Theorem 3.1

Let C and Q be nonempty closed convex subsets of real Hilbert spaces (H_{1}) and (H_{2}), respectively, and let (S:Crightarrow C) be a nonexpansive mapping. Let (D_{1},D_{2},D_{3}:Crightarrow H_{1}) be (d_{1},d_{2},d_{3})inverse strongly monotone, respectively, with (d^{*}=operatorname{min} {d_{1},d_{2},d_{3}} ). Let (bar{D_{1}},bar{D_{2}},bar{D_{3}}:Qrightarrow H_{2}) be (bar{d_{1}},bar{d_{2}},bar{d_{3}})inverse strongly monotone, respectively, with (hat{d}=operatorname{min} {bar{d_{1}},bar{d_{2}},bar{d_{3}}}). Let (A:H_{1}rightarrow H_{2}) be a bounded linear operator with adjoint (A^{*}) and (eta in (0,frac{1}{L})) with L being the spectral radius of the operator (A^{*}A). Define (M_{C}:H_{1}rightarrow C) by

$$begin{aligned} M_{C}(x)=P_{C}(I-zeta D_{1}) bigl(ax+(1-a)P_{C}(I-zeta D_{2}) bigl(ax+(1-a)P_{C}(I- zeta D_{3})x bigr) bigr), end{aligned}$$

(forall xin H_{1}), where (ain [0,1)), (zeta in (0,2d^{*})), and define (M_{Q}:H_{2}rightarrow Q) by

$$begin{aligned} M_{Q}(x)=P_{Q}(I-bar{zeta }bar{D_{1}}) bigl(ahat{x}+(1-a)P_{Q}(I- bar{zeta }bar{D_{2}}) bigl(a hat{x}+(1-a)P_{Q}(I-bar{zeta }bar{D_{3}}) hat{x} bigr) bigr), end{aligned}$$

(forall hat{x}in H_{1}), where (ain [0,1)), (bar{zeta }in (0,2hat{d})). Let the sequences ({x_{n}}) and ({y_{n}}) be generated by (x_{1}in H_{1}) and

$$begin{aligned} y_{n}=M_{C} W_{n} =P_{C}(I-zeta D_{1})T_{n}, end{aligned}$$

where (W_{n}=(I-eta A^{*}(I-M_{Q})A)x_{n}) and (T_{n}=aW_{n}+(1-a)P_{C}(I-zeta D_{2})(aW_{n}+(1-a)P_{C}(I-zeta D_{3})W_{n}))).

$$begin{aligned} &Q_{n}= bigl{ zin H: bigllangle (I-zeta D_{1})T_{n}-y_{n},y_{n}-z bigrrangle geq 0 bigr} ,\ &x_{n+1}=alpha _{n}T_{n}+(1-alpha _{n})SP_{Q_{n}} bigl(T_{n}-zeta D_{1}(y_{n}) bigr) end{aligned}$$

for all (nin mathbb{N}).

Assume that the following conditions hold:

  1. (i)

    (Im =F(S)bigcap bigcap_{i=1}^{3}Phi _{i}neq emptyset ), where (Phi _{i}={win VI(C,D_{i})|Awin VI(Q,bar{D}_{i})}) for all (i=1,2,3).

  2. (ii)

    (alpha _{n}in [c,d]subset (0,1)).

Then ({x_{n}}) converges weakly to (x_{0}=P_{Im }{x_{n}}), which ((x_{0},y_{0},z_{0})in Omega ^{D_{1},D_{2},D_{3}}_{bar{D_{1}}, bar{D_{2}},bar{D_{3}}}), (y_{0}=P_{C}(I-zeta D_{2})(ax_{0}+(1-a)z_{0})), and (z_{0}=P_{C}(I-zeta D_{3})x_{0}) with (bar{x_{0}}=Ax_{0}), (bar{y_{0}}=Ay_{0}) and (bar{z_{0}}=Az_{0}).

Proof

Denote (k_{n}:=P_{Q_{n}}(T_{n}-zeta D_{1}(y_{n}))) for all (ngeq 0). Let (x^{*}in Im ). From the definition of (P_{Q_{n}}), we have (y_{n}=P_{Q_{n}}(I-zeta D_{1})T_{n}). Let (M_{n}=T_{n}-zeta D_{1}(y_{n})). From (Csubseteq Q_{n}), and applying (6), we have

$$begin{aligned} biglVert k_{n}-x^{*} bigrVert ^{2}={}& biglVert P_{Q_{n}}M_{n}-x^{*} bigrVert ^{2} \ leq{}& biglVert M_{n}-x^{*} bigrVert ^{2}- Vert M_{n}-P_{Q_{n}}M_{n} Vert ^{2} \ ={}& biglVert T_{n}-zeta D_{1}(y_{n})-x^{*} bigrVert ^{2}- biglVert T_{n}-zeta D_{1}(y_{n})-P_{Q_{n}}M_{n} bigrVert ^{2} \ ={}& biglVert T_{n}-x^{*} bigrVert ^{2}-2 zeta bigllangle T_{n}-x^{*},D_{1}(y_{n}) bigrrangle + zeta ^{2} biglVert D_{1}(y_{n}) bigrVert ^{2} \ &{} – Vert T_{n}-P_{Q_{n}}M_{n} Vert ^{2}+2zeta bigllangle T_{n}-P_{Q_{n}}M_{n},D_{1}(y_{n}) bigrrangle -zeta ^{2} biglVert D_{1}(y_{n}) bigrVert ^{2} \ ={}& biglVert T_{n}-x^{*} bigrVert ^{2}- Vert T_{n}-P_{Q_{n}}M_{n} Vert ^{2}-2 zeta bigllangle P_{Q_{n}}M_{n}-x^{*},D_{1}(y_{n}) bigrrangle . end{aligned}$$

(8)

