Let $\varnothing \ne C\subset H$ and $P_C$ be the metric projection from H onto C with C being convex and closed in a real Hilbert space H. Suppose that the $\langle ·,·\rangle$ and $\parallel ·\parallel$ are the inner product and induced norm in H, respectively. Given a nonlinear operator $S:C\to C$. We denote by $\mathrm{Fix}\left(S\right)$ the fixed-point set of S. Also, the $\mathbf{R},\rightharpoonup$ and → are used to stand for the real-number set, the weak convergence and the strong convergence, respectively. A self-mapping S on C is known as being of asymptotical nonexpansivity if ∃ (nonnegative real sequence) $\left\{{\theta }_n\right\}$ s.t. with ${\theta }_n\to 0$. In particular, in case ${\theta }_n=0\forall n\ge 1$, S reduces to a nonexpansive mapping. Let $A:H\to H$ be a mapping. Recall that so-called variational inequality problem (VIP) is to find $v\in C$ such that

Here $\mathrm{VI}(C,A)$ denotes the set of solutions of the VIP. In 1976, under weaker assumptions, Korpelevich [24] put forward the extragradient rule for approximating an element of $\mathrm{VI}(C,A)$, i.e., for any starting $t_0\in C$, $\left\{t_n\right\}$ is the sequence generated by

with $ϵ\in (0,\frac{1}{L})$. If $\mathrm{VI}(C,A)\ne \varnothing$, then $\left\{t_n\right\}$ converges weakly to an element in $\mathrm{VI}(C,A)$. To the best of our understanding, the Korpelevich extragradient rule is one of the most effective approaches for solving the VIP till now. The literature on the VIP is vast and the Korpelevich extragradient rule has acquired the extensive attention paid by numerous scholars, who ameliorated it in various ways; see e.g., [1-6, 8-9, 13-16, 19, 21-23, 25-28, 31, 34].

Recently, Thong and Hieu [21] first put forth the inertial subgradient extragradient rule, i.e., for any starting $t_0,t_1\in H$, $\left\{t_n\right\}$ is the sequence generated by

with constant $\ell \in (0,\frac{1}{L})$. Under suitable assumptions, they proved the weak convergence of $\left\{t_n\right\}$ to an element of $\mathrm{VI}(C,A)$. Subsequently, Thong and Hieu [15] introduced two inertial subgradient extragradient algorithms with linear-search process for solving the VIP with Lipschitz continuous monotone mapping A and the fixed-point problem (FPP) of a quasi-nonexpansive mapping S with demiclosedness property in H.

Algorithm 1.1 (see [15, Algorithm 1]). Initialization: Given $\gamma >0,l\in (0,1),\mu \in (0,1)$. Choose any initial $t_0,t_1\in H$.

Iterations: Compute $t_{n+1}$ below:

Step 1. Put $s_n=t_n+{\alpha }_n(t_n-t_{n-1})$ and calculate $y_n=P_C(s_n-{\ell }_{n}A{s}_n)$, wherein ${\ell }_n$ is picked as the largest $\ell \in \{\gamma ,\gamma l,\gamma l^2,...\}$ s.t. $\ell \parallel A{s}_n-A{y}_n\parallel \le \mu \parallel s_n-y_n\parallel$.

Step 2. Calculate $u_n=P_{C_n}(s_n-{\ell }_{n}A{y}_n)$, where $C_n:=\{u\in H:\langle s_n-{\ell }_{n}A{s}_n-y_n,u-y_n\rangle \le 0\}$.

Step 3. Calculate $t_{n+1}=(1-{\beta }_n)s_n+{\beta }_{n}S{u}_n$. When $s_n=u_n=t_{n+1}$, one has $s_n\in \mathrm{Fix}\left(T\right)\cap \mathrm{VI}(C,A)$. Put $n:=n+1$ and return to Step 1.

Algorithm 1.2 (see [15, Algorithm 2]). Initialization: Given $\gamma >0,l\in (0,1),\mu \in (0,1)$. Choose any initial $t_0,t_1\in H$.

Iterative steps: Compute $t_{n+1}$ below:

Step 1. Putt $s_n=t_n+{\alpha }_n(t_n-t_{n-1})$ and calculate $y_n=P_C(s_n-{\ell }_{n}A{s}_n)$, wherein ${\ell }_n$ is picked as the largest $\ell \in \{\gamma ,\gamma l,\gamma l^2,...\}$ s.t. $\ell \parallel A{s}_n-A{y}_n\parallel \le \mu \parallel s_n-y_n\parallel$.

