×

The existence of cone critical point and common fixed point with applications. (English) Zbl 1304.54077

In the first part of this paper, the authors establish some fixed point theorems and critical point theorems in cone metric spaces. The abstract setting corresponds to multivalued maps and key ingredients in the proofs are the Ekeland variational principle and the fuzzy fixed point theorem. Some equivalent properties are established in the final section of this paper.

MSC:

54H25 Fixed-point and coincidence theorems (topological aspects)
54C60 Set-valued maps in general topology
54A40 Fuzzy topology
54E40 Special maps on metric spaces
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] S. Danc\vs, M. Heged\Hus, and P. Medvegyev, “A general ordering and fixed-point principle in complete metric space,” Acta Scientiarum Mathematicarum, vol. 46, no. 1-4, pp. 381-388, 1983. · Zbl 0532.54030
[2] D. H. Hyers, G. Isac, and T. M. Rassias, Topics in Nonlinear Analysis and Applications, World Scientific, River Edge, NJ, USA, 1997. · Zbl 0878.47040 · doi:10.1142/9789812830432
[3] L.-J. Lin and W.-S. Du, “From an abstract maximal element principle to optimization problems, stationary point theorems and common fixed point theorems,” Journal of Global Optimization, vol. 46, no. 2, pp. 261-271, 2010. · Zbl 1209.90320 · doi:10.1007/s10898-009-9423-1
[4] M. Amemiya and W. Takahashi, “Fixed point theorems for fuzzy mappings in complete metric spaces,” Fuzzy Sets and Systems, vol. 125, no. 2, pp. 253-260, 2002. · Zbl 0987.54051 · doi:10.1016/S0165-0114(01)00046-X
[5] S. S. Chang and Q. Luo, “Caristi’s fixed point theorem for fuzzy mappings and Ekeland’s variational principle,” Fuzzy Sets and Systems, vol. 64, no. 1, pp. 119-125, 1994. · Zbl 0842.54041 · doi:10.1016/0165-0114(94)90014-0
[6] W.-S. Du, “On some nonlinear problems induced by an abstract maximal element principle,” Journal of Mathematical Analysis and Applications, vol. 347, no. 2, pp. 391-399, 2008. · Zbl 1148.49013 · doi:10.1016/j.jmaa.2008.06.020
[7] B. G. Kang and S. Park, “On generalized ordering principles in nonlinear analysis,” Nonlinear Analysis, Theory, Methods & Applications, vol. 14, no. 2, pp. 159-165, 1990. · Zbl 0712.54022 · doi:10.1016/0362-546X(90)90021-8
[8] L.-J. Lin and W.-S. Du, “Ekeland’s variational principle, minimax theorems and existence of nonconvex equilibria in complete metric spaces,” Journal of Mathematical Analysis and Applications, vol. 323, no. 1, pp. 360-370, 2006. · Zbl 1101.49022 · doi:10.1016/j.jmaa.2005.10.005
[9] L.-J. Lin and W.-S. Du, “Some equivalent formulations of the generalized Ekeland’s variational principle and their applications,” Nonlinear Analysis, Theory, Methods & Applications, vol. 67, no. 1, pp. 187-199, 2007. · Zbl 1111.49013 · doi:10.1016/j.na.2006.05.006
[10] L.-J. Lin and W.-S. Du, “On maximal element theorems, variants of Ekeland’s variational principle and their applications,” Nonlinear Analysis, Theory, Methods & Applications, vol. 68, no. 5, pp. 1246-1262, 2008. · Zbl 1133.58006 · doi:10.1016/j.na.2006.12.018
[11] L.-J. Lin and W.-S. Du, “Systems of equilibrium problems with applications to new variants of Ekeland’s variational principle, fixed point theorems and parametric optimization problems,” Journal of Global Optimization, vol. 40, no. 4, pp. 663-677, 2008. · Zbl 1218.49014 · doi:10.1007/s10898-007-9146-0
[12] W. Takahashi, Nonlinear Functional Analysis, Yokohama Publishers, Yokohama, Japan, 2000. · Zbl 0997.47002
[13] A. I. Perov, “The Cauchy problem for systems of ordinary di erential equations,” in Approximate Methods of Solving di Erential Equations, pp. 115-134, Naukova Dumka, Kiev, Ukraine, 1964.
[14] B. V. Kvedaras, A. V. Kibenko, and A. I. Perov, “On certain bundary value problems,” Litovskiĭ Matematicheskiĭ Sbornik, vol. 5, pp. 69-84, 1965.
[15] A. I. Perov and A. V. Kibenko, “On a certain general method for investigation of boundary value problems,” Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya, vol. 30, pp. 249-264, 1966.
[16] E. M. Mukhamadiev and V. J. Stetsenko, “Fixed point principle in generalized metric space,” Izvestiya Akademii Nauki Tadzhikskoi SSR, vol. 10, pp. 9-19, 1969.
