Exploring the existence of a solution of nonlinear Fredholm integral equations: Novel approaches to estimating common fixed points

2.1. Definition

Gallagher (2016) κ is said to be demiclosed with p ∈ B if, for each sequence {ρ_n} belonging to A and u ∈ A, {ρ_n} weakly converges to u and κρ_n strongly converges to p which implies κu = p.

2.2. Definition

Opial (1967) When each sequence {ρ_n} converges weakly to u^∗ ∈ B with below mentioned satisfied:

$\underset{n\to \infty }{\text{liminf}} ‖{\rho }_n-u*‖<\underset{n\to \infty }{\text{liminf}}‖{\rho }_n-p‖,$

∀p ∈ B and u^∗ $\ne$ p, then Banach space B meets Opial’s condition.

Remark 2.1. A Banach space B is considered uniformly smooth if the following limit exists.

where ${\rho }_B$ represents unit sphere of B.

When the limit (2) is uniformly obtained for all $\psi$ ∈ B and b ∈ ρ_B, then the norm of B is called Frechet differentiable (Wataru et al., 1998). Then, we get

where g denotes an increasing function over non-negative numbers belonging to R with lim _t→0 $\frac{f(t)}{t}$ = 0 and K($\tilde{x}$) denotes the Frechet derivative of the functional $\frac{1}{2}\parallel .{\parallel }^2$ at $\tilde{x}\in B$.

Here is concept for a mapping meeting Condition(I) that is given by Sentor and Dotson (1974):

2.3. Definition

κ : A → A meets Condition(I) if s : [0,∞) → [0,∞) exists such that

d(ρ,κρ) ≥ s(d(ρ,F(κ))),∀ρ ∈ A, where following holds:

where s is a non-decreasing function.

• i.s(0) = 0;

• ii.s(t) > 0,∀t > 0;

• iii.d(ρ,F(κ)) = inf(ρ,ℓ) : ℓ ∈ F(κ).

2.4. Definition

Shahzad and Al-Dubiban (2006) κ₁,κ₂ : A → A satisfy Condition(I”) when a non-decreasing function s : [0,∞) → [0,∞) exists such that

where the following conditions hold:

• i.s(0) = 0;

• ii.s(t) > 0,∀t > 0;

• iii.d(u_n,F) = inf ∥u,ℓ∥ : ℓ ∈ F(κ).

Definitions of asymptotic center and radius are provided below, details in Edelstein (1974).

2.5. Definition

Let R(.,{ρ_n}) : B → R⁺ be mapping defined as:

$R\left(d,\left\{{\rho }_n\right\}\right)=\underset{n\to \infty }{\text{limsup}}\parallel {\rho }_n-d\parallel ,$

for all d ∈ B, then R(d,{ρ_n}) is known as asymptotic radius of {ρ_n} at d. Asymptotic radius of {ρ_n}

corresponding to A, where A ⊂ B is:

$$R(A,\{{\rho }_n\})=\inf R(d,\{{\rho }_n\}),$$

where d ∈ A. The set is defined as:

is called asymptotic center of {ρ_n} relative to A, where d $\in$ A.

The asymptotic center contains one element whenever B is a Banach space with uniformly convexity (Maleknejad et al., 2020).

2.6. Definition

Yambangwai and Thianwan (2024) Let us consider a vector space V along with a subset W $\ne$∅ of V ×V. When we have all (q,ρ),(q,v),(ρ,q),(v,q) ∈ D and all s ∈ [0,1], we can say that G is coordinate-convex if

$$s(q,\rho )+(1-s)(q,v)\in D\text{and}s(\rho ,q)+(1-s)(v,q)\in D.$$

Remark 2.2. G is coordinate-convex in V × V when G is convex in V × V. But its converse is not always true.

Proposition 2.1 (García-Falset et al., 2011). Suppose a mapping κ : A → B is satisfying Garcia-Falset property on A. κ is quasi-nonexpansive whenever κ has at least one fixed point.

