The fixed point index and asymptotic fixed point theorems for kkk-set-contractions (original) (raw)
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Using a variational method introduced in [D. Azé and J.-N. Corvellec, ‘A variational method in fixed point results with inwardness conditions’, Proc. Amer. Math. Soc.134(12) (2006), 3577–3583], deriving directly from the Ekeland principle, we give a general result on the existence of a fixed point for a very general class of multifunctions, generalizing the recent results of [Y. Feng and S. Liu, ‘Fixed point theorems for multi-valued contractive mappings and multi-valued Caristi type mappings’, J. Math. Anal. Appl.317(1) (2006), 103–112; D. Klim and D. Wardowski, ‘Fixed point theorems for set-valued contractions in complete metric spaces’, J. Math. Anal. Appl.334(1) (2007), 132–139]. Moreover, we give a sharp estimate for the distance to the fixed-points set.
Fixed point results for set-valued contractions by altering distances in complete metric spaces
2009
Nadler's contraction principle has led to fixed point theory of set-valued contraction in non-linear analysis. Inspired by the results of Nadler, the fixed point theory of setvalued contraction has been further developed in different directions by many authors, in particular, by Reich, Mizoguchi-Takahashi, Feng-Liu and many others. In the present paper, the concept of generalized contractions for set-valued maps in metric spaces is introduced and the existence of fixed point for such a contraction are guaranteed by certain conditions. Our first result extends and generalizes the Nadler, Feng-Liu and Klim-Wardowski theorems and the second result is different from the Reich and Mizoguchi-Takahashi results. As a consequence, we derive some results related to fixed point of set-valued maps satisfying certain conditions of integral type.
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Fixed Point Theory and Applications, 2012
In this paper, we continue the study of contractive conditions for mappings in complete partial metric spaces. Concretely, we present fixed point results for weakly contractive and weakly Kannan mappings in such a way that the classical metric counterpart results are retrieved as a particular case. Special attention to the cyclical case is paid. Moreover, the well-posedness of the fixed point problem associated to weakly (cyclic) contractive and weakly (cyclic) Kannan mappings is discussed, and it is shown that these contractive mappings are both good Picard operators and special good Picard operators. 1 Introduction Throughout this paper, the letters R, R + , N and Z + will denote the set of real numbers, the set of nonnegative real numbers, the set of positive integer numbers and the set of nonnegative integer numbers, respectively. The celebrated fixed point theorem of Banach asserts the following. Theorem If (X, d) is a complete metric space and f : X → X is a mapping such that d f (x), f (y) ≤ αd(x, y) (.) for all x, y ∈ X and some α ∈ [, [, then f has a unique fixed point x * ∈ X. Moreover, the Picard sequence of iterates {f n (x)} n∈N converges, for every x ∈ X, to x *. In [], Kannan obtained the following extension of the aforementioned fixed point theorem of Banach to a larger class of mappings, now known as Kannan mappings. Theorem Let (X, d) be a complete metric space and let f : X → X be a mapping such that d f (x), f (y) ≤ α d x, f (x) + d y, f (y) (.) for all x, y ∈ X and some α ∈ [, [, then f has a unique fixed point x * ∈ X. Moreover, the Picard sequence of iterates {f n (x)} n∈N converges, for every x ∈ X, to x *. Another extensions of Banach's fixed point theorem were given by Kirk, Srinivasan and Veeramani in []. They obtained general fixed point theorems for mappings satisfying cyclical contractive conditions. Among other results, the following one was proven in [].