Fixed points of multivalued maps under local Lipschitz conditions and applications

2019 ◽  
Vol 150 (3) ◽  
pp. 1467-1494
Author(s):  
Claudio A. Gallegos ◽  
Hernán R. Henríquez
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Banach Spaces ◽  
Fixed Point Theory ◽  
Mild Solutions ◽  
Abstract Cauchy Problem ◽  
Point Theory ◽  
Vector Subspace ◽  
Multivalued Maps ◽  
Lipschitz Continuous

AbstractIn this work we are concerned with the existence of fixed points for multivalued maps defined on Banach spaces. Using the Banach spaces scale concept, we establish the existence of a fixed point of a multivalued map in a vector subspace where the map is only locally Lipschitz continuous. We apply our results to the existence of mild solutions and asymptotically almost periodic solutions of an abstract Cauchy problem governed by a first-order differential inclusion. Our results are obtained by using fixed point theory for the measure of noncompactness.

scholarly journals Erratum: Abdou, A. A. N. and Khamsi, M.A. Fixed Points of Kannan Maps in the Variable Exponent Sequence Spaces ℓp(·). Mathematics 2020, 8, 76

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 578
Author(s):  
Afrah A. N. Abdou ◽  
Mohamed Amine Khamsi
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Banach Spaces ◽  
Variable Exponent ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Sequence Spaces ◽  
Vector Spaces ◽  
Nonexpansive Maps ◽  
Banach Contraction

Kannan maps have inspired a branch of metric fixed point theory devoted to the extension of the classical Banach contraction principle. The study of these maps in modular vector spaces was attempted timidly and was not successful. In this work, we look at this problem in the variable exponent sequence spaces lp(·). We prove the modular version of most of the known facts about these maps in metric and Banach spaces. In particular, our results for Kannan nonexpansive maps in the modular sense were never attempted before.

scholarly journals Coincidences and fixed points of reciprocally continuous and compatible hybrid maps

2002 ◽  
Vol 30 (10) ◽  
pp. 627-635 ◽  
Author(s):  
S. L. Singh ◽  
S. N. Mishra
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Metric Space ◽  
Fixed Point Theorems ◽  
Metric Spaces ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Multivalued Maps

It is proved that a pair of reciprocally continuous and nonvacuously compatible single-valued and multivalued maps on a metric space possesses a coincidence. Besides addressing two historical problems in fixed point theory, this result is applied to obtain new general coincidence and fixed point theorems for single-valued and multivalued maps on metric spaces under tight minimal conditions.

scholarly journals Instantaneous and Noninstantaneous Impulsive Integrodifferential Equations in Banach Spaces

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Saïd Abbas ◽  
Mouffak Benchohra ◽  
Gaston M. N'Guérékata
Keyword(s):  
Fixed Point ◽  
Banach Spaces ◽  
Resolvent Operator ◽  
Measure Of Noncompactness ◽  
Fixed Point Theory ◽  
Mild Solutions ◽  
Point Theory ◽  
Integrodifferential Equations ◽  
The Resolvent Operator

This paper deals with some existence of mild solutions for two classes of impulsive integrodifferential equations in Banach spaces. Our results are based on the fixed point theory and the concept of measure of noncompactness with the help of the resolvent operator. Two illustrative examples are given in the last section.

scholarly journals Fixed Points of Kannan Maps in the Variable Exponent Sequence Spaces ℓp(·)

Mathematics ◽  
2020 ◽  
Vol 8 (1) ◽  
pp. 76 ◽  
Author(s):  
Afrah Abdou ◽  
Mohamed Khamsi
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Banach Spaces ◽  
Variable Exponent ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Sequence Spaces ◽  
Vector Spaces ◽  
Nonexpansive Maps ◽  
Banach Contraction

Kannan maps have inspired a branch of metric fixed point theory devoted to the extension of the classical Banach contraction principle. The study of these maps in modular vector spaces was attempted timidly and was not successful. In this work, we look at this problem in the variable exponent sequence spaces ℓ p ( · ) . We prove the modular version of most of the known facts about these maps in metric and Banach spaces. In particular, our results for Kannan nonexpansive maps in the modular sense were never attempted before.

scholarly journals Fixed Points Theorems for Unsaturated and Saturated Classes of Contractive Mappings in Banach Spaces

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 713
Author(s):  
Vasile Berinde ◽  
Mădălina Păcurar
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Banach Spaces ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Contractive Mappings ◽  
Contractive Type