Lemma 2.1 (Schu, 1991). Suppose B is a uniformly convex Banach space and 0 < s_n < 1 for all n ∈${\mathbb{Z}}^{+}$. Consider {p_n} and {q_n} are two sequences satisfying $\underset{n\to \infty }{\text{limsup}}\parallel p_n\parallel \le ı$, $\underset{n\to \infty }{\text{limsup}}$∥q_n∥ ≤ ı and $\underset{n\to \infty }{\text{limsup}}$∥(1− s_n)p_n + s_nq_n∥ = ı for any ı ≥ 0, then $\underset{n\to \infty }{\text{lim}}$∥p_n − q_n∥ = 0.

Lemma 2.2 ( Şoltuz and Otrocol, 2007). Define a sequence {β_n} on ${ℝ}^{+}$that fulfills the inequality as:

if ${\alpha }_n\in$ (0,1) and ${\displaystyle \sum }_{n=0}^{\infty }{\beta }_n=\infty$, then lim _n→∞ $\parallel {\rho }_n-\iota \parallel$= 0.

3.1. Definition

Assume that a mapping κ : A → A satisfies the following:

1. κ preserves edges in η(G) i.e. if (a,b) ∈ η(G) then (κa,κb) ∈ η(G).

where ∅ ̸$\ne$ A ⊂ B with B is a Banach space and λ ≥ 1 then κ is said to be non-symmetric graphical Garcia-Falset (G -F) mapping.

In this section set A,B and the graphical G -F mappings κ₁,κ₂ satisfy following conditions:

• 1. ∅ ̸$\ne$A ⊂ B

• 2.G = (ω(G),η(G)) be a digraph and transitive graph. Also, ω(G) = A and η(G) is satisfying coordinate convexity condition.

• 3. κ₁,κ₂ are edge-preserving graphical G - F mappings and F(κ₁) ∩ F(κ₂) ≠ ∅.

• 4. For all q ∈ F(κ₁)∩F(κ₂), the sequence ρ_n is generated as in algorithm (1) so that (ρ₁,q),(q,ρ₁) ∈ η(G).

Proposition 3.1. If the conditions (Beg et al., 2022; Abed and Hasan, 2019; Ameer et al., 2020; Aoyama and Kohsaka, 2011) satisfy then, we say

$$\begin{array}{l}({\rho }_n,q),(b_n,q),(c_n,q),(d_n,q),(q,{\rho }_n),(q,b_n),(q,c_n),(q,d_n),({\rho }_n,b_n),({\rho }_n,c_n),\\ ({\rho }_n,d_n),({\rho }_n,{\rho }_{n+1}\text{)}\in \eta (G),\forall n\in N.\end{array}$$

Proof. We first show via mathematical induction that

It is obvious from assumption that (q,ρ₁) ∈ η(G).

Suppose (4.1) is satisfied for n = l, we have to prove (q,ρ_l+1) ∈ η(G).

In fact, as κ₂ is edge-preserving, so (q,ρ_l) ∈ η(G) implies (q,κ₂ρ_l) ∈ η(G). Consider

Since (q,ρ_k),(q,κ₂ρ_k) ∈ η(G) and also by coordinate convexity (q,d_k) ∈ η(G) that implies (q,κ₂d_k) ∈ η(G), which is due to edge preserving property of κ₂. Also, by taking c_k = κ₂(x_k) where x_k = (1−q_k)d_k+q_kκ₁(d_k),

As (q,d_k),(q,κ₁d_k) ∈ η(G) and also by coordinate convexity (q,x_k) ∈ η(G) that implies (q,κ₁x_k),(q,κ₂x_k) ∈ η(G). It gives (q,κ₂c_k) ∈ η(G).

Furthermore, let y_k = (1 − s_k)κ₁(d_k) + s_kκ₂(c_k)).

