Novel fixed point results for a class of enriched nonspreading mappings in real Banach spaces

1.   Introduction

Throughout this research work, $X$ will denote a real Banach space which is endowed with norm $\Vert.\Vert$ and the dual space $X^{\star}$; $\emptyset\neq C\subset \Omega$, $\mathbb{N}$, and $\mathbb{R}$ will represent a closed convex subset of $X$ with at least one member, the set of positive integers, and the set of real numbers, respectively. Let $\{b_{n}\}_{n = 1}^{\infty}$ be a sequence in $C$ and $T:C\longrightarrow C$ be a given mapping. We use $b_{n}\rightharpoonup b$ and $b_{n}\rightarrow b$ to denote that the sequence $\{b_{n}\}_{n = 1}^{\infty}$ converges weakly and strongly to a point $b$, respectively. The generalized duality map is the operator $J_{\phi}:C\longrightarrow2^{C^{\star}}$ associated with the gauge function $\phi$ given by the mapping. Then, the following identity holds:

where $\phi(\ell) = \ell^{q-1}$ for all $\ell\geq0$ and $1 < q < \infty$. Specifically, if $q = 2$, $J_{\phi} = J_{2}$ is called the normalized duality map (NDM) represented as $J$ which is defined by

It is known (see ^[1]) that if $\Omega$ is a real Hilbert space $\mathcal{H}$, the NDM becomes an identity, i.e., $\mathcal{J}\omega = \{\omega\}$.

Let $T:C\longrightarrow C$ be a nonlinear map. The fixed point problem is to search for a point $a\in C$ that assures

We represent with $F(T)$ the set of fixed points of $T$, i.e., $F(T) = \{b\in C:Tb = b\}$. We use $\omega_{\omega}(b_{n}) = \{b:\exists b_{n}\rightharpoonup b\}$ to represent the weak $\omega$-limit set of the sequence $\{b_{n}\}_{n = 1}^{\infty}$. A nonlinear map $T:C\longrightarrow C$ is known as nonexpansive if it satisfies the inequality

The mapping $T$ is known as quasi-nonexpansive (QN), if $F(T)\neq\emptyset$ and (1.3) holds for all $b\in C$ and $a\in F(T)$.

The notion of nonexpansive operators (NM) stands as an indispensable part of the investigation of the Mann-type iterative technique for evaluating invariant points of an operator $T:C\longrightarrow C$, where $C$ is as described above. Recall that the Mann-type iterative technique ^[2], developed from an arbitrary $b_{1}\in C$, is given as follows:

where $\{\delta_{n}\}^{\infty}_{n = 1}\subset [0, 1]$ satisfies some mild conditions.

The problem of investigating fixed points of NM with respect to strong convergence has been widely studied by several authors. In this regard, Halpern ^[3] gave the following general iterative technique:

where $\{\alpha_{n}\}^{\infty}_{n = 1}, \{\delta_{n}\}^{\infty}_{n = 1} \subset [0, 1]$ satisfy appropriate conditions and $u\in C$ is fixed. In particular, if $\alpha_{n} = 0$, (1.5) reduces to the standard Mann iteration (1.4). Moreover, Halpern ^[3] proved the strong convergence result of (1.5) when $\delta_{n} = 0$ and for appropriate conditions on $\{\alpha_{n}\}^{\infty}_{n = 1}$. He further established that the control parameters

are necessary for convergence of (1.5) to the fixed point of $\Im$. Thereafter, several investigations have been done to ascertain the implications of conditions $\mathcal{D}_{1}$ and $\mathcal{D}_{2}$ on the convergence of (1.5); see ^[4,5] for further reading.

In ^[6], Osilike and Isogugu studied the Halpern-type fixed point algorithm for $k$-strictly pseudononspreading mappings $T$, which includes the class of nonspreading mappings (NSM) as a special case. To achieve strong convergence results, they substituted an averaged-type mapping $T_{\delta}$ for the mapping $T$, where

Recently, Kohasaka and Takahashi ^[7,8] studied an important class of nonlinear operators which they referred to as NMS. Let $\Omega$ be a real, smooth, strictly convex (SC) and reflexive Banach space (RBS) and denote by $j:x \longrightarrow 2^{x^{\star}}$ the duality mapping of $x$.

Let $\emptyset\neq C\subset X$ be closed and convex. A mapping $T:C\longrightarrow C$ is called nonspreading if

for all $b, a\in C$, where

for all $b, a\in X.$

Kohasaka and Takahashi considered the class of NSM to study the resolvent of a maximum monotone operator in real, smooth, SC, and RBS. These mappings originate from another group of operators called firmly nonexpansive mappings (see, for example, ^[7,9]). In a real Hilbert space ($\mathcal{H}$), (1.8) reduces to the following identity:

Consequently, if $\mathcal{H}$ and $C$ are as described above, then $T$ is nonspreading if

It is established in ^[10] that (1.9) is equivalent to the inequality

Remark. If $T$ is nonspreading (resp. nonexpansive) and $F(T)\neq\emptyset$, then $T$ is QN.

