On general quasi-variational inequalities

Definition 2.1

Definition 2.1 Cristescu and Lupsa, 2002; Noor, 2008b

Let K be any set in H. The set K is said to be g-convex, if there exists a function $g:H\to H$ such that $$g(u)+t(v-g(u))\in K\text{,}\forall u\text{,}v\in H:g(u)\text{,}v\in K\text{,}t\in [0\text{,}1]\text{.}$$ Note that every convex set is g-convex, but the converse is not true, see Noor (2010a, 2009a,b,c).

Definition 2.2

Definition 2.2 Noor, 2008b

The function $F:K\to H$ is said to be g-convex, if there exists a function g such that $$F(g(u)+t(v-g(u)))⩽(1-t)F(g(u))+\mathit{tF}(v)\text{,}\forall u\text{,}v\in H:g(u)\text{,}v\in K\text{,}t\in [0\text{,}1]\text{.}$$ Clearly every convex function is g-convex, but the converse is not true, see Noor (2008a), Noor (2010a), Noor (2009a).

Lemma 2.1

Lemma 2.1 Noor, 2008b

Let $F:K\to H$ be a differentiable g-convex function. Then $u\in H:g(u)\in K$ is the minimum of g-convex function F on K, if and only if, $u\in H:g(u)\in K$ satisfies the inequality

$$\langle F^{\prime }(g(u))\text{,}v-g(u)\rangle ⩾0\text{,}\forall v\in K\text{,}$$ where $F^{\prime }(u)$ is the differential of F at $g(u)\in K$ .

Proof

Let $u\in H:g(u)\in K$ be a minimum of g-convex function F on K. Then

$$F(g(u))⩽F(v)\text{,}\forall v\in K\text{.}$$ Since K is a g-convex set, so, for all $u\text{,}v\in H:g(u)\text{,}v\in K\text{,}t\in [0\text{,}1]\text{,}g(v_t)=g(u)+t(v-g(u))\in K$ . Setting $v=g(v_t)$ in (2.5), we have $$F(g(u))⩽F(g(u)+t(v-g(u)))\text{.}$$ Dividing the above inequality by t and taking $t\to 0$ , we have $$\langle F^{\prime }(g(u))\text{,}v-g(u)\rangle ⩾0\text{,}\forall v\in H:v\in K\text{,}$$ which is the required result (2.4).

Conversely, let $u\in H:g(u)\in K$ satisfy the inequality (2.4). Since F is a g-convex function, $\forall u\text{,}v\in H:g(u)\text{,}v\in K\text{,}t\in [0\text{,}1]\text{,}g(u)+t(v-g(u))\in K$ and $$F(g(u)+t(v-g(u)))⩽(1-t)F(g(u))+\mathit{tF}(v)\text{,}$$ which implies that $$F(g(v))-F(g(u))⩾\frac{F(g(u)+t(v-g(u)))-F(g(u))}{t}\text{.}$$ Letting $t\to 0$ , and using (2.4), we have $$F(g(v))-F(g(u))⩾\langle F^{\prime }(h(u))\text{,}v-g(u)\rangle ⩾0\text{,}$$ which implies that $$F(g(u))⩽F(v)\text{,}\forall v\in K$$ showing that $u\in H:g(u)\in K$ is the minimum of F on K in H.  □

Lemma 2.2

Let $K(u)$ be a closed and convex set in H. Then, for a given $z\in H\text{,}u\in K(u)$ satisfies the inequality $$\langle u-z\text{,}v-u\rangle ⩾0\text{,}\forall v\in K(u)\text{,}$$ if and only if $$u=P_{K(u)}z\text{,}$$ where $P_{K(u)}$ is the projection of Honto the closed convex set $K(u)$ in H.

We would like to point out that the implicit projection operator $P_{K(u)}$ is not non-expansive. We shall assume that the implicit projection operator $P_{K(u)}$ satisfies the Lipschitz type continuity, which plays an important and fundamental role in the existence theory and in developing numerical methods for solving the quasi- variational inequalities.

Assumption 2.1

For all $u\text{,}v\text{,}w\in H$ , the implicit projection operator $P_{K(u)}$ satisfies the condition

$$\Vert P_{K(u)}w-P_{K(v)}w|\Vert ⩽\nu \Vert u-v\Vert \text{,}$$ where $\nu >0$ is a positive constant.

