Next Article in Journal
Probabilistic Linguistic Multiple Attribute Group Decision-Making Based on a Choquet Operator and Its Application in Supplier Selection
Previous Article in Journal
Frequency-Based Finite Element Updating Method for Physics-Based Digital Twin
Previous Article in Special Issue
Convergence of Infinite Products of Uniformly Locally Nonexpansive Mappings
 
 
Font Type:
Arial Georgia Verdana
Font Size:
Aa Aa Aa
Line Spacing:
Column Width:
Background:
Article

A Novel Fixed-Point Iterative Process for Multivalued Mappings Applied in Solving a HIV Model of Fractional Order

by
Rubayyi T. Alqahtani
1,*,
Godwin Amechi Okeke
2 and
Cyril Ifeanyichukwu Ugwuogor
2
1
Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh P.O. Box 90950, Saudi Arabia
2
Functional Analysis and Optimization Research Group Laboratory (FANORG), Department of Mathematics, School of Physical Sciences, Federal University of Technology Owerri, Owerri P.M.B. 1526, Imo State, Nigeria
*
Author to whom correspondence should be addressed.
Mathematics 2025, 13(5), 739; https://doi.org/10.3390/math13050739
Submission received: 4 January 2025 / Revised: 9 February 2025 / Accepted: 10 February 2025 / Published: 25 February 2025
(This article belongs to the Special Issue Applied Functional Analysis and Applications: 2nd Edition)

Abstract

:
We introduce a new and a faster iterative method for the approximation of the fixed point of multivalued nonexpansive mappings in the setting of uniformly convex Banach spaces. We prove some stability and data-dependence results for this novel iterative scheme. A series of numerical illustrations and examples was constructed to validate our results. As an application, we propose a novel method for solving a certain fractional differential equation using our newly developed iterative scheme. Our results extend, unify, and improve several of the known results in the literature.

1. Introduction

Well-known mathematicians have extensively studied multivalued nonlinear mappings and proved some interesting fixed-point theorems in the framework of Banach spaces. Consequently, many problems in some areas of mathematics such as game theory and market economy, among others, can be transformed into fixed-point problems of multivalued nonlinear mathematical models. It is well known that if the fixed point of a given multivalued nonlinear map T exists, then finding the value of the fixed point p of the multivalued nonlinear map T is demanding. As a remedy to this task, mathematicians developed a fruitful way of solving this problem by using the fixed-point iteration method. An important way in determining the choice of a fixed-point iteration scheme to be used for the approximation of the fixed point of a given multivalued nonlinear mapping is the rate of convergence of the iteration process under consideration. For instance, for a comparison of two iterative processes in a one-dimensional case, we refer the reader to Rhoades [1]. Several mathematicians have developed certain iterative processes for the approximation of the fixed point of some nonlinear mappings in the literature (see, e.g., Agarwal [2,3,4], Mann [5], Ishikawa [6], Noor [7], Okeke [8], and Khan [9,10,11], among others). Next, we present some well-known fixed-point iterative schemes in the literature. The Picard [12] or successive iterative process is defined by the sequence { u n } n = 0 as follows:
u 1 = u C , u n + 1 = T u n , n N .
The Mann [5] iterative process is defined by the sequence { v n } n = 0 as follows:
v 1 = v C , v n + 1 = ( 1 α n ) v n + α n T v n , n N ,
where { α n } n = 0 ( 0 , 1 ) .
The Ishikawa [6] iterative process is given by the sequence { z n } n = 0 defined as follows:
z 1 = z C , y n = ( 1 β n ) z n + β n T z n z n + 1 = ( 1 α n ) z n + α n T y n , n N ,
where { α n } n = 0 and { β n } n = 0 are real sequences in ( 0 , 1 ) .
The Krasnosel’skii [13] iterative process is defined by the sequence { s n } as follows:
s 1 = s C s n + 1 = ( 1 λ ) s n + λ T s n , n N .
where { λ } is a real sequences in (0, 1).
Ullah and Arshad [14] introduced the iterative sequence { x n } defined by
x 1 = x C z n = ( 1 α n ) x n + α n T x n y n = T z n x n + 1 = T y n , n N
where { α n } are real sequences in (0, 1). They established that this iterative scheme converges faster than several existing iterative processes.
Noor [7] introduced the iterative process { u n } defined by
u 1 = u C y n = ( 1 γ n ) u n + γ n T u n w n = ( 1 β n ) u n + β n T y n u n + 1 = ( 1 α n ) u n + α n T w n , n N .
where { α n } , { β n } and { γ n } are sequences in (0, 1).
Okeke [8] developed a novel iterative process called the Picard–Ishikawa hybrid scheme, defined by the sequence { x n } n = 0 as follows:
x 1 = x C z n = ( 1 α n ) x n + α n T x n y n = ( 1 β n ) x n + β n T z n x n + 1 = T y n , n N .
where { α n } and { β n } are real sequences in ( 0 , 1 ) . The author established that this iterative process converges faster than several existing iterative schemes. Next, we pose the following crucial question.
Problem 1.
Is it possible to develop an iteration process whose rate of convergence is even faster than those defined above?
The essence of this paper is to propose a faster and more efficient iterative process for approximating the fixed point of some classes of generalized nonlinear mappings. This novel iterative scheme converges faster than some of the iterative schemes listed above in the sense of Berinde [15]. To answer this natural question, as posed in Problem 1 above, we introduce the following iterative method, called the IA-iterative scheme and defined by the sequence { x n } n = 0 as follows:
x 1 = x C z n = α n x n + β n x n + γ n T ( x n ) y n = T [ ( 1 λ n ) T ( z n ) + λ n T ( x n ) ] x n + 1 = T ( y n ) n N ,
where { α n } , { β n } , { γ n } and { λ n } are real sequences in ( 0 , 1 ) and α n + β n + γ n = 1 .
Several mathematical models for addressing the issue of infection disease especially HIV transmission disease have been developed (see [16,17]). As a result of this, several fractional derivatives in the sense of Caputo were developed as one of the novel formulations that have found widespread usage in the simulation of many kinds of infection models. It was suggested as part of the method of preventing and treating infection disease. One of the differential equations was developed by Caputo–Fabrizio and was named the Caputo–Fabrizio fractional differential equation. Consequently, the Krasnoselskii’s and Banach’s fixed-point approach, in combination with the kernels, were part of the methods adopted in solving the Caputo–Fabrizio fractional differential equation (CFFD) [16]. Readers interested in studies involving Riemann–Liouville fractional differential equations, Delta fractional differential equations, and conformable fractional differential equations may consult [18,19,20] and the references therein. Several works of literature involve several other methods for solving fractional differential equations, such as the Laplace differential transform method, hyperbolic non-polynomial spline method, homotopy perturbation method, fractional non-polynomial spline method, and Pade differential transform method, among others.
In the present paper, we prove that our newly developed fixed-point iterative process, called the IA-iterative scheme, is highly efficient in solving the Caputo–Fabrizio fractional differential equation (CFFD). We present several numerical experiments to validate our results. Our results generalize and extend several known results.

