Bareq B. Salman* | Salwa Salman Abed
© 2023 IIETA. This article is published by IIETA and is licensed under the CC BY 4.0 license (http://creativecommons.org/licenses/by/4.0/).
OPEN ACCESS
The fixed point theory is of great importance as it is used as an application for solving differential equations for different types of equations and various applications in physical, engineering, and statistical sciences. This investigation aims to define (λ, ρ) -firmly nonexpansive multivalued mappings in modular function spaces and to introduce a new iterative algorithm. Accordingly, some results of approximating fixed points for these mappings are proved with an example. Further, the concept of stability is discussed and supported by an example.
firmly nonexpansive mappings, applications in differential equations, iterative sequences, stability, fixed point
A lot of academics have worked on the fixed point since it has numerous applications in a wide range of industries. Over the years, many researchers have introduced iterative processes to solve fixed point problems, but the search for more effective and quick methods continues [1]. Modular function spaces were first introduced by Nakano in (1950), and Musielak and Orlicz greatly expanded on them [2]. One of the most important characteristics of the Modular function spaces is that considered a space separate from other spaces and is dealt with by functions.
Khamisi and Kozlowski [3] were the first to discuss the fixed point in modular function spaces in 1990. Researchers in this subject, which is thought to be growing, have tried to approximate the fixed point in modular function spaces. In Mann and Ishikawa iterative processes, Dehaish and Kozlowski [4] proved certain conclusions of approximation fixed point by modular function spaces. For monotone asymptotically nonexpansive mapping in modular function spaces, Alfuraidan and Khamsi [5] developed the Fibonacci-Mann iteration with studding. Hussain Khan [6] developed the notion of a strongly nonexpansive mapping from Banach spaces to modular function spaces. Additionally, Panwar [7] recently presented some findings in this area as shown in Figure 1. Only the fixed point theory for single-valued mappings acting in modular function spaces was addressed by Kozlowski [8]. Berinde [9] introduced and studied a tighter idea of nearly stability for fixed point iteration algorithms. Besides, Abed and Abduljabbar [10] demonstrated a general two theorem for the two step iterative sequence of multivalued mappings in a complete convex real modular space. In order to achieve the best approximation in modular spaces, Abed and Sada [11] used fixed point theorems of compact -nonexpansive multivalued mapping. For iteration schemes in multivalued mappings in modular function spaces, Abdul Jabbar and Abed [12] studied convergence. For shared fixed points and convergence, Morwal and Panwar [13] presented a three-step iterative technique in three multivalued -nonexpansive mappings and studded approximation. Okeke and Khan [14] subsequently extended the proof to the class of multivalued -quasi-contractive mappings with studding of stability. Abed and Jabbar [15] developed the idea of normalized duality mapping in actual convex modular spaces. Then, a few of its characteristics have emerged, enabling the handling of outcomes connected to the idea of uniformly smooth convex real modular spaces. In this regards, Abed and Abduljabbar [16] demonstrated convergence for iteration algorithms in multivalued mappings in modular function spaces. Using the Picard-Krasnoselskii hybrid iterative process in these spaces, Okeke et al. [17] proved theorems for -quasi-nonexpansive mappings.
There are many iterative schemes presented by researchers, the aim of the paper to find a new iterative scheme to approximate the fixed point that is faster than the previous iterative scheme.
With the introduction of securely multivalued mappings and a new iterative technique and some comparison, the goal of this work is to extend the findings of prior studies, The following Figure in the study [7]:
Figure 1. Generated of ρ-converges to the fixed point by taking different initial values
Here, we construct an iterative sequence of a four-step for (λ, ρ)- firmly nonexpansive multi-valued mappings and study it is convergence and stabile to fixed point in the framework of modular function spaces confirming the results an example and tables are provided. Further, we mention to utilize our proposed algorithm in solve differential equation as an application.
This section includes some fundamentals, significant definitions, and certain lemmas. Assume that Ω is a nonempty set and that Σ is a nontrivial σ -algebra of L p subsets. Considering that ρ is closed in terms of constructing a finite union and has countable crossings and differences, let ρ be nontrivial ring subsets of Ω. Additionally, let us assume that $E \cap A \in \rho$ for any $E \in \rho$ and $A \in \Sigma$, there exists an increasing series of sets $K_n \in \rho$ such that $\Omega= \cup K_n$. With help from, we denote by E the linear space of all simple functions. The space of all extended measurable functions, or all functions, is denoted by $M_{\infty}$.
$f: \Omega \rightarrow[-\infty, \infty] \quad$ Then, $\quad\left\{g_n\right\} \subset E, \quad\left|g_n\right| \leq|f|$ and $g_n(w) \rightarrow f$ for all $w \in \Omega$, by $1_A$ the characteristic function of the set $A[8]$.
Definition 1 [8]:
Let $\rho: M_{\infty} \longrightarrow[0, \infty]$ be a nontrivial, convex, and even function.
Then, ρ is a regular convex function pseudo modular if:
(a) ρ(0)=0.
(b) ρ is considered as a monotone, and, |f(w)|≤|g(w)| for all $w \in \Omega$ implies ρ(f)≤ρ(g), where $f, g \in M_{\infty}$.
(c) ρ is considered as an orthogonally sub additive, and, $\rho\left(f_{1_{A \cup B}}\right) \leq \rho\left(f_{1_A}\right)+\rho\left(f_{1_B}\right)$ for any $A, B \in \Sigma$ as A∩B nonempty, where $f \in M_{\infty}$.
(d) $\rho$ has the Fatou property: $\left|f_n(w)\right| \uparrow|f(w)|$ for all $w \in$ $\Omega$ implies $\rho\left(f_n\right) \uparrow \rho(f)$, where $f \in M_{\infty}$.
(e) $\rho$ is considered as an order continuous in $E$, and, $g_n \in E$ and $\left|g_n(w)\right| \downarrow 0$ implies $\rho\left(g_n\right) \downarrow 0$.
$M=\left\{f \in M_{\infty}:|f(w)|<\infty, \rho-a . e\right\}$
where, each $f \in M$ is actually an equivalence class of functions equal ρ-a.e. rather than an individual function.
Definition 2 [18]:
Let ρ:M⟶[0,∞] possesses the following properties:
1- ρ(0)=0, f=0, ρ-a.e
2- ρ(αf)=ρ(f), for every scalar $\alpha$
3- ρ(αx+βy)≤ρ(x)+ρ(y) For every α,β≥0 with α+β=1
where: convex modular.
Definition 3 [3]:
According to definition 2 ρ is considered as a convex modular in X, as well as is named modular function spaces:
$L_p=\{f \in M: \rho(\lambda f) \rightarrow 0$ as $\lambda \rightarrow 0\}$
Definition 4 [8]:
Let $\rho \in \Re$ then ρ has $\Delta_2$-condition if $\sup \rho\left(2 f_n, D\right) \rightarrow 0$ as k⟶∞ and $D \rightarrow \varnothing$, and $\sup \rho\left(f_n, D\right) \longrightarrow 0$
Here, ρ is considered regular convex function modular if ρ(f)=0 then f=0, a-e the class of all nonzero regular convex function in modular Ω is denoted by $\mathfrak{R}$.
Definition 5 [8]:
Here, we considered the above definition of ρ on Ω let r>0, ϵ>0 define $D(r, \epsilon)=\left\{(f, g): f, g \in L_P, \rho f \leq r, \rho f-g \geq \epsilon r\right\}$.
