Sentence examples for a unique solution by from inspiring English sources

Hence, problem (5.1) has a unique solution by Theorem 3.1.

Therefore, this problem has a unique solution by Theorem 3.3.

This suggests that the problem (4.5) has a unique solution by the conclusion of Theorem 4.4.

Then we regularize it to find a unique solution by gradient-projection algorithm.

We conclude that BVP (4.3) has a unique solution by Theorem 3.3.

Therefore, BVP (1.3) has a unique solution by Banach's contraction mapping principle.

It turns out that such a formulation leads to infinite number of solutions; one can obtain an unique solution by introducing an additional requirement inherent in vibration tailoring: namely, designing the system that possesses the pre-specified natural frequency.

When the initial data is given, we can obtain a unique solution to (E) by using the method of steps.

If conditions (2.16) and (2.21) hold, then the abstract operator equation (2.10) has a unique solution in bounded by.

If conditions (2.16) and (2.21) hold, then the discrete equivalent logistic equation (2.11), has a unique solution in bounded by.

Finally we consider the existence of a unique solution for problem (1.1) by applying the contraction mapping principle.