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Research article

The error analysis for the Cahn-Hilliard phase field model of two-phase incompressible flows with variable density

  • Received: 20 September 2023 Revised: 10 November 2023 Accepted: 15 November 2023 Published: 23 November 2023
  • MSC : 35K51, 35K55, 65M12, 65M70

  • In this paper, we consider the numerical approximations of the Cahn-Hilliard phase field model for two-phase incompressible flows with variable density. First, a temporal semi-discrete numerical scheme is proposed by combining the fractional step method (for the momentum equation) and the convex-splitting method (for the free energy). Second, we prove that the scheme is unconditionally stable in energy. Then, the L2 convergence rates for all variables are demonstrated through a series of rigorous error estimations. Finally, by applying the finite element method for spatial discretization, some numerical simulations were performed to demonstrate the convergence rates and energy dissipations.

    Citation: Mingliang Liao, Danxia Wang, Chenhui Zhang, Hongen Jia. The error analysis for the Cahn-Hilliard phase field model of two-phase incompressible flows with variable density[J]. AIMS Mathematics, 2023, 8(12): 31158-31185. doi: 10.3934/math.20231595

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  • In this paper, we consider the numerical approximations of the Cahn-Hilliard phase field model for two-phase incompressible flows with variable density. First, a temporal semi-discrete numerical scheme is proposed by combining the fractional step method (for the momentum equation) and the convex-splitting method (for the free energy). Second, we prove that the scheme is unconditionally stable in energy. Then, the L2 convergence rates for all variables are demonstrated through a series of rigorous error estimations. Finally, by applying the finite element method for spatial discretization, some numerical simulations were performed to demonstrate the convergence rates and energy dissipations.



    In 1922, Banach first presented the Banach contraction principle [1] in metric spaces, which is a powerful and classical means to solve problems about fixed point. Subsequently, it has been generalized in many aspects. One vital generalization is to promote the concept of metric spaces. b-metric spaces is regarded as a well-known generalization of metric spaces. In 1993, Czerwik [2] first introduced the concept of b-metric spaces by modifying the third condition of the metric function. The author also provided fixed point results for contraction conditions in this type space. In the sequel, several papers have been published on the fixed point theory of various classes of single-valued and multi-valued operators in b-metric spaces (see [3,4,5,6]).

    In 1969, Boyd and Wong [7] gave a definition of ϕ-contraction in metric spaces for the first time. Afterward, Alber and Guerre [8] defined the concept of weak contraction and got some fixed point results in Hilbert space. In [9], Rhoades generalized Alber and Guerre's results to more general forms. Alutn [10] proved the common fixed point theorem for weakly contraction mappings of integral type. Later, more scholars [11,12,13,14] presented some fixed point theorems for weakly contractive mappings in different spaces.

    In particular, Perveen [15] obtained the θ-weak contraction principle in metric spaces as follows:

    Theorem 1.1. [15] Suppose (Ω,) is a complete metric space and S:ΩΩ is a θ-weak contraction. If θ is continuous, then

    (a) S has unique fixed point (say, zΩ),

    (b)limn+Snz=z, zΩ.

    Motivated and inspired by results in [15], in this paper we give some fixed point theorems for contractive mappings of the integral type in b-metric spaces. Furthermore, two examples are given to prove the feasibility of the theorems. Also, the solvability of a functional equation arising in dynamic programming is considered by means of obtained results.

    We introduce the following definitions and lemmas, which will be used to obtain our main results.

    Definition 2.1. [2] Let be a nonempty set and s1 be a given real number. A mapping ϖ:×[0,+) is said to be a b-metric if, and only if, for all κ,λ,μ, the following conditions are satisfied:

    (ⅰ) ϖ(κ,λ)=0 if, and only if, κ=λ;

    (ⅱ) ϖ(κ,λ)=ϖ(λ,κ);

    (ⅲ) ϖ(κ,λ)s(ϖ(κ,μ)+ϖ(λ,μ)).

    In general, (,ϖ) is called a b-metric space with parameter s1.

    Remark 2.2. Visibly, every metric space is a b-metric space with s=1. There are several examples of b-metric spaces that are not metric spaces (see [16]).

    Example 2.3. [17] Let (,d) be a metric space, and ϖ(κ,λ)=(d(κ,λ))p, where p>1 is a real number, then ϖ(κ,λ) is a b-metric with s=2p1.

    Definition 2.4. [18] Let (,ϖ) be a b-metric space with parameter s1, then a sequence {κι}+ι=1 in is said to be:

    (ⅰ) b-convergent if there exists κ such that ϖ(κι,κ)0 as ι+;

    (ⅱ) a Cauchy sequence if ϖ(κι,κυ)0 when ι,υ+.

