A Modified Iterative Algorithm for Split Feasibility Problems of Right Bregman Strongly Quasi-Nonexpansive Mappings in Banach Spaces with Applications

Let $E_1$, $E_2$ be Banach spaces and C, Q be nonempty closed convex subsets of $E_1$ and $E_2$, respectively. Let A$:E_1\to E_2$ be a bounded linear operator. The split feasibility problem (shortly, (SFP)) is as follows:

We denote the solution set of the problem (SFP) by $\mathcal{F}:=\{x\in C:Ax\in Q\}=C\cap A^{-1}(Q).$ It is worth mentioning that (SFP) in finite-dimensional spaces was first introduced by Censor and Elfving [1] for modelling inverse problems which arise from phase retrievals and medical image reconstruction.

Note that, in finite dimensional Hilbert spaces, the strong convergence of a sequence is equivalent to the weak convergence and the boundedness of a sequence implies that there exists a strongly convergent subsequence. However, in infinite dimensional Hilbert spaces, the strong convergence of a sequence is not equivalent to the weak convergence and the boundedness of a sequence implies that there exists a weakly convergent subsequence. So, for some algorithms, we can prove only strong convergence theorems in finite dimensional Hilbert spaces, but we can prove weak and strong convergence theorems in infinite dimensional Hilbert spaces.

In [2], Byrne presented a new method $\{x_n\}$, which is called the CQ-algorithm for solving the problem (SFP) that does not involve matrix inverses, defined as follows:

For any $x_0\in C$ and $n\ge 1,$ where $P_C$ and $P_Q$ is the orthogonal projections onto C and $Q,$ respectively, $\gamma \in (0,\frac{2}{L}),$ L is the largest eigenvalue of the matrix $A^{T}A$ and I is the identity matrix.

After that many authors [3,4,5,6,7] study extend some iterative algorithms from Hilbert spaces to Banach spaces by using Bregman’s technic as follows:

In solving the problem (SFP) in p-uniformly convex real Banach spaces which are also uniformly smooth, Schopfer et al. [8] proposed the following algorithm $\{x_n\}$ defined as follows:

For any $x_1\in E_1$ and $n\ge 1,$ where ${\Pi }_C$ denotes the Bregman projection and J the duality mapping.

Clearly, the algorithm (3) covers Byrne’s CQ algorithm (2), which is a gradient-projection method (GPM) in convex minimization as a special case. The duality mapping of $E_1$ is sequentially weak-to-weak-continuous (see [8]) in Banach spaces such as the classical $L_p(2<p<\infty )$ spaces.

In [9], Wang modified the algorithm (3) and proved strong convergence theorems for the following multiple-sets split feasibility problem (MSSFP): where $r,s$ are two given integers, $C_i,i=1,2,3,\cdots ,r$, is a closed convex subset in $E_1$ and $Q_j,j=r+1,\cdots ,r+s,$ is a closed convex subset in $E_2.$ He defined the following: for each $n\in \mathbb{N},$ where $i:\mathbb{N}\to \mathcal{I}$ is the cyclic control mapping $i(n)=n\mathrm{mod}(r+s)+1$ and $t_n$ satisfies with a constant $C_q$ and proposed the following algorithm $\{x_n\}$ defined as follows: For any $x_1=\overline{x}$ and $n\ge 1$,

Recently, Zegeye and Shahzad [10] proved a strong convergence theorem for a common fixed point of a finite family of right Bregman strongly nonexpansive mappings in the framework of real reflexive Banach spaces. Furthermore, they applied their method to approximate a common zero of a finite family of maximal monotone operators and a solution of a finite family of convex feasibility problems in reflexive real Banach spaces.

Let $f:E\to \mathbb{R}$ be a cofinite function which is bounded, uniformly Fŕechet differentiable and totally convex on bounded subsets of $E.$ Let C be a nonempty closed convex subset of int(dom f) and let $T_i:C\to C,$ for $i=1,2,\cdots ,N,$ be a finite family of right Bregman strongly nonexpansive mappings such that $F(T_i)=\widehat{F}(T_i)$ for each $i\in \{1,2,\cdots ,N\}.$ Assume that $F:=\widehat{F}(T_i)$ is nonempty. For any $u,x_1\in C,$ let $\{x_n\}$ be a sequence generated by for each $n\ge 1$, where $T=T_N\circ T_{N-1}\circ \cdots \circ T_1$ and $\{{\alpha }_n\}\subset (0,1)$ satisfy the following conditions:

(i)

$\underset{n\to \infty }{lim}{\alpha }_n=0;$

(ii)

${\displaystyle \sum _{n=1}^{\infty }}{\alpha }_n=\infty .$

Then $\{x_n\}$ converges strongly to a point $\widehat{x}.$

In this paper, we modify the Halpern-Mann iterative method for split feasibility problems and fixed point problems concerning right Bregman strongly quasi-nonexpansive mappings in p-uniformly convex and uniformly smooth Banach spaces. We prove strong convergence theorem of the sequences generated by our scheme under some appropriate conditions in real p-uniformly convex and uniformly smooth Banach spaces. Also, we give numerical examples of our result to study its efficiency and implementation. Our results extend and improve the recent ones of some others in the literature.

Let $E_1$, $E_2$ be real Banach spaces and A: $E_1\to E_2$ be a bounded linear operator. The dual (adjoint) operator of A, denoted by $A^{*}$, is a bounded linear operator defined by $A^{*}$: $E_2^{*}\to E_1^{*}$ for all $x\in E_1$ and $\overline{y}\in E_2^{*}$ and the equalities $\parallel A^{*}\parallel =\parallel A\parallel$, $\mathcal{N}(A^{*})=\mathcal{R}{(A)}^{\perp }$ are valid, where

For more details on bounded linear operators and their duals, see [11,12].