Consider

We have (κ₁d_k),(q,κ₂c_k) ∈ η(G) and also by coordinate convexity (q,y_k) ∈ η(G), we get (q,κ₂y_k) ∈ η(G), which is due to edge preserving property of κ₂. Also, by taking b_k = κ₂(y_k) where y_k = (1 − s_k)κ₁(d_k) + s_kκ₂(c_k)), we conclude that (q,b_k) ∈ η(G). We know ρ_k+1 = κ₁b_k, and also we have (q,b_k) ∈ η(G), thus edge-preserving property implies,

(q,κ₁b_k) ∈ η(G), and hence (q,ρ_k+1) ∈ η(G). Further,

Similarly, (q,ρ_n),(q,κ₂ρ_n) ∈ η(G) and also by coordinate convexity (q,d_n) ∈ η(G) that implies (q,κ₂d_n) ∈ η(G), which is due to edge preserving property of κ₂. Take c_n = κ₂(x_n) where x_n = (1 − q_n)d_n + q_nκ₁(d_n)

Proceeding in the same way as above, we have (q,d_n),(q,κ₁d_n) ∈ η(G) and coordinate convexity implies (q,x_n) ∈ η(G) which gives (q,κ₁x_n),(q,κ₂x_n) ∈ η(G). Hence (q,κ₂c_n) ∈ η(G). Consider y_n = (1 − s_n)κ₁(d_n) + s_nκ₂(c_n)).

And

Again (κ₁d_n),(q,κ₂c_n) ∈ η(G) and by coordinate convexity (q,y_n) ∈ η(G) we get (q,κ₂y_n) ∈ η(G), which is due to edge preserving property of κ₂. Also, by taking b_n = κ₂(y_n) where y_n = (1 − s_n)κ₁(d_n) + s_nκ₂(c_n)), we conclude that (q,b_n) ∈ η(G).

Further ρ_n+1 = κ₁b_n, we prove already that (q,b_n) ∈ η(G). So by edge-preserving property, (q,κ₁b_n) ∈ η(G), which gives (q,ρ_n+1) ∈ η(G).

From the above, we can conclude that

$$(q,{\rho }_n),(q,b_n),(q,c_n),(q,d_n),(q,{\rho }_{n+1})\in \eta (G),\forall n\in \text{N}.$$

In the same way, we have

$$({\rho }_n,q),(b_n,q),(c_n,q),(d_n,q),({\rho }_{n+1},q)\in \eta (G),\forall n\in \text{N}.$$

By using transitive property of D, we conclude

$$({\rho }_n,b_n),({\rho }_n,c_n),({\rho }_n,d_n),({\rho }_n,{\rho }_{n+1})\in \eta (G),\forall n\in N.$$

Lemma 3.1. Assume that conditions (Beg et al., 2022; Abed and Hasan, 2019; Ameer et al., 2020; Aoyama and Kohsaka, 2011) are satisfied then the following statements are true:

• i. lim _n→∞ ∥ρ_n − ι∥ exists where ι ∈ F(κ).

• ii. {ρ_n} is bounded and lim _n→∞ ∥κ₁ρ_n − ρ_n∥ = 0 = lim _n→∞ ∥κ₂ρ_n − ρ_n∥.

Proof. Since p ∈ F(κ₁) ∩ F(κ₂) and (q,ρ₁),(ρ₁,q) ∈ η(G), by Proposition 3.1, we have

(ρ_n,q),(c_n,q),(d_n,q),(b_n,q) ∈ η(G). Since (ρ_n,q) ∈ η(G) and κ₂p = p and κ₂ is a Garcia-Falset mapping with graph. Also, let ι ∈ F(κ). Since κ₁ and κ₂ are two quasi-nonexpansive mappings, by Proposition 2.1, (7) and using algorithm (1), we have

Using (15), since (d_n,ι) ∈ η(G) and κ₁ι = κ₂ι = ι and κ₁,κ₂ are two Garcia-Falset mappings with graph, we obtain