Let $\xi_1(r, \epsilon)=\inf \left\{1-\frac{1}{r} \rho\left(\frac{f+g}{2}\right):(f, g) \in D(r, \epsilon)\right\}$
If $D(r, \epsilon) \neq \varnothing$ and $\xi_1(r, \epsilon)=1$, If $D(r, \epsilon)=\emptyset$
Also, $\rho$ satisfy (UC1) if for every $\mathrm{r}>0, \epsilon>0 \xi_1(r, \epsilon)>0$ then $D(r, \epsilon) \neq \varnothing$.
Definition 6 [8]:
Let $\rho$ be a nonzero regular convex function modular defined on $\Omega \rho$ satisfy (UUC1) $\delta \geq 0, \epsilon>0$ there exists $\eta_1(r, \epsilon)>0$ depending only on $\delta$ and $\epsilon$ such that $\xi_1(r, \epsilon)>\eta_1(r, \epsilon)>0$ for any $r>\delta$.
Definition 7 [10, 15]:
Let $\rho \in \mathfrak{R}$
1- We say that $\left\{f_n\right\}$ is $\rho$-convergent to $f$ if $\rho\left(f_n-f\right) \rightarrow 0$.
2- A sequence $\left\{f_n\right\}$ is $\rho$-Cauchy sequence if $\rho\left(f_n-f_m\right) \rightarrow$ 0 as $n, m \rightarrow \infty$.
3- $B \sqsubset L_p$ is named ρ-closed if for any $f_n \in L_p$ the convergence $\rho\left(f_n-f\right) \rightarrow 0$ and f belongs to B.
4- $B \sqsubset L_p$ is namedρ-bounded if ρ- diameter is finite. ρ- diameter define as $\mathfrak{H}_p(B)=\sup \{\rho(f-g), f \in B, g \in B\}<\infty$.
5- A set $B \sqsubset L_p$ is named strongly ρ-bounded if $\beta>1$ and $M_p(B)=\sup \{\rho(\beta(f-g)), f \in B, g \in B\}<\infty$.
6- A set $B \sqsubset L_p$ is called ρ-compact if every $f_n \in B$, there exists a subsequence {$f_{n_k}$} and f in $\rho\left(f_{n_k}-f\right) \rightarrow 0$.
7- A set $B \sqsubset L_p$ is called ρ-a.e, closed if every $f_n \in B$, which $\rho-a . e$, converges to some f, then f in B.
8- A set $B \sqsubset L_p$ is called ρ-a.e, -compact if every $f_n \in B$, there exists a subsequence {$f_{n_k}$} $\rho-a . e$-converges to some f in B.
9- Let f in $L_p$ and $B \sqsubset L_p$, the ρ-distance between f and B is defined as:
$\operatorname{dist}_p(f, B)=\inf \{\rho(f-g), g \in B\}$.
Note that, ρ dose note satisfy triangle inequality so ρ-convergence dose not ρ-Cauchy, it is possible that this relationship can be realized if and only if ρ satisfies $\Delta_2$-condition [6].
Definition 8 [6]:
The sequence $\left\{t_n\right\}$ is considered to be bounded away from 0 if α>0 and t $t_n>\alpha$ for all $n \in N$. Also, the sequence $\left\{t_n\right\}$ is considered to be bounded away from 1 if $b<1$ and $t_n \leq b$ for all $n \in N$.
Definition 9 [6]:
$E \subset L_p$, let $T: E \rightarrow 2^E$ said to be satisfy condition (I) if no decreasing function $\emptyset:[0, \infty) \rightarrow[0, \infty)$ with $\emptyset(0)=$ $0, \emptyset(r)>0$ for all $r \in[0, \infty]$ such that $\rho(f-T f) \geq$ $\emptyset\left(\right.$ dist $\left._\rho\left(f, F_p(t)\right)\right)$ for all $f \in E$.
Definition 10 [13]:
A set $E \subset L_p$ is named $\rho$ - proximinal if for each $f \in L_p$ there exists an element $g$ in $E$. And, $\rho(f-g)=$ $\operatorname{dist}_p(f, E)=\inf \{\rho(f-h): h$ in $E$.
Lemma 1 [6]:
Let $\rho \in \mathfrak{R}$ satisfy (UUC1) and let $\left\{t_n\right\}$ in $(0,1)$ be bounded away from 0 and 1 , if $m>0$ Then, $\lim \sup _{n \rightarrow \infty} \rho\left(f_n\right) \leq$ $m$, lim $\sup _{n \rightarrow \infty} \rho\left(g_n\right) \leq m$, and $\lim _{n \rightarrow \infty} \rho\left(t_n f_n+(1-\right.$ $\left.\left.t_n\right) g_n\right)=m$, then $\lim _{n \rightarrow \infty} \rho\left(f_n-g_n\right)=0$.
Lemma 2 [9]:
Let $\left\{c_n\right\}_{n=0}^{\infty},\left\{d_n\right\}_{n=0}^{\infty}$ be sequence of nonnegative number and $0 \leq r<1$, such that $c_{n+1} \leq r c_n+d_n$ for all $n \geq 0$
1- if $\lim _{n \rightarrow \infty} d_n=0$, then $\lim _{n \rightarrow \infty} c_n=0$
2- if $\sum_{n=0}^{\infty} d_n<\infty$, then $\sum_{n=0}^{\infty} c_n<\infty$
Here, $P_p(E)$ denotes the family of nonempty $\rho$-proximinal, $\rho$-bounded subset of $E, C_p(E)$ denotes the family of nonempty $\rho$-closed, $\rho$-bounded subset of $E$, and $H_p(.,.) \rho$ - Hausdorff distance on $C_p(E)$
$H_p(A, B)=\max \left\{\sup _{f \in A} \operatorname{dist}_p(f, B), \quad \sup _{g \in B} \operatorname{dist}_p(g, A)\right\} A, B \in C_p\left(L_p\right)$
where, $\operatorname{dist}_p(f, B)=\inf \{\rho(f-g), g \in B\}$.
Lemma 3 [13]:
Let $\rho \in \mathcal{R}$ and satisfy $A, B \in P_p\left(L_p\right)$ for each $f$ in $A$ there exists $g$ in $B$ such that $\rho(f-g) \leq H_p(A, B)$.
Definition 11 [13, 14]:
Let $T: E \rightarrow 2^E$ is multivalued mapping said to be $\rho$ nonexpansive mapping if $H_p(T f, T g) \leq \rho(f-g)$ said to be $\rho$ - quasi nonexpansive mapping if for $s \in F_p(T)$ of $T$ in modular spaces:
$H_p(T f, s) \leq \rho(f-s)$
Finally, we can consider to be ρ-contraction mapping if there exists constant:
$0 \leq k<1$
$H_p(T f-T g) \leq k \rho(f-g)$
For all f, g in E.
Lemma 4 [13]:
Let $T: E \rightarrow 2^E$ be a multivalued mapping and $P_p^T(f)=\left\{g \in T: \rho(f-g)=\operatorname{dist}_\rho(f, T f)\right\}$ then:
1- $f \in F_p(T)$, and $f \in T(f)$
2- $P_p^T=\{f\}$ and f=g for each $g \in P_p^T(f)$
3- $f \in F_p\left(P_p^T(f)\right)$, and $f \in P_p^T(f)$, further $F_p(T)=F\left(P_p^T(f)\right)$ where $F\left(P_p^T(f)\right)$ denotes the set of fixed points of $P_p^T(f)$.
