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A hybrid inertial algorithm for approximating solution of convex feasibility problems with applications
Fixed Point Theory and Applications volume 2020, Article number: 12 (2020)
Abstract
An inertial iterative algorithm for approximating a point in the set of zeros of a maximal monotone operator which is also a common fixed point of a countable family of relatively nonexpansive operators is studied. Strong convergence theorem is proved in a uniformly convex and uniformly smooth real Banach space. This theorem extends, generalizes and complements several recent important results. Furthermore, the theorem is applied to convex optimization problems and to Jfixed point problems. Finally, some numerical examples are presented to show the effect of the inertial term in the convergence of the sequence of the algorithm.
Introduction
An inertialtype algorithm was first introduced and studied by Polyak [35], as a method of speeding up the convergence of the sequence of an algorithm. This algorithm is a two step iterative procedure in which the successive iterates are obtained by using two previous iterates. Numerical experiments have shown that an algorithm with an inertial extrapolation term converges faster than an algorithm without it. Thus, one can see an increasing interest in the class of inertialtype algorithms (see, for example, the following papers [12, 26, 44] and the references therein).
Let X be a real normed space with dual space \(X^{*}\). Let \(T:X\to 2^{X^{*}}\), be a setvalued operator with domain \(D(T):=\{p\in X :Tp \neq\emptyset\}\), range \(R(T):=\bigcup_{p\in D(T)}\{ Tp\}\) and graph \(G(T):=\{(p,p^{*}) : p^{*}\in Tp\}\). Then T is called monotone if
T is said to be maximal monotone if \(G(T)\) is not properly contained in the graph of any other monotone operator. Monotone maps were first introduced by Minty [29] to aid in the abstract study of electrical networks and later studied by Browder [4] in the setting of partial differential equations. Later, Kačurovskii [19], Minty [30], Zarantonello [48] and many other authors studied this class of operators in Hilbert spaces. Interest in monotone operators stems mainly from their various applications (see e.g., the following monographs [2, 5, 17] and the references therein).
A fundamental problem of interest in the study of monotone operators in Banach spaces is the following:
For the prove of existence of solutions of (1.2) see, for example, Browder [3], and Martin [27]. Many problems in applications can be transformed into the form of the inclusion (1.2). For example, problems arising from convex minimization, variational inequality, Hammerstein equations, and evolution equations can be transformed into the form of the inclusion (1.2) (see, e.g., Chidume et al. [8, 14], Rockafellar [37]).
Iterative methods for approximating solutions of the inclusion (1.2) have been studied extensively by various authors in Hilbert spaces and in more general Banach spaces. One of the classical methods for approximating solution(s) of (1.2) in Hilbert spaces is the celebrated proximal point algorithm (PPA) introduced by Martinet [28] and studied extensively by Rockafellar [37] and a host of other authors. Concerning the iterative approximation of solution(s) of (1.2) in more general Banach space, see, e.g., [6, 11, 14, 20, 32].
Let \(S: X \to X\) be a map and let \(p\in X\), p be called an asymptotic fixed point of S if X contains a sequence \(\{ p_{n}\}\) which converges weakly to p and \(\lim_{n \to \infty} \p_{n}Sp_{n}\=0\). We denote the set of asymptotic fixed points of S by \(\widehat{F}(S)\). The map S is said to be relatively nonexpansive if \(\widehat{F}(S)=F(S)\neq\emptyset\) and \(\psi(p,Sq)\leq\psi(p,q)\), for all \(p\in F(S)\) and \(q\in X\), where \(F(S)=\{p\in X : Sp=p\}\) and ψ is the Lyapunov function (see, e.g., Alber [1]).
One of the motivations for the study of relatively nonexpansive self or nonself mappings in Banach spaces is the fact that they are an extension of nonexpansive mappings with nonempty fixed point sets in Hilbert spaces. In 2018, Chidume et al. [12] introduced and studied an inertialtype algorithm in a uniformly convex and uniformly smooth real Banach space. They proved the following theorem.
