# An abstract point of view on iterative approximation schemes of fixed points for multivalued operators

Volume 6, Issue 2, pp 97--107
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### Authors

Adrian Petruşel - Department of Mathematics, Babeş-Bolyai University, Kogalniceanu Street no. 1, 400084 Cluj-Napoca, Romania. Ioan A. Rus - Department of Mathematics, Babeş-Bolyai University, Kogalniceanu Street no. 1, 400084 Cluj-Napoca, Romania.

### Abstract

In this paper we will present an abstract point of view on iterative approximation schemes of fixed points for multivalued operators. More precisely, we suppose that the algorithms are convergent and we will study the impact of this hypothesis in the theory of operatorial inclusiosns: data dependence, stability and Gronwall type lemmas. Some open problems are also presented.

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##### ISRP Style

Adrian Petruşel, Ioan A. Rus, An abstract point of view on iterative approximation schemes of fixed points for multivalued operators, Journal of Nonlinear Sciences and Applications, 6 (2013), no. 2, 97--107

##### AMA Style

Petruşel Adrian, Rus Ioan A., An abstract point of view on iterative approximation schemes of fixed points for multivalued operators. J. Nonlinear Sci. Appl. (2013); 6(2):97--107

##### Chicago/Turabian Style

Petruşel, Adrian, Rus, Ioan A.. "An abstract point of view on iterative approximation schemes of fixed points for multivalued operators." Journal of Nonlinear Sciences and Applications, 6, no. 2 (2013): 97--107

### Keywords

• multivalued operator
• fixed point
• strict fixed point
• iterative scheme
• multivalued Picard operator
• multivalued weakly Picard operator.

•  47H10
•  54H25

### References

• [1] J. Andres and L. Górniewicz, Topological Fixed Point Principles for Boundary Value Problems, Kluwer Acad. Publ., Dordrecht, 2003. , , , ()

• [2] J.-P. Aubin and A. Cellina, Differential Inclusions, Springer Verlag, Berlin, 1984. , , , ()

• [3] V. Berinde, Iterative Approximations of Fixed Points, Springer Verlag, Berlin, 2007., , , ()

• [4] [4] L.M. Blumenthal, Theory and Applications of Distance Geometry, Oxford University Press, 1953. , , , ()

• [5] S. Carl and S. Heikkilä, Fixed Point Theory in Ordered Sets and Applications, Springer, Berlin, 2011., , , ()

• [6] [6] S.S. Chang and K.-K. Tan, Iteration processes for approximating fixed points of operators by monotone type, Bull. Austral. Math. Soc., 57(1998), 433-445. , , , (),

• [7] C. Chidume, Geometric Properties of Banach Spaces and Nonlinear Iterations, Springer Verlag, Berlin, 2009. [, , , ()

• [8] 8] H. Covitz and S.B. Nadler jr., Multivalued contraction mappings in generalized metric spaces, Israel J. Math., 8(1970), 5-11. , , , (),

• [9] R. Espínola, A. Petruşel, Existence and data dependence of fixed points for multivalued operators on gauge spaces, J. Math. Anal. Appl., 309(2005), 420-432., , , (),

• [10] [10] Y. Feng and S. Liu, Fixed point theorems for multivalued increasing operators in partial ordered spaces, Soochow J. Math., 30(2004), No.4, 461-469. , , , (),

• [11] M. Fréchet, Les espaces abstraits, Gauthier-Villars, Paris, 1928., , , ()

• [12] [12] V. Glăvan and V. Guţu, Shadowing and stability in set-valued dynamics (Preprint). , , , ()

• [13] L. Górniewicz, Topological Fixed Point Theory of Multivalued Mappings, Kluwer Acad. Publ., Dordrecht, 1999., , , ()

• [14] [14] S. Gudder and F. Schroeck, Generalized convexity, SIAM J. Math. Anal. 11(1980), 984-1001. , , , (),

• [15] S. Hu, N. S. Papageorgiou, Handbook of Multivalued Analysis, Vol. I and II, Kluwer Acad. Publ., Dordrecht, 1997 and 1999. , , , ()

• [16] [16] J.R. Jachymski, Fixed point theorems in metric and uniform spaces via the Knaster-Tarski principle, Nonlinear Anal., 32(1998) No.2, 225-233. , , , (),

• [17] M.A. Khamsi and W.A. Kirk, An Introduction to Metric Spaces and Fixed Point Theory, Pure and Applied Mathematics, Wiley-Interscience, New York, 2001. [, , , ()

• [18] 18] W.A. Kirk and B. Sims (Editors), Handbook of Metric Fixed Point Theory, Kluwer Acad. Publ., Dordrecht, 2001. , , , ()

• [19] S.B. Nadler jr., Multivalued contraction mappings, Pacific J. Math., 30(1969), 475-488. , , , (),

• [20] [20] K. Neammanee, A. Kaewkhao, On multivalued weak contraction mappings, J. Math. Research, 3(2011), No. 2, 151-156. , , , (),

• [21] K. Palmer, Shadowing in Dynamical Systems. Theory and Applications, Kluwer Acad. Publ., Dordrecht, 2000. , , , ()

• [22] [22] B. Panyanak, Mann and Ishikawa iterative processes for multivalued mappings in Banach spaces, Comput. Math. Appl., 54(2007), No. 6, 872-877. , , , (),

• [23] A. Petruşel, Starshaped and fixed points, Seminar on Fixed Point Theory, Babes-Bolyai Univ., 1987, 19-24. , , , (),

