For given real Banach spaces (X_{1}) and (X_{2}), and the mappings (F:X_{1}times X_{2}rightarrow X_{1}), (G:X_{1}times X_{2}rightarrow X_{2}), (widehat{H}_{1}:X_{1}rightarrow X_{1}), (widehat{H}_{2}:X_{2}rightarrow X_{2}), (M:X_{1}rightrightarrows X_{1}), and (N:X_{2}rightrightarrows X_{2}), we consider the problem of finding ((a,b)in X_{1}times X_{2}) such that

$$begin{aligned} begin{aligned} textstylebegin{cases} 0in F(a,b)+M(a), \ 0in G(a,b)+N(b), end{cases}displaystyle end{aligned} end{aligned}$$

(3.1)

which is called a system of variational inclusions ((operatorname{SVI})) involving Ĥ-accretive mappings.

It is important to emphasize that by taking different choices of the operators F, G, (widehat{H}_{i}), M, N and the underlying spaces (X_{i}) ((i=1,2)) in the SVI (3.1), one can easily obtain the problems studied in [1214, 22, 29, 58] and the references therein.

The following conclusion, which tells us that SVI (3.1) is equivalent to a fixed-point problem, provides us with a characterization of the solution of the SVI (3.1).

Lemma 3.1

Let (X_{1}) and (X_{2}) be two real smooth Banach spaces, and (widehat{H}_{1}:X_{1}rightarrow X_{1}) and (widehat{H}_{2}:X_{2}rightarrow X_{2}) be strictly accretive mappings. Suppose further that (M:X_{1}rightrightarrows X_{1}) is an (widehat{H}_{1})accretive operator and (N:X_{2}rightrightarrows X_{2}) is an (widehat{H}_{2})accretive operator. Then, the following statements are equivalent:

  1. (i)

    ((a,b)in X_{1}times X_{2}) is a solution of the SVI (3.1);

  2. (ii)

    For any (lambda ,rho >0), ((a,b)) satisfies

    $$begin{aligned} begin{aligned} textstylebegin{cases} a=R^{widehat{H}_{1}}_{M,lambda}[widehat{H}_{1}(a)-lambda F(a,b)], \ b=R^{widehat{H}_{2}}_{N,rho}[widehat{H}_{2}(b)-rho G(a,b)]; end{cases}displaystyle end{aligned} end{aligned}$$

  3. (iii)

    For some (lambda _{0},rho _{0}>0), ((a,b)) satisfies

    $$begin{aligned} begin{aligned} textstylebegin{cases} a=R^{widehat{H}_{1}}_{M,lambda _{0}}[widehat{H}_{1}(a)-lambda _{0} F(a,b)], \ b=R^{widehat{H}_{2}}_{N,rho _{0}}[widehat{H}_{2}(b)-rho _{0} G(a,b)]. end{cases}displaystyle end{aligned} end{aligned}$$

Proof

“(i) (ii)” Let us first assume that ((a,b)in X_{1}times X_{2}) is a solution of the SVI (3.1). Then, using Definition 2.11, it yields

$$begin{aligned}& begin{aligned} textstylebegin{cases} 0in F(a,b)+M(a), \ 0in G(a,b)+N(b), end{cases}displaystyle end{aligned} \& Rightarrow \& begin{aligned} textstylebegin{cases} widehat{H}_{1}(a)-lambda F(a,b)in widehat{H}_{1}(a)+lambda M(a)=( widehat{H}_{1}+lambda M)(a), \ widehat{H}_{2}(b)-rho G(a,b)in widehat{H}_{2}(b)+rho N(b)=( widehat{H}_{2}+rho N)(b), end{cases}displaystyle end{aligned} \& Rightarrow \& begin{aligned} textstylebegin{cases} a=(widehat{H}_{1}+lambda M)^{-1}(widehat{H}_{1}(a)-lambda F(a,b)), \ b=(widehat{H}_{2}+rho N)^{-1}(widehat{H}_{2}(b)-rho G(a,b)), end{cases}displaystyle end{aligned} \& Rightarrow \& begin{aligned} textstylebegin{cases} a=R^{widehat{H}_{1}}_{M,lambda}[widehat{H}_{1}(a)-lambda F(a,b)], \ b=R^{widehat{H}_{2}}_{N,rho}[widehat{H}_{2}(b)-rho G(a,b)], end{cases}displaystyle end{aligned} end{aligned}$$

where (R^{widehat{H}_{1}}_{M,lambda}=(widehat{H}_{1}+lambda M)^{-1}) and (R^{widehat{H}_{2}}_{N,rho}=(widehat{H}_{2}+rho N)^{-1}).

The proof of “(ii) (iii)” is obvious.

“(iii) (i)” Suppose that for some (lambda _{0},rho _{0}>0), ((a,b)) satisfies

$$begin{aligned} begin{aligned} textstylebegin{cases} a=R^{widehat{H}_{1}}_{M,lambda _{0}}[widehat{H}_{1}(a)-lambda _{0} F(a,b)], \ b=R^{widehat{H}_{2}}_{N,rho _{0}}[widehat{H}_{2}(b)-rho _{0} G(a,b)]. end{cases}displaystyle end{aligned} end{aligned}$$

Then, in the light of Definition 2.11, we obtain

$$begin{aligned} begin{aligned} textstylebegin{cases} a=(widehat{H}_{1}+lambda _{0}M)^{-1}(widehat{H}_{1}(a)-lambda _{0}F(a,b)], \ b=(widehat{H}_{2}+rho _{0}N)^{-1}(widehat{H}_{2}(b)-rho _{0}G(a,b)], end{cases}displaystyle end{aligned} end{aligned}$$

which implies that

$$begin{aligned} begin{aligned} textstylebegin{cases} widehat{H}_{1}(a)-lambda _{0}F(a,b)in widehat{H}_{1}(a)+lambda _{0}M(a), \ widehat{H}_{2}(b)-rho _{0}G(a,b)in widehat{H}_{2}(b)+rho _{0}N(b), end{cases}displaystyle end{aligned} end{aligned}$$

and hence,

$$begin{aligned} begin{aligned} textstylebegin{cases} 0in F(a,b)+M(a), \ 0in G(a,b)+N(b), end{cases}displaystyle end{aligned} end{aligned}$$

i.e., ((a,b)in X_{1}times X_{2}) is a solution of the SVI (3.1). The proof is completed. □

Before proceeding to the main result of this section, we need to recall the following notion that will be used efficiently in its proof.

Definition 3.2

A mapping (F:Xtimes Xrightarrow X) is said to be

  1. (i)

    ς-Lipschitz continuous with respect to its first argument if there exists a constant (varsigma >0) such that

    $$begin{aligned} biglVert F(x_{1},y)-F(x_{2},y) bigrVert leq varsigma Vert x_{1}-x_{2} Vert ,quad forall x_{1},x_{2},yin X; end{aligned}$$

  2. (ii)

    ξ-Lipschitz continuous with respect to its second argument if there exists a constant (xi >0) such that

    $$begin{aligned} biglVert F(x,y_{1})-F(x,y_{2}) bigrVert leq xi Vert y_{1}-y_{2} Vert , quad forall x,y_{1},y_{2} in X. end{aligned}$$

Theorem 3.3

Let (X_{1}) and (X_{2}) be two real smooth Banach spaces with norms (Vert cdotVert _{1}) and (Vert cdotVert _{2}), respectively, (widehat{H}_{1}:X_{1}rightarrow X_{1}) be a (varrho _{1})strongly accretive and rLipschitz continuous mapping, (widehat{H}_{2}:X_{2}rightarrow X_{2}) be a (varrho _{2})strongly accretive and kLipschitz continuous mapping, (M:X_{1}rightrightarrows X_{1}) be an (widehat{H}_{1})accretive setvalued mapping, and (N:X_{2}rightrightarrows X_{2}) be an (widehat{H}_{2})accretive setvalued mapping. Suppose further that the mapping (F:X_{1}times X_{2}rightarrow X_{1}) is (tau _{1})Lipschitz continuous with respect to its first argument and (tau _{2})Lipschitz continuous with respect to its second argument, and the mapping (G:X_{1}times X_{2}rightarrow X_{2}) is (theta _{1})Lipschitz continuous with respect to its first argument and (theta _{2})Lipschitz continuous with respect to its second argument. If (r<varrho _{1}) and (k<varrho _{2}) then, the SVI (3.1) admits a unique solution.