In ^[10], the authors studied the iterative estimation of common invariant points of NM ($\eth$) and NSM ($\Im$) of $\Lambda$ into itself in $\mathcal{H}$. They studied a technique akin to the one employed by Moudafi in ^[11]. To be precise, they established the following result:

Theorem 1.1 (^[10], Theorem 4.1). Let $\mathcal{H}$ and $C$ be as described above. Let $T, S:C\longrightarrow C$ be as described in Remark 1 with the property that $F(T)\cap F(S)\neq\emptyset$. Let $b_{n}\}_{n = 1}^{\infty}$ be a real sequence generated by

where $\{\alpha_{n}\}_{n = 1}^{\infty}, \{\mu_{n}\}_{n = 1}^{\infty} \subset [0, 1]$. Then, we have:

(1) If $\underset{n = 1}{\overset{\infty}{\sum}}\alpha_{n}(1-\alpha_{n}) = \infty$ and $\underset{n = 1}{\overset{\infty}{\sum}}(1-\mu_{n}) < \infty$, then $b_{n}\rightharpoonup\nu\in F(T).$

(2) If $\underset{n\rightarrow \infty}{\liminf}\alpha_{n}(1-\alpha_{n}) > 0$ and $\underset{n = 1}{\overset{\infty}{\sum}}\mu_{n} < \infty$, then $b_{n}\rightharpoonup\nu\in F(T).$

(3) If $\underset{n\rightarrow \infty1}{\liminf}\alpha_{n}(1- \alpha_{n}) > 0$ and $\underset{n\rightarrow \infty1}{\liminf}\mu_{n}(1-\mu_{n}) > 0$, then $b_{n}\rightharpoonup\nu\in F(T)\cap F(S).$

Finding the fixed points of NM, NSM and some other related mappings have remained invaluable topics in fixed point theory, and have been shown to be fundamental in the applied areas of signal processing ^[12], the split feasibility problems ^[13], and convex feasibility problem ^[14]. In subsequent works, Berinde ^[15,16] came up with the notion of enriched nonlinear mappings as a generalization of the class of NM in the setup of $\mathcal{H}$. This concept was later studied in a more general Banach space by Saleem, Agwu and Igbokwe ^[17,18].

Definition 1.2. A mapping $T:C\longrightarrow C$ is referred to as $\psi_{T}$-enriched Lipschitzian (or $(\sigma, \psi_{T})$-enriched Lipshitzian) (see ^[17,18]) (shortly, $(\sigma, \psi_{T})$-ELM) if for all $b, a\in Ca$, there exist $\sigma\in [0, +\infty)$ and a continuous nondecreasing function $\psi_{T}:\mathcal{R}^{+}\longrightarrow \mathcal{R}^{+}$, with $\psi_{T}(0) = 0$, such that

Observe from inequality (1.11) that:

$(a)$ If $\sigma = 0$, then the class of mappings called $\psi_{T}$-Lipschitzian emerges.

$(b)$ If $\sigma = 0$ and $\psi(t) = Lt$, for $L > 0$, then (1.11) reduces to a class of mappings known as $L$-Lipschitzian, $L$ represents the Lipschitz constant. In particular, if $\sigma = 0, \Psi_{\Im}(t) = Lt,$ and $L = 1$, then $(\sigma, \psi_{T})$-ELM immediately reduces to the class of NM on $C$.

$(c)$ If $\psi_{T}(t) = t$, then inequality (1.11) becomes

and it is called a $\sigma$-enriched nonexpansive mapping. The class of mappings defined by (1.12) was first studied by Berinde ^[15,16] as a generalization of a well-known class of nonlinear mappings called NM.

Closer observation reveals that if $\psi_{T}$ is not necessarily nondecreasing and guarantees the condition

then we have the class of $\sigma$-enriched contraction mappings.

In view of the papers studied, particularly, the results obtained by Lemoto and Takahashi ^[10], Berinde^[16], and other related results in this direction, we consider the following questions:

Question 1.3. (1) Could there be a nonlinear mapping that contains the class of mappings defined by (1.9) for which we would obtain the results in ^[10] as special cases?

(2) Could it be possible to obtain a strong convergence result for an averaged mapping in a more general Banach space?

Lemoto and Takahashi considered the class of nonspreading mappings and proved the weak convergence theorem as their main result in ^[10] in the setup of $\mathcal{H}$. Their results together with those of Kohasaka and Takahashi ^[8] opened a new direction in metric fixed point theory. In the current paper, we shall consider a new class of nonlinear mapping called $\sigma$-enriched nonspreading mappings (($\sigma$)-ENSM) in the setup of $\Omega$. Further, we present some nontrivial examples to demonstrate its existence (and its independency on the class of $\sigma$-enriched nonexpansive mappings (($\sigma$)-ENEM)). By modifying the iterative method studied in ^[10], we established strong convergence theorems which solve the problems raise in Question 1.3.

The rest of the paper is organized as follows: Section 2 will consider preliminary results which will be needed in establishing our main results. Proposition 3.7, and Theorems 3.6, 3.8, 3.9, and 3.11, which will serve as our main results (including some of their consequences) and the conclusion of the results obtained in this paper, will be considered in Section 3.

2.   Preliminaries

The convexity of a Banach space $X$ is characterized by the function $\delta(\epsilon):(0, 2]\longrightarrow [0, 1]$, known as the modulus of convex of $X$, defined by

The space $C$ is regarded as uniformly convex if and only if $\delta(\epsilon) > 0$ for every $\epsilon$ such that $0\leq\epsilon\leq2$. Let $S_{X} = \{b\in X:\Vert b\Vert = 1\}$. For each $b, a\in S_{X}$, the norm of $X$ is referred to as Gateaux differentiable provided the limit

exists. For the above case, $X$ is called smooth. It is known as uniformly smooth (US) if the limit described by (2.1) is achieved uniformly in $X$; whereas it is called strictly convex (SC) if

whenever $b, a\in S_{X}$ and $b\neq a$. It is an established fact that $X$ is uniformly convex (UC) if and only if $X^{\star}$ is US; for further details, see ^[19].

The smoothness of a Banach space $X$ is characterized by the function $\rho:[0, \infty)\longrightarrow [0, \infty)$, known as the modulus of convexity of $X$, defined by

It is known that $X$ is US provided

Set $1 < q\leq2$. Then, $X$ is called $q$-uniformly smooth if we can find a constant $c_{q} > 0$ with the property that $\rho(t)\leq c_{q}t^{q}$ for all $t > 0$. It is worth noting that $X$ assumes smoothness if the mapping $J$ (called a sequentially continuous duality mapping) dwells in $X$ with its domain in the weak topology and the range in the weak-star topology. In this case, $b_{n}\rightharpoonup b\in X \Rightarrow Jb_{n} \overset{\star}{\rightharpoonup} Jb$; see, for example, ^[20] for further details. $X$ enjoys the Opial property ^[21] if for any sequence $\{b_{n}\}_{n = 1}^{\infty}$ which converges weakly in $X$ with a weak limit $\wp$,

for all $a\in X$ with $b\neq a$. It is a known fact that all $X$ with finite dimension, all Hilbert spaces and all spaces accredited to $\ell^{p}(1\leq p < \infty)$ admit the Opial property; see ^[20,21] for more details. It is also on record that if $X$ recognizes $J$, then it is smooth and also assures the employment of the Opial property; see ^[20].