2. Preliminaries

Throughout this paper, we denote F ( T ) as the set of all fixed points of the mapping T . A point p is called a fixed point of T if p T p .
The following definitions will be useful in this study.
Definition 1
([21]). Suppose E is a Banach space. A subset K is said to be proximinal if for every x E there exists a k K such that
d ( x , k ) = inf { x y : y K } = d ( x , K )
It has been proven in the literature that every closed convex subset of a uniformly convex Banach space is proximinal. We denote P ( K ) as the family of a nonempty bounded proximinal subset of K.
Definition 2
([21]). The Hausdorff distance H ( . , . ) on P ( K ) is defined by
H ( A , B ) = max { sup a A d ( a , B ) , sup b B d ( b , B ) } ,
where A , B P ( K ) and d ( a , B ) = inf { a b : b B } .
Definition 3
([21]). Suppose K is a nonvoid convex subset of normed space E. A mapping T : K K is said to be a contraction if for every x , y K there exists a δ ( 0 , 1 ) such that
T x T y     δ x y .
Definition 4
([21]). A multivalued mapping T : K P ( K ) is said to be nonexpansive if for each x , y K we have
H ( T x , T y )     x y .
Definition 5
([15]). Suppose that E is a Banach space and T , T ˜ : E E is self mapping on E. Then, T ˜ is said to be approximate mapping of T if for each x E and for fixed ϵ > 0 , we have
T x T ˜ x     ϵ .
Definition 6
([15]). Let { a n } and { b n } be two sequences of real numbers converging to x and y, respectively. The sequence { a n } is said to converge faster than the sequence { b n } if
lim n | a n x | | b n y | = 0 .
Definition 7
([15]). Let { u n } and { v n } be two fixed-point iteration schemes that converge to a certain fixed point p of a given operator T. Suppose that the error estimates
u n p     a n , n N
v n p     b n , n N
exist, where { a n } and { b n } are two sequences of nonnegative numbers converging to 0. If { a n } converges faster than { b n } , then { u n } converges faster than { v n } to p .
Lemma 1
([22]). Let λ n be a positive sequence, such that there exist n 0 N , , such that for all n n 0 satisfying the inequality
λ n + 1 = ( 1 μ n ) ν n + μ n σ n ,
where { μ n } ( 0 , 1 ) for all n N , Σ n = 1 μ n and σ n 0 for all n N . Then, the inequality below holds
lim sup n ν n lim sup n σ n .
Lemma 2
([23]). Let { x n } n = 0 be a sequence of nonnegative real numbers, including zero, satisfying
s n + 1 ( 1 λ n ) s n .
If { λ n } ( 0 , 1 ) and Σ n = 0 λ n = then lim n s n = 0 .
Lemma 3
([24]). Let { α n } n = 0 , { β n } n = 0 , and { γ n } n = 0 be real sequences, such that
(i) 
0 α n , β n < 1 and γ n < 1
(ii) 
β n 0 , γ n 0 as n 0 .
Let { θ n } be a positive real sequence, such that Σ β n γ n ( 1 γ n ) θ n is bounded. Then, { θ n } has a subsequence which converges to zero.
Lemma 4.
Suppose that E is a Banach space. Then, E is uniformly convex if for any given number ρ > 0 , there exists a continuous strictly increasing function φ : [ 0 , ) [ 0 , ) with φ ( 0 ) = 0 , such that for each x , y B ρ , we have the following:
α n x n + ( 1 α n ) y n 2 α n x n 2 + ( 1 α n ) y n 2 α n ( 1 α n ) φ ( x n y n ) .
Lemma 5
([16]). For n 1 < k n , ζ > 1 , ρ 0 , and ω R , we have:
(1) 
D σ k σ ω = Γ ( k + 1 ) Γ ( ω k + 1 ) σ ω k
(2) 
D σ k ρ = 0
(3) 
D σ k R σ k ψ ( ν , σ ) = ψ ( ν , σ )
(4) 
R σ k D σ k ψ ( ν , σ ) = ψ ( ν , σ ) Σ i = 0 n 1 i ψ ( ν , 0 ) σ i i ! .