3.1 Convergence results for (λ, ρ)- firmly nonexpansive multivalued mappings
Starting with the definition below:
Definition 12:
Let $T: E \rightarrow 2^E$ said to be (λ, ρ)- firmly nonexpansive multivalued mapping if for λ in (0,1), $H_p(T f, T g) \leq \rho[(1-\lambda)(f-g)+\lambda(u-v)]$, $u \in T f, v \in T g$ said to be (λ, ρ)- quasi firmly nonexpansive multivalued mapping if for λ in (0,1) and $s \in F_p(T)$ is the set of fixed point of T in modular spaces:
$\begin{gathered}H_p(T f, s) \leq \rho[(1-\lambda)(f-s)+\lambda(u-s)] \\ u \in T f\end{gathered}$
Clearly, (λ, ρ)- quasi firmly nonexpansive mapping is quasi nonexpansive mapping.
Lemma 5:
Every (λ, ρ) - firmly nonexpansive mapping is ρ-nonexpansive mapping:
Proof:
By Definition 12, convexity of ρ and Lemma 3, obtaining:
$\begin{aligned} H_p(T f, T g) \leq & \rho[(1-\lambda)(f-g)+\lambda(u-v)], u \in P_p^T(f), v \in P_p^T(g) \\ & \leq(1-\lambda) \rho(f-g)+\lambda \rho(u-v) \\ & \leq(1-\lambda) \rho(f-g)+\lambda H_p(T f, T g)\end{aligned}$
Hence $H_p(T f, T g) \leq \rho(f-g)$.
Lemma 6:
Let $\rho \in \mathfrak{R}$ and $E$ be nonempty $\rho$-bounded, $\rho$-closed and $E \subset L_p$ let $T: E \rightarrow 2^E$ be $(\lambda, \rho)$ - firmly nonexpansive multivalued mapping, then the $F_p(T)$ is convex and closed.
Proof:
Let {$f_n$} is a sequence in fixed point set $F_p(T)$ is ρ-converges to some f in E, to prove f fixed point.
$\begin{array}{ll}\rho\left(\frac{f-T f}{2}\right) \leq \frac{1}{2} \rho\left(f-f_n\right)+\frac{1}{2} \rho\left(T f-f_n\right) \leq \frac{1}{2} \rho\left(f-f_n\right)+ \frac{1}{2} H_p\left(T f, T f_n\right) \leq \frac{1}{2} \rho\left(f-f_n\right)+\frac{1}{2} \rho\left(f-f_n\right) =\rho\left(f-f_n\right) \rightarrow 0 \text { as } n \rightarrow \infty\end{array}$.
$\rho\left(\frac{f-T f}{2}\right)=0$, then $T f=f$, and $f \in F_p(T)$, by define ρ – closed $F_p(T)$ is closed.
To prove $F_p(T)$ convex, let f, g in $F_p(T)$ and $h=\frac{f+g}{2}$:
$\rho(f-T h)=\rho(T h-f) \leq H_p(T h, T f) \leq \rho(h-f)=\rho\left(\frac{f-g}{2}\right)$ (1)
$\rho(g-T h)=\rho(T h-g) \leq H_p(T h, T g) \leq \rho(h-g)=\rho\left(\frac{f-g}{2}\right)$ (2)
$\rho(f-h)=\rho\left(\frac{f-g}{2}\right), \rho(g-h)=\rho\left(\frac{f-g}{2}\right)$ (3)
$\rho\left(f-\frac{h+T h}{2}\right)=\rho\left(\frac{1}{2}(f-h)+\frac{1}{2}(f-T h)\right)$
By $h=\frac{f+g}{2}$ and convexly $\leq \frac{1}{2} \quad \rho\left(\frac{f-g}{2}\right)+\frac{1}{2}$ $\rho\left(\frac{f-g}{2}\right)=\rho\left(\frac{f-g}{2}\right) \rho\left(g-\frac{h+T h}{2}\right)=\rho\left(\frac{1}{2}(g-h)+\frac{1}{2}(g-T h)\right.$
By $h=\frac{f+g}{2}$ and convexly $\leq \frac{1}{2} \quad \rho\left(\frac{\mathrm{f}-\mathrm{g}}{2}\right)+\frac{1}{2}$ $\rho\left(\frac{\mathrm{f}-\mathrm{g}}{2}\right)$=$\rho \left(\frac{f-g}{2}\right)$$\rho\left(\frac{f-g}{2}\right) \leq \frac{1}{2} \rho\left(f-\frac{h+T h}{2}\right)+\frac{1}{2} \rho\left(\frac{h+T h}{2}-g\right)$
$\rho\left(\frac{f-g}{2}\right) \leq \rho\left(f-\frac{h+T h}{2}\right)$
And by above:
$\rho\left(f-\frac{h+T h}{2}\right) \leq \rho\left(\frac{f-g}{2}\right)$, then $\rho\left(f-\frac{h+T h}{2}\right)=\rho\left(\frac{f-g}{2}\right)$ (4)
By (1), (2), (3), (4), and Lemma 1, ρ(h-Th)=0, h in $F_p(T)$, then $F_p(T)$ is convex.
Below, we introduce a new iterative algorithm and then prove convergence results.
Let $T: E \longrightarrow 2^E$, and let E nonempty convex subset of $L_p$ sequence {$f_n$} by the following iterative process:
$\begin{gathered}f_1 \in E \\ h_n=\left(1-\beta_n\right) f_n+\beta_n u_n \\ g_n=v_n \\ J_n=\left(1-\alpha_n\right) g_n+\alpha_n w_n \\ f_{n+1}=m_n, n \in N\end{gathered}$ (5)
where, {$\alpha_n$}and {$\beta_n$} in (0,1), $u_n \in P_\rho^T\left(f_n\right), v_n \in P_\rho^T\left(h_n\right), w_n \in P_\rho^T\left(g_n\right)$, $m_n \in P_\rho^T\left(J_n\right)$.
Theorem 1:
Let $\rho \in \mathfrak{R}$ satisfy (UUC1) and $\Delta_2$-condition, let E be nonempty ρ-bounded, ρ-closed and convex $E \subset L_p$ and $T: E \rightarrow 2^E$, be (λ, ρ)- firmly nonexpansive multivalued mapping, let {$f_n$} in E define by (5) then $\lim _{n \rightarrow \infty} \rho\left(f_n-s\right)$ exists for all s fixed point.
Proof:
Let $s \in F_p(T)$. To prove $\lim _{n \rightarrow \infty} \rho\left(f_n-s\right)$ exists.