Step 2. Calculate $u_n=P_{C_n}(s_n-{\ell }_{n}A{y}_n)$, where $C_n:=\{u\in H:\langle s_n-{\ell }_{n}A{s}_n-y_n,u-y_n\rangle \le 0\}$.

Step 3. Calculate $t_{n+1}=(1-{\beta }_n)t_n+{\beta }_{n}S{u}_n$. When $s_n=u_n=t_n=t_{n+1}$, one has $t_n\in \mathrm{Fix}\left(T\right)\cap \mathrm{VI}(C,A)$. Put $n:=n+1$ and return to Step 1.

With the help of suitable assumptions, it was proved in [15] that the sequences generated by the suggested algorithms converge weakly to a point in $\mathrm{VI}(C,A)\cap \mathrm{Fix}\left(S\right)$. In addition, combining the subgradient extragradient method and the Halpern’s iteration rule, Kraikaew and Saejung [22] proposed the Halpern subgradient extragradient rule for solving the VIP, and showed that the sequence generated by the proposed rule converges strongly to a point in $\mathrm{VI}(C,A)$. Recently, Reich et al. [27] put forward two gradient-projection algorithms for solving the VIP for uniformly continuous pseudomonotone mapping. In particular, they used a novel Armijo-type line search to acquire a hyperplane which strictly separates the current iterate from the solutions of the VIP under consideration. They proved that the sequences generated by two algorithms converge weakly and strongly to a point in $\mathrm{VI}(C,A)$, respectively.

On the other hand, let $\varnothing \ne C\subset E$ where C is convex and closed in a uniformly smooth and p-uniformly convex Banach space E for $p,q>1$ satisfying $\frac{1}{p}+\frac{1}{q}=1$. Let $J_E^p$ be the duality mapping of E, and let $E^{*}$ be the dual of E with the duality $J_{E^{*}}^q$. Suppose that the norm and the duality pairing between E and $E^{*}$ are denoted by $\parallel ·\parallel$ and $\langle ·,·\rangle$, respectively. Let $f_p\left(u\right)={\parallel u\parallel }^p/p\forall u\in E$, $D_{f_p}$ be the Bregman distance with respect to (w.r.t) $f_p$ and ${\Pi }_C$ be Bregman’s projection w.r.t. $f_p$ from E onto C, and presume that $\left\{{\alpha }_n\right\},\left\{{\beta }_n\right\}\subset (0,1)$ s.t. ${\sum }_{n=1}^{\infty }{\alpha }_n=\infty$, ${lim}_{n\to \infty }{\alpha }_n=0$ and $0<{lim\; inf}_{n\to \infty }{\beta }_n\le {lim\; sup}_{n\to \infty }{\beta }_n<1$. Assume that $A:E\to E^{*}$ is uniformly continuous and pseudomonotone operator and S is Bregman relatively nonexpansive self-mapping on C. Very recently, inspired by the research works in [27], Eskandani et al. [31] proposed the hybrid projection approach with linesearch process for approximating a point in $\mathrm{VI}(C,A)\cap \mathrm{Fix}\left(S\right)$.

Algorithm 1.3 (see [31]). Initialization: Given $l\in (0,1),\nu >0,\lambda \in (0,\frac{1}{\nu })$ and choose $u,t_1\in C$ randomly.

Iterations: Compute $t_{n+1}(n\ge 1)$ below:

Step 1. Calculate $y_n={\Pi }_C\left(J_{E^{*}}^q(J_E^p{t}_n-\lambda A{t}_n)\right)$ and $r_{\lambda }\left(t_n\right):=t_n-y_n$. If $r_{\lambda }\left(t_n\right)=0$ and $S{t}_n=t_n$, then stop; $t_n\in \Omega =\mathrm{VI}(C,A)\cap \mathrm{Fix}\left(S\right)$. If this case does not occur, then,

Step 2. Calculate $s_n=t_n-{ϵ}_n{r}_{\lambda }\left(t_n\right)$, with both ${ϵ}_n:=l^{k_n}$ and $k_n$ being the smallest $k\ge 0$ s.t. $\langle A{t}_n-A(t_n-l^k{r}_{\lambda }\left(t_n\right)),r_{\lambda }\left(t_n\right)\rangle \le \frac{\nu }{2}{D}_{f_p}(t_n,y_n)$.