Using (15), we have

Applying (17), (c_n,ι) ∈ η(G) and κ₁ι = κ₂ι = ι, also κ₁,κ₂ are two Garcia-Falset mappings endowed with graph, this implies

$$\begin{array}{l}‖b_n-\iota ‖=‖{\kappa }_2((1-s_n){\kappa }_1{d}_n+s_n{\kappa }_2(c_n))-\iota ‖\\ \le (1-s_n)‖d_n-\iota ‖+s_n‖c_n-\iota ‖.\end{array}$$

From equations (17) and (15), we get

Consequently, from (18), as (b_n,ι) ∈ η(G) and κ₁ι = ι and κ₁ is Garcia-Falset mapping with graph, we get

$$\begin{array}{l}‖{\rho }_{n+1}-\iota ‖=‖{\kappa }_1{b}_n-\iota ‖\\ \le ‖b_n-\iota ‖\end{array}$$

Equation (18) implies

which means {∥ρ_n − ι∥} is decreasing and bounded for each ι ∈ F(κ).

This yields that lim _n→∞ ∥ρ_n − ι∥ exists.

Proof of part (ii):

Since F(κ₁) ∩ F(κ₂) ̸= ∅ and (ρ₁,q),(q,ρ₁) ∈ η(G). Thus, from Proposition 3.1, we have

$$({\rho }_n,q),(c_n,q),(d_n,q),(b_n,q),({\rho }_n,b_n),({\rho }_n,c_n),({\rho }_n,d_n)\in \eta (G).$$

We can see from part (i) that limit of $\underset{n\to \infty }{\text{lim}}$ ∥ρ_n − ι∥ exist and {ρ_n} is bounded. Let

$\underset{n\to \infty }{\text{lim}}‖{\rho }_n-\iota ‖=l.$

Also

Similarly,

$\parallel {\kappa }_2{\rho }_n-\iota \parallel \le \parallel {\rho }_n-\iota \parallel$

From (15), (17), (18) and (20), we have

Taking $\underset{n\to \infty }{\text{liminf},}$ we have

Using (25) and (27)

Thus we have,

$\underset{n\to \infty }{\text{lim}}‖{\kappa }_2((1-s_n){\kappa }_1{d}_n+s_n{\kappa }_2{c}_n)-\iota ‖=1,$

Using modified AA-iterative scheme

$‖d_n-\iota ‖=‖(1-r_n){\rho }_n+r_n{\kappa }_2({\rho }_n)-\iota ‖\le ‖{\rho }_n-\iota ‖.$

As κ₁ is graphical Garcia-Falset mapping, so

$‖{\kappa }_1{d}_n-\iota ‖\le ‖d_n-\iota ‖\le ‖{\rho }_n-\iota ‖.$

Moreover, κ₂ is the graphical Garcia-Falset mapping, so

By using (29), (30), (31) on Lemma 2.1, we have

Using a modified AA-iterative scheme and graphical Garcia-Falset mapping,

$\begin{array}{l}\underset{n\to \infty }{\text{lim}}‖{\rho }_{n+1}-k_1{d}_n‖=\underset{n\to \infty }{\text{lim}}‖{\kappa }_1{b}_n-k_1{d}_n‖\\ \le \underset{n\to \infty }{\text{lim}}‖d_n-\iota ‖+\underset{n\to \infty }{\text{lim}}‖d_n-\iota ‖\\ \le 2\underset{n\to \infty }{\text{lim}}‖{\rho }_n-\iota ‖\le l.\end{array}$

Similarly,

$\begin{array}{l}l\le \underset{n\to \infty }{\text{liminf}}‖c_n-\iota ‖\\ \underset{n\to \infty }{\text{lim}}‖c_{n-1}-\iota ‖=l\\ l\le \underset{n\to \infty }{\text{liminf}}‖d_n-\iota ‖\\ \underset{n\to \infty }{\text{lim}}‖d_n-\iota ‖=l.\end{array}$

Moreover,

$\begin{array}{l}l\le \underset{n\to \infty }{\text{lim}}‖d_n-\iota ‖\\ \le \underset{n\to \infty }{\text{lim}}‖{\rho }_n-\iota ‖\le l.\end{array}$

