This paper presents an initial investigation into the dynamic properties of a well-known iterative method for solving nonlinear equations: Schröder's method. We characterize the degree of the rational map induced by applying the method to polynomial equations, along with other dynamical features such as the nature of extraneous fixed points and the presence of attracting cycles. Particular attention is given to the significant dynamical differences between Schröder's method and other iterative methods, notably Newton's method, with a focus on the behavior at infinity.
Citation: Víctor Galilea, José Manuel Gutiérrez. Schröder's method and the infinity point[J]. AIMS Mathematics, 2025, 10(6): 12919-12934. doi: 10.3934/math.2025581
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This paper presents an initial investigation into the dynamic properties of a well-known iterative method for solving nonlinear equations: Schröder's method. We characterize the degree of the rational map induced by applying the method to polynomial equations, along with other dynamical features such as the nature of extraneous fixed points and the presence of attracting cycles. Particular attention is given to the significant dynamical differences between Schröder's method and other iterative methods, notably Newton's method, with a focus on the behavior at infinity.
A self-mapping F on a convex, closed, and bounded subset K of a Banach space U is known as nonexpansive if ‖Fu−Fv‖ ≤ ‖u−v‖, u,v∈U and need not essentially possess a fixed point. It is widely known that a point u∈U is a fixed point or an invariant point if Fu=u. However, some researchers ensured the survival of a fixed point of nonexpansive mapping in Banach spaces utilizing suitable geometric postulates. Numerous mathematicians have extended and generalized these conclusions to consider several nonlinear mappings. One such special class of mapping is Suzuki generalized nonexpansive mapping (SGNM). Many extensions, improvements and generalizations of nonexpansive mappings are given by eminent researchers (see [8,9,10,13,15,17,19,21,22,25], and so on). On the other hand, Krasnosel'skii [16] investigated a novel iteration of approximating fixed points of nonexpansive mapping. A sequence {ui} utilizing the Krasnosel'skii iteration is defined as: u1=u,ui+1=(1−α)ui+αFui, where α∈(0,1) is a real constant. This iteration is one of the iterative methods which is the extension of the celebrated Picard iteration [24], ui+1=Fui. The convergence rate of the Picard iteration [24] is better than the Krasnosel'skii iteration although the Picard iterative scheme is not essentially convergent for nonexpansive self-mappings. It is interesting to see that the fixed point of a self-mapping F is also a fixed point of the iteration Fn (n∈N), of the self-mapping F but the reverse implication is not feasible. Recently several authors presented extended and generalized results for better approximation of fixed points (see [1,3,11,23,26,27]).
We present convergence and common fixed point conclusions for the associated α-Krasnosel'skii mappings satisfying condition (E) in the current work. Also, we support these with nontrivial illustrative examples to demonstrate that our conclusions improve, generalize and extend comparable conclusions of the literature.
We symbolize F(F), to be the collection of fixed points of a self-mapping F, that is, F(F) = {u∈U:Fu=u}. We begin with the discussion of convex Banach spaces, α-Krasnosel'skii mappings and the condition (E) (see [12,18,20,23]).
Definition 2.1. [14] A Banach space U is uniformly convex if, for ϵ∈(0,2] ∃ δ>0 satisfying, ‖u+v2‖ ≤1−δ so that ‖u−v‖>ϵ and ‖u‖=‖v‖=1, u,v∈U.
Definition 2.2. [14] A Banach space U is strictly convex if, ‖u+v2‖<1 so that u≠v,‖u‖=‖v‖=1, u,v∈U.
Theorem 2.1. [5] Suppose U is a uniformly convex Banach space. Then ∃ a γ>0, satisfying ‖12(u+v)‖≤[1−γϵδ]δ for every ϵ,δ>0 so that ‖u−v‖≥ϵ, ‖u‖≤δ and ‖v‖≤δ, for u,v∈U.
Theorem 2.2. [14] The subsequent postulates are equivalent in a Banach space U:
(i) U is strictly convex.
(ii) u=0 or v=0 or v=cu for c>0, whenever ‖u+v‖ = ‖u‖+‖v‖,u,v∈U.
Definition 2.3. Suppose F is a self-mapping on a non-void subset V of a Banach space U.
(i) Suppose for u∈U, ∃ v∈V so that for all w∈V, ‖v−u‖ ≤‖w−u‖. Then v is a metric projection [6] of U onto V, and is symbolized by PV(.). The mapping PV(u):U→V is the metric projection if PV(x) exists and is determined uniquely for each x∈U.
