(Annual membership, which costs $60 for an individual and $90 for a couple, lowers the ticket price to $5 from $9). "We're both real movie fans, and most of the movies we want to see are in the city," Mr. McLoughlin said.
Using Lemma 2.4 iv), one can show easily that ( u −, u + ) is a coupled lower and upper solution to (12 - 13 12 - 13
In this article, the existence of solution for the first-order nonlinear coupled system of ordinary differential equations with nonlinear coupled boundary condition (CBC for short) is studied using a coupled lower and upper solution approach.
They applied their result to a periodic boundary value problem for a first order differential equation under a mixed monotone condition and under the existence of a coupled lower and upper solutions.
where η : [ 0, + ∞ ) → [ 0, + ∞ ), β : u ↦ u − η ( u ) ∈ Φ. Suppose also that (1 - 2) has a coupled lower and upper solution ( u −, u + ) ∈ P × P. Then the boundary value problem (1 - 2) has a unique positive solution u ∗ ∈ C ( [ 0, 1 ] ).
Now, let be a coupled lower and upper solution of the integral equations (4.1), then we have α ( t ) ≤ F ( t ) and β ( t ) ≤ F ( t ) for all t ∈ [ 0, T ], that is, α ≤ F and β ≥ F.
Now, let ((u, v inOmegatimesOmega) be a coupled lower-upper solution of (4.5).
A pair (u, vin OmegatimesOmega) is called a coupled lower-upper solution of (4.5) if left { textstylebegin{array}{l} u_{t} x,t leq u_{xx} (x,t)+f x,t,u,u_{x})+g x,t,v,v_{x}),quad -infty< x< infty, 0< tleq T, u x,0 leqvarphi(x), quad -infty< x< infty end{array}displaystyle right.
In tandem, Leon's descriptions of his fellow passengers broaden the focus of the narrative – a priest drumming his nails on a black briefcase, a young couple lowering the window to lean and gaze at another train blurry in the hazy rain of early evening, are framed in his middle aged eyes.
"As a hitter, it's really tough when you see a ball in a certain spot and you go to swing at it and it ends up being a couple inches lower than you thought it was," Berkman said.
