Generalized Differential Inclusions in Banach Spaces

Abstract

We study a new type of solutions to differential inclusions in Banach spaces, which we call directional solutions. The idea is based on the observation that for a differentiable function $u$ and a closed set $V$

$$u\prime {\left( t \right)} \in V\,{\text{iff}}\,{\mathop {\lim }\limits_{h \to 0} }d{\left( {\frac{{u{\left( {t + h} \right)} - u{\left( t \right)}}}{h},V} \right)} = 0.$$

The above formula, which ‘makes sense’ also for non-differentiable functions, allows us to investigate nowhere differentiable solutions to differential inclusions. Thus we say that $u$ is a directional solution to $u\prime = F{\left( {t,u} \right)}$ if

$${\mathop {\lim }\limits_{h \to 0} }d{\left( {\frac{{u{\left( {t + h} \right)} - u{\left( t \right)}}}{h},F{\left( {t,u{\left( t \right)}} \right)}} \right)} = 0\,{\text{for}}\,{\text{all}}\,t.$$

We show that directional solutions have better properties than classical ones, in particular a limit of a convergent sequence of approximate solutions is an exact solution. We also prove that $u$ is a directional solution to $u\prime \in F{\left( {t,u} \right)}$ if

$$u{\left( {t_{2} } \right)} \in u{\left( {t_{1} } \right)} + {\int_{t_{1} }^{t_{2} } {F{\left( {t,u{\left( t \right)}} \right)}{\rm d}t\,{\text{for}}\,{\text{all}}\,t_{1} ,t_{2} .} }$$