zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Weak condition for generalized multi-valued (f,α,β)-weak contraction mappings. (English) Zbl 1206.54064
Summary: In 2007, T. Kamran [Nonlinear Anal., Theory Methods Appl. 67, No. 7, A, 2289–2296 (2007; Zbl 1128.54024)] extended the notion of multi-valued mapping from weak contraction and generalized (α,L)-weak contraction to f-weak contraction and generalized multi-valued f-weak contraction. He also obtained some common fixed point theorems with the notion of T-weakly commuting at a coincidence point of a hybrid pair. In this paper, we can drop the condition of T-weakly commuting in Theorems 2.9 and 3.5 in [loc. cit.]. We further extend the notion of generalized multi-valued f-weak contraction and introduce the notion of generalized multi-valued (f,α,β)-weak contraction. We also establish some coincidence and common fixed point theorems with generalized multi-valued (f,α,β)-weak contraction mappings. Our results extend and generalize several common fixed point theorems of many authors.
MSC:
54H25Fixed-point and coincidence theorems in topological spaces
47H10Fixed point theorems for nonlinear operators on topological linear spaces
References:
[1]Banach, S.: Sur LES opérations dans LES ensembles abstraits et leurs applications aux équations intégrales, Fund. math. 3, 133-181 (1922) · Zbl 48.0201.01
[2]Pathak, H. K.; Khan, M. S.: Fixed and coincidence points of hybrid mappings, Arch. math. (Brno) 38, 201-208 (2002) · Zbl 1068.47073 · doi:emis:journals/AM/02-3/index.htm
[3]Markin, J. T.: Continuous dependence of fixed point sets, Proc. amer. Math. soc. 38, 545-547 (1973) · Zbl 0278.47036 · doi:10.2307/2038947
[4]Kamran, T.: Coincidence and fixed points for hybrid strict contractions, J. math. Anal. appl. 299, 235-241 (2004) · Zbl 1064.54055 · doi:10.1016/j.jmaa.2004.06.047
[5]Kamran, T.: Multivalued f-weakly Picard mappings, Nonlinear anal. 67, 2289-2296 (2007) · Zbl 1128.54024 · doi:10.1016/j.na.2006.09.010
[6]Chatterjea, S. K.: Fixed point theorems, C. R. Acad. bulgare sci. 25, 727-730 (1972) · Zbl 0274.54033
[7]Ciric, Lj.B.: Fixed point theory, (2003)
[8]Petrusel, A.: On frigongranas-type multifunctions, Nonlinear anal. Forum 7, 113-121 (2002)
[9]Reich, S.: Kannans fixed point theorem, Boll. unione mat. Ital. 4, 1-11 (1971) · Zbl 0219.54042
[10]Reich, S.: A fixed point theorem for locally contractive multi-valued functions, Rev. roumaine math. Pures appl. 17, 569-572 (1972) · Zbl 0239.54033
[11]Reich, S.: Fixed points of contractive functions, Boll. unione mat. Ital. 5, 26-42 (1972) · Zbl 0249.54026
[12]Rus, I. A.: Generalized contractions and applications, (2001)
[13]Rus, I. A.; Petrusel, A.; Sintamarian, A.: Data dependence of fixed point set of some multi-valued weakly Picard operators, Nonlinear anal. 52, 1947-1959 (2003)
[14]Berinde, V.: Generalized contractions and applications, Generalized contractions and applications 22 (1997) · Zbl 0885.47022
[15]Berinde, V.: Iterative approximation of fixed points, (2002) · Zbl 1034.47038
[16]Berinde, V.: On approximation of fixed points of weak φ-contractive operators, Fixed point theory 4, 131-142 (2003)
[17]Zamfirescu, T.: Fixed point theorems in metric spaces, Arch. math. (Basel) 23, 292-298 (1972) · Zbl 0239.54030 · doi:10.1007/BF01304884
[18]Reich, S.: Some problems and results in fixed point theory, Contemp. math. 21, 179-187 (1983) · Zbl 0531.47048
[19]Dugundji, J.; Granas, A.: Weakly contractive maps and elementary domain invariance theorem, Bull. Greek math. Soc. 19, 141-151 (1978) · Zbl 0417.54010
[20]Mizoguchi, N.; Takahashi, W.: Fixed point theorems for multi-valued mappings on complete metric space, J. math. Anal. appl. 141, 177-188 (1989) · Zbl 0688.54028 · doi:10.1016/0022-247X(89)90214-X
[21]Daffer, P. Z.; Kaneko, H.: Fixed points of generalized contractive multi-valued mappings, J. math. Anal. appl. 192, 655-666 (1995) · Zbl 0835.54028 · doi:10.1006/jmaa.1995.1194
[22]Akbar, F.; Khan, A. R.: Common fixed point and approximation results for noncommuting maps on locally convex spaces, Fixed point theory appl. (2009) · Zbl 1168.47044 · doi:10.1155/2009/207503
[23]Abbas, M.; Rhoades, B. E.: Common fixed point theorems for hybrid pairs of occasionally weakly compatible mappings satisfying generalized contractive condition of integral type, Fixed point theory appl. (2007)
[24]Imdad, M.; Khan, L.: Fixed point theorems for a family of hybrid pairs of mappings in metrically convex spaces, Fixed point theory appl., No. 3, 281-294 (2005) · Zbl 1096.54503 · doi:10.1155/FPTA.2005.281
[25]Latif, A.; Abdou, Afrah A. N.: Fixed points of generalized contractive maps, Fixed point theory appl. (2009)
[26]Nashine, H. K.: Coincidence points and invariant approximation results on (q)-normed spaces, Filomat 21, No. 2, 45-53 (2007) · Zbl 1224.41094 · doi:10.2298/FIL0702045N
[27]Pathak, H. K.; Rodriguez-Lopez, R.; Verma, R. K.: A common fixed point theorem using implicit relation and property (E.A) in metric spaces, Filomat 21, No. 2, 211-234 (2007) · Zbl 1141.54018 · doi:10.2298/FIL0702211P
[28]Sintunavart, W.; Kumam, P.: Coincidence and common fixed points for hybrid strict contractions without the weakly commuting condition, Appl. math. Lett. 22, 1877-1881 (2009) · Zbl 1225.54028 · doi:10.1016/j.aml.2009.07.015
[29]Xiao, J. -Z.; Zhu, X. -H.: Common fixed point theorems on weakly contractive and nonexpansive mappings, Fixed point theory appl. (2008) · Zbl 1144.54321 · doi:10.1155/2008/469357
[30]Beg, I.; Abbas, M.: Coincidence point and invariant approximation for mappings satisfying generalized weak contractive condition, Fixed point theory appl. (2006) · Zbl 1133.54024 · doi:10.1155/FPTA/2006/74503
[31]Kamran, T.: Fixed points of asymptotically regular noncompatible maps, Demonstratio math. 38, 485-494 (2005) · Zbl 1070.54020
[32]Jr., S. B. Nadler: Multivalued contraction mappings, Pacific J. Math. 30, 475-488 (1969) · Zbl 0187.45002
[33]Assad, N. A.; Kirk, W. A.: Fixed point theorems for setvalued mappings of contractive type, Pacific J. Math. 43, 553-562 (1972) · Zbl 0239.54032