Let $X, C, \rightharpoonup$ and $\rightarrow$ be as described in section one. Let $\emptyset\neq M$ and $\emptyset\neq N$ be two subsets of $X$ with $M\subset N$. An operator $Q_{N}:M\longrightarrow N$ is called sunny if

for each $b\in X$ and $\xi\geq0$. A mapping $Q_{N}:m\longrightarrow N$ is said to be a retraction if $Q_{N}b = b$ for each $b\in C$.

Lemma 2.1. ^[22] Let $\Omega$ be as described above and $\emptyset\neq M, N\subset \Omega$ be such that $M\subset N$. Let $Q_{N}:M\longrightarrow N$ be a retraction of $M$ onto $N$. Then $Q_{N}$ is sunny and nonexpansive if and only if

for all $b\in M$ and $a\in N$, where $j(a-Q_{N}(b)) \in J(a-Q_{N}(b))$ retains its usual meaning in $X$.

Lemma 2.2. ^[22] Let $\Omega$ and $J$ be as described above. Then,

for all $b, a\in X$ and for all $j(b+ a) \in J(b+\ a)$.

Proposition 2.3. Let $\mathcal{H}$ be a real Hilbert space, $\emptyset\neq C\subset \mathcal{H}$ and $T: C\longrightarrow C$ be a $\sigma$-enriched nonspreading mapping. Then, $F(T)$ is closed and convex.

Proof. Let $\{b_{n}\}_{n = 1}^{\infty}$ be a sequence in $F(T)$ which converges to $b.$ We want to show that $b\in F(T)$. Now, since

and since $T$ is a $\sigma$-enriched nonspreading mapping, we have

Equations (2.2) and (2.3) imply that

Hence, $b\in F(T)$.

Next, let $\vartheta_{1}, \vartheta_{2}\in F(T)$ and $\lambda\in [0, 1]$. We prove that $\lambda\vartheta_{1}+(1-\lambda)\vartheta_{2}\in F(\Im)$. Let $\psi = \lambda\vartheta_{1}+(1-\lambda)\vartheta_{2}$. Then, $\vartheta_{1}b = (1-\lambda)(\vartheta_{1}-\vartheta_{2})$ and $\vartheta_{2}-b = \lambda(\vartheta_{2}-\vartheta_{1})$. Since

it follows that

Therefore, $b = Tb$ implies that $b\in F(T)$ as required. □

Definition 2.4. Let $X$ be as described above, $\emptyset\neq C\subset X$ be closed and convex, and $\{b_{n}\}_{n = 1}^{\infty}$ be a bounded sequence in $X$. For any $b\in C$, we set

The asymptotic radius of $\{b_{n}\}_{n = 1}^{\infty}$ with respect to $C$ is given as

The asymptotic center of $\{b_{n}\}_{n = 1}^{\infty}$ with respect to $C$ is the set

It is an established fact that if $X$ is UC, then $A(C, \{b_{n}\}_{n = 1}^{\infty})$ is fixed at a point (see, for instance, ^[3,22]).

Lemma 2.5. ^[23] Let $\{\nu_{n}\}_{n = 1}^{\infty}$ be a sequence of non-negative real real numbers validating the following inequality:

where $\{\pi_{n}\}_{n = 1}^{\infty}$ and $\{\mu_{n}\}_{n = 1}^{\infty}$ satisfy the conditions:

$(i)\; \{\pi_{n}\}_{n = 1}^{\infty}\subset [0, 1]$ and $\underset{n = 0}{\overset{\infty}{\sum}}\pi_{n} = \infty$ or, equivalently, $\underset{n = 1}{\overset{\infty}{\prod}}(1-\pi_{n}) = 0$;

$(ii)\; \limsup_{n\rightarrow \infty}\mu_{n}\leq 0$ or $\underset{n = 0}{\overset{\infty}{\sum}}\pi_{n}\mu_{n} < \infty$.

Then, $\underset{n\rightarrow \infty}{\lim}\nu_{n} = 0$.

Lemma 2.6. ^[19] Let $\{\gamma_{n}\}_{n = 0}^{\infty}\subset\mathbb{R}$ be such that we can find a subsequence $\{\gamma_{n_{k}}\}_{k = 0}^{\infty}$ such that $\gamma_{k} < \gamma_{k+1}$ for all $k\in \mathbb{N}$. Consider the sequence of integers $\{\tau(n)\}_{n = 1}^{\infty}$ given by

Then, $\{\tau(n)\}_{n = 1}^{\infty}$ is a nondecreasing sequence, for all $n\geq n_{0}$, validating the following requirements:

$(i)\; \underset{n\rightarrow \infty}{\lim}\tau(n) = \infty;$

$(ii)\; \gamma_{\tau(n)} < \gamma_{\tau(n)}, \quad \forall n\geq n_{0}$;

$(iii)\; \gamma_{n} < \gamma_{\tau(n)}, \quad \forall n\geq n_{0}$.

Lemma 2.7. ^[1,22] Let $X$ be a UC and $B_{\varrho}\{\wp\in X:\Vert\wp\Vert\leq \varrho\}, \varrho > 0$. Then, we can find a continuous, strictly increasing function $g:[0, \infty)\longrightarrow [0, \infty)$ with $g(0) = 0$ such that

for all $b, a, c\in B_{\varrho}$ and for all $r, s, t \in [0, 1, ]$ with $r+s+t = 1$.

