Obviously, [f.sub.1](x) is a

strictly increasing function for x [greater than or equal to] 2 and m > 0.

Suppose that [phi] [member of] C([R.sub.0], [R.sub.0]) is a

strictly increasing function with [phi](0) = 0 and [psi](u) is a nondecreasing continuous function with [psi](u) > 0 for u [member of] [R.sub.+].

Note that for v [less than or equal to] 30, u(v) given by (3) can be viewed as a

strictly increasing function of v, we therefore have

In particular, the above definition implies that if [F.sub.T] is a continuous and

strictly increasing function then [F.sub.T] has a unique inverse and [Q.sub.T](u) = [F.sup.-1.sub.T] (u), 0 < u < 1.

There exists some [t.sub.q] ([less than or equal to] [infinity]) such that for 0 [less than or equal to] t < [t.sub.q], the integral equation (2.1) has a unique nonnegative continuous solution u, and u (vt,t) is a

strictly increasing function of t.

Moreover, [x.sub.n,k]([lambda], c) (for a fixed value of k and n > 0) is a

strictly increasing function of [lambda].

A mapping T is called strongly [phi]-pseudocontractive if for all x, y [member of] C, there exist j(x--y) [member of] J(x - y) and a

strictly increasing function [phi] : [0, [infinity]) [right arrow] [0, [infinity]) with [PHI](0) = 0 such that

where F is a

strictly increasing function and h([x.sub.1], ..., [x.sub.n]) is a homogeneous function of any given degree d.

Let [psi] : [I.sub.T] [right arrow] R be continuous and

strictly increasing function with f (0) = 0 and let the function L e Crd ([I.sub.T] x [R.sub.+], [R.sub.+]) satisfies the condition

(ii) If u, v [member of] (-[infinity], 0], then f (u, v) isa strictly decreasing function in u, and a

strictly increasing function in v.

The function E(G) is a

strictly increasing function of G such that E(G) [right arrow] + [infinity] as G [right arrow] +[infinity].

(d) f (x, x) is a

strictly increasing function in (-[infinity], + [infinity]).