(ii) F satisfies condition (Eμ) [23] on V if ∃ μ≥1, satisfying ‖u−Fv‖≤μ‖u−Fu‖+‖u−v‖,u,v∈V. Moreover, F satisfies condition (E) on V, if F satisfies (Eμ).
(iii) F satisfies condition (E) [23] and F(F)≠0, then F is quasi-nonexpansive.
(iv) F is a generalized α-Reich-Suzuki nonexpansive [21] if for an α∈[0,1), 12‖u−Fu‖≤‖u−v‖⟹‖Fu−Fv‖≤ max {α‖Fu−u‖+α‖Fv−v‖+(1−2α)‖u−v‖,α‖Fu−v‖+α‖Fv−u‖+(1−2α)‖u−v‖}, ∀u,v∈V.
(v) A self-mapping Fα:V→V is an α-Krasnosel'skii associated with F [2] if, Fαu=(1−α)u+αFu, for α∈(0,1), u∈V.
(vi) F is asymptotically regular [4] if limn→∞‖Fnu−Fn+1u‖=0.
(vii) F is a generalized contraction of Suzuki type [2], if ∃ β∈(0,1) and α1,α2,α3∈[0,1], where α1+2α2+2α3=1, satisfying β‖u−Fu‖≤‖u−v‖ implies
‖Fu−Fv‖≤α1‖u−v‖+α2(‖u−Fu‖+‖v−Fv‖)+α3(‖u−Fv‖+‖v−Fu‖) ,u,v∈U. |
(viii) F is α-nonexpansive [7] if ∃ an α<1 satisfying
‖Fu−Fv‖≤α‖Fu−v‖+α‖Fv−u‖+(1−2α)‖u−v‖,u,v∈U. |
Theorem 2.3. [5] A continuous mapping on a non-void, convex and compact subset V of a Banach space U has a fixed point in V.
Pant et al.[23] derived a proposition that if β=12, then a generalized contraction of Suzuki type is a generalized α-Reich-Suzuki nonexpansive. Moreover, the reverse implication may not necessarily hold.
Lemma 2.1. [2] Let F be a generalized contraction of the Suzuki type on a non-void subset V of a Banach space U. Let β∈[12,1), then
‖u−Fv‖≤(2+α1+α2+3α31−α2−α3)‖u−Fu‖+‖u−v‖. |
Proposition 2.1. [23] Let F be a generalized contraction of the Suzuki type on a non-void subset V of a Banach space U, then F satisfies condition (E).
The converse of this proposition is not true, which can be verified by the following example.
Example 2.1. Suppose U=(R2,‖.‖) with the Euclidean norm and V=[−1,1]×[−1,1] be a subset of U. Let F:V→V be defined as
F(u1,u2)={(u12,u2),if|u1|≤12(−u1,u2),if|u1|>12. |
Case I. Let x=(u1,u2),y=(v1,v2) with |u1|≤12, |v1|≤12. Then,
‖Fx−Fy‖=‖(u12,u2)−(v12,v2)‖=√(u1−v1)24+(u2−v2)2≤√(u1−v1)2+(u2−v2)2=‖x−y‖, |
which implies
‖x−Fy‖≤‖x−Fx‖+‖Fx−Fy‖≤‖x−Fx‖+‖x−y‖. |
Case II. If |u1|≤12, |v1|>12
‖x−Fy‖=√(u1+v1)2+(u2−v2)2‖x−y‖=√(u1−v1)2+(u2−v2)2‖x−Fx‖=|u1|2. |
Consider
‖x−Fy‖=√(u1−v1)2+(u2−v2)2+4u1v1≤√(u1−v1)2+(u2−v2)2+4|u1|≤√(u1−v1)2+(u2−v2)2+4|u1|. |
Hence,
‖x−Fy‖≤8‖x−Fx‖+‖x−y‖. |
Here μ=8 satisfies the inequality.
