An inertial self-adaptive iterative algorithm for finding the common solutions to split feasibility and fixed point problems in specific Banach spaces

Abstract

In this paper, we propose a new self adaptive iterative algorithm with an inertial technique for solving split feasibility problems and fixed point problems of relative nonexpansive mappings in $p$-uniformly convex and uniformly smooth Banach spaces. Under some appropriate assumptions imposed on the parameters and operators, we prove that the iterative sequence generated by our proposed algorithm converges strongly to a common solution of split feasibility problems and fixed point problems. Our algorithm does not require a prior knowledge of the norm of the bounded linear operator as we use a new self adaptive step size. Moreover, we give some numerical examples to show the effectiveness and feasibility of our algorithm.

Introduction

Let $E_1$ and $E_2$ be two real Banach spaces and let $E_1^{\ast }$ and $E_2^{\ast }$ be the dual spaces of $E_1$ and $E_2$, respectively. Let $C$ and $Q$ be nonempty, closed and convex subsets of $E_1$ and $E_2$, respectively, let $A:E_1\to E_2$ be a bounded linear operator and $A^{\ast }:E_2^{\ast }\to E_1^{\ast }$ be its adjoint. It is well known that the split feasibility problem (in short, SFP) is:

$$\text{Find }{x}^{\ast }\in C\text{ such that }A{x}^{\ast }\in Q.$$

The set of solutions of SFP is denoted by $\Omega ≔C\cap A^{-1}\left(Q\right)=\{x^{\ast }\in C:A{x}^{\ast }\in Q\}$.

SFP plays an important role in solving many problems appeared in applied science, such as, signal processing, radiotherapy, data compression, medical image reconstruction and so on. Many iterative algorithms have been proposed and studied by some authors for solving the SFP and related problems, see [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27] and the references therein. For example, Schöpfer et al. [9] introduced the following iterative algorithm for solving the problem (1.1) in $p$-uniformly smooth Banach space:

$$x_{n+1}={\Pi }_C{J}_{E_1^{\ast }}^q[J_{E_1}^p\left(x_n\right)-t{A}^{\ast }{J}_{E_2}^p(A{x}_n-P_Q\left(A{x}_n\right))],$$

where $x_0\in E_1,n\ge 0$, ${\Pi }_C$ is the Bregman projection and $J_{E_1}^p$ is the duality mapping of $E_1$. They obtained a weak convergence theorem about the sequence generated by (1.2) under the assumption that $J_{E_2}^p$ is sequentially weak-to-weak continuous. Moreover, strong convergence result was given when $E_1$ is finite dimensional or $C$ is bounded and compact.

Recently, Shehu [10] constructed another iterative scheme for solving problem (1.1) in real $p$-uniformly convex Banach spaces which are also uniformly smooth:

$$\left\{\begin{array}{c}{y}_n=J_{E_1^{\ast }}^q[J_{E_1}^p\left(x_n\right)-t_n{A}^{\ast }{J}_{E_2}^p(A{x}_n-P_Q\left(A{x}_n\right))],\hfill \\ x_{n+1}={\Pi }_C{J}_{E_1^{\ast }}^q({\alpha }_n{J}_{E_1}^p\left(u\right)+(1-{\alpha }_n)J_{E_1}^p\left(y_n\right)),n\ge 1.\hfill \end{array}\right.$$

Under the assumptions that ${lim}_{n\to \infty }{\alpha }_n=0$, ${\sum }_{n=1}^{\infty }{\alpha }_n=\infty$ and $0<t\le t_n\le k<{\left(\frac{q}{C_q{\Vert A\Vert }^q}\right)}^{\frac{1}{q-1}}$, Shehu proved that the sequence $\left\{x_n\right\}$ generated by (1.3) converges strongly to an element $\bar{x}\in \Omega$, where $\bar{x}={\Pi }_{\Omega }u$. Furthermore, Shehu et al. [17] studied split feasibility problems and fixed point problems concerning left Bregman strongly relatively nonexpansive mappings in $p$-uniformly convex and uniformly smooth Banach spaces. Precisely, they proposed the following algorithm:

$$\left\{\begin{array}{c}{x}_n={\Pi }_C{J}_{E_1^{\ast }}^q[J_{E_1}^p\left(u_n\right)-t_n{A}^{\ast }{J}_{E_2}^p(A{u}_n-P_Q\left(A{u}_n\right))],\hfill \\ u_{n+1}={\Pi }_C{J}_{E_1^{\ast }}^q({\alpha }_n{J}_{E_1}^p\left(u\right)+(1-{\alpha }_n)J_{E_1}^p\left(T{x}_n\right)),n\ge 1.\hfill \end{array}\right.$$

Suppose that the conditions ${lim}_{n\to \infty }{\alpha }_n=0$, ${\sum }_{n=1}^{\infty }{\alpha }_n=\infty$ and $0<t\le t_n\le k<{\left(\frac{q}{C_q{\Vert A\Vert }^q}\right)}^{\frac{1}{q-1}}$ are satisfied, they showed that the sequence $\left\{x_n\right\}$ generated by (1.4) converges strongly to an element $x^{\ast }\in F\left(T\right)\cap \Omega$, where $x^{\ast }={\Pi }_{F\left(T\right)\cap \Omega }u$.

On the other hand, the inertial methods have received great attention by many authors(see [15], [28], [29], [30], [31], [32], [33], [34], [35], [36] and the references therein). For examples, Bot et al. [29] proposed an inertial Douglas–Rachford splitting algorithm for finding the set of zeros of the sum of two maximally monotone operators in Hilbert spaces and investigated its convergence properties. Mainge [37] proposed the following inertial Krasnosel’skii–Mann algorithm:

$$\left\{\begin{array}{c}{w}_n=x_n+{\theta }_n(x_n-x_{n-1}),\hfill \\ x_{n+1}=(1-{\alpha }_n)w_n+{\alpha }_{n}T{w}_n,\forall n\ge 1,\hfill \end{array}\right.$$

and proved that the sequence $\left\{x_n\right\}$ generated by above algorithm converges weakly to a fixed point of $T$ under some appropriate conditions imposed on the parameters. Recently, Ma and Liu [38] modified the inertial relaxed CQ algorithm in [39] with the Halpern-type method and a simple stepsize strategy and obtained a strong convergence result for the split feasibility problems with non-Lipschitz continuity of the gradient operator in real Hilbert spaces.

Inspired and motivated by the above problems and methods, we introduce a new self adaptive iterative algorithm with an inertial technique for solving split feasibility problems and fixed point problems of relative nonexpansive mappings in $p$-uniformly convex and uniformly smooth Banach spaces. Under some suitable assumptions imposed on the parameters, we prove that the iterative sequence generated by our proposed algorithm converges strongly to a common solution of split feasibility problems and fixed points problems. In addition, we give some numerical examples to show the effectiveness of our proposed algorithm.