## 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 $$\sqrt{\varphi(d(x, y))}d(x,y)\leq d (x, Tx)$$ and $$d(y, Ty)\leq\varphi(d(x, y))d(x, y)$$, then $$z\in Tz$$ for some $$z\in\mathbb{Z}$$.
This theorem generalizes results of D. Klim and D. Wiatrowski [J. Math. Anal. Appl. 334, No. 1, 132–139 (2007; Zbl 1133.54025)], Y. Feng and S. Liu [ibid., 317, No. 1, 103–112 (2006; Zbl 1094.47049)], N. Mizoguchi and W. Takahashi [ibid., 141, No. 1, 177–188 (1989; Zbl 0688.54028)].

### MSC:

 54H25 Fixed-point and coincidence theorems (topological aspects) 54C60 Set-valued maps in general topology

### Citations:

Zbl 1133.54025; Zbl 1094.47049; Zbl 0688.54028
Full Text:

### References:

 [1] Banach, S., Sur LES opérations dans LES ensembles abstraits et leur application aux équations intégrales, Fund. math., 3, 133-181, (1922) · JFM 48.0201.01 [2] Nadler, S.B., 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, 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, 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) · Zbl 0329.47019 [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) · Zbl 1055.47047 [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
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