    As usual, a b-metric space is called complete if, and only if, each Cauchy sequence in this space is b-convergent.

    The following lemma plays a key role in our conclusion.

    Lemma 2.5. [17] Let (,ϖ) be a b-metric space with parameter s1. Assume that {κι}+ι=1 and {λι}+ι=1 are b-convergent to κ and λ, respectively, then we have

    1s2ϖ(κ,λ)lim infι+ϖ(κι,λι)lim supι+ϖ(κι,λι)s2ϖ(κ,λ).

    In particular, if κ=λ, then we have limι+ϖ(κι,λι)=0. Moreover, for each μ, we have

    1sϖ(κ,μ)lim infι+ϖ(κι,μ)lim supι+ϖ(κι,μ)sϖ(κ,μ).

    Lemma 2.6. [19] Let φ and {κι}ιN be a nonnegative sequence with limn+κι=κ, then

    limn+κι0φ(ω)dω=κ0φ(ω)dω.

    Lemma 2.7. [19] Let φ and {κι}ιN be a nonnegative sequence, then

    limι+κι0φ(ω)dω=0

    if, and only if, limι+κι=0.

    Throughout this paper, we assume that R+=[0,+), N0=N{0}, where N stands for the set of positive integers,

    ={ξ|ξ:R+R+ satisfies that ξ is Lebesgue integrable, and δ0ξ(ω)dω>0 for each δ>0}.

    Let (,ϖ) be a b-metric space with parameter s1 and S be a self-mapping on . For any u,v, set

    (u,v)=max{ϖ(u,v),ϖ(u,Su),ϖ(v,Sv),ϖ(u,Sv)+ϖ(v,Su)2s}.

    In this part, we introduce the new concept of αsp-admissible mapping and other definitions, which will be used to prove the fixed point theorems of the integral type in b-metric space. Moreover, we also provide two examples to support our results.

    Let

    Θ1={θ|θ:(0,+)(1,+) satisfies the following conditions (1) and (3)},

    Θ2={θ|θ:(0,+)(0,1) satisfies the following conditions (2) and (3)},

    where

    (1) θ is nondecreasing and continuous;

    (2) θ is nonincreasing and continuous;

    (3) for each sequence {βι}+ι=1(0,+), limι+θ(βι)=1limι+βι=0.

    Definition 3.1. Let (,ϖ) be a b-metric space with parameter s1 and p1 be an integer. A mapping S: is said to be αspadmissible if for all z,w, one has

    α(z,w)spα(Sz,Sw)sp

    where α:×[0,+) is a given function.

    Lemma 3.2. Let φ and {κι}ιN be a nonnegative sequence. If lim supι+κι=κ, then

    κ0φ(ω)dωlim supι+κι0φ(ω)dω.

    If lim infι+κι=κ, then

    lim infι+κι0φ(ω)dωκ0φ(ω)dω.

    Proof. It follows from lim supι+κι=κ that there exists a subsequence {κις} of {κι} such that

    limς+κις=κ.

    In view of Lemma 2.6, we deduce that

    κ0φ(ω)dω=limς+κις0φ(ω)dωlim supι+κι0φ(ω)dω.

    Similarly, one can prove another inequality.

    Theorem 3.3. Let (,ϖ) be a complete b-metric space with parameter s1 and S: be a given self-mapping. Assume that α:×[0,+) and p3. If

    (ⅰ) S is αsp-admissible,

    (ⅱ) there is p0 satisfying α(p0,Sp0)sp,

    (ⅲ) α satisfies transitive property, i.e., for ξ,η,ζ if

    α(ξ,η)sp and α(η,ζ)spα(ξ,ζ)sp,

    (ⅳ) if {pι} is a sequence in satisfying pιp as ι+, then there exists a subsequence {pι(k)}+k=1 of {pι}+ι=1 with α(pι(k),p)sp,

    (ⅴ) S is a θ-weak contraction, that is, there exists (0,1), φ, θΘ1 such that: for any u,v,

    α(u,v)sp,ϖ(Su,Sv)0φ(ω)dω>0θ(α(u,v)ϖ(Su,Sv)0φ(ω)dω)[θ( (u,v)0φ(ω)dω)], (3.1)

    then S has a fixed point in . Furthermore, if

    (ⅵ) for p,qFix(S), one can get the conditions of α(p,q)sp and α(q,p)sp, where Fix(S) represents the collection of all fixed points of S,

    then the fixed point is unique.