Theorem 1.1
LetBbe a uniformly convex and uniformly smooth real Banach space. Let\(T_{i} :B\rightarrow B\), \(i=1,2,3,\ldots\)be a countable family of relatively nonexpansive maps such that\(\bigcap_{i=1}^{\infty}F(T_{i})\neq \emptyset\). Suppose\({\lbrace\eta_{i} \rbrace} \subset(0,1)\)and\({\lbrace\beta_{i} \rbrace} \subset(0,1)\)are sequences such that\(\sum_{i=1}^{\infty}\eta_{i}=1\)and\(T :B \rightarrow B\)is defined by\(Tp=J^{1} (\sum_{i=1}^{\infty}\eta_{i}(\beta_{i}Jp+(1\beta _{i})JT_{i}p) )\)for each\(p\in B\). Let\(\lbrace x_{n}\rbrace\)be generated by the following algorithm:
\(n\geq0\), where\(\alpha_{n}\in[0,1)\), \(\beta\in(0,1)\). Then\(\{x_{n}\}\)converges strongly to\(p=\varPi_{F(T)}x_{0}\).
Several iterative algorithms for approximating fixed points of self maps satisfying certain contractive conditions and zeros of monotone and monotone type operators has recently been studied extensively by various authors; see e.g., [24, 33, 34, 39–42]. In 2009, Inoue et al. [18] introduced and studied a hybrid algorithm in a uniformly convex and uniformly smooth Banach space. They proved the following theorem.
Theorem 1.2
LetBbe a uniformly convex and uniformly smooth Banach space and letCbe a nonempty closed and convex subset ofB. Let\(A: B\to 2^{B^{*}}\)be a maximal monotone operator satisfying\(D(A) \subset C\)and let\(J_{r}=(J+rA)^{1}J\)for all\(r>0\). Let\(S: C \to C\)be a relatively nonexpansive mapping such that\(F(S)\cap A^{1}0 \neq\emptyset\). Let\(\{x_{n}\}\)be a sequence generated by\(x_{0}=x\in C\)and
for all\(n\in \Bbb {N}\cup\{0\}\), whereJis the duality mapping onB, \(\{\beta_{n}\}\subset[0,1]\), and\(\{r_{n}\}\subset[a,\infty)\)for some\(a>0\). If\(\liminf_{n\to\infty}(1\beta_{n})>0\), then\(\{x_{n}\}\)converges strongly to\(\varPi_{F(S)\cap A^{1}0}x_{0}\).
In 2009, Klin et al. [21] extended the results of Inoue et al. [18]. They proved the following theorem.
Theorem 1.3
LetBbe a uniformly convex and uniformly smooth Banach space and letCbe a nonempty closed and convex subset ofB. Let\(A: B\to 2^{B^{*}}\)be a maximal monotone operator satisfying\(D(A) \subset C\)and let\(J_{r}=(J+rA)^{1}J\)for all\(r>0\). LetSandTbe relatively nonexpansive mappings fromCinto itself such that\(\varOmega =F(S)\cap F(T)\cap A^{1}0 \neq\emptyset\). Let\(\{x_{n}\}\)be a sequence generated by\(x_{0}\in C\)and
for all\(n\in \Bbb {N}\cup\{0\}\), whereJis the duality mapping onB, \(\{\alpha_{n}\},\{\beta_{n}\}\subset[0,1]\), and\(\{r_{n}\}\subset [a,\infty)\)for some\(a>0\). If\(\liminf_{n\to\infty}(1\alpha _{n})>0\)and\(\liminf_{n\to\infty}(1\beta_{n})>0\), then\(\{x_{n}\}\)converges strongly to\(\varPi_{\varOmega}x_{0}\).
Motivated by the results of Chidume et al. [12] and Klin et al. [21], in this paper we introduce and study an inertial iterative algorithm in a uniformly convex and uniformly smooth real Banach space and prove a strong convergence theorem for approximating a common element in the set of zeros of a maximal monotone operator and the sets of fixed points of countable family of relatively nonexpansive mappings. Furthermore, we give applications of our theorem to convex optimization and Jfixed point. Finally, we present numerical examples to demonstrate the effect of the inertial term on the convergence of the sequence of our algorithm.
Preliminaries
The following definitions and lemmas will be needed in the sequel.
Definition 2.1
Let X be a real normed space. The normalized duality map J from X to \(2^{X^{*}}\) is defined by \(Jp := \lbrace p^{*}\in X^{*} :\langle p,p^{*} \rangle=\p\^{2}=\p^{*}\ ^{2}, \forall p\in X \rbrace\), where \(\langle\cdot, \cdot \rangle\) denotes the value of \(p^{*}\) at p and \(X^{*}\) is the dual space of X. It is well known that if X is smooth then J is singlevalued and if X is uniformly smooth, then J is uniformly continuous on bounded subsets of X.