Step 3. Calculate $u_n=J_{E^{*}}^q({\beta }_n{J}_E^p{t}_n+(1-{\beta }_n)J_E^p\left(S{\Pi }_{C_n}{t}_n\right))$ and $t_{n+1}={\Pi }_C\left(J_{E^{*}}^q({\alpha }_n{J}_E^{p}u+(1-{\alpha }_n)J_E^p{u}_n)\right)$, with $C_n:=\{y\in C:0\ge h_n\left(y\right)\}$ and $h_n\left(y\right)=\langle A{s}_n,y-t_n\rangle +\frac{{ϵ}_n}{2\lambda }{D}_{f_p}(t_n,y_n)$.

Again put $n:=n+1$ and return to Step 1.

With the help of suitable conditions, it was proven in [31] that $\left\{t_n\right\}$ converges strongly to ${\Pi }_{\Omega }u$.

This article designs two parallel subgradient-like extragradient algorithms with inertial effect for resolving a pair of variational inequality and fixed point problems (VIFPPs) in uniformly smooth and p-uniformly convex Banach space E. Here two variational inequality problems (VIPs) involve two uniformly continuous pseudomonotone operators and two fixed point problems implicate two uniformly continuous Bregman relatively asymptotically nonexpansive mappings. Moreover, each algorithm consists of two parts which are of symmetric structure mutually. With the help of appropriate registrations, it is proven that the sequences generated by the suggested algorithms converge weakly and strongly to a solution of this pair of VIFPPs, respectively. Lastly, an illustrative instance is furnished to check the implementability and applicability of the proposed approaches.

The structure of the article is described as follows: Section 2 releases certain terminologies and preliminary results for later applications. Section 3 is focused on discussing the convergence of the suggested algorithms. In Section 4, the major outcomes are utilized to deal with the CFPP and VIPs in an illustrative instance. Our results improve and develop the revelent ones obtained previously in [15, 27, 31].

Let ($E,\parallel ·\parallel$) be a real Banach space, whose dual is denoted by $E^{*}$. We use the $y_n\to y$ and $y_n\rightharpoonup y$ to represent the strong and weak convergence of $\left\{y_n\right\}$ to $y\in E$, respectively. Moreover, the set of weak cluster points of $\left\{y_n\right\}$ is denoted by ${\omega }_w\left(y_n\right)$, i.e., ${\omega }_w\left(y_n\right)=\{y^{†}\in E:y_{n_k}\rightharpoonup y^{†}\mathrm{for}\mathrm{some}\left\{y_{n_k}\right\}\subset \left\{y_n\right\}\}$. Let $U=\{y\in E:\parallel y\parallel =1\}$ and $1<q\le 2\le p$ with $\frac{1}{p}+\frac{1}{q}=1$. A Banach space E is referred to as being strictly convex if for each $y,x\in U$ with $y\ne x$, one has $\parallel y+x\parallel /2<1$. E is referred to as being uniformly convex if $\forall \varsigma \in (0,2]$, $\exists \delta >0$ s.t. $\forall y,x\in U$ with $\parallel y-x\parallel \ge \varsigma$, one has $\parallel y+x\parallel /2\le 1-\delta$. It is known that a uniformly convex Banach space is reflexive and strictly convex. The modulus of convexity of E is the function $\delta :[0,2]\to [0,1]$ defined by $\delta \left(\varsigma \right)=inf\{1-\parallel y+x\parallel /2:y,x\in U\mathrm{with}\parallel y-x\parallel \ge \varsigma \}$. It is also known that E is uniformly convex if and only if $\delta \left(\varsigma \right)>0\forall \varsigma \in (0,2]$. Moreover, E is referred to as being p-uniformly convex if $\exists c>0$ s.t. $\delta \left(\varsigma \right)\ge c{\varsigma }^p$ for $0\le \varsigma \le 2$.