    Proof. Under condition (ⅱ), there is a p0 satisfying α(p0,Sp0)sp. Define sequence {pn} in by pn+1=Spn for nN. If pn0=Spn0 for some n0, then pn0 is a fixed point of S. Suppose that pn+1pn for nN. It follows from condition (ⅰ) that

    α(p0,Sp0)spα(Sp0,S2p0)sp,
    α(p1,p2)spα(Sp1,Sp2)sp,
    α(p2,p3)spα(Sp2,Sp3)sp,

    Thus, for all nN, we have α(pn1,pn)sp. Using (3.1) by u=pn1 and v=pn, one gets

    θ(α(pn1,pn)ϖ(Spn1,Spn)0φ(ω)dω)[θ( (pn1,pn)0φ(ω)dω)] (3.2)

    where

    (pn1,pn)=max{ϖ(pn1,pn),ϖ(pn1,Spn1),ϖ(pn,Spn),ϖ(pn1,Spn)+ϖ(pn,Spn1)2s}=max{ϖ(pn1,pn),ϖ(pn1,pn),ϖ(pn,pn+1),ϖ(pn1,pn+1)+ϖ(pn,pn)2s}=max{ϖ(pn1,pn),ϖ(pn,pn+1)}. (3.3)

    If ϖ(pn,pn+1)ϖ(pn1,pn) for some nN, in view of (3.2) and (3.3), we have (pn1,pn)=ϖ(pn,pn+1), so

    θ(ϖ(pn,pn+1)0φ(ω)dω)<θ(spϖ(pn,pn+1)0φ(ω)dω)θ(α(pn1,pn)ϖ(Spn1,Spn)0φ(ω)dω)[θ((pn1,pn)0φ(ω)dω)]=[θ(ϖ(pn,pn+1)0φ(ω)dω)]

    which is impossible. Hence,

    ϖ(pn1,pn)>ϖ(pn,pn+1). (3.4)

    (3.4) implies that (pn1,pn)=ϖ(pn1,pn) is decreasing. Thus, we have

    θ(ϖ(pn,pn+1)0φ(ω)dω)<[θ(ϖ(pn1,pn)0φ(ω)dω)]<[θ(ϖ(pn2,pn1)0φ(ω)dω)]2<<[θ(ϖ(p0,p1)0φ(ω)dω)]n.

    Letting n+ in the above inequality, we get

    1limn+θ(ϖ(pn,pn+1)0φ(ω)dω)limn+[θ(ϖ(p0,p1)0φ(ω)dω)]n=1

    i.e., limn+θ(ϖ(pn,pn+1)0φ(ω)dω)=1, which by the definition of θ yields that

    limn+ϖ(pn,pn+1)0φ(ω)dω=0

    which implies

    limn+ϖ(pn,pn+1)=0.

    Now, we prove {pn} is a Cauchy sequence. Suppose {pn} is not Cauchy, then there exists ε>0 for which we can choose sequences {pn(k)} and {pm(k)} of {pn}, such that n(k) is the smallest index for which n(k)>m(k)>k,

    εϖ(pm(k),pn(k)),ϖ(pm(k),pn(k)1)<ε. (3.5)

    Under the triangle inequality and (3.5), we get

    εϖ(pm(k),pn(k))sϖ(pm(k),pn(k)1)+sϖ(pn(k)1,pn(k))<sε+sϖ(pn(k)1,pn(k)).

    Taking the superior limit and inferior limit as k+, we get

    εlim infk+ϖ(pm(k),pn(k))lim supk+ϖ(pm(k),pn(k))sε. (3.6)

    Similarly, one can deduce the following inequalities:

    ϖ(pm(k),pn(k))sϖ(pm(k),pm(k)1)+s2ϖ(pm(k)1,pn(k)1)+s2ϖ(pn(k)1,pn(k)), (3.7)
    ϖ(pm(k)1,pn(k)1)sϖ(pm(k)1,pm(k))+s2ϖ(pm(k),pn(k))+s2ϖ(pn(k),pn(k)1), (3.8)
    ϖ(pm(k),pn(k))sϖ(pm(k),pm(k)1)+sϖ(pm(k)1,pn(k)), (3.9)
    ϖ(pm(k)1,pn(k))sϖ(pm(k)1,pm(k))+sϖ(pm(k),pn(k)), (3.10)
    ϖ(pm(k),pn(k))sϖ(pm(k),pn(k)1)+sϖ(pn(k)1,pn(k)), (3.11)
    ϖ(pm(k),pn(k)1)sϖ(pm(k),pn(k))+sϖ(pn(k),pn(k)1). (3.12)

    By (3.6)–(3.8), we have

    εs2lim infk+ϖ(pm(k)1,pn(k)1)lim supk+ϖ(pm(k)1,pn(k)1)s3ε. (3.13)

    It follows from (3.6), (3.9), and (3.10) that

    εslim infk+ϖ(pm(k)1,pn(k))lim supk+ϖ(pm(k)1,pn(k))s2ε. (3.14)

    According to (3.6), (3.11), and (3.12), one can obtain

    εslim infk+ϖ(pm(k),pn(k)1)lim supk+ϖ(pm(k),pn(k)1)s2ε. (3.15)

    Thus, there exists NN0 such that for m(k),n(k)N, ϖ(pm(k)1,pn(k)1)0φ(ω)dω>0.

