0
votes
0answers
22 views

Lebesgue integrable of composite functions

Let $a, b\in R$ be such that $a<b$ and $u\in L([a, +\infty);R)$. Prove that $$ v(t)=\frac{b-a}{(b-t)^2}u\left(\frac{b-a}{b-t}+a-1\right)\in L([a, b];R). $$ Thank you for all helping and guidance. ...
0
votes
3answers
90 views

Nonnegative function satisfying integral constraints

Find a real function $w(t)\in L_2[0,1]$ such that: $w(t)\geq 0 \quad \forall t\in [0,1];$ $\displaystyle\int_0^{s}w(t)dt\leq s \quad \forall s\in [0,1];$ $\displaystyle \int_0^1 w(t)dt\leq 2;$ ...
1
vote
4answers
151 views

Finding a differentiable function satisfying some given conditions

Finding a differentiable function $g:\mathbb{R}\rightarrow \mathbb{R}$ satisfying the following condiotions: $\displaystyle g(0)=0, g(1)=1, g(-1)=-1;$ $\displaystyle ...