By Definitions (11, 12), convexity of ρ, and Lemmas (3,5), we get:
$\rho\left(f_{n+1}-s\right)=\rho\left(m_n-s\right) \leq H_p\left(P_p^T\left(J_n\right), P_p^T(s)\right) \leq \rho\left(J_n-s\right)$ (6)
$\left.\rho\left(J_n-s\right) \leq \rho\left(\left(1-\alpha_n\right) g_n+\alpha_n w_n\right)-s\right) \leq\left(1-\alpha_n\right) \rho\left(g_n-s\right)+\alpha_n H_p\left(P_p^T\left(g_n\right), P_p^T(s)\right) \leq \rho\left(g_n-s\right)$ (7)
Also,
$\rho\left(g_n-s\right)=\rho\left(v_n-s\right) \leq H_p\left(P_p^T\left(h_n\right), P_p^T(s)\right) \leq \rho\left(h_n-s\right)$ (8)
Similarly,
$\rho\left(h_n-s\right)=\rho\left(\beta_n u_n+\left(1-\beta_n\right) f_n-s\right) \leq \beta_n H_p\left(P_p^T\left(f_n\right), P_p^T(s)\right)+\left(1-\beta_n\right) \rho\left(f_n-s\right) \leq \rho\left(f_n-s\right)$ (9)
By (6), (7), (8) and (9), $\rho\left(f_{n+1}-s\right) \leq \rho\left(f_n-s\right)$, so, $\lim _{n \rightarrow \infty} \rho\left(f_n-s\right)$ exists for all $s \in F_p(T)$.
Theorem 2:
Let $\rho \in \mathfrak{R}$ satisfy (UUC1) and $\Delta_2$-condition, let E be nonempty ρ-bounded, ρ-closed and convex $E \subset L_p$ and $: E \rightarrow 2^E$, be (λ,ρ)- firmly nonexpansive multivalued mapping, let {$f_n$} in E define by (5) then $\lim _{n \rightarrow \infty} \operatorname{dist}_\rho \rho\left(f_n, P_p^T\left(f_n\right)\right)=0$
Proof:
By Theorem 1 $\lim _{n \rightarrow \infty} \rho\left(f_n-s\right)$ exists
Let
$\lim _{n \rightarrow \infty} \rho\left(f_n-s\right)=k$, where $k \geq 0$ (10)
By (7), (8), and (9) the following hold:
$\rho\left(h_n-s\right) \leq \rho\left(f_n-s\right) \Rightarrow \lim _{n \rightarrow \infty} \rho\left(h_n-s\right) \leq k$ (11)
$\lim _{n \rightarrow \infty} \rho\left(g_n-s\right) \leq k$ (12)
$\lim _{n \rightarrow \infty} \rho\left(J_n-s\right) \leq k$ (13)
$\rho\left(v_n-s\right) \leq H_p\left(P_p^T\left(h_n\right), P_p^T(s)\right) \leq \rho\left(h_n-s\right) \leq \rho\left(f_n-s\right)$
$\lim _{n \rightarrow \infty} \rho\left(v_n-s\right) \leq \lim _{n \rightarrow \infty} \rho\left(f_n-s\right) \leq k$ (14)
$\rho\left(u_n-s\right) \leq H_p\left(P_p^T\left(f_n\right), P_p^T(s)\right) \leq \rho\left(f_n-s\right)$
Then $\lim _{n \rightarrow \infty} \rho\left(u_n-s\right) \leq k$ (15)
$\rho\left(w_n-s\right) \leq H_p\left(P_p^T\left(g_n\right), P_p^T(s)\right) \leq \rho\left(g_n-s\right) \leq \rho\left(f_n-s\right)$
Then $\lim _{n \rightarrow \infty} \rho\left(w_n-s\right) \leq k$ (16)
$\rho\left(m_n-s\right) \leq H_p\left(P_p^T\left(J_n\right), P_p^T(s)\right) \leq \rho\left(J_n-s\right) \leq \rho\left(f_n-s\right)$
Then $\lim _{n \rightarrow \infty} \rho\left(m_n-s\right) \leq k$ (17)
Let $\lim _{n \rightarrow \infty} \alpha_n=\alpha$, $\rho\left(f_{n+1}-s\right)=\rho\left(m_n-s\right) \leq H_p\left(P_p^T\left(J_n\right), P_p^T(s)\right) \leq \rho\left(J_n-s\right) \leq \rho\left(\alpha_n w_n+\left(1-\alpha_n\right) g_n-s\right) \leq \alpha_n \rho\left(w_n-s\right)+\left(1-\alpha_n\right) \rho\left(g_n-s\right)$.
So, $\lim _{n \rightarrow \infty} \inf \rho\left(f_{n+1}-s\right) \leq \lim _{n \rightarrow \infty} \inf \left[\alpha_n \rho\left(w_n-s\right)+\left(1-\alpha_n\right) \rho\left(g_n-s\right)\right]$.
Then, $k \leq \lim _{n \rightarrow \infty} \inf \alpha_n \rho\left(w_n-s\right)+(1-\alpha) k \Rightarrow \alpha k \leq \alpha \lim _{n \rightarrow \infty} \inf \rho\left(w_n-s\right)$.
Hence,
$k \leq \lim _{n \rightarrow \infty} \inf \rho\left(w_n-s\right)$ (18)
By (16) and (18),
$\lim _{n \rightarrow \infty} \rho\left(w_n-s\right)=k$
$\rho\left(w_n-s\right) \leq H_p\left(P_p^T\left(g_n\right), P_p^T(s)\right) \leq \rho\left(g_n-s\right)$ (19)
Then,
$k \leq \rho\left(g_n-s\right)$ (20)
By (12) and (20),
$\lim _{n \rightarrow \infty} \rho\left(g_n-s\right)=k$ (21)
Since,
$\rho\left(g_n-s\right)=\rho\left(v_n-s\right)$, so, $\lim _{n \rightarrow \infty} \rho\left(v_n-s\right)=k$
$\rho\left(v_n-s\right) \leq H_p\left(P_p^T\left(h_n\right), P_p^T(s)\right) \leq \rho\left(h_n-s\right) \Rightarrow \lim _{n \rightarrow \infty} \rho\left(v_n-s\right) \leq \lim _{n \rightarrow \infty} \rho\left(h_n-s\right)$ (22)
so,
$k \leq \lim _{n \rightarrow \infty} \rho\left(h_n-s\right)$ (23)
By (11) and (23), then:
$\lim _{n \rightarrow \infty} \rho\left(h_n-s\right)=k$ (24)
By (24),
$\lim _{n \rightarrow \infty} \rho\left(h_n-s\right)=k \Rightarrow \lim _{n \rightarrow \infty} \rho\left(\beta_n u_n+\left(1-\beta_n\right) f_n-s\right)=k$
$\lim _{n \rightarrow \infty} \rho\left(\beta_n\left(u_n-s\right)+\left(1-\beta_n\right)\left(f_n-s\right)=k\right.$ (25)
By (10), (15), (25) and Lemma1, $\lim _{n \rightarrow \infty} \rho\left(f_n-u_n\right)=0$ then $u_n \in P_p^T\left(f_n\right)$. Since $\operatorname{dist}_\rho \rho\left(f_n, P_p^T\left(f_n\right)\right) \leq \lim _{n \rightarrow \infty} \rho\left(f_n-u_n\right)$, $\lim _{n \rightarrow \infty} \operatorname{dist}_\rho \rho\left(f_n, P_p^T\left(f_n\right)\right)$=0. This completes the proof.
Theorem 3:
Let $\rho \in \mathfrak{R}$ satisfy (UUC1) and $\Delta_2$-condition, let $E$ be nonempty $\rho$-bounded, $\rho$-closed and convex $E \subset L_p$ and $T: E \rightarrow 2^E$, be $(\lambda, \rho)$ - firmly nonexpansive multivalued mapping, let $\left\{f_n\right\}$ in $E$ define by (5), $f_0$ unique fixed point in $T$, then $f_n$ converge to fixed point in $T$.
