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Article

Novel Method for Approximating Fixed Point of Generalized α-Nonexpansive Mappings with Applications to Dynamics of a HIV Model

by
Godwin Amechi Okeke
1,*,†,
Akanimo Victor Udo
1,† and
Rubayyi T. Alqahtani
2,†
1
Functional Analysis and Optimization Research Group Laboratory (FANORG), Department of Mathematics, School of Physical Sciences, Federal University of Technology Owerri, Owerri 460114, Nigeria
2
Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11623, Saudi Arabia
*
Author to whom correspondence should be addressed.
These authors contributed equally to this work.
Mathematics 2025, 13(4), 550; https://doi.org/10.3390/math13040550
Submission received: 15 January 2025 / Revised: 3 February 2025 / Accepted: 5 February 2025 / Published: 7 February 2025

Abstract

:
In this paper, we use an existing fixed point iterative scheme to approximate a class of generalized α -nonexpansive mapping in Banach spaces. We also prove weak and strong convergence results for the mapping using the AG iterative scheme. An example of a generalized α -nonexpansive mapping is given to show the validity of the claims. We apply the main results to the approximation of solution of a mixed type Voltera–Fredholm functional nonlinear integral equation and to the spread of HIV modeled in terms of a fractional differential equation of the Caputo type.

1. Introduction

Suppose H is a Banach space and B is a uniformly convex subset of H . A mapping G : B B is said to be a contraction if for some α [ 0 , 1 ) ,
G x G y α x y , x , y B .
If α = 1 , such that (1) becomes
G x G y x y ,
then the mapping is known as nonexpansive.
As a generalization of the nonexpansive mapping, Suzuki [1], in 2008, introduced a new class of mapping which is said to be a mapping satisfying condition (C) and defined as follows:
if
1 2 x G x x y implies G x G y x y ,
for all x , y B . This mapping is also known as Suzuki’s generalized nonexpansive mapping.
In 2011, García-Falset et al. [2], further extended the class of mapping satisfying condition (C) to a new class of mapping known to satisfy condition ( E μ ), defined thus:
if for all x , y B ,
x G y μ x G x + x y ,
for μ 1 .
Again, in 2011, Aoyama and Kohsaka [3] introduced the class of α -nonexpansive mapping in Banach spaces. They defined it thus:
the mapping G : B B is said to be α -nonexpansive if for a real number α < 1 ,
G x G y 2 α G x y 2 + α x G y 2 + ( 1 2 α ) x y 2 ,
for all, x , y B .
The concept of α -nonexpansive mapping is trivial for α < 0 [4].
As a way to generalize the class of α -nonexpansive mapping, Pant and Shukla [5], in 2017, introduced the class of generalized α -nonexpansive mapping as presented in the definition below.
 Definition 1.
Let B be a uniformly convex nonempty subset of a Banach space H . A mapping G : B B is called a generalized α-nonexpansive mapping if for any real number α [ 0 , 1 ) ,
1 2 x G x x y implies G x G y α G x y + α G y x + ( 1 2 α ) x y
for all x , y B .
 Remark 1.
Every class of generalized α-nonexpansive mapping fully contains the mapping satisfying condition (C) and partially contains the class of α-nonexpansive mapping.
Fixed point theory is a fundamental area of mathematics with profound implications in various branches of mathematics and sciences. It has not only provided deep theoretical insights but has also been instrumental in addressing real-world problems across disciplines. Over the years, fixed point theory has evolved into a vibrant area of research, drawing significant attention due to its versatility and applicability. Fixed point theory has established itself as a cornerstone in mathematical analysis, with broad applications that extend far beyond its foundational theorems. Its ability to address problems in fractional differential equation, ordinary differential equations, boundary value problems, integral equations, delay differential equations, and other advanced mathematical constructs underscores its significance in contemporary research and applied mathematics (see, for example, Refs. [6,7,8,9,10] and other references therein).
Modeling the spread of HIV and other biological phenomena using fractional differential equations (FDEs) has become significant in offering a more nuanced understanding of the disease’s dynamics by incorporating memory effects and hereditary properties inherent in biological systems. The application of fixed point theory, particularly through iterative schemes, plays a crucial role in approximating solutions to these complex models. FDEs extend classical differential equations by allowing derivatives of non-integer orders, effectively capturing the memory and hereditary characteristics of HIV transmission and progression. This approach provides a more accurate representation of the disease’s dynamics compared to integer-order models. Recent studies have employed various fractional derivatives, such as the Caputo and Caputo–Fabrizio operators, to model HIV/AIDS transmission dynamics (see [11] and references therein).
The aim of this paper is to use an existing fixed point iterative scheme by Okeke et al. [12] to approximate the fixed point of a class of generalized α -nonexpansive mapping in a Banach space with a numerical example and apply the result in approximating the solution of a mixed type Voltera–Fredholm functional nonlinear integral equation and in solving the problem of the spread of HIV, modeled as a fractional differential equation of the Caputo type.
The rest of this paper is arranged as follows: Section 2 is for the preliminaries which contain the basic definitions, lemmas, and propositions. Section 3 is dedicated to the main results. Meanwhile, the numerical example and rate of convergence with a table and graphs are in Section 4. Application of the main results to the mixed type integral equation is presented in Section 5. Section 6 is for application of the main result to the spread of HIV, modeled as a fractional differential equation of the Caputo type. The concluding remark of the paper is presented in Section 7.

