If (g(t)) (>0) is a strictly decreasing continuous function in ((frac{1}{2},infty )), which is strictly convex satisfying (int_{frac{1}{2}}^{infty }g(t),dtin mathbf{R}_), then we have int_{1}^{infty }g(t),dt< sum _{n=1}^{infty }g(n)< int_{frac{1}{2}}^{infty }g(t),dt.
But there are two fallacies in the water policies adopted by all sprawling communities in the American sunbelt that are as true today as they were in Mulholland's time.First, the constant pursuit of additional supplies of water has never been strictly about satisfying a demand from communities that were supposedly running dry.
(i For any, is strictly monotone, satisfying (2.1).
where is defined as in the beginning of Section 2, is strictly increasing odd function satisfying for, satisfies (4.2). for given functions and satisfying for, and functions and satisfy that.
Moreover, if n is even and (F|_{I_) has a regular fixed point, then F also has a strictly decreasing nth iterative root of the form (3.1), in which (f_{1}) is a strictly decreasing function satisfying ({f_{1}}^{n} =F|_{I _).
Proposition 3.1 Let f be a strictly convex function satisfying the PDE (1.1).
Here the existence of strictly increasing solutions satisfying is proved under the assumption that has two zeros 0 and and a superlinear behaviour near.
We solve problems of the form [detgi][det(g)]−1=F x, ∇ϕ; ϕ) inXandϕ=uon ∂X, whereF∈C∞(TX×R) is an everywhere strictly positive function satisfying some assumptions andu∈C∞(∂X).
Moreover, if n is even and (F|_{I_) has a regular fixed point, then F has also a nonmonotonic nth iterative root of the form (3.2), in which (f_{2}) is a strictly decreasing function satisfying ({f_{2}}^{n} =F|_{I _).
Theorem 2.5 Let ( X, d ) be a complete metric space and let G : X → C L X X ) be a strictly -contractive mapping satisfying the following assumptions: (i) G is an α ∗ -admissible mapping; (ii) there exist x 0 ∈ X and x 1 ∈ G x 0 such that α ( x 0, x 1 ) ≥ 1 ; (iii) G is continuous. .
Thus (3.22) yields f ( x ) = r 1 x by Theorem 2 of [14], which is the C 0 strictly increasing solution satisfying 0 < f ( x ) < x for x > 0. The other case f ( x ) > x can be reduced to the first one by considering the dual equation (1.5).
