Multivariate fixed point theorems for contractions and nonexpansive mappings with applications

详细信息    查看全文

• 作者：Yongfu Su ; Adrian Petruşel ; Jen-Chih Yao
• 关键词：contraction mapping principle ; complete metric spaces ; multivariate fixed point ; multiply metric function ; multivariate mapping ; differential equation ; strong and weak convergence
• 刊名：Fixed Point Theory and Applications
• 出版年：2016
• 出版时间：December 2016
• 年：2016
• 卷：2016
• 期：1
• 全文大小：1,702 KB
• 参考文献：1. Browder, FE: On the convergence of successive approximations for nonlinear functional equations. Nederl. Akad. Wetensch. Proc. Ser. A 71=Indag. Math. 30, 27-35 (1968) MathSciNet CrossRef
  2. Boyd, DW, Wong, JSW: On nonlinear contractions. Proc. Am. Math. Soc. 20, 458-464 (1969) MATH MathSciNet CrossRef
  3. Jachymski, J: Equivalence of some contractivity properties over metrical structures. Proc. Am. Math. Soc. 125, 2327-2335 (1997) MATH MathSciNet CrossRef
  4. Jachymski, J, Jóźwik, I: Nonlinear contractive conditions: a comparison and related problems. In: Fixed Point Theory and Its Applications. Polish Acad. Sci., vol. 77, pp. 123-146. Banach Center Publ., Warsaw (2007) CrossRef
  5. Geraghty, M: On contractive mappings. Proc. Am. Math. Soc. 40, 604-608 (1973) MATH MathSciNet CrossRef
  6. Samet, B, Vetro, C, Vetro, P: Fixed point theorem for α-ψ-contractive type mappings. Nonlinear Anal. 75, 2154-2165 (2012) MATH MathSciNet CrossRef
  7. Su, Y, Yao, J-C: Further generalized contraction mapping principle and best proximity theorem in metric spaces. Fixed Point Theory Appl. 2015, 120 (2015) CrossRef
  8. Kir, M, Kiziltunc, H: The concept of weak \((\psi,\alpha,\beta)\) contractions in partially ordered metric spaces. J. Nonlinear Sci. Appl. 8(6), 1141-1149 (2015) MathSciNet
  9. Asgari, MS, Badehian, Z: Fixed point theorems for α-β-ψ-contractive mappings in partially ordered sets. J. Nonlinear Sci. Appl. 8(5), 518-528 (2015) MathSciNet
  10. Amini-Harandi, A, Emami, H: 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) MATH MathSciNet CrossRef
  11. Harjani, J, Sadarangni, K: Fixed point theorems for weakly contraction mappings in partially ordered sets. Nonlinear Anal. 71, 3403-3410 (2009) MATH MathSciNet CrossRef
  12. Gnana Bhaskar, T, Lakshmikantham, V: Fixed point theorems in partially ordered metric spaces and applications. Nonlinear Anal. 65, 1379-1393 (2006) MATH MathSciNet CrossRef
  13. Lakshmikantham, V, Ćirić, L: Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces. Nonlinear Anal. 70, 4341-4349 (2009) MATH MathSciNet CrossRef
  14. Nieto, JJ, Rodriguez-López, R: Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. Order 22, 223-239 (2005) MATH MathSciNet CrossRef
  15. Nieto, JJ, Rodriguez-López, R: Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations. Acta Math. Sin. 23, 2205-2212 (2007) MATH CrossRef
  16. O’Regan, D, Petruşel, A: Fixed point theorems for generalized contractions in ordered metric spaces. J. Math. Anal. Appl. 341, 1241-1252 (2008) MATH MathSciNet CrossRef
  17. Harjani, J, Sadarangni, K: Generalized contractions in partially ordered metric spaces and applications to ordinary differential equations. Nonlinear Anal. 72, 1188-1197 (2010) MATH MathSciNet CrossRef
  18. Yan, FF, Su, Y, Feng, Q: A new contraction mapping principle in partially ordered metric spaces and applications to ordinary differential equations. Fixed Point Theory Appl. 2012, 152 (2012) MathSciNet CrossRef
  19. Khan, MS, Swaleh, M, Sessa, S: Fixed point theorems by altering distances between the points. Bull. Aust. Math. Soc. 30(1), 1-9 (1984) MATH MathSciNet CrossRef
  20. Lee, H, Kim, S: Multivariate coupled fixed point theorems on ordered partial metric spaces. J. Korean Math. Soc. 51(6), 1189-1207 (2014) MATH MathSciNet CrossRef
  21. Prešić, SB: Sur une classe d’inéquations aux différences finies et sur la convergence de certaines suites. Publ. Inst. Math. (Belgr.) 5, 75-78 (1965)
  22. Ćirić, LB, Presić, SB: On Prešić type generalization of the Banach contraction principle. Acta Math. Univ. Comen. 76(2), 143-147 (2007) MATH
  23. Tasković, MR: Monotonic mappings on ordered sets, a class of inequalities with finite differences and fixed points. Publ. Inst. Math. (Belgr.) 17, 163-172 (1974)
  24. Wittmann, R: Approximation de points of nonexpansive mappings. Arch. Math. 58, 486-491 (1992) MATH MathSciNet CrossRef
  25. Lions, P: Approximation de contractions. C. R. Acad. Sci. Paris Sér. A-B 284, 1357-1359 (1997)
  26. Xu, HK: Another control condition in an iterative method for nonexpansive mappings. Bull. Aust. Math. Soc. 65, 109-113 (2002) MATH CrossRef
  27. Reich, S: Weak convergence theorems for nonexpansive mappings in Banach spaces. J. Math. Anal. Appl. 67, 274-276 (1979) MATH MathSciNet CrossRef
  28. Browder, FE, Petryshyn, WV: Construction of fixed points of nonlinear mappings in Hilbert spaces. J. Math. Anal. Appl. 20, 197-228 (1967) MATH MathSciNet CrossRef
  29. Halpern, B: Fixed points of nonexpanding maps. Bull. Am. Math. Soc. 73, 957-961 (1967) MATH CrossRef
  30. Kim, TH, Xu, HK: Strong convergence of modified Mann iterations. Nonlinear Anal. 61, 51-60 (2005) MATH MathSciNet CrossRef
  31. Kim, TH, Xu, HK: Strong convergence of modified Mann iterations for asymptotically nonexpansive mappings and semigroups. Nonlinear Anal. 64, 1140-1152 (2006) MATH MathSciNet CrossRef
  32. Lions, PL: Approximation de points fixes de contractions. C. R. Acad. Sci. Paris Sér. A-B 284, 1357-1359 (1977) MATH
  33. Matinez-Yanes, C, Xu, HK: Strong convergence of the CQ method for fixed point processes. Nonlinear Anal. 64, 2400-2411 (2006) MathSciNet CrossRef
  34. Nakajo, K, Takahashi, W: Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups. J. Math. Anal. Appl. 279, 372-379 (2003) MATH MathSciNet CrossRef
  35. O’Hara, JG, Pillay, P, Xu, HK: Iterative approaches to finding nearest common fixed points of nonexpansive mappings in Hilbert spaces. Nonlinear Anal. 54, 1417-1426 (2003) MATH MathSciNet CrossRef
  36. O’Hara, JG, Pillay, P, Xu, HK: Iterative approaches to convex feasibility problems in Banach spaces. Nonlinear Anal. 64, 2022-2042 (2006) MATH MathSciNet CrossRef
  37. Reich, S: Strong convergence theorems for resolvents of accretive operators in Banach spaces. J. Math. Anal. Appl. 75, 287-292 (1980) MATH MathSciNet CrossRef
  38. Scherzer, O: Convergence criteria of iterative methods based on Landweber iteration for solving nonlinear problems. J. Math. Anal. Appl. 194, 911-933 (1991) MathSciNet CrossRef
  39. Shioji, N, Takahashi, W: Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces. Proc. Am. Math. Soc. 125, 3641-3645 (1997) MATH MathSciNet CrossRef
  40. Tan, KK, Xu, HK: Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process. J. Math. Anal. Appl. 178(2), 301-308 (1993) MATH MathSciNet CrossRef
  41. Tan, KK, Xu, HK: Fixed point iteration processes for asymptotically nonexpansive mappings. Proc. Am. Math. Soc. 122, 733-739 (1994) MATH MathSciNet CrossRef
  42. Wittmann, R: Approximation of fixed points of nonexpansive mappings. Arch. Math. 58, 486-491 (1992) MATH MathSciNet CrossRef
  43. Xu, HK: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. 66, 240-256 (2002) MATH CrossRef
  44. Xu, HK: Remarks on an iterative method for nonexpansive mappings. Commun. Appl. Nonlinear Anal. 10(1), 67-75 (2003) MATH
  45. Xu, HK: Strong convergence of an iterative method for nonexpansive mappings and accretive operators. J. Math. Anal. Appl. 314, 631-643 (2006) MATH MathSciNet CrossRef
  