The nonnegative function ${\rho }_E(·)$ on $[0,\infty )$ is called the modulus of smoothness of E if ${\rho }_E\left(\varsigma \right):=sup\left\{\right(\parallel y+\varsigma x\parallel +\parallel y-\varsigma x\parallel )/2-1:y,x\in U\}\forall \varsigma \in [0,\infty )$. E is said to be uniformly smooth if ${lim}_{\varsigma \to 0}{\rho }_E\left(\varsigma \right)/\varsigma =0$, and q-uniformly smooth if $\exists C_q>0$ s.t. ${\rho }_E\left(\varsigma \right)\le C_q{\varsigma }^q\forall \varsigma >0$. Recall that E is of p-uniform convexity iff $E^{*}$ is of q-uniform smoothness; see e.g., [32] for more details. Putting $B(0,r)=\{y\in E:\parallel y\parallel \le r\}$ for each $r>0$, we say that $f:E\to \mathbf{R}$ is uniformly convex on bounded sets (see [31]) if ${\rho }_r\left(\varsigma \right)>0\forall r,\varsigma >0$, where ${\rho }_r\left(\varsigma \right):[0,\infty )\to [0,\infty ]$ is specified below

The ${\rho }_r$ is known as the gauge function of f with uniform convexity. It is clear that the gauge ${\rho }_r$ is nondecreasing.

Let $f:E\to \mathbf{R}$ be a convex function. If the limit ${lim}_{\varsigma \to 0^{+}}\frac{f(y+\varsigma x)-f\left(y\right)}{\varsigma }$ exists for each $x\in E$, then f is referred to as being Gâteaux differentiable at y. In this case, the gradient $\nabla f\left(y\right)$ of f at y is of linearity, and is formulated as $\langle \nabla f\left(y\right),x\rangle :={lim}_{\varsigma \to 0^{+}}\frac{f(y+\varsigma x)-f\left(y\right)}{\varsigma }\forall x\in E$. The f is referred to as being of Gâteaux differentiablility if it is of Gâteaux differentiablility at any $y\in E$. In case ${lim}_{\varsigma \to 0^{+}}\frac{f(y+\varsigma x)-f\left(y\right)}{\varsigma }$ is achieved uniformly for any $x\in U$, we say that f is of Fréchet differentiablility at y. Besides, f is referred to as being of uniform Fréchet differentiablility on a subset $K\subset E$ if ${lim}_{\varsigma \to 0^{+}}\frac{f(y+\varsigma x)-f\left(y\right)}{\varsigma }$ is achieved uniformly for $(y,x)\in K\times U$. When the norm of E is of Gâteaux differentiablility, E is said to be of smoothness.

Let $\frac{1}{p}+\frac{1}{q}=1$ for $p,q>1$. The $J_E^p:E\to E^{*}$ is specified below

It is known that E is of smoothness iff $J_E^p$ is of single value from E into $E^{*}$. Also, E is of reflexivity iff $J_E^p$ is of surjectivity, and E is strictly convex iff $J_E^p$ is of one-to-one property. So it follows that, when the smooth Banach space E is of both strict convexity and reflexivity, $J_E^p$ is the bijection and in this case, $J_{E^{*}}^q={\left(J_E^p\right)}^{-1}$. Also, recall that E is of uniform smoothness iff the function $f_p\left(y\right)={\parallel y\parallel }^p/p$ is of uniform Fréchet differentiablility on bounded sets iff $J_E^p$ is of uniform continuity on bounded sets. Moreover, E is of uniform convexity iff the function $f_p$ is of uniform convexity (see [32]).

Let the function $f:E\to \mathbf{R}$ be of both Gâteaux differentiablility and convexity. Bregman’s distance w.r.t. f is specified below

It is worth mentioning that the $D_f(·,·)$ is not a metric in the common sense of the terminology. Evidently, $D_f(t,t)=0$ but $D_f(t,s)=0$ can not lead to $t=s$. Generally, $D_f$ is not of symmetry and fulfill no triangle inequality. However, $D_f$ fulfills the three point equequality

See [20] for many details.

It is remarkable that the $J_E^p$ on the smooth E is Gâteaux’s derivative of $f_p$. Thus, Bregman’s distance w.r.t. $f_p$ is specified below

In the p-uniformly convex and smooth Banach space E for $2\le p<\infty$, there holds the following relationship between the metric and Bregman distance: where $\tau >0$ is some fixed number (see [12]). Via (2.1) it can be easily seen that for each $\left\{y_n\right\}\subset E$ of boundedness, the relation is valid:

Let $\varnothing \ne C\subset E$ with C being convex and closed in a strictly convex, smooth and reflexive Banach space E. Bregman’s projection is formulated as minimizers of Bregman’s distance. Bregman’s projection of $y\in E$ onto C w.r.t. $f_p$ indicates a unique point ${\Pi }_{C}y\in C$ s.t. $D_{f_p}({\Pi }_{C}y,y)={min}_{x\in C}{D}_{f_p}(x,y)$. In the case of Hilbert space, Bregman’s projection w.r.t. $f_2$ reduces to the metric projection. Using [30, Theorem 2.1] and [18, Corollary 4.4], in a uniformly convex Banach space, the characterization of Bregman’s projection is formulated by:

Meantime, (2.2) is equivalent to the descent property

When $p=2$, $J_E^p$ reduces to the normalized duality mapping and is written as J. The $\varphi :E^2\to \mathbf{R}$ is formulated below and ${\Pi }_C\left(t\right)={\mathrm{argmin}}_{s\in C}\varphi (s,t)\forall t\in E$.

