Strong convergence of a self-adaptive method for the split feasibility problem in Banach spaces

Abstract

In signal processing and image reconstruction, the split feasibility problem (SFP) has been now investigated extensively because of its applications. A classical way to solve the SFP is to use Byrne’s CQ-algorithm. However, this method requires the computation of the norm of the bounded linear operator or the matrix norm in a finite-dimensional space. In this work, we aim to propose an iterative scheme for solving the SFP in the framework of Banach spaces. We also introduce a new way to select the step-size which ensures the convergence of the sequences generated by our scheme. We finally provide examples including its numerical experiments to illustrate the convergence behavior. The main results are new and complements many recent results in the literature.

1 Introduction

Let E and F be two p-uniformly convex real Banach spaces which are also uniformly smooth. Let C and Q be nonempty, closed and convex subsets of E and F, respectively. Let $A:E\rightarrow F$ be a bounded linear operator and $A^{*}:F^{*}\rightarrow E^{*}$ be the adjoint of A which is defined by

$$\begin{aligned} \langle A^{*}\bar{y},x\rangle := \langle \bar{y},Ax\rangle , \quad \forall x \in E,\bar{y}\in F^{*}. \end{aligned}$$

The split feasibility problem (SFP) is to find a point $x\in C$ such that $Ax\in Q$. We denote by $\Omega =C\cap A^{-1}(Q)=\{y\in C: Ay\in Q\}$ the solution set of SFP. Then we have that $\Omega$ is a closed and convex subset of E.

The SFP in finite-dimensional Hilbert spaces was first introduced by Censor and Elfving [15] for modeling inverse problems which arise from phase retrievals, medical image reconstruction and recently in modeling of intensity modulated radiation therapy. The SFP attracts the attention of many authors due to its application in signal processing. Various algorithms and some interesting results have been studied in order to solve it (see, for example [3,4,5, 13, 22, 24,25,26, 33])).

In Hilbert spaces, a classical way to solve the SFP is to employ the CQ-algorithm which was introduced by Byrne [12], which is defined in the following manner:

$$\begin{aligned} x_{n+1}=P_C(x_n-\mu _nA^{*}(I-P_Q)Ax_n), \quad n\ge 1, \end{aligned}$$

(1.1)

where the step-size $\mu _n\in (0,\frac{2}{\Vert A\Vert ^{2}})$ and $P_C, P_Q$ are the metric projections on C and Q, respectively. We note that this algorithm is found to be a gradient-projection method in convex minimization as a spacial case. It was proved that $\{x_n\}$ generated by (1.1) converges weakly to a solution of SFP.

However, it is noted that the operator norm $\Vert A\Vert$ may not be calculated easily in general. To overcome this difficulty, López et al. [22] suggested the following self-adaptive method, which permits step-size $\mu _n$ being selected self-adaptively in such a way:

$$\begin{aligned} \mu _n=\frac{\rho _nf(x_n)}{\Vert \nabla f(x_n)\Vert ^{2}}, \quad n\ge 1, \end{aligned}$$

(1.2)

where $\rho _n\in (0,4)$, $f(x_n)=\frac{1}{2}\Vert (I-P_Q)Ax_n\Vert ^{2}$ and $\nabla f(x_n)=A^{*}(I-P_Q)Ax_n$ for all $n\ge 1$. It was proved that the sequence $\{x_n\}$ defined by (1.2) converges weakly to a solution of SFP.

Also, employing the idea of Halpern’s iteration, López et al. [22] proposed the following iteration method:

$$\begin{aligned} x_{n+1}=\alpha _nu+(1-\alpha _n)P_C(x_n-\mu _n\nabla f(x_n)), \quad n\ge 1, \end{aligned}$$

(1.3)

where u is fixed in C, $\{\alpha _n\}\subset [0,1]$ and the step-size $\mu _n$ is chosen as above. It was shown that $\{x_n\}$ defined by (1.3) converges strongly to a solution of SFP provided $\lim _{n \rightarrow \infty }\alpha _n=0$ and $\sum _{n=1}^{\infty }{{{\alpha }_{n}}}=\infty$. Subsequently, there have been many modifications of the CQ-algorithm and the self-adaptive method established in the recent years (see also [37, 38]).

For solving the SFP, in the framework of p-uniformly convex and uniformly smooth real Banach spaces, Schöpfer [29] proposed the following algorithm: $x_1\in E$ and

$$\begin{aligned} x_{n+1}=\Pi _C{J^{*}_{E}}[J_E(x_n)-\mu _nA^{*}J_F(Ax_n-P_Q(Ax_n))], \quad n\ge 1, \end{aligned}$$

(1.4)

where $\Pi _C$ denotes the Bregman projection and J the duality mapping.Clearly, the above algorithm covers the CQ-algorithm as a special case. It was proved that the sequence $\{x_n\}$ defined by (1.4) converges weakly to a solution of SFP provided the duality mapping J is weak-to-weak continuous and $\mu _n\in \Big (0,(\frac{q}{C_q\Vert A\Vert ^{q}})^{\frac{1}{q-1}}\Big )$ where $\frac{1}{p}+\frac{1}{q}=1$ and $C_{q}$ is the uniform smoothness coefficient of E . See some modifications in [30, 31].

In this work, motivated by the previous works, we introduce a Halpern-type iteration process and prove its strong convergence of the sequence generated by our scheme for solving the SFP without prior knowledge of the operator norm in the framework of Banach spaces. Numerical experiments are included to illustrate the convergence behavior. Our main results complement the results of López et al. [22] (from Hilbert spaces to Banach spaces) and Schöpfer [29]. Moreover, our results improve many other results in the literature. We note that the obtained results seem to be new in this direction.

2 Preliminaries and lemmas

Let E be a real Banach space with norm $\Vert \cdot \Vert$, and $E^*$ denotes the Banach dual of E endowed with the dual norm $\Vert \cdot \Vert _*$. We write $\langle x,j \rangle$ for the value of a functional j in $E^*$ at x in E. As usual, $x_{\nu }\rightarrow x$ and $x_{\nu }\rightharpoonup x$ stand for the norm and weak convergence of a net ${\{x_\nu }\}$ to x in E, respectively. The modulus of convexity $\delta _E : [0,2]\rightarrow [0,1]$ is defined as

$$\begin{aligned} \delta _E(\epsilon )=\inf \left\{ 1-\frac{\Vert x+y\Vert }{2}: \Vert x\Vert =1=\Vert y\Vert , \Vert x-y\Vert \ge \epsilon \right\} . \end{aligned}$$

E is called uniformly convex if $\delta _E(\epsilon )>0$ for any $\epsilon \in (0,2]$ and p-uniformly convex if there is a $C_p>0$ such that $\delta _E(\epsilon )\ge C_p\epsilon ^{p}$ for any $\epsilon \in (0,2]$. The modulus of smoothness $\rho _E (\tau ) : [0,\infty )\rightarrow [0,\infty )$ is defined by

$$\begin{aligned} \rho _E(\tau )= \left\{ \frac{\Vert x+\tau y\Vert +\Vert x-\tau y\Vert }{2}-1: \Vert x\Vert =\Vert y\Vert =1 \right\} . \end{aligned}$$

Then E is called uniformly smooth if $\lim _{\tau \rightarrow 0}\frac{\rho _E(\tau )}{\tau }=0$ and q-uniformly smooth if there is a $C_q>0$ such that $\rho _E(\tau )\le C_q\tau ^{q}$ for any $\tau >0$. Let $1<q\le 2\le p$ with $\frac{1}{p}+\frac{1}{q}=1$. It is known (see, for example, [1, 18]) that E is p-uniformly convex if and only if its dual $E^{*}$ is q-uniformly smooth. Furthermore, Hilbert spaces, $L_p(or \ \ l_p)$ spaces, $1<p<\infty$, and the Sobolev spaces, $W^p_m$, $1<p<\infty$, are q-uniformly smooth. Hilbert spaces are uniformly smooth while

$$\begin{aligned} L_p (or\;\ell _p)\; \mathrm{or}\; W_m^p \hbox { is } \left\{ \begin{array}{ll} &{} p-\hbox {uniformly smooth if }\; 1 < p \le 2\\ &{} 2-\hbox {uniformly smooth if }\; p\ge 2. \end{array} \right. \end{aligned}$$

A continuous strictly increasing function $\varphi : \mathbb {R}^{+}\rightarrow \mathbb {R}^{+}$ is said to be a gauge if

$$\begin{aligned} \varphi (0) = 0, \quad \underset{t\rightarrow +\infty }{\mathop {\lim }}{\varphi (t)} = +\infty . \end{aligned}$$

