# Fixed point theorems for nonlinear non-self mappings in Hilbert spaces and applications

- Wataru Takahashi
^{1, 2}, - Ngai-Ching Wong
^{2, 3}and - Jen-Chih Yao
^{3, 4}Email author

**2013**:116

https://doi.org/10.1186/1687-1812-2013-116

© Takahashi et al.; licensee Springer 2013

**Received: **30 January 2013

**Accepted: **15 April 2013

**Published: **29 April 2013

## Abstract

Recently, Kawasaki and Takahashi (J. Nonlinear Convex Anal. 14:71-87, 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. 14:2497-2511, 2010) and strict pseudo-contractive mappings (Browder and Petryshyn in J. Math. Anal. Appl. 20:197-228, 1967). They proved fixed point theorems for such mappings. In this paper, we prove fixed point theorems for widely more generalized hybrid non-self mappings in a Hilbert space by using the idea of Hojo *et al.* (Fixed Point Theory 12:113-126, 2011) and Kawasaki and Takahashi fixed point theorems (J. Nonlinear Convex Anal. 14:71-87, 2013). Using these fixed point theorems for non-self mappings, we proved the Browder and Petryshyn fixed point theorem (J. Math. Anal. Appl. 20:197-228, 1967) for strict pseudo-contractive non-self mappings and the Kocourek *et al.* fixed point theorem (Taiwan. J. Math. 14:2497-2511, 2010) for super hybrid non-self mappings. In particular, we solve a fixed point problem.

**MSC:**Primary 47H10; secondary 47H05.

## Keywords

## 1 Introduction

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*?

*H*be a real Hilbert space and let

*C*be a non-empty 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

*i.e.*,

*i.e.*,

*i.e.*,

*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 pseudo-contractive 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 quasi-nonexpansive. However, a widely more generalized hybrid mapping is not quasi-nonexpansive 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 pseudo-contractive 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 non-self 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 non-self mappings, we prove the Browder and Petryshyn fixed point theorem [8] for strict pseudo-contractive non-self mappings and the Kocourek, Takahashi and Yao fixed point theorem [1] for super hybrid non-self mappings. In particular, we solve the above problem by using one of our fixed point theorems.

## 2 Preliminaries

*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

*C*be a non-empty, 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 non*-*empty 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 non*-*empty*, *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*.

*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 non*-

*empty*,

*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 non*-

*empty*,

*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 non*-

*empty*,

*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 non-self 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 non-self mappings in a Hilbert space.

**Theorem 3.1**

*Let*

*C*

*be a non*-

*empty*,

*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 $Tx=x+t(y-x)$. From this, we have $Tx=ty+(1-t)x$ and hence

*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

*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 non*-

*empty*,

*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

*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

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

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 well-known mappings

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*$\alpha ,\beta \in \mathbb{R}$

*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 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*$(\alpha ,\beta ,\gamma ,\delta ,\epsilon ,\zeta )$-

*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 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*. □

for any $x,y\in C$. 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*$r,k\in \mathbb{R}$

*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 pseudo-contractive mapping

*T*from

*C*into

*H*is a $(1,0,0,-r,0,0,-k)$-widely more generalized hybrid mapping. Furthermore, $1+0+0+(-r)\ge 0$, $1+0+0+(-k)=1-k>0$, $1+0+0+(-k)=1-k>0$ and $[0,1)\cap \{\lambda \mid (1+0)\lambda +0-k\ge 0\}=[k,1)\ne \mathrm{\varnothing}$, 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. □

*r*with $0<r<1$ such 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 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*$\alpha ,\beta ,\gamma \in \mathbb{R}$

*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

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].

## Declarations

### Acknowledgements

Dedicated to Professor Hari M Srivastava.

The first author was partially supported by Grant-in-Aid 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 99-2115-M-110-007-MY3 and the grant NSC 99-2115-M-037-002-MY3, respectively.

## Authors’ Affiliations

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