Sentence examples for given the unique solution from inspiring English sources

For a given the unique solution of the boundary value problem (2.4). is given by (2.5).

Section 4 gives the unique solution of (1.1) and an example is introduced.

In Section 4 we established an existence theorem for the solution of an initial value problem, which not only gives the unique solution but also locates the domain for the solution.

Here, we give the unique solution of BVP (1.1) under the conditions that and is mixed nonmonotone in and does not need to be separable by using the cone theory and the Banach contraction mapping principle.

Proof Let the mapping f a ( t ) = t a e − t k k be defined on ( 0, ∞ ), then f a ′ ( t ) = t a − 1 e − t k k [ a − t k ] ; f a ′ ( t ) = 0 gives the unique solution t 0 = a 1 k, which implies that f a is an increasing function on ( 0, t 0 ) and decreasing on ( t 0, ∞ ).

Given and, the unique solution of (2.5). is given by (2.6).

Proposition 2.3 Let x α be given as the unique solution of (2.2).

Theorem 2.4 Let w α be given as the unique solution of (2.2).

Proposition 2.3 Let w α be given as the unique solution of (2.2).

The final parameter estimate is given by the unique solution (up to a constant factor) of Equation 9.

Then, for arbitrary given (0<sigma<1), the unique solution (x t)) of BVP (2) satisfies x t geqgamma e(t)|x|,quad tin[0,sigma], where 0< gamma=frac{asigma+b}{a+b}-sigma^{alpha-1}< 1 quadtextit{and} quad e(t)= frac{at+b}{a+b},quad tin[0,1].