2. Preliminaries

Fixed point theory has been very remarkable in mathematics and its application, especially in the aspect of the approximation of the fixed point of operators and by extension to the approximation of solutions of certain mathematical problems in applications modeled as ordinary differential equations, partial differential equations, integral equations, boundary value problems, fractional differential equation, etc.
Several fixed point iterative schemes have been developed and used accordingly for approximation of the fixed point of certain classes of mappings.
In 2014, Abbas and Nazir [13] introduced a three-step iterative process called the AK iterative scheme and defined it as follows:
x 0 = x B z n = ( 1 γ n ) x n + γ n G x n y n = ( 1 β n ) G x n + β n G z n x n + 1 = ( 1 α n ) G y n + α n G z n
for { α n } , { β n } , { γ n } ( 0 , 1 )
In 2014, Gürsoy et al. [14] introduced an iterative scheme called the Picard-S defined thus:
x 0 = x B z n = ( 1 β n ) x n + β n G x n y n = ( 1 α n ) G x n + α n G z n x n + 1 = G y n , n N .
In 2016, Thakur et al. [15] introduced an iterative scheme defined thus:
x 0 = x B z n = ( 1 γ n ) x n + γ n G x n y n = G [ ( 1 β n ) x n + β n z n ] x n + 1 = G y n , n N
for { β n } , { γ n } B .
In 2018, Hussain et al. [16] introduced a K iterative scheme which is defined as follows:
x 0 = x B z n = ( 1 γ n ) x n + γ n G x n y n = G [ ( 1 β n ) G x n + β n G z n ] x n + 1 = G y n , n N
In 2018, Piri et al. [17] introduced the following iterative scheme,
x 0 = x B , z n = G [ ( 1 β n ) x n + β n G x n ] , y n = G z n , x n + 1 = ( 1 α n ) G y n , n N ,
which they used to approximate the fixed point of the class of generalized α -nonexpansive mapping in Banach spaces for { α n } , { β n } ( 0 , 1 ) .
Recently, Ullah et al. [18] used the K iterative scheme (6) to approximate the fixed point of a class of generalized α -nonexpansive mapping (2).
Motivated by the foregoing, we use the existing AG iterative scheme introduced by Okeke et al. [6] and defined as follows
u 0 = u B u n + 1 = G v n v n = G [ ( 1 α n ) w n + α n G w n ] w n = ( 1 β n ) G u n + β n G x n x n = ( 1 γ n ) u n + γ n G u n , n N ,
where { α n } , { β n } and { γ n } are sequences of real numbers in [ 0 , 1 ] to answer the following question:
Question: Can the AG (8) iterative scheme be used to approximate the fixed point of the class of generalized α -nonexpansive mapping?
The answer is in the affirmative.
Next, we outline some important basic definitions and lemmas.
 Definition 2
([19]). A mapping G : B B with F ( G ) is said to satisfy condition (I) if there exists a nondecreasing function h : [ 0 , ) [ 0 , ) with h ( 0 ) = 0 , h ( s ) > 0 for s ( 0 , ) , such that u G u h ( ρ ( u , F ( G ) ) ) for all u D , where ρ ( u , F ( G ) ) = inf { u τ : τ F ( G ) } .
 Definition 3
([20]). A Banach space H is said to satisfy the Opial condition [21] if for each sequence { u n } in H , converging weakly to p H , we have
lim sup n u n p < lim sup n u n q ,
for all q H such that p q .
 Definition 4
([20,22]). Let B be a nonempty closed convex subset of a Banach space H and { u n } be a bounded sequence in H . For each u H , we define
(a)
asymptotic radius of { u n } at u as a functional, R ( · , { u n } ) : X R + defined by R ( u , { u n } ) = lim sup n u u n ,
(b)
asymptotic radius of { u n } relative to the set B by
R ( B , { u n } ) = inf { R ( u , { u n } ) : u B } , and
(c)
asymptotic center of { u n } relative to the set H by
A ( B , { u n } ) = { u H : R ( u , { u n } ) = R ( B , { u n } ) } .
 Lemma 1
([23]). Let H be a uniformly convex Banach space and { α n } n = 0 , { β n } n = 0 and { γ n } n = 0 be sequences of numbers such that 0 < a α n b < 1 , n 1 , for a , b R . Let { s n } n = 0 and { t n } n = 0 be sequences in H such that lim sup n s n λ , lim sup n t n λ and
lim sup n α n s n + ( 1 α n ) t n = λ
for some λ 0 . Then, lim n s n t n = 0 .
 Proposition 1
([5]). Let B be a nonempty subset of a Banach space H and G : B B be a self mapping, and the following will hold.
  • G is generalized α-nonexpansive whenever it satisfies condition ( C ) .
  • G is quasi-nonexpansive whenever it is generalized α-nonexpansive.
  • F ( G ) is closed in B whenever G is generalized α-nonexpansive.
  • For each point x , y B , x G y 3 + α 1 α x G x + x y holds, whenever G is generalized α-nonexpansive.
  • If G is generalized α-nonexpansive, { u n } a sequence in H such that H has Opial’s condition and { u n } is weakly convergent with weak limit ϖ, then ϖ F ( G ) whenever lim n u n G u n = 0 .

3. Main Results

In the sequel, the following lemmas will be useful.
 Lemma 2.
Let B be a nonempty closed convex subset of a Banach space H . Let G : B B be a generalized α-nonexpansive mapping with F ( G ) and { u n } represents a sequence generated by the AG iterative scheme (8). Then for each * F ( G ) , lim n u n * exists.
 Proof. 
Let * F ( G ) be a fixed point. By axiom (2) of Proposition 1 together with (8), we have,
x n * = ( 1 γ n ) u n + γ n G u n * ( 1 γ n ) u n * + γ n G u n G * ( 1 γ n ) u n * + γ n u n * = u n * .
Again,
w n * = ( 1 β n ) G u n + β n G x n * ( 1 β n ) G u n * + β n G x n G * ( 1 β n ) G u n G * + β n G x n G * ( 1 β n ) u n * + β n x n * = u n * .
Using (8) and (11),
v n * = G [ ( 1 α n ) w n + α n G w n ] * G [ ( 1 α n ) w n + α n G w n ] G * [ ( 1 α n ) w n + α n G w n ] * ( 1 α n ) w n * + α n G w n G * ( 1 α n ) w n * + α n w n * = w n * = u n * .
Using (8) and (12), we have
u n + 1 * = G v n * G v n G * v n * = u n * .
Here, we can say that { u n * } is bounded and nonincreasing. Therefore, for all * F ( G ) , lim n u n * exists. □
 Lemma 3.
Suppose B is a nonempty closed convex subset of a uniformly convex Banach space and consider a mapping G : B B which is a generalized α-nonexpansive mapping. Let { u n } be a sequence generated by the AG iterative scheme (8). Then, F ( G ) if and only if { u n } is bounded and lim n G u n u n = 0 holds.
 Proof. 
Assume that F ( G ) and * F ( G ) is a fixed point. Then by using Lemma 2, we have that lim n u n * exists and { u n } is bounded. Let d be any real value.
Set
lim n u n * = d .
From Lemma 2 and (14), we have the following
lim sup n x n * lim sup n u n * = d
lim sup n w n * lim sup n u n * = d
and
lim sup n v n * lim sup n u n * = d .
By condition (2) of Proposition 1, and (14), we have
lim sup n G u n * lim sup n u n * = d .
Now, from Lemma 2, we recall that
u n + 1 * = G v n * v n * .
Again from condition (2) of Proposition 1, together with (14), we have
d = lim inf n u n + 1 * lim inf n v n * .
From (17) and (20), it follows that
lim inf n v n * lim sup n v n * = d ,
so that we have
lim n v n * = d .
From (21), we have
d = lim inf n v n * lim inf n w n * ,
which follows that
d = lim inf n w n * lim sup n w n * = d ,
so that
lim n w n * = d
and
w n * = ( 1 β n ) G u n + β n G x n * ( 1 β n ) G u n * + β n G x n * ( 1 β n ) u n * + β n x n * .
We have that
w n * u n * w n * u n * β n x n * u n *
and therefore,
w n * x n * .
Here,
d = lim inf n w n * lim inf n x n * ,
d lim inf n x n * ,
so that by combining (15) and (24), we have
d = lim n x n * .
Using (25), we have
d = lim n x n * = lim n ( 1 γ n ) u n + γ n G u n * lim n ( 1 γ n ) ( u n * ) + γ n ( G u n * ) ,
that is,
d = lim n ( 1 γ n ) ( u n * ) + γ n ( G u n * ) .
Applying Lemma 1 to (26), we have that
lim n G u n u n = 0 .
Conversely, we need to show that F ( G ) is nonempty whenever { u n } is bounded with lim n G u n u n = 0 .
To achieve that, let * A ( B , { u n } ) .
Applying condition (4) of Proposition 1, we have
R ( G * , { u n } ) = lim sup n u n G * 3 + α 1 α lim sup n u n G u n + lim sup n u n * = lim sup n u n * = R ( * , { u n } ) .
Hence, we have shown that G * A ( B , { u n } ) . However, A ( B , { u n } ) is a singleton, thus it follows that G * = * and F ( G ) is nonempty. Therefore, the proof is complete. □

