Multivalued second order differential problem

Ewelina Mainka-Niemczyk

Abstract


Let K be a closed convex cone with nonempty interior in a real Banach space and let F,G,H: K → cc(K) be three given continuous additive set-valued functions. We study the existence and uniqueness of a solution of the second order differential problem

D2(t,x) = Φ(t,H(x))

Φ(0,x) = F(x),           DΦ(t,x)|t=0 = G(x)

for t ≥ 0 and x ∈ K, where DΦ(t,x) and D2Φ(t,x) denote the Hukuhara derivative and the second Hukuhara derivative of Φ(t,x) with respect to t.


Keywords


set-valued sine and cosine families; Riemann integral for set-valued functions; Hukuhara derivative of a set-valued function

Mathematics Subject Classification


26E25; 47H04; 47D09; 39B52

References


Berge, C. "Topological Spaces." Edinburgh and London: Oliver and Boyd, 1963.

Castaing, C. and M. Valadier "Convex Analysis and Measurable Multifunctions." Lecture Notes in Mathematics, 580. Berlin-New York: Springer-Verlag, 1977.

Dinghas, A. "Zum Minkowskischen Integralbegriff abgeschlossener Mengen." Math. Z. 66 (1956): 173-188.

Hukuhara, M. "Intégration des applications mesurables dont la valeur est un compact convexe." Funkcial. Ekvac. 10 (1967): 205-223.

Mainka-Niemczyk, E. "Some properties of set-valued sine families." Opuscula Math. 32 (2012): 157-168.

---. "Integral representation of set-valued sine families." J. Appl. Anal. 18.2 (2012): 243-258.

Nikodem, K. "K-convex and K-concave set-valued functions." Zeszyty Nauk. Politech. Łódz., Mat. 559. Rozprawy Nauk. 114, 1989.

Piszczek, M. "Second Hukuhara derivative and cosine family of linear set-valued functions." Annales Acad. Paed. Crac. Stud. Math. 5 (2006): 87-98.

---. "On multivalued cosine families." J. Appl. Anal. 13 (2007): 57-76.

---. "On a multivalued second order differential problem with Hukuhara derivative." Opuscula Math. 28 (2008): 151-161.

---. "On multivalued iteration semigroups." Aequationes Math. 81 (2011): 97-108.

Rådström, H. "An embedding theorem for spaces of convex sets." Proc. Amer. Math. Soc. 3 (1952): 165-169.

Smajdor, A. "On regular multivalued cosine families." Ann. Math. Sil. 13 (1999): 271-280.

---. "Hukuhara’s derivative and concave iteration semigroups of linear setvalued functions." J. Appl. Anal. 8 (2002): 297-305.

---. "On a multivalued differential problem." Internat. J. Bifur. Chaos Appl. Sci. Engrg. 13 (2003): 1877-1882.

Smajdor, W. "Superadditive set-valued functions and Banach-Steinhaus theorem." Rad. Mat. 3 (1987): 203-214.


Full Text: PDF

Download statistics: 2337



e-ISSN: 2300-133X, ISSN: 2081-545X

Since 2017 Open Access in De Gruyter and CrossCheck access cofinanced by The Ministry of Science and Higher Education - Republic of Poland - DUN 775/P-DUN/2017 see more

The Journal is indexed in:
and others see Abstracting and Indexing list

AUPC SM is on the List of the Ministry’s scored journals with 20 points for 2019

Deklaracja dostępności cyfrowej