The proposition below assures some essential properties of generalized duality mapping ($J_{\phi}$).

Proposition 2.8. ^[18,24] Let $X$ and $X^{\star}$ be as described above. For $q\in(1, \infty)$, $J_{\phi}: X\longrightarrow2^{X^{\star}}$ has the following fundamental properties:

$(1)\; J_{\phi}(b)\neq\emptyset \forall b\in X$ and $D(J_{\phi}) (:the\; dormain\; of\; J_{\phi}) = X$;

$(2)\; J_{\phi}(b) = \Vert b\Vert^{\phi-1} J_{2}(b), \quad \forall b\in X (b\neq0)$;

$(3)\; Jb_{ \phi}(\alpha b) = \alpha^{\phi-1} J_{\phi}(b), \quad \alpha\in [0, \infty)$;

$(4)\; J_{ \phi}(-b) = - J_{\phi}(b)$.

3.   Results and discussion

Definition 3.1. Let $X$ be as described above. A mapping $T$ with domain $D(T)$ and range $R(T)$ in $X$ is known as $\sigma$-enriched nonspreading ($\sigma$-ENSM, for short) in the sense of Kurokawa and Takahashi^[25] if there exists $\sigma\in [0, \infty)$ and $j(a-Ta)\in J(a-Ta)$ such that for all $b, a\in D(T)$, the following inequality holds:

Now, by setting $\sigma = \dfrac{1}{\beta}-1$, for some $\beta\in (0, 1]$, it follows from Proposition 2.8 (3) and (3.1) that

Remark. Observe that if $\sigma = 0$ in (3.1) (or $\beta = 1$ in (3.2)), we obtain an important class of nonspreading mappings studied in ^[25]. Again, if we take $T_{\beta} = (I-\beta)I+\beta T$, then (3.2) reduces to the inequality

Therefore, the averaged operator $T_{\beta}$ is a nonspreading mapping whenever $T$ is a $\sigma$-enriched nonspreading mapping.

Remark. Any nonspreading mapping $T$ validating (3.1) with $\sigma = 0$ is known as $0$-enriched nonspreading.

Example 3.2. Let $B_{\rho}\{b\in \mathcal{H}:\Vert b\Vert\leq\rho\}$ for $\rho > 0$ and $C = B_{2}\subset\mathcal{H}$. Define an operator $T:C\longrightarrow C$ by

where $P_{A}$ is a projection map of $\mathcal{H}$ onto $\mathcal{A}$. Then, $T$ is an enriched nonspreading mapping which does not admit continuity. Obviously, $F(\Im) = B_{2}$. Let $b, a\in C$. It suffices to examine the situation for which $b\in \mathcal{C}\setminus B_{2}, a\in B_{2}.$ Now, since $P_{B_{1}}$ is nonexpansive (and hence $0$-enriched nonexpansive) and $b- Tb = 0$, it follows that

Therefore, $T$ is a $\sigma$-ENSM. Clearly, $T$ is not continuous. In fact, for $b_{0}\in\partial B_{2}, \omega_{0}\in \partial C,$ consider $c_{n} = \big(1-\dfrac{1}{n}\big) b_{0}+\dfrac{1}{n} b_{0}\in C$ for each $n\geq1$. Then, $b_{n}\rightarrow b_{0}$ but $Tb_{n} = P_{B_{1}} b_{n}\nrightarrow Tb_{0}$ because $\Vert P_{B_{1}}b_{n}\Vert = 1$ and $\Vert b_{0}\Vert = 2$.

Remark. Note that $T$ is not continuous in the last example; hence, $T$ is not uniformly continuous. In other words, the class of $\sigma$-ENSM is generally not Lipschitzian.

The following examples demonstrate the fact that the class of $\sigma$-ENEM and the class of $\sigma$-ESNM are independent.

Example 3.3. Let $\mathbb{R}\supset \mathcal{C} = \Big[\dfrac{1}{2}, 2\Big]$ be endowed with the usual norm and let $\Im:\mathcal{C}\longrightarrow\mathcal{C}$ be define by $\Im\psi = \dfrac{1}{\wp}$ for all $\psi\in \mathcal{C}$. Then,

$(i)\; T$ is not nonexpansive.

$(ii)\; T$ is $\dfrac{3}{2}$-enriched nonexpansive.

$(iii)\; F(T) = \{1\}$.

$(iv)\; T$ is not a $\dfrac{3}{2}$-ESNM.

To validate $(i)-(iv)$:

$(i)$ Assume $T$ is NE. Then, by the definition of NE, we should have

which, when $b = \dfrac{1}{2}$ and $a = 1$, yields a contradiction.

$(ii)$ For all $\forall b, a\in C$,

Observe that for any $\sigma\geq\dfrac{3}{2}$, the last identity becomes

and as such validates our conclusion that $T$ is $\dfrac{3}{2}$-enriched nonexpansive.

$(iii)\; F(T) = \{1\}$ is not difficult to see.

$(iv)$ Since every $\sigma$-enriched nonexpansive mapping satisfies the $\sigma$-enriched Lipschitz condition (see, for instance, ^[17]),

where $L$ is the Lipschitz constant, and since every $\sigma$-enriched nonspreading mapping is generally not Lipschitzian (see Example 3.2 and Remark 3 above), it follows from $(ii)$ that $\Im$ is not a $\sigma$-enriched nonspreading mapping.

