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Generalized Differential Inclusions in Banach Spaces

Authors:

Jacek Tabor

Affiliation:

(1) Institute of Mathematics, Jagiellonian University, Reymonta 4, 30-059 Kraków, Poland

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 $$V$$

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

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 $$uprime = F{left( {t,u} right)}$$

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

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 $$uprime in F{left( {t,u} right)}$$

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

Keywords:

differential inclusion directional inclusion Radon– Nikodym property

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