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Fixed point theorems for nonlinear nonself mappings in Hilbert spaces and applications
Fixed Point Theory and Applications volume 2013, Article number: 116 (2013)
Abstract
Recently, Kawasaki and Takahashi (J. Nonlinear Convex Anal. 14:7187, 2013) defined a broad class of nonlinear mappings, called widely more generalized hybrid, in a Hilbert space which contains generalized hybrid mappings (Kocourek et al. in Taiwan. J. Math. 14:24972511, 2010) and strict pseudocontractive mappings (Browder and Petryshyn in J. Math. Anal. Appl. 20:197228, 1967). They proved fixed point theorems for such mappings. In this paper, we prove fixed point theorems for widely more generalized hybrid nonself mappings in a Hilbert space by using the idea of Hojo et al. (Fixed Point Theory 12:113126, 2011) and Kawasaki and Takahashi fixed point theorems (J. Nonlinear Convex Anal. 14:7187, 2013). Using these fixed point theorems for nonself mappings, we proved the Browder and Petryshyn fixed point theorem (J. Math. Anal. Appl. 20:197228, 1967) for strict pseudocontractive nonself mappings and the Kocourek et al. fixed point theorem (Taiwan. J. Math. 14:24972511, 2010) for super hybrid nonself mappings. In particular, we solve a fixed point problem.
MSC:Primary 47H10; secondary 47H05.
1 Introduction
Let ℝ be the real line and let [0,\frac{\pi}{2}] be a bounded, closed and convex subset of ℝ. Consider a mapping T:[0,\frac{\pi}{2}]\to \mathbb{R} defined by
for all x\in [0,\frac{\pi}{2}]. Such a mapping T has a unique fixed point z\in [0,\frac{\pi}{2}] such that cosz=z. What kind of fixed point theorems can we use to find such a unique fixed point z of T?
Let H be a real Hilbert space and let C be a nonempty subset of H. Kocourek, Takahashi and Yao [1] introduced a class of nonlinear mappings in a Hilbert space which covers nonexpansive mappings, nonspreading mappings [2] and hybrid mappings [3]. A mapping T:C\to H is said to be generalized hybrid if there exist \alpha ,\beta \in \mathbb{R} such that
for all x,y\in C. We call such a mapping an (\alpha ,\beta )generalized hybrid mapping. An (\alpha ,\beta )generalized hybrid mapping is nonexpansive for \alpha =1 and \beta =0, i.e.,
for all x,y\in C. It is nonspreading for \alpha =2 and \beta =1, i.e.,
for all x,y\in C. Furthermore, it is hybrid for \alpha =\frac{3}{2} and \beta =\frac{1}{2}, i.e.,
for all x,y\in C. They proved fixed point theorems and nonlinear ergodic theorems of Baillon type [4] for generalized hybrid mappings; see also Kohsaka and Takahashi [5] and Iemoto and Takahashi [6]. Very recently, Kawasaki and Takahashi [7] introduced a broader class of nonlinear mappings than the class of generalized hybrid mappings in a Hilbert space. A mapping T from C into H is called widely more generalized hybrid if there exist \alpha ,\beta ,\gamma ,\delta ,\epsilon ,\zeta ,\eta \in \mathbb{R} such that
for all x,y\in C. Such a mapping T is called an (\alpha ,\beta ,\gamma ,\delta ,\epsilon ,\zeta ,\eta )widely more generalized hybrid mapping. In particular, an (\alpha ,\beta ,\gamma ,\delta ,\epsilon ,\zeta ,\eta )widely more generalized hybrid mapping is generalized hybrid in the sense of Kocourek, Takahashi and Yao [1] if \alpha +\beta =\gamma \delta =1 and \epsilon =\zeta =\eta =0. An (\alpha ,\beta ,\gamma ,\delta ,\epsilon ,\zeta ,\eta )widely more generalized hybrid mapping is strict pseudocontractive in the sense of Browder and Petryshyn [8] if \alpha =1, \beta =\gamma =0, \delta =1, \epsilon =\zeta =0, \eta =k, where 0\le k<1. A generalized hybrid mapping with a fixed point is quasinonexpansive. However, a widely more generalized hybrid mapping is not quasinonexpansive in general even if it has a fixed point. In [7], Kawasaki and Takahashi proved fixed point theorems and nonlinear ergodic theorems of Baillon type [4] for such widely more generalized hybrid mappings in a Hilbert space. In particular, they proved directly the Browder and Petryshyn fixed point theorem [8] for strict pseudocontractive mappings and the Kocourek, Takahashi and Yao fixed point theorem [1] for super hybrid mappings by using their fixed point theorems. However, we cannot use Kawasaki and Takahashi fixed point theorems to solve the above problem. For a nice synthesis on metric fixed point theory, see Kirk [9].
