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Research article Special Issues

Schröder's method and the infinity point

  • Received: 12 December 2024 Revised: 24 April 2025 Accepted: 09 May 2025 Published: 05 June 2025
  • MSC : 37F10, 65S05

  • 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 FuFv uv, u,vU and need not essentially possess a fixed point. It is widely known that a point uU 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=(1α)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 (nN), 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) = {uU: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 1δ so that uv>ϵ and u=v=1, u,vU.

    Definition 2.2. [14] A Banach space U is strictly convex if, u+v2<1 so that uv,u=v=1, u,vU.

    Theorem 2.1. [5] Suppose U is a uniformly convex Banach space. Then a γ>0, satisfying 12(u+v)[1γϵδ]δ for every ϵ,δ>0 so that uvϵ, uδ and vδ, for u,vU.

    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,vU.

    Definition 2.3. Suppose F is a self-mapping on a non-void subset V of a Banach space U.

    (i) Suppose for uU, vV so that for all wV, vu wu. Then v is a metric projection [6] of U onto V, and is symbolized by PV(.). The mapping PV(u):UV is the metric projection if PV(x) exists and is determined uniquely for each xU.

    (ii) F satisfies condition (Eμ) [23] on V if μ1, satisfying uFvμuFu+uv,u,vV. 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), 12uFuuvFuFv max {αFuu+αFvv+(12α)uv,αFuv+αFvu+(12α)uv}, u,vV.

    (v) A self-mapping Fα:VV is an α-Krasnosel'skii associated with F [2] if, Fαu=(1α)u+αFu, for α(0,1), uV.

    (vi) F is asymptotically regular [4] if limnFnuFn+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 βuFuuv implies

    FuFvα1uv+α2(uFu+vFv)+α3(uFv+vFu) ,u,vU.

    (viii) F is α-nonexpansive [7] if an α<1 satisfying

    FuFvαFuv+αFvu+(12α)uv,u,vU.

    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

    uFv(2+α1+α2+3α31α2α3)uFu+uv.

    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:VV 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,

    FxFy=(u12,u2)(v12,v2)=(u1v1)24+(u2v2)2(u1v1)2+(u2v2)2=xy,

    which implies

    xFyxFx+FxFyxFx+xy.

    Case II. If |u1|12, |v1|>12

    xFy=(u1+v1)2+(u2v2)2xy=(u1v1)2+(u2v2)2xFx=|u1|2.

    Consider

    xFy=(u1v1)2+(u2v2)2+4u1v1(u1v1)2+(u2v2)2+4|u1|(u1v1)2+(u2v2)2+4|u1|.

    Hence,

    xFy8xFx+xy.

    Here μ=8 satisfies the inequality.

    Case III. If |u1|>12, |v1|12

    xFy=(u1v12)2+(u2v2)2xy=(u1+v1)2+(u2v2)2xFx=2|u1|.

    Consider

    xFy=(u1v12)2+(u2v2)2(u1v1)2+(u2v2)2(u1v1)2+(u2v2)2+|u1|(u1v1)2+(u2v2)2+2|u1|.

    So,

    xFyxFy+xy.

    Case IV. If |u1|>12 and |v1|>12, then

    xFy=(u1+v1)2+(u2v2)2xy=(u1v1)2+(u2v2)2xFx=2|u1|.

    Since |u1|>12 and |v1|>12, by simple calculation as above, we attain

    xFyμxFx+xy.

    Thus, F satisfies condition (E) for μ=4.

    Now, suppose x=(12,1) and y=(1,1), so

    βxFx=β(1214)=β4xy=12.

    Clearly, FxFy=(54)2+(11)2=54.

    Consider

    α1xy+α2(xFx+yFy)+α3(xFy+yFx)=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(1α12)(by Definition 2.3 (vii))=α12+989α18=985α18.

    Since α1,α2,α30, therefore

    FxFy>α1xy+α2(xFy+yFy)+α3(xFy+yFx),

    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(F1F2)ϕ. Then the α-Krasnosel'skii mappings Fiα, α(0,1) and i{1,2} are asymptotically regular.

    Proof. Let v0V. Define vn+1=Fiαvn for i{1,2} and nN{0}. Thus,

    Fiαvn=yn+1=(1α)vn+αFivnfori{1,2},

    and

    Fiαvnvn=FiαvnFiαvn1=α(Fivnvn)fori{1,2}.

    It is sufficient to show that limnFivnvn=0 to prove Fiα is asymptotically regular.