Proof

For any given (lambda ,rho >0), define (T_{lambda}:X_{1}times X_{2}rightarrow X_{1}) and (S_{rho}:X_{1}times X_{2}rightarrow X_{2}) for all ((x,y)in X_{1}times X_{2}), by

$$begin{aligned} T_{lambda}(x,y)=R^{widehat{H}_{1}}_{M,lambda}bigl[ widehat{H}_{1}(x)- lambda F(x,y)bigr] end{aligned}$$

(3.2)

and

$$begin{aligned} S_{rho}(x,y)=R^{widehat{H}_{2}}_{N,rho}bigl[ widehat{H}_{2}(y)-rho G(x,y)bigr], end{aligned}$$

(3.3)

respectively. At the same time, for any given (lambda ,rho >0), define (Q_{lambda ,rho}:X_{1}times X_{2}rightarrow X_{1}times X_{2}) by

$$begin{aligned} Q_{lambda ,rho}(x,y)=bigl(T_{lambda}(x,y),S_{rho}(x,y) bigr),quad forall (x,y)in X_{1}times X_{2}. end{aligned}$$

(3.4)

Making use of (3.2) and Lemma 2.12, it follows that for all ((x_{1},y_{1}),(x_{2},y_{2})in X_{1}times X_{2}),

$$begin{aligned} begin{aligned} & biglVert T_{lambda}(x_{1},y_{1})-T_{lambda}(x_{2},y_{2}) bigrVert _{1} \ &quad = biglVert R^{widehat{H}_{1}}_{M,lambda}bigl[widehat{H}_{1}(x_{1})- lambda F(x_{1},y_{1})bigr] -R^{widehat{H}_{1}}_{M,lambda} bigl[widehat{H}_{1}(x_{2})- lambda F(x_{2},y_{2}) bigr] bigrVert _{1} \ & quad leq frac{1}{varrho _{1}} biglVert widehat{H}_{1}(x_{1})- widehat{H}_{1}(x_{2})- lambda bigl(F(x_{1},y_{1})-F(x_{2},y_{2}) bigr) bigrVert _{1} \ & quad leq frac{1}{varrho _{1}} bigl( biglVert widehat{H}_{1}(x_{1})- widehat{H}_{1}(x_{2}) bigrVert _{1}+lambda biglVert F(x_{1},y_{1})-F(x_{2},y_{2}) bigrVert _{1} bigr). end{aligned} end{aligned}$$

(3.5)

Taking into account that (widehat{H}_{1}) is r-Lipschitz continuous, and F is (tau _{1})-Lipschitz continuous with respect to its first argument and (tau _{2})-Lipschitz continuous with respect to its second argument, we obtain

$$begin{aligned} biglVert widehat{H}_{1}(x_{1})- widehat{H}_{1}(x_{2}) bigrVert _{1}leq r Vert x_{1}-x_{2} Vert _{1} end{aligned}$$

(3.6)

and

$$begin{aligned} begin{aligned} biglVert F(x_{1},y_{1})-F(x_{2},y_{2}) bigrVert _{1}leq{}& biglVert F(x_{1},y_{1})-F(x_{2},y_{1}) bigrVert _{1} \ & {}+ biglVert F(x_{2},y_{1})-F(x_{2},y_{2}) bigrVert _{1} \ leq{}& tau _{1} Vert x_{1}-x_{2} Vert _{1}+tau _{2} Vert y_{1}-y_{2} Vert _{2}. end{aligned} end{aligned}$$

(3.7)

Combining (3.5)–(3.7), we deduce that for all ((x_{1},y_{1}),(x_{2},y_{2})in X_{1}times X_{2}),

$$begin{aligned} biglVert T_{lambda}(x_{1},y_{1})-T_{lambda}(x_{2},y_{2}) bigrVert _{1} leq frac{r+lambda tau _{1}}{varrho _{1}} Vert x_{1}-x_{2} Vert _{1}+ frac{lambda tau _{2}}{varrho _{1}} Vert y_{1}-y_{2} Vert _{2}. end{aligned}$$

(3.8)

By arguments analogous to the previous inequalities (3.5)–(3.8), employing the assumptions, for all ((x_{1},y_{1}),(x_{2},y_{2})in X_{1}times X_{2}), we obtain

$$begin{aligned} biglVert S_{rho}(x_{1},y_{1})-S_{rho}(x_{2},y_{2}) bigrVert _{2}leq frac{k+rho theta _{2}}{varrho _{2}} Vert y_{1}-y_{2} Vert _{2}+ frac{rho theta _{1}}{varrho _{2}} Vert x_{1}-x_{2} Vert _{1}. end{aligned}$$

(3.9)

Define the function (Vert cdotVert _{*}) on (X_{1}times X_{2}) by

$$begin{aligned} biglVert (x_{1},x_{2}) bigrVert _{*}= Vert x_{1} Vert _{1}+ Vert x_{2} Vert _{2}, quad forall (x_{1},x_{2}) in X_{1}times X_{2}. end{aligned}$$

(3.10)

It can be easily seen that ((X_{1}times X_{2},Vert cdotVert _{*})) is a Banach space. Then, applying (3.4), (3.8), and (3.9), yields

$$begin{aligned} begin{aligned} & biglVert Q_{lambda ,rho}(x_{1},y_{1})-Q_{lambda ,rho}(x_{2},y_{2}) bigrVert _{*} \ &quad = biglVert bigl(T_{lambda}(x_{1},y_{1}),S_{rho}(x_{1},y_{1}) bigr) -bigl(T_{lambda}(x_{2},y_{2}),S_{ rho}(x_{2},y_{2}) bigr) bigrVert _{*} \ &quad = biglVert bigl(T_{lambda}(x_{1},y_{1})-T_{lambda}(x_{2},y_{2}),S_{rho}(x_{1},y_{1})-S_{ rho}(x_{2},y_{2}) bigr) bigrVert _{*} \ & quad = biglVert T_{lambda}(x_{1},y_{1})-T_{lambda}(x_{2},y_{2}) bigrVert _{1}+ biglVert S_{rho}(x_{1},y_{1})-S_{rho}(x_{2},y_{2}) bigrVert _{2} \ &quad leq biggl(frac{r+lambda tau _{1}}{varrho _{1}}+ frac{rho theta _{1}}{varrho _{2}}biggr) Vert x_{1}-x_{2} Vert _{1} +biggl( frac{k+rho theta _{2}}{varrho _{2}}+ frac{lambda tau _{2}}{varrho _{1}}biggr) Vert y_{1}-y_{2} Vert _{2} \ & quad =vartheta _{lambda ,rho}bigl( Vert x_{1}-x_{2} Vert _{1}+ Vert y_{1}-y_{2} Vert _{2}bigr) \ & quad =vartheta _{lambda ,rho} biglVert (x_{1},y_{1})-(x_{2},y_{2}) bigrVert _{*}, end{aligned} end{aligned}$$