Assumption 2.1 has been used to prove the existence of a solution of the quasi-variational inequalities as well as in analyzing convergence of the iterative methods. One can easily show that the Assumption 2.1 holds for certain cases.

Remark 2.1

In many important applications (Glowinski et al., 1981; Noor, 1975, 2000, 2004) the convex-valued set $K(u)$ can be written as

$$K(u)=m(u)+K\text{,}$$ where $m(u)$ is a point-point mapping and K is a convex set. In this case, we have

(2.10)

$$P_{K(u)}w=P_{m(u)+K}(w)=m(u)+P_K[w-m(u)]\text{,}\forall u\text{,}v\in H\text{.}$$ We note that if $K(u)$ is defined by (2.9), and $m(u)$ is a Lipschitz continuous mapping with constant $\gamma >0$ , then $$\Vert P_{K(u)}w-P_{K(v)}w\Vert =\Vert m(u)-m(v)+P_K[w-m(u)]-P_K[w-m(v)\Vert ⩽2\Vert m(u)-m(v)\Vert ⩽2\gamma \Vert u-v\Vert \text{,}\forall u\text{,}v\text{,}w\in H\text{.}$$ which shows that Assumption 2.1 holds with $\nu =2\gamma$ .

Definition 2.3

For all $u\text{,}v\in H$ , an operator $T:H\to H$ is said to be:

1. strongly monotone, if there exists a constant $\alpha >0$ such that $$\langle T(u)-T(v)\text{,}u-v\rangle ⩾\alpha \Vert u-v{\Vert }^2$$

2. Lipschitz continuous, if there exists a constant $\beta >0$ such that $$\Vert T(u)-T(v)\Vert ⩽\beta \Vert u-v\Vert \text{.}$$

If T verifies (i) and (ii), then $\alpha ⩽\beta$ .

Lemma 3.1

$u\in H:g(u)\in K(u)$ is a solution of the general quasi-variational inequality (2.1) if and only if $u\in H:g(u)\in K(u)$ satisfies the relation

$$g(u)=P_{K(u)}[u-\rho T(u)]\text{,}$$ where $P_{K(u)}$ is the projection operator and $\rho >0$ is a constant.

Proof

Let $u\in H:g(u)\in K(u)$ be solution of (2.1). Then, from (2.1), we have $$\langle g(u)-(u-\rho T(u))\text{,}v-g(u)\rangle ⩾0\text{,}\forall v\in K(u)\text{,}$$ which is equivalent to finding $u\in H:g(u)\in K(u)$ such that $$g(u)=P_{K(u)}[u-\rho T(u)]\text{,}$$ using Lemma 2.1, the required result (3.1).

Lemma 3.1 implies that the general quasi-variational inequality (2.1) is equivalent to the fixed-point problem (3.1). This alternative equivalent formulation is very useful from numerical and theoretical points of view. We use this alternative fixed-point formulation to discuss the existence of a solution of the general quasi-variational inequality (2.1) and this is the main motivation of our next result. □

Theorem 3.1

Let the operators $T\text{,}g:H\to H$ be both strongly monotone with constants $\alpha >0\text{,}\sigma >0$ and Lipschitz continuous with constants with $\beta >0\text{,}\delta >0$ , respectively. If Assumption 2.1 holds and there exists a constant $\rho >0$ such that

$$|\rho -\frac{\alpha }{{\beta }^2}|<\frac{\sqrt{{\alpha }^2-{\beta }^{2}k(2-k)}}{{\beta }^2}\text{,}\alpha >\beta \sqrt{k(2-k)}\text{,}k<1\text{,}$$ where

(3.3)

$$k=\sqrt{1-2\sigma +{\delta }^2}+\nu \text{,}$$ then there exists a solution $u\in K(u)$ satisfying the general quasi-variational inequality (2.1).

Proof

Let $u\in K(u)$ be a solution of the general quasi-variational inequality (2.1). Then, using Lemma 3.1, we have $$g(u)=P_{K(u)}[u-\rho T(u)]\text{.}$$ Thus we can define the mapping $F(u)$ as:

$$F(u)=u-g(u)+P_{K(u)}[u-\rho T(u)]\text{.}$$ In order to prove the existence of a solution of (2.1), it is enough to show that the mapping $F(u)$ , defined by (3.4), is a contraction mapping.