Proof:
By convexity of ρ, Lemma 3, Definitions (11,12) and Lemma 5, implies that:
$\begin{gathered}\left.\rho\left(h_n-f_0\right)=\rho\left(\left(1-\beta_n\right) f_n+\beta_n u_n\right)-f_0\right) \\ \leq\left(1-\beta_n\right) \rho\left(f_n-f_0\right)+\beta_n H_p\left(P_p^T\left(f_n\right), P_p^T\left(f_0\right)\right) \\ \leq\left(1-\beta_n\right) \rho\left(f_n-f_0\right)+\beta_n \rho\left(f_n-f_0\right) \\ \leq \rho\left(f_n-f_0\right)\end{gathered}$ (26)
Again,
$\rho\left(g_n-f_0\right) \leq \rho\left(v_n-f_0\right)$
$\leq H_p\left(P_p^T\left(h_n\right), P_p^T\left(f_0\right)\right)$
$\leq \rho\left(h_n-f_0\right)$
$\leq \rho\left(f_n-f_0\right)$ (27)
Similarity,
$\begin{gathered}\rho\left(J_n-f_0\right)=\rho\left(\left(1-\alpha_n\right) g_n+\alpha_n w_n-f_0\right) \\ \leq\left(1-\alpha_n\right) \rho\left(g_n-f_0\right)+\alpha_n H_p\left(P_p^T\left(g_n\right), P_p^T\left(f_0\right)\right) \\ \leq\left(1-\alpha_n\right) \rho\left(g_n-f_0\right)+\alpha_n \rho\left(g_n-f_0\right) \\ \leq \rho\left(f_n-f_0\right)\end{gathered}$ (28)
Similarity,
$\rho\left(f_{n+1}-f_0\right)=\rho\left(m_n-f_0\right) \leq H_p\left(P_p^T\left(J_n\right), P_p^T\left(f_0\right)\right) \leq \rho\left(J_n-f_0\right)$
$\rho\left(f_n-f_0\right) \leq \rho\left(f_{n-1}-f_0\right)$ (29)
Since $\rho\left(f_1-f_0\right) \leq \rho\left(f_0-f_0\right)$, so, $\rho\left(f_n-f_0\right) \leq \rho\left(f_0-f_0\right)$, $\rho\left(f_n-f_0\right) \leq \rho(0)=0$, then $f_n \rightarrow f_0$.
Theorem 4:
Let $\rho \in \mathfrak{R}$ satisfy (UUC1) and $\Delta_2$-condition, let E be nonempty ρ-compact, ρ-closed and convex $E \subset L_p$ and $T: E \longrightarrow 2^E$, be (λ, ρ)- firmly nonexpansive multivalued mapping, let {$f_n$} n E define by (5) then $f_n$ converge to fixed point of T.
Proof:
Since E is ρ-compact there exists subsequence $f_{n_k}$ of $f_n$ and $\rho\left(f_{n_k}-f\right)=0$.
$\lim _{n \rightarrow \infty} \rho\left(f_{n_k}-f\right)=0$, to prove f fixed point.
Let g fixed point $g \in P_p^T(f), g_k \in P_p^T\left(f_{n_k}\right)$.
By Lemma 3, Definitions (7, 11), and Lemma 5, we get
$\rho\left(\frac{f-g}{3}\right)=\rho\left[\frac{f-f_{n_k}}{3}+\frac{f_{n_k}-g_k}{3}+\frac{g_k-g}{3}\right]$
$\leq \frac{1}{3} \rho\left(f-f_{n_k}\right)+\frac{1}{3} \rho\left(f_{n_k}-g_k\right)+\frac{1}{3} \rho\left(g_k-g\right)$
$\leq \frac{1}{3} \rho\left(f-f_{n_k}\right)+\frac{1}{3} \operatorname{dist}_\rho\left(f_{n_{k^{\prime}}} P_p^T\left(f_{n_k}\right)\right)$
$+\frac{1}{3} H_p\left(P_p^T\left(f_{n_k}\right), P_p^T(f)\right)$
$\leq \frac{1}{3} \rho\left(f-f_{n_k}\right)+\frac{1}{3} \operatorname{dist}_\rho\left(f_{n_k}, P_p^T\left(f_{n_k}\right)\right)$
$+\frac{1}{3}\left(f_{n_k}-f\right) \operatorname{dist}_\rho\left(f_{n_k}, P_p^T\left(f_{n_k}\right)\right)=0$
by Theorem 2, $\rho\left(\frac{f-g}{3}\right)=0$, therefore f=g, then T has unique fixed point f, $f_n$ converge to fixed point of T.
Theorem 5:
Let $\rho \in \Re$ satisfy (UUC1) and $\Delta_2$-condition, let E be nonempty ρ-bounded, ρ-closed and convex $E \subset L_p$ and $T: E \longrightarrow 2^E$, be (λ, ρ)- firmly nonexpansive multivalued mapping, let {$f_n$} in E define by (5) then $f_n$ converge to fixed point s of T if and only if $\lim \inf _{n \rightarrow \infty} \operatorname{dist}_p\left(f_n, F(T)\right)=0$, where $\operatorname{dist}_p\left(f_n, F(T)\right)=\inf \left\{\rho(f-s), s \in F_p(T)\right\}$.
Proof:
Let $f_n$ converge to fixed point s of T, to prove $\lim \inf _{n \rightarrow \infty} \operatorname{dist}_p\left(f_n, F(T)\right)=0$.
Since $f_n \rightarrow s$, then $\lim _{n \rightarrow \infty} \operatorname{dist}_p\left(f_n, s\right)=0$.
Since $\operatorname{dist}_p\left(f_n, F_p(T)\right) \leq \operatorname{dist}_p\left(f_n, s\right)$, then $\lim \inf _{n \rightarrow \infty} \operatorname{dist}_p\left(f_n, F_p(T)\right)=0$.
If $\lim \inf _{n \rightarrow \infty} \operatorname{dist}_p\left(f_n, F_p(T)\right)=0$, to prove $\rho\left(f_n-s\right)=0$.
By Theorem 1 $\lim _{n \rightarrow \infty} \rho\left(f_n-s\right)$ exists, then $\lim _{n \rightarrow \infty} \rho\left(f_n-F_p(T)\right)$ exists and $\mathrm{s} \in F_p(T)$.
Suppose $f_{n_k}$ s any subsequence of $f_n$, and $u_k$ sequence in $F_p(T)$.
$\rho\left(f_{n_k}-u_k\right) \leq \frac{1}{2^k}$, Since, $\lim \inf _{n \rightarrow \infty} \operatorname{dist}_p\left(f_n, F_p(T)\right)=0$.
$\rho\left(f_{n+1}-u_k\right) \leq \rho\left(f_n-u_k\right) \leq \frac{1}{2^k}$.
$\rho\left(u_{k+1}-u_k\right) \leq \rho\left(u_{k+1}-f_{n+1}\right)+\rho\left(f_{n+1}-u_k\right) \leq \frac{1}{2^{k+1}}+\frac{1}{2^k} \leq \frac{1}{2^{k-1}}$.
$\rho\left(u_{k+1}-u_k\right) \rightarrow 0$, as $k \rightarrow \infty$.
$u_k$ is ρ-cauchy in $F_p(T)$, since $\Delta_2$ condition, so ρ-cauchy $\Leftrightarrow \rho$-converge.
