Some Notes on the Paper [Further Discussion on Modified Multivalued α_*-Ψ-Contractive Type Mappings]

Document Type: Original Article

Authors

1 Department of Mathematics, Marand Branch, Islamic Azad University, Marand, Iran.

2 Department of Mathematics, Gilan-E-Gharb Branch, Islamic Azad University, Gilan-E-Gharb, Iran

Abstract

In this paper, we show that the claim of the paper [Ali et al., Further discussion on modifi ed multivalued
α_*-Ψ-contractive type mappings, Filomat 29 (2015)] which says that the notion of α_*-η-Ψ-contractive multivalued
mappings can not be reduced into α_*-Ψ-contractive multi-valued mappings, is not true. Also, we
provide a common fixed point result for an α_*-admissible countable family of multi-valued mappings. Finally,
we show that the common fixed point result of Ali et al. for a countable family of multi-valued mappings
using α_*-admissible mappings with respect to η can be reduced to α_*-admissible mappings without using
the auxiliary function].

Keywords


[1]M. U. Ali, T. Kamran, Multivalued F -contractions and related fixed point theorems with an application, Filomat,
4 (2016), 3779-3793.


[2]M. U. Ali, T. Kamran, E. Karapinar, (α,Ψ,ξ)-contractive multi-valued mappings, Fixed Point Theory Applications,
2014 2014: 7.


[3]M. U. Ali, T. Kamran, E. Karapinar, A new approach to (α,Ψ)-contractive nonself multivalued mappings, Journal
Inequalities and Applications, 2014 2014 :71.


[4]M. U. Ali, T. Kamran, On (α_*,Ψ)-contractive multi-valued mappings, Fixed Point Theory, and Applications,
2013 2013 :137.


[5]M. U. Ali, T. Kamran,W. Sintunavarat, P. Katchang, Mizoguchi-Takahashis fixed point theorem with α, η functions,
Abstract and Applied Analysis, 2013 (2013), Article ID 418798.


[6]M. U. Ali, T. Kamran, E. Karapinar, Further discussion on modi fied multivalued α_*-Ψ-contractive type mappings,
Filomat, 29 (2015), 1893-1900.


[7]H. Alikhani, Sh. Rezapour, N. Shahzad, Fixed points of a new type of contractive mappings and multifunctions,
Filomat, 27 (2013), 1315-1319.


[8]P. Amiri, S. Rezapour, N. Shahzad,Fixed points of generalized α-Ψ-contractions, Revista de la Real Academia de
Ciencias Exactas Fisicas y Naturales Serie A Mate doi: 10.1007/s13398-013-0123-9.


[9]J. H. Asl, S. Rezapour, N. Shahzad, On fi xed points of α-Ψ-contractive multifunctions, Fixed Point Theory and
Applications, 2012 2012 :212.


[10]M. Berzig, E. Karapinar, Note on "Modi ed α-Ψ -contractive mappings with application", Thai Journal of Mathematics
(2014) In press.


[11]N. Hussain, P. Salimi, A. Latif, Fixed point results for single and set-valued α-η-Ψ-contractive mappings, Fixed
Point Theory and Applications, 2013 2013: 212.


[12]E. Karapinar, α-Ψ-Geraghty contraction type mappings and some related fixed point results, Filomat, 28 (2014)
3748.


[13]E. Karapinar, B. Samet, Generalized α-Ψ-contractive type mappings and related fixed point theorems with applications,
Abstract Applied Analysis, 2012 (2012) Article id: 793486.


[14]G. Minak, I. Altun, Some new generalizations of Mizoguchi-Takahashi type fixed point theorem, Journal Inequalities
, and Applications, 2013 2013: 493.


[15]B. Mohammadi, S. Rezapour, N Shahzad, Some results on fixed points of α-Ψ-Ciric generalized multifunctions,
Fixed Point Theory and Applications, 2013 2013: 24.


[16]B. Mohammadi, S. Rezapour, On modi fied α-φ-contractions, Journal of Advanced Mathematical Studies, 6 (2013),
162-166.


[17]P. Salimi, A. Latif, N. Hussain, Modifi ed α-Ψ-contractive mappings with applications, Fixed Point Theory and
Applications, 2013 2013:151.


[18]P. Salimi, C. Vetro, P. Vetro, Fixed point theorems for twisted (α,β)-Ψ-contractive type mappings and applications,
Filomat, 27 (2013), 605-615.


[19]B. Samet, C. Vetro, P. Vetro, Fixed point theorems for α-Ψ-contractive type mappings, Nonlinear Analysis, 75
(2012), 2154-2165.