3. Convergence Analysis

We begin this section by proving the following results.
Theorem 1.
Suppose that K is a nonvoid compact convex subset of a uniformly convex Banach space E and the map T : K P ( K ) is a nonexpansive mapping with F ( T ) . Let { α n } , { β n } , { γ n } and { λ n } be four sequences in (0, 1). Suppose { x n } is the sequence defined by (8), satisfying
(i) 
0 α n , λ n < 1 , γ n < 1
(ii) 
β n 0 , γ n 0 , λ n 0 and
(iii) 
n = 1 α n β n γ n ( 1 γ n ) ( 1 λ n ) = .
Then, the sequence { x n } converges strongly to fixed point p F ( T ) .
Proof. 
Let p F ( T ) 0 and n 0 . Using the fact that α n + β n + γ n = 1 and w n T x n is such that w n p = d ( p , T x n ) , we obtain
z n p 2 = α n x n + β n x n + γ n w n p 2 = ( 1 γ n ) ( x n p ) + γ n ( w n p ) 2 ( 1 γ n ) x n p 2 + γ n w n p 2 α n β n γ n ( 1 γ n ) φ ( x n w n ) ( 1 γ n ) x n p 2 + γ n H 2 ( T x n , T p ) α n β n γ n ( 1 γ n ) φ ( x n w n ) ( 1 γ n ) x n p 2 + γ n x n p 2 γ n ( 1 γ n ) φ ( x n w n ) = x n p 2 α n β n γ n ( 1 γ n ) φ ( x n w n ) .
Secondly, let d n = ( 1 λ n ) v n + λ n w n , where v n T z n and w n T x n are such that v n p = d ( p , T z n ) and w n p = d ( p , T x n ) . Therefore, considering (14), we obtain
d n p 2 = ( 1 λ n ) ( v n p ) + λ n ( z n p ) 2 ( 1 λ n ) v n p 2 + λ n w n p 2 λ n ( 1 λ n ) φ ( v n w n ) ( 1 λ n ) H 2 ( T z n , T p ) + λ n H 2 ( T x n , T p ) λ n ( 1 λ n ) φ ( v n w n ) ( 1 λ n ) z n p 2 + λ n x n p 2 λ n ( 1 λ n ) φ ( v n w n ) ( 1 λ n ) [ x n p 2 α n β n γ n ( 1 γ n ) φ ( x n w n ) ] + λ n x n p 2 λ n ( 1 λ n ) φ ( v n w n ) x n p 2 α n β n γ n ( 1 γ n ) ( 1 λ n ) φ ( x n w n ) .
Considering that y n = T ( d n ) where f n T ( d n ) , such that f n p = d ( p , T d n ) and, using (15), we have the following:
y n p 2 = f n p 2 H 2 ( T d n , T p ) d n p 2 x n p 2 α n β n γ n ( 1 γ n ) ( 1 λ n ) φ ( x n w n ) .
Thus, x n + 1 = T ( y n ) where g n T ( y n ) , such that g n p = d ( p , T y n ) and, using (16), we have the following:
x n + 1 p 2 = g n p 2 H 2 ( T y n , T p ) y n p 2 x n p 2 α n β n γ n ( 1 γ n ) ( 1 λ n ) φ ( x n w n ) .
From (17), we have
x n + 1 p 2 x n p 2 α n β n γ n ( 1 γ n ) ( 1 λ n ) φ ( x n w n ) .
Hence, we obtain the following
α n β n γ n ( 1 γ n ) ( 1 λ n ) φ ( x n w n ) x n p 2 x n + 1 p 2 .
Consequently, it implies that
Σ n = 1 α n β n γ n ( 1 γ n ) ( 1 λ n ) φ ( x n w n ) x 1 p 2 < .
Using Lemma 4, there exists a subsequence { x n k w n k } of { x n w n } such that lim k φ ( x n w n ) = 0 , by considering the continuity and strictly increasing property of φ . By the compactness of K, we assume that x n k q for some q K , Therefore,
d ( q , T q ) q x n k + d ( x n k , T x n k ) + H ( T x n k , q ) .
Furthermore,
d ( q , T q ) q x n k + x n k w n k + x n k q 0 as n .
Consequently, q is a fixed point of mapping T. Now, replacing q with p, we obtain { x n q } , which is a decreasing sequence by (19). Since { x n q } 0 as n , it then follows that { x n q } decreases to 0. □

4. Rate of Convergence

In this section, we show that the IA-iterative scheme converges better than some other iterative processes mentioned above.
Proposition 1.
Suppose K is a nonvoid closed convex subset of a normed space E and T : K K is a contraction map satisfying (11) with δ ( 0 , 1 ) . Suppose that each of the iterative schemes (1)–(7) converge to the same fixed point p of T, where { α n } , { β n } , { γ n } , and { λ n } are four sequences in (0, 1). Then, the IA-iterative process (8) converges faster than iterative schemes (1)–(7).
Proof
Suppose p is a fixed point of T and also that δ ( 0 , 1 ) . Using (11) and the Picard iterative scheme (1), we have the following
u n + 1 p = T u n p δ u n p . . . δ n u 1 p .
Now, let
θ n = δ n u 1 p .
Considering Mann iterative scheme (2) and (3), we obtain the following
v n + 1 p = ( 1 α n ) v n + α n T v n p ( 1 α n ) v n p + T v n p ( 1 α n ) v n p + δ v n p ( 1 α n ( 1 δ ) ) v n p . . . ( 1 α ( 1 δ ) ) n v 1 p .
Therefore, consider
ϑ n = ( 1 α ( 1 δ ) ) n v 1 p .
Considering Picard–Mann (5) and (11), we obtain
s n + 1 p = T t n p δ t n p .
Secondly, we obtain
t n p = ( 1 α n ) s n + α n T s n p ( 1 α n ) s n p + α n T s n p ( 1 α n ) s n p + δ α n s n p ( 1 α n ( 1 δ ) ) s n p .
Combining (25) and (26), we have the following
s n + 1 p = T t n p δ t n p δ ( 1 α n ( 1 δ ) ) s n p δ ( 1 α n ( 1 δ ) ) s n p . . . ( δ ( 1 α ( 1 δ ) ) ) n s 1 p .
Moreover, take
ι n = ( δ ( 1 α ( 1 δ ) ) ) n s 1 p .
Now, we consider the new explicit iterative process named the IA-iterative scheme. Also, using (11) and (8)
z n p = α n x n + β n x n + γ n T x n p α n x n p + β n x n p + γ n T x n p ( 1 γ n ) x n p + δ γ n x n p ( 1 γ n ( 1 δ ) ) x n p .
Using (11), (8), and (29), we have
y n p = T [ ( 1 λ n ) T z n + λ n T x n ] p δ [ ( 1 λ n ) T z n + λ n T x n p ] δ [ ( 1 λ n ) T z n p + λ n T x n p ] [ 1 γ n λ n ( 1 δ n ) ] x n p .
Moreover, considering (11), (8), and (30), we obtain
x n + 1 p = T ( y n ) p δ y n p δ 3 ( 1 γ n λ n ( 1 δ ) ) x n p . . . ( δ 3 ( 1 γ n λ n ( 1 δ ) ) ) n x 1 p .
Now, we have
ϖ n = ( δ 3 ( 1 γ n λ n ( 1 δ ) ) n x 1 p .
We now justify the rate of convergence of our iterative process (8) as follows:
(i)
ϖ n θ n = ( δ 3 ( 1 γ n λ n ( 1 δ ) ) n x 1 p δ n u 1 p = δ 2 n ( ( 1 γ n λ n ( 1 δ ) ) n x 1 p u 1 p 0
as n . This implies that IA-iterative process (8) converges faster to p than the Picard iterative process (1).
(ii)
ϖ n ϑ n = ( δ 3 ( 1 γ n λ n ( 1 δ ) ) n ( 1 α ( 1 δ ) ) n x 1 p v 1 p = δ 3 n ( ( 1 γ n λ n ( 1 δ ) ) n ( 1 α ( 1 δ ) ) n x 1 p v 1 p 0
as n . This implies that IA-iterative process (8) converges faster to p than the Mann iterative process (2).
(iii)
ϖ n ι n = ( δ 3 ( 1 γ n λ n ( 1 δ ) ) n ( δ ( 1 α ( 1 δ ) ) ) n x 1 p s 1 p = δ 2 n ( ( 1 γ n λ n ( 1 δ ) ) n ( 1 α ( 1 δ ) ) n x 1 p v 1 p 0
as n . This implies that IA-iterative process (8) converges faster to p than the Picard–Mann iterative process (5). Similarly, from (3)–(7), following the same method as above, the IA-iterative process (8) then converges faster to p than from (3)–(7).
ϖ n κ n = ( δ 3 ( 1 γ n λ n ( 1 δ ) ) n x 1 p ( 1 α ( 1 δ ) ) n u 1 p = δ 3 n ( ( 1 γ n λ n ( 1 δ ) ) n ( 1 α ( 1 δ ) ) n x 1 p u 1 p 0
as n . This implies that IA-iterative process (8) converges faster to p than from (3)–(7). □

