Our main interest in this manuscript is to explore the main positive solutions (PS) and the first implications of their existence and uniqueness for a type of fractional pantograph differential equation using Caputo fractional derivatives with a kernel depending on a strictly increasing function Ψ (shortly Ψ-Caputo). Such function-dependent kernel fractional operators unify and generalize several types of fractional operators such as Riemann-Liouvile, Caputo and Hadamard etc. Hence, our investigated qualitative concepts in this work generalise and unify several existing results in literature. Using Schauder's fixed point theorem (SFPT), we prove the existence of PS to this equation with the addition of the upper and lower solution method (ULS). Furthermore using the Banach fixed point theorem (BFPT), we are able to prove the existence of a unique PS. Finally, we conclude our work and give a numerical example to explain our theoretical results.
Citation: Ridha Dida, Hamid Boulares, Bahaaeldin Abdalla, Manar A. Alqudah, Thabet Abdeljawad. On positive solutions of fractional pantograph equations within function-dependent kernel Caputo derivatives[J]. AIMS Mathematics, 2023, 8(10): 23032-23045. doi: 10.3934/math.20231172
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Our main interest in this manuscript is to explore the main positive solutions (PS) and the first implications of their existence and uniqueness for a type of fractional pantograph differential equation using Caputo fractional derivatives with a kernel depending on a strictly increasing function Ψ (shortly Ψ-Caputo). Such function-dependent kernel fractional operators unify and generalize several types of fractional operators such as Riemann-Liouvile, Caputo and Hadamard etc. Hence, our investigated qualitative concepts in this work generalise and unify several existing results in literature. Using Schauder's fixed point theorem (SFPT), we prove the existence of PS to this equation with the addition of the upper and lower solution method (ULS). Furthermore using the Banach fixed point theorem (BFPT), we are able to prove the existence of a unique PS. Finally, we conclude our work and give a numerical example to explain our theoretical results.
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,ℏ) is a complete metric space and S:Ω→Ω is a θ∗-weak contraction. If θ is continuous, then
(a) S has unique fixed point (say, z∗∈Ω),
(b)limn→+∞Snz=z∗, ∀z∈Ω.
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 ℵ be a nonempty set and s≥1 be a given real number. A mapping ϖ:ℵ×ℵ→[0,+∞) is said to be a b-metric if, and only if, for all κ,λ,μ∈ℵ, 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, (ℵ,ϖ) is called a b-metric space with parameter s≥1.
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 (ℵ,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=2p−1.
Definition 2.4. [18] Let (ℵ,ϖ) be a b-metric space with parameter s≥1, then a sequence {κι}+∞ι=1 in ℵ is said to be:
(ⅰ) b-convergent if there exists κ∈ℵ 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 (ℵ,ϖ) be a b-metric space with parameter s≥1. Assume that {κι}+∞ι=1⊂ℵ and {λι}+∞ι=1⊂ℵ 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∈ℵ, we have
1sϖ(κ,μ)≤lim infι→+∞ϖ(κι,μ)≤lim supι→+∞ϖ(κι,μ)≤sϖ(κ,μ). |
Lemma 2.6. [19] Let φ∈ℑ and {κι}ι∈N be a nonnegative sequence with limn→+∞κι=κ, then
limn→+∞∫κι0φ(ω)dω=∫κ0φ(ω)dω. |
Lemma 2.7. [19] Let φ∈ℑ and {κι}ι∈N 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,
ℑ={ξ|ξ:R+→R+ satisfies that ξ is Lebesgue integrable, and ∫δ0ξ(ω)dω>0 for each δ>0}. |
Let (ℵ,ϖ) be a b-metric space with parameter s≥1 and S be a self-mapping on ℵ. For any u,v∈ℵ, 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)=1⇒limι→+∞βι=0.
Definition 3.1. Let (ℵ,ϖ) be a b-metric space with parameter s≥1 and p≥1 be an integer. A mapping S:ℵ→ℵ is said to be αsp−admissible if for all z,w∈ℵ, one has
α(z,w)≥sp⇒α(Sz,Sw)≥sp |
where α:ℵ×ℵ→[0,+∞) is a given function.
