Opuscula Math. 28, no. 2 (2008), 151-161
Opuscula Mathematica
On a multivalued second order differential problem with Hukuhara derivative
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.
- 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.