$u_k$ is $\rho$-convergence in $F_p(T)$, so $\rho\left(u_k-s\right) \rightarrow 0$.
Now, $\rho\left(f_{n_k}-s\right) \leq \rho\left(f_{n_k}-u_k\right)+\rho\left(u_k-s\right), \rho\left(f_{n_k}-\right.$ $\left.u_k\right) \rightarrow 0$, and $\rho\left(u_k-s\right) \rightarrow 0, f_n$ Converges to fixed point $s$ in $F_p(T)$.
Theorem 6:
Let $\rho \in \mathfrak{R}$ satisfy (UUC1) and $\Delta_2$-condition, let E be nonempty ρ-bounded, ρ-closed and convex $E \subset L_p$ and $T: E \longrightarrow 2^E$ be (λ,ρ)- firmly nonexpansive multivalued mapping, and T satisfied condition (I), let {$f_n$} in E define by (5) then $f_n$ converge to fixed point s of T.
Proof:
By Theorem 1 $\lim _{n \rightarrow \infty} \rho\left(f_n-s\right)$ exists, s is fixed point.
If $\lim _{n \rightarrow \infty} \rho\left(f_n-s\right)=0$, nothing to prove, if $\lim _{n \rightarrow \infty} \rho\left(f_n-s\right)=k, k \geq 0$.
By Theorem 1 $\rho\left(f_{n+1}-s\right) \leq \rho\left(f_n-s\right)$
Then, $\operatorname{dist}_\rho\left(f_{n+1}, F_p(T)\right) \leq \operatorname{dist}_\rho\left(f_n, F_p(T)\right)$, hence $\lim _{n \rightarrow \infty} \operatorname{dist}_\rho\left(f_n, F_p(T)\right)$ exists.
By applying condition (I) and Theorem 2 $\lim _{n \rightarrow \infty} \emptyset\left(\operatorname{dist}_\rho\left(f_n, F_p(T)\right) \leq \lim _{n \rightarrow \infty}\left(f_n, P_p^T\left(f_n\right)\right)=0\right.$.
Since $\varnothing(0)=0$, so, $\lim _{n \rightarrow \infty} \operatorname{dist}_\rho\left(f_n, F_p(T)\right)$=0, by Theorem 5, $f_n$ is ρ -converge to fixed point s.
3.2 Stability
In this section, firstly we reform the definition of stability as in the study [9], then give results and an example.
Definition 13 [9]:
Let E be anon empty convex subset of modular function space $L_p$, and T:E$\longrightarrow$E, let $x_1$ in E, and $x_{n+1}=f\left(T, x_n\right)$ define the iterative schemes which given sequence $x_n$ in E, suppose that $\left\{x_n\right\}_{n=1}^{\infty}$ converge to $x \in F_p(T) \neq \emptyset$, let $\left\{y_n\right\}_{n=1}^{\infty}$ be any bounded sequence in E and put $\varepsilon_n=\rho\left(y_{n+1}-f\left(T, x_n\right)\right)$.
1- The iteration scheme $\left\{x_n\right\}_{n=1}^{\infty}$ define by $x_{n+1}=f\left(T, x_n\right)$ is displayed to be T-stable on $E$ if $\lim _{n \rightarrow \infty} \varepsilon_n=0$ implies that $\lim _{n \rightarrow \infty} y_n=x$
2- The iteration scheme $\left\{x_n\right\}_{n=1}^{\infty}$ define by $x_{n+1}=f\left(T, x_n\right)$ is displayed to be almost T-stable on $E$ if $\sum_{n=1}^{\infty} \varepsilon_n<\infty$ implies that $\lim _{n \rightarrow \infty} y_n=x$.
3- The iteration scheme $\left\{x_n\right\}_{n=1}^{\infty}$ define by $x_{n+1}=f\left(T, x_n\right)$ is considered to be summably almost T-stable on $E$ if and only if $\sum_{n=1}^{\infty} \varepsilon_n<\infty$ implies that $\sum_{n=1}^{\infty} \rho\left(y_n-x\right)<\infty$.
Theorem 7:
Let $\rho \in \mathfrak{R}$ satisfy (UUC1) and $\Delta_2$-condition, let $E$ be nonempty $\rho$-bounded, $\rho$-closed and convex $E \subset L_p$ and $T: E \rightarrow 2^E$, be $(\lambda, \rho)$ - firmly nonexpansive multivalued mapping, and $T$ satisfied condition (I), let $\left\{f_n\right\}$ in $E$ define by (5) then $f_n$ is summably almost T-stable.
Proof:
Let $s$ is fixed point of $T$, and $\varepsilon_n=\rho\left(f_{n+1}-m_n\right)$, by (5), convexity of $\rho$, Lemma 3, Definitions $(11,12)$, and Lemma 5, implies that:
$\begin{aligned} \rho\left(f_{n+1}-s\right)=\rho( & \left.\left(f_{n+1}-m_n\right)+\left(m_n-s\right)\right) \\ & \leq \rho\left(f_{n+1}-m_n\right)+\rho\left(m_n-s\right) \\ & \leq \varepsilon_n+H_p\left(P_p^T\left(J_n\right), P_p^T(s)\right) \\ & \leq \varepsilon_n+\rho\left(J_n-s\right)\end{aligned}$
Substituting $J_n$ and similarity above:
$\left.\left.\leq \varepsilon_n+p\left(\left(1-\alpha_n\right) g_n+\alpha_n w_n\right)-s\right)\right)$
$\leq \varepsilon_n+\left(1-\alpha_n\right) \rho\left(g_n-s\right)+\alpha_n H_p\left(P_p^T\left(g_n\right), P_p^T(s)\right)$
$\leq \varepsilon_n+\rho\left(g_n-s\right)$
Substituting $g_n$ and similarity above:
$=\varepsilon_n+\rho\left(v_n-s\right)$
$\leq \varepsilon_n+H_p\left(P_p^T\left(h_n\right), P_p^T(s)\right)$
$\leq \varepsilon_n+\rho\left(h_n-s\right)$
Substituting $h_n$ and similarity above:
$\leq \varepsilon_n+\rho\left(\beta_n u_n+\left(1-\beta_n\right) f_n-s\right)$
$\leq \varepsilon_n+\rho\left(f_n-s\right)$
$\rho\left(f_{n+1}-s\right) \leq \varepsilon_n+\rho\left(f_n-s\right)$
So, by Lemma 2 and Definition 13, implies that $f_n$ is summably almost T-stable.
Example 1:
The set of real number $\Re$ by the space $\rho(f)=|f|, \rho$ is satisfy (UUC1) and $\Delta_2$-condition, $E=\left\{f \in L_p: 0 \leq f<\infty\right\}$, $T: E \rightarrow E(\lambda, \rho)$ - firmly nonexpansive mapping and $T f=\frac{f}{4}$, with $F_p(T)=\{0\}$, and let $\alpha_n, \beta_n=0.5$ for all $n$.