Lemma 3.2. Let φ∈ℑ and {κι}ι∈N be a nonnegative sequence. If lim supι→+∞κι=κ, then
∫κ0φ(ω)dω≤lim supι→+∞∫κι0φ(ω)dω. |
If lim infι→+∞κι=κ, then
lim infι→+∞∫κι0φ(ω)dω≤∫κ0φ(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)dω≤lim supι→+∞∫κι0φ(ω)dω. |
Similarly, one can prove another inequality.
Theorem 3.3. Let (ℵ,ϖ) be a complete b-metric space with parameter s≥1 and S:ℵ→ℵ be a given self-mapping. Assume that α:ℵ×ℵ→[0,+∞) and p≥3. If
(ⅰ) S is αsp-admissible,
(ⅱ) there is p0∈ℵ satisfying α(p0,Sp0)≥sp,
(ⅲ) α satisfies transitive property, i.e., for ξ,η,ζ∈ℵ if
α(ξ,η)≥sp and α(η,ζ)≥sp⇒α(ξ,ζ)≥sp, |
(ⅳ) if {pι} is a sequence in ℵ satisfying pι→p 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 ℓ∈(0,1), φ∈ℑ, θ∈Θ1 such that: for any u,v∈ℵ,
α(u,v)≥sp,∫ϖ(Su,Sv)0φ(ω)dω>0⇒θ(∫α(u,v)ϖ(Su,Sv)0φ(ω)dω)≤[θ( ∫△(u,v)0φ(ω)dω)]ℓ, | (3.1) |
then S has a fixed point in ℵ. Furthermore, if
(ⅵ) for p,q∈Fix(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∈ℵ satisfying α(p0,Sp0)≥sp. Define sequence {pn} in ℵ by pn+1=Spn for n∈N. If pn0=Spn0 for some n0, then pn0 is a fixed point of S. Suppose that pn+1≠pn for n∈N. It follows from condition (ⅰ) that
α(p0,Sp0)≥sp⇒α(Sp0,S2p0)≥sp, |
α(p1,p2)≥sp⇒α(Sp1,Sp2)≥sp, |
α(p2,p3)≥sp⇒α(Sp2,Sp3)≥sp, |
……… |
Thus, for all n∈N, we have α(pn−1,pn)≥sp. Using (3.1) by u=pn−1 and v=pn, one gets
θ(∫α(pn−1,pn)ϖ(Spn−1,Spn)0φ(ω)dω)≤[θ( ∫△(pn−1,pn)0φ(ω)dω)]ℓ | (3.2) |
where
△(pn−1,pn)=max{ϖ(pn−1,pn),ϖ(pn−1,Spn−1),ϖ(pn,Spn),ϖ(pn−1,Spn)+ϖ(pn,Spn−1)2s}=max{ϖ(pn−1,pn),ϖ(pn−1,pn),ϖ(pn,pn+1),ϖ(pn−1,pn+1)+ϖ(pn,pn)2s}=max{ϖ(pn−1,pn),ϖ(pn,pn+1)}. | (3.3) |
If ϖ(pn,pn+1)≥ϖ(pn−1,pn) for some n∈N, in view of (3.2) and (3.3), we have △(pn−1,pn)=ϖ(pn,pn+1), so
θ(∫ϖ(pn,pn+1)0φ(ω)dω)<θ(∫spϖ(pn,pn+1)0φ(ω)dω)≤θ(∫α(pn−1,pn)ϖ(Spn−1,Spn)0φ(ω)dω)≤[θ(∫△(pn−1,pn)0φ(ω)dω)]ℓ=[θ(∫ϖ(pn,pn+1)0φ(ω)dω)]ℓ |
which is impossible. Hence,
ϖ(pn−1,pn)>ϖ(pn,pn+1). | (3.4) |
(3.4) implies that △(pn−1,pn)=ϖ(pn−1,pn) is decreasing. Thus, we have
θ(∫ϖ(pn,pn+1)0φ(ω)dω)<[θ(∫ϖ(pn−1,pn)0φ(ω)dω)]ℓ<[θ(∫ϖ(pn−2,pn−1)0φ(ω)dω)]ℓ2<⋯<[θ(∫ϖ(p0,p1)0φ(ω)dω)]ℓn. |
Letting n→+∞ in the above inequality, we get
1≤limn→+∞θ(∫ϖ(pn,pn+1)0φ(ω)dω)≤limn→+∞[θ(∫ϖ(p0,p1)0φ(ω)dω)]ℓn=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,
ε≤ϖ(pm(k),pn(k)),ϖ(pm(k),pn(k)−1)<ε. | (3.5) |
Under the triangle inequality and (3.5), we get
ε≤ϖ(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