The mapping $J_{\varphi } : E\rightarrow 2^{E^*}$ defined by

$$\begin{aligned} J_{\varphi }(x) = {\{j\in E^* : \langle x,j \rangle = \Vert x\Vert \Vert j\Vert _* , \Vert j\Vert _* = \varphi (\Vert x\Vert ) }\}, \quad x\in E, \end{aligned}$$

is called the duality mapping with gauge $\varphi$. When $\varphi (t) = t$, the duality mapping $J_\varphi = J$ is the normalized duality mapping. In the case $\varphi (t) = t^{p-1}$ where $p>1$, the duality mapping $J_\varphi = J_p$ is called the generalized duality mapping and it is defined by

$$\begin{aligned} J_{p}(x) = {\{j\in X^* : \langle x,j \rangle = \Vert x\Vert \Vert j\Vert _* , \Vert j\Vert _* = \Vert x\Vert ^{p-1} }\}, \quad x\in E. \end{aligned}$$

Example 2.1

Let $x=(x_{1},x_{2},\ldots )\in \ell _{p}$ $(1<p<\infty )$. Then the generalized duality mapping $J_{p}$ in $\ell _{p}$ is given by

$$\begin{aligned} J_{p}(x)=(|x_{1}|^{p-1}\hbox {sgn}(x_{1}), |x_{2}|^{p-1}\hbox {sgn}(x_{2}),\ldots ). \end{aligned}$$

Example 2.2

Let $f\in L_{p}([\alpha ,\beta ])$ $(1<p<\infty )$. Then the generalized duality mapping $J_{p}$ is given by

$$\begin{aligned} J_{p}(f)(t)=|f(t)|^{p-1}\hbox {sgn}(f(t)). \end{aligned}$$

For a gauge $\varphi$, the function $\Phi : \mathbb {R}^{+}\rightarrow \mathbb {R}^{+}$ defined by

$$\begin{aligned} \Phi (t) = \int _{0}^{t}\varphi (s)\mathrm{d}s \end{aligned}$$

is a continuous convex strictly increasing differentiable function on $\mathbb {R}^{+}$ with $\Phi ^{'}(t) = \varphi (t)$ and $\lim _{t\rightarrow +\infty }\Phi (t)/t = +\infty$. When E is uniformly smooth, the duality mapping $J_{\varphi }$ on E is norm to norm uniformly continuous on bounded subsets of E (see [1, 18]).

We know the following inequality which was proved by Xu [34].

Lemma 2.3

[34] Let $x, y \in E$. If E is q-uniformly smooth, then there exists $C_q>0$ such that

$$\begin{aligned} \Vert x-y\Vert ^{q}\le \Vert x\Vert ^{q}-q\langle y, J^{q}_E(x)\rangle +C_q\Vert y\Vert ^{q}. \end{aligned}$$

Let C be a nonempty, closed and convex subset of E. The metric projection $P_{C} : E\rightarrow C$ is defined by

$$\begin{aligned} P_{C}x = \mathrm{argmin}_{y \in C}\frac{1}{2}\Vert x-y\Vert ^2,\quad x\in E. \end{aligned}$$

It has been employed successfully in optimization, optimal control, approximation theory, and fixed point theory. In the framework of Hilbert spaces, the metric projection $P_C$ is nonexpansive (i.e., $\Vert P_{C}x-P_{C}y\Vert \le \Vert x-y\Vert$ for all x, y in H). However, we note that this is no longer true in the framework of Banach spaces. Let E be a smooth, strictly convex and reflexive Banach space. Let C be a nonempty, closed and convex subset of E, and let $x\in E$ and $z\in C$. Then $z = P_{C}x$ if and only if $\langle z-y, J(x-z)\rangle \ge 0$ for all $y\in C$; see [32].

We next recall the definition of Bregman distance studied in [6]. Let E be a real smooth Banach space. The Bregman distance $D_{\varphi }(x,y)$ between x and y in E is defined by

$$\begin{aligned} D_{\varphi }(x,y)= \Phi (\Vert x\Vert )-\Phi (\Vert y\Vert )-\langle x-y,J_{\varphi }(y)\rangle . \end{aligned}$$

We note that $D_{\varphi }(x,y) \ge 0$ and $D_{\varphi }(x,y)=0$ if and only if $x=y$ (see [21]). It is easily seen by definition that

$$\begin{aligned} D_{\varphi }(x,y)+D_{\varphi }(y,z)-D_{\varphi }(x,z)=\langle x-y,J_{\varphi }(z)-J_{\varphi }(y)\rangle \end{aligned}$$

(2.1)

and

$$\begin{aligned} D_{\varphi }(x,y)+D_{\varphi }(y,x)=\langle x-y,J_{\varphi }(x)-J_{\varphi }(y)\rangle \end{aligned}$$

(2.2)

for all $x,y,z\in E$. In the case $\varphi (t) = t^{p-1}$ where $p>1$, the distance $D_{\varphi }= D_{p}$ is called the p-Lyapunov functional studied in [7] and it is given by

$$\begin{aligned} D_{p}(x,y) = \frac{\Vert x\Vert ^p}{p}+\frac{\Vert y\Vert ^p}{q}-\langle x,J_{p}(y)\rangle , \end{aligned}$$

(2.3)

where $\frac{1}{p}+\frac{1}{q}=1$. Note that

$$\begin{aligned} \phi (x,y) := 2D_{2}(x,y) = \Vert x\Vert ^2 - 2\langle x,Jy\rangle +\Vert y\Vert ^2 \end{aligned}$$

is the Lyapunov functional. See [8, 9, 16]. Let E be a strictly convex, smooth and reflexive Banach space. Following [2], we make use of the function $V_p : E\times E^{*} \rightarrow [0,+\infty )$, which is defined by

$$\begin{aligned} V_p(x,\bar{x}) = \frac{1}{p}\Vert x\Vert ^{p}-\langle x,\bar{x}\rangle +\frac{1}{q}\Vert \bar{x}\Vert ^{q},\quad \forall x \in E, \bar{x}\in E^{*}, \end{aligned}$$

where $\frac{1}{p}+\frac{1}{q}=1$. Then $V_p$ is nonnegative and

$$\begin{aligned} V_p(x,\bar{x}) = D_p(x,J_{p}^{-1}(\bar{x})) \end{aligned}$$

for all $x\in E$ and $\bar{x}\in E^{*}$. For a proper, lower semicontinuous and convex function $f:E\rightarrow (-\infty ,\infty ]$, the subdifferential $\partial f$ of f at $x\in E$ is defined by

$$\begin{aligned} \partial f(x)=\{\bar{x}\in E^*: f(x)+\langle y-x,\bar{x}\rangle \le f(y)\quad \forall y\in E\}. \end{aligned}$$

We see that, for each $x\in E$, the mapping g defined by $g(\bar{x})=V_{p}(x,\bar{x})$ for all $\bar{x}\in E^*$ is a continuous and convex function from $E^*$ into $\mathbb {R}$. So, by the subdifferential of g, we obtain the following inequality:

$$\begin{aligned} V_p(x,\bar{x})+\langle \bar{y}, J_{p}^{-1}(\bar{x})-x\rangle \le V_p(x,\bar{x}+\bar{y}) \end{aligned}$$

(2.4)

for all $x\in E$ and $\bar{x},\bar{y}\in E^*$ (see also [20]). Indeed, we have

$$\begin{aligned} \partial g(\bar{x})= & {} \partial \left( -\langle x,\cdot \rangle +\frac{1}{q}\Vert \cdot \Vert ^{q}\right) (\bar{x})\\= & {} -x+J_{p}^{-1}(\bar{x}) \end{aligned}$$

for all $\bar{x}\in E^*$. So we obtain

$$\begin{aligned} g(\bar{x})+\langle J_{p}^{-1}(\bar{x})-x,\bar{y}\rangle \le g(\bar{x}+\bar{y}), \end{aligned}$$

for all $x\in E$ and $\bar{x},\bar{y}\in E^*$ which consequently implies (2.4).

Proposition 2.4

[10, 21] Let E be a smooth and uniformly convex Banach space. Let ${\{x_n}\}$ and ${\{y_n}\}$ be two sequences in E such that $D_{\varphi }(x_n,y_n)\rightarrow 0$. If ${\{y_n}\}$ is bounded, then $\Vert x_n-y_n\Vert \rightarrow 0.$

Proposition 2.5

[11, 21] Let C be a nonempty, closed and convex subset of a reflexive, strictly convex and smooth Banach space E. Let $x\in E$. Then there exists a unique element $x_0$ in C such that

$$\begin{aligned} D_{\varphi }(x_0,x) =\inf {\{D_{\varphi }(z,x) : z \in C}\}. \end{aligned}$$

In this case, we denote the generalized projection from E onto C by $\Pi _{C}^{\varphi }(x)=x_0$. When $\varphi (t)=t$, we have $\Pi _{C}^{\varphi }$ coincides with the generalized projection studied in [2]. Let $p>1$ and $\varphi (t)=t^{p-1}$. Then $\Pi _{C}^{\varphi }$ becomes the generalized projection with respect to p and is also denoted by $\Pi _C$.