For $u_1\ne u_2\in H$ , and using Assumption 2.1, we have

$$\Vert F(u_1)-F(u_2)\Vert =\Vert u_1-u_2-(g(u_1)-g(u_2))\Vert +\Vert P_{K(u_1)}[u_1-\rho T(u_1)]-P_{K(u_2)}[u_2-\rho T(u_2)]\Vert ⩽\Vert u_1-u_2-(g(u_1)-g(u_2))\Vert +\Vert P_{K(u_1)}[u_1-\rho T(u_1)]-P_{K(u_2)}[u_1-\rho T(u_1)]\Vert +\Vert P_{K(u_2)}[u_1-\rho T(u_1)]-P_{K(u_2)}[u_2-\rho T(u_2)]\Vert =\nu \Vert u_1-u_2\Vert +\Vert u_1-u_2-(g(u_1)-g(u_2))\Vert +\Vert u_1-u_2-\rho (T(u_1)-T(u_2))\Vert \text{.}$$ Since the operator T is strongly monotone with constant $\alpha >0$ and Lipschitz continuous with constant $\beta >0$ , it follows that:

(3.6)

$$\Vert u_1-u_2-\rho (T(u_1)-T(u_2)){\Vert }^2⩽\Vert u_1-u_2{\Vert }^2-2\rho \langle T(u_1)-T(u_2)\text{,}{u}_1-u_2\rangle +{\rho }^2\Vert T(u_1)-T(u_2){\Vert }^2⩽(1-2\rho \alpha +{\rho }^2{\beta }^2)\Vert u_1-u_2{\Vert }^2\text{.}$$ In a similar way, we have

(3.7)

$$\Vert u_1-u_2-(g(u_1)-g(u_2)){\Vert }^2⩽(1-2\sigma +{\delta }^2)\Vert u_1-u_2{\Vert }^2\text{,}$$ where we have used the fact that g is strongly monotone with constant $\sigma >0$ and Lipschitz continuous with constant $\delta >0$ .

From (3.3), (3.5)–(3.7), we have $$‖F(u_1)-F(u_2)‖⩽\left\{\nu +\sqrt{(1-2\sigma +{\delta }^2)}+\sqrt{(1-2\rho \alpha +{\rho }^2{\beta }^2)}\right\}\Vert u_1-u_2\Vert =\left\{k+\sqrt{(1-2\rho \alpha +{\rho }^2{\beta }^2)}\right\}\Vert u_1-u_2\Vert =\theta \Vert u_1-u_2\Vert \text{,}$$ where

$$\theta =k+\sqrt{(1-2\rho \alpha +{\rho }^2{\beta }^2)}\text{.}$$ From (3.2), we see that $\theta <1$ . Thus it follows that the mapping $F(u)$ , defined by (3.4), is a contraction mapping and consequently it has a fixed point, which belongs to $K(u)$ satisfying the general quasi-variational inequality (2.1), the required result. □

Remark 3.1

We would like to emphasize that the conditions that ensure the existence of the constant $\rho >0$ , which satisfies (3.2) in Theorem 3.1 have been verified in Noor (1988a, 2000, 2004, 2007, 2008a, 2010a, 2009a,b,c) for the quasi-variational inequalities and their related optimization problems. In several cases, these conditions have been used in the existing results and also in the studies of the convergence criteria of the iterative methods for solving the general quasi-variational inequalities. For the examples of the function g, see Noor (2000, 2009c) and the references therein.

Algorithm 4.1

For a given $u_0\in H$ , find the approximate solution $u_{n+1}$ by the iterative schemes

$$u_{n+1}=(1-{\alpha }_n)u_n+{\alpha }_n\{u_n-g(u_n)+P_{K(u_n)}[u_n-\rho T(u_n)]\}\text{,}n=0\text{,}1\text{,}\dots$$ which is known as the Mann iteration process for solving the general quasi-variational inequality (2.1).

Note that if $g=I$ , then the Algorithm 4.1 reduces to the following iterative method for solving the quasi-variational inequality (2.10) and appears to be a new one.

Algorithm 4.2

For a given $u_0\in H$ , find the approximate solution $u_{n+1}$ by the iterative schemes $$u_{n+1}=(1-{\alpha }_n)u_n+{\alpha }_n{P}_{K(u_n)}[u_n-\rho T(u_n)]\text{,}n=0\text{,}1\text{,}\dots$$ If $K(u)\equiv K$ , that is, the convex set $K(u)$ is independent of the solution u, then Algorithm 4.1 reduces to the following algorithm for solving the general variational inequalities (2.2), which was suggested by Noor (2008a).

