klim, D. Wardowski, Fixed point theorems for set-valued contractions in complete metric spaces. J. Math. Anal. Appl. 334, 132–139 (2007)
K. Kuratowski, Topology, vol. I (Academic Press, New York, 1966)
S. Leader, Fixed points for general contractions in metric spaces. Math. Jpn. 24, 17–24 (1979)
Z. Liu, W. Sun, S.M. Kang, J.S. Ume, On fixed point theorems for multivalued contractions. Fixed Point Theory Appl. 2010, Article ID: 870980 (2010)
Z. Liu, Z. Wu, S.M. Kang, S. Lee, Some fixed point theorems for nonlinear set-valued contractive mappings. J. Appl. Math. 2012, Article ID: 786061 (2012)
W.A.J. Luxemburg, On the convergence of successive approximations in the theory of ordinary differential equations (II). Indag. Math. 20, 540–546 (1958)
J. Matkowski, Integrable Solutions of Functional Equations. Dissertationes Math., vol. 127 (Polish Sci. Publ., Warsaw, 1975)
A. Meir, E. Keeler, A theorem on contraction mappings. J. Math. Anal. Appl. 28, 326–329 (1969)
N. Mizoguchi, W. Takahashi, Fixed point theorems for multivalued mappings on complete metric spaces. J. Math. Anal. Appl. 141, 177–188 (1989)
G.H. Moore, Zermelo’s Axiom of Choice: Its Origin, Development and Influence (Springer, New York, 1982)
Y. Moskhovakis, Notes on Set Theory (Springer, New York, 2006)
S.B. Nadler Jr., Multi-valued contraction mappings. Pac. J. Math. 30, 475–488 (1969)
J.J. Nieto, R. Rodriguez-Lopez, Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. Order 22, 223–239 (2005)
A.C.M. Ran, M.C. Reurings, A fixed point theorem in partially ordered sets and some applications to matrix equations. Proc. Am. Math. Soc. 132, 1435–1443 (2004)
S. Reich, Fixed points of contractive functions. Boll. Un. Mat. Ital. 5, 26–42 (1972)
B.E. Rhoades, A comparison of various definitions of contractive mappings. Trans. Am. Math. Soc. 226, 257–290 (1977)
I.A. Rus, Generalized Contractions and Applications (Cluj University Press, Cluj-Napoca, 2001)
B. Samet, M. Turinici, Fixed point theorems on a metric space endowed with an arbitrary binary relation and applications. Commun. Math. Anal. 13, 82–97 (2012)
E. Schechter, Handbook of Analysis and Its Foundation (Academic Press, New York, 1997)
T. Suzuki, A generalized Banach contraction principle that characterizes metric completeness. Proc. Am. Math. Soc. 136, 1861–1869 (2008)
A. Tarski, Axiomatic and algebraic aspects of two theorems on sums of cardinals. Fundam. Math. 35, 79–104 (1948)
V. Timofte, New tests for positive iteration series. Real Anal. Exchange 30, 799–812 (2004/2005)
M. Turinici, Nonlinear contractions and applications to Volterra functional equations. An. Şt. Univ. “Al. I. Cuza” Iaşi (S I-a, Mat) 23, 43–50 (1977)
M. Turinici, Multivalued contractions and applications to functional differential equations. Acta Math. Acad. Sci. Hung. 37, 147–151 (1981)
M. Turinici, Fixed points for monotone iteratively local contractions. Demonstratio Math. 19, 171–180 (1986)
M. Turinici, Abstract comparison principles and multivariable Gronwall-Bellman inequalities. J. Math. Anal. Appl. 117, 100–127 (1986)
M. Turinici, Function pseudometric VP and applications. Bul. Inst. Polit. Iaşi (S. Mat. Mec. Teor. Fiz.) 53(57), 393–411 (2007)
M. Turinici, Wardowski implicit contractions in metric spaces. arXiv 1211-3164-v2, 15 Sept 2013
M. Turinici, Contraction maps in pseudometric structures, in Essays in Mathematics and Its Applications, ed. by T.M. Rassias, P.M. Pardalos (Springer Intl. Publ., Switzerland, 2016), pp. 513–562
D. Wardowski, Fixed points of a new type of contractive mappings in complete metric spaces. Fixed Point Theory Appl. 2012, 94 (2012)
E.S. Wolk, On the principle of dependent choices and some forms of Zorn’s lemma. Can. Math. Bull. 26, 365–367 (1983)
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Turinici, M. (2020). Nadler-Liu Functional Contractions in Metric Spaces. In: Daras, N., Rassias, T. (eds) Computational Mathematics and Variational Analysis. Springer Optimization and Its Applications, vol 159. Springer, Cham. https://doi.org/10.1007/978-3-030-44625-3_28
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