In this paper, motivated by such a problem, we prove fixed point theorems for widely more generalized hybrid nonself mappings in a Hilbert space by using the idea of Hojo, Takahashi and Yao [10] and Kawasaki and Takahashi fixed point theorems [7]. Using these fixed point theorems for nonself mappings, we prove the Browder and Petryshyn fixed point theorem [8] for strict pseudocontractive nonself mappings and the Kocourek, Takahashi and Yao fixed point theorem [1] for super hybrid nonself mappings. In particular, we solve the above problem by using one of our fixed point theorems.
2 Preliminaries
Throughout this paper, we denote by ℕ the set of positive integers. Let H be a (real) Hilbert space with the inner product \u3008\cdot ,\cdot \u3009 and the norm , respectively. From [11], we know the following basic equality: For x,y\in H and \lambda \in \mathbb{R}, we have
Furthermore, we know that for x,y,u,v\in H,
Let C be a nonempty, closed and convex subset of H and let T be a mapping from C into H. Then we denote by F(T) the set of fixed points of T. A mapping S:C\to H is called super hybrid [1, 12] if there exist \alpha ,\beta ,\gamma \in \mathbb{R} such that
for all x,y\in C. We call such a mapping an (\alpha ,\beta ,\gamma )super hybrid mapping. An (\alpha ,\beta ,0)super hybrid mapping is (\alpha ,\beta )generalized hybrid. Thus the class of super hybrid mappings contains generalized hybrid mappings. The following theorem was proved in [12]; see also [1].
Theorem 2.1 ([12])
Let C be a nonempty subset of a Hilbert space H and let α, β and γ be real numbers with \gamma \ne 1. Let S and T be mappings of C into H such that T=\frac{1}{1+\gamma}S+\frac{\gamma}{1+\gamma}I. Then S is (\alpha ,\beta ,\gamma )super hybrid if and only if T is (\alpha ,\beta )generalized hybrid. In this case, F(S)=F(T). In particular, let C be a nonempty, closed and convex subset of H and let α, β and γ be real numbers with \gamma \ge 0. If a mapping S:C\to C is (\alpha ,\beta ,\gamma )super hybrid, then the mapping T=\frac{1}{1+\gamma}S+\frac{\gamma}{1+\gamma}I is an (\alpha ,\beta )generalized hybrid mapping of C into itself.
In [1], Kocourek, Takahashi and Yao also proved the following fixed point theorem for super hybrid mappings in a Hilbert space.
Theorem 2.2 ([1])
Let C be a nonempty, bounded, closed and convex subset of a Hilbert space H and let α, β and γ be real numbers with \gamma \ge 0. Let S:C\to C be an (\alpha ,\beta ,\gamma )super hybrid mapping. Then S has a fixed point in C. In particular, if S:C\to C is an (\alpha ,\beta )generalized hybrid mapping, then S has a fixed point in C.
