Proving the existence of a solution for integral equation.

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Let $u(x)$ is a continuous monotone strictly increasing function of $x$ on a finite interval $[0,b]$ and the equation $1-delta,u(x)=0$ has solution in $(0,b]$, where $delta>0$. The function $v(x)$ defined as
$$
v(x)=frac1xintlimits_0^xfracu(xi),xi1-delta,u(xi),dxi,quad xin (0,b].
$$
Prove that there is a point $x^*in (0,b]$ in which $v(x)=1$.




real-analysis integration convergence




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    Let $u(x)$ is a continuous monotone strictly increasing function of $x$ on a finite interval $[0,b]$ and the equation $1-delta,u(x)=0$ has solution in $(0,b]$, where $delta>0$. The function $v(x)$ defined as
    $$
    v(x)=frac1xintlimits_0^xfracu(xi),xi1-delta,u(xi),dxi,quad xin (0,b].
    $$
    Prove that there is a point $x^*in (0,b]$ in which $v(x)=1$.




    real-analysis integration convergence




    share|cite|improve this question





















      up vote
      0
      down vote

      favorite









      up vote
      0
      down vote

      favorite











      Let $u(x)$ is a continuous monotone strictly increasing function of $x$ on a finite interval $[0,b]$ and the equation $1-delta,u(x)=0$ has solution in $(0,b]$, where $delta>0$. The function $v(x)$ defined as
      $$
      v(x)=frac1xintlimits_0^xfracu(xi),xi1-delta,u(xi),dxi,quad xin (0,b].
      $$
      Prove that there is a point $x^*in (0,b]$ in which $v(x)=1$.




      real-analysis integration convergence




      share|cite|improve this question











      Let $u(x)$ is a continuous monotone strictly increasing function of $x$ on a finite interval $[0,b]$ and the equation $1-delta,u(x)=0$ has solution in $(0,b]$, where $delta>0$. The function $v(x)$ defined as
      $$
      v(x)=frac1xintlimits_0^xfracu(xi),xi1-delta,u(xi),dxi,quad xin (0,b].
      $$
      Prove that there is a point $x^*in (0,b]$ in which $v(x)=1$.





      real-analysis integration convergence





      share|cite|improve this question










      share|cite|improve this question




      share|cite|improve this question









      asked Jul 25 at 8:46









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