Abstract
We discuss the set-valued dynamics related to the theory of functional equations. We look for selections of convex set-valued functions satisfying set-valued Euler-Lagrange inclusions. We improve and extend upon some of the results in [13, 20], but under weaker assumptions. Some applications of our results are also provided.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Aubin J P, Frankowska H. Set-valued analysis//Modern Birkhäuser Classics. Boston: Birkhäuser, 2008
Bae J H, Park W G. A functional equation having monomials as solutions. Appl Math Comput, 2010, 216: 87–94
Brzdȩk J, Pietrzyk A. A note on stability of the general linear equation. Aequationes Math, 2008, 75: 267–270.
Brzdȩk J, Piszczek M. Selections of set-valued maps satisfying some inclusions and the Hyers-Ulam stability//Handbook of Functional Equations. Springer Optim Appl 96. New York: Springer, 2014: 83–100
Brzdȩk J, Piszczek M. Fixed points of some nonlinear operators in spaces of multifunctions and the Ulam stability. J Fixed Point Theory Appl, 2017, 19: 2441–2448
Brzdȩk J, Piszczek M. Ulam stability of some functional inclusions for multi-valued mappings. Filomat, 2017, 31: 5489–5495
Brzdȩk J, Popa D, Raşa I, Xu B. Ulam Stability of Operators. Oxford: Academic Press, Elsevier, 2018
Brzdȩk J, Popa D, Xu B. Selections of set-valued maps satisfying a linear inclusion in a single variable. Nonlinear Anal, 2011, 74: 324–330
Chang I S, Kim H M. On the Hyers-Ulam stability of quadratic functional equations. J Ineq Pure Appl Math, 2002, 3: Art. 33
Czerwik S. Functional Equations and Inequalities in Several Variables. World Scientific London, 2002
Gordji M E, Alizadeh Z, Khodaei H, Park C. On approximate homomorphisms: A fixed point approach. Math Sci, 2012, 6: Art No 59
Hyers D H. On the stability of the linear functional equation. Proc Natl Acad Sci, 1941, 27: 222–224
Khodaei H, Rassias Th M. Set-valued dynamics related to generalized Euler-Lagrange functional equations. J Fixed Point Theory Appl, 2018, 20: Art No 32
Kim H M. On the stability problem for a mixed type of quartic and quadratic functional equation. J Math Anal Appl, 2006, 324: 358–372
Lu G. Park C. Hyers-Ulam stability of additive set-valued functional equations. Appl Math Lett, 2011, 24: 1312–1316
Nikodem K. On quadratic set-valued functions. Publ Math Debrecen, 1984, 30: 297–301
Nikodem K. K-Convex and K-Concave Set-Valued Functions. Zeszyty Naukowe, Politech, Krakow, Poland, 1989
Nikodem K, Popa D. On single-valuedness of set-valued maps satisfying linear inclusions. Banach J Math Anal, 2009, 3: 44–51
Nikodem K, Popa D. On selections of general linear inclusions. Publ Math Debrecen, 2009, 75: 239–249
Park C, O’Regan D, Saadati R. Stabiltiy of some set-valued functional equations. Appl Math Lett, 2011, 24: 1910–1914
Piszczek M. On selections of set-valued inclusions in a single variable with applications to several variables. Results Math, 2013, 64: 1–12
Piszczek M. The properties of functional inclusions and Hyers-Ulam stability. Aequations Math, 2013, 85: 111–118
Popa D. Additive selections of (α, β)-subadditive set valued maps. Glas Mat Ser III, 2001, 36: 11–16
Popa D. A stability result for general linear inclusion. Nonlinear Funct Anal Appl, 2004, 3: 405–414
Rådström H. An embedding theorem for space of convex sets. Proc Amer Math Soc, 1952, 3: 165–169
Smajdor A. Additive selections of superadditive set-valued functions. Aequationes Math, 1990, 39: 121–128
Smajdor A, Szczawińska J. Selections of set-valued functions satisfying the general linear inclusion. J Fixed Point Theory Appl, 2016, 18: 133–145
Ulam S M. A Collection of Mathematical Problems. New York: Interscience Publishers, 1960; Reprinted as: Problems in Modern Mathematics. New York: John Wiley & Sons Inc, 1964
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Khodaei, H., El-Fassi, Ii. & Hayati, B. On selections of set-valued Euler-Lagrange inclusions with applications. Acta Math Sci 40, 1105–1115 (2020). https://doi.org/10.1007/s10473-020-0416-y
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10473-020-0416-y