Algorithm 4.3

For a given $u_0\in H$ , find the approximate solution $u_{n+1}$ by the iterative schemes $$u_{n+1}=(1-{\alpha }_n)u_n+{\alpha }_n\{u_n-g(u_n)+P_K[g(u_n)-\rho T(u_n)]\}\text{,}n=0\text{,}1\text{,}\dots$$ For the convergence analysis of Algorithm 4.3, see Noor (2008a).

We now consider the convergence analysis of Algorithm 4.1 and this is the main motivation of our next result.

Theorem 4.1

Let the operators $T\text{,}g:H\to H$ be both strongly monotone with constants $\alpha >0\text{,}\sigma >0$ and Lipschitz continuous with constants with $\beta >0\text{,}\delta >0$ , respectively. Let Assumption 2.1 hold and $\theta$ be as in the proof of Theorem 3.1. If (3.1) holds and $0⩽{\alpha }_n⩽1$ , for all $n⩾0$ and ${\sum }_{n=0}^{\infty }{\alpha }_n=\infty$ , then the approximate solution $u_n$ obtained from Algorithm 4.1 converges to a solution $u\in K(u)$ satisfying the general quasi-variational inequality (2.1).

Proof

From Theorem 3.1, it follows that there does exist a unique solution of the general quasi-variational inequality (2.1). Let $u\in K(u)$ be a solution of the general quasi-variational inequality (2.1). Then, from Lemma 3.1, we have

$$u=(1-{\alpha }_n)u+{\alpha }_n\{u-g(u)+P_{K(u)}[u-\rho T(u)]\}\text{,}$$ where $0⩽{\alpha }_n⩽1$ is a constant.

From (4.1) and (4.2), we have

$$\Vert u_{n+1}-u\Vert =\Vert (1-{\alpha }_n)(u_n-u)+{\alpha }_n\Vert u_n-u-(g(u_n)-g(u))\Vert {\alpha }_n\{P_{K(u)}[u_n-\rho T(u_n)]-P_{K(u)}[u-\rho T(u)]\}\Vert ⩽(1-{\alpha }_n)\Vert u_n-u\Vert +{\alpha }_n\Vert u_n-u-(g(u_n)-g(u))\Vert +{\alpha }_n\Vert P_{K(u_n)}[u_n-\rho T(u_n)]-P_{K(u_n)}[u-\rho T(u)]\Vert +{\alpha }_n\Vert P_{K(u_n)}[u-\rho T(u)]-P_{K(u)}[u-\rho T(u)]\Vert ⩽(1-{\alpha }_n)\Vert u_n-u\Vert +{\alpha }_n\Vert u_n-u-(g(u_n)-g(u))\Vert +{\alpha }_n\nu \Vert u_n-u\Vert +{\alpha }_n\Vert u_n-u-\rho (T(u_n)-T(u))\Vert \text{.}$$ From (3.6), (3.7) and (4.3), we have $$\Vert u_{n+1}-u\Vert ⩽(1-{\alpha }_n)\Vert u_n-u\Vert +{\alpha }_n\{\nu +\sqrt{1-2\sigma +{\delta }^2}+\sqrt{1-2\alpha \rho +{\beta }^2{\rho }^2}\}\Vert u_n-u\Vert =(1-{\alpha }_n)\Vert u_n-u\Vert +(k+t(\rho ))\Vert u_n-u\Vert \text{,}\text{using}\text{(3.3)}.=(1-{\alpha }_n)\Vert u_n-u\Vert +\theta \Vert u_n-u\Vert \text{,}$$ where

(4.4)

$$t(\rho )=\sqrt{1-2\alpha \rho +{\rho }^2{\beta }^2.}$$ From (3.2), it follows that $\theta <1$ .

Thus $$\Vert u_{n+1}-u\Vert ⩽(1-{\alpha }_n)\Vert u_n-u\Vert +{\alpha }_n\theta \Vert u_n-u\Vert =[1-(1-\theta ){\alpha }_n]\Vert u_n-u\Vert ⩽{\prod }_{i=0}^n[1-(1-\theta ){\alpha }_i]\Vert u_0-u\Vert \text{.}$$ Since ${\sum }_{n=0}^{\infty }{\alpha }_n$ diverges and $1-\theta >0$ , we have ${\lim }_{n\to \infty }{\prod }_{i=0}^n[1-(1-\theta ){\alpha }_i]=0$ . Consequently the sequence $\{u_n\}$ convergences strongly to u. This completes the proof. □