We also know the following results.

Proposition 2.6

[21] Let C be a nonempty, closed and convex subset of a reflexive, strictly convex and smooth Banach space E. Let $x_0 \in C$ and $x \in E$. Then the following assertions are equivalent:

• (a) $x_0 = \Pi _{C}^{\varphi }(x)$;

• (b) $\langle z-x_0 , J_{\varphi }(x_0) - J_{\varphi }(x)\rangle \ge 0, \ \forall z \in C$.

Moreover, we have

$$\begin{aligned} D_{\varphi }(y,\Pi _{C}^{\varphi }(x))+D_{\varphi }(\Pi _{C}^{\varphi }(x),x) \le D_{\varphi }(y,x),\quad \forall y\in C. \end{aligned}$$

We also need the following tools in analysis which will be used in the sequel.

Lemma 2.7

[23] Let $\{s_n\}$ be a sequence of real numbers that does not decrease at infinity in the sense that there exists a subsequence ${\{s_{n_i}}\}$ of ${\{s_n}\}$ which satisfies $s_{n_i} < s_{{n_i}+1}$ for all $i\in \mathbb {N}$. Define the sequence ${\{\tau (n)}\}_{n\ge n_0}$ of integers as follows:

$$\begin{aligned} \tau (n) = \max {\{k\le n : s_{k}<s_{k+1}}\}, \end{aligned}$$

where $n_{0}\in \mathbb {N}$ such that ${\{k\le n_{0} : s_{k}<s_{k+1}}\}\ne \emptyset$. Then, the following hold:

• (i) $\tau (n_0)\le \tau (n_{0}+1)\le \cdots$ and $\tau (n)\rightarrow \infty$;

• (ii) $s_{\tau (n)}\le s_{\tau (n)+1}$ and $s_n\le s_{\tau (n)+1}$, $\forall n\ge n_0$.

Lemma 2.8

[35] Let $\{a_n\}$ be a sequence of nonnegative real numbers satisfying the following relation:

$$\begin{aligned} a_{n+1}\le (1-\alpha _n)a_n+\alpha _n\sigma _n+\gamma _n,\ n\ge 1, \end{aligned}$$

where (i) $\{\alpha _n\}\subset [0,1]$, $\sum _{n=1}^{\infty }\alpha _n=\infty$; (ii) $\limsup _{n\rightarrow \infty }\sigma _n\le 0$; (iii) $\gamma _n\ge 0$, $\sum _{n=1}^{\infty }\gamma _n<\infty$. Then, $a_n\rightarrow 0$ as $n\rightarrow \infty$.

3 Main results

In this section, we prove strong convergence of the sequence generated by our scheme for solving the split feasibility problem in Banach spaces. Throughout this paper, let $1<q\le 2\le p<\infty$ and $\frac{1}{p}+\frac{1}{q}=1$ and denote by $J_{X}^{p}$ and $J_{X^*}^{q}$ the duality mappings of a smooth Banach space X and its dual space, respectively.

Employing the method of proof given by Xu in [36], we prove the following fixed point formulation of SFP in a reflexive, strictly convex and smooth Banach space.

Lemma 3.1

Let E and F be two reflexive, strictly convex and smooth Banach spaces. Let C and Q be nonempty, closed and convex subsets of E and F, respectively. Let $A:E\rightarrow F$ be a bounded linear operator and $A^*:F^*\rightarrow E^*$ be the adjoint of A. Let $x^* \in E$. Then $x^*$ solves the SFP (i.e., $x^* \in C\cap A^{-1}(Q)$) if and only if $x^*$ solves the fixed point equation

$$\begin{aligned} x^*=\Pi _CJ^q_{E^*}\Big [J^p_E(x^*)-\gamma A^*J^p_F(Ax^*-P_Q(Ax^*))\Big ]. \end{aligned}$$

(3.1)

Proof

Suppose $x^*$ solves the SFP. We show that $x^*$ solves (3.1). Now, $x^*$ solves SFP implies that $x^* \in C$ and $Ax^* \in Q$. Therefore,

$$\begin{aligned} Ax^*=P_Q(Ax^*) \Rightarrow Ax^*-P_Q(Ax^*)=0. \end{aligned}$$

Thus,

$$\begin{aligned} J^p_F(Ax^*-P_Q(Ax^*)=0 \end{aligned}$$

and this implies

$$\begin{aligned} \gamma A^*J^p_F(Ax^*-P_Q(Ax^*))=0. \end{aligned}$$

So

$$\begin{aligned} J^q_{E^*}\Big [J^p_E(x^*)-\gamma A^*J^p_F(Ax^*-P_Q(Ax^*))\Big ]=J^q_{E^*}\Big (J^p_E(x^*)\Big )=x^*. \end{aligned}$$

Hence,

$$\begin{aligned} \Pi _CJ^q_{E^*}\Big [J^p_E(x^*)-\gamma A^*J^p_F(Ax^*-P_Q(Ax^*))\Big ]=\Pi _Cx^*=x^*. \end{aligned}$$

Therefore, $x^*$ solves (3.1).

Conversely, assume that $x^*$ solves the fixed point equation (3.1). We next show that $x^* \in C,~~Ax^* \in Q.$ Now, if

$$\begin{aligned} x^*=\Pi _CJ^q_{E^*}\Big [J^p_E(x^*)-\gamma A^*J^p_F(Ax^*-P_Q(Ax^*))\Big ], \end{aligned}$$

then by Proposition 2.6 (b) we have

$$\begin{aligned} \langle J^p_E(x^*)-\gamma A^*J^p_F(Ax^*-P_Q(Ax^*))-J^p_E(x^*), z-x^* \rangle \le 0,\quad \forall z \in C. \end{aligned}$$

That is,

$$\begin{aligned} \langle \gamma A^*J^p_F(Ax^*-P_Q(Ax^*)), z-x^* \rangle \ge 0,\quad \forall z \in C. \end{aligned}$$

Hence,

$$\begin{aligned} \langle J^p_F(Ax^*-P_Q(Ax^*)), Ax^*-Az \rangle \le 0,\quad \forall z \in C. \end{aligned}$$

(3.2)

On the other hand, we have from the characterization of metric projection $P_Q$ that

$$\begin{aligned} \langle J^p_F(Ax^*-P_Q(Ax^*)), v-Ax^*\rangle \le 0,\quad \forall v \in Q. \end{aligned}$$

(3.3)

Adding up (3.2) and (3.3), we obtain

$$\begin{aligned} \langle J^p_F(Ax^*-P_Q(Ax^*)), v-Az\rangle \le 0,\quad \forall v \in Q, z \in C. \end{aligned}$$

Putting $z=x^* \in C$ and $v=P_Q(Ax^*) \in Q$ gives us $Ax^*=P_Q(Ax^*) \in Q$. This completes the proof. $\square$

Remark 3.2

Our Lemma 3.1 extends the fixed point equivalence of SFP given by Xu in [36] from real Hilbert spaces to reflexive, strictly convex and smooth Banach spaces. This fixed point formulation of SFP allows us to construct a fixed point iteration method to solve the SFP in Banach spaces and this iterative method is given below in (3.4).

Theorem 3.3

Let E be a p-uniformly convex and uniformly smooth Banach space and F a reflexive, strictly convex and smooth Banach space. Let C and Q be nonempty, closed and convex subsets of E and F, respectively. Let $A:E\rightarrow F$ be a bounded linear operator and $A^*:F^*\rightarrow E^*$ be the adjoint of A. Suppose that $\Omega =C\cap A^{-1}(Q)\ne \emptyset$. Define a sequence $\{x_n\}$ by $u, x_1\in E$ and

$$\begin{aligned} x_{n+1}=\Pi _C{J^{q}_{{E}^{*}}}\Big [\alpha _n{J^{p}_{E}}(u)+(1-\alpha _n)\Big ({J^{p}_{E}}(x_n)-\rho _n\frac{f^{p-1}(x_n)}{\Vert \nabla f(x_n)\Vert ^{p}}\nabla f(x_n)\Big )\Big ],\quad n\ge 1, \end{aligned}$$

(3.4)

where $f(x_n)=\frac{1}{p}\Vert (I-P_Q)Ax_n\Vert ^{p}$. If $\alpha _n\rightarrow 0$, $\sum _{n=1}^{\infty }\alpha _n=\infty$ and $\{\rho _n\}\subset (0,\infty )$ satisfies

$$\begin{aligned} \underset{n}{\inf }{\rho _n}\Big (pq-C_q{\rho ^{q-1}_{n}}\Big )>0, \end{aligned}$$

(3.5)

then $x_n\rightarrow \Pi _\Omega u$.