Example 3.4. Let $X = \mathbb{R}$ denote the reals with the usual norm. For each $\psi\in \mathbb{R}$, let the mapping $\Im$ be given by

Then, for all $b, a\in (-\infty, 2]$ and for all $\sigma\in [0, \infty)$, we have

Also, for all $b, a\in (2, \infty)$ and for all $\sigma\in [0, \infty)$, we have

Finally, if $b\in (-\infty, 2]$ and $a\in (2, \infty)$, then for all $\sigma\in [0, \infty)$, we get

Thus, for all $b, a\in X$ and for all $\sigma\in [0, \infty)$, we obtain

Hence, $T$ is $\sigma$-enriched nonspreading. Since every $\sigma$-enriched nonexpansive mapping $T$ must satisfy $\sigma$-enriched Lipschitz condition (see, for instance, ^[17])

where $L$ is the Lipschitz constant. It is not difficult to see that $T$ is not a $\sigma$-enriched nonexpansive mapping.

The next example shows that a $\sigma$-ENSM is different from a NSM thereby leading to the conclusion that the class of a $\sigma$-ENSM properly contains the class of NSM.

Example 3.5. Let $\mathbb{R}$ be as described above with the usual norm and suppose the mapping $T:\mathbb{R}\longrightarrow\mathbb{R}$ is given by

Then, $T$ is a $\sigma$-enriched nonspreading mapping. Indeed, for all $b, a\in\mathbb{R}$ with

and

there exists a $\sigma\in [1, \infty)$ such that

However, $T$ is not nonspreading, for if $b\neq0$ and $a = -b$, then

Remark. If $F(T)\neq\emptyset$ in (3.1), then we obtain a class of mapping called $\sigma$-enriched quasi-nonexpansive mappings.

Now, we give the existence theorems of invariant points of $\sigma$-ENSM in $X$.

Theorem 3.6. Let $X$ be a UC and $\emptyset\neq C \subset X$ be closed and convex. Let $T:C\longrightarrow C$ be a $\sigma$-ENSM. Then, the statements below are equivalent:

$(i)$ The invariant point set of $F(T)\neq\emptyset$.

$(ii)\; \exists \{b_{n}\}_{n = 1}^{\infty}\subset C,$ with $\{b_{n}\}_{n = 1}^{\infty}$ bounded, such that $\underset{n\rightarrow \infty}{\liminf} \Vert b_{n}-Tb_{n}\Vert = 0$.

Proof. We can see that $(i)\Rightarrow (ii)$ is quite obvious. To establish the opposite implication, we assume $\exists \{b_{n}\}_{n = 1}^{\infty}\subset C,$ with $\{b_{n}\}_{n = 1}^{\infty}$ bounded, such that $\underset{n\rightarrow \infty}{\liminf} \Vert b_{n}-Tb_{n}\Vert = 0$. As a consequence, $\exists\{Tb_{n_{k}}\}_{k = 1}^{\infty}\subset \{Tb_{n}\}_{n = 1}^{\infty}$ for which $\underset{k\rightarrow \infty}{\liminf} \Vert b_{n_{k}}-Tb_{n_{k}}\Vert = 0$. Suppose $A(C, \{b_{n_{k}}\}_{k = 1}^{\infty}) = \{\varrho\}$. Let $\Theta_{1} = \sup\{\Vert b_{n_{k}}\Vert, \Vert Tb_{n_{k}}\Vert, \Vert\varrho\Vert, \Vert T\varrho\Vert:k\in \mathbb{N}\} < \infty$.

Since the mapping $T$ is $\sigma$-nonspreading, it follows that

It, therefore, follows from the last inequality that

As a consequence, we obtain

This, by implication, entails that $T\varrho\in A(C, \{b_{n}\}_{n = 1}^{\infty})$. In view of the uniform convexity of $C$, we conclude that $T\varrho = \varrho$ as required. □

The result below is an immediate consequence of Theorem 3.6.

Proposition 3.7. Let $X$ and $C$ be as described in Theorem 3.6. Let $T:C\longrightarrow X$ be a $\sigma$-ESNM with $F(T) \neq\emptyset$. If $b_{n}\rightharpoonup\varrho\in C$ and $(I-T)b_{n}\rightarrow0$, then $\varrho\in F(T)$.

Theorem 3.8. Let $T$ and $C$ be as in Theorem 3.6 with $X$ admitting the Opial property. Let $T:C\longrightarrow C$ be a $\sigma$-ESNM such that $F(T) \neq\emptyset$. If $\{\gamma_{n}\}_{n = 1}^{\infty}$ is a sequence in $(0, 1)$ with $0 < \alpha\leq\gamma_{n}\leq1-\alpha < 1$, and $\{b_{n}\}_{n = 1}^{\infty}$ is a sequence in $C$ developed from

where $T_{\beta} = (I-\beta)I+\beta T$, then (3.4) converges weakly to an element of $F$.

Proof. Let $\varrho\in F(T) = F(T_{\beta})$ be arbitrarily chosen. Then, by Lemma 2.7, we can find a strictly increasing function $g:[0, \infty)\longrightarrow[0, \infty)$, characterized by convexity and the continuity property, with $g(0) = 0$ such that

Since $\alpha > 0$ and $\sigma\in [0, \infty)$, it follows from (3.5) that

This implies that $\underset{n\rightarrow \infty}{\lim} \Vert b_{n}-\varrho\Vert$ exists. Therefore, $\{b_{n}\}_{n = 1}^{\infty}$ is bounded. By setting

we obtain from (3.5) that

which yields that

But, $\{b_{n}\}_{n = 1}^{\infty}$ is bounded. Therefore, $\exists\{b_{n_{k}}\}_{k = 1}^{\infty} \subset\{b_{n}\}_{n = 1}^{\infty}$ such that $b_{n_{k}}\rightharpoonup\varrho$. Also, $\underset{n\rightarrow \infty}{\lim}\Vert b_{n}-Tb_{n}\Vert = 0$ implies that $\underset{k\rightarrow \infty}{\lim} \Vert b_{n_{k}}-Tb_{n_{k}}\Vert = 0$. From Proposition 3.7, $(I-T)b_{n}\rightarrow0$ as $n\rightarrow \infty$. Consequently, $\varrho\in F(\Im)$. To conclude, it suffices to establish the fact that for another subsequence $\{b_{n_{i}}\}_{i = 1}^{\infty}\subseteq \{b_{n}\}_{n = 1}^{\infty}$ which is characterized by the weak convergence property (i.e., $b_{n_{i}}\rightharpoonup\nu$ as $n\rightarrow \infty$), we have $\varrho = \nu$. Suppose otherwise and let $\varrho \neq\nu$. Then, we get from Opial's theorem that

