Opuscula Math. 28, no. 2 (2008), 151-161

Opuscula Mathematica

# On a multivalued second order differential problem with Hukuhara derivative

Magdalena Piszczek

Abstract. Let $$K$$ be a closed convex cone with the nonempty interior in a real Banach space and let $$cc(K)$$ denote the family of all nonempty convex compact subsets of $$K$$. Assume that continuous linear multifunctions $$H,\Psi : K \to cc(K)$$ are given. We consider the following problem \begin{aligned}D^2\Phi(t,x) =& \Phi(t,H(x)),\\ D\Phi(t,x)|_{t=0} =& \{0\},\\ \Phi(0,x) =& \Psi(x)\end{aligned} for $$t \geq 0$$ and $$x \in K$$, where $$D\Phi(t,x)$$ denotes the Hukuhara derivative of $$\Phi(t,x)$$ with respect to $$t$$.

Keywords: Hukuhara's derivative, multivalued cosine families, Riemann integral for multifunctions, Cauchy problem for a set-valued differential equation.

Mathematics Subject Classification: 26E25, 39B52, 47D09.

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