**Theorem 2.1**
*Let*
*E*
*be an ordered Banach space with lattice structure*,
*be bounded*, *and*
*be a decreasing and condensing operator*. *Then the operator*
*A*
*has a fixed point in*
*D*.

*Proof* For any
, since
, we have
.

Since

*E* is a Banach space with lattice structure and

is bounded, there exists

such that

Since

*A* is a decreasing operator, we have

(2.1) and (2.2) show that

Similar to the proof of (2.3), there exists

such that

Since

*A* is a decreasing operator, we have

(2.4) and (2.5) show that

(2.3) and (2.6) together with

show that

For any

, since

*A* is a decreasing operator, we have

It is easy to know that
is a closed convex set. Since
is bounded, we have
is bounded. Hence,
is a closed bounded convex set. Thus, Lemma 1.1 implies that the operator *A* has a fixed point in *D*. □

**Theorem 2.2**
*Let*
*E*
*be an ordered Banach space with lattice structure*,
*be a normal cone*, *and*
*be a decreasing and condensing operator*. *Then the operator*
*A*
*has a fixed point in*
*E*.

*Proof* For any
, since
, we have
.

Since

*E* is a Banach space with lattice structure, there exists

such that

Since

*A* is a decreasing operator, we have

(2.8) and (2.9) show that

Similar to the proof of (2.10), there exist

such that

Since

*A* is a decreasing operator, we have

(2.11) and (2.12) show that

(2.10) and (2.13) together with

show that

For any

, since

*A* is a decreasing operator, we have

It is easy to know that
is a closed convex set. Since *P* is a normal cone of *E*, we have
is bounded. Hence,
is a closed bounded convex set. Thus, Lemma 1.1 implies that the operator *A* has a fixed point in *D*. □