This is a contradiction. Consequently, $\{b_{n}\}_{n = 1}^{\infty}$ converges weakly to $\varrho\in F(T)$. □

Theorem 3.9. Let $X$ be a UC which admits a weakly sequentially continuous duality mapping $J$, $\emptyset\neq C \subset X$ be closed and convex, and $T:C\longrightarrow C$ be a $\sigma$-enriched nonspreading mapping such that $F(T) \neq\emptyset$. Let $\{\gamma_{n}\}_{n = 1}^{\infty}$ and $\{\delta_{n}\}_{n = 1}^{\infty}$ be two sequences in $(0, 1)$ such that the following requirements are validated:

$(a)\; \underset{n\rightarrow \infty}{\lim}\gamma_{n} = 0$;

$(b)\; \underset{n = 1}{\overset{\infty}{\sum}} \gamma_{n} = \infty;$

$(c)\; \underset{k\rightarrow \infty}{\liminf} \delta_{n}(1-\delta_{n}) > 0.$

Let the sequence $\{b_{n}\}_{n = 1}^{\infty}$ be developed from

where $T_{\beta} = (I-\beta)I+\beta T$. Then, $\{b_{n}\}_{n = 1}^{\infty}$ given (3.6) admits strong convergence to a point $\varrho \in Q_{F}u$, where $Q_{F}$ denotes a sunny nonexpansive retraction from $X$ onto $F(T)$.

Proof. Since $T_{\beta}$ is quasi-nonexpansive, we have that $F(T)$ is closed and convex. Set

We shall divide the rest of the proof into several steps.

Step 1. We demonstrate that $\{b_{n}\}_{n = 1}^{\infty}$, $\{\vartheta_{n}\}_{n = 1}^{\infty}$, and $\{Tb_{n}\}_{n = 1}^{\infty}$ are bounded. First, we establish that $\{b_{n}\}_{n = 1}^{\infty}$ admits boundedness.

Fix $\varrho\in F(T_{\beta}) = F(T)$. Using Lemma 2.7, we can find a strictly increasing function $g:[0, \infty) \longrightarrow[0, \infty)$ which is characterized by continuity, convexity, and $g(0) = 0$ properties such that the following estimates hold:

Again, from (3.6), we have

Using induction, we get

The last inequality yields that $\{\Vert b_{n}-\varrho\Vert\}_{n = 1}^{\infty}$ is bounded and as a consequence, $\{b_{n}\}_{n = 1}^{\infty}$ is bounded. The boundedness of $\{\vartheta_{n}\}_{n = 1}^{\infty}$ and $\{Tb_{n}\}_{n = 1}^{\infty}$ follows from the above result and (3.6).

Step 2. Now, for any $n\in\mathbb{N}$, we want to show that

To do this, note that for each $n\in\mathbb{N}$, (3.7) (with $\varrho = \xi$) gives

This, together with (3.6), gives

Set $\Theta_{2} = \sup\Big\{\Vert u-\xi\Vert -\Vert b_{n}-\xi\Vert^{2} +\delta(1-\delta_{n})g\Big(\dfrac{1}{\sigma+1}\Vert b_{n} -Tb_{n}\Vert\Big):n\in \mathbb{N}\Big\}$. Then, we obtain from (3.9) that

Now, from Lemma 2.2 and (3.6), we get

Step 3. Now, we demonstrate that $\underset{n\rightarrow \infty}{\lim}b_{n} = \xi$.

To do this, we consider the two cases below:

Case A. If the sequence $\{\Vert b_{n}-\xi\Vert\}_{n = 1}^{\infty}$ is monotonically decreasing, then there exists an $n_{0}\in\mathbb{N}$ for which $\{\Vert b_{n}-\xi\Vert\}_{n = n_{0}}^{\infty}$ is decreasing. Consequently, $\{\Vert b_{n}-\xi\Vert\}_{n = 1}^{\infty}$ is convergent and as such $\underset{n\rightarrow \infty}{\lim} (\Vert b_{n}-\xi\Vert^{2} -\Vert b_{n+1}-\xi\Vert^{2}) = 0$. This, in view of condition (c) and (3.10), yields

From the property of $g$, we have

Since from (3.6)

it follows from (3.11) and condition (a) that

Therefore, using the triangular inequality and (3.12), we get

as $n\rightarrow \infty \quad (\text{by}\quad (3.12))$.

Since $\{b_{n}\}_{n = 1}^{\infty}$ is bounded, there exists a subsequence $\{b_{n_{k}}\}_{k = 1}^{\infty}$ of $\{b_{n}\}_{n = 1}^{\infty}$ such that $b_{n_{k}+1}\rightharpoonup \varpi\in \Lambda$ as $k\rightarrow \infty$. It, therefore, follows from Proposition 3.7 and (3.11) that $\varpi\in F(T)$. This, together with Lemma 2.1, implies that

Thus, by Lemma 2.5, the result follows immediately.