Case III. If |u1|>12, |v1|≤12
‖x−Fy‖=√(u1−v12)2+(u2−v2)2‖x−y‖=√(u1+v1)2+(u2−v2)2‖x−Fx‖=2|u1|. |
Consider
‖x−Fy‖=√(u1−v12)2+(u2−v2)2≤√(u1−v1)2+(u2−v2)2≤√(u1−v1)2+(u2−v2)2+|u1|≤√(u1−v1)2+(u2−v2)2+2|u1|. |
So,
‖x−Fy‖≤‖x−Fy‖+‖x−y‖. |
Case IV. If |u1|>12 and |v1|>12, then
‖x−Fy‖=√(u1+v1)2+(u2−v2)2‖x−y‖=√(u1−v1)2+(u2−v2)2‖x−Fx‖=2|u1|. |
Since |u1|>12 and |v1|>12, by simple calculation as above, we attain
‖x−Fy‖≤μ‖x−Fx‖+‖x−y‖. |
Thus, F satisfies condition (E) for μ=4.
Now, suppose x=(12,1) and y=(1,1), so
β‖x−Fx‖=β(12−14)=β4≤‖x−y‖=12. |
Clearly, ‖Fx−Fy‖=√(54)2+(1−1)2=54.
Consider
α1‖x−y‖+α2(‖x−Fx‖+‖y−Fy‖)+α3(‖x−Fy‖+‖y−Fx‖)=α1‖(12,1)−(1,1)‖+α2(‖(12,1)−(14,1)‖+‖(1,1)−(−1,1)‖)+α3(‖(12,1)−(−1,1)‖+‖(1,1)−(14,1)‖)=α12+α24+2α2+3α32+3α34=α12+94(α2+α3)=α12+94(1−α12)(by Definition 2.3 (vii))=α12+98−9α18=98−5α18. |
Since α1,α2,α3≥0, therefore
‖Fx−Fy‖>α1‖x−y‖+α2(‖x−Fy‖+‖y−Fy‖)+α3(‖x−Fy‖+‖y−Fx‖), |
which is a contradiction.
Thus, F is not a generalized contraction of the Suzuki type.
Now, we establish results for a pair of α-Krasnosel'skii mappings using condition (E).
Theorem 3.1. Let Fi, for i∈{1,2}, be self-mappings on a non-void convex subset V of a uniformly convex Banach space U and satisfy condition (E) so that F(F1∩F2)≠ϕ. Then the α-Krasnosel'skii mappings Fiα, α∈(0,1) and i∈{1,2} are asymptotically regular.
Proof. Let v0∈V. Define vn+1=Fiαvn for i∈{1,2} and n∈N∪{0}. Thus,
Fiαvn=yn+1=(1−α)vn+αFivnfori∈{1,2}, |
and
Fiαvn−vn=Fiαvn−Fiαvn−1=α(Fivn−vn)fori∈{1,2}. |
It is sufficient to show that limn→∞‖Fivn−vn‖=0 to prove Fiα is asymptotically regular.
By definition, for u0∈F(F1∩F2), we have
‖u0−Fivn‖≤‖u0−vn‖fori∈{1,2} | (3.1) |
and for i∈{1,2},
‖u0−vn+1‖=‖u0−Fiαvn‖=‖u0−(1−α)vn−αFivn‖≤(1−α)‖u0−vn‖+α‖u0−Fivn‖=(1−α)‖u0−vn‖+α‖u0−vn‖=‖u0−vn‖. | (3.2) |
Thus, the sequence {‖u0−vn‖} is bounded by s0=‖u0−v0‖. From inequality (3.2), vn→u0 as n→∞, if vn0=u0, for some n0∈N. So, assume vn≠u0, for n∈N, and
wn=u0−vn‖u0−vn‖anden=u0−Fivn‖u0−vn‖,fori∈{1,2}. | (3.3) |
If α≤12 and using Eq (3.3), we obtain
‖u0−vn+1‖=‖u0−Fiαvn‖,fori∈{1,2}=‖u0−(1−α)vn−αFivn‖,fori∈{1,2}=‖u0−vn+αvn−αFivn−2αu0+2αu0+αvn−αvn‖,fori∈{1,2}=‖(1−2α)u0−(1−2α)vn+(2αu0−αvn−αFivn)‖,fori∈{1,2}≤(1−2α)‖u0−vn‖+α‖2u0−vn−Fivn‖=2α‖u0−vn‖‖wn+en2‖+(1−2α)‖u0−vn‖. | (3.4) |
As the space U is uniformly convex with ‖wn‖≤1, ‖en‖≤1 and ‖wn−en‖=‖vn−Fivn‖‖u0−vn‖≥‖vn−Fivn‖s0=ϵ (say) for i∈{1,2}, we obtain
‖wn+en‖2≤1−δ‖vn−Fivn‖sofori∈{1,2}. | (3.5) |
From inequalities (3.4) and (3.5),