5. Data-Dependence Results

In this section, we establish some data dependence results for the newly introduced IA-iterative scheme (8).
Theorem 2.
Let T ˜ be an approximate operator of T satisfying (11). Let { x n } n = 1 be an iterative sequence generated by the new explicit iterative process named the IA-iterative scheme (8) for mapping T and define an iterative sequence { x ˜ n } n = 1 as follows
x ˜ 1 = x ˜ C , z ˜ n = α n x ˜ n + β n x ˜ n + γ n T ˜ ( x ˜ n ) y ˜ n = T ˜ [ ( 1 λ n ) T ˜ ( z ˜ n ) + λ n T ˜ ( x ˜ n ) ] x ˜ n + 1 = T ˜ ( y n ˜ ) , n N ,
where { α n } , { β n } and { γ n } are real sequences in (0,1) satisfying the following conditions:
(i) 
Σ n = 1 γ n = ,
(ii) 
1 2 [ γ n ( 1 δ ) ] for all n N , and
(iii) 
n = 1 [ γ n ( 1 δ ) ] = .
If T x * = x * and T ˜ x ˜ * = x ˜ * such that x ˜ n x ˜ * as n , then we have x * x ˜ * 7 ϵ 1 δ . where ϵ > 0 is fixed number.
Proof. 
Using (8), (11), (13), and (33), we obtain
z n z ˜ n = α n x n + β n x n + γ n T x n [ α n x n ˜ + β n x n ˜ + γ n T ˜ x n ˜ ] ( 1 γ n ) x n x n ˜ + γ n T x n T ˜ x n ˜ ( 1 γ n ) x n x n ˜ + γ n T x n T x n ˜ + γ n T x n ˜ T ˜ x n ˜ ( 1 γ n ) x n x n ˜ + γ n δ x n x n ˜ + γ n ϵ = ( 1 γ n ( 1 δ ) ) x n x n ˜ + γ n ϵ .
However, considering (8), (11), (13), (33) and (34) we have the following:
y n y n ˜ = T [ ( 1 λ n ) T ( z n ) + λ n T ( x n ) ] T ˜ [ ( 1 λ n ) T ˜ ( z ˜ n ) + λ n T ˜ ( x ˜ n ) ] δ ( 1 λ n ) T ( z n ) + λ n T ( x n ) [ ( 1 λ n ) T ˜ ( z ˜ n ) + λ n T ˜ ( x ˜ n ) ] δ [ ( 1 λ n ) T ( z n ) T ˜ ( z ˜ n ) + λ n T ( x n ) T ˜ ( x ˜ n ) ] ( 1 λ n ) T ( z n ) T ( z ˜ n ) + ( 1 λ n ) T ( z ˜ n ) T ˜ ( z ˜ n ) + λ n T ( x n ) T ( x ˜ n ) + λ n T ( x ˜ n ) T ˜ ( x ˜ n ) ( 1 λ n ) δ z n z ˜ n + λ n δ x n x ˜ n + ϵ ( 1 λ n ) δ [ ( 1 γ n ( 1 δ ) ) x n x n ˜ + γ n ϵ ] + λ n δ x n x ˜ n + ϵ ( 1 γ n ( 1 δ ) ) x n x n ˜ + ( 1 + γ n ) ϵ .
Using (8), (11), (13), (33) and (35), we obtain
x n + 1 x n + 1 ˜ = T ( y n ) T ˜ ( y n ˜ ) T ( y n ) T ( y n ˜ ) + T ( y n ˜ ) T ˜ ( y n ˜ ) δ y n y n ˜ + ϵ ( 1 γ n ( 1 δ ) ) x n x n ˜ + ( 2 + γ n ) ϵ .
From the conditions (i) and (ii), we observe that ( 36 ) gives
x n + 1 x n + 1 ˜ ( 1 γ n ( 1 δ ) ) x n x n ˜ + ( 2 + γ n ) ϵ × 7 ϵ 1 δ .
Let ν n : = x n x ˜ n , Ω n : = [ ( 1 γ n ( 1 δ ) ) ] ( 0 , 1 ) , σ n : = 7 ϵ ( 1 δ ) .
Moreover, using Lemma (1), it shows that
0 lim sup n x n x ˜ n lim sup n 7 ϵ ( 1 δ ) .
Therefore,
lim sup n x n x ˜ n lim sup n 7 ϵ ( 1 δ ) .
Considering the results of Theorem 1, we know that lim n x n = x * . Using the above, together with the conditions that lim n x ˜ n = x ˜ * , we obtain
x * x ˜ * 7 ϵ 1 δ .
The proof of Theorem 2 is completed. □