Case B. If the sequence $\{\Vert b_{n}-\xi\Vert\}_{n = 1}^{\infty}$ is not eventually decreasing, then there exists a subsequence $\{n_{k}\}_{k = 1}^{\infty}$ of $\{n\}_{n = 1}^{\infty}$such that

for all $k\in \mathbb{N}$. Using Lemma 2.6, we can find a nondecreasing sequence $\{m_{j}\}_{j = 1}^{\infty}\subset\mathbb{N}$ such that $m_{j}\rightarrow \infty$ and

for all $j\in \mathbb{N}$. This, together with (3.10), yields

From the requirements of (a) and (c) and the property of $g$, it follows that

Using the same method employed in Case A, we obtain

Since from (3.8)

and $\Vert b_{m_{j}}-\xi\Vert < \Vert b_{m_{j}+1}-\xi\Vert$, it follows that

In particular, since $\gamma_{m_{j}} > 0$, it follows from (3.17) that

and hence

The last identity, together with (3.16), yields

On the other hand, we have that $\Vert\wp_{m_{j}}-\xi\Vert < \Vert b_{m_{j}+1}-\xi\Vert \; \forall j\in\mathbb{N}$, which yields $b_{j}\rightarrow \xi$ as $j\rightarrow \infty$. Hence, $b_{n}\rightarrow \xi$ as $n\rightarrow \infty$ and the proof is complete. □

Corollary 3.10. Let $\emptyset\neq C \subset X$ be closed and convex, where $X$ is a real Hilbert space. Let $T:C\longrightarrow C$ be a $\sigma$-enriched nonspreading mapping such that $F(T) \neq\emptyset$, and $\{\gamma_{n}\}_{n = 1}^{\infty}$ be a sequence in $(0, 1)$ which validates the requirements that:

(a) $\underset{n\rightarrow \infty}{\lim}\gamma_{n} = 0$;

(b) $\underset{n = 1}{\overset{\infty}{\sum}}\gamma_{n} = \infty$.

Then $\{\wp_{n}\}_{n = 1}^{\infty}$ developed from

where $T_{\beta} = (I-\beta)I+\beta T$ admits strong convergence to a point $\varrho\in Q_{F}u$, where $P_{F}$ is the metric projection from $X$ onto $F$.

Theorem 3.11. Let $X$ and $C$ be as described in Theorem 3.9. Let $T_{\beta, 1}:C\longrightarrow C$ be a $\sigma$-enriched nonspreading mapping and $T_{\beta, 2}:C\longrightarrow C$ be a $\sigma$-enriched nonexpansive mapping such that $F(T_{1}) \cap F(T_{2})\neq\emptyset$. Let $\{\gamma_{n}\}_{n = 1}^{\infty}, \{\delta_{n, 1}\}_{n = 1}^{\infty}, \{\delta_{n, 2}\}_{n = 1}^{\infty}$, and $\{\delta_{n, 3}\}_{n = 1}^{\infty}$ be four sequences in $[0, 1]$ which validate the requirements that:

$(a)\; \underset{n\rightarrow \infty}{\lim}\gamma_{n} = 0$;

$(b)\; \underset{n = 1}{\overset{\infty}{\sum}} \gamma_{n} = \infty;$

$(c)\; \delta_{n, 1}+\delta_{n, 2}+\delta_{n, 3} = 1;$

$(d)\; \underset{k\rightarrow \infty}{\liminf} \delta_{n}(1-\delta_{n}) > 0.$

Then, $\{b_{n}\}_{n = 1}^{\infty}$ developed from

admits strong convergence to a point $\varrho\in Q_{F}u$, where $T_{\beta, 1} = (I-\beta)I +\beta T_{1},$ $T_{\beta, 2} = (I-\beta)I+\beta T_{2},$ and $Q_{F}$ denotes a sunny nonexpansive retraction from $X$ onto $F$.

Proof. Since $T_{\beta, 1}$ and $T_{\beta, 2}$ are quasi-nonexpansive, we have that $F(T)\cap F(S)$ is closed and convex. Set

We shall divide the rest of the proof into several steps.

Step 1. We demonstrate that $\{b_{n}\}_{n = 1}^{\infty}$, $\{\vartheta_{n}\}_{n = 1}^{\infty}$, $\{Tb_{n}\}_{n = 1}^{\infty}$, and $\{Sb_{n}\}_{n = 1}^{\infty}$ are bounded. First, we establish that $\{b_{n}\}_{n = 1}^{\infty}$ is bounded.

Let $\varrho\in F = F(T_{1})\cap F(T_{2})$ be fixed. Using Lemma 2.7, we can find a strictly increasing function $g:[0, \infty) \longrightarrow[0, \infty)$ which is characterized by continuity, convexity, and $g(0) = 0$ properties such that the following estimates hold:

Also, from (3.19), we have

which by induction yields

and as a consequence, it follows that $\{\Vert\wp_{n}-\varrho\Vert\}_{n = 1}^{\infty}$ is bounded. The boundedness of $\{b_{n}\}_{n = 1}^{\infty}, \{\vartheta_{n}\}_{n = 1}^{\infty}, \{T_{1}b_{n}\}_{n = 1}^{\infty}$, and $\{\Im_{2}b_{n}\}_{n = 1}^{\infty}$ follows directly from the boundedness of $\{b_{n}\}_{n = 1}^{\infty}$ and (3.19).

Step 2. We establish that

for any $n\in\mathbb{N}$. To do this, note that for each $n\in\mathbb{N}$ and $i = 1, 2$, (3.20) (with $\varrho = \xi$) gives

This, together with (3.19), gives

Set $\Theta_{3} = \sup\Big\{\Vert u-\xi\Vert -\Vert b_{n}-\xi\Vert^{2} +\delta_{n, i}\delta_{n, 3}g\Big(\dfrac{1}{\sigma+1}\Vert b_{n} -T_{i}b_{n}\Vert\Big):n\in \mathbb{N}, i = 1, 2\Big\}$. Then, we obtain from (3.22) that

Now, from Lemma 2.2 and (3.19), we get

Step 3. We demonstrate that $\wp_{n}\rightarrow \xi$ as $n\rightarrow \infty$.