Weak and Strong Convergence

 Theorem 1.
Assume B is a nonempty closed subset of a uniformly convex banach space H . Let G : B B be a generalized α-nonexpansive mapping with F ( G ) and { u n } n = 0 be a sequence generated by the iterative scheme (8). Then { u n } n = 0 converges weakly to a fixed point * F ( G ) if H satisfies the Opial condition (9).
 Proof. 
In Lemma 3, it was shown that the sequence { u n } is bounded and lim n G u n u n = 0 holds.
Since the Banach space, H is convex, then we claim that it is reflexive.
Consequently, we find a convergent subsequence { u n i } of { u n } such that { u n i } converges weakly to a point r 1 B (i.e., u n i r 1 ).
By condition (5) of Proposition 1, we can say that r 1 F ( G ) and we may conclude that the proof is complete.
However, suppose in the contrary that our claim fails, then there exists another subsequence { u n j } of { u n } that converges weakly to a point r 2 B , such that r 1 r 2 .
Again, using condition (5) of Proposition 1, we have that r 2 F ( G ) .
By Lemma 2 and using the Opial condition (9) for the Banach space, H , we have
lim n u n r 1 = lim i u n i r 1 < lim i u n i r 2 = lim n u n r 2 = lim j u n j r 2 < lim j u n j r 1 = lim n u n r 1 .
Clearly, we obtain lim n u n r 1 < lim n u n r 1 which is a contradiction. Hence, r 1 = r 2 , implying that there is only one limit point. It can be concluded that the sequence { u n } converges to a fixed point in F ( G ) . Thereby completing the proof. □
 Theorem 2.
Let B be a nonempty closed convex subset of a Banach space H . Assume G : B B is a generalized α-nonexpansive mapping with F ( G ) . If { u n } n = 0 is a sequence generated by the AG iterative scheme (8). Then { u n } n = 0 converges strongly to a fixed point * F ( G ) .
 Proof. 
As proved in Lemma 3, lim n G u n u n = 0 . Suppose B is compact and convex. Then, there exist a convergent subsequence { u n i } i = 0 of { u n } n = 0 having a strong limit k for some k B .
By condition (4) of Proposition 1, we have
u n i G k 3 + α 1 α u n i G u n i + u n i k .
If we consider setting i , then G k = k .
By Lemma 2, lim n u n k exist which follows that { u n } converges strongly to the limit k. Hence, the proof is complete. □
 Theorem 3.
Let B be a nonempty closed and convex subset of a Banach space H . Assume G : B B be a generalized α-nonexpansive mapping with F ( G ) and the sequence { u n } generated by the AG iterative scheme (8). Then the sequence { u n } converges strongly to some fixed point of G if and only if lim inf n ρ ( u n , F ( G ) ) = 0 , where ρ ( u n , F ( G ) ) = inf u F ( G ) ρ ( u n , u ) and ρ ( u n , F ( G ) ) denotes a distance function.
 Proof. 
Let * F ( G ) be a fixed point. As proved earlier in Lemma 2, we have that lim n u n * exists.
It follows that lim n ρ ( u n , F ( G ) ) exists.
Next, suppose that
lim n ρ ( u n , F ( G ) ) = 0 .
With regards to (27), we want to construct a Cauchy sequence in F ( G ) . We begin by constructing a subsequence { j k } in F ( G ) and { u n k } of { u n } with the property that
u n k j k 1 2 k , k N .
On the other hand, if { j k } is nonincreasing, then we have
u n k + 1 j k u n k j k 1 2 k .
Consequently, we have
j k + 1 j k j k + 1 u n k + 1 + u n k + 1 j k 1 2 k + 1 + 1 2 k 1 2 k 1 0 as k .
The above shows that { j k } is a Cauchy sequence in F ( G ) .
Again, by condition (3) of Proposition 1, we have that F ( G ) is closed in B . Therefore, { j k } is a convergent sequence in F ( G ) and it converges to * F ( G ) .
Hence, since
u n k * u n k j k + j k * 0 as n ,
we have lim n u n k * = 0 and so { u n k } converges strongly to * F ( G ) .
Again, by Lemma 2, lim n u n * exists; it follows that { u n } converges strongly to the fixed point * F ( G ) . Thereby completing the proof. □
 Theorem 4.
Let B be a nonempty closed convex subset of a Banach space H . G : B B be a mapping satisfying the condition for a α-nonexpansive mapping and { u n } is a sequence generated by the AG iterative scheme (8). Then { u n } converges strongly to a limit in F ( G ) if G satisfies condition ( I ) .
 Proof. 
From 3, we saw that lim n G u n u n = 0 . It follows that
lim inf n G u n u n = 0 .
Since G satisfies condition (I), we have that
G u n u n h ( ρ ( u n , F ( G ) ) ) .
From (28), we have
lim inf n h ( ρ ( u n , F ( G ) ) ) = 0 .
Again, from condition (28), we infer that h : [ 0 , ) [ 0 , ) is a nondecreasing function with h ( 0 ) = 0 and h ( s ) > 0 for all s ( 0 , ) , hence we infer that
lim inf n ρ ( u n , F ( G ) ) = 0 .
Since all claims of Theorem 3 are satisfied, then { u n } converges strongly to a fixed point of G . □

4. Numerical Example and Rate of Convergence

In this section, we consider a numerical example of a generalized α -nonexpansive mapping G .
 Example 1.
Consider the set B = [ 1 , 1 ] endowed with the usual norm, · and let G : B B be defined as
G x = x 2 , if x [ 1 , 0 ) x , if x [ 0 , 1 ] { 1 2 } 0 , if x = 1 2 .
Then G is a generalized α-nonexpansive mapping. Here, we choose α = 1 2 and the proof continues in different cases.
Case 1:If u , v [ 1 , 0 ) , we have
α G u v + α G v u + ( 1 2 α ) u v = 1 2 | u 2 v | + 1 2 | v 2 u | = 1 2 | u 2 v | + 1 2 | u v 2 | 1 2 | u 2 v + u v 2 | = 3 4 | u v | 1 2 | u v | = G u G v
Case 2:If u [ 1 , 0 ) , v [ 0 , 1 ] { 1 2 } .
Since u < 0 and v 0 , we have
G u G v = | x 2 + y | = x 2 + y , if | x | 2 < y x 2 y , if | x | 2 y .
At first, we have that
α G u v + α G v u + ( 1 2 α ) u v = 1 2 | u 2 v | + 1 2 | v u | 1 2 ( v u 2 ) + 1 2 ( v + u ) = u 4 + v u 2 + v = G u G v
Second, we have
α G u v + α G v u + ( 1 2 α ) u v = 1 2 | u 2 v | + 1 2 | v u | 1 2 ( v u 2 ) + 1 2 ( v u ) = 3 4 u u 2 v = G u G v
Case 3:If u [ 1 , 0 ) and v = 1 2 , then
α G u v + α G v u + ( 1 2 α ) u v = 1 2 | u 2 1 2 | + 1 2 | 0 u | = 1 2 ( 1 2 u 2 ) 1 2 u = 3 4 u + 1 4 u 2 = G u G v
Case 4:If u , v [ 0 , 1 ] { 1 2 } and u v , then
α G u v + α G v u + ( 1 2 α ) u v = 1 2 | u v | + 1 2 | v u | = 1 2 ( u + v ) + 1 2 ( v + u ) = ( v + u ) v u = G u G v .
Case 5:If u [ 0 , 1 ] { 1 2 } , v = 1 2 , then
α G u v + G v u + ( 1 2 α ) u v = 1 2 | u 11 2 | + | 0 u | = 1 2 ( u + 1 2 ) + 1 2 u = u + 1 4 u = G u G v .
The expected fixed point of G is 0.
Next, we use the example above to compare the rate of convergence of the AG iterative scheme with some other existing iterative schemes in the literature for α n = 18 n + 3 20 n + 4 , β n = 3 n + 2 4 n + 3 , and γ n = 18 n + 3 20 n + 4 with a maximum value achieved when n . This can be shown in the table and graph below.