(3.11)

where

$$begin{aligned} vartheta _{lambda ,rho}=max biggl{ frac{r+lambda tau _{1}}{varrho _{1}}+ frac{rho theta _{1}}{varrho _{2}}, frac{k+rho theta _{2}}{varrho _{2}}+ frac{lambda tau _{2}}{varrho _{1}} biggr} . end{aligned}$$

Since (r<varrho _{1}) and (k<varrho _{2}), we can choose (lambda _{0},rho _{0}>0) small enough such that

$$begin{aligned} frac{r+lambda _{0}tau _{1}}{varrho _{1}}+ frac{rho _{0}theta _{1}}{varrho _{2}}< 1quad text{and}quad frac{k+rho _{0}theta _{2}}{varrho _{2}}+ frac{lambda _{0}tau _{2}}{varrho _{1}}< 1. end{aligned}$$

(3.12)

From (3.12) it follows that

$$begin{aligned} vartheta _{lambda _{0},rho _{0}}=max biggl{ frac{r+lambda _{0}tau _{1}}{varrho _{1}}+ frac{rho _{0}theta _{1}}{varrho _{2}}, frac{k+rho _{0}theta _{2}}{varrho _{2}}+ frac{lambda _{0}tau _{2}}{varrho _{1}} biggr} in (0,1) end{aligned}$$

(3.13)

and so (Q_{lambda _{0},rho _{0}}) is a contraction mapping. Then, the Banach Fixed-Point Theorem ensures the existence of a unique ((a,b)in X_{1}times X_{2}) such that (Q_{lambda ,rho}(a,b)=(a,b)). Thereby, making use of (3.2)–(3.4) we conclude that for some (lambda _{0},rho _{0}>0),

$$begin{aligned} begin{aligned} textstylebegin{cases} a=R^{widehat{H}_{1}}_{M,lambda _{0}}[widehat{H}_{1}(a)-lambda _{0} F(a,b)], \ b=R^{widehat{H}_{2}}_{N,rho _{0}}[widehat{H}_{2}(b)-rho _{0} G(a,b)]. end{cases}displaystyle end{aligned} end{aligned}$$

Accordingly, Lemma 3.1 guarantees that ((a,b)in X_{1}times X_{2}) is the unique solution of the SVI (3.1). This completes the proof. □

Given a real normed space X with a norm (Vert cdotVert ), we recall that a nonlinear mapping (T:Xrightarrow X) is called nonexpansive if (Vert T(x)-T(y)Vert leq Vert x-yVert ) for all (x,yin X). It is well known that the class of nonexpansive mappings has a deep and close relation with the classes of monotone and accretive operators that arise naturally in the theory of differential equations. On the other hand, the fixed-point theory is an attractive and interesting subject with a large number of applications in various fields of mathematics and other branches of science. At the same time, the study of nonexpansive mappings is a very interesting research area in fixed-point theory. These facts have motivated many researchers to extend the notion of nonexpansive mapping and several interesting generalized nonexpansive mappings in the framework of different spaces have appeared in the literature. For example, two classes of generalized nonexpansive mappings are recalled in the next definition.

Definition 3.4

A nonlinear mapping (T:Xrightarrow X) is said to be

  1. (i)

    L-Lipschitzian if there exists a constant (L>0) such that

    $$begin{aligned} biglVert T(x)-T(y) bigrVert leq L Vert x-y Vert ,quad forall x,yin X; end{aligned}$$

  2. (ii)

    uniformly L-Lipschitzian if there exists a constant (L>0) such that for each (nin mathbb{N}),

    $$begin{aligned} biglVert T^{n}(x)-T^{n}(y) bigrVert leq L Vert x-y Vert ,quad forall x,y in X. end{aligned}$$

It is significant to emphasize that every uniformly L-Lipschitzian mapping is L-Lipschitzian but the converse need not be true. The following example illustrates that the class of L-Lipschitzian mappings contains properly the class of uniformly L-Lipschitzian mappings.

Example 3.5

Consider (X=mathbb{R}) with the Euclidean norm (Vert cdotVert =|cdot|) and let the self-mapping T of X be defined by (T(x)=kx) for all (xin X), where (k>1) is an arbitrary real constant. Taking into account that for all (x,yin X), (|T(x)-T(y)|=k|x-y|), it follows that T is a k-Lipschitzian mapping. However, thanks to the fact that (k>1), for any (x,yin X) and (nin mathbb{N}backslash {1}), we obtain (|T^{n}(x)-T^{n}(y)|=k^{n}|x-y|>k|x-y|). This fact ensures that T is not a uniformly k-Lipschitzian mapping.

The introduction and study of the notion of asymptotically nonexpansive mapping as a generalization of the concept of nonexpansive mapping was first made by Goebel and Kirk [47].

Definition 3.6

([47])

A nonlinear mapping (T:Xrightarrow X) is said to be asymptotically nonexpansive if, there exists a sequence ({a_{n}}subset (0,+infty )) with (lim_{nrightarrow infty}a_{n}=0) such that for each (nin mathbb{N}),

$$begin{aligned} biglVert T^{n}(x)-T^{n}(y) bigrVert leq (1+a_{n}) Vert x-y Vert ,quad forall x,yin X. end{aligned}$$

Equivalently, we say that the mapping T is asymptotically nonexpansive if there exists a sequence ({k_{n}}subset [1,+infty )) with (lim_{nrightarrow infty}k_{n}=1) such that for each (nin mathbb{N}),

$$begin{aligned} biglVert T^{n}(x)-T^{n}(y) bigrVert leq k_{n} Vert x-y Vert ,quad forall x,y in X. end{aligned}$$

In recent decades, successful attempts in this direction have continued and several other interesting generalizations of nonexpansive mappings and asymptotically nonexpansive mappings are presented. For instance, in 2006, Alber et al. [49] succeeded in introducing a class of generalized nonexpansive mappings, the so-called total asymptotically nonexpansive mappings, which are more general than the classes of asymptotically nonexpansive mappings and nearly asymptotically nonexpansive mappings.

Definition 3.7

([49])

A nonlinear mapping (T:Xrightarrow X) is said to be total asymptotically nonexpansive (also referred to as (({a_{n}},{b_{n}},phi ))-total asymptotically nonexpansive) if there exist nonnegative real sequences ({a_{n}}) and ({b_{n}}) with (a_{n},b_{n}rightarrow 0) as (nrightarrow infty ) and a strictly increasing continuous function (phi :mathbb{R}^{+}rightarrow mathbb{R}^{+}) with (phi (0)=0) such that for all (x,yin X),

$$begin{aligned} biglVert T^{n}(x)-T^{n}(y) bigrVert leq Vert x-y Vert +a_{n}phi bigl( Vert x-y Vert bigr)+b_{n},quad forall nin mathbb{N}. end{aligned}$$

Using a modified Mann iteration process, they also studied the iterative approximation of the fixed point of total asymptotically nonexpansive mappings under some appropriate conditions. Note, in particular, that every asymptotically nonexpansive mapping is total asymptotically nonexpansive with (b_{n}=0) (or equivalently (b_{n}=0) and (a_{n}=k_{n}-1)) for all (nin mathbb{N}) and (phi (t)=t) for all (tgeq 0), but the converse need not be true. In other words, the class of total asymptotically nonexpansive mappings is more general than the class of asymptotically nonexpansive mappings. This fact is shown in the next example.

