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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  • Magdalena Piszczek
  • Pedagogical University, Institute of Mathematics, ul. Podchorążych 2, 30-084 Cracow, Poland
  • Received: 2007-06-13.
  • Revised: 2007-11-10.
  • Accepted: 2007-11-13.
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Cite this article as:
Magdalena Piszczek, On a multivalued second order differential problem with Hukuhara derivative, Opuscula Math. 28, no. 2 (2008), 151-161

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