ε≤lim 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
εs2≤lim 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
εs≤lim 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
εs≤lim infk→+∞ϖ(pm(k),pn(k)−1)≤lim supk→+∞ϖ(pm(k),pn(k)−1)≤s2ε. | (3.15) |
Thus, there exists N∈N0 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)]ℓ≤[θ(∫s3ε0φ(ω)dω)]ℓ |
which is a contradiction. So, {pn} is Cauchy. As ℵ is complete, there exists p∗∈ℵ such that pn→p∗ as n→+∞.
Next, we prove the point p∗ to be a fixed point of S. So, we think about a set, say Q={n∈N0: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 N∗∈N such that ∫ϖ(pn,Sp∗)0φ(ω)dω>0 for any n≥N∗. 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, ∀k≥N∗. 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)]ℓ | (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(∫s3⋅1sϖ(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)]ℓ=[θ(∫ϖ(p∗,Sp∗)0φ(ω)dω)]ℓ |
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)]ℓ |
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(∫s3⋅ϖ(p∗,q∗)0φ(ω)dω)≤θ(∫α(p∗,q∗)ϖ(Sp∗,Sq∗)0φ(ω)dω)≤[θ(∫△(p∗,q∗)0φ(ω)dω)]ℓ=[θ(∫ϖ(p∗,q∗)0φ(ω)dω)]ℓ |
a contradiction. Thus, \mathfrak{p}^* = \mathfrak{q}^* , which proves the uniqueness of the fixed point. This completes the proof.
Example 3.4. Let \aleph = [0, 1] and \varpi(\mathfrak{p}, \mathfrak{q}) = (\mathfrak{p}-\mathfrak{q})^2. It is easy to show that (\aleph, \varpi) is a b -metric space with parameter s = 2 . Define mappings \mathbb{S}:\aleph\rightarrow \aleph by
\mathbb{S} \mathfrak{p} = \left\{\begin{aligned} &-\frac{\mathfrak{p}}{4}+1, \ \ \mathfrak{p}\in [0, 1), \\ &\frac{7}{8}, \quad \quad \quad \ \mathfrak{p} = 1 \end{aligned}\right.\\ |
and \alpha:\aleph \times \aleph\rightarrow [0, +\infty) by
\alpha(\mathfrak{p}, \mathfrak{q}) = 2^3, \forall\mathfrak{p}, \mathfrak{q}\in \aleph. |
Define \theta:[0, +\infty)\rightarrow (1, +\infty) and \varphi:[0, +\infty)\rightarrow [0, +\infty) by
\theta(\omega) = e^{256\omega+\sin \omega} \text{ and } \varphi(\omega) = 2\omega. |
It is easy to get that \alpha(\mathfrak{u}, \mathfrak{v})\geq 2^3 , \int_0^{\varpi(\mathbb{S}\mathfrak{u}, \mathbb{S}\mathfrak{v})}\varphi(\omega)d\omega > 0 \Leftrightarrow \mathfrak{u}, \mathfrak{v}\in [0, 1] and \mathfrak{u}\neq \mathfrak{v} . We consider the two following cases:
Case 1. \mathfrak{u}, \mathfrak{v}\in [0, 1) . It follows that