6. Stability Results

In this section, we prove some stability results of our novel iterative process.
Theorem 3.
Let ( E , . ) be an arbitrary Banach space and T : C C be a contraction mapping defined by (11). Suppose there exists x * F ( T ) such that the IA-iterative process (8) satisfies Σ n = 0 γ n = and γ n γ ( 0 , 1 ) for each n N , { x n } n = 1 converges to x * . Then, the IA-iterative process (8) is T-stable.
Proof. 
Suppose that { a n } n = 1 E is an arbitrary sequence; put
ε n = a n T x n + 1 ,
where
x n + 1 = T ( y n ) .
Therefore, with the IA-iterative scheme (8), and using contractive condition (11) with δ ( 0 , 1 ) , we have
a n x * a n T x n + 1 + T x n + 1 x * ε n + δ [ x n + 1 x * ] = ε n + δ [ T ( y n ) x * ] ε n + δ 2 [ y n x * ] .
Using (8), (11) and (40)
y n x * = T [ ( 1 λ n ) T ( z n ) + λ n T ( x n ) ] x * δ [ ( 1 λ n ) δ z n x * + λ n T x n x * ] ( 1 λ n ) δ z n x * + λ n δ x n x * .
Using (8), (11) and (41), we have
z n x * = α n x n + β n x n + γ n T ( x n ) x * ( α n + β n ) x n x * + γ n T ( x n ) x * ( 1 γ n ) x n x * + γ n δ x n x * [ 1 γ n ( 1 δ ) ] x n x * .
Inserting (42) into (41), we obtain
y n x * ( 1 λ n ) δ z n x * + δ λ n x n x * ( 1 λ n ) δ [ 1 γ n ( 1 δ ) ] x n x * + λ n δ x n x * .
Putting (43) into (40), we obtain
a n x * ε n + δ 2 [ y n x * ] ε n + δ 2 [ ( 1 λ n ) δ [ 1 γ n ( 1 δ ) ] x n x * + λ n δ x n x * ] ε n + δ 3 [ 1 γ n ( 1 δ ) ] x n x * .
Considering that γ n γ ( 0 , 1 ) for all n N , . It follows by Lemma 2 and (44) that
lim n a n = x * .
Conversely, let lim n a n = x * . Using (11), (39) and (40), we obtain
ε n = a n T g n + 1 a n x * + x * T x n + 1 a n x * + δ [ x * x n + 1 ] a n x * + δ [ T y n x * ] a n x * + δ 2 [ y n x * ] a n x * + δ 5 [ 1 γ n ( 1 δ ) ] x n x * .
Therefore, we have
ε n a n x * + δ 5 [ 1 γ n ( 1 δ ) ] x n x * .
So,
lim n ε n = 0 .
It implies that IA-iterative process (8) is T-stable. □