Lemma 5.1

The solution $u\in H:g(u)\in K(u)$ satisfies the general quasi-variational inequality (2.1) if and only if $z\in H\text{,}u\in K(u)$ is a solution of the general Wiener–Hopf equation (5.1), where

$$g(u)=P_{K(u)}z$$

(5.3)

$$z=u-\rho T(u)\text{,}\rho >0\text{,}\text{a constant.}$$

Proof

Let $u\in H:g(u)\in K(u)$ be a solution of (2.1). Then, from Lemma 3.1, we have

$$g(u)=P_{K(u)}[u-\rho T(u)]\text{.}$$ Let $$z=u-\rho T(u)\text{.}$$ Then

(5.5)

$$g(u)=P_{K(u)}z\text{.}$$ Combining (5.3)–(5.5), we have $$z=u-\rho T(u)=g^{-1}{P}_{K(u)}z-\rho T(g^{-1}{P}_{K(u)}z)\text{,}$$ from which it follows that $z\in H$ is a solution of the general Wiener–Hopf equation (5.1), the required result. □

Algorithm 5.1

For a given $z_0\in H$ , compute the approximate solution $z_{n+1}$ by the iterative schemes

$$g(u_n)=P_{K(u_n)}{z}_n\text{,}$$

(5.7)

$$z_{n+1}=(1-{\alpha }_n)z_n+{\alpha }_n\{u_n-\rho T(u_n)\}\text{,}n=0\text{,}1\text{,}\dots \text{,}$$ where $0⩽{\alpha }_n⩽1$ , for all $n⩾0$ and ${\sum }_{n=0}^{\infty }{\alpha }_n=\infty$ .

Algorithm 5.2

For a given $z_0\in H$ , compute the approximate solution $z_{n+1}$ by the iterative schemes $$\begin{array}{cc}\hfill & g(u_n)=P_{K(u_n)}{z}_n\hfill \\ \hfill & z_{n+1}=u_n-\rho T(u_n)+(1-{\rho }^{-1})Q_{K(u_n)}{z}_n\text{,}n=0\text{,}1\text{,}\dots \text{.}\hfill \end{array}$$ (III) If T is linear and $T^{-1}$ exists, then the Wiener–Hopf equations (5.1) can be written as: $$z=\left(I-{\rho }^{-1}{\mathit{gT}}^{-1}\right)Q_{K(u)}z\text{.}$$ This fixed-point formulation allows us to suggest the following iterative method for solving the general quasi-variational inequality (2.1).

Algorithm 5.3

For a given $z_0\in H$ , compute the approximate solution $z_{n+1}$ by the iterative schemes $$z_{n+1}=\left(I-{\rho }^{-1}{\mathit{gT}}^{-1}\right)Q_{K(u_n)}{z}_n\text{,}n=0\text{,}1\text{,}\dots \text{.}$$ If $K(u)\equiv K$ , then Algorithms 5.1, 5.2, 5.3 reduce to the following iterative methods for solving the general variational inequality (2.2), which are due to Noor (2008a).

Algorithm 5.4

For a given $z_0\in H$ , compute the approximate solution $z_{n+1}$ by the iterative schemes $$\begin{array}{cc}\hfill & g(u_n)=P_K{z}_n\text{,}\hfill \\ \hfill & z_{n+1}=(1-{\alpha }_n)z_n+{\alpha }_n\{u_n-\rho T(u_n)\}\text{,}n=0\text{,}1\text{,}\dots \text{,}\hfill \end{array}$$ where $0⩽{\alpha }_n⩽1$ , for all $n⩾0$ and ${\sum }_{n=0}^{\infty }{\alpha }_n=\infty$ .

Algorithm 5.5

For a given $z_0\in H$ , compute the approximate solution $z_{n+1}$ by the iterative schemes $$\begin{array}{cc}\hfill & g(u_n)=P_K{z}_n\text{,}\hfill \\ \hfill & z_{n+1}=u_n-\rho T(u_n)+(1-{\rho }^{-1})Q_K{z}_n\text{,}n=0\text{,}1\text{,}n=0\text{,}1\text{,}\dots \text{,}\hfill \end{array}$$ where $0⩽{\alpha }_n⩽1$ , for all $n⩾0$ and ${\sum }_{n=0}^{\infty }{\alpha }_n=\infty$ .