In terms of [31], the function $V_{f_p}:E\times E^{*}\to [0,\infty )$ associated with $f_p$ is specified below

So, $V_{f_p}(y,y^{*})=D_{f_p}(y,J_{E^{*}}^q\left(y^{*}\right))\forall (y,y^{*})\in E\times E^{*}$. Moreover, by the subdifferential inequality, we obtain

In addition, $V_{f_p}$ is convex in the second variable. Hence one has

Lemma 2.1 ([30]). Let E be a uniformly convex Banach space and $\left\{s_n\right\},\left\{t_n\right\}$ be two sequences in E such that the first one is bounded. If ${lim}_{n\to \infty }{D}_{f_p}(t_n,s_n)=0$, then ${lim}_{n\to \infty }\parallel t_n-s_n\parallel =0$.

Assume that S is a self-mapping on C. Let the $\mathrm{Fix}\left(S\right)$ indicate the set of fixed points of S, that is, $\mathrm{Fix}\left(S\right)=\{y\in C:y=Sy\}$. A point $y^{†}\in C$ is referred to as an asymptotic fixed point of S if $\exists \left\{y_n\right\}\subset C$ s.t. $y_n\rightharpoonup y^{†}$ and $(I-S)y_n\to 0$. Let the $\widehat{\mathrm{Fix}}\left(S\right)$ denote the asymptotic fixed point set of S. The terminology of asymptotic fixed points was invented in [11]. A self-mapping S on C is known as being the one of Bregman’s relatively asymptotical nonexpansivity w.r.t. $f_p$ if $\mathrm{Fix}\left(S\right)=\widehat{\mathrm{Fix}}\left(S\right)\ne \varnothing$, and $\exists \left\{{\theta }_n\right\}\subset [0,\infty )$ with both ${\theta }_n\to 0(n\to \infty )$ and

In particular, if ${\theta }_n=0\forall n\ge 1$, then S reduces to the one of Bregman’s relatively nonexpansivity w.r.t. $f_p$, that is, $\mathrm{Fix}\left(S\right)=\widehat{\mathrm{Fix}}\left(S\right)\ne \varnothing$ and $D_{f_p}(y,Sx)\le D_{f_p}(y,x)\forall y\in \mathrm{Fix}\left(S\right),x\in C$. In addition, a mapping $A:C\to E^{*}$ is known as being

(i) monotone on C if $\langle Av-Ay,v-y\rangle \ge 0\forall v,y\in C$;

(ii) pseudo-monotone if $\langle Ay,v-y\rangle \ge 0\Rightarrow \langle Av,v-y\rangle \ge 0\forall v,y\in C$;

(iii)ℓ-Lipschitz continuous or ℓ-Lipschitzian if $\exists \ell >0$ s.t. $\parallel At-Ay\parallel \le \ell \parallel t-y\parallel \forall t,y\in C$;

(iv) of weakly sequential continuity if $\forall \left\{t_n\right\}\subset C$,the relation holds: $t_n\rightharpoonup t\Rightarrow A{t}_n\rightharpoonup At$.

Lemma 2.2 ([31]). Let $r>0$ be a constant and suppose that $f:E\to \mathbf{R}$ is a uniformly convex function on any bounded subset of a Banach space E. Then

$\forall i,j\in \{1,2,...,n\},{\left\{t_k\right\}}_{k=1}^n\subset B(0,r)$ and ${\left\{{\alpha }_k\right\}}_{k=1}^n\subset (0,1)$ for ${\sum }_{k=1}^n{\alpha }_k=1$, with ${\rho }_r$ being the gauge of f with uniform convexity.

Proof. It is easy to show the conclusion.

Lemma 2.3 ([28]). Let $E_i$ be a Banach space for $i=1,2$ and suppose that $A:E_1\to E_2$ is of uniform continuity on any bounded subset of $E_1$ and $D\subset E_1$ is of boundedness. Then $A\left(D\right)\subset E_2$ is of boundedness.