Example 3.8

For (1leq p<infty ), consider

$$begin{aligned} l^{p}= Biggl{ x={x_{n}}_{nin mathbb{N}}:sum _{n=1}^{infty} vert x_{n} vert ^{p}< infty , x_{n}in mathbb{F}=mathbb{R} text{ or } mathbb{C} Biggr} , end{aligned}$$

the classical space consisting of all p-power summable sequences, with the p-norm (Vert cdotVert _{p}) defined on it by

$$begin{aligned} Vert x Vert _{p}=Biggl(sum_{n=1}^{infty} vert x_{n} vert ^{p}Biggr)^{ frac{1}{p}}, qquad x= {x_{n}}_{nin mathbb{N}}in l^{p}. end{aligned}$$

Furthermore, let B denote the closed unit ball in the Banach space (l^{p}) and consider (X:=mathbb{R}times B) with the norm (Vert cdotVert _{X}=|cdot|_{mathbb{R}}+Vert cdotVert _{p}) and define the self-mapping T of X by

$$ T(u,x)=textstylebegin{cases} frac{1}{beta}(1,x^{dagger}), & text{if } uin [0,beta ], \ (0,frac{x^{dagger}}{beta}), & text{if } uin (-infty ,0)cup ( beta ,+infty ), end{cases} $$

where

$$begin{aligned} begin{aligned} x^{dagger}={}&bigl(underbrace{0,0,dots ,0} _{lambda text{ times}}, gamma sin^{k_{1}} vert x_{1} vert ,0, gamma vert x_{2} vert ^{s_{1}},0,gamma sin^{k_{2}} vert x_{3} vert ,0, gamma vert x_{4} vert ^{s_{2}}, \ &{} dots ,0,gamma sin^{k_{frac{m+1}{2}}} vert x_{m} vert ,0,gamma vert x_{m+1} vert ^{s_{ frac{m+1}{2}}},0,gamma x_{m+2},dots bigr), end{aligned} end{aligned}$$

(gamma in (0,1)) and (beta >1) are arbitrary real constants, (lambda in mathbb{N}) is an arbitrary constant, m is an arbitrary but fixed odd natural number, (lambda geq m+1) is an arbitrary but fixed natural number and (k_{i},s_{i}in mathbb{N}backslash {1}) ((i=1,2,dots ,frac{m+1}{2})) are arbitrary constants. Indeed, the element (x^{dagger}) of (l^{p}) can be written as (x^{dagger}={x_{n}^{dagger}}_{n=1}^{infty}), where (x_{i}^{dagger}=0) for all (1leq ileq lambda ), (x_{lambda +2i}^{dagger}=0) for all (iin mathbb{N}),

$$ x_{lambda +2i-1}^{dagger}=textstylebegin{cases} gamma sin^{k_{frac{i+1}{2}}} vert x_{i} vert , & text{if } iin {2t-1|t=1,2, dots ,frac{m+1}{2}}, \ gamma vert x_{i} vert ^{s_{frac{i}{2}}}, & text{if } iin {2sigma | sigma =1,2,dots ,frac{m+1}{2}}, end{cases} $$

and (x_{lambda +2m+j}^{dagger}=gamma x_{m+frac{j+1}{2}}) for all (jin {2l+1|lin mathbb{N}}). Taking into account that the mapping T is not continuous at the points ((beta ,x)) for all (xin B), we conclude that T is not Lipschitzian and so it is not an asymptotically nonexpansive mapping. For all ((u,x),(v,y)in [0,beta ]times B) and ((u,x),(v,y)in ((-infty ,0)cup (beta ,+infty ) )times B), one can show that

$$begin{aligned} begin{aligned} & biglVert T(u,x)-T(v,y) bigrVert _{X} \ &quad =biggVert (0,frac{1}{beta}bigl(bigl(underbrace{0,0,dots ,0} _{ lambda text{ times}},gamma bigl(sin^{k_{1}} vert x_{1} vert -sin^{k_{1}} vert y_{1} vert bigr),0, \ &qquad {} gamma bigl( vert x_{2} vert ^{s_{1}}- vert y_{2} vert ^{s_{1}}bigr),0, gamma bigl(sin^{k_{2}} vert x_{3} vert -sin^{k_{2}} vert y_{3} vert bigr),0, \ &qquad {} gamma bigl( vert x_{4} vert ^{s_{2}}- vert y_{4} vert ^{s_{2}}bigr),dots ,0,gamma bigl( sin^{k_{ frac{m+1}{2}}} vert x_{m} vert -sin^{k_{frac{m+1}{2}}} vert y_{m} vert bigr),0, \ &qquad {} gamma bigl( vert x_{m+1} vert ^{s_{frac{m+1}{2}}}- vert y_{m+1} vert ^{s_{ frac{m+1}{2}}}bigr),0,gamma (x_{m+2}-y_{m+2}), dots bigr) bigr)biggVert _{X} \ &quad =frac{1}{beta} Biggl(gamma ^{p}sum _{i=1}^{frac{m+1}{2}} biglvert sin^{k_{i}} bigrvert x_{2i-1} biglvert -sin^{k_{i}} vert y_{2i-1} vert bigrvert ^{p} \ & qquad {}{}+gamma ^{p}sum_{i=1}^{frac{m+1}{2}} biglvert vert x_{2i} vert ^{s_{i}}- vert y_{2i} vert ^{s_{i}} bigrvert ^{p} +gamma ^{p}sum_{i=m+2}^{infty} vert x_{i}-y_{i} vert ^{p} Biggr)^{frac{1}{p}} \ & quad leq gamma Biggl(sum_{i=1}^{frac{m+1}{2}}Biggl( sum_{j=1}^{k_{i}} vert x_{2i-1} vert ^{k_{i}-j} vert y_{2i-1} vert ^{j-1} Biggr)^{p} vert x_{2i-1}-y_{2i-1} vert ^{p} \ &qquad {} {}+sum_{i=1}^{frac{m+1}{2}}Biggl(sum _{r=1}^{s_{i}} vert x_{2i} vert ^{s_{i}-r} vert y_{2i} vert ^{r-1} Biggr)^{p} vert x_{2i}-y_{2i} vert ^{p} +sum_{i=m+2}^{infty} vert x_{i}-y_{i} vert ^{p} Biggr)^{frac{1}{p}}. end{aligned} end{aligned}$$

(3.14)