Definition 2.2
Let B be a smooth real Banach space; the Lyapunov functional \(\psi: B\times B \to \Bbb {R}\) is defined by
The mapping ψ was introduced by Alber [1]. Since its introduction, one can notice an increasing interest in the functional see e.g., [7, 10, 38, 43, 45, 46, 49]. Observe that, in a real Hilbert space H, Eq. (2.1) reduces to \(\psi(p,y)=\py\^{2}\), \(\forall p,y\in H\). Furthermore, the following properties of ψ can be verified easily from its definition:

(P1)
\((\p\\q\)^{2}\leq\psi(p,q) \leq(\p\+\q\)^{2}\),

(P2)
\(\psi(p,q)= \psi(p,z)+\psi(z,q)+2\langle pz, JzJq \rangle\),

(P3)
\(\psi(p,q) \leq\p\\JpJq\+\qp\\q\\),
for all \(p,q,z \in B\).
Definition 2.3
Let B be a strictly convex, smooth and reflexive real Banach space and let C be a nonempty, closed and convex subset of B. The map \(\varPi_{C}:B\rightarrow C\) defined by \(\tilde{t}:=\varPi_{C} (t) \) such that \(\psi(\tilde{t},t)= \inf_{y\in C}\psi(y,t)\) is called the generalized projection of t onto C. Observe that in a real Hilbert space, the generalized projection \(\varPi_{C}\) and the metric projection \(P_{C}\) are equivalent.
Lemma 2.4
(Rockafellar, [36])
LetBbe a smooth, strictly convex and reflexive real Banach space and\(A: B\to2^{B^{*}}\)be a monotone mapping. ThenAis maximal if and only if\(R(J+rA)=B^{*}\), \(\forall r>0\).
Lemma 2.5
(Alber, [1])
LetCbe a nonempty closed and convex subset of a smooth, strictly convex and reflexive real Banach spaceB. Then:

(1)
given\(t\in B\)and\(y \in C\), \(\tilde{t} =\varPi_{C}t\)if and only if\(\langle\tilde{t}y, JtJ\tilde{t} \rangle\geq0\), for all\(y\in C\),

(2)
\(\psi(y,\tilde{t})+\psi(\tilde{t},t)\leq\psi(y,t)\), for all\(t\in B, y \in C\).
Lemma 2.6
(Nilsrakoo and Saejung, [31])
Let B be a smooth Banach space. Then
Remark 1
Let B be a smooth, strictly convex and reflexive real Banach space, let C be a nonempty closed and convex subset of B and let \(A: B\to 2^{B^{*}}\) be a monotone operator satisfying
Then we can define the resolvent \(J_{r}: C\to D(A)\) of A by
It is well known that \(J_{r}t\) is singlevalued. For \(r>0\), the Yosida approximation \(A_{r}:C\to B^{*}\) is defined by \(A_{r}t=(JtJJ_{r}t)/r\) for all \(t\in C\).
Lemma 2.7
(Kohsaka and Takahashi, [22])
LetBbe a smooth, strictly convex and reflexive real Banach space, letCbe a nonempty closed convex subset ofBand let\(A: B\to 2^{B^{*}}\)be a monotone operator satisfying (2.2). Let\(r>0\)and let\(J_{r}\)and\(A_{r}\)be the resolvent and the Yosida approximation ofA, respectively. Then the following hold:

(i)
\(\psi(u,J_{r}t)+\psi(J_{r}t,t)\leq\psi(u,t)\), \(\forall t\in C\), \(u\in A^{1}0\);

(ii)
\((J_{r}t, A_{r}t)\in A\), \(\forall t\in C\), where\((t,t^{*})\in A\)denotes the value of\(t^{*}\)att (\(t^{*}\in At\)).

(iii)
\(F(J_{r})=A^{1}0\).
Lemma 2.8
(Xu, [47])
LetBbe a uniformly convex Banach space and let\(r>0\). Then there exists a strictly increasing, continuous, and convex function\(g:[0,\infty) \to[0,\infty)\)such that\(g(0)=0\)and
for all\(t,y\in B_{r}(0)\)and\(\tau\in[0,1]\).