5. Application to Mixed Type Integral Equation

In this section, we consider the application of the AG iterative scheme (8) to the approximation of the solution of a mixed type integral equation.
Our aim is to show the convergence of the sequence { u n } generated by the iterative scheme to a fixed point for a mixed type Volterra–Fredholm functional nonlinear integral equation:
u ( x ) = F t , u ( t ) , k 1 l 1 k n l n K ( t , s , u ( s ) ) d s , k 1 η 1 k n η n H ( t , s , u ( s ) ) d s ,
where [ k 1 ; η 1 ] × × [ k n ; η n ] is an interval in R n , K , H : [ k 1 ; η 1 ] × × [ k n ; η n ] × [ k 1 ; η 1 ] × × [ k n ; η n ] × R R continuous functions and F : [ k 1 ; η 1 ] × × [ k n ; η n ] × R 3 R . The following lemma will be very useful in proving the main result.
 Lemma 4.
Suppose that the following conditions are satisfied.
(C1)
K , H C ( [ k 1 ; η 1 ] × × [ k n ; η n ] × [ k 1 ; η 1 ] × × [ k n ; η n ] × R ) ;
(C2)
F ( [ k 1 ; η 1 ] × × [ k n ; η n ] × R 3 ) ;
(C3)
There exists nonnegative constants α, β, γ such that
| F ( t , f 1 , p 1 , q 1 ) F ( t , f 2 , p 2 , q 2 ) | α | f 1 f 2 | + β | p 1 p 2 | + γ | q 1 q 2 | ,
for all t [ k 1 ; η 1 ] × × [ k n ; η n ] , f 1 , p 1 , q 1 , f 2 , p 2 , q 2 R ;
(C4)
There exist nonnegative constants L K and L H such that
| K ( t , s , f ) K ( t , s , p ) | L K | f p | , | H ( t , s , f ) H ( t , s , p ) | L H | f p | ,
for all t , s [ k 1 ; η 1 ] × × [ k n ; η n ] , f , p R ;
(C5)
α + ( β L K + γ L H ) ( η 1 k 1 ) ( η n k n ) < 1 .
Then, the nonlinear integral Equation (29) has a unique solution u C ( [ k 1 ; η 1 ] × × [ k n ; η n ] ) .
Next is the main result.
 Theorem 5.
Suppose that all the conditions ( C 1 ) ( C 5 ) in Lemma 4 are satisfied. Let { u n } be defined by the AG iterative scheme (8) with real sequences α n , β n , γ n ( 0 , 1 ) , satisfying n = 1 α n = . Then (29) has a unique solution and the AG iterative scheme (8) converges strongly to the unique solution of the mixed type Voltera–Fredholm functional nonlinear integral Equation (29), say u C ( [ k 1 ; η 1 ] × × [ k n ; η n ] ) .
 Proof. 
Let X = C ( [ k 1 ; η 1 ] × × [ k n ; η n ] , · C ) , where · C is the Chebychev norm. Let { u n } be the iterative sequence generative by the AG iterative scheme (8) for the operator,
G ( u ) ( t ) = F t , u ( t ) , k 1 l 1 k n l n K ( t , s , u ( s ) ) d s , k 1 η 1 k n η n H ( t , s , u ( s ) ) d s .
Our aim is to show that lim n u n = u * .
Using (8), (29), (30) and the assumptions ( C 1 ) ( C 5 ) , we have that
u n + 1 u * = G v n u * = | G v n G u * | = | F t , v n ( t ) , k 1 l 1 k n l n K ( t , s , v n ( s ) ) d s , k 1 η 1 k n η n H ( t , s , v n ( s ) ) d s F t , u * ( t ) , k 1 l 1 k n l n K ( t , s , u * ( s ) ) d s , k 1 η 1 k n η n H ( t , s , u * ( s ) ) d s | α | v n ( t ) u * ( t ) | + β | k 1 l 1 k n l n K ( t , s , v n ( s ) ) d s k 1 l 1 k n l n K ( t , s , u * ( s ) ) d s | + γ | k 1 l 1 k n l n H ( t , s , v n ( s ) ) d s k 1 l 1 k n l n H ( t , s , u * ( s ) ) d s | α | v n ( t ) u * ( t ) | + β k 1 l 1 k n l n | K ( t , s , v n ( s ) ) K ( t , s u * ( s ) ) | d s + γ k 1 η 1 k n η n | H ( t , s , v n ( s ) ) H ( t , s u * ( s ) ) | d s α | v n ( t ) u * ( t ) | + β k 1 l 1 k n l n L K | v n ( s ) u * ( s ) | d s + γ k 1 η 1 k n η n L H | v n ( s ) u * ( s ) | d s α v n u * + β j = 1 m ( η j k j ) L K v n u * + γ j = 1 m ( η j k j ) L H v n u * = [ α + ( β L K + γ L H ) j = 1 m ( η j k j ) ] v n u *
v n u * = G [ ( 1 α n ) w n + α n G w n ] u * = | G [ ( 1 α n ) w n + α n G w n ] G u * | = | F ( t , [ ( 1 α n ) w n + α n G w n ] ( t ) , k 1 l 1 k n l n K ( t , s , [ ( 1 α n ) w n + α n G w n ] ( s ) ) d s , k 1 η 1 k n η n H ( t , s , [ ( 1 α n ) w n + α n G w n ] ( s ) ) d s ) F t , u * ( t ) , k 1 l 1 k n l n K ( t , s , u * ( s ) ) d s , k 1 η 1 k n η n H ( t , s , u * ( s ) ) d s | α | [ ( 1 α n ) w n + α n G w n ] ( t ) u * ( t ) | + β | k 1 l 1 k n l n K ( t , s , [ ( 1 α n ) w n + α n G w n ] ( s ) ) d s k n l n K ( t , s , u * ( s ) ) d s | + γ | k 1 η 1 k n η n H ( t , s , [ ( 1 α n ) w n + α n G w n ] ( s ) ) d s k 1 η 1 k n η n H ( t , s , u * ( s ) ) d s | α | [ ( 1 α n ) w n + α n G w n ] ( t ) u * ( t ) | + β k 1 l 1 k n l n | K ( t , s , [ ( 1 α n ) w n + α n G w n ] ( s ) ) K ( t , s , u * ( s ) ) | d s + γ k 1 η 1 k n η n | H ( t , s , [ ( 1 α n ) w n + α n G w n ] ( s ) ) H ( t , s , u * ( s ) ) | d s α | [ ( 1 α n ) w n + α n G w n ] u * | + β k 1 l 1 k n l n L K | [ ( 1 α n ) w n + α n G w n ] u * | d s + γ k 1 η 1 k n η n L H | [ ( 1 α n ) w n + α n G w n ] u * | d s α [ ( 1 α n ) w n + α n G w n ] u * + β j = 1 m ( η j k j ) L K [ ( 1 α n ) w n + α n G w n ] u * + γ j = 1 m ( η j k j ) L H [ ( 1 α n ) w n + α n G w n ] u *
= α + ( L K β + L H γ ) j = 1 m ( η j k j ) [ ( 1 α n ) w n + α n G w n ] u * .
However,
[ ( 1 α n ) w n + α n G w n ] u * ( 1 α n ) w n u * + α n G w n u * ( 1 α n ) w n u * + α n | G w n G u * | ( 1 α n ) w n u * + α n | F t , w n ( t ) , k 1 l 1 k n l n K ( t , s , w n ( s ) ) d s , k 1 η 1 k n η n H ( t , s , w n ( s ) ) d s F t , u * ( t ) , k 1 l 1 k n l n K ( t , s , u * ( s ) ) d s , k 1 η 1 k n η n H ( t , s , u * ( s ) ) d s | ( 1 α n ) w n u * + α n { α | w n ( t ) u * ( t ) | + β | k 1 l 1 k n l n K ( t , s , w n ( s ) ) d s k 1 l 1 k n l n K ( t , s , u * ( s ) ) d s | + γ | k 1 l 1 k n l n H ( t , s , w n ( s ) ) d s k 1 η 1 k n η n H ( t , s , u * ( s ) ) d s | } ( 1 α n ) w n u * + α n { α | w n ( t ) u * ( t ) | + β k 1 l 1 k n l n | K ( t , s , w n ( s ) ) K ( t , s , u * ( s ) ) | d s + γ k 1 η 1 k n η n | H ( t , s , w n ( s ) ) H ( t , s , u * ( s ) ) | d s } ( 1 α n ) w n u * + α n { α | w n ( t ) u * ( t ) | + β k 1 l 1 k n l n L K | w n ( s ) ) u * ( s ) | d s + γ k 1 η 1 k n η n L H | w n ( s ) ) u * ( s ) | d s } ( 1 α n ) w n u * + α n { α w n u * + β j = 1 m ( η j k j ) L K w n u * + γ j = 1 m ( η j k j ) L H w n u * } = ( 1 α n ) w n u * + α n α + ( β L K + γ L H ) j = 1 m ( η j k j ) w n u * .
Putting (34) back in (32), we have
