Engine delay start

When I turn the key to start the engine I have to hold it for 2-3 seconds, while there is no noise from the starter (initially I thought the battery is dead), and then it starts. When I bought it, a month ago, when I was turning the key all the way it started, without the 2-3 seconds of delay. What seems to be the problem? Thanks.

Make sure your battery and ground connections are cleaned of corrosion. Then look to the starter relay. Is there a CLICK sound at the starter or no sound at all?

What year Lexus and how many miles?

It might be a problem with the anti-theft RF link to the key.

Thanks for answers.
2003 Lexus LS 430 with 84000 miles.
There is no click, or cranking when I turn the key all the way; it’s completely silence, like when the battery is dead; but if I hold the key in that position for a few seconds, it starts. This happened in the last 10 days. Before it started normally.