Sentence examples for we establish a solution from inspiring English sources

In this paper, we establish a solution to the following integral equation: u ( t ) = ∫ 0 T G ( t, s ) f ( s, u ( s ) ) d s for all  t ∈ [ 0, T ], (1).

We establish a result concerning the existence of solutions for the following implicit integral equation:, where is not supposed continuous with respect to the second variable.

Again, we should establish a solution ( x 1 ( k ), x 2 ( k ) ) of system (7) as in the form of (13).

By some fixed theorems of a sum operator on cone, we establish a unique solution and set up two iterative schemes to approximate the solution.

In this paper, we establish a new solution of the problem (C).

In this section, we establish an optimized solution for the BLT problem based on the hybrid algorithm scheme.

Furthermore, we establish a blow-up solution and obtain both upper and lower bounds for the maximum blow-up time under some appropriate hypotheses.

Hence, we need first to establish a solution of the left problem.

Under suitable assumptions on the initial data and the relaxation function, we establish a blow-up result of a solution for negative initial energy and some positive initial energy if the influence of the source term is greater than the dissipation.

We establish a general existence result for solutions lying in an admissible bounding set for the system of ordinary differential equations (1) satisfying boundary conditions (2) and impulses (3).

By using the concavity method, we establish a blow-up result of the solutions under suitable assumptions on the initial energy.