• [24] [24] A. Petruşel, Operatorial Inclusions, House of the Book of Science, Cluj-Napoca, 2002. , , , ()

• [25] A. Petruşel, Multivalued weakly Picard operators and applications, Sci. Math. Jpn., 59(2004), 169-202., , , (),

• [26] [26] A. Petruşel and I.A. Rus, Multivalued Picard and weakly Picard operators, Proc. 6th International Conference on Fixed Point Theory and Applications, Valencia, Spain, July 19-26, 2003 (E. Llorens Fuster, J. Garcia Falset, B. Sims-Eds.), Yokohama Publ., 2004, 207-226. , , , (),

• [27] A. Petruşel and I.A. Rus, The theory of a metric fixed point theorem for multivalued operators, Proc. 9th International Conference on Fixed Point Theory and its Applications, Changhua, Taiwan, July 16-22, 2009, (L.J. Lin, A. Petruşel, H.K. Xu-Eds.), Yokohama Publ. 2010, 161-175., , , (),

• [28] [28] A. Petruşel and G. Petruşel, Multivalued Picard operators, J. Nonlinear Convex Anal., 13(2012), No. 1, 157-171. , , , (),

• [29] A. Petruşel, I.A. Rus and J.-C. Yao, Well-posedness in the generalized sense of the fixed point problems, Taiwanese J. Math., 11(2007), No.3, 903-914. , , , (),

• [30] [30] G. Petruşel and A. Petruşel, Existence and data dependence of the strict fixed points for multivalued $\delta$-contractions on graphic, Pure Math. Appl., 17(2006), No. 3-4, 413-418. , , , (),

• [31] S.Yu. Pilyugin, Shadowing in Dynamical Systems, Springer Verlag, Berlin, 1999., , , ()

• [32] [32] T. Puttasantiphat, Mann and Ishikawa iteration schemes for multivalued mappings in CAT(0) spaces, Appl. Math. Sci., 4(2010), No.61, 3005-3018. , , , (),

• [33] S. Reich, Fixed point of contractive functions, Boll. Un. Mat. Ital., 5(1972), 26-42., , , (),

• [34] [34] S. Reich and A.J. Zaslavski, Convergence of inexact iterative schemes for nonexpansive set-valued mappings, Fixed Point Theory Appl., 2010(2010), Article ID 518243, 10 p. , , , (),

• [35] I.A. Rus, Picard operators and applications, Sci. Math. Jpn., 58(2003), 191-219. , , , (),

• [36] [36] I.A. Rus, The theory of a metrical fixed point theorem: theoretical and applicative relevance, Fixed Point Theory, 9(2008), 541-559. , , , (),

• [37] I.A. Rus, Strict fixed point theory, Fixed Point Theory, 4(2003), 177-183., , , (),

• [38] [38] I.A. Rus, Generalized Contractions and Applications, Cluj University Press, Cluj-Napoca, 2001. , , , ()

• [39] I.A. Rus, Fixed Point Structure Theory, Cluj University Press, Cluj-Napoca, 2006., , , ()

• [40] [40] I.A. Rus, An abstract point of view on iterative approximation of fixed points: impact on the theory of fixed point equations, Fixed Point Theory, 13(2012), No.1, 179-192. , , , (),

• [41] I.A. Rus, A. Petruşel and A. Sîntămărian, Data dependence of the fixed point set of some multivalued weakly Picard operators, Nonlinear Anal., 52(2003), no. 8, 1947-1959., , , (),

• [42] [42] I.A. Rus, A. Petruşel and G. Petruşel, Fixed Point Theory, Cluj University Press, 2008. , , , ()

• [43] N. Shahzad and H. Zegeye, On Mann and Ishikawa schemes for multivalued maps in Banach spaces, Nonlinear Anal., 71(2009), 838-844. , , , (),

• [44] [44] A. Sîntămărian, Metrical strict fixed point theorems for multivalued mappings, Sem. on Fixed Point Theory, 1997, 27-30. , , , (),

• [45] S.L. Singh, C. Bhatnagar and A.M. Hashim, Round-off stability of Picard iterative procedure for multivalued operators, Nonlinear Anal. Forum, 10(2005), No. 1, 13-19. , , , (),

• [46] [46] R.E. Smithson, Fixed point of order preserving multifunction, Proc. Amer. Math. Soc., 28(1971), 304-310. , , , (),

• [47] Y. Song and Y.J. Cho, Some notes on Ishikawa iteration for multi-valued mappings, Bull. Korean Math. Soc., 48(2011), No. 3, 575-584., , , (),

• [48] [48] Y. Song and H. Wang, Convergence of iterative algorithms for multivalued mappings in Banach spaces, Nonlinear Anal. Theory Methods Appl., 70(2009), No. 4-A, 1547-1556. , , , (),

• [49] W. Takahashi, Nonlinear Functional Analysis. Fixed Point Theory and its Applications, Yokohama Publishers, Yokohama, 2000. , , , ()

• [50] [50] W. Takahashi, A convexity in metric spaces and nonexpansive mapping I, Kodai Math. Sem. Rep., 22(1970), 142-149. , , , (),

• [51] E. Tarafdar and G.X.-Z. Yuan, Set-valued contraction mapping principle, Applied Math. Letters, 8(1995), 79-81., , , (),

• [52] [52] G.X.-Z. Yuan, KKM Theory and Applications in Nonlinear Analysis, Marcel Dekker, New York, 1999., , , ()