zbMATH — the first resource for mathematics

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.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
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)
Multi-valued nonlinear contraction mappings. (English) Zbl 1179.54053
Among three fixed point theorems established in the paper there is the following: Let $(X, d)$ be a complete metric space and let $\varphi: [0,\infty)\to[a, 1)$, $0< a< 1$, be such that $\varlimsup_{r\to t+}(r)< 1$ for all $t\in[0,\infty)$. If $T: X\to\text{Cl}(X)$ (= all nonempty closed sets of $X$) is such that $x\mapsto d(x,Tx)$ is lower-semicontinuous and for any $x\in X$ there is $y\in Tx$ with $\sqrt{\varphi(d(x, y))}d(x,y)\le d (x, Tx)$ and $d(y, Ty)\le\varphi(d(x, y))d(x, y)$, then $z\in Tz$ for some $z\in\bbfZ$. This theorem generalizes results of {\it D. Klim} and {\it D. Wiatrowski} [J. Math. Anal. Appl. 334, No. 1, 132--139 (2007; Zbl 1133.54025)], {\it Y. Feng} and {\it S. Liu} [ibid., 317, No. 1, 103--112 (2006; Zbl 1094.47049)], {\it N. Mizoguchi} and {\it W. Takahashi} [ibid., 141, No. 1, 177--188 (1989; Zbl 0688.54028)].

54H25Fixed-point and coincidence theorems in topological spaces
54C60Set-valued maps (general topology)
Full Text: DOI
[1] Banach, S.: Sur LES opérations dans LES ensembles abstraits et leur application aux équations intégrales. Fund. math. 3, 133-181 (1922) · Zbl 48.0201.01
[2] Jr., S. B. Nadler: Multi-valued contraction mappings. Pacific J. Math. 30, 475-488 (1969) · Zbl 0187.45002
[3] Markin, J. T.: A fixed point theorem for set-valued mappings. Bull. amer. Math. soc. 74, 639-640 (1968) · Zbl 0159.19903
[4] Ćirić, L. B.; Ume, J. S.: Multi-valued non-self mappings on convex metric spaces. Nonlinear anal. TMA 60, 1053-1063 (2005) · Zbl 1078.47015
[5] Ćirić, L. B.; Ume, J. S.: Some common fixed point theorems for weakly compatible mappings. J. math. Anal. appl. 314, No. 2, 488-499 (2006) · Zbl 1086.54027
[6] Ćirić, L. B.: Fixed point theorems for multi-valued contractions in complete metric spaces. J. math. Anal. appl. 348, No. 1, 499-507 (2008) · Zbl 1213.54063
[7] Daffer, P. Z.; Kaneko, H.; Li, W.: On a conjecture of S. Reich. Proc. amer. Math. soc. 124, 3159-3162 (1996) · Zbl 0866.47040
[8] Eldred, A. A.; Anuradha, J.; Veeramani, P.: On equivalence of generalized multi-valued contractions and nadler’s fixed point theorem. J. math. Anal. appl. 336, 751-757 (2007) · Zbl 1128.47051
[9] Feng, Y.; Liu, S.: Fixed point theorems for multi-valued contractive mappings and multi-valued caristi type mappings. J. math. Anal. appl. 317, 103-112 (2006) · Zbl 1094.47049
[10] Jachymski, J.: On reich’s question concerning fixed points of multimaps. Boll. unione mat. Ital. (7) 9, 453-460 (1995) · Zbl 0863.54042
[11] Klim, D.; Wardowski, D.: Fixed point theorems for set-valued contractions in complete metric spaces. J. math. Anal. appl. 334, 132-139 (2007) · Zbl 1133.54025
[12] Mizoguchi, N.; Takahashi, W.: Fixed point theorems for multivalued mappings on complete metric spaces. J. math. Anal. appl. 141, 177-188 (1989) · Zbl 0688.54028
[13] Naidu, S. V. R.: Fixed-point theorems for a broad class of multimaps. Nonlinear anal. TMA 52, 961-969 (2003) · Zbl 1029.54049
[14] Reich, S.: Fixed points of contractive functions. Boll. unione mat. Ital. 5, 26-42 (1972) · Zbl 0249.54026
[15] Reich, S.: Some fixed point problems. Atti acad. Naz. lincei 57, 194-198 (1974)
[16] Reich, S.: Some problems and results in fixed point theory. Contemp. math. 21, 179-187 (1983) · Zbl 0531.47048
[17] Zhong, C. K.; Zhu, J.; Zhao, P. H.: An extension of multi-valued contraction mappings and fixed points. Proc. amer. Math. soc. 128, 2439-2444 (2000) · Zbl 0948.47058
[18] 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)
[19] Suzuki, T.: Mizoguchi--takahashi’s fixed point theorem is a real generalization of nadler’s. J. math. Anal. appl. 340, 752-755 (2008) · Zbl 1137.54026