The weak Ekeland variational principle and fixed points

Beer, Gerald; Dontchev, Asen L.

doi:10.1016/j.na.2014.01.022

Cited by 9 publications

(3 citation statements)

References 13 publications

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“…The remarkable fact is that (with obvious set-valued modification) the proof of the theorem repeats word for word the classical Banach proof based on simple Picard's iterations. 3 In fact we prove a stronger result in Section 3 (Theorem 3.3). The idea of the improvement is the concept of a regularity horizon (introduced in [13] under the name "reach function") in the context of nonlocal regularity-see Section 2.…”

Section: Introductionmentioning

confidence: 73%

“…[4]) in existence theorems of metric analysis. As far as regularity properties (e.g., in covering theorems) 3 It seems appropriate to slightly rephrase [4, p. 12] and state that "in essence the whole history of the generalizations of 'the contraction mapping principle' reduces to finding new formulations from the standard process of proof." In [4] the authors spoke about the theorem of Lyusternik.…”

Section: Introductionmentioning

confidence: 99%

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Metric regularity, fixed points and some associated problems of variational analysis

Ioffe

2014

J. Fixed Point Theory Appl.

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The paper is basically concerned with interrelations between metric regularity and metric critical point theories. At the early stage of developments of regularity theory there was a feeling that every regularity criterion can be obtained with the help of a suitable (set-valued) contraction mapping principle. On the other hand, the assumptions of the well-known metric fixed point theorems (e.g., of Nadler (1969) or of Dontchev and Hager (1994)) imply metric regularity of the inverse mapping. The message of the paper is that, nonetheless, the theories are independent although the fundamental principles on which they are based have much in common. The first main result of the paper guarantees the existence of a fixed point if the contraction property holds only along the orbits of the mapping (rather than for arbitrary pairs of points in the feasible area) and actually only on some bounded pieces of the orbits determined by a certain "regularity horizon" condition. The bulk of the paper is devoted to the study of "two maps paradigm" when we have two set-valued mappings acting in opposite directions and we are interested in the existence of the so-called double fixed point. We consider two types of iteration procedures to get a fixed point, one based on simple Picard's iterations and the other on the Lyusternik-Graves version of Newton's method, that in principle may lead to different fixed points, although the impression is that estimates provided by the first procedure may be better. The paper is concluded by a discussion of the regularity problem for compositions of set-valued mappings. There are also quite a few examples in the paper.Mathematics Subject Classification. 49J53, 54C60, 58C06, 47H04, 49K40, 46A30, 90C31.

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Section: Introductionmentioning

confidence: 73%

Section: Introductionmentioning

confidence: 99%

Metric regularity, fixed points and some associated problems of variational analysis

Ioffe

2014

J. Fixed Point Theory Appl.

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“…For the next result, we give a version of the fixed-point theorems proved by Beer and Dontchev [14] (see Theorem 4) and Dontchev and Hager [15] in a strong b-metric space. Hence, we obtain a partial answer to the question raised by Kirk and Shahzad [8] (p. 128).…”

Section: Local Version Of the Covitz-nadler Theoremmentioning

confidence: 99%

Multivalued Contraction Fixed-Point Theorem in b-Metric Spaces

Slimani,

Graef,

Ouahab

2024

Mathematics

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The authors explore fixed-point theory in b-metric spaces and strong b-metric spaces. They wish to prove some new extensions of the Covitz and Nadler fixed-point theorem in b-metric spaces. In so doing, they wish to answer a question proposed by Kirk and Shahzad about Nadler’s theorem holding in strong b-metric spaces. In addition, they offer an improvement to the fixed-point theorem proven by Dontchev and Hager.

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b-Metric Spaces

Kirk

Shahzad

2014

Fixed Point Theory in Distance Spaces

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The weak Ekeland variational principle and fixed points

Cited by 9 publications

References 13 publications

Metric regularity, fixed points and some associated problems of variational analysis

Metric regularity, fixed points and some associated problems of variational analysis

Multivalued Contraction Fixed-Point Theorem in b-Metric Spaces

b-Metric Spaces

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