    In view of the definition of (u,v), we have

    (pm(k)1,pn(k)1)=max{ϖ(pm(k)1,pn(k)1),ϖ(pm(k)1,Spm(k)1),ϖ(pn(k)1,Spn(k)1),ϖ(pm(k)1,Spn(k)1)+ϖ(pn(k)1,Spm(k)1)2s}=max{ϖ(pm(k)1,pn(k)1),ϖ(pm(k)1,pm(k)),ϖ(pn(k)1,pn(k)),ϖ(pm(k)1,pn(k))+ϖ(pn(k)1,pm(k))2s}. (3.16)

    Letting k+ in (3.16), we get

    lim infk+(pm(k)1,pn(k)1)lim supk+(pm(k)1,pn(k)1)max{s3ε,0,0,s2ε+s2ε2s}=s3ε. (3.17)

    The transitivity property of α yields that α(pm(k)1,pn(k)1)sp. Choosing u=pm(k)1 and v=pn(k)1 in (3.1), by Lemma 3.2, one can deduce

    θ(s3ε0φ(ω)dω)lim infk+θ(spϖ(pm(k),pn(k))0φ(ω)dω)lim infk+θ(α(pm(k)1,pn(k)1)ϖ(Spm(k)1,Spn(k)1)0φ(ω)dω)lim infk+[θ((pm(k)1,pn(k)1)0φ(ω)dω)][θ(s3ε0φ(ω)dω)]

    which is a contradiction. So, {pn} is Cauchy. As is complete, there exists p such that pnp as n+.

    Next, we prove the point p to be a fixed point of S. So, we think about a set, say Q={nN0:pn=Sp}, then it has two situations. One, if Q is an infinite set, then there exists a subsequence {pn(k)}{pn}, which converges to Sp. By the uniqueness of limit, we have Sp=p. The other, if Q is a finite set, then there is NN such that ϖ(pn,Sp)0φ(ω)dω>0 for any nN. By (iv), we obtain that there exists a subsequence {pn(k)}{pn} such that α(pn(k)1,p)sp and ϖ(pn(k),Sp)0φ(ω)dω>0, kN. Taking u=pn(k)1 and v=p in (3.1), we get

    θ(α(pn(k)1,p)ϖ(Spn(k)1,Sp)0φ(t)dt)[θ( (pn(k)1,p)0φ(ω)dω)] (3.18)

    where

    (pn(k)1,p)=max{ϖ(pn(k)1,p),ϖ(pn(k)1,Spn(k)1),ϖ(p,Sp),ϖ(pn(k)1,Sp)+dϖ(p,Spn(k)1)2s}=max{ϖ(pn(k)1,p),ϖ(pn(k)1,pn(k)),ϖ(p,Sp),ϖ(pn(k)1,Sp)+ϖ(p,pn(k))2s}. (3.19)

    Putting the limit as k+ in (3.19), we get

    limk+(pn(k)1,p)=max{0,0,ϖ(p,Sp),ϖ(p,Sp)2}=ϖ(p,Sp).

    According to (3.18), (3.19), and Lemma 2.5, we get

    θ(ϖ(p,Sp)0φ(ω)dω)<θ(s31sϖ(p,Sp)0φ(ω)dω)lim supn+θ(spϖ(Spn(k)1,Sp)0φ(ω)dω)lim supn+θ(α(pn(k)1,p)ϖ(Spn(k)1Sp)0φ(ω)dω)lim supn+[θ((pn(k)1,p)0φ(ω)dω)]=[θ(ϖ(p,Sp)0φ(ω)dω)]

    which is contradiction. Hence, Sp=p.

    For the uniqueness, let q be one more fixed point of S, then (vi) yields α(p,q)sp. Using (3.1), one can arrive at

    θ(α(p,q)ϖ(Sp,Sq)0φ(ω)dω)[θ( (p,q)0φ(ω)dω)]

    where

    (p,q)=max{ϖ(p,q),ϖ(p,Sp),ϖ(q,Sq),ϖ(p,Sq)+ϖ(q,Sp)2s}=max{ϖ(p,q),0,0,ϖ(p,q)+ϖ(q,p)2s,0,0}=ϖ(p,q).