A super hybrid mapping is not quasinonexpansive in general even if it has a fixed point. There exists a class of nonlinear mappings in a Hilbert space defined by Kawasaki and Takahashi [13] which covers contractive mappings and generalized hybrid mappings. A mapping T from C into H is said to be widely generalized hybrid if there exist \alpha ,\beta ,\gamma ,\delta ,\epsilon ,\zeta \in \mathbb{R} such that
for any x,y\in C. Such a mapping T is called (\alpha ,\beta ,\gamma ,\delta ,\epsilon ,\zeta )widely generalized hybrid. Kawasaki and Takahashi [13] proved the following fixed point theorem.
Theorem 2.3 ([13])
Let H be a Hilbert space, let C be a nonempty, closed and convex subset of H and let T be an (\alpha ,\beta ,\gamma ,\delta ,\epsilon ,\zeta )widely generalized hybrid mapping from C into itself which satisfies the following conditions (1) and (2):

(1)
\alpha +\beta +\gamma +\delta \ge 0;

(2)
\epsilon +\alpha +\gamma >0, or \zeta +\alpha +\beta >0.
Then T has a fixed point if and only if there exists z\in C such that \{{T}^{n}z\mid n=0,1,\dots \} is bounded. In particular, a fixed point of T is unique in the case of \alpha +\beta +\gamma +\delta >0 under the condition (1).
Very recently, Kawasaki and Takahashi [7] also proved the following fixed point theorem which will be used in the proofs of our main theorems in this paper.
Theorem 2.4 ([7])
Let H be a Hilbert space, let C be a nonempty, closed and convex subset of H and let T be an (\alpha ,\beta ,\gamma ,\delta ,\epsilon ,\zeta ,\eta )widely more generalized hybrid mapping from C into itself, i.e., there exist \alpha ,\beta ,\gamma ,\delta ,\epsilon ,\zeta ,\eta \in \mathbb{R} such that
for all x,y\in C. Suppose that it satisfies the following condition (1) or (2):

(1)
\alpha +\beta +\gamma +\delta \ge 0, \alpha +\gamma +\epsilon +\eta >0 and \zeta +\eta \ge 0;

(2)
\alpha +\beta +\gamma +\delta \ge 0, \alpha +\beta +\zeta +\eta >0 and \epsilon +\eta \ge 0.
Then T has a fixed point if and only if there exists z\in C such that \{{T}^{n}z\mid n=0,1,\dots \} is bounded. In particular, a fixed point of T is unique in the case of \alpha +\beta +\gamma +\delta >0 under the conditions (1) and (2).
In particular, we have the following theorem from Theorem 2.4.
Theorem 2.5 Let H be a Hilbert space, let C be a nonempty, bounded, closed and convex subset of H and let T be an (\alpha ,\beta ,\gamma ,\delta ,\epsilon ,\zeta ,\eta )widely more generalized hybrid mapping from C into itself which satisfies the following condition (1) or (2):

(1)
\alpha +\beta +\gamma +\delta \ge 0, \alpha +\gamma +\epsilon +\eta >0 and \zeta +\eta \ge 0;

(2)
\alpha +\beta +\gamma +\delta \ge 0, \alpha +\beta +\zeta +\eta >0 and \epsilon +\eta \ge 0.
Then T has a fixed point. In particular, a fixed point of T is unique in the case of \alpha +\beta +\gamma +\delta >0 under the conditions (1) and (2).
3 Fixed point theorems for nonself mappings
In this section, using the fixed point theorem (Theorem 2.5), we first prove the following fixed point theorem for widely more generalized hybrid nonself mappings in a Hilbert space.
