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I have a this functional: associated to the impulsive problem : $$ \begin{cases} -(p(t)u'(t))'=f(t,u(t))\\ u(0)=u(+\infty)=0\\ \Delta(p(t_j)u'(t_j))=h(t_j)I_j(u(t_j)) \end{cases} $$

$J(u)=\frac12||u||^2+\sum_{j=0}^{+\infty} h(t_j)\int_0^{u(t_j)}I_j(s) ds-\int_0^{\infty}F(t,u(t))dt$

and the integral equation associated is given by: $$Au(t)=\int_0^{+\infty}G(t,s) f(s,u(s))-\sum_{j=0}^{\infty} G(t,t_j)h(t_j)I_j(u(t_j))$$ and i calculate:

$(J'(u),v)=\int_0^{+\infty} p(t) u'(t) v'(t)dt +\sum_{j=0}^{\infty}h(t_j)I_j(u(t_j))v(t_j)-\int_0^{\infty} f(t,u(t)) v(t) dt$

$(J''(u)v,w)=\int_0^{+\infty}p(t) v'(t) w'(t) dt +\sum_{j=0}^{\infty} h(t_j)I'(u)t_j))v(t_j)w(t_j)-\int_0^{\infty}f'_u(t,u(t))v(t)w(t)dt$

I want to write $J(u)$ and $(J''(u)v,w)$ using the operator $A$, i do it for $(J'(u),v)$ and i found that $(J'(u),v)=(u-Au,v)$

such that$(u,v)=\int_0^{+\infty} p(t) u'(t)v'(t) dt$

Please help me

Thank you.

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