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Abstract

We prove fixed point theorems for Suzuki type multi-functions on complete metric spaces. An example is constructed to illustrate that our results are new.

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... Moreover, we shall also prove generalization of Theorem 4 in the setting of a compact normed space. A multivalued fixed point theorem is also proved which generalizes Theorem 2 in [4], and Theorem 2.3 in [2] in the setting of Banach space. ...
... Clearly satisfies all the conditions of Theorem 2.1 of [2]. Hence ( ) ̸ = ∅. ...
... In this case, we have = 1. Then ( ) ̸ = ∅ by Theorem 2.1 of [2]. ...
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... Beg and Aleomraninejad [6] proved the following result in this direction. Theorem 1.10. ...
... Theorem 1.10. [6] Let (X, d) be a complete metric space and T : X −→ CB(X). Assume that there are α, r ∈ (0, 1) such that α(r + 1) ≤ 1 and ...
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In this paper, we introduce a class of Suzuki-type contractive multivalued mappings and establish the existence of fixed points of such mappings in the setup of b-metric spaces. Some examples are presented to support the results proved herein. An estimate of Hausdorff distance between the fixed point sets of two Suzuki-type contractive multivalued mappings is obtained. As an application of results thus obtained, existence and uniqueness of periodic solution of delay differential equations are shown.
... It has numerous real-world applications in game theory, constrained optimization, differential inclusions, optimal control problems, energy management problems, image reconstruction and so on. Over the years the results of Nadler [17] have been extended and generalized in terms of spaces and nonlinear mappings (see [18,19] and the references therein). ...
... Later on, Berinde and Barinde [5] generalized the contraction principle used by Nadler [19]. Sadded et al. [30], Beg and Aleomraninejad [4] also proved fixed point theorems for mutivalued mapping using different contractive conditions. Singh et al. [32] presented the development of the Hutchinson Barnsley (HB) theory for a system of single valued and multi valued contractions on metric spaces. ...
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... It has numerous real-world applications in game theory, constrained optimization, differential inclusions, optimal control problems, energy management problems, image reconstruction and so on. Over the years the results of Nadler [17] have been extended and generalized in terms of spaces and nonlinear mappings (see [18,19] and the references therein). ...
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The purpose of this work is to introduce the notion of a multivalued strictly (α, β)-admissible mappings and a multivalued (α, β)-Meir-Keeler contractions with respect to the partial Hausdorff metric Hp in the framework of partial metric spaces. In addition, we present fixed points and endpoints results for a multivalued (α, β)-Meir-Keeler contraction mappings in the framework of the complete partial metric spaces. The results obtained in this work provides extension as well as substantial generalizations and improvements of several well-known results on fixed point theory and its applications. MSC: 47H09; 47H10; 49J20; 49J40
... In 2012, Singh et al. [29] proved a common fixed point theorem for a pair multivalued maps on a complete metric space. Many fixed point theorems have been proved by various authors as generalizations of Suzuki's theorem (see [1,4,7,8,13]). ...
... Afterward an interesting and rich fixed point theory for such mappings has been developed. Inspired from the work of Nadler [35], the fixed point theory of multi-valued contraction was further developed in different directions by many authors, in particular, by Beg et al. [4,5], Reich [39,40], Mizoguchi and Takahashi [34], Kaneko [21], Lim [31], Lami Dozo [30], Feng and Liu [18], Klim and Wardowski and [28], Suzuki [45], Pathak and Shahzad [36] and many others. For further details and more references, see [38,44]. ...
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In order to generalize the well-known Banach contraction theorem, many authors have introduced various types of contraction inequalities. In 2008, Suzuki introduced a new method (Suzuki (2008) [4]) and then his method was extended by some authors (see for example, Dhompongsa and Yingtaweesittikul (2009), Kikkawa and Suzuki (2008) and Mot and Petrusel (2009) , , and ). Kikkawa and Suzuki extended the method in (Kikkawa and Suzuki (2008) [5]) and then Mot and Petrusel further generalized it in (Mot and Petrusel (2009) [6]). In this paper, we shall provide a new condition for T which guarantees the existence of its fixed point. Our results generalize some old results.
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In this paper, we are concerned with mathematical properties of the previous contractive types of set-valued maps or fuzzy mappings on a metric space (see, for instance, Nadler, Pacific J. Math. 30 (1969) 475–488; Heilpern, J. Math. Anal. Appl. 83 (1981) 566–569; Papageorgiou, Nonlinear Anal. Theory Methods Appl. 7 (1983) 763–770; Bose and Sahani, Fuzzy Sets and Systems 21 (1987) 53–58; Som and Mukherjee, Fuzzy Sets and Systems 33 (1989) 213–219; Park and Jeong, Fuzzy Sets and Systems 59 (1993) 231–235; 87 (1997) 111–116; Lee et al., Fuzzy Sets and Systems 101 (1999) 143–152) and we prove some crucial theorems relating fixed points of fuzzy mappings on complete metric spaces. First, we prove a theorem by using the concept of w-distance (Kada et al., Math. Japonica 44 (1996) 381–391), which generalizes the results of Amemiya and Takahashi (Fuzzy Sets and Systems 114 (2000) 469–476), and Kada et al. (1996). Next, we prove two theorems for fuzzy mappings, which are connected with fixed point theorems for set-valued maps. Then we shall observe that our theorems can be used to obtain almost all results proved so far concerning fixed points of fuzzy mappings or set-valued maps on complete metric spaces and therefore, that our results imply the existence of fixed points of various contractive types of fuzzy mappings or set-valued maps on complete metric spaces.
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In this paper a new fixed point theorem is proved for contraction mappings in a complete metric space by observing that if the space is metrically convex, then significant weakenings may be made concerning the domain and range of the mapping considered. While the main theorem is formulated for set-valued mappings, its point-to-point analogue is also a new result. This result, proved in § 1, is the following: Suppose M is a complete, metrically convex, metric space, K a nonempty closed subset of M, and tp a contraction mapping from K into the family of nonempty closed bounded subsets of M supplied with the Hausdorff metric. Then if ϕ maps the boundary of K into subsets of K, ϕ has a fixed point in K, i.e., there is a point x0∈ K such that x0∈ ϕ (x0).
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Let (M, d) be a complete metric space and S(M) the set of all nonempty bounded closed subsets of M. A set-valued mapping f: M → S(M) will be called (uniformly) locally contractive if there exist ε and λ(ε > 0, 0 < λ < 1) such that D(f(x), f(y)) ≤ λd (x, y) whenever d(x, y) < ε and where D(f(x), f(y)) is the distance between f(x) and f(y) in the Hausdorff metric induced by d on S(M). It is shown in the first theorem that if M is “well-chained,” then has a fixed point is, that is, a point x ε M such that x ε f(x). This fact, in turn, yields a fixed-point theorem for locally nonexpansive set-valued mappings on a compact star-shaped subset of a Banach space. Both theorems are extensions of earlier results.
On fixed point generalizations of Suzuki's method
  • S M A Aleomraninejad
  • Rezapour
  • Sh
  • N Shahzad
Aleomraninejad, S.M.A., Rezapour, Sh, Shahzad, N.: On fixed point generalizations of Suzuki's method. Appl. Math. Lett. 24, 1037-1040 (2011)