• 作者单位：Yongfu Su (1)
  Adrian Petruşel (2)
  Jen-Chih Yao (3)
  
  1. Department of Mathematics, Tianjin Polytechnic University, Tianjin, 300387, China
  2. Department of Mathematics, Babeş-Bolyai University, Cluj-Napoca, 400084, Romania
  3. Center for General Education, China Medical University, Taichung, 40402, Taiwan
  
• 刊物主题：Analysis; Mathematics, general; Applications of Mathematics; Differential Geometry; Topology; Mathematical and Computational Biology;
• 出版者：Springer International Publishing
• ISSN：1687-1812

文摘

The first purpose of this paper is to prove an existence and uniqueness result for the multivariate fixed point of a contraction type mapping in complete metric spaces. The proof is based on the new idea of introducing a convenient metric space and an appropriate mapping. This method leads to the changing of the non-self-mapping setting to the self-mapping one. Then the main result of the paper will be applied to an initial-value problem related to a class of differential equations of first order. The second aim of this paper is to prove strong and weak convergence theorems for the multivariate fixed point of a N-variables nonexpansive mapping. The results of this paper improve several important works published recently in the literature. Keywords contraction mapping principle complete metric spaces multivariate fixed point multiply metric function multivariate mapping differential equation strong and weak convergence