Thus,

By part (i) and Equations (20), (22) and (33), we get

$\underset{n\to \infty }{\text{lim}}‖{\rho }_n-{\kappa }_2{\rho }_n‖=0.$

Similarly by part (i) and Equations (20), (21) and (33), we get

$\underset{n\to \infty }{\text{lim}}‖{\rho }_n-{\kappa }_1{\rho }_n‖=0.$

Theorem 3.1. Suppose B is satisfying conditions (Beg et al., 2022; Abed and Hasan, 2019; Ameer et al., 2020; Aoyama and Kohsaka, 2011) and Opial’s condition then {ρ_n} converges weakly to common fixed point of κ₁ and κ₂ with ∅$\ne$ F(κ) = F(κ₁) ∩ F(κ₂).

Proof. Since ∅ $\ne$F(κ). Suppose ι is fixed point of κ. From Lemma 3.1, $\underset{n\to \infty }{\text{lim}}$∥ρ_n−ι∥ exists. Let {ρ_ni} and

{ρ_nj} be subsequences weakly converging to ρ₁ and ρ₂ respectively. By Lemma 3.1, $\underset{n\to \infty }{\text{lim}}$∥ρ_n − κ₁ρ_n∥ =

0 = ∥ρ_n − κ₂ρ_n∥, I − κ₁ and I − κ₂ at zero yields,

(I −κ₁)ρ₁ = 0, i.e. κ₁ρ₁ = ρ₁ and κ₂ρ₁ = ρ₁. This implies ρ₁ ∈ F(κ). In similar manner, (I −κ₁)ρ₂ = 0, i.e. κ₁ρ₂ = ρ₂ and κ₂ρ₂ = ρ₂ i.e. ρ₂ ∈ F(κ). Therefore ρ₁ and ρ₂ are fixed points of κ₁ and κ₂. To show uniqueness, when ρ₁ $\ne$ ρ₂. By Opial’s condition, we get

$$\begin{array}{l}\underset{n\to \infty }{\text{lim}}‖{\rho }_n-{\rho }_1‖=\underset{n\to \infty }{\text{lim}}‖\left\{{\rho }_{ni}\right\}-{\rho }_1‖\\ <\underset{n\to \infty }{\text{lim}}‖{\rho }_{ni}-{\rho }_2‖\\ =\underset{n\to \infty }{\text{lim}}‖{\rho }_n-{\rho }_2‖\\ =\underset{n_j\to \infty }{\text{lim}}‖{\rho }_{nj}-{\rho }_2‖\\ <\underset{n_j\to \infty }{\text{lim}}‖{\rho }_{nj}-{\rho }_1‖\\ =\underset{n\to \infty }{\text{lim}}‖{\rho }_n-{\rho }_1‖,\end{array}$$

a contradiction. Hence ρ₁ = ρ₂. This implies, {ρ_n} converges weakly to a fixed point of κ.

Next, we prove some weak convergence results of 1 to common fixed points of two graphical Garcia-Falset mappings in B by replacing Opial’s condition with the Frechet differential norm.

Theorem 3.2. Suppose B has Frechet differential norm and conditions (Beg et al., 2022; Abed and Hasan, 2019; Ameer et al., 2020; Aoyama and Kohsaka, 2011) are satisfied. Let lim _n→∞ ∥tρ_n+ (1−t)a−b∥ be exists for each a,b ∈ F(κ). In addition, consider demiclosedness of I −κ₁ and I −κ₂ at zero. Then {ρ_n} converges weakly to a fixed point of κ₁ and κ₂ with ∅ ̸= F(κ) = F(κ₁) ∩ F(κ₂).

