Enriched Multivalued Contractions with Applications to Differential Inclusions and Dynamic Programming

The purpose of this paper is to introduce the class of enriched multivalued contraction mappings. Both single-valued and multivalued enriched contractions are defined by means of symmetric inequalities. Our main result extends and generalizes the recent result of Berinde and Păcurar (Approximating fixed points of enriched contractions in Banach spaces, Journal of Fixed Point Theory and Applications, 22 (2), 1–10, 2020). We also study a data dependence problem of the fixed point set and Ulam–Hyers stability of the fixed point problem for enriched multivalued contraction mappings. Applications of the results obtained to the problem of the existence of a solution of differential inclusions and dynamic programming are presented.

[1]  V. Berinde,et al.  Fixed point theorems for enriched Ćirić-Reich-Rus contractions in Banach spaces and convex metric spaces , 2021, Carpathian Journal of Mathematics.

[2]  Vasile Berinde,et al.  Fixed Points Theorems for Unsaturated and Saturated Classes of Contractive Mappings in Banach Spaces , 2021, Symmetry.

[3]  William A. Kirk,et al.  Fixed point theorems for set-valued mappings of contractive type , 1972 .

[4]  V. Berinde.  Approximating fixed points of enriched nonexpansive mappings by Krasnoselskij iteration in Hilbert spaces , 2019, Carpathian Journal of Mathematics.

[5]  Suthep Suantai,et al.  A Hybrid Method for a Countable Family of Multivalued Maps, Equilibrium Problems, and Variational Inequality Problems , 2010 .

[6]  Teck-Cheong Lim,et al.  On fixed point stability for set-valued contractive mappings with applications to generalized differential equations , 1985 .

[7]  R. Baskaran,et al.  A note on the solution of a class of functional equations , 1986 .

[8]  Wataru Takahashi,et al.  Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces , 2007 .

[9]  V. Berinde,et al.  Approximating fixed points of enriched contractions in Banach spaces , 2019, Journal of Fixed Point Theory and Applications.

[10]  V. Berinde,et al.  Existence and Approximation of Fixed Points of Enriched Contractions and Enriched φ-Contractions , 2021, Symmetry.

[11]  Teck-Cheong Lim,et al.  A fixed point theorem for multivalued nonexpansive mappings in a uniformly convex Banach space , 1974 .

[12]  S. Banach.  Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales , 1922 .

[13]  Convergence theorems for fixed point iterative methods defined as admissible perturbations of a nonlinear operator , 2013 .

[14]  Fixed point theory for a new type of contractive multivalued operators , 2009 .

[15]  P. C. Bhakta,et al.  Some existence theorems for functional equations arising in dynamic programming, II , 1984 .

[16]  V. Berinde.  Approximating fixed points of enriched nonexpansive mappings in Banach spaces by using a retraction-displacement condition , 2020 .

[17]  J. T. Markin.  Continuous dependence of fixed point sets , 1973 .

[18]  Vasile Berinde,et al.  On a general class of multi-valued weakly Picard mappings , 2007 .

[19]  V. Berinde.  Weak and strong convergence theorems for the Krasnoselskij iterative algorithm in the class of enriched strictly pseudocontractive operators , 2018, Annals of West University of Timisoara - Mathematics and Computer Science.

[20]  A. Petruşel,et al.  The Retraction-Displacement Condition in the Theory of Fixed Point Equation with a Convergent Iterative Algorithm , 2016 .

[21]  Vasile Berinde,et al.  Kannan's fixed point approximation for solving split feasibility and variational inequality problems , 2021, J. Comput. Appl. Math..

[22]  L. Dey,et al.  On some enriched contractions in Banach spaces , 2020, 2006.11500.

[23]  Richard Bellman,et al.  Functional equations in dynamic programming , 1978 .

[24]  V. Berinde,et al.  Iterative Methods for the Class of Quasi-Contractive Type Operators and Comparsion of their Rate of Convergence in Convex Metric Spaces , 2016 .

[25]  Vasile Berinde,et al.  Enhancing Ant-Based Algorithms for Medical Image Edge Detection by Admissible Perturbations of Demicontractive Mappings , 2021, Symmetry.

[26]  D. H. Hyers.  On the Stability of the Linear Functional Equation. , 1941, Proceedings of the National Academy of Sciences of the United States of America.

[27]  S. Nadler.  Multi-valued contraction mappings. , 1969 .

[28]  L. Ciric,et al.  ON THE CONVERGENCE OF THE ISHIKAWA ITERATES TO A COMMON FIXED POINT OF MULTIVALUED MAPPINGS , 2003 .