Lemma 2.4 ([10]). Assume $\varnothing \ne C\subset E$ with C being convex and closed, and let $A:C\to E^{*}$ be of both pseudomonotonicity and continuity. Given $y^{†}\in C$. Then $\langle A{y}^{†},y-y^{†}\rangle \ge 0\forall y\in C\iff \langle Ay,y-y^{†}\rangle \ge 0\forall y\in C$.

Lemma 2.5. Suppose that E is a smooth and p-uniformly convex Banach space for $p\ge 2$, where $J_E^p$ is of weakly sequential continuity. Assume $\left\{s_n\right\}\subset E$ and $\varnothing \ne \Omega \subset E$. If ${\omega }_w\left(s_n\right)\subset \Omega$, and $\left\{D_{f_p}(z,s_n)\right\}$ converges for each $z\in \Omega$. Then one has the weak convergence of $\left\{s_n\right\}$ to an element of $\Omega$.

Proof. First, one has $\tau \parallel z-s_n{\parallel }^p\le D_{f_p}(z,s_n)\forall z\in \Omega$ by (2.1). Thus we obtain that $\left\{s_n\right\}$ is of boundedness. Since E is reflexive, we get ${\omega }_w\left(s_n\right)\ne \varnothing$. Also, one claims that $\left\{s_n\right\}$ converges weakly to an element of $\Omega$. Indeed, let $\overline{s},\widehat{s}\in {\omega }_w\left(s_n\right)$ with $\overline{s}\ne \widehat{s}$. Then, $\exists \left\{s_{n_k}\right\}\subset \left\{s_n\right\}$ and $\exists \left\{s_{m_k}\right\}\subset \left\{s_n\right\}$ s.t. $s_{n_k}\rightharpoonup \overline{s}$ and $s_{m_k}\rightharpoonup \widehat{s}$. Since $J_E^p$ is weakly sequentially continuous, we obtain both $J_E^p\left(s_{n_k}\right)\rightharpoonup J_E^p\overline{s}$ and $J_E^p\left(s_{m_k}\right)\rightharpoonup J_E^p\widehat{s}$. Note that $D_{f_p}(\overline{s},\widehat{s})+D_{f_p}(\widehat{s},s_n)=D_{f_p}(\overline{s},s_n)-\langle J_E^p\widehat{s}-J_E^p{s}_n,\overline{s}-\widehat{s}\rangle$. So, utilizing the convergence of the sequences $\left\{D_{f_p}(\overline{s},s_n)\right\}$ and $\left\{D_{f_p}(\widehat{s},s_n)\right\}$, we conclude that which hence yields $\langle J_E^p\overline{s}-J_E^p\widehat{s},\overline{s}-\widehat{s}\rangle =0$. From (2.1) we get $0<\tau \parallel \overline{s}-\widehat{s}{\parallel }^p\le D_{f_p}(\overline{s},\widehat{s})\le \langle J_E^p\overline{s}-J_E^p\widehat{s},\overline{s}-\widehat{s}\rangle =0$. This arrives at a contradiction. Thereupon, this means that $\left\{s_n\right\}$ converges weakly to an element of $\Omega$.

The lemma below was put forth in ${\mathbf{R}}^m$ by [29]. It is easy to verify that the proof of the lemma in Banach spaces is actually the same as in ${\mathbf{R}}^m$.

Lemma 2.6. Assume $\varnothing \ne C\subset E$ with C being convex and closed. Suppose that $K:=\{y\in C:h(y)\le 0\}$ where $h:E\to \mathbf{R}$ is defined on E. If $K\ne \varnothing$ and h is Lipschitz continuous on C with modulus $\theta >0$, then $\theta \mathrm{dist}(x,K)\ge max\left\{h\right(x),0\}\forall x\in C$, where $\mathrm{dist}(x,K)$ stands for the distance of x to K.