7. Application to Disease Model

Many interested models have been developed to obtain the solutions of infectious diseases. These models have actually become various differential equations and the case remains how to solve the various differential equations. In this application, we study a special case of infection disease model, i.e, the mathematical modeling of acquired immune deficiency syndrome (AIDS). Consequently, comprehending the dynamics of acquired immune deficiency syndrome (AIDS) and evaluating prospective methods of preventing and treatment have both outstandingly benefited from the use of mathematical modeling. Based on this, a fractional differential equation emanated from the mathematical modeling of the prevention and treatment of AIDS, which was developed by Caputo–Fabrizio and was named the Caputo–Fabrizio fractional differential equation [16]. Moreover, solving the Caputo–Fabrizio differential equation is taxing, and many authors have suggested ways of solving the Caputo–Fabrizio differential equation. Krasnoselskii’s and Banach’s fixed-point approach in combination with the kernels were part of the methods adopted in solving the Caputo–Fabrizio fractional differential equation (CFFD). In this study, we suggest that the best approach is to use our newer and faster fixed-point iterative process called the IA-iterative scheme (8) in solving the Caputo–Fabrizio fractional differential equation (CFFD). Furthermore, this novel iterative scheme can be applied in solving wider classes of diseases models, such as diseases with direct transmission among individuals, as in the case of HIV with long latency phases. The scheme can also be applied in solving models of diseases with indirect transmission via vectors (see, e.g., [25]).
Consider the following Caputo–Fabrizio fractional differential equation.
D t ϑ 0       L C h t = F ( t , h ( t ) ) , 0 < ϱ 1 h ( 0 ) = h 0 .
The solution to Equation (48) is provided by Lemma 5 if and only if the right side vanishes at 0, i.e.,
h ( t ) + h 0 + G F ( t , h ( t ) ) + G 0 t F ( ξ , h ( Θ ) ) d Θ ,
where G = 1 ϱ K ( ϱ ) and G = ϱ K ( ϱ ) .
Moreover, define the Banach space D = L [ 0 , T ] and the norm of D = L [ 0 , T ] on 0 < t T < by
h = sup t [ 0 , T ] { | h ( t ) | : h D } .
Assume that the following assumptions are met.
  • ( C 1 ) Let k F > 0 be a constant, then
  • | F ( t , h ( t ) ) F ( t , h ( t ) ) | K F | h h | .
  • ( C 2 ) Assume that γ n ( 1 ( λ n + G K F ( 1 + T ) ) ) < 1 .
We now prove the next theorem as follows
Theorem 4.
Assume that the conditions ( C 1 ) ( C 2 ) are met. Then, problems (48) and (49) have a unique solution; say x * , in ( D [ 0 , T ] , R ) D 1 ( [ 0 , T ] , R ) , then the new and faster iterative process named IA-iterative scheme (8) with α n , β n , γ n and λ n are real sequences in ( 0 , 1 ) , such that α n + β n + γ n = 1 and Σ n = 1 γ n ( 1 ( λ n + G K F ( 1 + T ) ) ) = , converges to x * .
Proof. 
We consider the Banach space D = ( L [ 0 , T ] , . ) , where . is the norm of D = L [ 0 , T ] . Let { x n } n = 1 be an iterative sequence generated by a newer and a faster iterative process named the IA-iterative scheme for an operator T : D D defined by
T ( x ) ( t ) = x 0 + G F ( t , x n ( t ) ) + G 0 t F ( t , x n ( s ) ) d s .
Consequently, we will show that x n converges to the solution x * as n tends to . Now, using (8), (50), and ( C 1 ) , we have the following
z n x * = α n x n + β n x n + γ n T x n x * = ( 1 γ n ) x n + γ n T x n x * ( 1 γ n ) x n x * + γ n T x n T x * = ( 1 γ n ) x n x * + γ n | x 0 + G F ( t , x n ( t ) ) + 0 t G F ( t , x n ( s ) ) d s [ x 0 + G F ( t , x * ( t ) ) + G 0 t F ( s , x * ( s ) ) d s ] | ( 1 γ n ) x n x * + γ n | G F ( t , x n ( t ) ) G F ( t , x * ( t ) ) | + γ n | G 0 t F ( t , x n ( s ) ) d s G 0 t F ( t , x * ( s ) ) d s | ( 1 γ n ) x n x * + γ n | G F ( t , x n ( t ) ) G F ( t , x * ( t ) ) | + γ n G 0 t [ | F ( t , x n ( s ) ) d s F ( t , x * ( s ) ) | ] d s ( 1 γ n ) x n x * + sup t [ 0 , T ] | G F ( t , x n ( t ) ) G F ( t , x * ( t ) ) | + γ n G sup t [ 0 , T ] [ 0 t | F ( t , x n ( s ) ) F ( t , x * ( s ) ) | d s ] ( 1 γ n ) x n x * + γ n G K F x n x * + γ n G K F T x n x * ( 1 γ n ) x n x * + γ n G K F x n x * + γ n G K F T x n x * [ 1 γ n ( 1 G K F ( 1 + T ) ) ] x n x * .
Let f n = ( 1 λ n ) T z n + λ n T x n ; then, using (8), (50), (51), and ( C 1 ) , we obtain
y n x * = T f n x * = T f n T x * | x 0 + G F ( t , f n ( t ) ) + G 0 t F ( t , f n ( s ) ) d s [ x 0 + G F ( t , x * ( t ) ) + G 0 t F ( s , x * ( s ) ) d s ] | | G F ( t , f n ( t ) ) G F ( t , x * ( t ) ) | + G 0 t [ | F ( t , f n ( s ) ) d s F ( t , x * ( s ) ) | ] d s sup t [ 0 , T ] | G F ( t , f n ( t ) ) G F ( t , x * ( t ) ) | + G sup t [ 0 , T ] [ 0 t | F ( t , f n ( s ) ) F ( t , x * ( s ) ) | d s ] G K F f n x * + G K F T f n x * G K F ( 1 + T ) f n x * f n x * .
Since f n = ( 1 λ n ) T z n + λ n T x n and considering (8), (50), (52), and ( C 1 ) , we have
y n x * f n x * = ( 1 λ n ) T z n x * + λ n T x n x * ( 1 λ n ) [ | x 0 + G F ( t , z n ( t ) ) + G 0 t F ( t , z n ( s ) ) d s [ x 0 + G F ( t , x * ( t ) ) + G 0 t F ( s , x * ( s ) ) d s ] | ] + λ n [ | x 0 + G F ( t , x n ( t ) ) + G 0 t F ( t , z n ( s ) ) d s [ x 0 + G F ( t , x * ( t ) ) + G 0 t F ( s , x * ( s ) ) d s ] | ] ( 1 λ n ) | G F ( t , z n ( t ) ) G F ( t , x * ( t ) ) | + ( 1 λ n ) G 0 t [ | F ( t , z n ( s ) ) d s F ( t , x * ( s ) ) | ] d s + λ n | G F ( t , x n ( t ) ) G F ( t , x * ( t ) ) | + λ n G 0 t [ | F ( t , x n ( s ) ) d s F ( t , x * ( s ) ) | ] d s ( 1 λ n ) sup t [ 0 , T ] | G F ( t , z n ( t ) ) G F ( t , x * ( t ) ) | + ( 1 λ n ) G sup t [ 0 , T ] [ 0 t | F ( t , z n ( s ) ) F ( t , x * ( s ) ) | d s ] + λ n sup t [ 0 , T ] | G F ( t , x n ( t ) ) G F ( t , x * ( t ) ) | + λ n G sup t [ 0 , T ] [ 0 t | F ( t , x n ( s ) ) F ( t , x * ( s ) ) | d s ] ( 1 λ n ) G K F z n x * + ( 1 λ n ) G K F T z n x * + λ n G K F x n x * + λ n G K F T x n x * ( 1 λ n ) G K F ( 1 + T ) z n x * + λ n G K F ( 1 + T ) x n x * ( 1 λ n ) G K F ( 1 + T ) [ 1 γ n ( 1 G K F ( 1 + T ) ) ] x n x * + λ n G K F ( 1 + T ) x n x * [ 1 γ n ( 1 ( λ n + G K F ( 1 + T ) ) ) ] x n x * .
Considering (8), (50), (53), and ( C 1 ) , we have
x n + 1 x * = T y n T x * | x 0 + G F ( t , y n ( t ) ) + 0 t G F ( t , y n ( s ) ) d s [ x 0 + G F ( t , x * ( t ) ) + G 0 t F ( s , x * ( s ) ) d s ] | | G F ( t , y n ( t ) ) G F ( t , x * ( t ) ) | + G 0 t [ | F ( t , y n ( s ) ) d s F ( t , x * ( s ) ) | ] d s sup t [ 0 , T ] | G F ( t , y n ( t ) ) G F ( t , x * ( t ) ) | + G sup t [ 0 , T ] [ 0 t | F ( t , y n ( s ) ) F ( t , x * ( s ) ) | d s ] G K F y n x * + G K F T y n x * = G K F ( 1 + T ) y n x * y n x * [ 1 γ n ( 1 ( λ n + G K F ( 1 + T ) ) ) ] x n x * .
Therefore, we have the following from (54)
x n + 1 x * [ 1 γ n ( 1 ( λ n + G K F ( 1 + T ) ) ) ] x n x * .
From (55) and considering Lemma 2, we take μ n = γ n ( 1 ( λ n + G K F ( 1 + T ) ) ) < 1
and x n x * = s n , ; then, the conditions of Lemma 2 are satisfied.
Hence, lim n x n x * = 0 .
It follows that the proof of Theorem 4 is completed. □