[17] J. S. Vandergraft, “Newton’s method for convex operators in partially ordered spaces,” SIAM Journal on Numerical Analysis, vol. 4, pp. 406-432, 1967. · Zbl 0161.35302 · doi:10.1137/0704037
[18] P. P. Zabrejko, “K-metric and K-normed linear spaces: survey,” Collectanea Mathematica, vol. 48, no. 4-6, pp. 825-859, 1997. · Zbl 0892.46002
[19] L.-G. Huang and X. Zhang, “Cone metric spaces and fixed point theorems of contractive mappings,” Journal of Mathematical Analysis and Applications, vol. 332, no. 2, pp. 1468-1476, 2007. · Zbl 1118.54022 · doi:10.1016/j.jmaa.2005.03.087
[20] M. Abbas and G. Jungck, “Common fixed point results for noncommuting mappings without continuity in cone metric spaces,” Journal of Mathematical Analysis and Applications, vol. 341, no. 1, pp. 416-420, 2008. · Zbl 1147.54022 · doi:10.1016/j.jmaa.2007.09.070
[21] M. Abbas and B. E. Rhoades, “Fixed and periodic point results in cone metric spaces,” Applied Mathematics Letters, vol. 22, no. 4, pp. 511-515, 2009. · Zbl 1167.54014 · doi:10.1016/j.aml.2008.07.001
[22] B. S. Choudhury and N. Metiya, “Fixed points of weak contractions in cone metric spaces,” Nonlinear Analysis, Theory, Methods & Applications, vol. 72, no. 3-4, pp. 1589-1593, 2010. · Zbl 1191.54036 · doi:10.1016/j.na.2009.08.040
[23] W.-S. Du, “A note on cone metric fixed point theory and its equivalence,” Nonlinear Analysis, Theory, Methods & Applications, vol. 72, no. 5, pp. 2259-2261, 2010. · Zbl 1205.54040 · doi:10.1016/j.na.2009.10.026
[24] S. Rezapour and R. Hamlbarani, “Some notes on the paper: “Cone metric spaces and fixed point theorems of contractive mappings”,” Journal of Mathematical Analysis and Applications, vol. 345, no. 2, pp. 719-724, 2008. · Zbl 1145.54045 · doi:10.1016/j.jmaa.2008.04.049
[25] S. Rezapour and R. H. Haghi, “Fixed point of multifunctions on cone metric spaces,” Numerical Functional Analysis and Optimization, vol. 30, no. 7-8, pp. 825-832, 2009. · Zbl 1171.54033 · doi:10.1080/01630560903123346
[26] B. Samet, “Common fixed point under contractive condition of Ćirić’s type in cone metric spaces,” Applicable Analysis and Discrete Mathematics, vol. 5, no. 1, pp. 159-164, 2011. · Zbl 1265.54192 · doi:10.2298/AADM110206007S
[27] B. Samet, “Common fixed point theorems involving two pairs of weakly compatible mappings in K-metric spaces,” Applied Mathematics Letters, vol. 24, pp. 1245-1250, 2011. · Zbl 1243.54075
[28] W.-S. Du, “Nonlinear contractive conditions for coupled cone fixed point theorems,” Fixed Point Theory and Applications, vol. 2010, Article ID 190606, 16 pages, 2010. · Zbl 1220.54022 · doi:10.1155/2010/190606
[29] W.-S. Du, “New cone fixed point theorems for nonlinear multivalued maps with their applications,” Applied Mathematics Letters, vol. 24, no. 2, pp. 172-178, 2011. · Zbl 1218.54037 · doi:10.1016/j.aml.2010.08.040
[30] R. K. Bose and D. Sahani, “Fuzzy mappings and fixed point theorems,” Fuzzy Sets and Systems, vol. 21, no. 1, pp. 53-58, 1987. · Zbl 0609.54032 · doi:10.1016/0165-0114(87)90152-7
[31] S. S. Chang, “Fixed point theorems for fuzzy mappings,” Fuzzy Sets and Systems, vol. 17, no. 2, pp. 181-187, 1985. · Zbl 0579.54034 · doi:10.1016/0165-0114(85)90055-7
[32] W.-S. Du, “Critical point theorems for nonlinear dynamical systems and their applications,” Fixed Point Theory and Applications, vol. 2010, Article ID 246382, 16 pages, 2010. · Zbl 1213.49023 · doi:10.1155/2010/246382
[33] G.-y. Chen, X. Huang, and X. Yang, Vector Optimization, vol. 541 of Lecture Notes in Economics and Mathematical Systems, Springer, Berlin, Germany, 2005. · Zbl 1104.90044
[34] A. E. Taylor and D. C. Lay, Introduction to Functional Analysis, John Wiley & Sons, New York, NY, USA, 2nd edition, 1980. · Zbl 0501.46003
[35] R. Kannan, “Some results on fixed points. II,” The American Mathematical Monthly, vol. 76, pp. 405-408, 1969. · Zbl 0179.28203 · doi:10.2307/2316437
[36] S. K. Chatterjea, “Fixed-point theorems,” Comptes Rendus de l’Académie Bulgare des Sciences, vol. 25, pp. 727-730, 1972. · Zbl 0274.54033
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.