In many nonlinear equations, the existence of a solution (but not its uniqueness) follows from a topological argument in the vein of BFP Theorem. The BFP is at work especially when the equation is posed in some finite dimensional vector space, and you can establish an a priori estimate of the size of the solution. This means that you know a ball $B_R$ containing all the solutions.
The most important example of this situation is the stationary Navier-Stokes equation, with Dirichlet condition $u=0$ on the boundary of the domain. Of course, the ambient space is infinite dimensional, so you first establish the existence of an approximate solution in a subspace of dimension $n$ (Galerkin procedure); this is where you use the BFP Theorem, or its equivalent form that a continuous vector field over $B_R$ which is outgoing on $\partial B_R$ must vanish somewhere. Then passing to the limit as $n\rightarrow\infty$ is pedestrian.
The BFP Theorem is a consequence of the fact that the Euler-PoincarĂ© characteristic of the ball is non-zero. There are counterparts when you work on a compact manifold (with boundary) whose EPC is non-zero. This happened to me in a very interesting way. I considered the free fall of a rigid body in water filling the entire space. The mathematical problem is a coupling between Navier-Stokes and the Euler equation for the top. I looked at a permanent regime, in which the solid body has a time-independent velocity field, and that of the fluid is time-independent as well, once you consider it in the moving frame attached to the solid. The difficulty is that you don't know a priori the direction of the vertical axis (the direction of gravity) in this frame. After a Galerkin procedure, the problem reduces to the search of a zero of a tangent vector field over $B_R\times S^2$. This vector field is outgoing on the boundary $\partial B_R\times S^2$. Because the EPC of $B_R\times S^2$ is non-zero, $EP(B_R\times S^2)=EP(B_R)\cdot EP(S^2)=1\cdot2\ne0,$$such a zero exists. Therefore the permanent regime does exist. Remark that because$EP=2$, we even expect an even number of solutions when counting multiplicities, at least at each level of the Galerkin approximation. 1 In many nonlinear equations, the existence of a solution (but not its uniqueness) follows from a topological argument in the vein of BFP Theorem. The BFP is at work especially when the equation is posed in some finite dimensional vector space, and you can establish an a priori estimate of the size of the solution. This means that you know a ball$B_R$containing all the solutions. The most important example of this situation is the stationary Navier-Stokes equation, with Dirichlet condition$u=0$on the boundary of the domain. Of course, the ambient space is infinite dimensional, so you first establish the existence of an approximate solution in a subspace of dimension$n$(Galerkin procedure); this is where you use the BFP Theorem. Then passing to the limit as$n\rightarrow\infty$is pedestrian. The BFP Theorem is a consequence of the fact that the Euler-PoincarĂ© characteristic of the ball is non-zero. There are counterparts when you work on a compact manifold (with boundary) whose EPC is non-zero. This happened to me in a very interesting way. I considered the free fall of a rigid body in water filling the entire space. The mathematical problem is a coupling between Navier-Stokes and the Euler equation for the top. I looked at a permanent regime, in which the solid body has a time-independent velocity field, and that of the fluid is time-independent as well once you consider it in the moving frame attached to the solid. The difficulty is that you don't know a priori the direction of the vertical axis (the direction of gravity) in this frame. After a Galerkin procedure, the problem reduces to the search of a zero of a tangent vector field over$B_R\times S^2$. This vector field is outgoing on the boundary$\partial B_R\times S^2$. Because the EPC of$B_R\times S^2\$ is non-zero, such a zero exists.