Clearly, (Vert f(x -g(x -grt leqwidehat{p}(f,g)) for all (x Vert.
end{cases} Define (F: Irightarrow2^{I}) by (F x)={0}cup{f(x)}) for each (xin I).
Writing down the fact for (n=1), in (10), we get (H x fgeq0), for (xin I) and (fin V).
Furthermore, (D_{q,omega_{0}}F x)) exists for every (xin I) and D_{q,omega }F x)=f(x).
Assume that (f:I to{mathrm{R}}) is α-forward integrable over I. Let (xin I) and define (F x)=int_{a}^{x} {f(t)Delta _{alpha}t} ).
(2) Then every positive solution of equation (1) with initial conditions in I converges to x̄. (f x,y)) is non-decreasing in each of its arguments; equation (1) has a unique positive equilibrium point x̄ and the function (f x, x)) satisfies the negative feedback condition (x - bar{x}) bigl(f x,x) - xbigr) < 0 quad textit{for every } <span class="lh">xin I-{bar {x}}.
(3) If f is nonexpansive with respect to the second variable, then from (4.9) we have (w_{0}=mathcal{N}_{f} ( theta,theta ) ) and (k=1), and hence the preceding inequalities become K ( t,x ) leq g ( t,0 ) leqfrac{rho}{1+2rho} and f bigl( t, ( 0,0 ) bigr) leqfrac{1}{rho}g ( t,0 ) for almost all (t,xin I). .
(3.11) Since (mu ( widehat{p} varphi,y ) )leqfrac{delta }{lambda (b-a)}), by taking into account (3.10), (3.11), and the last inequality, we obtain widehat{p} varphi,y leqfrac{1}{1-alpha widehat{p} varphi,y leqfrac{1}{1-alpha widehat{pi)leqfrac{epsilon} varphi,y, which implies biglVert varphi(x)-y(x)bigrVert leqfrac{epsilon}{1-delta} quadmbox{for all }xin I.
If f is nonexpansive with respect to the second variable, then from (4.9) we have (w_{0}=mathcal{N}_{f} ( theta,theta ) ) and (k=1), and hence the preceding inequalities become K ( t,x ) leq g ( t,0 ) leqfrac{rho}{1+2rho} and f bigl( t, ( 0,0 ) bigr) leqfrac{1}{rho}g ( t,0 ) for almost all (t,xin I).
On the other hand, if (mathscr{M}) is a symmetric and Jensen convex repetition invariant mean on I, then (1.1) holds with reversed inequality for all (nin mathbb {N}) and (xin I^{n}).