Lemma 2.9
(Kamimura and Takahashi, [20])
LetBbe a uniformly convex and smooth real Banach space, and let\(\{ x_{n}\}\)and\(\{y_{n}\}\)be two sequences ofB. If either\(\{x_{n}\}\)or\(\{y_{n}\}\)is bounded and\(\psi(x_{n},y_{n} )\to0\), then\(\Vert x_{n}y_{n}\Vert\to0 \).
Lemma 2.10
(Kohsaka and Takahash, [23])
LetCbe a closed convex subset of a uniformly smooth and uniformly convex Banach spaceBand let\((S_{i})_{i=1}^{\infty}\), \(S_{i}: C \to B\), for each\(i \ge1\), be a family of relatively nonexpansive maps such that\(\bigcap_{i=1}^{\infty}F(S_{i}) \neq\emptyset\). Let\((\eta_{i})_{i=1}^{\infty}\subset(0,1)\)and\((\mu_{i})_{i=1}^{\infty}\subset(0,1)\)be sequences such that\(\sum_{i=1}^{\infty}\eta_{i} =1\). Consider the map\(T:C \to B\)defined by
ThenTis relatively nonexpansive and\(F(T)=\bigcap_{i=1}^{\infty}F(S_{i})\).
Main result
Theorem 3.1
LetBbe a uniformly convex and uniformly smooth real Banach space. Let\(A: B\to2^{B^{*}}\)be a maximal monotone operator and let\(J_{r}=(J+rA)^{1}J\), for all\(r>0\). Let\(S:B\to B\)and\(T:B \to B\)be relatively nonexpansive mappings such that\(\varOmega =F(S)\cap F(T)\cap A^{1} 0 \neq\emptyset\). Define inductively the sequence\(\{x_{n}\}\)by: \(x_{0},x_{1} \in B\)
for all\(n\in \Bbb {N}\cup\{0\}\), \(\{\alpha_{n}\} \subset[0,1)\), \(\beta, \gamma\in(0,1)\)and\(\{r_{n}\} \subset[a,\infty)\), for some\(a>0\). Then\(\{x_{n}\}\)converges strongly to\(\varPi_{\varOmega}x_{0}\).
Proof
We divide the proof into four steps.
Step 1. We show that \(\{x_{n}\}\) is well defined and \(\varOmega \subset C_{n}\), \(\forall n\geq0\). Observe that by definition, \(C_{n+1}\) is closed and convex, \(\forall n\geq0\). We now show that \(\varOmega \subset C_{n}\). Let \(y_{n}=J_{r_{n}}w_{n}\) and \(u\in\varOmega\). Using Lemma 2.6, the fact that S is relatively nonexpansive and Lemma 2.7(i), we obtain
Similarly, using Lemma 2.6, the fact that T is relatively nonexpansive and inequality (3.3), we have
which implies \(u\in C_{n+1}\). So, by induction, \(\varOmega\subset C_{n}\), \(\forall n\geq 0\). Thus, \(\{x_{n}\}\) is well defined.
Step 2. We show that \(\{x_{n}\}\), \(\{w_{n}\}\), \(\{z_{n}\}\), \(\{u_{n}\}\) are bounded and \(\{x_{n}\}\) is Cauchy. We observe that \(x_{n}=\varPi_{C_{n}}x_{0}\) and \(C_{n+1}\subset C_{n}\), \(\forall n\geq0\). So, by Lemma 2.5(2)
Thus, \(\{\psi(x_{n},x_{0})\}\) is nondecreasing. Furthermore, we have
which implies that \(\{\psi(x_{n},x_{0})\}\) is bounded and by (P1), \(\{x_{n}\} \) is also bounded. Since \(\{\psi(x_{n},x_{0})\}\) is nondecreasing, \(\{\psi (x_{n},x_{0})\}\) is convergent. Furthermore, \(\{x_{n}\}\) bounded implies \(\{ w_{n}\}\) is bounded which also imply that \(\{z_{n}\}\) and \(\{u_{n}\}\) are bounded (by using inequalities (3.3) and (3.4), respectively and (P1)).
Next we show that \(\{x_{n}\}\) is Cauchy. Using Lemma 2.5(2)
Hence, \(\{x_{n}\}\) is Cauchy and this implies that \(\x_{n+1}x_{n}\\to 0\), as \(n\to\infty\).