Theorem 3.1 Let C be a nonempty, bounded, closed and convex subset of a Hilbert space H and let \alpha ,\beta ,\gamma ,\delta ,\epsilon ,\zeta ,\eta \in \mathbb{R}. Let T:C\to H be an (\alpha ,\beta ,\gamma ,\delta ,\epsilon ,\zeta ,\eta )widely more generalized hybrid mapping. Suppose that it satisfies the following condition (1) or (2):

(1)
\alpha +\beta +\gamma +\delta \ge 0, \alpha +\gamma +\epsilon +\eta >0, \alpha +\beta +\zeta +\eta \ge 0 and \zeta +\eta \ge 0;

(2)
\alpha +\beta +\gamma +\delta \ge 0, \alpha +\beta +\zeta +\eta >0, \alpha +\gamma +\epsilon +\eta \ge 0 and \epsilon +\eta \ge 0.
Assume that there exists a positive number m>1 such that for any x\in C,
for some y\in C and t with 0<t\le m. Then T has a fixed point in C. In particular, a fixed point of T is unique in the case of \alpha +\beta +\gamma +\delta >0 under the conditions (1) and (2).
Proof We give the proof for the case of (1). By the assumption, we have that for any x\in C, there exist y\in C and t with 0<t\le m such that Tx=x+t(yx). From this, we have Tx=ty+(1t)x and hence
Define Ux\in C as follows:
Taking \lambda >0 with m=1+\lambda, we have that
and hence
Since T:C\to H is an (\alpha ,\beta ,\gamma ,\delta ,\epsilon ,\zeta ,\eta )widely more generalized hybrid mapping, we have from (3.1) and (2.1) that for any x,y\in C,
This implies that U is widely more generalized hybrid. Since \alpha +\beta +\gamma +\delta \ge 0, \alpha +\gamma +\epsilon +\eta >0, \alpha +\beta +\zeta +\eta \ge 0 and \zeta +\eta \ge 0, we obtain that
By Theorem 2.5, we obtain that F(U)\ne \mathrm{\varnothing}. Therefore, we have from F(U)=F(T) that F(T)\ne \mathrm{\varnothing}. Suppose that \alpha +\beta +\gamma +\delta >0. Let {p}_{1} and {p}_{2} be fixed points of T. We have that
and hence {p}_{1}={p}_{2}. Therefore, a fixed point of T is unique.
Similarly, we can obtain the desired result for the case when \alpha +\beta +\gamma +\delta \ge 0, \alpha +\beta +\zeta +\eta >0, \alpha +\gamma +\epsilon +\eta \ge 0 and \epsilon +\eta \ge 0. This completes the proof. □
The following theorem is a useful extension of Theorem 3.1.
Theorem 3.2 Let H be a Hilbert space, let C be a nonempty, bounded, closed and convex subset of H and let T be an (\alpha ,\beta ,\gamma ,\delta ,\epsilon ,\zeta ,\eta )widely more generalized hybrid mapping from C into H which satisfies the following condition (1) or (2):

(1)
\alpha +\beta +\gamma +\delta \ge 0, \alpha +\gamma +\epsilon +\eta >0, \alpha +\beta +\zeta +\eta \ge 0 and [0,1)\cap \{\lambda \mid (\alpha +\beta )\lambda +\zeta +\eta \ge 0\}\ne \mathrm{\varnothing};

(2)
\alpha +\beta +\gamma +\delta \ge 0, \alpha +\beta +\zeta +\eta >0, \alpha +\gamma +\epsilon +\eta \ge 0 and [0,1)\cap \{\lambda \mid (\alpha +\gamma )\lambda +\epsilon +\eta \ge 0\}\ne \mathrm{\varnothing}.
Assume that there exists m>1 such that for any x\in C,
for some y\in C and t with 0<t\le m. Then T has a fixed point. In particular, a fixed point of T is unique in the case of \alpha +\beta +\gamma +\delta >0 under the conditions (1) and (2).