v n u * = [ α + ( β L K + γ L H ) j = 1 m ( η j k j ) ] { ( 1 α n ) w n u * + α n [ α + ( β L K + γ L H ) j = 1 m ( η j k j ) ] w n u * } = [ α + ( β L K + γ L H ) j = 1 m ( η j k j ) ] × 1 α n + α n [ α + ( β L K + γ L H ) j = 1 m ( η j k j ) ] w n u * = [ α + ( β L K + γ L H ) j = 1 m ( η j k j ) ] × 1 α n ( 1 [ α + ( β L K + γ L H ) j = 1 m ( η j k j ) ] ) w n u * ,
w n u * = ( 1 β n ) G u n + β n G x n u * ( 1 β n ) | G u n u * | + β n | G x n u * | = ( 1 β n ) | G u n G u * | + β n | G x n G u * | = ( 1 β n ) | F t , u n ( t ) , k 1 l 1 k n l n K ( t , s , u n ( s ) ) d s , k 1 η 1 k n η n H ( t , s , u n ( s ) ) d s F t , u * ( t ) , k 1 l 1 k n l n K ( t , s , u * ( s ) ) d s , k 1 η 1 k n η n H ( t , s , u * ( s ) ) d s | + β n | F t , x n ( t ) , k 1 l 1 k n l n K ( t , s , x n ( s ) ) d s , k 1 η 1 k n η n H ( t , s , x n ( s ) ) d s F t , u * ( t ) , k 1 l 1 k n l n K ( t , s , u * ( s ) ) d s , k 1 η 1 k n η n H ( t , s , u * ( s ) ) d s | ( 1 β n ) { α | u n ( t ) u * ( t ) | + β | k 1 l 1 k n l n K ( t , s , u n ( s ) ) d s k 1 l 1 k n l n K ( t , s , u * ( s ) ) d s | + γ | k 1 η 1 k n η n H ( t , s , u n ( s ) ) d s k 1 η 1 k n η n H ( t , s , u * ( s ) ) d s | } + β n { α | x n ( t ) u * ( t ) | + β | k 1 l 1 k n l n K ( t , s , x n ( s ) ) d s k 1 l 1 k n l n K ( t , s , u * ( s ) ) d s | + γ | k 1 η 1 k n η n H ( t , s , x n ( s ) ) d s k 1 η 1 k n η n H ( t , s , u * ( s ) ) d s | } ( 1 β n ) { α | u n ( t ) u * ( t ) | + β k 1 l 1 k n l n | K ( t , s , u n ( s ) ) K ( t , s , u * ( s ) ) | d s + γ k 1 η 1 k n η n | H ( t , s , u n ( s ) ) d s H ( t , s , u * ( s ) ) | d s } + β n { α | x n ( t ) u * ( t ) | + β k 1 l 1 k n l n | K ( t , s , x n ( s ) ) d s K ( t , s , u * ( s ) ) | d s + γ k 1 η 1 k n η n | H ( t , s , x n ( s ) ) d s H ( t , s , u * ( s ) ) | d s } ( 1 β n ) { α | u n ( t ) u * ( t ) | + β k 1 l 1 k n l n L K | u n ( s ) u * ( s ) | d s + γ k 1 η 1 k n η n L H | u n ( s ) u * ( s ) | d s } + β n { α | x n ( t ) u * ( t ) | + β k 1 l 1 k n l n L K | x n ( s ) u * ( s ) | d s + γ k 1 η 1 k n η n L H | x n ( s ) u * ( s ) | d s } ( 1 β n ) { α u n u * + β j = 1 m ( η j k j ) L K u n u * + γ j = 1 m ( η j k j ) L H u n u * } + β n α x n u * + β j = 1 m ( η j k j ) L K x n u * + γ j = 1 m ( η j k j ) L H x n u * = ( 1 β n ) [ α + ( β L K + γ L H ) j = 1 m ( η j k j ) ] u n u * + β n [ α + ( β L K + γ L H ) j = 1 m ( η j k j ) ] x n u * ,
x n u * = ( 1 γ n ) u n + γ n G u n u * ( 1 γ n ) | u n u * | + γ n | G u n G u * | ( 1 γ n ) | u n u * | + γ n | F t , u n , k 1 l 1 k n l n K ( t , s , u n ( s ) ) d s , k 1 η 1 k n η n H ( t , s , u n ( s ) ) d s F t , u * ( t ) , k 1 l 1 k n l n K ( t , s , u * ( s ) ) d s , k 1 η 1 k n η n H ( t , s , u * ( s ) ) d s | ( 1 γ n ) | u n u * | + γ n { α | u n u * | + β | k 1 l 1 k n l n K ( t , s , u n ( s ) ) d s k 1 l 1 k n l n K ( t , s , u * ( s ) ) d s | + | k 1 η 1 k n η n H ( t , s , u n ( s ) ) d s k 1 η 1 k n η n H ( t , s , u * ( s ) ) d s | } ( 1 γ n ) | u n u * | + γ n { α | u n u * | + β k 1 l 1 k n l n | K ( t , s , u n ( s ) ) K ( t , s , u * ( s ) ) | d s + γ k 1 η 1 k n η n | H ( t , s , u n ( s ) ) H ( t , s , u * ( s ) ) | d s } ( 1 γ n ) | u n u * | + γ n { α | u n u * | + β k 1 l 1 k n l n L K | u n ( s ) u * ( s ) | d s + γ k 1 η 1 k n η n L H | u n ( s ) u * ( s ) | d s } ( 1 γ n ) u n u * + γ n { α u n u * + β j = 1 m ( η j k j ) L K u n u * + γ j = 1 m ( η j k j ) L H u n u * } = ( 1 γ n ) u n u * + γ n [ α + ( β L K + γ L H ) j = 1 m ( η j k j ) ] u n u * = [ 1 γ n ( 1 [ α + ( β L K + γ L H ) j = 1 m ( η j k j ) ] ) ] u n u * .
Putting (37) in (36), we have,
w n u * = ( 1 β n ) [ α + ( β L K + γ L H ) j = 1 m ( η j k j ) ] u n u * + β n [ α + ( β L K + γ L H ) j = 1 m ( η j k j ) ] × [ 1 γ n ( 1 [ α + ( β L K + γ L H ) j = 1 m ( η j k j ) ] ) ] u n u * = { ( 1 β n ) [ α + ( β L K + γ L H ) j = 1 m ( η j k j ) ] + β n [ α + ( β L K + γ L H ) j = 1 m ( η j k j ) ] × [ 1 γ n ( 1 [ α + ( β L K + γ L H ) j = 1 m ( η j k j ) ] ) ] } u n u * .
Putting (38) in (35), we have
v n u * = [ α + ( β L K + γ L H ) j = 1 m ( η j k j ) ] × [ 1 α n ( 1 [ α + ( β L K + γ L H ) j = 1 m ( η j k j ) ] ) ] × { ( 1 β n ) [ α + ( β L K + γ L H ) j = 1 m ( η j k j ) ] + β n [ α + ( β L K + γ L H ) j = 1 m ( η j k j ) ] × [ 1 γ n ( 1 [ α + ( β L K + γ L H ) j = 1 m ( η j k j ) ] ) ] } u n u *
Since [ α + ( β L K + γ L H ) j = 1 m ( η j k j ) ] < 1 from ( C 5 ) of Lemma 4.
v n u * [ 1 α n ( 1 [ α + ( β L K + γ L H ) j = 1 m ( η j k j ) ] ) ] × [ 1 γ n ( 1 [ α + ( β L K + γ L H ) j = 1 m ( η j k j ) ] ) ] u n u * .
Putting (40) in (31), we have
u n + 1 u * [ α + ( β L K + γ L H ) j = 1 m ( η j k j ) ] [ 1 α n ( 1 [ α + ( β L K + γ L H ) j = 1 m ( η j k j ) ] ) ] × [ 1 γ n ( 1 [ α + ( β L K + γ L H ) j = 1 m ( η j k j ) ] ) ] u n u * .
Again, since [ α + ( β L K + γ L H ) j = 1 m ( η j k j ) ] < 1 , we have that (41) reduces to
u n + 1 u * [ 1 γ n ( 1 [ α + ( β L K + γ L H ) j = 1 m ( η j k j ) ] ) ] u n u * .
From (42), we inductively generate the following inequalities,
u n + 1 u * [ 1 γ n ( 1 [ α + ( β L K + γ L H ) j = 1 m ( η j k j ) ] ) ] u n u * u n u * [ 1 γ n 1 ( 1 [ α + ( β L K + γ L H ) j = 1 m ( η j k j ) ] ) ] u n 1 u * v 1 u * [ 1 γ 0 ( 1 [ α + ( β L K + γ L H ) j = 1 m ( η j k j ) ] ) ] u 0 u * ,
from (43), we have
u n + 1 u * u 0 u * k = 0 n [ 1 γ k ( 1 [ α + ( β L K + γ L H ) j = 1 m ( η j k j ) ] ) ] .
Since γ k [ 0 , 1 ] for k N and [ α + ( β L K + γ L H ) j = 1 m ( η j k j ) ] < 1 , we have that
1 γ k ( 1 [ α + ( β L K + γ L H ) j = 1 m ( η j k j ) ] ) < 1 .
From elementary analysis, we can recall that 1 x < e x for all x [ 0 , 1 ] , then from (44), we can have that
u n + 1 u * u 0 u * e ( 1 [ α + ( β L K + γ L H ) j = 1 m ( η j k j ) ] ) k = 0 n γ k .
Taking the limit of both sides of (46) as n , we have that lim n u n u * = 0 . Therefore, we can say that (8) converges strongly to the solution of a mixed type Voltera–Fredholm nonlinear functional integral equation. □

