General existence results for nonconvex third order differential inclusions

M. Bounkhel, King Saud University, Riyadh, Saudi Arabia
B. Al-Senan, King Saud University, Riyadh, Saudi Arabia

E. J. Qualitative Theory of Diff. Equ., No. 21. (2010), pp. 1-10.

Abstract: In this paper we prove the existence of solutions to the following third order differential inclusion:
$$\left\{
\begin{array}{ll} x^{(3)}(t)\in
F(t,x(t),\dot{x}(t),\ddot{x}(t))+G(x(t),\dot{x}(t),\ddot{x}(t)),
\mbox{ a.e. on } [0,T]\cr x(0)=x_0, \dot x(0)=u_0, \ddot
x(0)=v_0, \mbox{ and }\ddot{x}(t)\in S, \forall t\in [0,T],
\end{array}\right.
$$
where $F:[0,T]\times \mathbb{H}\times \mathbb{H} \times \mathbb{H}\rightarrow \mathbb{H}$ is a continuous set-valued mapping, $G:\mathbb{H}\times \mathbb{H} \times \mathbb{H}\rightarrow \mathbb{H}$ is an upper semi-continuous set-valued mapping with $G(x,y,z)\subset \partial^C g(z)$ where $g: \mathbb{H}\rightarrow \mathbb{R}$ is a uniformly regular function over $S$ and locally Lipschitz and $S$ is a ball compact subset of a separable Hilbert space $\mathbb{H}$.

You can download the full text of this paper in DVI, PostScript or PDF format.