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Exact(51)
Then they are fixed points of the operator Φ.
Clearly, the fixed points of the operator are solution of the problem (1.1).
Observe that the fixed points of the operator ℱ correspond to the solutions of problem (1.1).
Obviously, fixed points of the operator A are solutions of problem (1.6 - 1.7 1.6 - 1.7
Clearly, the fixed points of the operator F are solutions of problem (1.1 - 1.2 1.1 - 1.2
Clearly, the fixed points of the operator T are the solutions of problem (1.1).
Similar(9)
with being the fixed point of the operator and.
Let be the fixed point of the operator at.
This solution is a fixed point of the operator ℱ. □.
that is, is a fixed point of the operator in.
The fixed point of the operator T is the solution of our boundary value problem (1 - 2).
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com