Proof Let \lambda \in [0,1)\cap \{\lambda \mid (\alpha +\beta )\lambda +\zeta +\eta \ge 0\} and define S=(1\lambda )T+\lambda I. Then S is a mapping from C into H. Since \lambda \ne 1, we obtain that F(S)=F(T). Moreover, from T=\frac{1}{1\lambda}S\frac{\lambda}{1\lambda}I and (2.1), we have that
Therefore S is an (\frac{\alpha}{1\lambda},\frac{\beta}{1\lambda},\frac{\gamma}{1\lambda},\frac{\lambda}{1\lambda}(\alpha +\beta +\gamma )+\delta ,\frac{\epsilon +\gamma \lambda}{{(1\lambda )}^{2}},\frac{\zeta +\beta \lambda}{{(1\lambda )}^{2}},\frac{\eta +\alpha \lambda}{{(1\lambda )}^{2}})widely more generalized hybrid mapping. Furthermore, we obtain that
Furthermore, from the assumption, there exists m>1 such that for any x\in C,
where y\in C and 0<t\le m. From 0\le \lambda <1, we have 0<t(1\lambda )\le m. Putting s=t(1\lambda ), we have that there exists m>1 such that for any x\in C,
for some y\in C and s with 0<s\le m. Therefore, we obtain from Theorem 3.1 that F(S)\ne \mathrm{\varnothing}. Since F(S)=F(T), we obtain that F(T)\ne \mathrm{\varnothing}.
Next, suppose that \alpha +\beta +\gamma +\delta >0. Let {p}_{1} and {p}_{2} be fixed points of T. As in the proof of Theorem 3.1, we have {p}_{1}={p}_{2}. Therefore a fixed point of T is unique.
In the case of \alpha +\beta +\gamma +\delta \ge 0, \alpha +\beta +\zeta +\eta >0, \alpha +\gamma +\epsilon +\eta \ge 0 and [0,1)\cap \{\lambda \mid (\alpha +\gamma )\lambda +\epsilon +\eta \ge 0\}\ne \mathrm{\varnothing}, we can obtain the desired result by replacing the variables x and y. □
Remark 1 We can also prove Theorems 3.1 and 3.2 by using the condition
instead of the condition
respectively. In fact, in the case of the condition \beta \delta +\epsilon +\eta >0, we obtain from \alpha +\beta +\gamma +\delta \ge 0 that
Thus we obtain the desired results by Theorems 3.1 and 3.2. Similarly, in the case of \gamma \delta +\epsilon +\eta >0, we can obtain the results by using the case of \alpha +\beta +\zeta +\eta >0.
4 Fixed point theorems for wellknown mappings
Using Theorem 3.1, we first show the following fixed point theorem for generalized hybrid nonself 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 nonempty, bounded, closed and convex subset of H and let T be a generalized hybrid mapping from C into H, i.e., there exist \alpha ,\beta \in \mathbb{R} such that
for any x,y\in C. Suppose \alpha \beta \ge 0 and assume that there exists m>1 such that for any x\in C,
for some y\in C and t with 0<t\le m. Then T has a fixed point.
Proof An (\alpha ,\beta )generalized hybrid mapping T from C into H is an (\alpha ,1\alpha ,\beta ,(1\beta ),0,0,0)widely more generalized hybrid mapping. Furthermore, \alpha +(1\alpha )\beta (1\beta )=0, \alpha +(1\alpha )+0+0=1>0, \alpha \beta +0+0=\alpha \beta \ge 0 and 0+0=0, that is, it satisfies the condition (2) in Theorem 3.1. Furthermore, since there exists m\ge 1 such that for any x\in C,
for some y\in C and t with 0<t\le m, 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 nonself mappings in a Hilbert space; see Kawasaki and Takahashi [13].
Theorem 4.2 Let H be a Hilbert space, let C be a nonempty, bounded, closed and convex subset of H and let T be an (\alpha ,\beta ,\gamma ,\delta ,\epsilon ,\zeta )widely generalized hybrid mapping from C into H which satisfies the following condition (1) or (2):

(1)
\alpha +\beta +\gamma +\delta \ge 0, \alpha +\gamma +\epsilon >0 and \alpha +\beta \ge 0;

(2)
\alpha +\beta +\gamma +\delta \ge 0, \alpha +\beta +\zeta >0 and \alpha +\gamma \ge 0.
