Alternate terms, or definition for functional logistic regression I have recently come upon a paper discussing "functional logistic regression."
I could not find literature related to functional logistic regression. Is there a different name for this kind of logistic regression which is more commonly used?
 A: These models are not common in general. Still much work is done on ordinary functional regression; let alone logistic functional regression. Having said that, you will be able to find better references if you look for "functional generalized linear model". 
A seminal paper on the matter is the Annnals of Statistics paper of Mueller & Stadtmueller's 2005 paper on "Generalized functional linear regression model". There they lay the foundations for Poisson and binomial functional regression models. 
The earliest (2002) applied paper I have come across is Ratcliffe et al.'s Functional data analysis with application to periodically stimulated foetal heart rate data. II: Functional logistic regression. Applications papers of similar nature has appeared more recently too (eg. Zhu & Cox (2009), Reiss & Ogden (2010), etc.) 
General word of caution: Notice that you are in completely different ball game if you deal with sparse instead of dense functional data.
A: To expand on the accepted answer, functional logistic regression refers to a model in which a functional covariate, $X(t)$ where $t\in T$ (e.g. time), is used to predict the response of a (scalar) binary outcome, $Y$. As mentioned in the above answer, this falls under the broader class of functional generalized linear models (FGLMs):
$$ E[Y_i|X_i(t)] = g^{-1}(\eta_i) \qquad \eta_i = \alpha + \int_T X_i(t)\beta(t)dt$$
where $g(\cdot)$ is a link function relating $X_i(t)$ to $Y_i$. In the case of logistic regression, $p_i = E[Y_i|X_i(t)] = P(Y_i = 1|X_i(t))$ and $g$ is the logit link function so that
$$ log\left(\frac{p_i}{1 - p_i}\right) = \alpha + \int_T X_i(t)\beta(t)dt.$$
The accepted answer provides excellent references. But since it mentioned the case of sparse functional data, I'll add one more paper that illustrates how to use functional PCA to accommodate such a sparse setting (Wei, Tang & Li, 2014).