Step 3. We show the following:

\(\lim_{n\to\infty} \x_{n}w_{n}\=0\), \(\lim_{n\to\infty} \ x_{n}u_{n}\=0\),

\(\lim_{n\to\infty} \z_{n}Tz_{n}\=0\), \(\lim_{n\to\infty} \ y_{n}Sy_{n}\=0\).
Using the definition of \(w_{n}\), we have
Now, using the fact that \(\{w_{n}\}\) is bounded, we have \(\psi (x_{n},w_{n})\to0\), as \(n\to\infty\). Since \(x_{n+1}\in C_{n}\), it follows that
Thus, \(\lim_{n\to\infty}\psi(x_{n+1},u_{n})=0\), which implies that \(\lim_{n\to\infty}\x_{n+1}u_{n}\=0\). Hence, \(\lim_{n\to\infty}\ x_{n}u_{n}\=0\). By the uniform continuity of J on bounded sets, we have
Observe that
which implies
Thus, \(\lim_{n\to\infty}\Jx_{n+1}JTz_{n}\=0\). By the uniform continuity of \(J^{1}\) on bounded sets, we have \(\lim_{n\to\infty} \x_{n+1}Tz_{n}\=0\). Furthermore,
Next we show that \(\lim_{n\to\infty} \z_{n}Tz_{n}\= \lim_{n\to\infty} \y_{n}Sy_{n}\=0\). Using Lemma 2.8 we have
This implies that
Let \(\{\w_{n_{k}}Sy_{n_{k}}\\}\) be an arbitrary subsequence of \(\{\ w_{n}Sy_{n}\\}\). Since \(\{w_{n_{k}}\}\) is bounded, there exists a subsequence \(\{w_{n_{k_{j}}}\}\) of \(\{w_{n_{k}}\}\) such that
Using (P2), (P3) and the fact that T is relatively nonexpansive, we obtain
Since \(\lim_{n\to\infty}\w_{n}Tz_{n}\=0\) and hence \(\lim_{n\to \infty}\Jx_{n}JTz_{n}\=0\) we obtain
We also have from inequality (3.3)
and hence
Thus, it follows from inequality (3.6) that \(\lim_{j\to \infty}g(\Jw_{n_{k_{j}}}JSy_{n_{k_{j}}}\)=0\). By the properties of g, we have \(\lim_{j \to\infty}\Jw_{n_{k_{j}}}JSy_{n_{k_{j}}}\=0\). By the uniform continuity of \(J^{1}\) on bounded sets, we obtain \(\lim_{j\to\infty} \w_{n_{k_{j}}}Sy_{n_{k_{j}}}\=0\). Hence, \(\lim_{n \to\infty}\w_{n}Sy_{n}\=0\). So, we have \(\lim_{n \to\infty} \ Jw_{n}JSy_{n}\=0\). Observe that
This implies that \(\lim_{n\to\infty}\Jz_{n}Jw_{n}\=0\), and hence \(\lim_{n\to\infty}\w_{n}z_{n}\=0\). Furthermore, from inequality (3.2), we have
Using \(y_{n}=J_{r_{n}}w_{n}\) and Lemma 2.7(i), we have
Thus, using inequality (3.8), we have
This implies that \(\lim_{n\to\infty} \psi(y_{n},w_{n})=0\). It follows from Lemma 2.9 that
Observe that
imply
Step 4. Finally, we show that \(\{x_{n}\}\) converges strongly to a point in Ω. Since \(\{w_{n}\}\) is bounded, there exists a subsequence \(\{w_{n_{k}}\}\) of \(\{w_{n}\}\) such that \(w_{n_{k}} \rightharpoonup p\). Furthermore, since \(\lim_{n\to\infty} \w_{n}y_{n}\=0\) and \(\lim_{n \to\infty} \w_{n}z_{n}\=0\), we have \(y_{n_{k}}\rightharpoonup p\) and \(z_{n_{k}}\rightharpoonup p\). Moreover, since S and T are relatively nonexpansive, we have \(p\in\widehat{F}(S)\cap\widehat {F}(T)=F(S)\cap F(T)\). Next, we show that \(p\in A^{1} 0\). By the uniform continuity of J on bounded sets, it follows from inequality (3.9) that
Since \(r_{n}\geq a\), we have \(\lim_{n\to\infty}\frac{1}{r_{n}} \Jw_{n}Jy_{n}\=0\). Therefore,