Proof. Let ι₁ and ι₂ be weak limits of ${\rho }_{n_i}$ and ${\rho }_{n_j}$ respectively. Since, lim _n→∞ ∥ρ_n − κρ_n∥ = 0. Also, we have (ρ₁,q),(q,ρ₁) ∈ η(G). Demiclosedness of I − κ at zero implies ι₁,ι₂ ∈ F(κ). Replacing p and q with a − b and t(ρ_n − a) with respectively in (3), then we have

$$\begin{array}{l}\frac{1}{2}\parallel a-b{\parallel }^2+t〈{\rho }_n-a,K(a-b)〉\le \frac{1}{2}\parallel t{\rho }_n+\left(1-t\right)a-b{\parallel }^{2 }\\ \le \frac{1}{2}\parallel a-b{\parallel }^2+t〈{\rho }_n-a,K(a-b)〉+f\left(t\parallel {\rho }_n-a\right)\parallel .\end{array}$$

From the given constraint, we have

$\begin{array}{l}\frac{1}{2}\parallel a-b{\parallel }^2+t\underset{\text{n}\to \infty }{\lim \sup }〈{\rho }_n-a,K\left(a-b\right)〉\le \frac{1}{2}\underset{n\to \infty }{\lim }\parallel t{\rho }_n+\left(1-t\right)a-b{\parallel }^2\\ \le \frac{1}{2}\parallel a-b{\parallel }^2+t\underset{\text{n}\to \infty }{\lim \inf }〈{\rho }_n-a,K\left(a-b\right)〉+O\left(t\right).\end{array}$

Therefore,

$\underset{n\to \infty }{\lim \sup }〈{\rho }_n-a,K\left(a-b\right)〉\le \underset{n\to \infty }{\lim \inf }〈{\rho }_n-a,K\left(a-b\right)〉+\frac{O\left(t\right)}{t}.$

Applying t → 0⁺, we obtain existence of $\underset{n\to \infty }{\text{lim}}$⟨ρ_n − a,K(a − b)⟩. Let ⟨ι₁ − a,K(a − b)⟩ = e and ⟨ι₂ − a,K(a − b)⟩ = e. Consequently, ⟨ι₁ − ι₂,K(a − b)⟩ = 0 ∀a,b ∈ F(κ). This implies

$${‖{\iota }_i-{\iota }_2‖}^2=<{\iota }_1-{\iota }_2,K({\iota }_1-{\iota }_2)>=0,$$

which cannot be true unless ι₁ = ι₂. Then ρ_n weakly converges to a common fixed point of κ₁ and κ₂.

Theorem 3.3. If conditions (Beg et al., 2022; Abed and Hasan, 2019; Ameer et al., 2020; Aoyama and Kohsaka, 2011) are satisfied then {ρ_n} converges to a fixed point of κ iff $\underset{n\to \infty }{\text{liminf }}$d(ρ_n,F(κ)) = 0 or $\underset{n\to \infty }{\text{limsup }}$d(ρ_n,F(κ)) = 0, with d(ρ_n,F(κ)) = inf (∥ρ_n − ι∥) and ι ∈ F(κ).

Proof. Suppose $\underset{n\to \infty }{\text{lim}}$ρ_n = ι where ι ∈ F(κ), then we are done.

Conversely, let $\underset{n\to \infty }{\text{liminf }}$ d(ρ_n,F(κ)) = 0. From Lemma 3.1, we have $\underset{n\to \infty }{\text{lim}}$∥ρ_n − ι∥ exists for every ι ∈ F(κ). By our supposition,

$\underset{n\to \infty }{\text{liminf }}d({\rho }_n,F(\kappa ))=0.$

Since $\underset{n\to \infty }{\text{liminf }}$d(ρ_n,F(κ)) = 0, for κ > 0, there exists n₀ ∈ $\mathbb{Z}$⁺, such that

$\begin{array}{l}d\left({\rho }_n,F\left(\kappa \right)\right)<\frac{\kappa }{2},\\ \text{inf }d\left({\rho }_n,\iota \right)<\frac{\kappa }{2},\text{where }\iota \in F\left(\kappa \right).\end{array}$