The fact that (x,yin B) implies that (0leq |x_{2i-1}|^{k_{i}-j}), (|y_{2i-1}|^{j-1}leq 1) for each (jin {1,2,dots ,k_{i}}) and (0leq |x_{2i}|^{s_{i}-r}), (|y_{2i}|^{r-1}leq 1) for each (rin {1,2,dots ,s_{i}}) and (iin {1,2,dots ,frac{m+1}{2}}). Relying on these facts, we conclude that (0leq sum_{j=1}^{k_{i}}|x_{2i-1}|^{k_{i}-j}|y_{2i-1}|^{j-1} leq k_{i}) and (0leq sum_{r=1}^{s_{i}}|x_{2i}|^{s_{i}-r}|y_{2i}|^{r-1} leq s_{i}) for each (iin {1,2,dots ,frac{m+1}{2}}). Thereby, making use of (3.14) it follows that for all ((u,x),(v,y)in [0,beta ]times B) and ((u,x),(v,y)in ((-infty ,0)cup (beta ,+infty ) )times B),

$$begin{aligned} begin{aligned} & biglVert T(u,x)-T(v,y) bigrVert _{X} \ &quad leq gamma Biggl(max Biggl{ Biggl(sum_{j=1}^{k_{i}} vert x_{2i-1} vert ^{k_{i}-j} vert y_{2i-1} vert ^{j-1}Biggr)^{p}, Biggl(sum _{r=1}^{s_{i}} vert x_{2i} vert ^{s_{i}-r} vert y_{2i} vert ^{r-1} Biggr)^{p},1: \ & qquad{} i=1,2,dots ,frac{m+1}{2} Biggr} sum_{i=1}^{infty} vert x_{i}-y_{i} vert ^{p} Biggr)^{frac{1}{p}} \ &quad =gamma max Biggl{ sum_{j=1}^{k_{i}} vert x_{2i-1} vert ^{k_{i}-j} vert y_{2i-1} vert ^{j-1}, sum_{r=1}^{s_{i}} vert x_{2i} vert ^{s_{i}-r} vert y_{2i} vert ^{r-1},1: \ &qquad {} i=1,2,dots ,frac{m+1}{2} Biggr} Vert x-y Vert _{p}. end{aligned} end{aligned}$$

(3.15)

If (uin [0,beta ]) and (vin (-infty ,0)cup (beta ,+infty )), then in a similar fashion to the preceding analysis, one can prove that for all (x,yin B),

$$begin{aligned} begin{aligned} & biglVert T(u,x)-T(v,y) bigrVert _{X} \ &quad = bigglVert frac{1}{beta}bigl(1,x^{dagger}bigr)-biggl(0, frac{1}{beta}y^{dagger}biggr) biggrVert _{X} = frac{1}{beta} biglVert bigl(1,x^{dagger}-y^{dagger}bigr) bigrVert _{X} \ &quad =frac{1}{beta} Biggl(1+gamma max Biggl{ sum _{j=1}^{k_{i}} vert x_{2i-1} vert ^{k_{i}-j} vert y_{2i-1} vert ^{j-1}, \ & qquad {}sum_{r=1}^{s_{i}} vert x_{2i} vert ^{s_{i}-r} vert y_{2i} vert ^{r-1},1:i=1,2, dots ,frac{m+1}{2} Biggr} Vert x-y Vert _{p} Biggr) \ & quad < vert u-v vert +gamma max Biggl{ sum_{j=1}^{k_{i}} vert x_{2i-1} vert ^{k_{i}-j} vert y_{2i-1} vert ^{j-1}, \ & qquad {}sum_{r=1}^{s_{i}} vert x_{2i} vert ^{s_{i}-r} vert y_{2i} vert ^{r-1},1:i=1,2, dots ,frac{m+1}{2} Biggr} Vert x-y Vert _{p}+ frac{1}{beta}. end{aligned} end{aligned}$$

(3.16)

Now, applying (3.15) and (3.16), for all ((u,x),(v,y)in X), we obtain

$$begin{aligned}& biglVert T(u,x)-T(v,y) bigrVert _{X} \& quad leq vert u-v vert +gamma max Biggl{ sum _{j=1}^{k_{i}} vert x_{2i-1} vert ^{k_{i}-j} vert y_{2i-1} vert ^{j-1}, \& qquad {}sum_{r=1}^{s_{i}} vert x_{2i} vert ^{s_{i}-r} vert y_{2i} vert ^{r-1},1:i=1,2, dots ,frac{m+1}{2} Biggr} Vert x-y Vert _{p}+ frac{1}{beta} \& quad leq vert u-v vert + Vert x-y Vert _{p}+gamma max Biggl{ sum_{j=1}^{k_{i}} vert x_{2i-1} vert ^{k_{i}-j} vert y_{2i-1} vert ^{j-1}, \& qquad {} sum_{r=1}^{s_{i}} vert x_{2i} vert ^{s_{i}-r} vert y_{2i} vert ^{r-1},1:i=1,2, dots ,frac{m+1}{2} Biggr} bigl( vert u-v vert + Vert x-y Vert _{p}bigr)+frac{1}{beta}. end{aligned}$$

(3.17)

For all (ngeq 2) and ((u,x)in X), we have

$$begin{aligned} begin{aligned} T^{n}(u,x)={}&frac{1}{beta} bigl(1,bigl( underbrace{0,0,dots ,0} _{(2^{n}-1)lambda text{ times}},gamma ^{n} sin^{k_{1}} vert x_{1} vert ,underbrace{0,0,dots ,0} _{(2^{n}-1) text{ times}}, gamma ^{n} vert x_{2} vert ^{s_{1}}, underbrace{0,0,dots ,0} _{(2^{n}-1) text{ times}}, \ &{} gamma ^{n} sin^{k_{2}} vert x_{3} vert , underbrace{0,0,dots ,0} _{(2^{n}-1) text{ times}},gamma ^{n} vert x_{4} vert ^{s_{2}},dots , underbrace{0,0,dots ,0} _{(2^{n}-1) text{ times}},gamma ^{n} sin^{k_{ frac{m+1}{2}}} vert x_{m} vert , \ &{} underbrace{0,0,dots ,0} _{(2^{n}-1) text{ times}},gamma ^{n} vert x_{m+1} vert ^{s_{ frac{m+1}{2}}}, underbrace{0,0,dots ,0} _{(2^{n}-1) text{ times}}, gamma ^{n} x_{m+2},dots bigr) bigr). end{aligned} end{aligned}$$

Then, by an argument analogous to those of (3.14) and (3.15), for all ((u,x),(v,y)in X) and (ngeq 2), one can deduce that

$$begin{aligned} begin{aligned} biglVert T^{n}(u,x)-T^{n}(v,y) bigrVert _{X}={}&frac{1}{beta} biglVert bigl(0, bigl( underbrace{0,0,dots ,0} _{(2^{n}-1)lambda text{ times}}, gamma ^{n} bigl( sin^{k_{1}} vert x_{1} vert -sin^{k_{1}} vert y_{1} vert bigr), \ & {}underbrace{0,0,dots ,0} _{(2^{n}-1) text{ times}}, gamma ^{n}bigl( vert x_{2} vert ^{s_{1}}- vert y_{2} vert ^{s_{1}}bigr), underbrace{0,0,dots ,0} _{(2^{n}-1) text{ times}}, \ &{} gamma ^{n}bigl(sin^{k_{2}} vert x_{3} vert -sin^{k_{2}} vert y_{3} vert bigr), underbrace{0,0,dots ,0} _{(2^{n}-1) text{ times}},gamma ^{n}bigl( vert x_{4} vert ^{s_{2}}- vert y_{4} vert ^{s_{2}}bigr), \ & {}dots , underbrace{0,0,dots ,0} _{(2^{n}-1) text{ times}}, gamma ^{n} bigl(sin^{k_{frac{m+1}{2}}} vert x_{m} vert -sin^{k_{frac{m+1}{2}}} vert y_{m} vert bigr), \ & {}underbrace{0,0,dots ,0} _{(2^{n}-1) text{ times}}, gamma ^{n}bigl( vert x_{m+1} vert ^{s_{ frac{m+1}{2}}}- vert y_{m+1} vert ^{s_{frac{m+1}{2}}}bigr), \ & {}underbrace{0,0,dots ,0} _{(2^{n}-1) text{ times}},gamma ^{n}(x_{m+2}-y_{m+2}), dots bigr) bigr) bigrVert _{X} \ leq{}& gamma ^{n}max Biggl{ sum_{j=1}^{k_{i}} vert x_{2i-1} vert ^{k_{i}-j} vert y_{2i-1} vert ^{j-1}, \ &{} sum_{r=1}^{s_{i}} vert x_{2i} vert ^{s_{i}-r} vert y_{2i} vert ^{r-1},1:i=1,2, dots ,frac{m+1}{2} Biggr} Vert x-y Vert _{p} \ leq{}& vert u-v vert + Vert x-y Vert _{p}+gamma ^{n}max Biggl{ sum_{j=1}^{k_{i}} vert x_{2i-1} vert ^{k_{i}-j} vert y_{2i-1} vert ^{j-1}, \ &{} sum_{r=1}^{s_{i}} vert x_{2i} vert ^{s_{i}-r} vert y_{2i} vert ^{r-1},1:i=1,2, dots ,frac{m+1}{2} Biggr} bigl( vert u-v vert \ & {}+ Vert x-y Vert _{p}bigr)+frac{1}{beta ^{n}}. end{aligned} end{aligned}$$