Using the fact that A is monotone and Lemma 2.7 (ii), we have
This implies that \(\lim_{k \to\infty}\langle vy_{n_{k}},v^{*}A_{r_{n_{k}}}w_{n_{k}} \rangle=\langle vp, v^{*}\rangle\geq 0\). Thus, \(p\in A^{1}0\), since A is maximal monotone. Therefore, \(p\in\varOmega\). From Step 3, there exists \(\{x_{n_{k}}\}\) a subsequence of \(\{x_{n}\}\), such that \(x_{n_{k}}\rightharpoonup p\), as \(k \to\infty\). We now show that \(p=\varPi_{\varOmega}x_{0}\). Set \(q=\varPi _{\varOmega}x_{0}\). Using the fact that \(x_{n}=\varPi_{C_{n}} x_{0}\) and \(\varOmega\subset C_{n}\), \(\forall n\geq0\), we have \(\psi(x_{n},x_{0})\leq \psi(q,x_{0})\). Using the fact that the norm is weakly lower semicontinuous, we obtain
But
Thus, \(\psi(p,x_{0})=\psi(q,x_{0})\). By uniqueness of \(\varPi_{\varOmega}x_{0}\), \(p=q\). Next, we show that \(x_{n_{k}}\to p\), as \(k\to\infty\). Using inequalities (3.10) and (3.11), we obtain \(\psi(x_{n_{k}},x_{0})\to\psi(p,x_{0})\), as \(k\to\infty\). Thus, \(\ x_{n_{k}}\\to\p\\), as \(k \to\infty\). By the Kadec–Klee property of B, we conclude that \(x_{n_{k}}\to p\) as \(k \to\infty\). Therefore, \(x_{n} \to\varPi_{\varOmega}x_{0}\). This completes the proof. □
Theorem 3.2
LetBbe a uniformly convex and uniformly smooth real Banach space. Let\(A: B\to2^{B^{*}}\)be a maximal monotone operator and let\(J_{r}=(J+rA)^{1}J\), for all\(r>0\). Let\(T: B \to B \)be a relatively nonexpansive and let\(\{ S_{i} \}_{i=1}^{\infty}\)be a countable family of relatively nonexpansive maps such that\(\bigcap_{i=1} ^{\infty}F(S_{i}) \neq\emptyset\), where\(S_{i}: B \to B \), ∀i. Let\(\{ \zeta_{i}\}_{i=1}^{\infty}\subset(0,1)\)and\(\{\tau_{i}\}_{i=1}^{\infty}\subset(0,1)\)be sequences such that\(\sum_{i=1}^{\infty}\zeta_{i} =1\). Assume\(\varOmega= (\bigcap_{i=1} ^{\infty}F(S_{i}) ) \cap F(T) \cap {A^{1} 0 \neq\emptyset}\). Define inductively the sequence\(\{x_{n}\}\)by: \(x_{0},x_{1} \in B\)
for all\(n\in \Bbb {N}\cup\{0\}\), where\(St=J^{1} (\sum_{i=1}^{\infty}\zeta_{i}(\tau_{i}Jt+(1\tau_{i})JS_{i}t) )\)for each\(t \in B\), \(\{\alpha_{n}\} \subset[0,1)\), \(\beta, \gamma\in(0,1)\)and\(\{r_{n}\} \subset[a,\infty)\), for some\(a>0\). Then\(\{x_{n}\}\)converges strongly to\(\varPi_{\varOmega}x_{0}\).
Proof
By Lemma 2.10, S is relatively nonexpansive and \(F(S)=\bigcap_{i=1} ^{\infty}F(S_{i})\). The conclusion follows from Theorem 3.1. □
Applications
Application to a convex optimization problem
Let X be a normed space and let \(f : X\to(\infty, \infty] \) be a convex, proper and lower semicontinuous function. The subdifferential of f is defined by
Observe that \(0\in\partial f(u) \) if and only if u is a minimizer of f. Furthermore, it is well known that the subdifferential of f, ∂f is maximal monotone (see, e.g., Rockafellar [37]). Set \(A=\partial f\) in Theorem 3.2.