From the monotonicity of (D_{1}), we have

$$begin{aligned} 0&leq bigllangle D_{1}y_{n}-D_{1}x^{*},y_{n}-x^{*} bigrrangle \ &= bigllangle D_{1}y_{n},y_{n}-x^{*} bigrrangle – bigllangle D_{1}x^{*},y_{n}-x^{*} bigrrangle \ &leq bigllangle D_{1}y_{n},y_{n}-x^{*} bigrrangle \ &=langle D_{1}y_{n},y_{n}-P_{Q_{n}}M_{n} rangle – bigllangle D_{1}y_{n},x^{*}-P_{Q_{n}}M_{n} bigrrangle , end{aligned}$$

which implies that

$$begin{aligned} bigllangle D_{1}y_{n},x^{*}-P_{Q_{n}}M_{n} bigrrangle &leq langle D_{1}y_{n},y_{n}-P_{Q_{n}}M_{n} rangle. end{aligned}$$

(9)

From (8) and (9), we have

$$begin{aligned} biglVert k_{n}-x^{*} bigrVert ^{2}&leq biglVert T_{n}-x^{*} bigrVert ^{2}- Vert T_{n}-P_{Q_{n}}M_{n} Vert ^{2}+2 zeta langle D_{1}y_{n},y_{n}-P_{Q_{n}}M_{n} rangle. end{aligned}$$

(10)

From (10) and Lemma 2.7, we have

$$begin{aligned} biglVert k_{n}-x^{*} bigrVert ^{2}leq{}& biglVert x_{n}-x^{*} bigrVert ^{2}-eta (1-eta L) biglVert (I-M_{Q})Ax_{n} bigrVert ^{2}- Vert P_{Q_{n}}M_{n}-T_{n} Vert ^{2} \ & {}+2zeta langle D_{1}y_{n},y_{n}-P_{Q_{n}}M_{n} rangle \ ={}& biglVert x_{n}-x^{*} bigrVert ^{2}- eta (1-eta L) biglVert (I-M_{Q})Ax_{n} bigrVert ^{2}- Vert P_{Q_{n}}M_{n}-y_{n} Vert ^{2} \ & {}- Vert y_{n}-T_{n} Vert ^{2} -2 langle P_{Q_{n}}M_{n}-y_{n},y_{n}-T_{n} rangle \ & {}+2zeta langle D_{1}y_{n},y_{n}-P_{Q_{n}}M_{n} rangle \ ={}& biglVert x_{n}-x^{*} bigrVert ^{2}- eta (1-eta L) biglVert (I-M_{Q})Ax_{n} bigrVert ^{2}- Vert P_{Q_{n}}M_{n}-y_{n} Vert ^{2} \ & {}- Vert y_{n}-T_{n} Vert ^{2}+2 langle P_{Q_{n}}M_{n}-y_{n},T_{n}-y_{n}- zeta D_{1}y_{n}rangle \ ={}& biglVert x_{n}-x^{*} bigrVert ^{2}- eta (1-eta L) biglVert (I-M_{Q})Ax_{n} bigrVert ^{2}- Vert P_{Q_{n}}M_{n}-y_{n} Vert ^{2} \ & {}- Vert y_{n}-T_{n} Vert ^{2}+2 bigllangle (I-zeta D_{1})T_{n}-y_{n},P_{Q_{n}}M_{n}-y_{n} bigrrangle \ &{} +2langle zeta D_{1}T_{n}-zeta D_{1}y_{n},P_{Q_{n}}M_{n}-y_{n} rangle \ leq{}& biglVert x_{n}-x^{*} bigrVert ^{2}- eta (1-eta L) biglVert (I-M_{Q})Ax_{n} bigrVert ^{2}- Vert P_{Q_{n}}M_{n}-y_{n} Vert ^{2} \ & {}- Vert y_{n}-T_{n} Vert ^{2} +2 zeta Vert D_{1}T_{n}-D_{1}y_{n} Vert Vert P_{Q_{n}}M_{n}-y_{n} Vert \ leq{}& biglVert x_{n}-x^{*} bigrVert ^{2}- eta (1-eta L) biglVert (I-M_{Q})Ax_{n} bigrVert ^{2}- Vert P_{Q_{n}}M_{n}-y_{n} Vert ^{2} \ &{} – Vert y_{n}-T_{n} Vert ^{2}+ frac{zeta }{d_{1}} bigl[ Vert T_{n}-y_{n} Vert ^{2}+ Vert P_{Q_{n}}M_{n}-y_{n} Vert ^{2} bigr] \ ={}& biglVert x_{n}-x^{*} bigrVert ^{2}- eta (1-eta L) biglVert (I-M_{Q})Ax_{n} bigrVert ^{2} \ &{} – biggl(1-frac{zeta }{d_{1}} biggr) Vert P_{Q_{n}}M_{n}-y_{n} Vert ^{2} – biggl(1- frac{zeta }{d_{1}} biggr)) Vert T_{n}-y_{n} Vert ^{2}. end{aligned}$$

(11)

By the definition of (x_{n+1}), (11), and Lemma 2.7, we have

$$begin{aligned} biglVert x_{n+1}-x^{*} bigrVert ^{2}={}& biglVert alpha _{n} bigl(T_{n}-x^{*} bigr)+(1-alpha _{n}) bigl(Sk_{n}-x^{*} bigr) bigrVert ^{2} \ leq {}&alpha _{n} biglVert T_{n}-x^{*} bigrVert ^{2}+(1-alpha _{n}) biglVert Sk_{n}-x^{*} bigrVert ^{2} \ ={}& alpha _{n} biglVert T_{n}-x^{*} bigrVert ^{2}+(1-alpha _{n}) biglVert Sk_{n}-x^{*} bigrVert ^{2} \ &{} -alpha _{n}(1-alpha _{n}) Vert T_{n}-Sk_{n} Vert ^{2} end{aligned}$$