Proof

We note that $\nabla f(x)=A^{*}{J^{p}_{F}}(I-P_Q)Ax$ for all $x\in E$ (see Proposition 5.7 in [19]). Set

$$\begin{aligned} y_n={J^{p}_{E}}(x_n)-\rho _{n}\frac{f^{p-1}(x_n)}{\Vert \nabla f(x_n)\Vert ^{p}}\nabla f(x_n) \end{aligned}$$

for all $n\in \mathbb {N}$. We see that $(p-1)q=p$. Then, by Lemma 2.3, we have

$$\begin{aligned} \Vert y_n\Vert ^{q}= & {} \Big \Vert {J^{p}_{E}}(x_n)-\rho _{n}\frac{f^{p-1}(x_n)}{\Vert \nabla f(x_n)\Vert ^{p}}\nabla f(x_n)\Big \Vert ^{q}\nonumber \\\le & {} \Vert x_n\Vert ^{p}-q\rho _{n}\frac{f^{p-1}(x_n)}{\Vert \nabla f(x_n)\Vert ^{p}}\langle x_n, \nabla f(x_n)\rangle +C_{q}{\rho ^{q}_{n}}\frac{f^{(p-1)q}(x_n)}{\Vert \nabla f(x_n)\Vert ^{pq}}\Vert \nabla f(x_n)\Vert ^{q}\nonumber \\= & {} \Vert x_n\Vert ^{p}-q\rho _{n}\frac{f^{p-1}(x_n)}{\Vert \nabla f(x_n)\Vert ^{p}}\langle x_n, \nabla f(x_n)\rangle +C_{q}{\rho ^{q}_{n}}\frac{f^{p}(x_n)}{\Vert \nabla f(x_n)\Vert ^{p}}. \end{aligned}$$

(3.6)

From Proposition 2.6 and (3.6), it follows that, for each $x^*\in \Omega$,

$$\begin{aligned} D_p(x^{*}, x_{n+1})\le & {} D_p(x^{*}, {J^{q}_{{E^{*}}}}(\alpha _n{J^{p}_{E}}(u)+(1-\alpha _n)y_n))\nonumber \\= & {} \frac{\Vert x^{*}\Vert ^{p}}{p}+\frac{1}{q}\Vert \alpha _n{J^{p}_{E}}(u)+(1-\alpha _n)y_n\Vert ^{q}-\alpha _n\langle x^{*}, {J^{p}_{E}}(u)\rangle \nonumber \\&-\,(1-\alpha _n)\langle x^{*}, y_n\rangle \nonumber \\\le & {} \frac{\Vert x^{*}\Vert ^{p}}{p}+\frac{1}{q}(\alpha _n\Vert u\Vert ^{p}+(1-\alpha _n)\Vert y_n\Vert ^{q})-\alpha _n\langle x^{*},{J^{p}_{E}}u\rangle \nonumber \\&-\,(1-\alpha _n)\langle x^{*},{J^{p}_{E}}(x_n)\rangle +(1-\alpha _n)\rho _{n}\frac{f^{p-1}(x_n)}{\Vert \nabla f(x_n)\Vert ^{p}}\langle x^{*},\nabla f(x_n)\rangle \nonumber \\= & {} \alpha _n\Big (\frac{\Vert x^{*}\Vert ^{p}}{p}+\frac{\Vert u\Vert ^{p}}{q}-\langle x^{*},{J^{p}_{E}}u\rangle \Big )\nonumber \\&+\,(1-\alpha _n)\Big (\frac{\Vert x^{*}\Vert ^{p}}{p}+\frac{\Vert y_n\Vert ^{q}}{q}-\langle x^{*},{J^{p}_{E}}(x_n)\rangle \Big )\nonumber \\&+\,(1-\alpha _n)\rho _{n}\frac{f^{p-1}(x_n)}{\Vert \nabla f(x_n)\Vert ^{p}}\langle x^{*},\nabla f(x_n)\rangle \nonumber \\\le & {} \alpha _nD_p(x^{*}, u)\nonumber \\&+\,(1-\alpha _n)\Big (\frac{\Vert x^{*}\Vert ^{p}}{p}+\frac{1}{q}\big (\Vert x_n\Vert ^{p}-q\rho _{n}\frac{f^{p-1}(x_n)}{\Vert \nabla f(x_n)\Vert ^{p}}\langle x_n, \nabla f(x_n)\rangle \nonumber \\&+\,C_{q}{\rho ^{q}_{n}}\frac{f^{p}(x_n)}{\Vert \nabla f(x_n)\Vert ^{p}}\Vert \big )\Big )\nonumber \\&-\,(1-\alpha _n)\langle x^{*}, {J^{p}_{E}}(x_n)\rangle +(1-\alpha _n)\rho _{n}\frac{f^{p-1}(x_n)}{\Vert \nabla f(x_n)\Vert ^{p}}\langle x^{*},\nabla f(x_n)\rangle \nonumber \\= & {} \alpha _nD_p(x^{*},u)+(1-\alpha _n)\Big (\frac{\Vert x^{*}\Vert ^{p}}{p}+\frac{\Vert x_n\Vert ^{p}}{q}-\langle x^{*},{J^{p}_{E}}(x_n)\rangle \Big )\nonumber \\&+\,(1-\alpha _n)\Big (\frac{{C_{q}}{\rho ^{q}_{n}}}{q}\frac{f^{p}(x_n)}{\Vert \nabla f(x_n)\Vert ^{p}}+\rho _n\frac{f^{p-1}(x_n)}{\Vert \nabla f(x_n)\Vert ^{p}}\langle x^{*}-x_n, \nabla f(x_n)\rangle \Big )\nonumber \\= & {} \alpha _nD_p(x^{*},u)+(1-\alpha _n)D_p(x^{*},x_n)\nonumber \\&+\,(1-\alpha _n)\Big (\frac{{C_{q}}{\rho ^{q}_{n}}}{q}\frac{f^{p}(x_n)}{\Vert \nabla f(x_n)\Vert ^{p}}+\rho _n\frac{f^{p-1}(x_n)}{\Vert \nabla f(x_n)\Vert ^{p}}\langle x^{*}-x_n, \nabla f(x_n)\rangle \Big ).\nonumber \\ \end{aligned}$$

(3.7)

On the other hand, we see that

$$\begin{aligned} \langle \nabla f(x_n), x^{*}-x_n\rangle= & {} \langle A^{*}{J^{p}_{E}}(I-P_Q)Ax_n, x^{*}-x_n\rangle \nonumber \\= & {} \langle {J^{p}_{E}}(I-P_Q)Ax_n, Ax^{*}-Ax_n\rangle \nonumber \\= & {} \langle {J^{p}_{E}}(I-P_Q)Ax_n, P_{Q}Ax_{n}-Ax_n\rangle \nonumber \\&+\,\langle {J^{p}_{E}}(I-P_Q)Ax_n, Ax^{*}-P_{Q}Ax_{n}\rangle \nonumber \\\le & {} -\Vert (I-P_Q)Ax_n\Vert ^{p}=-pf(x_n). \end{aligned}$$

(3.8)

Using (3.7) and (3.8), we obtain

$$\begin{aligned} D_p(x^{*},x_{n+1})\le & {} \alpha _nD_p(x^{*},u)+(1-\alpha _n)D_p(x^{*},x_n)\\&+\,(1-\alpha _n)\Big (\frac{{C_q}{\rho ^{q}_{n}}}{q}-\rho _{n}p\Big )\frac{f^{p}(x_n)}{\Vert \nabla f(x_n)\Vert ^{p}}, \end{aligned}$$

which implies, by (3.5)

$$\begin{aligned} D_p(x^{*},x_{n+1})\le \alpha _nD_p(x^{*},u)+(1-\alpha _n)D_p(x^{*},x_n). \end{aligned}$$

