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I was studying about green's functions and was rather excited by the prospect that I could use it to maybe look at non-linear differential equations, but the complex point is that is it possible to have an operator, like a simple differential operator and be able to guess which functions would lead to scaled delta functions as their solutions? Or is there a branch of mathematics which promotes this search?

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    A differential equation is a special case of convolution equations, and the Dirac delta is the identity of the convolution. – 2017-01-05
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    @user1952009: Is there considerable text available for convolution equations?? – 2017-01-05
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    The linear case is well known : Fourier/Laplace transform, distribution theory, functional analysis. The non-linear case is much more complicated in general. – 2017-01-05
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    Ok, thank you for your reply. Will check them out. Indeed I know about Fourier and Laplace Transforms myself but I never learned them as convolution equations. All I know about convolution is that they adhere to the integral $f*g=\int_{-\infty}^{t}f(\tau)h(t-\tau)\mathrm{d}\tau$. I also recognize that the integral solution to an operator using green's function is a convolution integral, but I never heard of a convolution equation. – 2017-01-05
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    $f''(t)+ 2 f'(t)+f(t) = g(t)$ with unknown $f$ is a convolution equation. Let $h(t ) =\delta''(t)+ 2 \delta'(t)+ \delta(t)$ it is a [distribution](https://en.wikipedia.org/wiki/Distribution_(mathematics)) and the equation is $f \ast h = g$, so it amounts to find $H(t)$ such that $h \ast H(t) = \delta(t)$ – 2017-01-05
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    @user1952009: Could you direct me to some literature? Also, for the above equation, I would normally use Laplace/Fourier Transform and then invert it. – 2017-01-05

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