On some fixed point theorems for multi-valued operators by altering distance techique

Abstract

The aim of this paper is to present some sufficient conditions for the existence and uniqueness of fixed points for (ϕ – ψ) type contractive multi-valued operators defined by altering distances. Furthermore, our main result consists of two theorems, one involving the convergence of the Picard successive approximation sequence to a fixed point of the multivalued ϕ -ψ operator, and a theorem concerning a more general form for a fixed point result for this type of mappings.

Authors

Cristian Daniel Alecsa
Babes-Bolyai University, Department of Mathematics, Cluj-Napoca, Romania
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, Cluj-Napoca, Romania

Adrian Petrusel
Babes-Bolyai University, Department of Mathematics, Cluj-Napoca, Romania
Academy of Romanian Scientists, Bucharest, Romania

Keywords

.Fixed point; (ϕ −ψ)-contraction; Altering distance; Multivalued operator; Weakly contractive mapping

Paper coordinates

C. D. Alecsa, A. Petrusel, On some fixed point theorems for multi-valued operators by altering distance technique, J. Nonlinear Var. Anal., 1 (2017) no. 2, pp. 237-251.

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J. Nonlinear Var. Anal.

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References

[1] Y. I. Alber, S. Guerre-Delabriere, Principle of weakly contractive maps in Hilbert spaces, New Results in Operator Theory and Its Applications: The Israel M. Glazman memorial volume, Oper. Theory, Adv. Appl. (I. Gohberg- Ed.), Birkhauser,  Basel, 1997, 7?2.

[2] A. Amini-Harandi, A. Petrusel, A fixed point theorem by altering distance technique in complete metric spaces, Miskolc Math. Notes, 14 (2013), 11-17.

[3] S. Dhompongsa, H. Yingtaweesittikul, Diametrically contractive multivalued mappings, Fixed Point Theory Appl. 2007 (2007), Article ID 19745.

[4] P. N. Dutta, B. S. Choudhury, A generalization of contraction principle in metric spaces, Fixed Point Theory Appl. 2008 (2008), Article ID 406368.

[5] T. Kamran, Q. Kiran, Fixed point theorems for multi-valued mappings obtained by altering distances, Math. Comput. Model. 54 (2011), 2772-2777.

[6] M. S. Khan, M. Swaleh, S. Sessa, Fixed point theorems by altering distances between the points, Bull. Aust. Math. Soc. 30 (1984), 1-9.

[7] V. L. Lazar, Fixed point theory for multivalued ϕ-contractions, Fixed Point Theory Appl. 2011 (2011), Article ID 50.

[8] T. Lazar, G. Mot¸, G. Petrusel, S. Szentesi, The theory of Reich’s fixed point theorem for multivalued operators, Fixed Point Theory Appl. 2010 (2010), Article ID 178421.

[9] Z. Liu, Z. Wu, S.M. Kang, S. Lee, Some fixed point theorems for nonlinear set-valued contractive mappings, J. Appl. Math. 2012 (2012) Article ID 786061.

[10] S. Moradi, A. Farajzadeh, On the fixed point of (ψ −ϕ)-weak and generalized (ψ −ϕ)-weak contraction mappings, Appl. Math. Lett. 25 (2012), 1257?262.

[11] S.B. Nadler Jr., Multi-valued contraction mappings, Pacific J. Math. 30 (1969), 475-488.

[12] T.P. Petru, M. Boriceanu, Fixed point results for generalized ϕ-contraction on a set with two metrics, Top. Method Nonlin. Anal. 33 (2009), 315-326

[13] A. Petrus¸el, Multivalued weakly Picard operators and applications, Sci. Math. Japon. 59 (2004), 169-202.

[14] A. Petrusel, peratorial Inclusions, House of the Book of Science, Cluj-Napoca, 2002.

[15] A. Petrusel, I.A. Rus, Multivalued Picard and weakly Picard operators, Fixed Point Theory Appl. (J. Garcia Falset, E. Llorens Fuster, B. Sims – Eds.), Yokohama Publ., 2004, 207-226.

[16] G. Petrusel, T. Lazar, V.L. Lazar, Fixed points and coupled fixed points for multi-valued (ψ −ϕ)-contractions in b-metric spaces, Applied Anal. Optimization, 1 (2017), 99-112.

[17] O. Popescu, Fixed points for (ψ −ϕ)-weak contractions, Appl. Math. Lett. 24 (2011), 1-4.

[18] O. Popescu, G. Stan, A generalization of Nadler’s fixed point theorem, Results Math. (2017), DOI 10.1007/s00025-017- 0694-4

[19] B. E. Rhoades, Some theorems on weakly contractive maps, Nonlinear Anal. 47 (2001), 2683?693.

[20] B. E. Rhoades, H.K. Pathak, S.N. Mishra, Some weakly contractive mappings theorems in partially ordered spaces and applications, Demonstratio Math. 45 (2012), 621-636.

[21] I.A. Rus, Generalized Contractions and Applications, Cluj University Press, Cluj-Napoca, 2001.

[22] I.A. Rus, A. Petrusel, G. Petrusel, Fixed Point Theory, Cluj University Press, Cluj-Napoca, 2008.

[23] Z. Xue, Fixed points theorems for generalized weakly contractive mappings, Bull. Aust. Math. Soc. 93 (2016), 321?29.

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