Sentence examples for a lower solution for from inspiring English sources

Finally, let α be a lower solution for (3.1).

Finally, let (alpha ) be a lower solution for (53).

Finally, let α ( t ) be a lower solution for (13).

If there exists a lower solution for the problem, then the problem has a unique solution.

Finally, let α ( t ) be a lower solution for (13); we claim that α ≤ F.

A function is said to be a lower solution for (3.2) if and (3.3).

Hypothesis implies that (hat{u}=0) is a lower solution of (1.3) for domain Ω.

It is clear that (hat{u}=0) is a lower solution of (1.3) for domain Ω by Hypothesis .

This implies that u ( n 0 ) is a lower solution of (3.6) for all s ≤ ν ¯ 4 n 0. Analogously, it follows that there exists n 1 ≤ − 1 such that u ( n 1 ) is an upper solution of (3.6) for all s ≥ ν ¯ 4 n 1.

From all of the above arguments, we see that (underline{Psi} underline{phi},underline{varphi},underline {psi})) is a lower solution of (15) for small enough (lambda in (0,hat{lambda}_{2})), where (hat{lambda}_{2}=min{lambda^{ast}_{4},lambda^{ast }_{5},lambda^{ast}_{6}}).

(v) Now, let (alphain C I,mathbb{R})) be a lower solution of (27), therefore, for all (tin I), begin{aligned} alpha(t) leq& v(t)+mu int_{a}^{b} K t,s alpha(s),ds, =& (Talpha) (t).