    So, we have

    θ(ϖ(p,q)0φ(ω)dω)<θ(s3ϖ(p,q)0φ(ω)dω)θ(α(p,q)ϖ(Sp,Sq)0φ(ω)dω)[θ((p,q)0φ(ω)dω)]=[θ(ϖ(p,q)0φ(ω)dω)]

    a contradiction. Thus, p=q, which proves the uniqueness of the fixed point. This completes the proof.

    Example 3.4. Let =[0,1] and ϖ(p,q)=(pq)2. It is easy to show that (,ϖ) is a b-metric space with parameter s=2. Define mappings S: by

    Sp={p4+1,  p[0,1),78, p=1

    and α:×[0,+) by

    α(p,q)=23,p,q.

    Define θ:[0,+)(1,+) and φ:[0,+)[0,+) by

    θ(ω)=e256ω+sinω and φ(ω)=2ω.

    It is easy to get that α(u,v)23, ϖ(Su,Sv)0φ(ω)dω>0 u,v[0,1] and uv. We consider the two following cases:

    Case 1. u,v[0,1). It follows that

    θ(α(u,v)ϖ(Su,Sv)0φ(ω)dω)=θ(23(u4+1+v41)202ωdω)=θ(14(uv)4)=e64(uv)4+sin(14(uv)4),
    [θ((u,v)0φ(ω)dω)]12[θ((uv)202ωdω)]12=[θ((uv)4]12=e128(uv)4+sin((uv)4)2.

    Case 2. u[0,1),v=1. One can deduce that

    θ(α(u,v)ϖ(Su,Sv)0φ(ω)dω)=θ(23(u4+178)202ωdω)=θ(14(u12)4)θ(14×16)=e4+sin164,
    [θ((u,v)0φ(ω)dω)]12[θ(ϖ(u,Sv)+ϖ(v,Su)2202ωdω)]12=[θ(14[(u78)2+u216]02ωdω)]12=[θ(141716[(u1417)2+(78)2(1417)2]02ωdω)]12[θ(141716[(78)2(1417)2]02ωdω)]12[θ(116)]12=e8+sin(116)2.

    Clearly, as =12, we have

    θ(α(u,v)ϖ(Su,Sv)0φ(ω)dω)[θ((u,v)0φ(ω)dω)].

    Hence, (3.1) holds. It follows that all conditions of Theorem 3.3 are satisfied with s=2 and p=3. Here, 45 is the fixed point of S.

    Remark 3.5. If (,ϖ) is a metric space and α(u,v)=1 in Theorem 3.3, then one can obtain Theorem 1.1 immediately.

    Theorem 3.6. Let (,ϖ) be a complete b-metric space with parameter s1 and S: be a given self-mapping. Assume that α:×[0,+) and p3. If

    (ⅰ) S is αsp-admissible,

    (ⅱ) there is p0 satisfying α(p0,Sp0)sp,

    (ⅲ) α satisfies transitive property, i.e., for ξ,η,ζ if

    α(ξ,η)sp and α(η,ζ)spα(ξ,ζ)sp,

    (ⅳ) if {pι} is a sequence in satisfying pιp as ι+, then there is a subsequence {pι(k)}+k=1 of {pι}+ι=1 with α(pι(k),p)sp,

    (ⅴ) S is a θ-ψ-weak contraction, that is, there exists φ, θΘ2 such that: for any u,vφ

    α(u,v)sp,ϖ(Su,Sv)0φ(ω)dω>0ψ(α(u,v)ϖ(Su,Sv)0φ(ω)dω)θ(ψ((u,v)0φ(ω)dω))ψ((u,v)0φ(ω)dω), (3.20)

    where ψ:[0,+)[0,+) is a continuous and increasing function with ψ(ω)=0 if, and only if, ω=0,

    then S has a fixed point in . Moreover, if

    (ⅵ) for p,qFix(S), one can get the conditions of α(p,q)sp and α(q,p)sp, where Fix(S) represents the collection of all fixed points of S,

    then the fixed point of S is unique.

    Proof. As in the proof of Theorem 3.3, we infer α(pn1,pn)sp. Using (3.16) with u=pn1 and v=pn, one can deduce that

    ψ(α(pn1,pn)ϖ(Spn1,Spn)0φ(ω)dω)θ(ψ((pn1,pn)0φ(ω)dω))ψ((pn1,pn)0φ(ω)dω) (3.21)

    where

    (pn1,pn)=max{ϖ(pn1,pn),ϖ(pn1,Spn1),ϖ(pn,Spn),ϖ(pn1,Spn)+ϖ(pn,Spn1)2s}=max{ϖ(pn1,pn),ϖ(pn1,pn),ϖ(pn,pn+1),ϖ(pn1,pn+1)+ϖ(pn,pn)2s}=max{ϖ(pn1,pn),ϖ(pn,pn+1)}. (3.22)