(12)

$$begin{aligned} ={}& alpha _{n} biglVert T_{n}-x^{*} bigrVert ^{2}+(1-alpha _{n}) biglVert k_{n}-x^{*} bigrVert ^{2} \ leq{}& alpha _{n} biglVert T_{n}-x^{*} bigrVert ^{2} +(1-alpha _{n}) biggl[ biglVert x_{n}-x^{*} bigrVert ^{2} \ &{} -eta (1-eta L) biglVert (I-M_{Q})Ax_{n} bigrVert ^{2} \ & {}- biggl(1-frac{zeta }{d_{1}} biggr) Vert P_{Q_{n}}M_{n}-y_{n} Vert ^{2}- biggl(1- frac{zeta }{d_{1}} biggr)) Vert T_{n}-y_{n} Vert ^{2} biggr] \ leq{}& alpha _{n} bigl[ biglVert x_{n}-x^{*} bigrVert ^{2}-alpha _{n}eta (1-eta L) biglVert (I-M_{Q})Ax_{n} bigrVert ^{2} bigr] \ & {}+(1-alpha _{n}) biggl[ biglVert x_{n}-x^{*} bigrVert ^{2} -eta (1-eta L) biglVert (I-M_{Q})Ax_{n} bigrVert ^{2} \ & {}- biggl(1-frac{zeta }{d_{1}} biggr) Vert P_{Q_{n}}M_{n}-y_{n} Vert ^{2} – biggl(1- frac{zeta }{d_{1}} biggr)) Vert T_{n}-y_{n} Vert ^{2} biggr] \ ={}& biglVert x_{n}-x^{*} bigrVert ^{2}- eta (1-eta L) (1+alpha _{n}) biglVert (I-M_{Q})Ax_{n} bigrVert ^{2} \ & {}-(1-alpha _{n}) biggl(1-frac{zeta }{d_{1}} biggr) bigl[ Vert T_{n}-y_{n} Vert ^{2}+ Vert y_{n}-k_{n} Vert ^{2} bigr]. end{aligned}$$

(13)

So,

$$begin{aligned} biglVert x_{n+1}-x^{*} bigrVert ^{2}&leq biglVert x_{n}-x^{*} bigrVert ^{2}. end{aligned}$$

Therefore (lim_{n rightarrow infty }|x_{n+1}-x^{*}|) exists, (forall x^{*}in Im ). So, we have ({x_{n}}^{infty }_{n=0}) and ({k_{n}}^{infty }_{n=0}) are bounded. From the last relations it follows that

$$begin{aligned} eta (1-eta L) (1+alpha _{n}) biglVert (I-M_{Q})Ax_{n} bigrVert ^{2}leq biglVert x_{n}-x^{*} bigrVert ^{2}- biglVert x_{n+1}-x^{*} bigrVert ^{2} end{aligned}$$

or

$$begin{aligned} biglVert (I-M_{Q})Ax_{n} bigrVert ^{2}& leq frac{ Vert x_{n}-x^{*} Vert ^{2}- Vert x_{n+1}-x^{*} Vert ^{2}}{eta (1-eta L)(1+alpha _{n})}. end{aligned}$$

Thus

$$begin{aligned} lim_{n rightarrow infty } biglVert (I-M_{Q})Ax_{n} bigrVert &=0. end{aligned}$$

(14)

By using the same method as above, we have

$$begin{aligned} lim_{n rightarrow infty } Vert T_{n}-y_{n} Vert &=0. end{aligned}$$

(15)

From (12), we get

$$begin{aligned} biglVert x_{n+1}-x^{*} bigrVert ^{2}leq{}& alpha _{n} biglVert T_{n}-x^{*} bigrVert ^{2}+(1-alpha _{n}) biglVert Sk_{n}-x^{*} bigrVert ^{2} \ & {}-alpha _{n}(1-alpha _{n}) Vert T_{n}-Sk_{n} Vert ^{2} \ leq{}& alpha _{n} biglVert T_{n}-x^{*} bigrVert ^{2}+(1-alpha _{n}) biglVert x_{n}-x^{*} bigrVert ^{2} \ &{} -alpha _{n}(1-alpha _{n}) Vert T_{n}-Sk_{n} Vert ^{2} \ leq{}& alpha _{n} biglVert x_{n}-x^{*} bigrVert ^{2}-alpha _{n}(1-alpha _{n}) Vert T_{n}-Sk_{n} Vert ^{2}, end{aligned}$$

so

$$begin{aligned} Vert T_{n}-Sk_{n} Vert ^{2}leq frac{ Vert x_{n}-x^{*} Vert ^{2}- Vert x_{n+1}-x^{*} Vert ^{2}}{alpha _{n}(1-alpha _{n})}, end{aligned}$$

which implies that

$$begin{aligned} lim_{n rightarrow infty } Vert T_{n}-Sk_{n} Vert &=0. end{aligned}$$

(16)

Consider

$$begin{aligned} W_{n}-x_{n}&=-eta A^{*}(I-M_{Q})Ax_{n}, end{aligned}$$

and by (14), we have

$$begin{aligned} lim_{n rightarrow infty } Vert W_{n}-x_{n} Vert &=0. end{aligned}$$

(17)

From the property of (P_{C}), we have

$$begin{aligned} & biglVert P_{C}(I-zeta D_{3})W_{n}-x^{*} bigrVert ^{2} \ &quad= biglVert P_{C}(I-zeta D_{3})W_{n}-P_{C}(I- zeta D_{3})x^{*} bigrVert ^{2} \ &quadleq biglVert (I-zeta D_{3})W_{n}-(I-zeta D_{3})x^{*} bigrVert ^{2} \ &quad= biglVert bigl(W_{n}-x^{*} bigr)-zeta bigl(D_{3}W_{n}-D_{3}x^{*} bigr) bigrVert ^{2} \ &quad= biglVert W_{n}-x^{*} bigrVert ^{2}-2 zeta bigllangle W_{n}-x^{*}, D_{3}W_{n}-D_{3}x^{*} bigrrangle \ &qquad{} +zeta ^{2} biglVert D_{3}W_{n}-D_{3}x^{*} bigrVert ^{2} \ &quadleq biglVert W_{n}-x^{*} bigrVert ^{2}-2zeta d_{3} biglVert D_{3}W_{n}-D_{3}x^{*} bigrVert ^{2} \ &qquad{} +zeta ^{2} biglVert D_{3}W_{n}-D_{3}x^{*} bigrVert ^{2} \ &quad= biglVert W_{n}-x^{*} bigrVert ^{2}- zeta (2d_{3}-zeta ) biglVert D_{3}W_{n}-D_{3}x^{*} bigrVert ^{2} \ &quadleq biglVert x_{n}-x^{*} bigrVert ^{2}- zeta (2d_{3}-zeta ) biglVert D_{3}W_{n}-D_{3}x^{*} bigrVert ^{2}. end{aligned}$$

