Take a look at gimp layer modes.  Maybe dodge mode on a new top layer and use a rectangle gradient to blend the burned edges inward.  Gradient mode might be fg color to transparent and use white.

how does one use a transparent layer? is it only for drawing things on? I have some old pictures that have darkened around the edges. How I think I can fix it is to duplicate the layer then  lighten the exposure value on the lower image, then paint black and uncover the lighter image over the section that has darkened. How do I do this with gradients and Darktable?

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:-)~MIKE~(-:

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