    If ϖ(pn,pn+1)ϖ(pn1,pn) for some nN, according to (3.22), one can obtain (pn1,pn)=ϖ(pn,pn+1). It follows that

    ψ(ϖ(pn,pn+1)0φ(ω)dω)ψ(α(pn1,pn)ϖ(Spn1,Spn)0φ(ω)dω)θ(ψ((pn1,pn)0φ(ω)dω))ψ((pn1,pn)0φ(ω)dω)=θ(ψ(ϖ(pn,pn+1)0φ(ω)dω))ψ(ϖ(pn,pn+1)0φ(ω)dω)

    which is a contradiction. Thus,

    ϖ(pn1,pn)>ϖ(pn,pn+1). (3.23)

    By (3.23), we get that (pn1,pn)=ϖ(pn1,pn) is a decreasing sequence. Hence, there exists ρ0 such that ϖ(pn1,pn)=ρ. If ρ>0, then

    ψ(ϖ(pn,pn+1)0φ(ω)dω)ψ(ϖ(pn1,pn)0φ(ω)dω)θ(ψ(ϖ(pn1,pn)0φ(ω)dω)).

    Taking n+, we obtain

    1limn+θ(ψ(ϖ(pn1,pn)0φ(ω)dω))1

    which implies limn+θ(ψ(ϖ(pn1,pn)0φ(ω)dω))=1. In view of the definition of θ and ψ, one can deduce that

    limn+ϖ(pn1,pn)0φ(ω)dω=0

    i.e.,

    limn+ϖ(pn,pn+1)=0,

    which is contradiction. It follows that limn+ϖ(pn,pn+1)=0.

    Next, we want to show {pn} is a Cauchy sequence. As in the proof of Theorem 3.3, we obtain that (3.13)–(3.17) hold. The transitivity property of α implies that α(pm(k)1,pn(k)1)sp. Putting u=pm(k)1 and v=pn(k)1 into (3.20), we get

    ψ(s3ε0φ(ω)dω)lim infk+ψ(spϖ(pm(k),pn(k))0φ(ω)dω)lim infk+ψ(α(pmk1,pnk1)ϖ(Spmk1,Spnk1)0φ(ω)dω)lim infk+[θ(ψ((pmk1,pnk1)0φ(ω)dω))ψ((pmk1,pnk1)0φ(ω)dω)]lim supk+θ(ψ((pmk1,pnk1)0φ(ω)dω)lim infk+ψ((pmk1,pnk1)0φ(ω)dω)=θ(lim infk+ψ((pmk1,pnk1)0φ(ω)dω))ψ(lim infk+(pmk1,pnk1)0φ(ω)dω)<ψ(s3ε0φ(ω)dω)

    which is a contradiction. Hence, {pn} is Cauchy. The completeness of ensures that there exists p such that {pn}p as n+.

    Next, we prove the point p to be a fixed point of S. Similar to the discussion related to Theorem 3.4, taking u=pn(k)1 and v=p in (3.20), we get

    ψ(α(pn(k)1,p)ϖ(Spn(k)1,Sp)0φ(ω)dω)θ(ψ((pn(k)1,p)0φ(ω)dω))ψ((pn(k)1,p)0φ(ω)dω) (3.24)

    where

    (pn(k)1,p)=max{ϖ(pn(k)1,p),ϖ(pn(k)1,Spn(k)1),ϖ(p,Sp),ϖ(pn(k)1,Sp)+ϖ(p,Spn(k)1)2s}=max{ϖ(pn(k)1,p),ϖ(pn(k)1,pn(k)),ϖ(p,Sp),ϖ(pn(k)1,Sp)+ϖ(p,pn(k))2s}. (3.25)

    Taking the limit as n+ in (3.25), we get

    limn+(pn(k)1,p)=max{0,0,ϖ(p,Sp),ϖ(p,Sp)2}=ϖ(p,Sp). (3.26)

    According to (3.24), (3.26), and Lemma 2.5, we get

    ψ(ϖ(p,Sp)0φ(t)dt)ψ(s31sϖ(p,Sp)0φ(t)dt)limn+ψ(α(pn(k)1,p)ϖ(Spn(k)1,Sp)0φ(ω)dω)limn+θ(ψ((pn(k)1,p)0φ(ω)dω))ψ((pn(k)1,p)0φ(ω)dω)=θ(ψ(ϖ(p,Sp)0φ(ω)dω))ψ(ϖ(p,Sp)0φ(ω)dω)<ψ(ϖ(p,Sp)0φ(ω)dω)

    which is impossible. It follows that Sp=p.