(18)

By the definition of (T_{n}), (7), Remark 1, and (18), we have

$$begin{aligned} biglVert T_{n}-x^{*} bigrVert ^{2}leq{}& a Vert W_{n}-W_{x^{*}} Vert ^{2}+a(1-a) Vert W_{n}-W_{x^{*}} Vert ^{2} \ &{} +(1-a)^{2} biglVert P_{C}(I-zeta D_{3})W_{n}-x^{*} bigrVert ^{2} \ leq{}& a biglVert x_{n}-x^{*} bigrVert ^{2}+a(1-a) biglVert x_{n}-x^{*} bigrVert ^{2} \ & {}+(1-a)^{2} biglVert P_{C}(I-zeta D_{3})W_{n}-x^{*} bigrVert ^{2} \ leq{}& bigl(2a-a^{2} bigr) biglVert x_{n}-x^{*} bigrVert ^{2}+(1-a)^{2} biglVert P_{C}(I- zeta D_{3})W_{n}-x^{*} bigrVert ^{2} \ leq{}& bigl(2a-a^{2} bigr) biglVert x_{n}-x^{*} bigrVert ^{2}+(1-a)^{2} bigl[ biglVert x_{n}-x^{*} bigrVert ^{2} \ &{} -zeta (2d_{3}-zeta ) biglVert D_{3}W_{n}-D_{3}x^{*} bigrVert ^{2} bigr] \ ={}& biglVert x_{n}-x^{*} bigrVert ^{2}- zeta (2d_{3}-zeta ) (1-a)^{2} biglVert D_{3}W_{n}-D_{3}x^{*} bigrVert ^{2}. end{aligned}$$

(19)

In addition, by the definition of (x_{n+1}) and (19), we have

$$begin{aligned} biglVert x_{n+1}-x^{*} bigrVert ^{2}leq{}& alpha _{n} biglVert T_{n}-x^{*} bigrVert ^{2}+(1-alpha _{n}) biglVert k_{n}-x^{*} bigrVert ^{2} \ leq{}& alpha _{n} bigl[ biglVert x_{n}-x^{*} bigrVert ^{2}-zeta (2d_{3}-zeta ) (1-a)^{2} biglVert D_{3}W_{n}-D_{3}x^{*} bigrVert ^{2} bigr] \ & {}+(1-alpha _{n}) biglVert k_{n}-x^{*} bigrVert ^{2} \ ={}& alpha _{n} biglVert x_{n}-x^{*} bigrVert ^{2}-alpha _{n}zeta (2d_{3}-zeta ) (1-a)^{2} biglVert D_{3}W_{n}-D_{3}x^{*} bigrVert ^{2} \ & {}+(1-alpha _{n}) biglVert x_{n}-x^{*} bigrVert ^{2} \ ={}& biglVert x_{n}-x^{*} bigrVert ^{2}- alpha _{n}zeta (2d_{3}-zeta ) (1-a)^{2} biglVert D_{3}W_{n}-D_{3}x^{*} bigrVert ^{2}, end{aligned}$$

so

$$begin{aligned} biglVert D_{3}W_{n}-D_{3}x^{*} bigrVert ^{2}leq frac{ Vert x_{n}-x^{*} Vert ^{2}- Vert x_{n+1}-x^{*} Vert ^{2}}{alpha _{n}zeta (2d_{3}-zeta )(1-a)^{2}}, end{aligned}$$

which implies that

$$begin{aligned} lim_{n rightarrow infty } biglVert D_{3}W_{n}-D_{3}x^{*} bigrVert =0. end{aligned}$$

(20)

From the property of (P_{C}), we have

$$begin{aligned} & biglVert P_{C}(I-zeta D_{3})W_{n}-x^{*} bigrVert ^{2} \ &quadleq bigllangle (I-zeta D_{3})W_{n}-(I-zeta D_{3})x^{*}, P_{C}(I- zeta D_{3})W_{n}-x^{*} bigrrangle \ &quad=frac{1}{2} bigl[ biglVert (I-zeta D_{3})W_{n}-(I- zeta D_{3})x^{*} bigrVert ^{2} bigr]+ biglVert P_{C}(I- zeta D_{3})W_{n}-x^{*} bigrVert ^{2} \ & qquad{}- biglVert (I-zeta D_{3})W_{n}-(I-zeta D_{3})x^{*}- bigl(P_{C}(I-zeta D_{3})W_{n}-x^{*} bigr) bigrVert ^{2} ] \ &quadleq frac{1}{2} bigl[ biglVert W_{n}-x^{*} bigrVert ^{2}+ biglVert P_{C}(I-zeta D_{3})W_{n}-x^{*} bigrVert ^{2} \ & qquad{}- biglVert (I-zeta D_{3})W_{n}-(I-zeta D_{3})x^{*}- bigl(P_{C}(I-zeta D_{3})W_{n}-x^{*} bigr) bigrVert ^{2} bigr] \ &quad=frac{1}{2} bigl[ biglVert W_{n}-x^{*} bigrVert ^{2}+ biglVert P_{C}(I-zeta D_{3})W_{n}-x^{*} bigrVert ^{2} \ & qquad{}- biglVert bigl(W_{n}-P_{C}(I-zeta D_{3})W_{n} bigr)-zeta bigl(D_{3}W_{n}-D_{3}x^{*} bigr) bigrVert ^{2} bigr] \ &quad=frac{1}{2} bigl[ biglVert W_{n}-x^{*} bigrVert ^{2}+ biglVert P_{C}(I-zeta D_{3})W_{n}-x^{*} bigrVert ^{2} \ &qquad{} – biglVert W_{n}-P_{C}(I-zeta D_{3})W_{n} bigrVert ^{2} -zeta ^{2} biglVert D_{3}W_{n}-D_{3}x^{*} bigrVert ^{2} \ &qquad{} +2zeta bigllangle W_{n}-P_{C}(I-zeta D_{3})W_{n},D_{3}W_{n}-D_{3}x^{*} bigrrangle bigr], end{aligned}$$