Hence, by induction, $\{D_p(x^{*},x_n)\}$ is bounded. So we can conclude that $\{x_n\}$ is bounded. Set $v_{n}={J^{q}_{{E}^{*}}}\Big [\alpha _n{J^{p}_{E}}(u)+(1-\alpha _n)\Big ({J^{p}_{E}}(x_n)-\rho _n\frac{f^{p-1}(x_n)}{\Vert \nabla f(x_n)\Vert ^{p}}\nabla f(x_n)\Big )\Big ]$ for all $n\in \mathbb {N}$. Using Proposition 2.6 and (2.4), we next consider the following estimation:

$$\begin{aligned} D_p(x^{*},x_{n+1})= & {} D_p(x^{*},\Pi _Cv_n)\le D_p(x^{*},v_n)-D_p(v_n, \Pi _Cv_n)\nonumber \\= & {} D_p(x^{*}, {J^{q}_{E^{*}}}(\alpha _n{J^{p}_{E}}(u)+(1-\alpha _n)y_n))-D_p(v_n,\Pi _Cv_n)\nonumber \\= & {} V_p(x^{*},\alpha _n{J^{p}_{E}}(u)+(1-\alpha _n)y_n)-D_p(v_n,\Pi _Cv_n)\nonumber \\\le & {} V_p(x^{*},\alpha _n{J^{p}_{E}}(u)+(1-\alpha _n)y_n-\alpha _n({J^{p}_{E}}(u)-{J^{p}_{E}}(x^{*}))\nonumber \\&+\,\alpha _n\langle {J^{p}_{E}}(u)-{J^{p}_{E}}(x^{*}),v_n-x^{*}\rangle -D_p(v_n,\Pi _Cv_n)\nonumber \\= & {} V_p(x^{*},\alpha _n{J^{p}_{E}}(x^{*})+(1-\alpha _n)y_n)\nonumber \\&+\,\alpha _n\langle {J^{p}_{E}}(u)-{J^{p}_{E}}(x^{*}),v_n-x^{*}\rangle -D_p(v_n,\Pi _Cv_n)\nonumber \\\le & {} (1-\alpha _n)V_p(x^{*},y_n)+\alpha _n\langle {J^{p}_{E}}(u)-{J^{p}_{E}}(x^{*}),v_n-x^{*}\rangle \nonumber \\&-\,D_p(v_n,\Pi _Cv_n)\nonumber \\= & {} (1\!-\!\alpha _n)D_p(x^{*},{J^{q}_{E^{*}}}(y_n))\!+\!\alpha _n\langle {J^{p}_{E}}(u)\!-\!{J^{p}_{E}}(x^{*}),v_n-x^{*}\rangle \nonumber \\&-\,D_p(v_n,\Pi _Cv_n)\nonumber \\= & {} (1-\alpha _n)\Big (\frac{\Vert x^{*}\Vert ^{p}}{p}+\frac{\Vert y_n\Vert ^{q}}{q}-\langle x^{*},y_n\rangle \Big )\nonumber \\&+\,\alpha _n\langle {J^{p}_{E}}(u)-{J^{p}_{E}}(x^{*}),v_n-x^{*}\rangle -D_p(v_n,\Pi _Cv_n)\nonumber \\\le & {} (1-\alpha _n)D_p(x^{*},x_n)+(1-\alpha _n)\rho _n\Big (\frac{{C_q}{\rho ^{q-1}_{n}}}{q}-p\Big )\frac{f^{p}(x_n)}{\Vert \nabla f(x_n)\Vert ^{p}}\nonumber \\&+\,\alpha _n\langle {J^{p}_{E}}(u)-{J^{p}_{E}}(x^{*}),v_n-x^{*}\rangle -D_p(v_n,\Pi _Cv_n). \end{aligned}$$

(3.9)

Let $s_n=D_p(\Pi _\Omega u,x_n)$ for all $n\in \mathbb {N}$. Then, by (3.9), we have

$$\begin{aligned} s_{n+1}\le & {} (1-\alpha _n)s_n+(1-\alpha _n)\rho _n\Big (\frac{{C_q}{\rho ^{q-1}_{n}}}{q}-p\Big )\frac{f^{p}(x_n)}{\Vert \nabla f(x_n)\Vert ^{p}}\nonumber \\&+\,\alpha _n\langle {J^{p}_{E}}(u)-{J^{p}_{E}}(\Pi _\Omega u),v_n-\Pi _\Omega u\rangle -D_p(v_n,\Pi _Cv_n). \end{aligned}$$

(3.10)

Case 1 If $\{s_n\}$ is decreasing, then $(1-\alpha _n)\rho _n\big (\frac{{C_q}{\rho ^{q-1}_{n}}}{q}-p\big )\frac{f^{p}(x_n)}{\Vert \nabla f(x_n)\Vert ^{p}}\rightarrow 0$ and $D_p(v_n,\Pi _Cv_n)\rightarrow 0.$ It follows that $f(x_n)\rightarrow 0$ by (3.5). Hence $\Vert Ax_n-P_QAx_n\Vert \rightarrow 0$ and $\Vert v_n-\Pi _Cv_n\Vert \rightarrow 0$ by Lemma 2.4. We also see that

$$\begin{aligned} \Vert {J^{p}_{E}}(v_n)-{J^{p}_{E}}(x_n)\Vert\le & {} \alpha _n\Vert {J^{p}_{E}}(u)-{J^{p}_{E}}(x_n)\Vert \\&+\,(1-\alpha _n)\Big \Vert \rho _n\frac{f^{p-1}(x_n)}{\Vert \nabla f(x_n)\Vert ^{p}}\nabla f(x_n)\Big \Vert \\\rightarrow & {} 0. \end{aligned}$$

Since $J^{q}_{E^*}$ is norm to norm uniformly continuous on bounded subsets of $E^*$, $\Vert v_n-x_n\Vert \rightarrow 0$ as $n\rightarrow \infty$. Since $\{v_n\}$ is bounded, there exists a subsequence $\{v_{n_i}\}$ of $\{v_n\}$ such that $v_{n_i}\rightharpoonup z$ in $\omega _w(v_n)$. Also, we have a subsequence $\{x_{n_i}\}$ of $\{x_n\}$ such that $x_{n_i}\rightharpoonup z\in \omega _w(x_n).$ From (2.2) we obtain

$$\begin{aligned} D_p(z,\Pi _{C}z)\le & {} \langle {J^{p}_{E}}(z)-{J^{p}_{E}}(\Pi _{C}z), z-\Pi _{C}z\rangle \nonumber \\= & {} \langle {J^{p}_{E}}(z)-{J^{p}_{E}}(\Pi _{C}z), z-v_{n_i}\rangle +\langle {J^{p}_{E}}(z)-{J^{p}_{E}}(\Pi _{C}z), v_{n_i}-\Pi _{C}v_{n_i}\rangle \nonumber \\&+\,\langle {J^{p}_{E}}(z)-{J^{p}_{E}}(\Pi _{C}z), \Pi _Cv_{n_i}-\Pi _{C}z\rangle \nonumber \\\le & {} \langle {J^{p}_{E}}(z)-{J^{p}_{E}}(\Pi _{C}z), z-v_{n_i}\rangle +\langle {J^{p}_{E}}(z)-{J^{p}_{E}}(\Pi _{C}z), v_{n_i}-\Pi _{C}v_{n_i}\rangle \nonumber \\\rightarrow & {} 0. \end{aligned}$$

(3.11)

It follows that $z\in C$. Since $x_{n_i}\rightharpoonup z$, $Ax_{n_i}\rightharpoonup Az$ and $\Vert Ax_{n_i}-P_QAx_{n_i}\Vert \rightarrow 0$ as $i\rightarrow \infty$, we have

$$\begin{aligned} \Vert Az-P_QAz\Vert ^{p}= & {} \langle {J^{p}_{F}}(Az-P_QAz), Az-P_QAz\rangle \nonumber \\= & {} \langle {J^{p}_{F}}(Az-P_QAz), Az-Ax_{n_i}\rangle \nonumber \\&+\,\langle {J^{p}_{F}}(Az-P_QAz), Ax_{n_i}-P_QAx_{n_i}\rangle \nonumber \\&+\,\langle {J^{p}_{F}}(Az-P_QAz), P_QAx_{n_i}-P_QAz\rangle \nonumber \\= & {} \langle {J^{p}_{F}}(Az-P_QAz), Az-Ax_{n_i}\rangle \nonumber \\&+\,\langle {J^{p}_{F}}(Az-P_QAz), Ax_{n_i}-P_QAx_{n_i}\rangle \nonumber \\\rightarrow & {} 0, \end{aligned}$$

as $i\rightarrow \infty$. Then we obtain $Az\in Q$ and therefore $z\in \Omega =C\cap A^{-1}Q$. We next show that

$$\begin{aligned} \underset{n\rightarrow \infty }{\limsup }\langle {J^{p}_{E}}(u)-{J^{p}_{E}}(\Pi _\Omega u), v_n-\Pi _\Omega (u)\rangle \le 0. \end{aligned}$$

To this end, we choose a subsequence $\{v_{n_i}\}$ of $\{v_n\}$ such that

$$\begin{aligned} \underset{n\rightarrow \infty }{\limsup }\langle {J^{p}_{E}}(u)-{J^{p}_{E}}(\Pi _\Omega u), v_n-\Pi _\Omega (u)\rangle =\underset{i\rightarrow \infty }{\lim }\langle {J^{p}_{E}}(u)-{J^{p}_{E}}(\Pi _\Omega u), v_{n_i}-\Pi _\Omega u\rangle . \end{aligned}$$

Since $v_{n_i}\rightharpoonup z\in \Omega$, it follows that

$$\begin{aligned} \underset{n\rightarrow \infty }{\limsup }\langle {J^{p}_{E}}(u)-{J^{p}_{E}}(\Pi _\Omega u), v_n-\Pi _\Omega u\rangle \le 0. \end{aligned}$$

Using Lemma 2.8, we conclude that $s_n\rightarrow 0$, that is, $D_p(\Pi _\Omega u,x_n)\rightarrow 0$ as $n\rightarrow \infty$. So, by Lemma 2.4, we obtain $x_n\rightarrow \Pi _\Omega u$ as $n\rightarrow \infty$.