Lemma 2.7 ([17]). Let $\left\{{\Gamma }_n\right\}$ be a sequence of real numbers that does not decrease at infinity in the sense that, $\exists \left\{{\Gamma }_{n_k}\right\}\subset \left\{{\Gamma }_n\right\}$ s.t. ${\Gamma }_{n_k}<{\Gamma }_{n_k+1}$ for all k. Assume that ${\left\{\phi \left(n\right)\right\}}_{n\ge n_0}$ is defined as $\phi \left(n\right)=max\{k\le n:{\Gamma }_k<{\Gamma }_{k+1}\}$, with integer $n_0\ge 1$ satisfying $\{k\le n_0:{\Gamma }_k<{\Gamma }_{k+1}\}\ne \varnothing$. Then the following hold:

(i) $\phi \left(n_0\right)\le \phi (n_0+1)\le \cdots$ and $\phi \left(n\right)\to \infty$;

(ii) ${\Gamma }_{\phi \left(n\right)}\le {\Gamma }_{\phi \left(n\right)+1}$ and ${\Gamma }_n\le {\Gamma }_{\phi \left(n\right)+1}\forall n\ge n_0$.

Lemma 2.8 ([7]). Let $\left\{{\sigma }_n\right\}$ be a sequence in $[0,\infty )$ satisfying ${\sigma }_{n+1}\le (1-{\mu }_n){\sigma }_n+{\mu }_n{c}_n\forall n\ge 1$, with $\left\{{\mu }_n\right\}$ and $\left\{c_n\right\}$ being real sequences satisfying the conditions: (i) $\left\{{\mu }_n\right\}\subset [0,1]$ and ${\sum }_{n=1}^{\infty }{\mu }_n=\infty$; and (ii) ${lim\; sup}_{n\to \infty }{c}_n\le 0$ or ${\sum }_{n=1}^{\infty }\left|{\mu }_n{c}_n\right|<\infty$. Then ${\sigma }_n\to 0$ as $n\to \infty$.

Lemma 2.9 ([33]). Let $\left\{a_n\right\},\left\{b_n\right\}$ and $\left\{{\delta }_n\right\}$ be sequences of nonnegative real numbers satisfying the inequality $a_{n+1}\le (1+{\delta }_n)a_n+b_n\forall n\ge 1$. If ${\sum }_{n=1}^{\infty }{\delta }_n<\infty$ and ${\sum }_{n=1}^{\infty }{b}_n<\infty$, then ${lim}_{n\to \infty }{a}_n$ exists.

In this section, let $\varnothing \ne C\subset E$ with C being convex and closed in uniformly smooth and p-uniformly convex Banach space E for $p\ge 2$. We are now in a position to present and analyze our iterative algorithms for approximating a common solution of a pair of VIFPPs, where each algorithm consists of two parts which are of symmetric structure mutually. Assume always that the following conditions hold:

(C1) $S_1,S_2:C\to C$ are the mappings of both uniform continuity and Bregman’s relatively asymptotical nonexpansivity with sequences ${\left\{{\theta }_n\right\}}_{n=1}^{\infty }$ and ${\left\{{\overline{\theta }}_n\right\}}_{n=1}^{\infty }$, respectively.

(C2) For $i=1,2$, $A_i:E\to E^{*}$ is of both uniform continuity and pseudomonotonicity on C, s.t. $\parallel A_i{y}^{†}\parallel \le {lim\; inf}_{n\to \infty }\parallel A_i{y}_n\parallel \forall \left\{y_n\right\}\subset C$ with $y_n\rightharpoonup y^{†}$.

(C3) $\Omega ={\bigcap }_{i=1}^2\mathrm{VI}(C,A_i)\cap \mathrm{Fix}\left(S_i\right)\ne \varnothing$.

Algorithm 3.1. Initialization: Given $x_0,x_1\in C$ arbitrarily and let $ϵ>0,{\mu }_i>0,{\lambda }_i\in (0,\frac{1}{{\mu }_i}),l_i\in (0,1)$ for $i=1,2$. Choose $\left\{{\alpha }_n\right\},\left\{{\beta }_n\right\}\subset (0,1)$ and $\left\{{\ell }_n\right\}\subset (0,\infty )$ s.t. $0<{lim\; inf}_{n\to \infty }{\alpha }_n(1-{\alpha }_n)$, $0<{lim\; inf}_{n\to \infty }{\beta }_n(1-{\beta }_n)$ and ${\sum }_{n=1}^{\infty }{\ell }_n<\infty$. Moreover, assume ${\sum }_{n=1}^{\infty }{\theta }_n<\infty$, and given the iterates $x_{n-1}$ and $x_n(n\ge 1)$, choose ${ϵ}_n$ s.t. $0\le {ϵ}_n\le \overline{{ϵ}_n}$, where

Iterations: Compute $x_{n+1}$ below:

