Sentence examples for a positive solution by from inspiring English sources

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.

We will obtain a positive solution by using the Nehari manifold method.

thus it follows that problem (7.1) has a positive solution by.

The authors obtained the uniqueness of a positive solution by using the fixed point theorem of the mixed monotone operator.

We also show the existence and uniqueness results of a positive solution by applying the Krasnoselskii fixed point theorem.

Under some appropriate conditions, we get the existence of a positive solution by combining the variational method with a Hardy-type inequality.

As an example, when λ = 24, the boundary value problem (3.29) has a positive solution given by y ( t ) = t 5 5 - t 4 2 + t 2 2. Case 2. q = 1.

(1) Since there exist parameters (a=bar{a}geq0) and (b=bar{b}geq 0) such that the boundary value problem (1.1) and (1.2) has a positive solution ū, by Lemma 2.8, we have bar{u}(t)= int_{0}^{1}G(t,s fbigl s,bar{u}(s bigr),mathrm{d}s +(bar{b}-bar{a} t^{delta-1}+bar{b}-bar{a} t2}.

If (3.3) has a positive solution x*, then by virtue of (3.2), y * ( t ) = ∫ 0 t x * ( s ) d s is a positive solution of (1.1).

Applying Theorem 2.7, we know that T has a fixed point in U ¯, which is a positive solution to (1.1) by Lemma 3.1.

Applying Lemma 2.6, we know that has a fixed point in, which is a positive solution to (1.1) by Lemma 3.1.