Case 2 Assume that $\{s_n\}$ is not monotonically decreasing and let $\tau :\mathbb {N}\rightarrow \mathbb {N}$ be a mapping for all $n\ge n_0$ (for some $n_0$ large enough) by

$$\begin{aligned} \tau (n)=\max \{k\in \mathbb {N} : k\le n, s_k\le s_{k+1}\}. \end{aligned}$$

Clearly, $\tau (n)$ is a nondecreasing sequence such that $\tau (n)\rightarrow \infty$ as $n\rightarrow \infty$ and $0\le s_{\tau (n)}\le s_{\tau (n)+1}, \forall n\ge n_0$. So from (3.10) we can show that $\Vert Ax_{\tau (n)}-P_QAx_{\tau (n)}\Vert \rightarrow 0$ and $\Vert v_{\tau (n)}-\Pi _Cv_{\tau (n)}\Vert \rightarrow 0$ as $n\rightarrow \infty$. By the similar argument as above in Case 1, we can also show that $\Vert v_{\tau (n)}-x_{\tau (n)}\Vert \rightarrow 0$ as $n\rightarrow \infty$ and

$$\begin{aligned} \underset{n\rightarrow \infty }{\limsup }\langle {J^{p}_{E}}(u)-{J^{p}_{E}}(\Pi _\Omega u), v_{\tau (n)}-\Pi _\Omega u\rangle \le 0. \end{aligned}$$

Also, from (3.10), we see that

$$\begin{aligned} s_{\tau (n)}\le \langle {J^{p}_{E}}(u)-{J^{p}_{E}}(\Pi _\Omega u), v_{\tau (n)}-\Pi _\Omega u\rangle . \end{aligned}$$

It follows that $\underset{n\rightarrow \infty }{\limsup }s_{\tau (n)}\le 0$ and thus $\underset{n\rightarrow \infty }{\lim }s_{\tau (n)}=0$. We next show that $\underset{n\rightarrow \infty }{\lim }s_{\tau (n)+1}=0$. To show this, it suffices to prove that $\Vert x_{\tau (n)+1}-x_{\tau (n)}\Vert \rightarrow 0$ as $n\rightarrow \infty$. Indeed, we observe that

$$\begin{aligned} \Vert x_{\tau (n)+1}-x_{\tau (n)}\Vert\le & {} \Vert x_{\tau (n)+1}-v_{\tau (n)}\Vert +\Vert v_{\tau (n)}-x_{\tau (n)}\Vert \nonumber \\= & {} \Vert \Pi _Cv_{\tau (n)}-v_{\tau (n)}\Vert +\Vert v_{\tau (n)}-x_{\tau (n)}\Vert \nonumber \\\rightarrow & {} 0. \end{aligned}$$

From (2.1), it follows that

$$\begin{aligned}&D_p(\Pi _\Omega u, x_{\tau (n)+1})+D_p(x_{\tau (n)+1}, x_{\tau (n)})-D_p(\Pi _\Omega u, x_{\tau (n)})\\&\quad =\langle \Pi _\Omega u-x_{\tau (n)+1},{J^{p}_{E}}(x_{\tau (n)})-{J^{p}_{E}}(x_{\tau (n)+1})\rangle . \end{aligned}$$

Hence

$$\begin{aligned} s_{\tau (n)+1}=D_p(\Pi _\Omega u, x_{\tau (n)+1})\le & {} D_p(\Pi _\Omega u, x_{\tau (n)})\\&+\,\langle \Pi _\Omega u-x_{\tau (n)+1},{J^{p}_{E}}(x_{\tau (n)})-{J^{p}_{E}}(x_{\tau (n)+1})\rangle \nonumber \\\rightarrow & {} 0. \end{aligned}$$

Thus, by Lemma 2.7, we obtain $s_n\le s_{\tau (n)+1}$, which implies that $\lim _{n\rightarrow \infty } s_n=0$. This shows that $x_n\rightarrow \Pi _\Omega u$ as $n\rightarrow \infty$. We thus complete the proof. $\square$

We consequently obtain the following result in Hilbert spaces which was studied by Yao et al. [37].

Theorem 3.4

(Yao et al. [37]) Let $H_1$ and $H_2$ be Hilbert spaces. Let C and Q be nonempty, closed and convex subsets of $H_1$ and $H_2$, respectively. Let $A:H_1\rightarrow H_2$ be a bounded linear operator and $A^*:H_2\rightarrow H_1$ be the adjoint of A. Suppose that $\Omega =C\cap A^{-1}(Q)\ne \emptyset$. Define a sequence $\{x_n\}$ by $u, x_1\in H_1$ and

$$\begin{aligned} x_{n+1}=P_{C}\Big [\alpha _{n}u+(1-\alpha _n)\Big (x_n-\rho _n\frac{f(x_n)}{\Vert \nabla f(x_n)\Vert ^{2}}\nabla f(x_n)\Big )\Big ],\quad n\ge 1, \end{aligned}$$

(3.12)

where $f(x_n)=\frac{1}{2}\Vert (I-P_Q)Ax_n\Vert ^{2}$. If $\alpha _n\rightarrow 0$, $\sum _{n=1}^{\infty }\alpha _n=\infty$ and $\{\rho _n\}\subset (0,\infty )$ satisfies

$$\begin{aligned} \underset{n}{\inf }{\rho _n}(4-\rho _{n})>0, \end{aligned}$$

(3.13)

then $x_n\rightarrow P_\Omega u$.

4 Applications

In this section, we apply our result on SFP to split equality problem (SEP) introduced by Moudafi [27, 28] in p-uniformly convex real Banach spaces which are also uniformly smooth. As far as we know, this is the first time SEP is being studied in higher Banach spaces outside real Hilbert spaces which has been studied by numerous authors in the literature.

Our interest here is to convert an SEP to SFP in p-uniformly convex real Banach spaces which are also uniformly smooth. To do this, we need the following important lemma.

Lemma 4.1

[17, 34]

• (i) A real Banach space X is p-uniformly convex if and only if there exists $c_{1}>0$ such that

  $$\begin{aligned} \Vert x+y\Vert ^{p}\ge \Vert x\Vert ^{p}+p\langle y,J^p_{X}(x)\rangle +c_{1}\Vert y\Vert ^{p},\quad \forall x,y\in X. \end{aligned}$$

• (ii) A real Banach space X is uniformly smooth if and only if there exists a continuous, strictly increasing and convex function

  $$\begin{aligned} g:\mathbb {R}^+ \rightarrow \mathbb {R}^+, g(0)=0 \end{aligned}$$

such that for all $x, y \in B_r:=\{x \in X:\Vert x\Vert \le r\}$, we have

$$\begin{aligned} \langle x-y,J^p_X(x)-J^p_X(y)\rangle \le g(\Vert x-y\Vert ). \end{aligned}$$

We now give the following lemma which is an analogue of Lemma 4.1 in product spaces. Furthermore, this lemma will be crucial in our application.

