Sentence examples for by a positive solution from inspiring English sources

By a positive solution of (1.3), we mean a vector such that satisfies (1.3) and, on.

By a positive solution to (1.1) we mean a mapping with positive components on such that (1.1) is satisfied.

By a positive solution of, we mean a positive function which is a solution of in the weak sense, that is such that (1.1).

By a positive solution y of (1.1), we mean a nontrivial y ∈ C [ 0, 1 ] satisfying (1.1) and y ( t ) ≥ 0 for t ∈ [ 0, 1 ].

By a positive solution of (1.2), we mean x is a solution of (1.2) with x > 0 (i.e., x ≥ 0 in ( 0, 1 ) and x ≢ 0 ).

By a positive solution for the fractional q-difference system (1.1) we mean a mapping with positive components on [ 0, 1 ] such that (1.1) is satisfied.

Then, by Theorem 2.2, the function u defined by (2.1) is a positive solution of (1.1).

While pointing out errors can be difficult, it can be easier when you temper it by also providing a positive solution.

In this section, we suppose that equation (1) is nonoscillatory, by h we denote a positive solution of this equation and suppose that both the coefficients rk, r k + r ̃ k are positive for large k.

As stated in [16], we are now in a position to present a sufficient condition for which system (1.3) possesses a positive solution by constructing suitable upper and lower solutions.

Thus by Lemma 2.2, (2.1) has a positive solution at ((lambda_{1}^,ldotsa_{2}^,lambda_{nbda_{n}^)), and by Lemma 3.2 we complete the proof.