Solution Space for a Fixed Point Problem

Question

Hi I need to find the criteria for which the following has a solution: $$X= K_1 (a_1-b_1X)^{c_1} (a_2-b_2 X)^{c_2} (a_3-b_3X)^{c_3} X^{c_4}$$ where $K_1>0; a_1>0; c_1>1; (b_2 b_3)\leq0; c_2<0; c_3<0; c_4<0;$ I am coding this, using fixed point iteration, but my code give weird results when it $X$ is around the roots ($\frac{a_i}{b_i}$), I would rather find the area in which there is a solution and code accordingly. I tried to see if I can use lipshitz continuity but I could not go much. Thanks.