Pseudo-monotone operator
Alright kiddo, let me tell you about pseudo-monotone operators. So you know how sometimes when you have two numbers and you compare them, you can tell which one is bigger or smaller, right? That's what we call monotone.
Now, imagine you have a machine that takes in different inputs and gives you different outputs. It's like a magic box that does different things depending on what you put in. Sometimes, when you put in a bigger input, you expect to get a bigger output. And sometimes, when you put in a smaller input, you expect to get a smaller output. That's also what we call monotone.
However, sometimes this magic box doesn't work exactly like that. It might be that if you increase the input just a little bit, the output might not increase or it might even decrease. But if you increase the input a lot more, then the output will start to increase again. That's what we call pseudo-monotone because it's not always increasing or always decreasing, but it sort of is.
So a pseudo-monotone operator is just a fancy way of saying that the machine that does different things depending on what you put in it doesn't always follow the same rules we expect from monotone operators. It's kind of like a quirky magic box!
Now, imagine you have a machine that takes in different inputs and gives you different outputs. It's like a magic box that does different things depending on what you put in. Sometimes, when you put in a bigger input, you expect to get a bigger output. And sometimes, when you put in a smaller input, you expect to get a smaller output. That's also what we call monotone.
However, sometimes this magic box doesn't work exactly like that. It might be that if you increase the input just a little bit, the output might not increase or it might even decrease. But if you increase the input a lot more, then the output will start to increase again. That's what we call pseudo-monotone because it's not always increasing or always decreasing, but it sort of is.
So a pseudo-monotone operator is just a fancy way of saying that the machine that does different things depending on what you put in it doesn't always follow the same rules we expect from monotone operators. It's kind of like a quirky magic box!