I'm using an alternate optimization scheme to minimize a function $F(a,b,c) = G(a,b,c) + H(c)$ where $G$ is continuous and differentiable, $H$ is not differentiable but lower semi-continuous and proximable. As such, proximal gradient descent is used as a step of the alternate optimization: $F$ is optimized in each variable separately (the two other variable being fixed), and the particular procedure used to optimize $F(c)$ is proximal descent.
I'm interested in proving the overall convergence of this scheme. In particular, knowing if the set-valued function $A_1 : (a,b) \mapsto \{c^*\vert c^* = prox(c^* - \gamma \nabla G(c^*))\}$ is itself lower semicontinuous would provide weak convergence guarantees, through the use of Corollary 3 from [3] for example.
However, I am just discovering lower semicontinuity and I can't manage to prove this. Is there a specific result I should use, or a paper or textbook I should read?
Sorry if this makes only little sense (especially the title), this is not my usual field of research.
[3]: Huard, P. (1979). Extensions of Zangwill’s theorem. In Point-to-Set Maps and Mathematical Programming (pp. 98-103). Springer, Berlin, Heidelberg.