Algorithm 5.6

For a given $z_0\in H$ , compute the approximate solution $z_{n+1}$ by the iterative schemes $$z_{n+1}=\left(I-{\rho }^{-1}{\mathit{gT}}^{-1}\right)Q_K{z}_n\text{,}n=0\text{,}1\text{,}\dots \text{.}$$ For $g=I$ , the identity operator, Algorithms 5.1, 5.1, 5.3 are due to Noor (1997a,b, 1998) for solving the quasi-variational inequality(2.4). In brief, for appropriate and suitable rearrangements of the terms of the general Wiener–Hopf equations (5.1), one can suggest and analyze a number of iterative methods for solving the general quasi-variational inequality (2.1) and the related optimization problems. For the investigation of such type of projection iterative methods and the verification of their numerical efficiency, further research efforts are needed.

We now consider the convergence analysis of Algorithm 5.1. In a similar way, one can study the convergence analysis of Algorithms 5.2 and 5.3.

Theorem 5.1

Let the operators $T\text{,}g$ satisfy all the assumptions of Theorem 3.1. If the condition (3.2) holds, then the approximate solution $\{z_n\}$ obtained from Algorithm 5.1 converges to a solution $z\in H$ satisfying the Wiener–Hopf equation (5.1) strongly.

Proof

Let $u\in H:g(u)\in K(u)$ be a solution of (2.1). Then, using Lemma 5.1, we have

$$g(u)=P_{K(u)}z$$

(5.9)

$$z=(1-{\alpha }_n)z+{\alpha }_n\{u-\rho T(u)\}\text{,}$$ where $0⩽{\alpha }_n⩽1$ , and ${\sum }_{n=0}^{\infty }{\alpha }_n=\infty$ .

From (5.7), (5.9), (3.6) and (3.7), we have

$$\Vert z_{n+1}-z\Vert ⩽(1-{\alpha }_n)\Vert z_n-z\Vert +{\alpha }_n\Vert u_n-u-\rho (T(u_n)-T(u))\Vert ⩽(1-{\alpha }_n)\Vert z_n-z\Vert +{\alpha }_n\sqrt{1-2\rho \alpha +{\beta }^2}\Vert u_n-u\Vert \text{.}$$ Also, from (5.6), (5.8) and Assumption 2.1, we have $$\Vert u_n-u\Vert ⩽\Vert u_n-u-(g(u_n)-g(u))\Vert +\Vert P_{K(u_n)}{z}_n-P_{K(u)}z\Vert ⩽\Vert u_n-u-(g(u_n)-g(u))\Vert +\Vert P_{K(u_n)}{z}_n-P_{K(u_n)}z\Vert +\Vert P_{K(u_n)}z-P_{K(u)}z\Vert ⩽\{\nu +\sqrt{1-2\delta +{\sigma }^2}\}\Vert u_n-u\Vert +\Vert z_n-z\Vert =k\Vert u_n-u\Vert +\Vert z_n-z\Vert \text{,}$$ from which it follows that:

(5.11)

$$\Vert u_n-u\Vert ⩽\frac{1}{1-k}\Vert z_n-z\Vert \text{.}$$ Combining (5.10) and (5.11), we have

(5.12)

$$\Vert z_{n+1}-z\Vert ⩽(1-{\alpha }_n)\Vert z_n-z\Vert +{\alpha }_n{\theta }_1\Vert z_n-z\Vert \text{,}$$ where

(5.13)

$${\theta }_1=\left\{\frac{\sqrt{1-2\alpha \rho +{\beta }^2{\rho }^2}}{1-k}\right\}\text{.}$$ From (3.2), we see that ${\theta }_1<1$ and consequently $$\Vert z_{n+1}-z\Vert ⩽(1-{\alpha }_n)\Vert z_n-z\Vert +{\alpha }_n{\theta }_1\Vert z_n-z\Vert =[1-(1-{\theta }_1){\alpha }_n]\Vert z_n-z\Vert ⩽\prod _{i=0}^n[1-(1-{\theta }_1){\alpha }_i]\Vert z_0-z\Vert \text{.}$$ Since ${\sum }_{n=0}^{\infty }{\alpha }_n$ diverges and $1-{\theta }_1>0$ , we have ${\lim }_{n\to \infty }{\prod }_{i=0}^n[1-(1-{\theta }_1){\alpha }_i]=0$ . Consequently the sequence $\{z_n\}$ convergences strongly to z in H, the required result. □