New differentiability concepts for set-valued functions and applications to set differential equations

Abstract In this paper, we propose new definitions of derivative for set-valued functions, based on the Pompeiu-Hausdorff distance between compact convex sets. The main properties of the new concepts introduced are also investigated. Moreover, using the differentiability concepts defined, we study the set differential equations and we present the connection between the solutions to set differential equations and their approximations.

[1]  S. Leela,et al.  Interconnection between set and fuzzy differential equations , 2003 .

[2]  H. Román-Flores,et al.  On new solutions of fuzzy differential equations , 2008 .

[3]  Luciano Stefanini,et al.  Some notes on generalized Hukuhara differentiability of interval-valued functions and interval differential equations , 2012 .

[4]  F. S. De Blasi,et al.  On the differentiability of multifunctions , 1976 .

[5]  Barnabás Bede,et al.  Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations , 2005, Fuzzy Sets Syst..

[6]  Barnabás Bede,et al.  A New gH-Difference for Multi-Dimensional Convex Sets and Convex Fuzzy Sets , 2019, Axioms.

[7]  Jean-Pierre Aubin,et al.  Introduction: Set-Valued Analysis in Control Theory , 2000 .

[8]  Sahin Emrah Amrahov,et al.  Solving a nonhomogeneous linear system of interval differential equations , 2018, Soft Comput..

[9]  Juan J. Nieto,et al.  Hybrid metric dynamical systems with impulses , 2006 .

[10]  Imre J. Rudas,et al.  First order linear fuzzy differential equations under generalized differentiability , 2007, Inf. Sci..

[11]  Yurilev Chalco-Cano,et al.  Generalized derivative and π-derivative for set-valued functions , 2011, Inf. Sci..

[12]  V. Lakshmikantham,et al.  Set Differential Equations and Flow Invariance , 2003 .

[13]  F. S. de Blasi.  Semifixed Sets of Maps in Hyperspaces with Application to Set Differential Equations , 2006 .

[14]  N. Gasilov,et al.  On differential equations with interval coefficients , 2019, Mathematical Methods in the Applied Sciences.

[15]  Didier Dubois,et al.  Interval linear systems as a necessary step in fuzzy linear systems , 2015, Fuzzy Sets Syst..

[16]  Luciano Stefanini,et al.  A generalization of Hukuhara difference and division for interval and fuzzy arithmetic , 2010, Fuzzy Sets Syst..

[17]  E.Stanley Lee,et al.  Approximate Solutions, Existence, and Uniqueness of the Cauchy Problem of Fuzzy Differential Equations☆ , 1996 .

[18]  Sahin Emrah Amrahov,et al.  Relationship between Bede-Gal differentiable set-valued functions and their associated support functions , 2016, Fuzzy Sets Syst..