Assume that there exists m>1 such that for any x\in C,
for some y\in C and t\in \mathbb{R} with 0<t\le m. Then T has a fixed point. In particular, a fixed point of T is unique in the case of \alpha +\beta +\gamma +\delta >0 under the conditions (1) and (2).
Proof Since T is (\alpha ,\beta ,\gamma ,\delta ,\epsilon ,\zeta )widely generalized hybrid, we obtain that
for any x,y\in C. In the case of \alpha +\gamma +\epsilon >0, from
we obtain that
that is, it is an (\alpha ,\beta ,\gamma ,\delta ,\epsilon ,0,0)widely more generalized hybrid mapping. Furthermore, we have that \alpha +\beta +\gamma +\delta \ge 0, \alpha +\gamma +\epsilon +0=\alpha +\gamma +\epsilon >0, \alpha +\beta +0+0=\alpha +\beta \ge 0 and 0+0=0, that is, it satisfies the condition (1) in Theorem 3.1. Furthermore, since there exists m\ge 1 such that for any x\in C,
for some y\in C and t with 0<t\le m, we obtain the desired result from Theorem 3.1. In the case of \alpha +\beta +\gamma +\delta \ge 0, \alpha +\beta +\zeta >0 and \alpha +\gamma \ge 0, we can obtain the desired result by replacing the variables x and y. □
We know that an (\alpha ,\beta ,\gamma ,\delta ,\epsilon ,\zeta ,\eta )widely more generalized hybrid mapping with \alpha =1, \beta =\gamma =\epsilon =\zeta =0, \delta =1 and \eta =k\in (1,0] is a strict pseudocontractive mapping in the sense of Browder and Petryshyn [8]. We also define the following mapping: T:C\to H is called a generalized strict pseudocontractive mapping if there exist r,k\in \mathbb{R} with 0\le r\le 1 and 0\le k<1 such that
for any x,y\in C. Using Theorem 3.2, we can show the following fixed point theorem for generalized strict pseudocontractive nonself mappings in a Hilbert space.
Theorem 4.3 Let H be a Hilbert space, let C be a nonempty, bounded, closed and convex subset of H and let T be a generalized strict pseudocontractive mapping from C into H, that is, there exist r,k\in \mathbb{R} with 0\le r\le 1 and 0\le k<1 such that
for all x,y\in C. Assume that there exists m>1 such that for any x\in C,
for some y\in C and t\in \mathbb{R} with 0<t\le m. Then T has a fixed point. In particular, if 0\le r<1, then T has a unique fixed point.
Proof A generalized strict pseudocontractive mapping T from C into H is a (1,0,0,r,0,0,k)widely more generalized hybrid mapping. Furthermore, 1+0+0+(r)\ge 0, 1+0+0+(k)=1k>0, 1+0+0+(k)=1k>0 and [0,1)\cap \{\lambda \mid (1+0)\lambda +0k\ge 0\}=[k,1)\ne \mathrm{\varnothing}, that is, it satisfies the condition (1) in Theorem 3.2. Furthermore, since there exists m\ge 1 such that for any x\in C,
for some y\in C and t with 0<t\le m, we obtain the desired result from Theorem 3.2. In particular, if 0\le r<1, then 1+0+0+(r)>0. We have from Theorem 3.2 that T has a unique fixed point. □
Let us consider the problem in the Introduction. A mapping T:[0,\frac{\pi}{2}]\to \mathbb{R} was defined as follows:
for all x\in [0,\frac{\pi}{2}]. We have that
Thus we have that for any x\in [0,\frac{\pi}{2}],
and hence
Using this, we also have from (2.1) that for any x,y\in [0,\frac{\pi}{2}],
and hence
Define a function f:[0,\frac{\pi}{2}]\to \mathbb{R} as follows:
for all x\in [0,\frac{\pi}{2}]. Then we have
and
Since
and {f}^{\mathrm{\prime}\mathrm{\prime}}(x)<0 for all x\in [0,\frac{\pi}{2}], we have from the mean value theorem that there exists a positive number r with 0<r<1 such that
for all x,y\in [0,\frac{\pi}{2}]. Therefore, we have from (4.2) that
for all x,y\in [0,\frac{\pi}{2}]. Furthermore, we have from (4.1) that
for all x\in [0,\frac{\pi}{2}]. Take m=1+\pi and let t=1+\frac{1}{2}x and y=cosx for all x\in [0,\frac{\pi}{2}]. Then we have that
Using Theorem 3.2, we have that T has a unique fixed point z\in [0,\frac{\pi}{2}]. We also know that z=Tz is equivalent to cosz=z. In fact,
Using Theorem 3.2, we can also show the following fixed point theorem for super hybrid nonself mappings in a Hilbert space; see [1].