(3.18)

Employing (3.17) and (3.18) and by virtue of the fact that for each (iin {1,2,dots ,frac{m+1}{2}}), (0leq sum_{j=1}^{k_{i}}|x_{2i-1}|^{k_{i}-j}|y_{2i-1}|^{j-1} leq k_{i}) and (0leq sum_{r=1}^{s_{i}}|x_{2i}|^{s_{i}-r}|y_{2i}|^{r-1} leq s_{i}), we conclude that for all ((u,x),(v,y)in X) and (nin mathbb{N}),

$$begin{aligned} begin{aligned} biglVert T^{n}(u,x)-T^{n}(v,y) bigrVert _{X}leq{}& vert u-v vert + Vert x-y Vert _{p}+gamma ^{n}max Biggl{ sum _{j=1}^{k_{i}} vert x_{2i-1} vert ^{k_{i}-j} vert y_{2i-1} vert ^{j-1}, \ & {}sum_{r=1}^{s_{i}} vert x_{2i} vert ^{s_{i}-r} vert y_{2i} vert ^{r-1},1:i=1,2, dots ,frac{m+1}{2} Biggr} bigl( vert u-v vert \ & {}+ Vert x-y Vert _{p}bigr)+frac{1}{beta ^{n}} \ leq{}& biglVert (u,x)-(v,y) bigrVert _{X}+gamma ^{n}xi biglVert (u,x)-(v,y) bigrVert _{X}+ frac{1}{beta ^{n}}, end{aligned} end{aligned}$$

where (xi =max {k_{i},s_{i}:i=1,2,dots ,frac{m+1}{2}}). Taking (a_{n}=gamma ^{n}) and (b_{n}=frac{1}{beta ^{n}}) for all (nin mathbb{N}), the fact that (0<gamma <1<beta ) implies that (a_{n},b_{n}rightarrow 0) as (nrightarrow infty ). Now, define the function (phi :[0,+infty )rightarrow [0,+infty )) by (phi (t)=xi t) for all (tin [0,+infty )). Then, for all ((u,x),(v,y)in X) and (nin mathbb{N}), we obtain

$$begin{aligned} biglVert T^{n}(u,x)-T^{n}(v,y) bigrVert _{X}leq biglVert (u,x)-(v,y) bigrVert _{X}+a_{n} phi bigl( biglVert (u,x)-(v,y) bigrVert _{X} bigr)+b_{n}, end{aligned}$$

that is, T is a (({gamma ^{n}},{frac{1}{beta ^{n}}},phi ))-total asymptotically nonexpansive mapping.

With the aim of presenting a unifying framework for generalized nonexpansive mappings available in the literature and verifying a general convergence theorem applicable to all these classes of nonlinear mappings, very recently, Kiziltunc and Purtas [50] introduced a new class of generalized nonexpansive mappings as follows.

Definition 3.9

([50])

A nonlinear mapping (T:Xrightarrow X) is said to be total uniformly L-Lipschitzian (or (({a_{n}},{b_{n}},phi ))-total uniformly L-Lipschitzian) if there exist a constant (L>0), nonnegative real sequences ({a_{n}}) and ({b_{n}}) with (a_{n},b_{n}rightarrow 0) as (nrightarrow infty ) and a strictly increasing continuous function (phi :mathbb{R}^{+}rightarrow mathbb{R}^{+}) with (phi (0)=0) such that for each (nin mathbb{N}),

$$begin{aligned} biglVert T^{n}(x)-T^{n}(y) bigrVert leq Lbigl[ Vert x-y Vert +a_{n}phi bigl( Vert x-y Vert bigr)+b_{n}bigr],quad forall x,yin X. end{aligned}$$

It is essential to note that, for given nonnegative real sequences ({a_{n}}) and ({b_{n}}) and a strictly increasing continuous function (phi :mathbb{R}^{+}rightarrow mathbb{R}^{+}), an (({a_{n}},{b_{n}},phi ))-total asymptotically nonexpansive mapping is (({a_{n}},{b_{n}},phi ))-total uniformly L-Lipschitzian with (L=1), but the converse may not be true. In the following example, the fact that the class of total uniformly L-Lipschitzian mappings contains properly the class of total asymptotically nonexpansive mappings is illustrated.

Example 3.10

Let (X=mathbb{R}) endowed with the Euclidean norm (Vert cdotVert =|cdot|) and let the self-mapping T of X be defined by

$$ T(x)=textstylebegin{cases} 0, & text{if } xin (-infty ,0), \ beta , & text{if } xin (0,frac{1}{beta})cup ( frac{1}{beta},alpha ), \ frac{1}{beta}, & text{if } xin [alpha ,+infty )cup {0, frac{1}{beta}}, end{cases} $$

where (alpha >0) and (beta >frac{alpha +sqrt{alpha ^{2}+4}}{2}) are arbitrary real constants such that (alpha beta >1). Since the mapping T is discontinuous at the points (x=0,alpha ,frac{1}{beta}), it follows that T is not Lipschitzian and so it is not an asymptotically nonexpansive mapping. Take (a_{n}=frac{gamma}{n}) and (b_{n}=frac{alpha}{k^{n}}) for each (nin mathbb{N}), where (gamma >0) and (k>1) are arbitrary constants such that (kneq alpha beta ). Let us now define the function (phi :mathbb{R}^{+}rightarrow mathbb{R}^{+}) by (phi (t)=theta t^{m}) for all (tin mathbb{R}^{+}), where (min mathbb{N}) and (theta in (0, frac{k^{m}(beta ^{2}-alpha beta -1)}{beta gamma (k-1)^{m}alpha ^{m}} )) are arbitrary constants. Selecting (x=alpha ) and (y=frac{alpha}{k}), we have (T(x)=frac{1}{beta}) and (T(y)=beta ). With the help of the fact that (0<theta < frac{k^{m}(beta ^{2}-alpha beta -1)}{beta gamma (k-1)^{m}alpha ^{m}}), it follows that

$$begin{aligned} begin{aligned} biglvert T(x)-T(y) bigrvert &=beta – frac{1}{beta} \ &>alpha + frac{gamma theta (k-1)^{m}alpha ^{m}}{k^{m}} \ & =frac{(k-1)alpha}{k}+ frac{gamma theta (k-1)^{m}alpha ^{m}}{k^{m}}+frac{alpha}{k} \ & = vert x-y vert +gamma theta vert x-y vert ^{m}+ frac{alpha}{k} \ & = vert x-y vert +a_{1}phi bigl( vert x-y vert bigr)+b_{1}, end{aligned} end{aligned}$$

which implies that T is not a (({frac{gamma}{n}},{frac{alpha}{k^{n}}},phi ))-total asymptotically nonexpansive mapping. However, for all (x,yin X), we obtain

$$begin{aligned} begin{aligned} biglvert T(x)-T(y) bigrvert &leq beta \ &leq frac{kbeta}{alpha}biggl( vert x-y vert + gamma theta vert x-y vert ^{m}+frac{alpha}{k}biggr) \ & =frac{kbeta}{alpha}bigl( vert x-y vert +a_{1}phi bigl( vert x-y vert bigr)+b_{1}bigr) end{aligned} end{aligned}$$