Lemma 4.2

For $p>1$, let X and Y be real p-uniformly convex Banach spaces which are also uniformly smooth. Let $E = X\times Y$ with norm

$$\begin{aligned} \Vert z\Vert _{E} = (\Vert u\Vert ^{p}_{X}+\Vert v\Vert ^{p}_{Y})^{\frac{1}{p}} \end{aligned}$$

for every arbitrarily $z = (u,v)\in E$. Let $E^{*} = X^{*}\times Y^{*}$ denote the dual space of E. For each $x = (x_{1},x_{2})\in E$, define the mapping $J^{p}_{E} : E \rightarrow E^*$ by

$$\begin{aligned} J^{p}_{E}(x) = J^{p}_{E}(x_{1},x_{2}) = (J^{p}_{X}(x_1),J^{p}_{Y}(x_2)), \end{aligned}$$

and for arbitrarily $z_{1} = (u_{1},v_{1})$, $z_{2} = (u_{2},v_{2})$ in E, the duality pair $\langle \cdot ,\cdot \rangle$ is given by

$$\begin{aligned} \langle z_{1},J^{p}_{E}(z_2)\rangle = \langle u_{1},J^{p}_{X}(u_{2})\rangle +\langle v_{1},J^{p}_{Y}(v_{2})\rangle . \end{aligned}$$

Then we have

• (a) $J^{p}_{E}$ is a duality mapping on E;

• (b) E is p-uniformly convex real Banach space which is also uniformly smooth.

Proof

(a) Observe that $J^{p}_{E}$ is single-valued if and only if E is smooth. For arbitrarily $x = (x_{1},x_{2})\in E$, let $J^{p}_{E}(x) = J^{p}_{E}(x_{1},x_{2}) = \psi _{p}$. Then $\psi _{p} = (J^{p}_{X}(x_1),J^{p}_{Y}(x_2))\in E^*$. Observe that for $q>1$ with $\frac{1}{p}+\frac{1}{q} = 1$,

$$\begin{aligned} \Vert \psi _{p}\Vert _{E^*}= & {} \Vert (J^{p}_{X}(x_1),J^{p}_{Y}(x_2))\Vert ^{\frac{1}{q}}\nonumber \\= & {} (\Vert J^p_{X}(x_1)\Vert ^{q}_{X^*}+\Vert J^p_{Y}(x_2)\Vert ^{q}_{Y^*})^{\frac{1}{q}}\nonumber \\= & {} (\Vert x_1\Vert ^{(p-1)q}_{X}+\Vert x_2\Vert ^{(p-1)q}_{Y})^{\frac{1}{q}}\nonumber \\= & {} (\Vert x_1\Vert ^{p}_{X}+\Vert x_2\Vert ^{p}_{Y})^{\frac{p-1}{p}}\nonumber \\= & {} \Vert x\Vert ^{p-1}_{E}. \end{aligned}$$

Hence $\Vert \psi _{p}\Vert _{E^*} = \Vert x\Vert ^{p-1}_{E}$. Also, we have

$$\begin{aligned} \langle x, \psi _{p}\rangle= & {} \langle (x_{1},x_{2}), (J^{p}_{X}(x_1),J^{p}_{Y}(x_2))\rangle \nonumber \\= & {} \langle (x_{1},J^{p}_{X}(x_1)), (x_{2},J^{p}_{Y}(x_2))\rangle \nonumber \\= & {} \Vert x_{1}\Vert ^{p}_{X}+\Vert x_{2}\Vert ^{p}_{Y}\nonumber \\= & {} (\Vert x_{1}\Vert _{X}^{p}+\Vert x_{2}\Vert _{Y}^{p})^{\frac{1}{p}} (\Vert x_{1}\Vert _{X}^{p}+\Vert x_{2}\Vert _{Y}^{p})^{\frac{p-1}{p}}\nonumber \\= & {} \Vert x\Vert _{E^*} \cdot \Vert \psi _{p}\Vert _{E^*}\nonumber \\= & {} \Vert x\Vert ^{p}_{E}. \end{aligned}$$

Hence $J^{p}_{E}$ is a single-valued normalized duality mapping on E.

(b) Let $x = (x_{1},x_{2}),y = (y_{1},y_{2})\in E$. Then

$$\begin{aligned} \Vert x+y\Vert ^{p}_{E}= & {} \Vert (x_{1}+y_{1},x_{2}+y_{2})\Vert ^{p}_{E}\nonumber \\= & {} \Vert x_{1}+y_{1}\Vert ^{p}_{X}+\Vert x_{2}+y_{2}\Vert ^{p}_{Y}\nonumber \\\ge & {} \Vert x_{1}\Vert ^{p}_{X}+\Vert x_{2}\Vert ^{p}_{Y}+c(\Vert y_{1}\Vert ^{p}_{X}+\Vert y_{2}\Vert ^{p}_{Y})\nonumber \\&+\,p\{\langle y_{1},J^{p}_{X}(x_1)\rangle + \langle y_{2},J^{p}_{Y}(x_2)\rangle \} \end{aligned}$$

for some $c>0$. Hence

$$\begin{aligned} \Vert x+y\Vert ^{p}_{E} \ge \Vert x\Vert ^{p}_{E}+p\langle y,J^{p}_{E}(x)\rangle +c\Vert y\Vert ^{p}_{E}. \end{aligned}$$

Therefore, E is p-uniformly convex from Lemma 4.1 (i) . We next show that E is uniformly smooth. Now,

$$\begin{aligned} \langle x-y, J^p_E(x)-J^p_E(y)\rangle= & {} \langle (x_1-y_1,x_2-y_2),(J^p_E(x_1)-J^p_E(y_1),J^p_E(x_2)-J^p_E(y_2)) \rangle \\= & {} \langle x_1-y_1,J^p_E(x_1)-J^p_E(y_1)\rangle + \langle x_2-y_2,J^p_E(x_2)-J^p_E(y_2)\rangle \\\le & {} g_1(\Vert x_1-y_1\Vert )+g_2(\Vert x_2-y_2\Vert ), \end{aligned}$$

where $g_1, g_2$ are strictly increasing continuous and convex functions on $\mathbb {R}^+$ and $g_1(0)=g_2(0) =0$. Therefore,

$$\begin{aligned} \langle x-y, J^p_E(x)-J^p_E(y)\rangle \le g(\Vert x-y\Vert ), \end{aligned}$$

where $g(\Vert x-y\Vert ) = g_1(\Vert x_1-y_1\Vert )+g_2(\Vert x_2-y_2\Vert )$. Hence the result follows from Lemma 4.1 (ii) that E is uniformly smooth. $\square$

Let $E_{1},E_{2}$ and $E_{3}$ be real p-uniformly convex which are also uniformly smooth Banach spaces. Suppose $C_{1}\subseteq E_{1}$ and $Q_{1}\subseteq E_{2}$ are nonempty closed and convex sets. Let $A : E_{1}\rightarrow E_{2}$ and $B : E_{2}\rightarrow E_{3}$ be bounded linear operators. The split equality problem (SEP) [27, 28] is defined by

$$\begin{aligned} \hbox {find } \ x \in C_{1} \quad \hbox {and} \quad y \in Q_{1} \ \hbox { such that }\ Ax = By. \end{aligned}$$

(4.1)

Our interest here is to transform (4.1) into the SFP. Now suppose $E = E_{1}\times E_{2}$, $F = E_{3}\times E_{3}$, $C = C_{1}\times Q_{1} \subset E$ and $Q = \{(z,w)\in F : z = w\}$. We know from Lemma 4.2 that E and F are p-uniformly convex real Banach spaces which are also uniformly smooth.

Define an operator $T : E\rightarrow F$ by

$$\begin{aligned} T(x,y) = (Ax,By) \end{aligned}$$

for all $(x,y)\in E$. Since if $z_{1} = (x_{1},y_{1}), z_{2} = (x_{2},y_{2})$, then

$$\begin{aligned} T(\alpha z_{1}+\beta z_{2})= & {} T[(\alpha x_{1},\alpha y_{1})+(\beta x_{2},\beta y_{2})]\nonumber \\= & {} T(\alpha x_{1}+\beta x_{2},\alpha y_{1}+\beta y_{2})\nonumber \\= & {} (A(\alpha x_{1}+\beta x_{2}),B(\alpha y_{1}+\beta y_{2}))\nonumber \\= & {} (\alpha Ax_{1}+\beta Ax_{2},\alpha By_{1}+\beta By_{2})\nonumber \\= & {} \alpha ( Ax_{1},By_{1}) +\beta ( Ax_{2},By_{2})\nonumber \\= & {} \alpha Tz_{1} +\beta Tz_{2}, \end{aligned}$$

which shows that T is linear. Also, it is easy to see that T is bounded from the boundedness of A and B. Set $S = \{(x,y)\in C : T(x,y)\in Q\}$. Hence $(x,y)\in E$ solves (4.1) (using Lemma 3.1) if and only if

$$\begin{aligned} (x,y) = \Pi _{C}J^{q}_{E^*}(J^{p}_{E}(x,y)+\gamma T^{*}J^{p}_{F}(P_{Q}-I)T(x,y)), \end{aligned}$$

where

$$\begin{aligned} P_{Q}(z,w)= & {} (\frac{z+w}{2},\frac{z+w}{2}) , (z,w)\in F ,\\ T^{*}J^{p}_{F}(z,w)= & {} (A^{*}J^{p}_{E_3}(z), B^{*}J^{p}_{E_3}(w)) ,\\ J^{p}_{E}(x,y)= & {} (J^{p}_{E_1}(x), J^{p}_{E_2}(y)) \quad \hbox {for all }\; (x,y)\in E , \end{aligned}$$

and

$$\begin{aligned} \Pi _{C}J^{q}_{E^*}(x,y) = (\Pi _{C_1}J^{q}_{E^{*}_{1}}(x), P_{Q_2}J^{q}_{E^{*}_{2}}(y)) \quad \hbox {for all } \; (x,y)\in E^*. \end{aligned}$$

Using the fixed point formulation (4.2), we construct an iterative method for solving SEP (4.1) and obtain the following convergence theorem for solving SEP (4.1) by applying the result of Theorem 3.3.