6. Application to the Spread of HIV Modeled as a Fractional Differential Equation of the Caputo Type

Consider a S , I a , I b and A-model for the spread of HIV comprising susceptible, acute, asymptotic and symptomatic, or acquired immunodeficiency syndrome (AIDS) populations, respectively.
The model is represented mathematically as follows:
d s d t = Υ β a I a S N β b I b S N μ S d I a d t = β a I a S N + β b I b S N ν I b ( α + μ ) I a d I b d t = α I a ( ν + γ + μ ) I b d A d t = γ I b ( μ + σ ) A
subject to the initial conditions,
S ( 0 ) 0 , I a ( 0 ) 0 , I b ( 0 ) 0 , a n d A ( 0 ) 0 .
where Υ is the recruitment rate, β a and β b are the effective contact rate of S with I a and I b , respectively, ν is the treatment rate from asymptomatic to acute class, γ is the rate at which I b transfers to the AIDS class, μ is the natural death rate, and σ is the AIDS-associated death. Expressing (47) as the Caputo fractional differential equation equivalent, thus
D t λ 0 C S ( t ) = Υ β a I a S N β b N μ S D t λ 0 C I a ( t ) = β a I a S N + β b I b S N ν I b ( α + μ ) I a D t λ 0 C I b ( t ) = α I a ( ν + γ + μ ) I b D t λ 0 C A ( t ) = γ I b ( μ + σ ) A ,
subject to the initial conditions,
S ( 0 ) 0 , I a ( 0 ) 0 , I b ( 0 ) 0 , a n d A ( 0 ) 0
Our aim here is to show the existence of a unique solution of the system (48).
To achieve that, let us express the right-hand side of (48) as follows:
g 1 ( t , S , I a , I b , A ) = Υ β a I a S N β b I b S N μ S g 2 ( t , S , I a , I b , A ) = β a I a S N + β b I b S N ν I b ( α + μ ) I a g 3 ( t , S , I a , I b , A ) = α I a ( ν + γ + μ ) I b g 4 ( t , S , I a , I b , A ) = γ I b ( μ + σ ) A ,
such that
S ( 0 ) = K 1 , I a ( 0 ) = K 2 , I b ( 0 ) = K 3 , A ( 0 ) = K 4 .
Hence, transforming (48) to
D t λ 0 C S ( t ) = g 1 ( t , S , I a , I b , A ) D t λ 0 C I a ( t ) = g 2 ( t , S , I a , I b , A ) D t λ 0 C I b ( t ) = g 3 ( t , S , I a , I b , A ) D t λ 0 C A ( t ) = g 4 ( t , S , I a , I b , A )
subject to
S ( 0 ) = K 1 , I a ( 0 ) = K 2 , I b ( 0 ) = K 3 , A ( 0 ) = K 4 .
Suppose, f ( t ) = S I a I b A and f 0 = K 1 K 2 K 3 K 4 such that G ( t , f ( t ) ) = g 1 ( t , f ( t ) ) g 2 ( t , f ( t ) ) g 3 ( t , f ( t ) ) g 4 ( t , f ( t ) ) .
Hence, (50) reduces to the following equation of the Caputo type.
D t λ 0 C f ( t ) = G ( t , f ( t ) ) , 0 < λ < 1 f ( 0 ) = f 0 .
Recall, our aim is to show the existence of the unique solution of (51).
To achieve that, we can express (51) in its integral equation equivalence as follows:
f ( t ) = f 0 + ( λ ) G ( t , f ( t ) ) + ( λ ) 0 t G ( s , f ( s ) ) d s
where ( λ ) = 1 λ Γ ( λ ) and ( λ ) = λ Γ ( λ ) .
Let 0 t T . Suppose N = [ 0 , T ] , then we can define a Banach space as H = ( N , R 4 ) endowed with the maximum norm; f = max t N { | f ( t ) | : f H } . The following lemma due to [24] will be useful in the sequel.
 Lemma 5.
Suppose the following conditions hold.
(D1)
There exists a constant L G > 0 such that
| G ( t , f 1 ( t ) ) G ( t , f 2 ( t ) ) | L G | f 1 f 2 | ,
for each f H and t [ 0 , T ] ,
(D2)
[ ( λ + T ( λ ) ) ] L G < 1 .
Then (51) has a unique solution.
 Theorem 6.
Assume that conditions ( D 1 ) and ( D 2 ) of Lemma 5 are satisfied. Let α n , β n , γ n [ 0 , 1 ] be arbitrary sequences as they appear in the AG iterative scheme (8) such that n = 0 α n = . Then (51) has a solution; ℓ and the sequence generated by the iterative scheme (8) converges ℓ.
 Proof. 
Let H = ( N , R 4 ) be a Banach space endowed with a maximum norm given as
f = max t N { | f ( t ) | : f H } .
Assume that { u n } is an iterative sequence generated by the AG iterative scheme (8) for the operator G : H H defined by
G f ( t ) = f 0 + ( λ ) G ( t , f ( t ) ) + ( λ ) a t G ( s , f ( s ) ) d s .
Our aim here is to show that u n converges to the point as n .
Using (8) and (53) together with the conditions ( D 1 ) ( D 2 ) of Lemma 5, we have
u n + 1 = G v n = G v n G max t [ 0 , T ] | G v n ( t ) G ( t ) | = max t [ 0 , T ] | ( λ ) G ( t , v n ( t ) ) + ( λ ) a t G ( s , v n ( s ) ) d s ( λ ) G ( t , ( t ) ) + ( λ ) a t G ( s , ( s ) ) d s | = max t [ 0 , T ] | ( λ ) G ( t , v n ( t ) ) ( λ ) G ( t , ( t ) ) | + max t [ 0 , T ] | ( λ ) a t G ( s , v n ( s ) ) d s ( λ ) a t G ( s , ( s ) ) d s | ( λ ) max t [ 0 , T ] | G ( t , v n ( t ) ) G ( t , ( t ) ) | + ( λ ) max t [ 0 , T ] | ( a t G ( s , v n ( s ) ) a t G ( s , ( s ) ) ) d s | ( λ ) max t [ 0 , T ] | G ( t , v n ( t ) ) g ( t , ( t ) ) | + ( λ ) max t [ 0 , T ] a t | G ( s , v n ( s ) ) G ( s , ( s ) ) | d s ( λ ) L G max t [ 0 , T ] | v n ( t ) ( t ) | + ( λ ) L G T max t [ 0 , T ] | v n ( t ) ( t ) | ( λ ) L G v n + ( λ ) L G T v n [ ( λ ) + ( λ ) T ] L G v n