Based on the technique of enriching contractive type mappings, a technique that has been used successfully in some recent papers, we introduce the concept of a saturated class of contractive mappings. We show that, from this perspective, the contractive type mappings in the metric fixed point theory can be separated into two distinct classes, unsaturated and saturated, and that, for any unsaturated class of mappings, the technique of enriching contractive type mappings provides genuine new fixed-point results. We illustrate the concept by surveying some significant fixed-point results obtained recently for five remarkable unsaturated classes of contractive mappings. In the second part of the paper, we also identify two important classes of saturated contractive mappings, whose main feature is that they cannot be enlarged by enriching the contractive mappings.

scholarly journals Retraction method in fixed point theory for monotone nonexpansive mappings

2016 ◽  
Vol 32 (3) ◽  
pp. 271-276
Author(s):  
M. R. ALFURAIDAN ◽  
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Banach Spaces ◽  
Fixed Point Theory ◽  
Nonexpansive Mappings ◽  
Point Theory ◽  
Common Fixed Points ◽  
The Common ◽  
Nonexpansive Retract

In this paper we study the properties of the common fixed points set of a commuting family of monotone nonexpansive mappings in Banach spaces endowed with a graph. In particular, we prove that under certain conditions, this set is a monotone nonexpansive retract.

Explicit algorithms for J-fixed points of some non linear mappings in certain Banach spaces

2020 ◽  
Vol 29 (1) ◽  
pp. 27-36
Author(s):  
M. M. GUEYE ◽  
M. SENE ◽  
M. NDIAYE ◽  
N. DJITTE
Keyword(s):  
Fixed Points ◽  
Banach Spaces ◽  
Fixed Point Theory ◽  
Maximal Monotone ◽  
Point Theory ◽  
Duality Mapping ◽  
Monotone Mappings ◽  
Normalized Duality Mapping ◽  
Maximal Monotone Mappings ◽  
Pseudocontractive Mappings

Let E be a real normed linear space and E∗ its dual. In a recent work, Chidume et al. [Chidume, C. E. and Idu, K. O., Approximation of zeros of bounded maximal monotone mappings, solutions of hammerstein integral equations and convex minimizations problems, Fixed Point Theory and Applications, 97 (2016)] introduced the new concepts of J-fixed points and J-pseudocontractive mappings and they shown that a mapping A : E → 2 E∗ is monotone if and only if the map T := (J −A) : E → 2 E∗ is J-pseudocontractive, where J is the normalized duality mapping of E. It is our purpose in this work to introduce an algorithm for approximating J-fixed points of J-pseudocontractive mappings. Our results are applied to approximate zeros of monotone mappings in certain Banach spaces. The results obtained here, extend and unify some recent results in this direction for the class of maximal monotone mappings in uniformly smooth and strictly convex real Banach spaces. Our proof is of independent interest.

scholarly journals Fixed poin sets in digital topology, 1

2020 ◽  
Vol 21 (1) ◽  
pp. 87 ◽  
Author(s):  
Laurence Boxer ◽  
P. Christopher Staecker
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Digital Images ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Point Sets ◽  
Fixed Point Set ◽  
Complete Computation ◽  
Point Set ◽  
Fixed Point Sets

<p>In this paper, we examine some properties of the fixed point set of a digitally continuous function. The digital setting requires new methods that are not analogous to those of classical topological fixed point theory, and we obtain results that often differ greatly from standard results in classical topology.</p><p>We introduce several measures related to fixed points for continuous self-maps on digital images, and study their properties. Perhaps the most important of these is the fixed point spectrum F(X) of a digital image: that is, the set of all numbers that can appear as the number of fixed points for some continuous self-map. We give a complete computation of F(C<sub>n</sub>) where C<sub>n</sub> is the digital cycle of n points. For other digital images, we show that, if X has at least 4 points, then F(X) always contains the numbers 0, 1, 2, 3, and the cardinality of X. We give several examples, including C<sub>n</sub>, in which F(X) does not equal {0, 1, . . . , #X}.</p><p>We examine how fixed point sets are affected by rigidity, retraction, deformation retraction, and the formation of wedges and Cartesian products. We also study how fixed point sets in digital images can be arranged; e.g., for some digital images the fixed point set is always connected.</p>

scholarly journals A renorming in some Banach spaces with applications to fixed point theory

2010 ◽  
Vol 258 (10) ◽  
pp. 3452-3468 ◽  
Author(s):  
Carlos A. Hernandez Linares ◽  
Maria A. Japon
Keyword(s):  
Fixed Point ◽  
Banach Spaces ◽  
Fixed Point Theory ◽  
Point Theory
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