    By definition, for u0F(F1F2), we have

    u0Fivnu0vnfori{1,2} (3.1)

    and for i{1,2},

    u0vn+1=u0Fiαvn=u0(1α)vnαFivn(1α)u0vn+αu0Fivn=(1α)u0vn+αu0vn=u0vn. (3.2)

    Thus, the sequence {u0vn} is bounded by s0=u0v0. From inequality (3.2), vnu0 as n, if vn0=u0, for some n0N. So, assume vnu0, for nN, and

    wn=u0vnu0vnanden=u0Fivnu0vn,fori{1,2}. (3.3)

    If α12 and using Eq (3.3), we obtain

    u0vn+1=u0Fiαvn,fori{1,2}=u0(1α)vnαFivn,fori{1,2}=u0vn+αvnαFivn2αu0+2αu0+αvnαvn,fori{1,2}=(12α)u0(12α)vn+(2αu0αvnαFivn),fori{1,2}(12α)u0vn+α2u0vnFivn=2αu0vnwn+en2+(12α)u0vn. (3.4)

    As the space U is uniformly convex with wn1, en1 and wnen=vnFivnu0vnvnFivns0=ϵ (say) for i{1,2}, we obtain

    wn+en21δvnFivnsofori{1,2}. (3.5)

    From inequalities (3.4) and (3.5),

    u0vn+1(2α(1δvnFivnso)+(12α))u0vn=(12αδ(vnFivns0) )u0vn. (3.6)

    By induction, it follows that

    u0vn+1nj=1(12αδ(vnFivns0))s0. (3.7)

    We shall prove that limnFivnvn=0 for i{1,2}. On the contrary, consider that {Fivnvn} for i{1,2} is not converging to zero, and we have a subsequence {vnk}, of {vn}, satisfying Fivnkvnk converges to ζ>1. As δ[0,1] is increasing and α12, 12αδvkFivks0[0,1], i{1,2}, for all kN. Since Fivnkvnkζ so, for sufficiently large k,Fivnkvnkζ2, from inequality (3.7), we have

    u0vnk+1s0(12αδ(ζ2s0))(nk+1). (3.8)

    Making k, it follows that vnku0. By inequality (3.1), we get Fivnku0 and vnkFivnk0 as k, which is a contradiction. If α>12, then 1α<12, because α(0,1). Now, for i{1,2}

    u0vn+1=u0(1α)vnαFivn=u0vn+αvnαFivn+(22α)u0(22α)u0+FivnFivn+αFivnαFivn=(2u0vnFivn)α(2u0vnFivn)+2α(u0Fivn)(u0Fivn)(1α)2u0vnFivn+(2α1)u0vn2(12α)u0vnwn+en2+(2α1)u0vn.

    By the uniform convexity of U, we attain, for i{1,2},

    x0yn+1(2(1α)2(1α)δynFiynso+(12α))x0yn. (3.9)

    By induction, we get

    u0vn+1nj=1(12(1α)δ(vjFivjs0))s0.

    Similarly, it can be easily proved that Fivnvn0 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={uR2:u1}, 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(F1F2)=(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)=(1α)(u1,u2)+αF2(u1,u2)=((1α)u1,(1α)u2)+α(u12,0)=(u1αu12,(1α)u2),
    F22α(u1,u2)=(1α)(u1αu12,(1α)u2)+αF2(u1αu12,(1α)u2)=(x1+α2u123αu12,(1α)2u2)+(αu2α2u14,0)=(u1αu1+α2u14,(1α)2x2).

    Continuing in this manner, we get

    fn2α(u1,u2)=((u1α2)n,(1α)nu2).

    Since (u1,u2)V and α(0,1), we get that limn(u1α2)n=0 and limn(1α)n=0. Now, consider

    limnFn2α(u1,u2)Fn+12α(u1,u2)=supuMlimn(u1α2)n(u1α2)n+1,((1α)n(1α)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(F1F2)0. Then, F(F1F2) is closed in V. Also, if U is strictly convex, then F(F1F2) is convex. Furthermore, if U is strictly convex, V is compact, and F is continuous, then for any s0V,α(0,1), the α-Krasnosel'skii sequence {Fniα(s0)}, converges to sF( F1F2) .

    Proof. (i) We assume {sn}F( F1F2)  so that snsF(F1F2) 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

    snFissnsfori{1,2},

    that is, Fis=s for i=1,2, hence F(F2F2) is closed.

    (ii) V is convex since U is strictly convex. Also fix γ( 0,1)  and u,vF(F1F2)  so that uv. Take s=γu+(1γ)vV. Since mapping Fi satisfy condition (E),

    uFisuFiu+us=usfori{1,2}.

    Similarly,

    vFisvsfori{1,2}.