‖u0−vn+1‖≤(2α(1−δ‖vn−Fivn‖so)+(1−2α))‖u0−vn‖=(1−2αδ(‖vn−Fivn‖s0) )‖u0−vn‖. | (3.6) |
By induction, it follows that
‖u0−vn+1‖≤n∏j=1(1−2αδ(‖vn−Fivn‖s0))s0. | (3.7) |
We shall prove that limn→∞‖Fivn−vn‖=0 for i∈{1,2}. On the contrary, consider that {‖Fivn−vn‖} for i∈{1,2} is not converging to zero, and we have a subsequence {vnk}, of {vn}, satisfying ‖Fivnk−vnk‖ converges to ζ>1. As δ∈[0,1] is increasing and α≤12, 1−2αδ‖vk−Fivk‖s0∈[0,1], i∈{1,2}, for all k∈N. Since ‖Fivnk−vnk‖→ζ so, for sufficiently large k,‖Fivnk−vnk‖≥ζ2, from inequality (3.7), we have
‖u0−vnk+1‖≤s0(1−2αδ(ζ2−s0))(nk+1). | (3.8) |
Making k→∞, it follows that vnk→u0. By inequality (3.1), we get Fivnk→u0 and ‖vnk−Fivnk‖→0 as k→∞, which is a contradiction. If α>12, then 1−α<12, because α∈(0,1). Now, for i∈{1,2}
‖u0−vn+1‖=‖u0−(1−α)vn−αFivn‖=‖u0−vn+αvn−αFivn+(2−2α)u0−(2−2α)u0+Fivn−Fivn+αFivn−αFivn‖=‖(2u0−vn−Fivn)−α(2u0−vn−Fivn)+2α(u0−Fivn)−(u0−Fivn)‖≤(1−α)‖2u0−vn−Fivn‖+(2α−1)‖u0−vn‖≤2(1−2α)‖u0−vn‖‖wn+en‖2+(2α−1)‖u0−vn‖. |
By the uniform convexity of U, we attain, for i∈{1,2},
‖x0−yn+1‖≤(2(1−α)−2(1−α)δ‖yn−Fiyn‖so+(1−2α))‖x0−yn‖. | (3.9) |
By induction, we get
‖u0−vn+1‖≤n∏j=1(1−2(1−α)δ(‖vj−Fivj‖s0))s0. |
Similarly, it can be easily proved that ‖Fivn−vn‖→0 as n→∞, which implies that Fiα for i∈{1,2}, is asymptotically regular.
Next, we demonstrate by a numerical experiment that a pair of α-Krasnosel'skii mappings are asymptotically regular for fix α∈(0,1).
Example 3.1. Assume U=(R2,||.||) with Euclidean norm and V={u∈R2:‖u‖≤1}, to be a convex subset of U. Fi for i∈{1,2} be self-mappings on V, satisfying
F1(u1,u2)=(u1,u2)F2(u1,u2)=(u12,0) |
Then, clearly both F1 and F2 satisfy the condition (E) and F(F1∩F2)=(0,0). Now, we will show that the α-Krasnosel'skii mappings Fiα for α∈(0,1) and i∈{1,2} are asymptotically regular.
Since F1 is the identity map, α- Krasnosel'skii mapping F1α is also identity and hence asymptotically regular.
Now, we show F2α is asymptotically regular, let u=(u1,u2)∈V
F2α(u1,u2)=(1−α)(u1,u2)+αF2(u1,u2)=((1−α)u1,(1−α)u2)+α(u12,0)=(u1−αu12,(1−α)u2), |
F22α(u1,u2)=(1−α)(u1−αu12,(1−α)u2)+αF2(u1−αu12,(1−α)u2)=(x1+α2u12−3αu12,(1−α)2u2)+(αu2−α2u14,0)=(u1−αu1+α2u14,(1−α)2x2). |
Continuing in this manner, we get
fn2α(u1,u2)=((u1−α2)n,(1−α)nu2). |
Since (u1,u2)∈V and α∈(0,1), we get that limn→∞(u1−α2)n=0 and limn→∞(1−α)n=0. Now, consider
limn→∞‖Fn2α(u1,u2)−Fn+12α(u1,u2)‖=supu∈Mlimn→∞‖(u1−α2)n−(u1−α2)n+1,((1−α)n−(1−α)n+1)x2‖=0. |
Hence, F2α is also asymptotically regular.