$h_n=\left(1-\beta_n\right) f_n+\beta_n T f_n$
$g_n=T h_n$
$J_n=\left(1-\alpha_n\right) g_n+\alpha_n T g_n$
$f_{n+1}=T J_n$
Let $f_n=\frac{n+1}{n+2}$
$h_n=\frac{1}{2} \frac{n+1}{n+2}+\frac{1}{2} \frac{n+1}{4(n+2)}=\frac{5}{8} \frac{(n+1)}{(n+2)}$, then $T h_n=\frac{5(n+1)}{32(n+2)}$
$f_{n+1}=T\left(\frac{1}{2}\left(\frac{5(n+1)}{32(n+2)}\right)+\frac{1}{2}\left(\frac{5(n+1)}{128(n+2)}\right)\right.$, so $f_{n+1}=\frac{25(n+1)}{1024(n+2)}$
$\varepsilon_n=\rho\left(f_{n+1}-f\left(T, f_n\right)\right)$
$=\rho\left(\frac{n+2}{n+3}-\frac{25(n+1)}{1024(n+2)}\right)=\left|\frac{n+2}{n+3}-\frac{25(n+1)}{1024(n+2)}\right|=\left|\left(1-\frac{25}{1024}\right)+\left(\frac{25}{1024(n+2)}-\frac{1}{n+3}\right)\right| \leq\left|1-\frac{25}{1024}\right|$
$+\left|\frac{25}{1024(n+2)}-\frac{1}{n+3}\right| \leq\left|\frac{1}{n+2}-\frac{1}{n+3}\right|=\frac{1}{n+2}-\frac{1}{n+3}$
Since $\sum_{n=1}^{\infty} \varepsilon_n \leq \sum_{n=1}^{\infty}\left(\frac{1}{n+2}-\frac{1}{n+3}\right)$, so $\sum_{n=1}^{\infty} \varepsilon_n<\infty$
$\sum_{n=1}^{\infty} \rho\left(f_n-s\right)$ s fixed point
$=\sum_{n=1}^{\infty}\left|\frac{n+1}{n+2}-0\right|=\sum_{n=1}^{\infty} \frac{n+1}{n+2}=\sum_{n=1}^{\infty}\left(1-\frac{n+1}{n+2}\right)<\infty$
The iterative scheme in (5) is summably almost T-stable.
Example 2:
The set of real number $\mathfrak{R}$ by the space ρ(f)=|f|, ρ is satisfy (UUC1) and $\Delta_2$-condition, E=[0, 3] define T:E$\longrightarrow$E a mapping, $\emptyset:[0, \infty) \rightarrow[0, \infty)$, $\emptyset(r)=\frac{r}{6}$
And $T f=\frac{f+4}{5}, F_p(T)=\{1\}$, to prove $\rho(f-T f) \geq$ $\emptyset\left(\operatorname{dist}_p\left(f, F_p(T)\right)\right.$ for all $f$ in $E$.
$\rho(f-T f)=\rho\left(f-\frac{f+4}{5}\right)=\frac{4 f+4}{5}$, while $\emptyset\left(\operatorname{dist}_p\left(f, F_p(T)\right)\right.$ $=\emptyset\left(\operatorname{dist}_p(f,\{1\})\right)=\phi[\rho(f-1)]=\frac{f-1}{6}$.
Now, prove T is (λ, ρ)- firmly nonexpansive mapping
$\rho(T f-T g)=\rho\left(\frac{f+4}{5}-\frac{g+4}{5}\right)=\left|\frac{1}{5}(f-g)\right| \leq\left|\frac{21}{25}(f-g)\right| \leq \rho\left(\frac{4}{5}(f-g)+\frac{1}{5}\left(\frac{1}{5}(f-g)\right)\right)$, T is (λ, ρ)-firmly nonexpansive mapping when $\lambda=\frac{1}{5}$.
Tables 1, 2, and 3 represent the corresponding results as shown below.
Table 1. Results of $f_n, h_n, g_n$, and $J_n$ where $\alpha_n=\beta_n=0.5$, with $f_1=2$
|
Step |
$\boldsymbol{f}_{\boldsymbol{n}}$ |
$\boldsymbol{h}_{\boldsymbol{n}}$ |
$\boldsymbol{g}_{\boldsymbol{n}}$ |
$\boldsymbol{J}_{\boldsymbol{n}}$ |
|
1 |
2 |
1.6 |
1.12 |
1.072 |
|
2 |
1.0144 |
1.00864 |
1.001728 |
1.0010368 |
|
3 |
1.00020736 |
1.000124416 |
1.00024883 |
1.00001493 |
|
4 |
1.000002986 |
1.000001701 |
1.00000034 |
1.000000204 |
|
5 |
1.000000041 |
1.000000025 |
1.000000005 |
1.000000003 |
|
6 |
1.000000001 |
1 |
1 |
1 |
|
7 |
1 |
1 |
1 |
1 |
Table 2. Results of $f_n, h_n, g_n$, and $J_n$ where $\alpha_n=\beta_n=0.2$, with $f_1=2$
|
Step |
$\boldsymbol{f}_{\boldsymbol{n}}$ |
$\boldsymbol{h}_{\boldsymbol{n}}$ |
$\boldsymbol{g}_{\boldsymbol{n}}$ |
$\boldsymbol{J}_{\boldsymbol{n}}$ |
|
1 |
2 |
1.84 |
1.168 |
1.14112 |
|
2 |
1.028224 |
1.02370816 |
1.004741632 |
1.003982971 |
|
3 |
1.000796594 |
1.000669138 |
1.000133828 |
1.000112415 |
|
4 |
1.000022483 |
1.000018885 |
1.000003777 |
1.000003173 |
|
5 |
1.00000635 |
1.000000533 |
1.000000107 |
1.000000090 |
|
6 |
1.000000018 |
1.000000014 |
1.000000003 |
1.000000002 |
|
7 |
1 |
1 |
1 |
1 |
Table 3. Results of $f_n, h_n, g_n$, and $J_n$ where $\alpha_n=\beta_n=0.8$, with $f_1=2$
|
Step |
$\boldsymbol{f}_{\boldsymbol{n}}$ |
$\boldsymbol{h}_{\boldsymbol{n}}$ |
$\boldsymbol{g}_{\boldsymbol{n}}$ |
$\boldsymbol{J}_{\boldsymbol{n}}$ |
|
1 |
2 |
1.36 |
1.072 |
1.02592 |
|
2 |
1.005184 |
1.00186624 |
1.000373248 |
1.000134369 |
|
3 |
1.000026874 |
1.0000009674 |
1.000001935 |
1.00000696 |
|
4 |
1.000000139 |
1.00000050 |
1.00000010 |
1.00000003 |
|
5 |
1.00000001 |
1 |
1 |
1 |
|
6 |
1 |
1 |
1 |
1 |
Through our study of the above tables, it becomes clear that the closer the value of $\alpha_n$ and $\beta_n$ o the fixed point, the faster the approximation to the fixed point.