We note here that $l_p(p>1)$ spaces has such a property, but the $L_p(p>2)$ space does not share this property. The domain of a convex function $f:E\to \mathbb{R}$ is defined by dom$f:=\{x\in E:f(x)<+\infty \}.$ When dom$f\ne \varnothing ,$ then we say that f is proper.

In the sequel, we adopt the following notations in this paper: $x_n\to x$ means that $x_n\to x$ strongly and $x_n\rightharpoonup x$ meansthat $x_n\to x$ weakly.

The duality mapping $J_E^p$ is actually the derivative of the function $f_p(x)=\frac{1}{p}{\parallel x\parallel }^p.$ If $f=f_p$, then the Bregman distance with respect to $f_p$ now becomes

The Bregman distance is not symmetric and so it is not a metric, but it posses the following important properties: for all $w,x,y\in E$, and

Let $1<q\le 2\le p$ with $\frac{1}{p}+\frac{1}{q}=1.$ The modulus of smoothness of E is the function ${\rho }_E:[0,\infty )\to [0,\infty )$ defined by

A Banach space E is said to be uniformly smooth if and, for any $q>1$, a Banach space E is said to be q-uniformly smooth if there exists $C_q>0$ such that ${\rho }_E(t)\le C_q{t}^q$ for any $t>0.$

Let $x,y\in E$ and $q>1$. If a Banach space E is q-uniformly smooth, then there exists $C_q>0$ such that

Let $\text{dim}(E)\ge 2.$ The modulus of convexity of E is the function ${\delta }_E(ϵ)$: $(0,2]\to [0,1]$ defined by

A Banach space E is said to be uniformly convex if ${\delta }_E(ϵ)>0$ for all $ϵ\in (0,2]$ and, for any $p>1$, a Banach space E is said to be p-uniformly convex if there is $C_p>0$ such that ${\delta }_E(ϵ)\ge C_p{ϵ}^p$ for any $ϵ\in (0,2].$ More information concerning uniformly convex spaces can be found, for example, in the book by Goebel and Reich [14].

It is known that a Banach space E is p-uniformly convex and uniformly smooth if and only if its dual $E^{*}$ is q-uniformly smooth and uniformly convex. It is also well known that the duality $J_E^p$ is one-to-one, single valued and satisfies $J_E^p={(J_{E^{*}}^q)}^{-1},$ where $J_{E^{*}}^q$ is the duality mapping of $E^{*}.$

For any p-uniformly convex Banach space E, the metric and the Bregman distance have the following relation: where $\tau >0$ is a fixed number.

Let C be a nonempty closed convex subset of $E.$ The metric projection for all $x\in E$ is the unique minimizer of the norm distance, which can be characterized by a variational inequality for all $z\in C$.

Similarly, the Bregman projection is defined as follows: for all $x\in E$, which is the unique minimizer of the Bregman distance. In addition, the Bregman projection can also be characterized by a variational inequality for all $z\in C$, from which one has for all $z\in C$.

Following [15,16], we will make use of the function $V_p$: $E^{*}\times E\to [0,+\infty )$ associated with $f_p,$ which is defined by for all $x\in E$ and $\overline{x}\in E^{*}$. Then $V_p$ is nonnegative and for all $x\in E$ and $\overline{x}\in E^{*}.$ Moreover, by the subdifferential inequality, we have for all $x\in E$ and $\overline{x},\overline{y}\in E^{*}$ (see also [17,18]). In addition, $V_p$ is convex in the first variable. Thus, for all $z\in E,$ where $\{x_i\}\subset E$ and $\{t_i\}\subset (0,1)$ with ${\displaystyle \sum _{i=1}^N}=1.$ For more details, see [19,20].

Let C be a nonempty, closed and convex subset of E. A mapping T: $C\to C$ is said to be nonexpansive if for all $x,y\in C$. We denote by $F(T)$ the set of fixed points of $T,$ that is, $F(T)=\{x\in C:Tx=x\}.$

Let C be a convex subset of int (dom $f_p$), where $f_p(x)=(\frac{1}{p}){\parallel x\parallel }^p,2\le p<\infty$, and T be a self-mapping of $C.$ A point $\widehat{x}\in C$ is called an asymptotic fixed point of T if C contains a sequence $\{x_n\}$ which converges weakly to $\widehat{x}$ and $\underset{n\to \infty }{lim}\parallel x_n-T{x}_n\parallel =0.$ The set of asymptotic fixed point of T is denoted by $\widehat{F}(T)$ (see [21]).

In general, the Bregman distance is not a metric due to the absence of symmetry, but it has some distance-like properties.

Now, we give our main results in this paper.

So, from (21) and (12), it follows that

By using $(c),$ we obtain

From (20), we have

Thus $\{d_p(u_n,w)\}$ is bounded and, consequently, we have that $\{d_p(x_n,w)\}$ is bounded. Hence the sequence $\{x_n\}$ and $\{u_n\}$ are bounded. Setting for each $n\ge 1$. Then we have

Now, we prove the strong convergence theorem by the two cases:

Setting $t_n=J_{E_1^{*}}^q(\frac{{\beta }_n}{1-{\alpha }_n}{J}_{E_1}^p(x_n)+\frac{{\gamma }_n}{1-{\alpha }_n}{J}_{E_1}^p(x_n)+\frac{{\delta }_n}{1-{\alpha }_n}{J}_{E_1}^p(T{x}_n)).$ Then we have

Therefore, we have as $n\to \infty$. Again, we obtain