How can I make sure that my values are actually the optimum?

I'm developing a dynamic stochastic general equilibrium model to investigate how the endogenous variables of my model would shift at the presence of a temporary shock. So first I need to find the steady state values of the variables in which there isn't any shock and the values do not change over time. To this end, I started with introducing some guess values to the system (it's nonlinear) and used fsolve to solve it. I have three guess values (shown by k_g, e_g and lambda_g) where lambda is the Lagrangian multiplier. Surprisingly, the system can be solved in the very first iteration regardless of the lambda_g value! I also found that the impulse response function that I received is very dependent to the guess values. This makes me suspected to a possible problem somewhere while I'm confident with the model itself. Specifically, I've got two questions: 1) Is that correct to get the same solution for any initial value of lambda (say for lambda_g equals to 1 to 1000)? 2)How can I make sure that the solution I find is the global (not local) optimum and stable. I have attached a part of my model here. Any help is much appreciated. Thanks a lot.