    Using strict convexity of U, there is a θ[ 0,1]  so that Fis=θu+(1θ)v for i=1,2

    (1θ)uv=FiuFisus=(1γ) uv,fori{1,2}, (3.10)

    and

    θuv=FivFisvs=γuv,fori{1,2}. (3.11)

    From inequalities (3.10) and (3.11), we obtain

    1θ1γandθγimplies thatθ=γ.

    Hence, Fis=s for i = 1, 2, implies sF(F1F2) .

    (iii) Let us define {sn} by sn=Fniαs0,s0V, where Fiαs0=(1α)s0+αFis0,α( 0,1) . We have a subsequence {snk} of {sn} converging to some sV, since V is compact. Using the Schauder theorem and the continuity of Fi, we have F(F1F2) ϕ. We shall demonstrate that sF(F1F2). Let w0F(F1F2), consider

    snw0=Fniαs0w0Fn1iαs0w0|=sn1w0.

    Therefore, {snw0} converges as it is a decreasing sequence that is bounded below by 0. Moreover, since Fiα for i=1,2 is continuous, we have

    w0s0=limksnk+1so=limkFiαsnks0=Fiαss0=(1α)s+αFiss0(1α)ss0+αFiss0fori{1,2}. (3.12)

    Since α>0, we get

    ss0Fiss0,fori{1,2}. (3.13)

    Since Fi are quasi-nonexpansive maps, we get

    Fiss0ss0,fori{1,2}, (3.14)

    and from inequalities (3.13) and (3.14), we get

    Fiss0=ss0,fori{1,2}. (3.15)

    Now, from inequality (3.12), we have

    ss0(1α)s+αFiss0,fori{1,2}(1α)ss0+αFiss0,fori{1,2}=ss0,

    which implies that

    (1α)s+αFiss0=(1α)ss0+αFiss0,fori{1,2}.

    Since U is strictly convex, either Fiss0=a(ss0) for some a0 or s=s0. From Eq (15), it follows that a=1, then, Fis=s for i=1,2 and sF(F1F2). Since limnsns0 exists and {snk} converges strongly to s. Hence, {sn} converges strongly to sF(F1F2).

    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(F1F2)ϕ. Let P:UF(F1F2) be the metric projection. Then, for every uU, the sequence {PFniu} for i={1,2}, converges to sF(F1F2).

    Proof. Let uV. For n,mN

    PFniuFniuPFmiuFniu,fornm,i{1,2}. (3.16)

    Since uF(F1F2) , nN and Fi are quasi-nonexpansive maps, fori{1,2} we have

    PFmiuFniu=PFmiuFiFn1iuPFmiuFn1iu.

    Therefore, for nm, it follows that

    PFmiuFniuPFmiuFmiu,fori{1,2}. (3.17)

    From inequalities (3.16) and (3.17), we have

    PFniuFniuPFmiuFmiu,fori{1,2},

    which implies that limnPFniuFniu exists. Taking limnPFniuFniu=l.

    If l=0, then we have an integer n0( ϵ)  for ϵ>0, satisfying

    PFniuFniu>ϵ4,fori{1,2}, (3.18)

    for nn0. Therefore, if nmn0 and using inequalities (3.17) and (3.18), we have, for i{1,2},

    PFniuPFmiuPFniuPFn0iu+PFn0iuFmiuPFniuFniu+FniuPFn0iu+PFmiuFmiu+FmiuPFn0iuPFniuFniu+Fn0iuPFn0iu+PFmiuFmiu+Fn0iuPFn0iuϵ4+ϵ4+ϵ4+ϵ4=ϵ.

    That is, {PFniu} for i={1,2} is a Cauchy sequence in F(F1F2). Using the completeness of U and the closedness of F(F1F2) from the above theorem, {PFnix} for i=1,2, converges in F(F1F2). 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 n0N, we have some r0,s0n0 satisfying

    PFr0iuPFs0iuϵ0,fori{1,2}.

    Now, we choose a θ>0

    (l+θ)(1δϵ0l+θ)<θ.

    Let m0 be as large as possible such that for qm0

    lPFqiuFqiul+θ.

    For this m0, there exist q1,q2 such that q1,q2>m0 and

    PFq1iuPFq2iuϵ0fori{1,2}.

    Thus, for q0max{q1,q2}, we attain

    PFq1ixFq0ixPFq1ixFq1ix<l+θ,

    and

    PFq2ixFq0ixPFq1ixFq1ix<l+θfori{1,2}.

    Now, using the uniform convexity of U, we attain

    lPFq0ixFq0ixPFq1ix+PFq2ix2Fq0ix,fori{1,2}( l+θ) (1δϵ0l+θ)<θ,

    a contradiction. Hence for every uV, the sequence {PFniu} for i=1,2, converges to some sF(F1F2).

    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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