Existence condition of solution for a system of nonlinear equations

Question

I have the following system of nonlinear equations:

$$\left\{ \begin{align} & \dfrac{-a}{w^2}\sin(wT) + \dfrac{T}{w}[a\cos(wT) - b\sin(wT)] + c = 0 \\ & \dfrac{b}{w}\sin(wT) + T[a\sin(wT) + b\cos(wT)] = 0 \\ & \int_{0}^{T}{h(a,b,c,w,t)dt} = 0 \\ & \sin(c+w) - w\cos(c+w) - c = 0\\ & \sin{c} - \int_{0}^{T}{h(a,b,c,w,t)tdt} = X, \text{ where}\\ \end{align} \right. $$

• $X$ is a given positive number.
• $a,b,c,w,T$ are real variables.
• $h(a,b,c,w,t)$ is a highly complicated function with many trigonometric terms.

For each $X$, I have tried to numerically solve this system of nonlinear equations by MATLAB and I always found a solution of $(a,b,c,w,T)$.

My concern is: I would like to prove the existence of solution for the above set of nonlinear equations for any value of $X$. Do you know what mathematical tools can we use for that purpose?

I very much appreciate any suggestion!

Answer 1

It really depends on $h$.

For example, if $h(a,b,c,w,t)>0$, then $T=0$ and if $X\neq 0$ there isn't a solution.