v n = G [ ( 1 α n ) w n + α n G w n ] = G [ ( 1 α n ) w n + α n G w n ] G max t [ 0 , T ] | G [ ( 1 α n ) w n + α n G w n ] G | = max t [ 0 , T ] | ( λ ) G ( t , [ ( 1 α n ) w n + α n G w n ] ( t ) ) + ( λ ) a t G ( s , [ ( 1 α n ) w n + α n G w n ] ( s ) ) d s ( λ ) G ( t , ( t ) ) + ( λ ) a t G ( s , ( s ) ) d s | = ( λ ) max t [ 0 , T ] | G ( t , [ ( 1 α n ) w n + α n G w n ] ( t ) ) G ( t , ( t ) ) | + ( λ ) max t [ 0 , T ] | a t G ( s , [ ( 1 α n ) w n + α n G w n ] ( s ) ) d s a t G ( s , ( s ) ) d s | ( λ ) L G max t [ 0 , T ] | ( 1 α ) w n + α n G w n | + ( λ ) L G T max t [ 0 , T ] | ( 1 α n ) w n + α n G w n | ( λ ) L G ( 1 α n ) w n + α n G w n + ( λ ) L G T ( 1 α n ) w n + α n G w n = L G ( ( λ ) + ( λ ) T ) ( 1 α n ) w n + α n G w n L G ( ( λ ) + ( λ ) T ) ( 1 α n ) w n + L G α n ( ( λ ) + ( λ ) T ) G w n L G ( 1 α n ) [ ( λ ) + ( λ ) T ] w n + L G α n [ ( λ ) + ( λ ) T ] G w n G L G ( 1 α n ) [ ( λ ) + ( λ ) T ] w n + L G α n [ ( λ ) + ( λ ) T ] max t [ 0 , T ] | G w n G | = L G ( 1 α n ) [ ( λ ) + ( λ ) T ] w n + L G α n [ ( λ ) + ( λ ) T ] × max t [ 0 , T ] | ( λ ) G ( t , w n ( t ) ) + ( λ ) a t G ( s , w n ( s ) ) d s ( ( λ ) G ( t , ( t ) ) + ( λ ) a t G ( s , ( s ) ) d s ) | = L G ( 1 α n ) [ ( λ ) + ( λ ) T ] w n + L G α n [ ( λ ) + ( λ ) T ] × ( λ ) max t [ 0 , T ] | g ( t , w n ( t ) ) G ( t , ( t ) ) | + max t [ 0 , T ] | a t G ( s , w n ( s ) ) d s a t G ( s , ( s ) ) d s | L G ( 1 α n ) [ ( λ ) + ( λ ) T ] w n + L G α n [ ( λ ) + ( λ ) T ] × L G ( λ ) max t [ 0 , T ] | w n ( t ) ( t ) | + ( λ ) L G T max t [ 0 , T ] | w n ( t ) ( t ) | L G ( 1 α n ) [ ( λ ) + ( λ ) T ] w n + L G α n [ ( λ ) + ( λ ) T ] × { L G ( λ ) w n + ( λ ) L G T w n } = L G ( 1 α n ) [ ( λ ) + ( λ ) T ] w n + L G α n [ ( λ ) + ( λ ) T ] × L G [ ( λ ) + ( λ ) T ] w n = L G ( 1 α n ) [ ( λ ) + ( λ ) T ] + L G α n [ ( λ ) + ( λ ) T ] L G [ ( λ ) + ( λ ) T ] w n
w n = ( 1 β n ) G u n + β n G x n ( 1 β n ) G u n + β n G x n = ( 1 β n ) G u n G + β n G x n G x n G ( 1 β n ) max t [ 0 , T ] | G u n G | + β n max t [ 0 , T ] | G x n G | = ( 1 β n ) max t [ 0 , T ] | ( λ ) G ( t , u n ( t ) ) + ( λ ) a t G ( s , u n ( s ) ) d s ( λ ) G ( t , ( t ) ) + ( λ ) a t G ( s , ( s ) ) d s | + β n max t [ 0 , T ] | ( λ ) G ( t , x n ( t ) ) + ( λ ) a t G ( s , x n ( s ) ) d s ( λ ) G ( t , ( t ) ) + ( λ ) a t G ( s , ( s ) ) | = ( 1 β n ) { max t [ 0 , T ] | ( λ ) G ( t , u n ( t ) ) ( λ ) G ( t , ( t ) ) | + max t [ 0 , T ] | ( λ ) a t G ( s , u n ( s ) ) d s ( λ ) a t G ( s , ( s ) ) d s | } + β n { max t [ 0 , T ] | ( λ ) G ( t , x n ( t ) ) ( λ ) G ( t , ( t ) ) | + max t [ 0 , T ] | ( λ ) a t G ( s , u n ( s ) ) d s ( λ ) a t G ( s , ( s ) ) | } = ( 1 β n ) { ( λ ) max t [ 0 , T ] | G ( t , u n ( t ) ) G ( t , ( t ) ) | + ( λ ) max t [ 0 , T ] | a t G ( s , u n ( s ) ) d s a t G ( s , ( s ) ) d s | } + β n { ( λ ) max t [ 0 , T ] | G ( t , x n ( t ) ) G ( t , ( t ) ) | + ( λ ) max t [ 0 , T ] | a t G ( s , x n ( s ) ) d s a t G ( s , ( s ) ) d s | } = ( 1 β n ) { ( λ ) max t [ 0 , T ] | G ( t , u n ( t ) ) G ( t , ( t ) ) | + ( λ ) max t [ 0 , T ] a t | G ( s , u n ( s ) ) G ( s , ( s ) ) | d s } + β n { ( λ ) max t [ 0 , T ] | G ( t , x n ( t ) ) G ( t , ( t ) ) | + ( λ ) max t [ 0 , T ] a t | G ( s , x n ( s ) ) G ( s , x n ( s ) ) | d s } ( 1 β n ) ( λ ) L G max t [ 0 , T ] | u n ( t ) ( t ) | + ( λ ) L G T max t [ 0 , T ] | u n ( t ) ( t ) | + β n ( λ ) L G max t [ 0 , T ] | x n ( t ) ( t ) | + ( λ ) L G T max t [ 0 , T ] | x n ( t ) ( t ) | ( 1 β n ) [ ( λ ) + ( λ ) T ] L G u n + β n [ ( λ ) + ( λ ) T ] L G x n
x n = ( 1 γ n ) u n + γ n G u n ( 1 γ n ) u n + γ n G u n = ( 1 γ n ) u n + γ n G u n G ( 1 γ n ) u n + γ n max t [ 0 , T ] | G u n G | = ( 1 γ n ) u n + γ n max t [ 0 , T ] | ( λ ) G ( t , u n ( t ) ) + ( λ ) a t G ( s , u n ( s ) ) d s ( λ ) G ( t , ( t ) ) + ( λ ) a t G ( s , ( s ) ) d s | = ( 1 γ n ) u n + γ n max t [ 0 , T ] | ( λ ) G ( t , u n ( t ) ) ( λ ) G ( t , ( λ ) ) | + γ n max t [ 0 , T ] | ( λ ) a t G ( s , u n ( s ) ) d s ( λ ) a t g ( s , ( s ) ) d s | = ( 1 γ n ) u n + γ n ( λ ) max t [ 0 , T ] | G ( t , u n ( t ) ) G ( t , ( t ) ) | + γ n ( λ ) max t [ 0 , T ] a t | G ( s , u n ( s ) ) G ( s , ( s ) ) | d s ( 1 γ n ) u n + γ n ( λ ) L G max t [ 0 , T ] | u n ( t ) ( t ) | + γ n ( λ ) L G T max t [ 0 , T ] | u n ( t ) ( t ) | ( 1 γ n ) u n + γ n ( λ ) L G u n + γ n ( λ ) L G T u n = ( 1 γ n ) u n + γ n L G [ ( λ ) + ( λ ) T ] u n .
Since L G [ ( λ ) + ( λ ) T ] < 1
x n u n ,
putting (57) in (56), we have,
w n ( 1 β n ) [ ( λ ) + ( λ ) T ] L G u n + β n L G [ ( λ ) + ( λ ) T ] u n = L G [ ( λ ) + ( λ ) T ] u n
Putting (58) in (55),
v n = 1 α n ( 1 L G [ ( λ ) + ( λ ) T ] ) [ ( λ ) + ( λ ) T ] 2 L G 2 u n .
Since L G [ ( λ ) + ( λ ) T ] < 1 , we have
v n { 1 α n ( 1 L G [ ( λ ) + ( λ ) T ] ) } u n .
Putting (59) in (54), we have,
u n + 1 [ ( λ ) + ( λ ) T ] L G { 1 α n ( 1 L G [ ( λ ) + ( λ ) T ] ) } u n .
Again, from ( D 2 ) , we have that [ ( λ ) + ( λ ) T ] L G < 1 such that (60) becomes
u n + 1 { 1 α n ( 1 L G [ ( λ ) + ( λ ) T ] ) } u n .
By induction, we have
u n + 1 = u 0 k = 1 n { 1 α k ( 1 L G [ ( λ ) + ( λ ) T ] ) } .
Since α k [ 0 , 1 ] for all k N , and furthermore from the condition ( D 2 ), we have that
1 α k ( 1 L G [ ( λ ) + ( λ ) T ] ) < 1 .
Recall from basic analysis that 1 x e x for all x [ 0 , 1 ] .
Therefore, (62) becomes
u n + 1 u 0 e ( 1 L G [ ( λ ) + ( λ ) T ] ) k = 0 α k .
Taking the limit as n , we have lim n u n = 0 , thereby completing the proof. □

