Homotopy between two morphisms in $Top(2)$ category Let $(X,A),(Y,B)$ be topological paris and $f,g:(X,A)\rightarrow (Y,B)$ be continuous functions. (i.e. $f(A)\subset B$ and $g(A)\subset B$)
Assume that $f,g$ are homotopic so that there exists a continuous function $F:X\times I \rightarrow Y$ such that $F(x,0)=f(x)$ and $F(x,1)=g(x)$. Then, does there exist a continuous function $G:X\times I \rightarrow Y$ such that $G(x,0)=f(x)$ and $G(x,1)=g(x)$ and $G(A\times I)\subset B$?
What would be a terminology to call such homotopy $G$ and the relation of $f,g$ that admits the existence of such $G$? $G$ is definitely not "a homotopy relative to $A$". Is there some good terminology to call this?
 A: Regarding the first question, the answer is (almost) clearly no: As a special case, consider $(X, A)=(I, \partial I)$ (unit interval and its boundary) and $(Y, B)=(\mathbb{C}\setminus \{1\}, \{0\})$ the punctured plane with a basepoint. Thus, all the cont. maps respecting the distinguished subspaces are loops in the punctured plane with a common basepoint ($=\{0\}$). 
In this case, the condition you impose on $G$ is being a homotopy relative to $\partial I$ (because $B=\{0\}$ is one element set). If the claim in your question holds, that would translate to the statement that the fundamental group of the punctured plane is trivial (since all the curves in the punctured plane are homotopic if no basepoint is fixed), which does not hold.
(As for the second question, I do not know a good terminology for it, but I agree that it seems (at least to me) more natural definition than the usual notion of relative homotopy.)
A: The existence of your $G$ is, I believe, the standard defintion of being homotopic in the category of pairs. See e.g. Definition 2.2.5 here. 
I don't think that it is equivalent to the existence of your $F$ in general. For a counter-example, think of $X=[0,1]$, $A=\{0,1\}$, $Y=S^1$, $B=\{1\}$, $f(x)=e^{2\pi i x}$, $g(x)=1$.
The notation "homotopy relative to $A$" usually refers to a situation where $G(a,t)$ is independent of $t$ for each $a\in A$. So, for your $G$, just call it "homotopy". Your $F$ is something nonstandard, I believe.
