Using Theorem 3.1, we first show the following fixed point theorem for generalized hybrid non-self mappings in a Hilbert space; see also Kocourek, Takahashi and Yao [1].
Theorem 4.1 Let H be a Hilbert space, let C be a non-empty, bounded, closed and convex subset of H and let T be a generalized hybrid mapping from C into H, i.e., there exist such that
for any . Suppose and assume that there exists such that for any ,
for some and t with . Then T has a fixed point.
Proof An -generalized hybrid mapping T from C into H is an -widely more generalized hybrid mapping. Furthermore, , , and , that is, it satisfies the condition (2) in Theorem 3.1. Furthermore, since there exists such that for any ,
for some and t with , we obtain the desired result from Theorem 3.1. □
Using Theorem 3.1, we can also show the following fixed point theorem for widely generalized hybrid non-self mappings in a Hilbert space; see Kawasaki and Takahashi [13].
Theorem 4.2 Let H be a Hilbert space, let C be a non-empty, bounded, closed and convex subset of H and let T be an -widely generalized hybrid mapping from C into H which satisfies the following condition (1) or (2):
-
(1)
, and ;
-
(2)
, and .
Assume that there exists such that for any ,
for some and with . Then T has a fixed point. In particular, a fixed point of T is unique in the case of under the conditions (1) and (2).
Proof Since T is -widely generalized hybrid, we obtain that
for any . In the case of , from
we obtain that
that is, it is an -widely more generalized hybrid mapping. Furthermore, we have that , , and , that is, it satisfies the condition (1) in Theorem 3.1. Furthermore, since there exists such that for any ,
for some and t with , we obtain the desired result from Theorem 3.1. In the case of , and , we can obtain the desired result by replacing the variables x and y. □
We know that an -widely more generalized hybrid mapping with , , and is a strict pseudo-contractive mapping in the sense of Browder and Petryshyn [8]. We also define the following mapping: is called a generalized strict pseudo-contractive mapping if there exist with and such that
for any . Using Theorem 3.2, we can show the following fixed point theorem for generalized strict pseudo-contractive non-self mappings in a Hilbert space.
Theorem 4.3 Let H be a Hilbert space, let C be a non-empty, bounded, closed and convex subset of H and let T be a generalized strict pseudo-contractive mapping from C into H, that is, there exist with and such that
for all . Assume that there exists such that for any ,
for some and with . Then T has a fixed point. In particular, if , then T has a unique fixed point.
Proof A generalized strict pseudo-contractive mapping T from C into H is a -widely more generalized hybrid mapping. Furthermore, , , and , that is, it satisfies the condition (1) in Theorem 3.2. Furthermore, since there exists such that for any ,
for some and t with , we obtain the desired result from Theorem 3.2. In particular, if , then . We have from Theorem 3.2 that T has a unique fixed point. □
Let us consider the problem in the Introduction. A mapping was defined as follows:
(4.1)
for all . We have that
Thus we have that for any ,
and hence
Using this, we also have from (2.1) that for any ,
and hence
(4.2)
Define a function as follows:
for all . Then we have
and
Since
and for all , we have from the mean value theorem that there exists a positive number r with such that
for all . Therefore, we have from (4.2) that
for all . Furthermore, we have from (4.1) that
for all . Take and let and for all . Then we have that
Using Theorem 3.2, we have that T has a unique fixed point . We also know that is equivalent to . In fact,
Using Theorem 3.2, we can also show the following fixed point theorem for super hybrid non-self mappings in a Hilbert space; see [1].
Theorem 4.4 Let H be a Hilbert space, let C be a non-empty, bounded, closed and convex subset of H and let T be a super hybrid mapping from C into H, that is, there exist such that
for all . Assume that there exists such that for any ,
for some and t with . Suppose that or . Then T has a fixed point.
Proof An -super hybrid mapping T from C into H is an -widely more generalized hybrid mapping. Furthermore, , and , that is, it satisfies the conditions , and in (2) of Theorem 3.2. Moreover, we have that
If , then
that is, it satisfies the condition in (2) of Theorem 3.2. If , then
that is, it satisfies the condition in (2) of Theorem 3.2. If and , then
that is, it again satisfies the condition in (2) of Theorem 3.2. Then we obtain the desired result from Theorem 3.2. Similarly, we obtain the desired result from Theorem 3.2 in the case of (1). □
We remark that some recent results related to this paper have been obtained in [14–17].