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Ordinary or partial differential equations. Delay differential equations, neutral equations, integro-differential equations. Well-posedness, asymptotic behavior, and related questions.

1 vote
0 answers
80 views

When is $|\int_a^b \exp(-izx)f(x) \, \mathrm{d}x| \leq |\int_a^b f(x) \, \mathrm{d}x|$ for a... [closed]

Obviously, if $f(x) \geq 0 $ on $(a,b)$, the above claim is trivial. We also see that if $a = -\infty, b = \infty$, above is claiming that $|\hat{f}(z)| \leq |\hat{f}(0)|$ for all real $z$, where $\ha …
1 vote
0 answers
113 views

Existence theorems Volterra Equation of second kind on unbounded domains

The general Volterra Equation of the second kind in convolution form can be described by: $$ \phi(x) = \int_a^x K(x-t)\phi(t)\, \mathrm{d}t + f(x), \text{ for } x\geq a $$ Suppose we wish to pr …