so

$$begin{aligned} biglVert P_{C}(I-zeta D_{3})W_{n}-x^{*} bigrVert ^{2}leq{}& biglVert W_{n}-x^{*} bigrVert ^{2} – biglVert W_{n}-P_{C}(I- zeta D_{3})W_{n} bigrVert ^{2} \ &{} +2zeta biglVert W_{n}-P_{C}(I-zeta D_{3})W_{n} bigrVert biglVert D_{3}W_{n}-D_{3}x^{*} bigrVert . end{aligned}$$

(21)

By the definition of (T_{n}), (7), Remark 1, and (21), we have

$$begin{aligned} & biglVert T_{n}-x^{*} bigrVert ^{2} \ &quadleq a Vert W_{n}-W_{x^{*}} Vert ^{2}+a(1-a) Vert W_{n}-W_{x^{*}} Vert ^{2} \ &qquad{} +(1-a)^{2} biglVert P_{C}(I-zeta D_{3})W_{n}-x^{*} bigrVert ^{2} \ &quadleq a biglVert x_{n}-x^{*} bigrVert ^{2}+a(1-a) biglVert x_{n}-x^{*} bigrVert ^{2} \ &qquad{} +(1-a)^{2} biglVert P_{C}(I-zeta D_{3})W_{n}-x^{*} bigrVert ^{2} \ &quadleq bigl(2a-a^{2} bigr) biglVert x_{n}-x^{*} bigrVert ^{2}+(1-a)^{2} biglVert P_{C}(I- zeta D_{3})W_{n}-x^{*} bigrVert ^{2} \ &quadleq bigl(2a-a^{2} bigr) biglVert x_{n}-x^{*} bigrVert ^{2}+(1-a)^{2} bigl[ biglVert W_{n}-x^{*} bigrVert ^{2} – biglVert W_{n}-P_{C} \ &qquad{}times(I- zeta D_{3})W_{n} bigrVert ^{2}+2 zeta biglVert W_{n}-P_{C}(I-zeta D_{3})W_{n} bigrVert biglVert D_{3}W_{n}-D_{3}x^{*} bigrVert bigr] \ &quad= bigl(2a-a^{2} bigr) biglVert x_{n}-x^{*} bigrVert ^{2}+(1-a)^{2} biglVert x_{n}-x^{*} bigrVert ^{2} \ &qquad{} -(1-a)^{2} biglVert W_{n}-P_{C}(I- zeta D_{3})W_{n} bigrVert ^{2} \ & qquad{}+2zeta (1-a)^{2} biglVert W_{n}-P_{C}(I- zeta D_{3})W_{n} bigrVert biglVert D_{3}W_{n}-D_{3}x^{*} bigrVert \ &quad= biglVert x_{n}-x^{*} bigrVert ^{2}-(1-a)^{2} biglVert W_{n}-P_{C}(I- zeta D_{3})W_{n} bigrVert ^{2} \ &qquad{} +2zeta (1-a)^{2} biglVert W_{n}-P_{C}(I- zeta D_{3})W_{n} bigrVert biglVert D_{3}W_{n}-D_{3}x^{*} bigrVert . end{aligned}$$

(22)

In addition, by the definition of (x_{n+1}), (11), and (22), we have

$$begin{aligned} biglVert x_{n+1}-x^{*} bigrVert ^{2}leq{}& alpha _{n} biglVert T_{n}-x^{*} bigrVert ^{2}+(1-alpha _{n}) biglVert k_{n}-x^{*} bigrVert ^{2} \ leq{}& alpha _{n} bigl[ biglVert x_{n}-x^{*} bigrVert ^{2}-(1-a)^{2} biglVert W_{n}-P_{C}(I- zeta D_{3})W_{n} bigrVert ^{2} \ &{} +2zeta (1-a)^{2} biglVert W_{n}-P_{C}(I- zeta D_{3})W_{n} bigrVert biglVert D_{3}W_{n}-D_{3}x^{*} bigrVert bigr] \ &{} +(1-alpha _{n}) biglVert k_{n}-x^{*} bigrVert ^{2} \ leq{}& alpha _{n} biglVert x_{n}-x^{*} bigrVert ^{2}-alpha _{n}(1-a)^{2} biglVert W_{n}-P_{C}(I- zeta D_{3})W_{n} bigrVert ^{2} \ &{} +2alpha _{n}zeta (1-a)^{2} biglVert W_{n}-P_{C}(I-zeta D_{3})W_{n} bigrVert biglVert D_{3}W_{n}-D_{3}x^{*} bigrVert \ & {}+(1-alpha _{n}) biglVert x_{n}-x^{*} bigrVert ^{2} \ ={}& biglVert x_{n}-x^{*} bigrVert ^{2}- alpha _{n}(1-a)^{2} biglVert W_{n}-P_{C}(I- zeta D_{3})W_{n} bigrVert ^{2} \ &{} +2alpha _{n}zeta (1-a)^{2} biglVert W_{n}-P_{C}(I-zeta D_{3})W_{n} bigrVert biglVert D_{3}W_{n}-D_{3}x^{*} bigrVert . end{aligned}$$

(23)

From (20) and (23), we get

$$begin{aligned} lim_{n rightarrow infty } biglVert W_{n}-P_{C}(I- zeta D_{3})W_{n} bigrVert &=0. end{aligned}$$

(24)

