I come to a statement that logistic regression is a non linear problem. How can one show this?
Is it possible to treat logistic discrimination in terms of equivalent linear regression problem?
Recall that the Logistic regression model is a non linear transformation of $\beta^Tx$
So to answer your question, Logistic regression is indeed non linear in terms of Odds and Probability, however it is linear in terms of Log Odds.
Fitting a logistic regression model on the following toy example gives the coefficients $\alpha = -5.05$ and $\beta = 1.3$
Plotting the probability $P(Y=1)$ as a function of $X$ clearly shows the non linear relationship
The Odds of $Y$ being 1 given $X$ is also non linear
Finally the log odds of $Y$ being 1 is a linear relationship
See here for some more details: Calculating confidence intervals for a logistic regression
For the first statement: logistic regression is used when a variable is dichotomous. Since the variable can assume only value 1 or 0, fitting a line assumes a linear relationship which cannot hold for dichotomous outcomes. It can be proved that the linear probability model will not be efficient and, furthermore, nothing ensures that the estimated dependent variable will be bounded between 0 and 1. The logit can solve these problem.
Please clarify your second statement.