Step 1. Put $g_n=J_{E^{*}}^q((1-{ϵ}_n)J_E^p{S}_1^n{x}_n+{ϵ}_n{J}_E^p(S_1^n{x}_n+x_n-x_{n-1}))$ and calculate $s_n=J_{E^{*}}^q({\beta }_n{J}_E^p{x}_n+(1-{\beta }_n)J_E^p{g}_n)$, $y_n={\Pi }_C\left(J_{E^{*}}^q(J_E^p{s}_n-{\lambda }_1{A}_1{s}_n)\right)$, $e_{{\lambda }_1}\left(s_n\right):=s_n-y_n$ and $t_n=s_n-{\tau }_n{e}_{{\lambda }_1}\left(s_n\right)$, with ${\tau }_n:=l_1^{k_n}$ and $k_n$ being the smallest $k\ge 0$ s.t.

Step 2. Calculate $w_n={\Pi }_{C_n}\left(s_n\right)$, with $C_n:=\{y\in C:h_n\left(y\right)\le 0\}$ and

Step 3. Calculate ${\overline{y}}_n={\Pi }_C\left(J_{E^{*}}^q(J_E^p{w}_n-{\lambda }_2{A}_2{w}_n)\right)$, $e_{{\lambda }_2}\left(w_n\right):=w_n-{\overline{y}}_n$ and ${\overline{t}}_n=w_n-{\overline{\tau }}_n{e}_{{\lambda }_2}\left(w_n\right)$, with ${\overline{\tau }}_n:=l_2^{j_n}$ and $j_n$ being the smallest $j\ge 0$ s.t.

Step 4. Calculate $v_n=J_{E^{*}}^q({\alpha }_n{J}_E^p{w}_n+(1-{\alpha }_n)J_E^p\left(S_2^n{w}_n\right))$ and $x_{n+1}={\Pi }_{{\overline{C}}_n\cap Q_n}\left(w_n\right)$, with $Q_n:=\{y\in C:(1+{\overline{\theta }}_n)D_{f_p}(y,w_n)\ge D_{f_p}(y,v_n)\}$, ${\overline{C}}_n:=\{y\in C:{\overline{h}}_n\left(y\right)\le 0\}$ and

Again set $n:=n+1$ and go to Step 1.

The following lemmas are used in the proofs of our main results in the sequel.

Lemma 3.1. Suppose that $\left\{x_n\right\}$ is the sequence constructed in Algorithm 3.1. Then the following hold: $\frac{1}{{\lambda }_1}{D}_{f_p}(s_n,y_n)\le \langle A_1{s}_n,e_{{\lambda }_1}\left(s_n\right)\rangle$ and $\frac{1}{{\lambda }_2}{D}_{f_p}(w_n,{\overline{y}}_n)\le \langle A_2{w}_n,e_{{\lambda }_2}\left(w_n\right)\rangle$.

Proof. Note that the former inequality is analogous to the latter. So it suffices to show that the latter holds. Indeed, using the definition of ${\overline{y}}_n$ and properties of ${\Pi }_C$, one has

Setting $y=w_n$ in the last inequality, from (2.1) we get which completes the proof.

Lemma 3.2. Linesearch rules (3.1), (3.3) of Armijo-type and sequence $\left\{x_n\right\}$ constructed in Algorithm 3.1 are well defined.

Proof. Observe that the rule (3.1) is analogous to the one (3.3). So it suffices to show that the latter is valid. Using the uniform continuity of $A_2$ on C, from $l_2\in (0,1)$ one gets ${lim}_{j\to \infty }\langle A_2{w}_n-A_2(w_n-l_2^j{e}_{{\lambda }_2}\left(w_n\right)),e_{{\lambda }_2}\left(w_n\right)\rangle =0$. In the case of $e_{{\lambda }_2}\left(w_n\right)=0$, it is explicit that $j_n=0$. In the case of $e_{{\lambda }_2}\left(w_n\right)\ne 0$, we obtain that $\exists j_n\ge 0$ s.t. (3.3) holds.

It is not hard to check that $Q_n$ and ${\overline{C}}_n$ are convex and closed for all n. Let us show that $Q_n\cap {\overline{C}}_n\supset \Omega$. Choose a $z\in \Omega$ arbitrarily. Since $S_2$ is Bregman’s relatively asymptotically nonexpansive mapping, by Lemma 2.2 one gets which hence leads to $z\in Q_n$. Meanwhile, from Lemma 2.4, we get $\langle A_2{\overline{t}}_n,{\overline{t}}_n-z\rangle \ge 0$. Thus,