8. Numerical Examples

Numerically, we compare our iteration process with seven existing iteration schemes. In Table 1 and Table 2 below, we compare the speed of convergence of various iterative schemes, viz: Picard, Mann, Ishikawa, Kransel’kii, Ullah and Arshad, Noor, Picard–Ishikawa, and the IA-iterative scheme. Choose α n = β n = λ = γ n = λ n = 1 2 with the initial value x 0 = 0.1 and the operator T ( x ) = 1 5 x + 3 , where F ( T ) = { 3.75 } . . Using Python 3.7, we obtain the following numerical results:
Moreover, the Table 1 and Table 2 show the numerical data obtained from the operator T defined above with the initial value x 0 = 0.1 . Clearly, we use the numerical data to compare the rate of convergence of the IA-iterative process among other existing iterative schemes viz: Picard, Ishikawa, Kransel’skii, Ullah & Arshad, Noor, and Picard–Ishikawa.
Remark 1.
Clearly, from Table 1 and Table 2, we compare the IA-iteration process numerically among other existing iterative processes, i.e., Picard, Mann, Ishikawa, Kransel’skii, Ullah & Arshad, Noor, and Picard–Ishikawa, to the fixed point of T.
Furthermore, we provide a graphical representation of the numerical data in Table 1 and Table 2 above as follows.
Remark 2.
Clearly, from Table 1 and Table 2 and Figure 1, we see that the IA-iteration process converges faster than Picard, Mann, Ishikawa, Kransel’skii, Ullah & Arshad, Noor, and Picard–Ishikawa to the fixed point of T , which is 3.75.

9. Conclusions

In this paper, we introduced a new and a faster iterative scheme. Moreover, we then proved that the constructed IA-type iterative process converges strongly to the fixed point of multivalued nonexpansive mapping T in uniformly convex Banach spaces. Consequently, we showed that the IA-iterative process is stable and data dependent. We showed that the newly iterative method converges faster and is more efficient than some of the iterative processes in the literature, i.e., Picard, Mann, Ishikawa, Kranselskii, Picard–Mann, Kranselskii, and Picard–Ishikawa, among others. In terms of application, we suggested a newer and faster method for solving one of the Caputo’s derivatives that emanated from the infection disease models, named the Caputo–Fabrizio fractional differential equation. Our results extend, unify, and improve several of the known results in the literature.

Author Contributions

Conceptualization, G.A.O. and C.I.U.; methodology, R.T.A. and C.I.U.; software, G.A.O. and C.I.U.; validation, R.T.A., G.A.O. and C.I.U.; formal analysis, R.T.A. and G.A.O.; investigation, R.T.A., G.A.O. and C.I.U.; resources, R.T.A.; data curation, C.I.U.; writing—original draft preparation, G.A.O. and C.I.U.; writing—review and editing, R.T.A. and G.A.O.; visualization, R.T.A. and G.A.O.; supervision, G.A.O.; project administration, R.T.A.; funding acquisition, R.T.A. All authors have read and agreed to the published version of the manuscript.

Funding

This work was supported and funded by the Deanship of Scientific Research at Imam Mohammad Ibn Saud Islamic University (IMSIU) (grant number IMSIU-DDRSP2501).

Data Availability Statement

The original contributions presented in this study are included in the article. Further inquiries can be directed to the corresponding author.

Acknowledgments

This work was supported and funded by the Deanship of Scientific Research at Imam Mohammad Ibn Saud Islamic University (IMSIU) (grant number IMSIU-DDRSP2501). The authors wish to thank the editor and the reviewers for their useful comments and suggestions.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