7. Conclusions

Clearly, it has been shown that the AG iterative scheme (8) converges faster than all of (3), (4), (5), (6), and (7) for the class of generalized α -nonexpansive mapping using Example 1 with pictorial evidence expressed in Table 1 and Figure 1 and Figure 2. The main result is also applied to guarantee its applicability.

Author Contributions

Conceptualization, G.A.O. and A.V.U.; Methodology, A.V.U. and R.T.A.; Software, R.T.A.; Validation, G.A.O. and A.V.U.; Formal analysis, G.A.O.; Investigation, A.V.U. and R.T.A.; Resources, R.T.A.; Data curation, A.V.U.; Writing—original draft, A.V.U.; Writing—review & editing, G.A.O. and R.T.A.; Visualization, A.V.U.; 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.

Conflicts of Interest

The authors declare that they have no conflict of interest.

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Figure 1. Graph corresponding to Table 1.
Figure 1. Graph corresponding to Table 1.
Mathematics 13 00550 g001
Figure 2. Three-dimensional surface plot corresponding to Table 1.
Figure 2. Three-dimensional surface plot corresponding to Table 1.
Mathematics 13 00550 g002
Table 1. Comparison of speed of convergence of some iterative scheme for Example 1.
Table 1. Comparison of speed of convergence of some iterative scheme for Example 1.
StepAGKThakurAbass–NasirPicard-SPiri
1−0.0485−0.1594−0.0469−0.3952−0.3187−0.2156
2−0.0023−0.0136−0.0049−0.1016−0.0543−0.0194
3−0.0001−0.0012−0.0005−0.0261−0.0092−0.0017
40.0000−0.0001−0.0001−0.0067−0.0016−0.0002
50.00000.00000.0000−0.0017−0.00030.0000
60.00000.00000.0000−0.00040.00000.0000
70.00000.00000.0000−0.00010.00000.0000
80.00000.00000.00000.00000.00000.0000
90.00000.00000.00000.00000.00000.0000
100.00000.00000.00000.00000.00000.0000
110.00000.00000.00000.00000.00000.0000
120.00000.00000.00000.00000.00000.0000
130.00000.00000.00000.00000.00000.0000
140.00000.00000.00000.00000.00000.0000
150.00000.00000.00000.00000.00000.0000
160.00000.00000.00000.00000.00000.0000
170.00000.00000.00000.00000.00000.0000
180.00000.00000.00000.00000.00000.0000
190.00000.00000.00000.00000.00000.0000
200.00000.00000.00000.00000.00000.0000
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Okeke, G.A.; Udo, A.V.; Alqahtani, R.T. Novel Method for Approximating Fixed Point of Generalized α-Nonexpansive Mappings with Applications to Dynamics of a HIV Model. Mathematics 2025, 13, 550. https://doi.org/10.3390/math13040550

AMA Style

Okeke GA, Udo AV, Alqahtani RT. Novel Method for Approximating Fixed Point of Generalized α-Nonexpansive Mappings with Applications to Dynamics of a HIV Model. Mathematics. 2025; 13(4):550. https://doi.org/10.3390/math13040550

Chicago/Turabian Style

Okeke, Godwin Amechi, Akanimo Victor Udo, and Rubayyi T. Alqahtani. 2025. "Novel Method for Approximating Fixed Point of Generalized α-Nonexpansive Mappings with Applications to Dynamics of a HIV Model" Mathematics 13, no. 4: 550. https://doi.org/10.3390/math13040550

APA Style

Okeke, G. A., Udo, A. V., & Alqahtani, R. T. (2025). Novel Method for Approximating Fixed Point of Generalized α-Nonexpansive Mappings with Applications to Dynamics of a HIV Model. Mathematics, 13(4), 550. https://doi.org/10.3390/math13040550

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