(3.19)

and for all (ngeq 2),

$$begin{aligned} begin{aligned} biglvert T^{n}(x)-T^{n}(y) bigrvert &< frac{kbeta}{alpha}biggl( vert x-y vert + frac{gamma theta}{n} vert x-y vert ^{m}+frac{alpha}{k^{n}}biggr) \ & =frac{kbeta}{alpha}bigl( vert x-y vert +a_{n}phi bigl( vert x-y vert bigr)+b_{n}bigr), end{aligned} end{aligned}$$

(3.20)

due to the fact that (T^{n}(z)=frac{1}{beta}) for all (zin X) and (ngeq 2). Making use of (3.19) and (3.20), we deduce that T is a (({frac{gamma}{n}},{frac{alpha}{k^{n}}},phi ))-total uniformly (frac{kbeta}{alpha})-Lipschitzian mapping.

Lemma 3.11

Let (X_{1}) and (X_{2}) be two real Banach spaces with norms (Vert cdotVert _{1}) and (Vert cdotVert _{2}), respectively, and let (S_{1}:X_{1}rightarrow X_{1}) and (S_{2}:X_{2}rightarrow X_{2}) be (({a_{i}}_{i=1}^{infty},{b_{i}}_{i=1}^{infty},phi _{1}))total uniformly (L_{1})Lipschitzian and (({c_{i}}_{i=1}^{infty},{d_{i}}_{i=1}^{infty},phi _{2}))total uniformly (L_{2})Lipschitzian mappings, respectively. Moreover, let Q and ϕ be selfmappings of (X_{1}times X_{2}) and (mathbb{R}^{+}), respectively, defined by

$$begin{aligned} Q(x_{1},x_{2})=(S_{1}x_{1},S_{2}x_{2}),quad forall (x_{1},x_{2}) in X_{1}times X_{2} end{aligned}$$

(3.21)

and

$$begin{aligned} phi (t)=max bigl{ phi _{j}(t):j=1,2bigr} ,quad forall tin mathbb{R}^{+}. end{aligned}$$

(3.22)

Then, Q is an (({a_{i}+c_{i}}_{i=1}^{infty}), ({b_{i}+d_{i}}_{i=1}^{infty}), ϕ)-total uniformly (max {L_{1},L_{2}})Lipschitzian mapping.

Proof

In view of the fact that for each (jin {1,2}), (phi _{j}:mathbb{R}^{+}rightarrow mathbb{R}^{+}) is a strictly increasing function, for all ((x_{1},x_{2}),(y_{1},y_{2})in X_{1}times X_{2}) and (iin mathbb{N}), yields

$$begin{aligned} biglVert Q^{i}(x_{1},x_{2})-Q^{i}(y_{1},y_{2}) bigrVert _{*} =& biglVert bigl(S_{1}^{i}x_{1},S_{2}^{i}x_{2} bigr) -bigl(S_{1}^{i}y_{1},S_{2}^{i}y_{2} bigr) bigrVert _{*} \ =& biglVert bigl(S_{1}^{i}x_{1}-S_{1}^{i}y_{1},S_{2}^{i}x_{2}-S_{2}^{i}y_{2} bigr) bigrVert _{*} \ =& biglVert S^{i}_{1}x_{1}-S^{i}_{1}y_{1} bigrVert _{1}+ biglVert S^{i}_{2}x_{2}-S^{i}_{2}y_{2} bigrVert _{2} \ leq& L_{1} bigl( Vert x_{1}-y_{1} Vert _{1}+a_{i}phi _{1}bigl( Vert x_{1}-y_{1} Vert _{1}bigr)+b_{i} bigr) \ & {}+L_{2} bigl( Vert x_{2}-y_{2} Vert _{2}+c_{i}phi _{2}bigl( Vert x_{2}-y_{2} Vert _{2}bigr)+d_{i} bigr) \ leq& max {L_{1},L_{2}} bigl( Vert x_{1}-y_{1} Vert _{1}+ Vert x_{2}-y_{2} Vert _{2} \ & {}+a_{i}phi _{1}bigl( Vert x_{1}-y_{1} Vert _{1}bigr)+c_{i}phi _{2}bigl( Vert x_{2}-y_{2} Vert _{2}bigr)+b_{i}+d_{i} bigr) \ leq& max {L_{1},L_{2}} bigl( Vert x_{1}-y_{1} Vert _{1}+ Vert x_{2}-y_{2} Vert _{2} \ & {}+a_{i}phi _{1}bigl( Vert x_{1}-y_{1} Vert _{1}+ Vert x_{2}-y_{2} Vert _{2}bigr) \ & {}+c_{i}phi _{2}bigl( Vert x_{1}-y_{1} Vert _{1}+ Vert x_{2}-y_{2} Vert _{2}bigr)+b_{i}+d_{i}bigr) \ leq& max {L_{1},L_{2}} bigl( biglVert (x_{1},x_{2})-(y_{1},y_{2}) bigrVert _{*} \ & {}+(a_{i}+c_{i})phi bigl( biglVert (x_{1},x_{2})-(y_{1},y_{2}) bigrVert _{*}bigr) +b_{i}+d_{i} bigr), end{aligned}$$

where (Vert cdotVert _{*}) is a norm on (X_{1}times X_{2}) defined by (3.10). This fact ensures that Q is an (({a_{i}+c_{i}}_{i=1}^{infty}), ({b_{i}+d_{i}}_{i=1}^{infty}), ϕ)-total uniformly (max {L_{1},L_{2}})-Lipschitzian mapping. The proof is completed. □