References

  1. Rhoades, B.E. Comments on two fixed point iteration methods. J. Math. Anal. Appl. 1976, 56, 741–750. [Google Scholar] [CrossRef]
  2. Agarwal, R.P.; O’Regan, D.; Sahu, D.R. Iteration construction of fixed points of nearly asymptotically nonexpansive mappings. J. Nonlinear Convex Anal 2007, 8, 61–79. [Google Scholar]
  3. Chugh, R.; Kumar, V.; Kumar, S. Strong convergence of a new step iterative scheme in Banach spaces. Am. J. Comput. Math. 2012, 2, 345–357. [Google Scholar] [CrossRef]
  4. Karahan, I.; Ozdemir, M. A general iterative method for approximation of fixed points and their applications. Adv. Fixed Point Theory 2013, 3, 510–526. [Google Scholar]
  5. Mann, W.R. Mean Value methods in iteration. Proc. Am. Math. Soc. 1953, 4, 506–510. [Google Scholar] [CrossRef]
  6. Ishikawa, S. Fixed points by a new iteration method. Proc. Am. Math. Soc. 1974, 44, 147–150. [Google Scholar] [CrossRef]
  7. Noor, M.A. New approximation scheme for general variation inequalities. J. Math. Anal. Appl. 2000, 251, 217–229. [Google Scholar] [CrossRef]
  8. Okeke, G.A. Convergence analysis of the Picard-Ishikawa hybrid iteration process with application. Afr. Math. 2019, 30, 817–835. [Google Scholar] [CrossRef]
  9. Khan, S.H. A Picard-Mann hybrid iterative process. Fixed Point Theory Appl. 2013, 2013, 69. [Google Scholar] [CrossRef]
  10. Sarivastava, V.; Rai, K.N. A multi-term fractional diffusion equation for oxygen delivery a capillary to tissues. Math. Comput. Model. 2010, 51, 616–624. [Google Scholar] [CrossRef]
  11. Tan, K.K.; Xu, H.K. Approximating fixed points of nonexpansive mappings by Ishikawa iteration process. J. Math. Anal. Appl. 1993, 178, 301–308. [Google Scholar] [CrossRef]
  12. Picard, E. Memoire sur la theorie des equations aux derivees partielles et la methode des approximations successives. J. Math. Pures Appl. 1890, 6, 145–210. [Google Scholar]
  13. Kransel, M.A. Two observations about the method of successive approximations. Uspekhi Mat. Nauk. 1957, 10, 131–140. [Google Scholar]
  14. Ullah, K.; Arshad, M. Numerical reckoning fixed points for Suzuki’s generalized nonexpansive mappings via new iteration process. Filomat 2018, 32, 187–196. [Google Scholar] [CrossRef]
  15. Berinde, V. Iterative Approximation of Fixed Points; Efemeride: Baia Mare, Romania, 2002. [Google Scholar]
  16. Alshehry, A.S.; Mukhtar, S.; Khan, H.S.; Shah, R. Fixed-point theory and numerical analysis of an epidemic model with fractional calculus: Exploring dynamical behavior. Open Phys. 2023, 21, 20230121. [Google Scholar] [CrossRef]
  17. Baleanu, D.; Arshad, S.; Jajarmi, A.; Shokat, W.; Ghassabzade, F.A.; Wali, M. Dynamical behaviors and stability analyzis of a generalied fractional model with a real case study. J. Adv. Res. 2023, 48, 157–173. [Google Scholar] [CrossRef] [PubMed]
  18. Agarwal, R.; Hristova, S.; O’Regan, D. Basic concepts on Riemann-Liouville fractional differential equations with non-instantaneous impulses. Symmetry 2019, 11, 614. [Google Scholar] [CrossRef]
  19. Kajouni, A.; Chafiki, A.; Hilal, K.; Oukessou, M. A new conformable fractional derivative and applications. Int. J. Differ. Equ. 2021, 2021, 6245435. [Google Scholar] [CrossRef]
  20. Vivas-Cortez, M.; Mohammed, P.O.; Guirao, J.L.G.; Yousif, M.A.; Ibrahim, I.S.; Chorfi, N. Improved fractional differences with kernels of delta Mittag-Leffler and exponential functions. Symmetry 2024, 16, 1562. [Google Scholar] [CrossRef]
  21. Okeke, G.A.; Abbas, M.; De la Sen, M. Approximation of the fixed point of multivalued quasi-nonexpansive mappings via a faster iterative process with applications. Discret. Dyn. Nat. Soc. 2020, 2020, 8634050. [Google Scholar] [CrossRef]
  22. Grosan, T.; Soltuz, S.M. Data dependence for Ishikawa iteration when dealing with contractive-like operators. Fixed Point Theory Appl. 2008, 2008, 242916. [Google Scholar]
  23. Soltuz, S.M.; Otrocol, D. Classical results via Mann-Ishikawa iteration. Rev. D’analyse Numer. L’approximation 2007, 36, 193–197. [Google Scholar]
  24. Shukka, D.P.; Tiwari, V.; Singh, R. Noor iterative processes for multivalued mappings in Banach Spaces. Int. J. Math. Anal. 2014, 8, 649–657. [Google Scholar] [CrossRef]
  25. Granger, T.; Michelitsch, T.M.; Bestehorn, M.; Riascos, A.P.; Collet, B.A. Stochastic compartment model with mortality and its application to epidemic spreading in complex networks. Entropy 2024, 26, 362. [Google Scholar] [CrossRef]
Figure 1. Comparison of rate of convergence among various iteration schemes.
Figure 1. Comparison of rate of convergence among various iteration schemes.
Mathematics 13 00739 g001
Table 1. A comparison of the speed of convergence among various iterative processes.
Table 1. A comparison of the speed of convergence among various iterative processes.
StepPicardMannIshikawaKrasnosel’kiiUllah and Arshad
00.1000000.10000000.10000000.10000000.1000000
13.020000001.560000001.7060000001.560000003.6624000000
23.6040000002.4360000002.605360000002.4360000003.7478976000000
33.72080000002.96160000003.10900160000002.96160000003.7499495424000000
43.744160000003.26960000003.3910408960000003.276960000003.7499987890176000
53.748832000003.466176000003.3.54898290175999963.46617600003.7499999709364227
63.749766400003.57970560003.6374304249855998003.57970560003.7499999993024744
73.749953280003.647823360003.686961037991936003.647823360003.74999999983200
83.499906560003.6886940160003.71469818127550003.688694016003.749999999999598
93.74999813123.7132164096003.73023098151427133.713164096003.74999999999902
103.749999626243.727929845763.7389293496479923.727929845763.7500000000000
Table 2. A comparison of the speed of convergence among various iterative processes.
Table 2. A comparison of the speed of convergence among various iterative processes.
StepNoorPicard–IshikawaIA-Iteration
00.10000000.10000000.1000000
11.7206000003.3141200003.7268400000
22.6216536000003.70421440000003.75730545600000
33.12263940160003.744872012800003.7575613658304
43.40118750728963.74942566543360003.7575635154730000
53.55606025405301733.74993567452856353.7575635335299733
63.6421695012534783.7499927955471993.757563533681652
73.690046242696933.74999919310128643.757563533682926
83.71666571093949473.7499999096273443.7575635336829367
93.7314661352823593.74999998987826283.7575635336829367
103.73969517121699153.74999999886636553.7575635336829367
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Share and Cite

MDPI and ACS Style

Alqahtani, R.T.; Okeke, G.A.; Ugwuogor, C.I. A Novel Fixed-Point Iterative Process for Multivalued Mappings Applied in Solving a HIV Model of Fractional Order. Mathematics 2025, 13, 739. https://doi.org/10.3390/math13050739

AMA Style

Alqahtani RT, Okeke GA, Ugwuogor CI. A Novel Fixed-Point Iterative Process for Multivalued Mappings Applied in Solving a HIV Model of Fractional Order. Mathematics. 2025; 13(5):739. https://doi.org/10.3390/math13050739

Chicago/Turabian Style

Alqahtani, Rubayyi T., Godwin Amechi Okeke, and Cyril Ifeanyichukwu Ugwuogor. 2025. "A Novel Fixed-Point Iterative Process for Multivalued Mappings Applied in Solving a HIV Model of Fractional Order" Mathematics 13, no. 5: 739. https://doi.org/10.3390/math13050739

APA Style

Alqahtani, R. T., Okeke, G. A., & Ugwuogor, C. I. (2025). A Novel Fixed-Point Iterative Process for Multivalued Mappings Applied in Solving a HIV Model of Fractional Order. Mathematics, 13(5), 739. https://doi.org/10.3390/math13050739

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

Article Metrics

Back to TopTop