    At last, we show the uniqueness of the fixed point of S. Suppose q is another fixed point of S. It follows from the condition (ⅳ) that α(p,q)sp. In light of (3.20), one can get

    ψ(α(p,q)ϖ(Sp,Sq)0φ(ω)dω)θ(ψ((p,q)0φ(ω)dω))ψ((p,q)0φ(ω)dω),
    (p,q)=max{ϖ(p,q),ϖ(p,Sp),ϖ(q,Sq),ϖ(p,Sq)+ϖ(q,Sp)2s}=max{ϖ(p,q),0,0,ϖ(p,q)+ϖ(q,p)2s}=ϖ(p,q).

    Then

    ψ(ϖ(p,q)0φ(ω)dω)ψ(α(p,q)ϖ(Sp,Sq)0φ(ω)dω)θ(ψ((p,q)0φ(ω)dω))ψ((p,q)0φ(ω)dω)<ψ(ϖ(p,q)0φ(ω)dω)

    a contradiction, which implies that p=q. This completes the proof.

    Example 3.7. Let =[0,1] and ϖ(p,q)=(pq)2. Define mappings S: by

    Sp={p3216e,  p[0,12],13216e,p(12,1]

    and α:×[0,+) by

    α(p,q)=24,p,q[0,1].

    Define θ:[0,+)(0,1) and ψ,φ:[0,+)[0,+) by

    θ(ω)=e4ω,ψ(ω)=ω  and  φ(ω)=2ω.

    One can deduce that α(u,v)24, ϖ(Su,Sv)0φ(ω)dω>0 u,v[0,1] with uv. It follows that we also consider two cases:

    Case 1. u,v[0,12], then

    ψ(α(u,v)ϖ(Su,Sv)0φ(ω)dω)=24(u3216ev3216e)202ωdω=1642×4e(uv)4,
    θ(ψ((u,v)0φ(ω)dω))ψ((u,v)0φ(ω)dω)=e4(u,v)02ωdω(u,v)02ωdω14e(uv)4.

    Case 2. u[0,12],v(12,1]. It is easy to obtain that

    ψ(α(u,v)ϖ(Su,Sv)0φ(ω)dω)=24(u3216e13216e)202ωdω=1642×4e(u1)41642×4e,
    θ(ψ((u,v)0φ(ω)dω))ψ((u,v)0φ(ω)dω)=e4(u,v)02ωdω(u,v)02ωdω1e4×116×(113216e)41642×4e.

    That is,

    ψ(α(u,v)ϖ(Su,Sv)0φ(ω)dω)θ(ψ((u,v)0φ(ω)dω))ψ((u,v)0φ(ω)dω).

    It follows that all conditions of Theorem 3.6 are satisfied with s=2 and p=4. It is easy to get that 0 is the unique fixed point of S.

    In this section, by using the fixed point theorems obtained in Section 3, we study the existence of solutions of the following functional Eq (4.2).

    Let O and P be two Banach spaces and SO and DP be the state and decision spaces. B(S) denotes the Banach space of all bounded real-valued functions on S with norm

    m∥=sup{|m(ξ)|:ξS} for any  mB(S). (4.1)

    Bellman [20] was the first to investigate the existence and uniqueness of solutions for the following functional equations arising in dynamic programming:

    f(x)=infyDmax{r(x,y),s(x,y),f(b(x,y))},
    f(x)=infyDmax{r(x,y),f(b(x,y))}

    in a complete metric space BB(S). As suggested in Bellman and Lee [21], the basic form of the functional equations in dynamic programming is as follows:

    f(x)=optyD{H(x,y,f(T(x,y)))},xS

    where the opt represents sup or inf. Bhakta and Mitra [22] obtained the existence and uniqueness of solutions for the functional equations

    f(x)=supyD{p(x,y)+A(x,y,f(a(x,y))}

    in a Banach space B(S) and

    f(x)=supyD{p(x,y)+f(a(x,y))}

    in BB(S), respectively. After that, many authors established the existence and uniqueness of solutions or common solutions for several classes of functional equations or systems of functional equations arising in dynamic programming by means of various fixed and common fixed point theorems (see [23,24,25]).

    It is easy to get that (B(S),ϖ) is a complete b-metric space with

    ϖ(ξ,η)=∥ξη2,ξ,ηB(S).