Theorem 3.2. Let Fi be quasi-nonexpansive self-mappings on a non-void and closed subset V of a Banach space U for i∈{1,2}, and satisfy condition (E) so that F(F1∩F2)≠0. Then, F(F1∩F2) is closed in V. Also, if U is strictly convex, then F(F1∩F2) is convex. Furthermore, if U is strictly convex, V is compact, and F is continuous, then for any s0∈V,α∈(0,1), the α-Krasnosel'skii sequence {Fniα(s0)}, converges to s∈F( F1∩F2) .
Proof. (i) We assume {sn}∈F( F1∩F2) so that sn→s∈F(F1∩F2) as n→∞. Hence, Fisn=sn for i∈{1,2}. Next, we show that Fis=s for i∈{1,2}. Since Fi are quasi-nonexpansive, we get
‖sn−Fis‖≤‖sn−s‖fori∈{1,2}, |
that is, Fis=s for i=1,2, hence F(F2∩F2) is closed.
(ii) V is convex since U is strictly convex. Also fix γ∈( 0,1) and u,v∈F(F1∩F2) so that u≠v. Take s=γu+(1−γ)v∈V. Since mapping Fi satisfy condition (E),
‖u−Fis‖≤‖u−Fiu‖+‖u−s‖=‖u−s‖fori∈{1,2}. |
Similarly,
‖v−Fis‖≤‖v−s‖fori∈{1,2}. |
Using strict convexity of U, there is a θ∈[ 0,1] so that Fis=θu+(1−θ)v for i=1,2
(1−θ)‖u−v‖=‖Fiu−Fis‖≤‖u−s‖=(1−γ) ‖u−v‖,fori∈{1,2}, | (3.10) |
and
θ‖u−v‖=‖Fiv−Fis‖≤‖v−s‖=γ‖u−v‖,fori∈{1,2}. | (3.11) |
From inequalities (3.10) and (3.11), we obtain
1−θ≤1−γandθ≤γimplies thatθ=γ. |
Hence, Fis=s for i = 1, 2, implies s∈F(F1∩F2) .
(iii) Let us define {sn} by sn=Fniαs0,s0∈V, where Fiαs0=(1−α)s0+αFis0,α∈( 0,1) . We have a subsequence {snk} of {sn} converging to some s∈V, since V is compact. Using the Schauder theorem and the continuity of Fi, we have F(F1∩F2) ≠ϕ. We shall demonstrate that s∈F(F1∩F2). Let w0∈F(F1∩F2), consider
‖sn−w0‖=‖Fniαs0−w0‖≤‖Fn−1iαs0−w0|=‖sn−1−w0‖. |
Therefore, {‖sn−w0‖} converges as it is a decreasing sequence that is bounded below by 0. Moreover, since Fiα for i=1,2 is continuous, we have
‖w0−s0‖=limk→∞‖snk+1−so‖=limk→∞‖Fiαsnk−s0‖=‖Fiαs−s0‖=‖(1−α)s+αFis−s0‖≤(1−α)‖s−s0‖+α‖Fis−s0‖fori∈{1,2}. | (3.12) |
Since α>0, we get
‖s−s0‖≤‖Fis−s0‖,fori∈{1,2}. | (3.13) |
Since Fi are quasi-nonexpansive maps, we get
‖Fis−s0‖≤‖s−s0‖,fori∈{1,2}, | (3.14) |
and from inequalities (3.13) and (3.14), we get
‖Fis−s0‖=‖s−s0‖,fori∈{1,2}. | (3.15) |
Now, from inequality (3.12), we have
‖s−s0‖≤‖(1−α)s+αFis−s0‖,fori∈{1,2}≤(1−α)‖s−s0‖+α‖Fis−s0‖,fori∈{1,2}=‖s−s0‖, |
which implies that
‖(1−α)s+αFis−s0‖=(1−α)‖s−s0‖+α‖Fis−s0‖,fori∈{1,2}. |
Since U is strictly convex, either Fis−s0=a(s−s0) for some a⪈0 or s=s0. From Eq (15), it follows that a=1, then, Fis=s for i=1,2 and s∈F(F1∩F2). Since limn→∞‖sn−s0‖ exists and {snk} converges strongly to s. Hence, {sn} converges strongly to s∈F(F1∩F2).