This implies there is some ι ∈ F(κ) such that

$\parallel {\rho }_{n_0}-\iota \parallel <\frac{\kappa }{2}.$

For m,n ≥ n₀,

$$\begin{array}{l}‖{\rho }_{m+n}-{\rho }_n‖\le ‖{\rho }_{m+n}-\iota ‖+‖{\rho }_{m+n}-\iota ‖\\ \le ‖{\rho }_{n0}-\iota ‖+‖{\rho }_{n0}-\iota ‖\\ =2‖{\rho }_{n0}-\iota ‖\\ =\kappa ,\end{array}$$

which means {ρ_n} is Cauchy sequence in A. Since A is closed so there exist s ∈ A, such that $\underset{n\to \infty }{\text{lim }}$ρ_n = s.

Also, $\underset{n\to \infty }{\text{liminf }}$d(ρ_n,F(κ)) = 0. Therefore, s ∈ F(κ).

Theorem 3.4. If conditions (Beg et al., 2022; Abed and Hasan, 2019; Ameer et al., 2020; Aoyama and Kohsaka, 2011) are satisfied then {ρ_n} converges strongly to ι ∈ F(κ).

Proof. Suppose F(κ) $\ne$ ∅, from Lemma 3.1, we have lim _n→∞ ∥ρ_n − κ₁ρ_n∥ = 0 = lim _n→∞ ∥ρ_n − κ₁ρ_n∥.

By compactness of A, there is a subsequence {${\rho }_n{}_k$ } of ρ_n such that ${\rho }_n{}_k$ → ι where ι ∈ A. Using the Garcia-

Falset property, we have

$$‖{\rho }_{nk}-{\kappa }_1\rho ‖\le v‖{\rho }_{nk}-k_1{\rho }_{nk}‖+‖{\rho }_{nk}-\iota ‖.$$

Applying limit when k → ∞, we have ∥${\rho }_n{}_k$ − κ₁ι∥ = 0.

So, we get ${\rho }_n{}_k$ → κ₁ι. Hence κ₁ι = ι, i.e. ι ∈ F(κ₁). Similarly, κ₂ι = ι, i.e. ι ∈ F(κ₂). Consequently, by Lemma 3.1, lim _n→∞ ∥ρ_n − ι∥ exists where ι ∈ F(κ) = F(κ₁) ∩ F(κ₂), thus {ρ_n} → ι.

The following result is a strong convergence result in which we use Condition(I^′′).

Theorem 3.5. If conditions (Beg et al., 2022; Abed and Hasan, 2019; Ameer et al., 2020; Aoyama and Kohsaka, 2011) satisfy then {ρ_n} converges strongly to ι ∈ F(κ) = F(κ₁)∩F(κ₂), where both Garcia-Falset mappings κ₁,κ₂ satisfy the Condition(I^′′).

Proof. Suppose ι ∈ F(κ). Using Lemma 3.1, lim _n→∞ ∥ρ_n − ι∥ exists ∀ι ∈ F(κ). Further

$$‖{\rho }_{n+1}-\iota ‖\le ‖{\rho }_n-\iota ‖,\forall n\in \text{N}.$$

This implies

$$d({\rho }_{n+1},F)\le d({\rho }_n,F).$$

Thus by Lemma 3.1, lim _n→∞ d(ρ_n,F) exists. From Lemma 3.1, we have

$${\lim }_{n\to \infty }‖{\rho }_n-{\kappa }_1{\rho }_n‖=0=\lim ‖{\rho }_n-{\kappa }_2{\rho }_n‖.$$

Using Condition (I^′′), we have

$${\lim }_{n\to \infty }h(d({\rho }_n,F))=0$$

Considering properties of h, we get

$${\lim }_{n\to \infty }d({\rho }_n,F({\kappa }_1))=0.$$

We can also show

$${\lim }_{n\to \infty }d({\rho }_n,F({\kappa }_2))=0.$$

Since Theorem 3.3 conditions are fully met, we deduce that {ρ_n} converges strongly to ι ∈ F(κ) = F(κ₁) ∩ F(κ₂).