\begin{align*} \theta(\int_{0}^{\alpha(\mathfrak{u}, \mathfrak{v})\varpi(\mathbb{S}\mathfrak{u}, \mathbb{S}\mathfrak{v})}\varphi(\omega)\, d\omega) = &\theta(\int_{0}^{2^3(-\frac{\mathfrak{u}}{4}+1+\frac{\mathfrak{v}}{4}-1)^2}2\omega\, d\omega)\\ = &\theta(\frac{1}{4}(\mathfrak{u}-\mathfrak{v})^4)\\ = &e^{64(\mathfrak{u}-\mathfrak{v})^4+\sin(\frac{1}{4}(\mathfrak{u}-\mathfrak{v})^4)}, \end{align*} |
\begin{align*} [\theta(\int_{0}^{\triangle(\mathfrak{u}, \mathfrak{v})}\varphi(\omega)\, d\omega)]^\frac{1}{2} \geq &[\theta(\int_{0}^{(\mathfrak{u}-\mathfrak{v})^2}2\omega\, d\omega)]^\frac{1}{2}\\ = &[\theta((\mathfrak{u}-\mathfrak{v})^4]^\frac{1}{2}\\ = &e^{128(\mathfrak{u}-\mathfrak{v})^4+\frac{\sin((\mathfrak{u}-\mathfrak{v})^4)}{2}}. \end{align*} |
Case 2. \mathfrak{u}\in [0, 1), \mathfrak{v} = 1 . One can deduce that
\begin{align*} \theta(\int_{0}^{\alpha(\mathfrak{u}, \mathfrak{v})\varpi(\mathbb{S}\mathfrak{u}, \mathbb{S}\mathfrak{v})}\varphi(\omega)\, d\omega) = &\theta(\int_{0}^{2^3(-\frac{\mathfrak{u}}{4}+1-\frac{7}{8})^2}2\omega\, d\omega)\\ = &\theta(\frac{1}{4}(\mathfrak{u}-\frac{1}{2})^4)\\ \leq&\theta(\frac{1}{4\times 16})\\ = &e^{4+\sin\frac{1}{64}}, \\ \end{align*} |
\begin{align*} [\theta(\int_{0}^{\triangle(\mathfrak{u}, \mathfrak{v})}\varphi(\omega)\, d\omega)]^\frac{1}{2} \geq&[\theta(\int_{0}^{\frac{\varpi(\mathfrak{u}, \mathbb{S}\mathfrak{v})+\varpi(\mathfrak{v}, \mathbb{S}\mathfrak{u})}{2\cdot 2}}2\omega\, d\omega)]^\frac{1}{2}\\ = &[\theta(\int_{0}^{\frac{1}{4}[(\mathfrak{u}-\frac{7}{8})^2+\frac{\mathfrak{u}^2}{16}]}2\omega\, d\omega)]^\frac{1}{2}\\ = &[\theta(\int_{0}^{\frac{1}{4}\cdot \frac{17}{16}[(\mathfrak{u}-\frac{14}{17})^2+(\frac{7}{8})^2-(\frac{14}{17})^2]}2\omega\, d\omega)]^\frac{1}{2}\\ \geq&[\theta(\int_{0}^{\frac{1}{4}\cdot \frac{17}{16}[(\frac{7}{8})^2-(\frac{14}{17})^2]}2\omega\, d\omega)]^\frac{1}{2}\\ \geq&[\theta(\frac{1}{16})]^\frac{1}{2}\\ = &e^{8+\frac{\sin(\frac{1}{16})}{2}}. \end{align*} |
Clearly, as \ell = \frac{1}{2} , we have
\theta(\int_{0}^{\alpha(\mathfrak{u}, \mathfrak{v})\varpi(\mathbb{S}\mathfrak{u}, \mathbb{S}\mathfrak{v})}\varphi(\omega)\, d\omega) \leq[\theta(\int_{0}^{\triangle(\mathfrak{u}, \mathfrak{v})}\varphi(\omega)\, d\omega)]^\ell. |
Hence, (3.1) holds. It follows that all conditions of Theorem 3.3 are satisfied with s = 2 and p = 3 . Here, \frac{4}{5} is the fixed point of \mathbb{S} .