Let (G_{n}=aW_{n}+(1-a)P_{C}(I-lambda _{3}D_{3})W_{n}). From the property of (P_{C}), we have

$$begin{aligned} & biglVert P_{C}(I-zeta D_{2})G_{n}-x^{*} bigrVert ^{2} \ &quad= biglVert P_{C}(I-zeta D_{2})G_{n}-P_{C}(I- zeta D_{2})x^{*} bigrVert ^{2} \ &quadleq biglVert (I-zeta D_{2})G_{n}-(I-zeta D_{2})x^{*} bigrVert ^{2} \ &quad= biglVert bigl(G_{n}-x^{*} bigr)-zeta bigl(D_{2}G_{n}-D_{2}x^{*} bigr) bigrVert ^{2} \ &quad= biglVert G_{n}-x^{*} bigrVert ^{2}-2 zeta bigllangle G_{n}-x^{*}, D_{2}G_{n}-D_{2}x^{*} bigrrangle \ &qquad{} +zeta ^{2} biglVert D_{2}G_{n}-D_{2}x^{*} bigrVert ^{2} \ &quadleq biglVert x_{n}-x^{*} bigrVert ^{2}-2zeta d_{2} biglVert D_{2}G_{n}-D_{2}x^{*} bigrVert ^{2} \ & qquad{}+zeta ^{2} biglVert D_{2}G_{n}-D_{2}x^{*} bigrVert ^{2} \ &quad= biglVert x_{n}-x^{*} bigrVert ^{2}- zeta (2d_{2}-zeta ) biglVert D_{2}G_{n}-D_{2}x^{*} bigrVert ^{2}. end{aligned}$$

(25)

By the definition of (T_{n}) and (25), we have

$$begin{aligned} biglVert T_{n}-x^{*} bigrVert ^{2}leq {}&a Vert W_{n}-W_{x^{*}} Vert ^{2}+(1-a) biglVert P_{C}(I- zeta D_{2})G_{n}-x^{*} bigrVert ^{2} \ leq{}& a biglVert x_{n}-x^{*} bigrVert ^{2}+(1-a) bigl[ biglVert x_{n}-x^{*} bigrVert ^{2} \ &{} -zeta (2d_{2}-zeta ) biglVert D_{2}G_{n}-D_{2}x^{*} bigrVert ^{2} bigr] \ ={}& biglVert x_{n}-x^{*} bigrVert ^{2}- zeta (1-a) (2d_{2}-zeta ) biglVert D_{2}G_{n}-D_{2}x^{*} bigrVert ^{2}. end{aligned}$$

(26)

In addition, by the definition of (x_{n+1}) and (26), we have

$$begin{aligned} biglVert x_{n+1}-x^{*} bigrVert ^{2}leq{}& alpha _{n} biglVert T_{n}-x^{*} bigrVert ^{2}+(1-alpha _{n}) biglVert k_{n}-x^{*} bigrVert ^{2} \ leq{}& alpha _{n} bigl[ biglVert x_{n}-x^{*} bigrVert ^{2}-zeta (1-a) (2d_{2}-zeta ) biglVert D_{2}G_{n}-D_{2}x^{*} bigrVert ^{2} bigr] \ & {}+(1-alpha _{n}) biglVert x_{n}-x^{*} bigrVert ^{2} \ ={}& biglVert x_{n}-x^{*} bigrVert ^{2}- zeta alpha _{n}(1-alpha _{n}) (2d_{2}-zeta ) biglVert D_{2}G_{n}-D_{2}x^{*} bigrVert ^{2}, end{aligned}$$

so

$$begin{aligned} biglVert D_{2}G_{n}-D_{2}x^{*} bigrVert ^{2}&leq frac{ Vert x_{n}-x^{*} Vert ^{2}- Vert x_{n+1}-x^{*} Vert ^{2}}{zeta alpha _{n}(1-alpha _{n})(2d_{2}-zeta )}. end{aligned}$$

It implies that

$$begin{aligned} lim_{n rightarrow infty } biglVert D_{2}G_{n}-D_{2}x^{*} bigrVert =0. end{aligned}$$

(27)

From the property of (P_{C}), we have

$$begin{aligned} & biglVert P_{C}(I-zeta D_{2})G_{n}-x^{*} bigrVert ^{2} \ &quad= bigllangle (I-zeta D_{2})G_{n}-(I-zeta D_{2})x^{*}, P_{C}(I-zeta D_{2})G_{n}-x^{*} bigrrangle \ &quad=frac{1}{2} bigl[ biglVert (I-zeta D_{2})G_{n}-(I- zeta D_{2})x^{*} bigrVert ^{2} + biglVert P_{C}(I- zeta D_{2})G_{n}-x^{*} bigrVert ^{2} \ & qquad{}- biglVert (I-zeta D_{2})G_{n}-(I-zeta D_{2})x^{*}- bigl((I-zeta D_{2})G_{n}-x^{*} bigr) bigrVert ^{2} bigr] \ &quadleq frac{1}{2} bigl[ biglVert G_{n}-x^{*} bigrVert ^{2}+ biglVert P_{C}(I-zeta D_{2})G_{n}-x^{*} bigrVert ^{2} \ &qquad{} – biglVert (I-zeta D_{2})G_{n}-(I-zeta D_{2})x^{*}- bigl((I-zeta D_{2})G_{n}-x^{*} bigr) bigrVert ^{2} bigr] \ &quad=frac{1}{2} bigl[ biglVert G_{n}-x^{*} bigrVert ^{2}+ biglVert P_{C}(I-zeta D_{2})G_{n}-x^{*} bigrVert ^{2} \ &qquad{} – biglVert bigl(G_{n}-P_{C}(I-zeta D_{2})G_{n} bigr)-zeta bigl(D_{2}G_{n}-D_{2}x^{*} bigr) bigrVert ^{2} bigr] \ &quad=frac{1}{2} bigl[ biglVert G_{n}-x^{*} bigrVert ^{2}+ biglVert P_{C}(I-zeta D_{2})G_{n}-x^{*} bigrVert ^{2} \ &qquad{} – biglVert G_{n}-P_{C}(I-zeta D_{2})G_{n} bigrVert ^{2} \ & qquad{}+2zeta bigllangle G_{n}-P_{C}(I-zeta D_{2})G_{n}, D_{2}G_{n}-D_{2}x^{*} bigrrangle \ & qquad{}-zeta ^{2} biglVert D_{2}G_{n}-D_{2}x^{*} bigrVert ^{2} bigr]. end{aligned}$$