Application to Jfixed point
The notion of Jfixed point (which has also been called semifixed point, Zegeye [49], duality fixed point, Liu [25]) has been defined and studied by Chidume and Idu [11], for maps from a space, say X, to its dual space \(X^{*}\).
Definition 4.1
Let \(T:X\to2^{X^{*}}\) be any map. A point \(u\in X\) is called a Jfixed point of T if \(Ju \in Tu\), where \(J:X\to X^{*}\) is the singlevalued normalized duality map on X.
Consider, for example, the evolution inclusion
where \(A : B \to2^{B^{*}}\) is monotone. At equilibrium, we have
and the solutions of Eq. (4.2) correspond to equilibrium states of (4.1). Define \(T:B\to2^{B^{*}}\) by \(T:=JA\). Then u is a Jfixed point of T if and only if u is a solution of (4.2). Consequently, approximating solutions of (4.2) is equivalent to approximating Jfixed points of maps \(T:X\to2^{X^{*}}\) defined by \(T:=JA\). This connection is now generating considerable research interest in the study of Jfixed points (see, e.g., Chidume and Idu [11], Chidume and Monday [13], Chidume et al. [15, 16], and the references contained in them). This notion turns out to be very useful and applicable in approximating solutions of Eq. (4.2). For example, Chidume and Idu [11], introduced the concept of Jpseudocontractive maps and proved a strong convergence theorem for approximating Jfixed points of a Jpseudocontractive map. As an application of their theorem, they proved a strong convergence theorem for approximating a zero of a maximal monotone operator.
Recently, Chidume et al. [9] introduced the concept of relativelyJnonexpansive maps in a uniformly smooth and uniformly convex real Banach spaces. They gave the following definitions.
Definition 4.2
Let \(T:B\to B^{*}\) be a map. A point \(x^{*}\in B\) is called an asymptoticJfixed point ofT if there exists a sequence \(\{x_{n}\}\subset B\) such that \(x_{n}\rightharpoonup x^{*}\) and \(\Jx_{n}Tx_{n}\ \to0\), as \(n \to\infty\). We shall denote the set of asymptotic Jfixed points of T by \(\widehat{F}_{J}(T)\).
Definition 4.3
A map \(T:B\to B^{*}\) is said to be relativelyJnonexpansive if

(i)
\(\widehat{F}_{J}(T)=F_{J}(T) \neq\emptyset\),

(ii)
\(\psi(p,J^{1}Tx)\leq\psi(p,x)\), \(\forall x\in B\), \(p\in F_{J}(T)\); where \(F_{J}(T)=\{x\in B: Tx=Jx \}\).
Chidume et al. [9] used these new definitions in approximating a common Jfixed point of a countable family of relatively Jnonexpansive mappings in a uniformly convex and uniformly smooth real Banach space. We now use these definitions to prove a similar result. The following remark is key in the proof of the theorem below.
Remark 2
Observe that in the definition above, a mapping T is relatively Jnonexpansive if and only if \(J^{1}T\) is relatively nonexpansive in the usual sense. Furthermore, \(x^{*}\in F_{J} (T ) \Leftrightarrow x^{*}\in F (J^{1} T )\).
Theorem 4.4
LetBbe a uniformly convex and uniformly smooth real Banach space. Let\(A: B\to2^{B^{*}}\)be a maximal monotone operator and let\(J_{r}=(J+rA)^{1}J\), for all\(r>0\). Let\(T: B \to B^{*} \)be a relatively nonexpansive and let\(\{ S_{i} \}_{i=1}^{\infty}\)be a countable family of relatively nonexpansive maps such that\(\bigcap_{i=1} ^{\infty}F(S_{i}) \neq\emptyset\), where\(S_{i}: B \to B^{*} \), ∀i. Let\(\{\zeta_{i}\}_{i=1}^{\infty}\subset(0,1)\)and\(\{\tau_{i}\} _{i=1}^{\infty}\subset(0,1)\)be sequences such that\(\sum_{i=1}^{\infty}\zeta_{i} =1\). Assume\(\varOmega= (\bigcap_{i=1} ^{\infty}F(S_{i}) ) \cap F(T) \cap {A^{1} 0 \neq\emptyset}\). Define inductively the sequence\(\{x_{n}\}\)by: \(x_{0},x_{1} \in B\)
for all\(n\in \Bbb {N}\cup\{0\}\), where\(Sx=J^{1} (\sum_{i=1}^{\infty}\zeta_{i}(\tau_{i}Jx+(1\tau_{i})JS_{i}x) )\)for each\(t \in B\), \(\{\alpha_{n}\} \subset[0,1)\), \(\beta, \gamma\in(0,1)\)and\(\{r_{n}\} \subset[a,\infty)\), for some\(a>0\). Then\(\{x_{n}\}\)converges strongly to\(\varPi_{\varOmega}x_{0}\).
