Can we Create Neural network(Simple one such as Multi Layer perceptron) that only contains positive weights only?
One way to do that is to customize the formula to update weights. 

Normally, the formula is something like this:

weight = weight - learning_rate * error


You can change it to:

weight = max(0, weight - learning_rate * error)


This will perform well or not? I have no idea. We can't say much about the guaranteed performance of neural network models in general.

However, if by good results, you mean a model with all positive weights, I believe this one will qualify.