Assume that (X_{1}) and (X_{2}) are two real smooth Banach spaces with norms (Vert cdotVert _{1}) and (Vert cdotVert _{2}), respectively, (S_{1}:X_{1}rightarrow X_{1}) is an (({a_{i}}_{i=1}^{infty},{b_{i}}_{i=1}^{infty},phi _{1}))-total uniformly (L_{1})-Lipschitzian mapping and (S_{2}:X_{2}rightarrow X_{2}) is a (({c_{i}}_{i=1}^{infty},{d_{i}}_{i=1}^{infty},phi _{2}))-total uniformly (L_{2})-Lipschitzian mapping. Furthermore, let Q be a self-mapping of (X_{1}times X_{2}) defined as (3.21). Denote by (operatorname{Fix}(S_{j})) ((j=1,2)) and (operatorname{Fix}(Q)) the sets of all the fixed points of (S_{j}) ((j=1,2)) and Q, respectively. At the same time, denote by (operatorname{SVI}(X_{j},widehat{H}_{j},M,N,F,G:j=1,2)) the set of all the solutions of the SVI (3.1), where for (j=1,2), the nonlinear mappings (widehat{H}_{j}:X_{j}rightarrow X_{j}) are strictly accretive, and the set-valued mappings (M:Xrightrightarrows X_{1}) and (N:X_{2}rightrightarrows X_{2}) are (widehat{H}_{1})-accretive and (widehat{H}_{2})-accretive, respectively. Using (3.21), we infer that for any ((x_{1},x_{2})in X_{1}times X_{2}), ((x_{1},x_{2})in operatorname{Fix}(Q)) if and only if for (j=1,2), (x_{j}in operatorname{Fix}(S_{j})), that is, (operatorname{Fix}(Q)=operatorname{Fix}(S_{1},S_{2})=operatorname{Fix}(S_{1}) times operatorname{Fix}(S_{2})). If ((a,b)in operatorname{Fix}(Q)cap operatorname{SVI}(X_{j},A_{j},M,N,F,G:j=1,2)), then with the help of Lemma 3.1 it can be easily observed that for each (iin mathbb{N}),

$$begin{aligned} begin{aligned} textstylebegin{cases} a=S^{i}_{1}a=R^{widehat{H}_{1}}_{M,lambda}[widehat{H}_{1}(a)- lambda F(a,b)]=S^{i}_{1}R^{widehat{H}_{1}}_{M,lambda}[widehat{H}_{1}(a)- lambda F(a,b)], \ b=S^{i}_{2}b=R^{widehat{H}_{2}}_{N,rho}[widehat{H}_{2}(b)-rho G(a,b)]=S^{i}_{2}R^{ widehat{H}_{2}}_{N,rho}[widehat{H}_{2}(b)-rho G(a,b)]. end{cases}displaystyle end{aligned} end{aligned}$$

(3.23)

Using the fixed-point formulation (3.23) we now are able to construct the following iterative algorithm for finding a common element of the two sets of (operatorname{SVI}(X_{j},widehat{H}_{j},M,N,F,G:j=1,2)) and (operatorname{Fix}(Q)=operatorname{Fix}(S_{1},S_{2})).

Algorithm 3.12

Assume that (X_{j}) ((j=1,2)), F and G are the same as in the SVI (3.1). Let for (igeq 0) and (j=1,2), (widehat{H}_{i,j}:X_{j}rightarrow X_{j}) be strictly accretive, (M_{i}:X_{1}rightrightarrows X_{1}) be an (widehat{H}_{i,1})-accretive set-valued mapping and (N_{i}:X_{2}rightrightarrows X_{2}) be an (widehat{H}_{i,2})-accretive set-valued mapping. Suppose further that for (j=1,2), (S_{j}:X_{j}rightarrow X_{j}) is a (({c_{i,j}}_{i=0}^{infty},{d_{i,j}}_{i=0}^{infty},phi _{j}))-total uniformly (L_{j})-Lipschitzian mapping. For an arbitrarily chosen initial point ((a_{0},b_{0})in X_{1}times X_{2}), compute the iterative sequence ({(a_{i},b_{i})}_{i=0}^{infty}) in (X_{1}times X_{2}) by the iterative schemes

$$begin{aligned} begin{aligned} textstylebegin{cases} a_{i+1}=alpha _{i}a_{i}+(1-alpha _{i})S^{i}_{1}R^{widehat{H}_{i,1}}_{M_{i},lambda _{i}}[widehat{H}_{i,1}(a_{i})-lambda _{i} F(a_{i},b_{i})],\ b_{i+1}=alpha _{i}b_{i}+(1-alpha _{i})S^{i}_{2}R^{widehat{H}_{i,2}}_{N_{i},rho _{i}}[widehat{H}_{i,2}(b_{i})-rho _{i} G(a_{i},b_{i})], end{cases}displaystyle end{aligned} end{aligned}$$

(3.24)

where (iin mathbb{N}cup {0}); (lambda _{i},rho _{i}>0) are real constants; and ({alpha _{i}}_{i=0}^{infty}) is a sequence in the interval ([0,1]) such that (limsup_{i}alpha _{i}<1).

If (S_{j}equiv I_{j}) ((j=1,2)), the identity mapping on (X_{j}), then Algorithm 3.12 reduces to the following algorithm.

Algorithm 3.13

Let (X_{j}), (widehat{H}_{i,j}), (M_{i}), (N_{i}), F, G ((j=1,2); (iin mathbb{N}cup {0})) be the same as in Algorithm 3.12. For any given ((a_{0},b_{0})in X_{1}times X_{2}), define the iterative sequence ({(a_{i},b_{i})}_{i=0}^{infty}) in (X_{1}times X_{2}) by the iterative processes

$$begin{aligned} begin{aligned} textstylebegin{cases} a_{i+1}=alpha _{i}a_{i}+(1-alpha _{i})R^{widehat{H}_{i,1}}_{M_{i},lambda _{i}}[widehat{H}_{i,1}(a_{i})-lambda _{i} F(a_{i},b_{i})],\ b_{i+1}=alpha _{i}b_{i}+(1-alpha _{i})R^{widehat{H}_{i,2}}_{N_{i},rho _{i}}[widehat{H}_{i,2}(b_{i})-rho _{i} G(a_{i},b_{i})], end{cases}displaystyle end{aligned} end{aligned}$$

where (iin mathbb{N}cup {0}); the constants (lambda _{i},rho _{i}>0) and the sequence ({alpha _{i}}_{i=0}^{infty}) are the same as in Algorithm 3.12.

If (widehat{H}_{i,j}=widehat{H}_{j}), (lambda _{i}=lambda ) and (rho _{i}=rho ) for each (igeq 0) and (jin {1,2}), then Algorithm 3.13 collapses to the following algorithm.

Algorithm 3.14

Suppose that (X_{j}) ((j=1,2)), F and G are the same as in Algorithm 3.12. Let for (j=1,2), (widehat{H}_{j}:X_{j}rightarrow X_{j}) be strictly accretive mappings and let for each (igeq 0), (M_{i}:X_{1}rightrightarrows X_{1}) be an (widehat{H}_{1})-accretive set-valued mapping and (N_{i}:X_{2}rightrightarrows X_{2}) be an (widehat{H}_{2})-accretive set-valued mapping. For any given ((a_{0},b_{0})in X_{1}times X_{2}), compute the iterative sequence ({(a_{i},b_{i})}_{i=0}^{infty}) in (X_{1}times X_{2}) by the iterative schemes

$$begin{aligned} begin{aligned} textstylebegin{cases} a_{i+1}=alpha _{i}a_{i}+(1-alpha _{i})R^{widehat{H}_{1}}_{M_{i},lambda}[widehat{H}_{1}(a_{i})-lambda F(a_{i},b_{i})],\ b_{i+1}=alpha _{i}b_{i}+(1-alpha _{i})R^{widehat{H}_{2}}_{N_{i},rho}[widehat{H}_{2}(b_{i})-rho G(a_{i},b_{i})], end{cases}displaystyle end{aligned} end{aligned}$$

where (iin mathbb{N}cup {0}); (lambda ,rho >0) are two constants; and the sequence ({alpha _{i}}_{i=0}^{infty}) is the same as in Algorithm 3.12.

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