Remark 3.5. If (\aleph, \varpi) is a metric space and \alpha(\mathfrak{u}, \mathfrak{v}) = 1 in Theorem 3.3, then one can obtain Theorem 1.1 immediately.
Theorem 3.6. Let (\aleph, \varpi) be a complete b -metric space with parameter s\geq 1 and \mathbb{S}:\aleph\rightarrow \aleph be a given self-mapping. Assume that \alpha:\aleph \times \aleph \rightarrow [0, +\infty) and p\geq3 . If
(ⅰ) \mathbb{S} is \alpha_{s^p} -admissible,
(ⅱ) there is \mathfrak{p}_0\in \aleph satisfying \alpha(\mathfrak{p}_0, \mathbb{S} \mathfrak{p}_0)\geq s^p ,
(ⅲ) \alpha satisfies transitive property, i.e., for \xi, \eta, \zeta\in \aleph if
\alpha(\xi, \eta)\geq s^p \text{ and } \alpha(\eta, \zeta)\geq s^p\Rightarrow \alpha(\xi, \zeta)\geq s^p, |
(ⅳ) if \{\mathfrak{p}_\iota\} is a sequence in \aleph satisfying \mathfrak{p}_\iota\rightarrow \mathfrak{p} as \iota\rightarrow +\infty , then there is a subsequence \{\mathfrak{p}_{\iota(k)}\}_{k = 1}^{+\infty} of \{\mathfrak{p}_{\iota}\}_{\iota = 1}^{+\infty} with \alpha(\mathfrak{p}_{\iota(k)}, \mathfrak{p})\geq s^p ,
(ⅴ) \mathbb{S} is a \theta - \psi -weak contraction, that is, there exists \varphi\in \Im , \theta\in \Theta_2 such that: for any \mathfrak{u}, \mathfrak{v} \in \varphi
\begin{align} \alpha(\mathfrak{u}, \mathfrak{v})\geq s^p, &\int_{0}^{\varpi(\mathbb{S}\mathfrak{u}, \mathbb{S}\mathfrak{v})}\varphi(\omega)\, d\omega > 0\\ \Rightarrow \psi(\int_{0}^{\alpha(\mathfrak{u}, \mathfrak{v})\varpi(\mathbb{S} \mathfrak{u}, \mathbb{S} \mathfrak{v})}\varphi(\omega)\, d\omega) \leq&\theta(\psi(\int_{0}^{\triangle(\mathfrak{u}, \mathfrak{v})}\varphi(\omega)\, d\omega)) \psi(\int_{0}^{\triangle(\mathfrak{u}, \mathfrak{v})}\varphi(\omega)\, d\omega), \end{align} | (3.20) |
where \psi:[0, +\infty)\rightarrow [0, +\infty) is a continuous and increasing function with \psi(\omega) = 0 if, and only if, \omega = 0 ,
then \mathbb{S} has a fixed point in \aleph . Moreover, if
(ⅵ) for \mathfrak{p}, \mathfrak{q}\in Fix(\mathbb{S}) , one can get the conditions of \alpha(\mathfrak{p}, \mathfrak{q})\geq s^p and \alpha(\mathfrak{q}, \mathfrak{p})\geq s^p , where Fix(\mathbb{S} ) represents the collection of all fixed points of \mathbb{S} ,
then the fixed point of \mathbb{S} is unique.