Theorem 4.4 Let H be a Hilbert space, let C be a nonempty, bounded, closed and convex subset of H and let T be a super hybrid mapping from C into H, that is, there exist \alpha ,\beta ,\gamma \in \mathbb{R} such that
for all x,y\in C. Assume that there exists m>1 such that for any x\in C,
for some y\in C and t with 0<t\le m. Suppose that \alpha \beta \ge 0 or \gamma \ge 0. Then T has a fixed point.
Proof An (\alpha ,\beta ,\gamma )super hybrid mapping T from C into H is an (\alpha ,1\alpha +\gamma ,\beta (\beta \alpha )\gamma ,1+\beta +(\beta \alpha 1)\gamma ,(\alpha \beta )\gamma ,\gamma ,0)widely more generalized hybrid mapping. Furthermore, \alpha +(1\alpha +\gamma )+(\beta (\beta \alpha )\gamma )+(1+\beta +(\beta \alpha 1)\gamma )=0, \alpha +(1\alpha +\gamma )+(\gamma )+0=1>0 and \alpha \beta (\beta \alpha )\gamma (\alpha \beta )\gamma +0=\alpha \beta \ge 0, that is, it satisfies the conditions \alpha +\beta +\gamma +\delta \ge 0, \alpha +\beta +\zeta +\eta >0 and \alpha +\gamma +\epsilon +\eta \ge 0 in (2) of Theorem 3.2. Moreover, we have that
If \alpha \beta >0, then
that is, it satisfies the condition [0,1)\cap \{\lambda \mid (\alpha +\gamma )\lambda +\epsilon +\eta \ge 0\}\ne \mathrm{\varnothing} in (2) of Theorem 3.2. If \alpha \beta =0, then
that is, it satisfies the condition [0,1)\cap \{\lambda \mid (\alpha +\gamma )\lambda +\epsilon +\eta \ge 0\}\ne \mathrm{\varnothing} in (2) of Theorem 3.2. If \alpha \beta <0 and \gamma \ge 0, then
that is, it again satisfies the condition [0,1)\cap \{\lambda \mid (\alpha +\gamma )\lambda +\epsilon +\eta \ge 0\}\ne \mathrm{\varnothing} 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].
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Acknowledgements
Dedicated to Professor Hari M Srivastava.
The first author was partially supported by GrantinAid for Scientific Research No. 23540188 from Japan Society for the Promotion of Science. The second and the third authors were partially supported by the grant NSC 992115M110007MY3 and the grant NSC 992115M037002MY3, respectively.
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Takahashi, W., Wong, NC. & Yao, JC. Fixed point theorems for nonlinear nonself mappings in Hilbert spaces and applications. Fixed Point Theory Appl 2013, 116 (2013). https://doi.org/10.1186/168718122013116
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DOI: https://doi.org/10.1186/168718122013116