To show this, consider the two cases below:

Case A. If the sequence $\{\Vert b_{n}-\xi\Vert\}_{n = 1}^{\infty}$ is monotonically decreasing, then $\exists n_{0}\in\mathbb{N}$ for which $\{\Vert b_{n}-\xi\Vert\}_{n = n_{0}}^{\infty}$ is decreasing. Consequently, $\{\Vert b_{n}-\xi\Vert\}_{n = 1}^{\infty}$ is convergent and $\underset{n\rightarrow \infty}{\lim}(\Vert b_{n}-\xi\Vert^{2} -\Vert\wp_{n+1}-\xi\Vert^{2}) = 0$. This, in view of requirement (c) and (3.23), yields

Employing the property of $g$, we have

Since from (3.19)

and $b_{n+1}-\vartheta_{n} = \gamma_{n}(u-\vartheta_{n})$, it follows from (3.24) and condition (a) that

Therefore, using the triangular inequality and (3.25), we get

Since $\{b_{n}\}_{n = 1}^{\infty}$ is bounded, we can find a subsequence $\{b_{n_{k}}\}_{k = 1}^{\infty}$ of $\{b_{n}\}_{n = 1}^{\infty}$ such that $b_{n_{k}+1}\rightharpoonup \varpi\in C$ as $k\rightarrow \infty$. It, therefore, follows from Proposition 3.7 and (3.24) that $\varpi\in F$. This, together with Lemma 2.1, implies that

Thus, by Lemma 2.5, the result follows immediately.

Case B. Using the approach employed in establishing Theorem 3.9, we can show that $\underset{n\rightarrow \infty}{\lim}b_{n} = \varrho$, and the proof is complete. □

Remark. (1) The result of this research work solves the question posed by Kurokawa and Takahashi; see Remark on page 1567 in ^[25].

(2) Theorem 4.1 of ^[10] admits only a weak convergent result while our Theorem 3.11 admits a strong convergence result. However, it is worth mentioning that the technique involved in proving Theorem 3.11 is very different from the one employed in proving Theorem 4.1.

(3) In most cases, strong convergence results are better than weak convergence results in applications.

4.   Rate of convergence

For a nonempty convex subset $C$ of a space $X$ and $T:\longrightarrow C$:

(1) The Mann ($M_{n}$) iteration method (see ^[2]) is defined by the following sequence $\{b_{n}\}$:

where $\{\gamma_{n}\}$ is a sequence in $(0, 1)$.

(2) The sequence $\{b_{n}\}$ given by

where $\{\gamma_{n}\}, \{\delta_{n}\}$ are sequences in $(0, 1)$, is called the Ishikawa ($Ish_{n}$) method (see ^[26]).

(3) Our method ($I_{n}$) is given by

where $\{\gamma_{n}\}, \{\delta_{n}\}$ are sequences in $(0, 1)$ and $T_{\beta} = (1-\beta)I+\beta T$ (with $\beta\in (0, 1)$).

Definition 4.1. ^[27] Suppose that $\{c_{n}\}$ and $\{d_{n}\}$ are two real convergent sequences with limits $c$ and $d$, respectively. Then, $\{c_{n}\}$ is said to converge faster than $\{d_{n}\}$ if

Now, using the example below, we prove that the iteration process $I_{n}$ used in obtaining our main result of Theorem 3.9 is faster than the Mann $M_{n}$ and Ishikawa $Ish_{n}$ methods for enriched nonspreading operators.

Example 4.2. Suppose $T:\mathbb{R}\longrightarrow \mathbb{R} = -b, \gamma_{n} = \dfrac{4}{\sqrt{n}}, \beta = \dfrac{5}{6}, \delta_{n} = \dfrac{1}{4},$ and $u = \dfrac{1}{2}$. It is clear that $T$ is an enriched nonspreading mapping with a unique fixed point of $0$ (see Example 3.5 above). Also, it is not difficult to see that Example 4.2 satisfies all the conditions of Theorem 3.9.

Proof. Since $\gamma_{n} = \dfrac{4}{\sqrt{n}}, \beta = \dfrac{5}{6},$ and $\delta_{n} = \dfrac{1}{4}$, it follows from $M_{n}, Ish_{n}$, and $I_{m}$ that for $b_{0}\neq0$,

and

Now, consider

It is not difficult to see that

Hence,

Thus, our iteration scheme converges faster than Mann's iteration method to the fixed point of $T$.

Similarly,

with

Therefore,

Thus, our iteration scheme converges faster than Ishikawa's iteration method to the fixed point of $T$. □

In general, we notice that for $x_{0} = b_{0} = -1, u = 0.5$, and $\gamma_{n} = \dfrac{2}{n+5}$, we can choose $\beta = \delta_{n} = \dfrac{5}{5}$. Thus, all the conditions of Theorem 3.9 are fulfilled and $\{x_{n}\} = \{b_{n}\}$ converges to $0 = P_{F(T)}u$ (see Figure 1 below). Similarly, for $x_{0} = b_{0} = 0.7, u = -1,$ and $\gamma_{n} = \dfrac{2}{n+5}$, the sequence $\{x_{n}\} = \{b_{n}\}$ converges to $0 = P_{F(T)}u$ (see Figure 1 below). A closer observation on Figure 1 shows that the convergence of the sequence $\{x_{n}\} = \{b_{n}\}$ to the fixed point of $T$ is independent of the numerical values of the initial point $x_{0} = b_{0}$ and $u$.

Author contributions

Asima Razzaque: Investigation, Writing review and editing; Imo Kalu Agwu: Conceptualization, Formal analysis, Investigation, Writing original draft preparation, Writing review and editing; Naeem Saleem: Conceptualization, Formal analysis, Investigation, Writing original draft preparation, Writing review and editing; Donatus Ikechi Igbokwe: Conceptualization; Maggie Aphane: Formal analysis, Writing review and editing. All authors have read and agreed to the published version of the manuscript. All the authors have read and approved the current version of this manuscript.

Use of Generative-AI tools declaration

The authors declare that AI was not involved in any manner during the writing of this manuscript.

Acknowledgements

This work was supported by the Deanship of Scientific Research, Vice Presidency for Graduate Studies and Scientific Research, King Faisal University, Saudi Arabia [KFU250094].

Conflict of interest

The authors declare that they do not have any competing interests.