The next conclusion for metric projection is slightly more fascinating.
Theorem 3.3. Let Fi be quasi-nonexpansive self-mappings on a non-void, closed, and convex subset V of a uniformly convex Banach space U for i∈{1,2}, and satisfies condition (E) so that F(F1∩F2)≠ϕ. Let P:U→F(F1∩F2) be the metric projection. Then, for every u∈U, the sequence {PFniu} for i={1,2}, converges to s∈F(F1∩F2).
Proof. Let u∈V. For n,m∈N
‖PFniu−Fniu‖≤‖PFmiu−Fniu‖,forn≥m,i∈{1,2}. | (3.16) |
Since u∈F(F1∩F2) , n∈N and Fi are quasi-nonexpansive maps, fori∈{1,2} we have
‖PFmiu−Fniu‖=‖PFmiu−FiFn−1iu‖≤‖PFmiu−Fn−1iu‖. |
Therefore, for n≥m, it follows that
‖PFmiu−Fniu‖≤‖PFmiu−Fmiu‖,fori∈{1,2}. | (3.17) |
From inequalities (3.16) and (3.17), we have
‖PFniu−Fniu‖≤‖PFmiu−Fmiu‖,fori∈{1,2}, |
which implies that limn→∞‖PFniu−Fniu‖ exists. Taking limn→∞‖PFniu−Fniu‖=l.
If l=0, then we have an integer n0( ϵ) for ϵ>0, satisfying
‖PFniu−Fniu‖>ϵ4,fori∈{1,2}, | (3.18) |
for n≥n0. Therefore, if n≥m≥n0 and using inequalities (3.17) and (3.18), we have, for i∈{1,2},
‖PFniu−PFmiu‖≤‖PFniu−PFn0iu‖+‖PFn0iu−Fmiu‖≤‖PFniu−Fniu‖+‖Fniu−PFn0iu‖+‖PFmiu−Fmiu‖+‖Fmiu−PFn0iu‖≤‖PFniu−Fniu‖+‖Fn0iu−PFn0iu‖+‖PFmiu−Fmiu‖+‖Fn0iu−PFn0iu‖≤ϵ4+ϵ4+ϵ4+ϵ4=ϵ. |
That is, {PFniu} for i={1,2} is a Cauchy sequence in F(F1∩F2). Using the completeness of U and the closedness of F(F1∩F2) from the above theorem, {PFnix} for i=1,2, converges in F(F1∩F2). Taking l>0, we claim that the sequence {PFniu} for i=1,2, is a Cauchy sequence in U. Also we have, an ϵ0>0 so that, for each n0∈N, we have some r0,s0≥n0 satisfying
‖PFr0iu−PFs0iu‖≥ϵ0,fori∈{1,2}. |
Now, we choose a θ>0
(l+θ)(1−δϵ0l+θ)<θ. |
Let m0 be as large as possible such that for q≥m0
l≤‖PFqiu−Fqiu‖≤l+θ. |
For this m0, there exist q1,q2 such that q1,q2>m0 and
‖PFq1iu−PFq2iu‖≥ϵ0fori∈{1,2}. |
Thus, for q0≥max{q1,q2}, we attain
‖PFq1ix−Fq0ix‖≥‖PFq1ix−Fq1ix‖<l+θ, |
and
‖PFq2ix−Fq0ix‖≥‖PFq1ix−Fq1ix‖<l+θfori∈{1,2}. |
Now, using the uniform convexity of U, we attain
l≤‖PFq0ix−Fq0ix‖≤‖PFq1ix+PFq2ix2−Fq0ix‖,fori∈{1,2}≤( l+θ) (1−δϵ0l+θ)<θ, |
a contradiction. Hence for every u∈V, the sequence {PFniu} for i=1,2, converges to some s∈F(F1∩F2).
We have proved some properties of common fixed points and also showed that if two mappings have common fixed points, then their α-Krasnosel'skii mappings are asymptotically regular. To show the superiority of our results, we have provided an example. Further, we have proved that the α-Krasnosel'skii sequence and its projection converge to a common fixed whose collection is closed.
Researchers would like to thank the Deanship of Scientific Research, Qassim University for funding publication of this project.
The authors declare no conflict of interest.
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