Since fixed point theory provides useful tools to solve many problems that have applications in different fields of sciences, the studying of iterative algorithms to approximate the solution of differential equations be one of most active studies area. Therefore, this section is devoted to applying the above results to differential equations in way similar to what is presented in the study [3]. In the following, we deal with especial case of algorithm (5):
Let $\rho \in \mathfrak{R}$, consider the initial value problem v: [0, B] $\longrightarrow$E and E in $L_p$:
$v(0)=g$
$v \backslash(t)+(I-T) v(t)=0$ (30)
where, $g \in E, B>0$, and $T: E \rightarrow E$ such that $P_\rho^T$ is $(\lambda, \rho)$-firmly nonexpansive mappings, and by define:
$Q(t)=1-e^{-t}=\int_0^t e^{s-t} d s$ (31)
For any $u:[0, B] \rightarrow L_p$ and $B>0$ then:
$S(u)(t)=\int_0^t e^{s-t} u(s) d s$ (32)
Also,
$S_t(u)(t)=\sum_{i=0}^{n-1}\left(t_{i+1}-t_i\right) e^{t_i-t} v\left(t_i\right)$ (33)
The following Lemma and Theorem in the study [3]:
Lemma 6:
Let $\rho \in \Re$ be separable let $m, l:[0, B] \rightarrow L_p$ by two Bochner-integrable $\|\cdot\|_\rho$-bounded functions where $B>0$ for $t \in[0, B]$ then:
$\rho\left(e^{-t} l(t)+\int_0^t e^{s-t} m(s) d s\right) \leq e^{-t} \rho(l(t))+Q(t) \sup \rho(m(s))$
$s \in[0, t]$
Theorem 8:
Let $\rho \in \mathfrak{R}$ be separable, let $D$ in $L_p$ be nonempty, convex, $\rho$-bounded, and $\rho$-closed set with vitali property. Let $T: D \rightarrow$ $P_\rho(D)$ be multivalued mapping such that $P_\rho^T$ is nonexpansive mappings. Let fixed $g \in E$ define sequence of functions $v_n:[0, B] \Rightarrow E$ by the following formula $v_0(0)=g$, $v_{n+1}(t)=e^{-t} g+\int_0^t e^{s-t} T\left(v_n(s)\right) d s$.
Then $t \in[0, B]$ there exists $v(t) \in E$ such that $\rho\left(v_n(t)-\right.$ $v(t)) \rightarrow 0$ and by the function $v:[0, B] \rightarrow E$ the $\rho\left(v_n(t)-\right.$ $v(t)) \rightarrow 0$ is solution to $(30)$, moreover $\rho\left(g-v_n(t)\right) \leq$ $Q^{n+1}(B) \delta_\rho(E)$.
The modular type conditions are more natural as assumptions of modular type can be checked more easily than their metric or modular counterparts especially in applications for differential operators, approximations and fixed point results. In the current investigation, the concept of (λ, ρ)-firmly nonexpansive mapping and its relationship with (λ, ρ)- quasi firmly nonexpansive mapping and nonexpansive mapping have been discussed. In addition, some convergence and stability results by using an iterative scheme in four steps of multivalued mapping in modular function space have been proved. The study suggests to the authors using the iterative scheme in other styles in modular function spaces or another spaces. This study is important as the iterative scheme that was presented is faster in reaching the fixed point than other iterative (see the research [19]). Lastly, we apply this algorithm to solving of a differential equation.
We look forward to employing our results (which relate to the convergence of algorithm 5 and equation 30) in practical application in one of the anther branches of sciences such as physical engineering or control. This aim may require cooperation with some colleagues.
[1] Akutsah, F., Narain, O.K., Afassinou, K., Mebawondu, A.A. (2021). An iterative scheme for fixed point problems. Advances in Mathematics: Scientific Journal, 10(5): 2295-2316. https://doi.org/10.37418/amsj.10.5.2
[2] Abdou, A.A.N., Khamsi, M.A., Khan, A.R. (2014). Convergence of Ishikawa iterates of two mappings in modular function spaces. Fixed Point Theory and Applications, 2014(1): 1-10. https://doi.org/10.1186/1687-1812-2014-74
[3] Khamsi, M.A., Kozlowski, W.M. (2014). Fixed Point Theory in Modular Function Spaces. ebook. https://doi.org/10.1007/978-3-319-14051-3
[4] Dehaish, B.A.B., Kozlowski, W.M. (2012). Fixed point iteration processes for asymptotic pointwise nonexpansive mapping in modular function spaces. Fixed Point Theory and Applications, 2012: 1-23. https://doi.org/10.1186/1687-1812-2012-118
[5] Alfuraidan, M.R., Khamsi, M.A. (2017). Fibonacci–Mann iteration for monotone asymptotically nonexpansive mappings. Bulletin of the Australian Mathematical Society, 96(2): 307-316. https://doi.org/10.1017/S0004972717000120
[6] Hussain Khan, S. (2018). Approximating fixed point of ({\lambda},{\rho})-firmly nonexpansive mappings in modular function spaces. arXiv e-prints, arXiv-1802. https://doi.org/10.48550/arXiv.1802.00681
[7] Panwar, A. (2021). Approximation of fixed points of (λ, ρ)-quasi firmly nonexpansive mappings. In Journal of Physics: Conference Series, 1849(1): 012020.
[8] Kozlowski, W.M. (2012). Advancements in fixed point theory in modular function spaces. Arabian Journal of Mathematics, 1: 477-494. https://doi.org/10.1007/s40065-012-0051-0
[9] Berinde, V. (2003). Summable almost stability of fixed point iteration procedures. Carpathian Journal of Mathematics, 81-88.
[10] Abed, S.S., Abduljabbar, M.F. (2020). Approximating fixed points in modular spaces. Karbala International Journal of Modern Science, 6(2): 3. https://doi.org/10.33640/2405-609X.1353
[11] Abed, S.S., Sada, K.E.A. (2018). Common fixed points in modular spaces. Ibn Al-Haitham Journal for Pure and Applied Science, 500-509. https://doi.org/10.30526/2017.IHSCICONF.1822
[12] Abduljabbar, M.F., Abed, S.S. (2019). The convergence of iteration scheme to fixed points in modular spaces. Iraqi Journal of Science, 60(10): 2196-2201. https://doi.org/10.24996/ijs.2019.60.10.13
[13] Morwal, R., Panwar, A. (2020). Common fixed point results for three multivalued\(\rho\)-nonexpansive mappings by using three steps iterative scheme. Communications in Mathematics and Applications, 11(2): 199-214. https://doi.org/10.26713/cma.v11i2.1335
[14] Okeke, G.A., Khan, S.H. (2020). Approximation of fixed point of multivalued ρ-quasi-contractive mappings in modular function spaces. Arab Journal of Mathematical Sciences, 26(1/2): 75-93. https://doi.org/10.1016/j.ajmsc.2019.02.001
[15] Abed, S.S., Jabbar, M.A. (2021). Some results on normalized duality mappings and approximating fixed points in convex real modular spaces. Baghdad Journal of Science, 1218-1225. http://dx.doi.org/10.21123/bsj.2021.18.4.1218
[16] Abed, S.S., Abduljabbar, M.F. (2019). Equivalence between iterative schemes in modular spaces. Journal of Interdisciplinary Mathematics, 22(8): 1529-1535. https://doi.org/10.1080/09720502.2019.1706850
[17] Okeke, G.A., Bishop, S.A., Khan, S.H. (2018). Iterative approximation of fixed point of multivalued ρ-quasi-nonexpansive mappings in modular function spaces with applications. Journal of Function Spaces, 2018: 1-9. https://doi.org/10.1155/2018/1785702
[18] Khan. S, Abbas. M. (2014). Approximating fixed points of multivalued ρ-nonexpansive mappings in modular function spaces. Fixed Point Theory and Applications, 2014(34). http://dx.doi.org/10.1186/1687-1812-2014-34
[19] Salman, B.B., Salman Abed, S. (2023). New accelerated iterative algorithm for (λ, ρ)-quasi firmly nonexpansive multiValued mappings. International Journal of Nonlinear Analysis and Applications, 14(1): 1825-1833. http://dx.doi.org/10.22075/ijnaa.2022.7059