It implies that

$$begin{aligned} & biglVert P_{C}(I-zeta D_{2})G_{n}-x^{*} bigrVert ^{2} \ &quadleq biglVert G_{n}-x^{*} bigrVert ^{2}- biglVert G_{n}-P_{C}(I- zeta D_{2})G_{n} bigrVert ^{2} \ &qquad{} +2zeta bigllangle G_{n}-P_{C}(I-zeta D_{2})G_{n}, D_{2}G_{n}-D_{2}x^{*} bigrrangle \ &quadleq biglVert G_{n}-x^{*} bigrVert ^{2}- biglVert G_{n}-P_{C}(I-zeta D_{2})G_{n} bigrVert ^{2} \ &qquad{} +2zeta biglVert G_{n}-P_{C}(I-zeta D_{2})G_{n} bigrVert biglVert D_{2}G_{n}-D_{2}x^{*} bigrVert . end{aligned}$$

(28)

By the definition of (T_{n}) and (28), we have

$$begin{aligned} biglVert T_{n}-x^{*} bigrVert ^{2}leq {}&a Vert W_{n}-W_{x^{*}} Vert ^{2}+(1-a) biglVert P_{C}(I- zeta D_{2})G_{n}-x^{*} bigrVert ^{2} \ leq{}& a biglVert x_{n}-x^{*} bigrVert ^{2}+(1-a) bigl[ biglVert G_{n}-x^{*} bigrVert ^{2}- biglVert G_{n}-P_{C} \ &{}times (I- zeta D_{2})G_{n} bigrVert ^{2} +2 zeta biglVert G_{n}-P_{C}(I-zeta D_{2})G_{n} bigrVert biglVert D_{2}G_{n}-D_{2}x^{*} bigrVert bigr] \ leq {}&a biglVert x_{n}-x^{*} bigrVert ^{2}+(1-a) biglVert x_{n}-x^{*} bigrVert ^{2} \ &{} -(1-a) biglVert G_{n}-P_{C}(I-zeta D_{2})G_{n} bigrVert ^{2} \ & {}+2zeta biglVert G_{n}-P_{C}(I-zeta D_{2})G_{n} bigrVert biglVert D_{2}G_{n}-D_{2}x^{*} bigrVert ] \ ={}& biglVert x_{n}-x^{*} bigrVert ^{2}-(1-a) biglVert G_{n}-P_{C}(I-zeta D_{2})G_{n} bigrVert ^{2} \ &{} +2zeta biglVert G_{n}-P_{C}(I-zeta D_{2})G_{n} bigrVert biglVert D_{2}G_{n}-D_{2}x^{*} bigrVert ]. end{aligned}$$

(29)

In addition, by the definition of (x_{n+1}) and (29), we have

$$begin{aligned} biglVert x_{n+1}-x^{*} bigrVert ^{2}leq {}&alpha _{n} biglVert T_{n}-x^{*} bigrVert ^{2}+(1-alpha _{n}) biglVert k_{n}-x^{*} bigrVert ^{2} \ leq{}& alpha _{n} bigl[ biglVert x_{n}-x^{*} bigrVert ^{2}-(1-a) biglVert G_{n}-P_{C}(I- zeta D_{2})G_{n} bigrVert ^{2} \ &{} +2zeta biglVert G_{n}-P_{C}(I-zeta D_{2})G_{n} bigrVert biglVert D_{2}G_{n}-D_{2}x^{*} bigrVert bigr] \ &{} +(1-alpha _{n}) biglVert x_{n}-x^{*} bigrVert ^{2} \ ={}& biglVert x_{n}-x^{*} bigrVert ^{2}- alpha _{n}(1-a) biglVert G_{n}-P_{C}(I-zeta D_{2})G_{n} bigrVert ^{2} \ &{} +2zeta alpha _{n}(1-a) biglVert G_{n}-P_{C}(I- zeta D_{2})G_{n} bigrVert biglVert D_{2}G_{n}-D_{2}x^{*} bigrVert , end{aligned}$$

(30)

by (30) and (27), we get

$$begin{aligned} lim_{n rightarrow infty } biglVert G_{n}-P_{C}(I- zeta D_{2})G_{n} bigrVert &=0. end{aligned}$$

(31)

Since

$$begin{aligned} T_{n}-W_{n}=(1-a) bigl(P_{C}(I-zeta D_{2}) bigl(aW_{n}+(1-a)P_{C}(I-zeta D_{3})W_{n} bigr)-W_{n} bigr). end{aligned}$$

From the property of norm, we have

$$begin{aligned} & biglVert P_{C}(I-zeta D_{2}) bigl(aW_{n}+(1-a)P_{C}(I- zeta D_{3})W_{n} bigr)-W_{n} bigrVert \ &quadleq biglVert P_{C}(I-zeta D_{2}) bigl(aW_{n}+(1-a)P_{C}(I-zeta D_{3})W_{n} bigr) \ &qquad{} – bigl(aW_{n}+(1-a)P_{C}(I-zeta D_{3})W_{n} bigr) bigrVert \ &qquad{} +bigr| (aW_{n}+(1-a)P_{C}(I-zeta D_{3})W_{n}-W_{n} bigr| \ &quad= biglVert P_{C}(I-zeta D_{2})G_{n}-G_{n} bigrVert +(1-a) biglVert P_{C}(I-zeta D_{3})W_{n}-W_{n} bigrVert . end{aligned}$$

(32)

Then we have

$$begin{aligned} Vert T_{n}-W_{n} Vert leq {}&(1-a) bigl[ biglVert P_{C}(I-zeta D_{2})G_{n}-G_{n} bigrVert \ &{} +(1-a) biglVert P_{C}(I-zeta D_{3})W_{n}-W_{n} bigrVert bigr]. end{aligned}$$