    Consider the functional equations arising in dynamic programming:

    f(x)=infyD{u(x,y)+H(x,y,f(T(x,y)))},xS (4.2)

    where u:S×DR, T:S×DS and H:S×D×RR are mappings. Let

    Sf(x)=infyD{u(x,y)+H(x,y,f(T(x,y)))},(x,f)S×B(S). (4.3)

    Theorem 4.1. Let u:S×DR, T:S×DS, H:S×D×RR, S:B(S)B(S), α:B(S)×B(S)R. If

    (ⅰ) u and H are bounded,

    (ⅱ) S is αsp-admissible,

    (ⅲ) there is p0B(S) satisfying α(p0,Sp0)sp,

    (ⅳ) α satisfies transitive property, i.e., for ξ,η,ζB(S) such that

    α(ξ,η)sp and α(η,ζ)spα(ξ,ζ)sp,

    (ⅴ) if {pn} is a sequence in B(S) satisfying pnp as n+, then there is a subsequence {pn(k)} of {pn} with α(pn(k),p)sp,

    (ⅵ) for p,qFix(S), one can get the condition of α(p,q)sp and α(q,p)sp, where Fix(S) represents the collection of all fixed points of S,

    (ⅶ) if there exists (0,1), φ such that

    α(u,v)sp,SuSv20φ(ω)dω>0exp(2α(u,v)|H(u,v,g(T(u,v)))H(u,v,h(T(u,v)))|20φ(ω)dω)[exp( (u,v)0φ(ω)dω)], (4.4)

    then the functional Eq (4.2) has a unique solution pB(S).

    Proof. It follows from (i) that there exists M>0 satisfying

    sup{|u(x,y)|,|H(x,y,t)|:(x,y,t)S×D×R}M.

    It is easy to see that S is a self-mapping in B(S). Define α:B(S)×B(S)[0,) by

    α(u,v)={sp,  ϖ(Su,Sv)>0,0,  otherwise.

    By (i) and φ, we have for each ε>0, there exists δ>0 such that

    Cφ(t)dt<ε,C[0,2M] with m(C)δ, (4.5)

    where m(C) denotes the Lebesgue measure of C.

    Let uS,h,gB(S). By (4.3), there exists v,wD satisfying

    Sg(u)>u(u,v)+H(u,v,g(T(u,v)))2δ2,
    Sh(u)>u(u,w)+H(u,w,h(T(u,w)))2δ2,
    Sg(u)u(u,w)+H(u,w,g(T(u,w))),
    Sh(u)u(u,v)+H(u,v,h(T(u,v))).

    Thus,

    Sg(u)Sh(u)<H(u,w,g(T(u,w)))H(u,w,h(T(u,w)))+2δ2|H(u,w,g(T(u,w)))H(u,w,h(T(u,w)))|+2δ2,
    Sh(u)Sg(u)<H(u,v,h(T(u,v)))H(u,v,g(T(u,v)))+2δ2|H(u,v,h(T(u,v)))H(u,v,g(T(u,v)))|+2δ2.

    It follows that

    ||SgSh||=supuS|Sg(u)Sh(u)|max{T1,T2}+2δ2, (4.6)

    where

    T1=|H(u,w,g(T(u,w)))H(u,w,h(T(u,w)))|,
    T2=|H(u,v,h(T(u,v)))H(u,v,g(T(u,v)))|.

    It is easy to get that ||SgSh||2max{2T12,2T22}+δ. Under (4.4) and (4.6), we have

    exp(sp||Sg(u)Sh(u)||20φ(ω)dω)exp(spmax{2T21,2T22}+δ0φ(ω)dω)=max{exp(2spT21+δ0φ(ω)dω),exp(2spT22+δ0φ(ω)dω)}=max{exp(2spT210φ(ω)dω)exp(2spT21+δ2spT21φ(ω)dω),exp(2spT220φ(ω)dω)exp(2spT22+δ2spT22φ(ω)dω)}max{exp(2spT210φ(ω)dω),exp(2spT220φ(ω)dω)}max{exp(2spT21+δ2spT21φ(ω)dω),exp(2spT22+δ2spT22φ(ω)dω)}[exp( (u,v)0φ(ω)dω)]exp(ε).

    Letting ε0+ in the above inequality, we get

    exp(α(u,v)SgSh20φ(ω)dω)[exp( (u,v)0φ(ω)dω)].

    Thus, the conditions of Theorem 3.3 are satisfied by taking θ(ω)=exp(ω), so the functional Eq (4.2) has a unique fixed sloution pB(S). This completes the proof.

    In this manuscript, we first defined two new types of weak contractions named θ-weak contraction and θ-ψ-weak contraction. Second, we presented the conditions of existence and uniqueness of fixed points for them in b-metric spaces. After that, two examples were given to demonstrate the practicability of our theorems. As an application, the existence and uniqueness of solutions for a class of functional equations arising in dynamic programming were discussed.

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    This work was financially supported by the Science and Research Project Foundation of Liaoning Province Education Department (No: JYTMS20231700).

    The authors declare that they have no conflicts of interest regarding the publication of this paper.



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