Theorem 4.3

Let $E_{1}$ and $E_{2}$ be two real p-uniformly convex which are also uniformly smooth Banach spaces and $E_{3}$ a reflexive, strictly convex andsmooth Banach space. Suppose $C_{1}\subseteq E_{1}$ and $Q_{1}\subseteq E_{2}$ are nonempty closed and convex sets. Let $A : E_{1}\rightarrow E_{3}$ and $B : E_{2}\rightarrow E_{3}$ be bounded linear operators. Let $A^*:E_3^*\rightarrow E_1^*$ and $B^*:E_3^*\rightarrow E_2^*$ be the adjoints of A and B respectively. Suppose that $\Omega$ denotes the set of solutions of SEP (4.1) and $\Omega \ne \emptyset$. Define a sequence $\{(x_n,y_n)\}$ by $x_1,y_1\in E_1$ and

$$\begin{aligned} \left\{ \!\!\! \begin{array}{llll} &{}x_{n+1}=\Pi _{C_{1}}{J^{q}_{{E_1}^{*}}}\Big [\alpha _n{J^{p}_{E_1}}(x_1)+(1-\alpha _n)\Big ({J^{p}_{E_1}}(x_n)+ \frac{\rho _n}{2}A^*J^p_{E_3}(By_n\!-\!Ax_n)\Big )\Big ],\quad \! n\ge 1,\\ &{}y_{n+1}=P_{Q_1}{J^{q}_{{E_2}^{*}}}\Big [\alpha _n{J^{p}_{E_2}}(y_1)+(1-\alpha _n)\Big ({J^{p}_{E_2}}(y_n)+ \frac{\rho _n}{2}B^*J^p_{E_3}(Ax_n-By_n)\Big )\Big ],\!\quad n\ge 1. \end{array} \right. \nonumber \\ \end{aligned}$$

(4.2)

If $\alpha _n\rightarrow 0$, $\sum _{n=1}^{\infty }\alpha _n=\infty$ and $\{\rho _n\}\subset (0,\infty )$ satisfies $\underset{n}{\inf }{\rho _n}\Big (pq-C_q{\rho ^{q-1}_{n}}\Big )>0,$ then $\{(x_n,y_n)\}$ converges strongly to $(x^*,y^*)$ which simultaneously solves SEP (4.1) and is the nearest point to the initial guess $(x_1,y_1)$.

5 Examples and numerical results

In this section, we present some numerical examples to illustrate the performance of our algorithm. All codes were written in Matlab 2012b and run on Hp $i-5$ Dual-Core 8.00 GB (7.78 GB usable) RAM laptop.

Example 5.1

We consider the problem in $(L_2([\alpha ,\beta ]),||\cdot ||_{L_2})$ and also give numerical examples using Theorem 3.3. Now take

$$\begin{aligned} C:=\{x \in L_2([\alpha ,\beta ]):\langle a,x\rangle \le b\}, \end{aligned}$$

where $0\ne a \in L_2([\alpha ,\beta ])$ and $b\in \mathbb {R}$, then (see [14])

$$\begin{aligned} \Pi _{C}(x)=P_{C}(x)= \left\{ \begin{array}{llll} &{} \frac{b-\langle a,x\rangle }{||a||_{L_2}^2}a+x, \quad \langle a,x\rangle > b\\ &{} x, \langle a,x\rangle \le b. \end{array} \right. \end{aligned}$$

Let

$$\begin{aligned} Q = \{x\in L_2([\alpha ,\beta ]):||x-d||_{L_2} \le r \} \end{aligned}$$

be a closed ball centered at $d\in L_2([\alpha ,\beta ])$ with radius $r > 0$, then

$$\begin{aligned} P_{Q}(x)= \left\{ \begin{array}{llll} &{} d+r\frac{x-d}{||x-d||}, x \notin Q\\ &{} x, x \in Q. \end{array} \right. \end{aligned}$$

Define an operator $A:L_2([0,2\pi ])\rightarrow L_2([0,2\pi ])$ by $Ax(t)=\frac{x(t)}{2},~~t \in [0,2\pi ]$ for all $x\in L_2([0,2\pi ])$. Then it can be easily verified that A is continuous and bounded linear operator.

Now, suppose

$$\begin{aligned} C=\left\{ x \in L_2([0,2\pi ]):\int _0^{2\pi } e^{t}x(t)\mathrm{d}t \le 1\right\} \end{aligned}$$

and

$$\begin{aligned} Q=\left\{ x\in L_2([0,2\pi ]):\int _0^{2\pi }|x(t)-\sin (t)|^2 \mathrm{d}t \le 16 \right\} . \end{aligned}$$

Let us consider the following problem:

$$\begin{aligned} \mathrm{find}~~x^* \in C ~~\mathrm{such~~ that}~~Ax^* \in Q. \end{aligned}$$

(5.1)

Observe that the set of solutions of problem (5.1) is nonempty (since $x(t)=0,~~a.e.$ is in the set of solutions). Take $\alpha _n=\frac{1}{n+1},~~\forall n \ge 1$, then our iterative scheme (3.4) becomes

$$\begin{aligned} x_{n+1}=P_{C}\left[ \frac{1}{n+1}(u)+\left( 1-\frac{1}{n+1}\right) \left( x_n-\rho _n\frac{f(x_n)}{\Vert \nabla f(x_n)\Vert ^{2}}\nabla f(x_n)\right) \right] ,\quad n\ge 1,\nonumber \\ \end{aligned}$$

(5.2)

where $f(x_n)=\frac{1}{2}\Vert Ax_n-P_QAx_{n}\Vert ^{2}$ for all $n\in \mathbb {N}$.

We now study the effect (in terms of convergence, stability, number of iterations required and the cpu time) of the sequence $\{\rho _n\}\subset (0,\infty )$ on the iterative scheme by choosing different $\rho _n$ such that $\underset{n}{\inf }{\rho _n}(4-\rho _{n})>0$ in the following cases (Figs. 1, 2, 3, 4; Table 1).

Fig. 1

Different cases with Choice 1

Fig. 2

Different cases with Choice 2

Fig. 3

Different cases with Choice 3

Fig. 4

Different cases with Choice 4

• Case 1: $\rho _n=\frac{n}{4n+1}$;

• Case 2: $\rho _n=\frac{n}{2n+1}$;

• Case 3: $\rho _n=\frac{n}{n+1}$;

• Case 4: $\rho _n=\frac{2n}{n+1}$;

• Case 5: $\rho _n=\frac{3n}{n+1}$.

  For each case mentioned above, using stopping criterion $\frac{||x_{n+1}-x_n||}{||x_2-x_1||}<10^{-4}$, we also consider different choices of $x_1$ and u as

• Choice 1: $x_1=2tcos(3t)e^{3t}$ and $3(t^7-1)e^{-5t}$;

• Choice 2: $x_1=2tsin(3t)e^{2t}$ and $t^2sin(5\pi t)$;

• Choice 3: $x_1=3t^2e^{4t-1}$ and $t^3-t^2+4t+1$;

• Choice 4: $x_1=2t^3e^{5t}$ and $t^4+3t^2+5$.

Table 1 Algorithm (5.2) with different cases of $\rho _n$ and different choices of $x_1$ and u

Remark 5.2

We make the following observations from the numerical results presented above.

• 1. The numerical results from different Cases and different Choices show that our proposed Algorithm (5.2) is fast, stable, efficient, easy to implement and required small number of iterations.

• 2. We have that the number of iterations and the cpu run time are decreasing starting from Case 1 to Case 5. However, there is no significant difference in both cpu run time and number of iterations for different Choices of $x_1$ and u. So, initial guess does not have any significant effect on the convergence of the algorithm.