From (24) and (31), it implies that

$$begin{aligned} lim_{n rightarrow infty } Vert T_{n}-W_{n} Vert &=0. end{aligned}$$

(33)

From (15), (17), (33), and

$$begin{aligned} Vert y_{n}-x_{n} Vert &leq Vert y_{n}-T_{n} Vert + Vert T_{n}-W_{n} Vert + Vert W_{n}-x_{n} Vert , end{aligned}$$

we have

$$begin{aligned} lim_{n rightarrow infty } Vert y_{n}-x_{n} Vert =0. end{aligned}$$

(34)

Moreover, from (16), (15), (34), and

$$begin{aligned} Vert x_{n}-Sk_{n} Vert &leq Vert x_{n}-y_{n} Vert + Vert y_{n}-T_{n} Vert + Vert T_{n}-Sk_{n} Vert , end{aligned}$$

we have

$$begin{aligned} lim_{n rightarrow infty } Vert x_{n}-Sk_{n} Vert &=0. end{aligned}$$

(35)

Since ({x_{n}}^{infty }_{n=0}) is bounded, it has a subsequence ({x_{n_{k}}}^{infty }_{k=0}) which weakly converges to some (bar{x}in C).

Assume (bar{x} notin F(S)). By the nonexpansiveness of S and Opial’s property and (35), we have

$$begin{aligned} lim_{k rightarrow infty }inf Vert x_{n_{k}}-bar{x} Vert &< lim _{k rightarrow infty }inf Vert x_{n_{k}}-Sbar{x} Vert \ &leq lim_{k rightarrow infty }inf bigl[ Vert x_{n_{k}}-Sk_{n_{k}} Vert + Vert Sk_{n_{k}}-Sbar{x} Vert bigr] \ &leq lim_{k rightarrow infty }inf bigl[ Vert x_{n_{k}}-Sk_{n_{k}} Vert + Vert k_{n_{k}}-bar{x} Vert bigr] \ &=lim_{k rightarrow infty }inf Vert k_{n_{k}}-bar{x} Vert \ &leq lim_{k rightarrow infty }inf Vert x_{n_{k}}-bar{x} Vert . end{aligned}$$

This is a contradiction, then we have

$$begin{aligned} bar{x}in F(S). end{aligned}$$

Assume (bar{x}notin bigcap_{i=1}^{3}Phi _{i}). From Lemma 2.6, we have (bar{x}notin F(M_{C}(I-eta A^{*}(I-M_{Q})A))). By Opial’s condition, (34), and Remark 1, we have

$$begin{aligned} lim_{k rightarrow infty }inf Vert x_{n_{k}}-bar{x} Vert < {}& lim_{k rightarrow infty }inf biglVert x_{n_{k}}-M_{C} bigl(I-eta A^{*}(I-M_{Q})A bigr) bar{x} bigrVert \ leq{}& lim_{k rightarrow infty }inf Vert x_{n_{k}}-y_{n_{k}} Vert +lim_{k rightarrow infty }inf biglVert M_{C} bigl(x_{n_{k}}-eta A^{*} \ &{}times (I- M_{Q})Ax_{n_{k}} bigr)-M_{C} bigl(I-eta A^{*}(I-M_{Q})A bigr)bar{x} bigrVert \ leq{}& lim_{k rightarrow infty }inf bigl( Vert x_{n_{k}}-y_{n_{k}} Vert + Vert x_{n_{k}}- bar{x} Vert bigr) \ ={}&lim_{k rightarrow infty }inf Vert x_{n_{k}}-bar{x} Vert . end{aligned}$$

(36)

This is a contradiction, then we have

$$begin{aligned} bar{x}in F bigl(M_{C} bigl(I-eta A^{*}(I-M_{Q})A bigr) bigr). end{aligned}$$

It implies that

$$begin{aligned} bar{x}in bigcap_{i=1}^{3}Phi _{i}. end{aligned}$$

Hence

$$begin{aligned} bar{x}in Im. end{aligned}$$

In order to show that the entire sequence ({x_{n}}) weakly converges to , assume ({x}_{n_{k}}rightharpoonup hat{x}) as (k rightarrow infty ), with (bar{x}neq hat{x}) and (hat{x}in Im ). By Opial’s condition, we have

$$begin{aligned} lim_{n rightarrow infty } Vert x_{n}-bar{x} Vert &=lim _{k rightarrow infty }inf Vert x_{n_{k}}-bar{x} Vert \ &< lim_{k rightarrow infty }inf Vert x_{n_{k}}-hat{x} Vert \ &=lim_{n rightarrow infty } Vert x_{n}-hat{x} Vert \ &=lim_{n rightarrow infty }inf Vert x_{n_{k}}-hat{x} Vert \ &< lim_{n rightarrow infty }inf Vert x_{n_{k}}-bar{x} Vert \ &=lim_{n rightarrow infty } Vert x_{n}-bar{x} Vert . end{aligned}$$

This is a contradiction, thus

$$begin{aligned} bar{x}doteq hat{x}. end{aligned}$$

It implies that the sequence ({x_{n}}^{infty }_{n=0}) weakly converges to (bar{x}in Im ).

From (34), we have ({y_{n}}^{infty }_{n=0}) weakly converges to (bar{x}in Im ).

Finally, if we take

$$begin{aligned} U_{n}&=P_{Im }x_{n}, end{aligned}$$

(37)

by Lemma 2.2, we see that ({P_{Im }x_{n}}^{infty }_{n=0}) converges strongly to some (zin Im ). From (37), we get

$$begin{aligned} langle bar{x}-U_{n},U_{n}-x_{n}rangle geq 0, quadforall bar{x} in Im. end{aligned}$$

Take (nrightarrow infty ), we also have

$$begin{aligned} langle bar{x}-z,z-bar{x}rangle geq 0, end{aligned}$$

and hence (bar{x}=z). Therefore (U_{n}) converges strongly to (bar{x}in Im ), this completes the proof. □

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