Proof
By Remark 2, \(J^{1}T\) is relatively nonexpansive and \(J^{1}S_{i}\) is relatively nonexpansive for each i. The conclusion follows from Theorem 3.2. □
Numerical illustrations
In this section, we give some examples to illustrate the effect of the inertial term in the fast convergence of the sequence of our algorithm. For simplicity, we consider an example in \(\Bbb {R}\) and choose A such that the resolvent can be easily computed.
Example 1
In Theorems 1.3 and 3.1, set \(B=C_{0}=\Bbb {R}\),
Clearly, A is maximal monotone and, T and S are relatively nonexpansive. Furthermore, \(\varOmega=\{0\}\). We choose \(\alpha _{n}=\beta_{n}=\frac{4n}{4n+5}\), \(r_{n}=\frac{2n+1}{n}\), \(\beta=\frac {1}{2}\), \(\gamma=\frac{1}{4}\) as the parameters. Obviously, these parameters satisfy the hypothesis of Theorems 1.3 and 3.1. We choose \(x_{0}=x_{1}=0.5\) and use a tolerance of 10^{−14} and set maximum number of iteration to be 2000 (see Tables 1 and 2 and Figs. 1 and 2).
Next, we give an example to show that Algorithm (3.12) is implementable.
Example 2
In Theorem 3.2, set \(C_{0}=\Bbb {R}\)
Clearly, A is maximal monotone, T is relatively nonexpansive and \(S_{i}\) is relatively nonexpansive for each i. Furthermore, \(\varOmega = (\bigcap_{i=1} ^{\infty}F(S_{i}) ) \cap F(T) \cap A^{1} 0 =\{0\}\). We choose \(\zeta_{i}=\tau_{i}=\frac{1}{2^{i}}\), \(i\geq1 \), and \(\alpha_{n}=\beta_{n}=\frac{4n}{4n+5}\), \(r_{n}=\frac{2n+1}{n}\), \(\beta =\frac{1}{2}\), \(\gamma=\frac{1}{4}\) as the parameters. Clearly, these parameters satisfy the hypothesis of Theorem 3.2. Observe that \(Sx=J^{1} (\sum_{i=1}^{\infty}\eta_{i}(\mu_{i}Jx+{(1\mu_{i})S_{i}x}) )=\frac{7x4\sin x}{21}\). We choose \(x_{0}=1\), \(x_{1}= 2.5\) and use a tolerance of 10^{−14} and set the maximum number of iterations to be 2000 (see Table 3 and Fig. 3).
Conclusion. From the numerical experiments above, we observe that indeed incorporating the inertial term in our algorithm speeds up the convergence of the sequence generated by our algorithm to the desired solution.
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Acknowledgements
The authors acknowledge the African Development Bank (AfDB), the Pan African Material Institute (PAMI), AUST and the Center of Excellence in Theoretical and Computational Science (TaCSCoE) for their financial support. The authors would like to thank the referees for their esteemed comments and suggestions.
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This work is supported from the African Development Bank (AfDB) research grant funds to AUST and the Center of Excellence in Theoretical and Computational Science (TaCSCoE) research grant to KMUTT.
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CEC and PK formulated the problem and suggested the method of proof of the theorem to AA. The computations using the method suggested by CEC and PK was carried out by AA. The analysis of the computations to arrive at the proof of the Theorem was done jointly by CEC, PK and AA. All authors read and approved the final manuscript.
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Chidume, C.E., Kumam, P. & Adamu, A. A hybrid inertial algorithm for approximating solution of convex feasibility problems with applications. Fixed Point Theory Appl 2020, 12 (2020). https://doi.org/10.1186/s1366302000678w
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Keywords
 Inertial
 Maximal monotone
 Fixed point
 Hybrid