Proof. As in the proof of Theorem 3.3, we infer \alpha(\mathfrak{p}_{n-1}, \mathfrak{p}_n)\geq s^p . Using (3.16) with \mathfrak{u} = \mathfrak{p}_{n-1} and \mathfrak{v} = \mathfrak{p}_{n} , one can deduce that
\begin{align} \psi(\int_{0}^{\alpha(\mathfrak{p}_{n-1}, \mathfrak{p}_n)\varpi(\mathbb{S} \mathfrak{p}_{n-1}, \mathbb{S} \mathfrak{p}_n)}\varphi(\omega)\, d\omega) \leq&\theta(\psi(\int_{0}^{\triangle(\mathfrak{p}_{n-1}, \mathfrak{p}_n)}\varphi(\omega)\, d\omega)) \psi(\int_{0}^{\triangle(\mathfrak{p}_{n-1}, \mathfrak{p}_n)}\varphi(\omega)\, d\omega) \end{align} | (3.21) |
where
\begin{align} \triangle(\mathfrak{p}_{n-1}, \mathfrak{p}_{n}) = \max&\{\varpi(\mathfrak{p}_{n-1}, \mathfrak{p}_{n}), \varpi(\mathfrak{p}_{n-1}, \mathbb{S}\mathfrak{p}_{n-1}), \varpi(\mathfrak{p}_{n}, \mathbb{S} \mathfrak{p}_{n}), \frac{\varpi(\mathfrak{p}_{n-1}, \mathbb{S}\mathfrak{p}_{n})+\varpi(\mathfrak{p}_{n}, \mathbb{S} \mathfrak{p}_{n-1})}{2s}\}\\ = \max&\{\varpi(\mathfrak{p}_{n-1}, \mathfrak{p}_n), \varpi(\mathfrak{p}_{n-1}, \mathfrak{p}_{n}), \varpi(\mathfrak{p}_{n}, \mathfrak{p}_{n+1}), \frac{\varpi(\mathfrak{p}_{n-1}, \mathfrak{p}_{n+1})+\varpi(\mathfrak{p}_n, \mathfrak{p}_n)}{2s}\}\\ = \max&\{\varpi(\mathfrak{p}_{n-1}, \mathfrak{p}_n), \varpi(\mathfrak{p}_{n}, \mathfrak{p}_{n+1})\}. \end{align} | (3.22) |
If \varpi(\mathfrak{p}_{n}, \mathfrak{p}_{n+1})\geq \varpi(\mathfrak{p}_{n-1}, \mathfrak{p}_n) for some n\in \mathbb{N} , according to (3.22), one can obtain \triangle(\mathfrak{p}_{n-1}, \mathfrak{p}_{n}) = \varpi(\mathfrak{p}_{n}, \mathfrak{p}_{n+1}) . It follows that
\begin{align} \psi(&\int_{0}^{\varpi(\mathfrak{p}_{n}, \mathfrak{p}_{n+1})}\varphi(\omega)\, d\omega)\\ \leq&\psi(\int_{0}^{\alpha(\mathfrak{p}_{n-1}, \mathfrak{p}_{n})\varpi(\mathbb{S} \mathfrak{p}_{n-1}, \mathbb{S}\mathfrak{p}_n)}\varphi(\omega)\, d\omega)\\ \leq&\theta(\psi(\int_{0}^{\triangle(\mathfrak{p}_{n-1}, \mathfrak{p}_n)}\varphi(\omega)\, d\omega)) \psi(\int_{0}^{\triangle(\mathfrak{p}_{n-1}, \mathfrak{p}_n)}\varphi(\omega)\, d\omega)\\ = &\theta(\psi(\int_{0}^{\varpi(\mathfrak{p}_{n}, \mathfrak{p}_{n+1})}\varphi(\omega)\, d\omega)) \psi(\int_{0}^{\varpi(\mathfrak{p}_{n}, \mathfrak{p}_{n+1})}\varphi(\omega)\, d\omega) \nonumber \end{align} |
which is a contradiction. Thus,
\begin{align} \varpi(\mathfrak{p}_{n-1}, \mathfrak{p}_{n}) > \varpi(\mathfrak{p}_{n}, \mathfrak{p}_{n+1}). \end{align} | (3.23) |
By (3.23), we get that \triangle(\mathfrak{p}_{n-1}, \mathfrak{p}_{n}) = \varpi(\mathfrak{p}_{n-1}, \mathfrak{p}_{n}) is a decreasing sequence. Hence, there exists \rho\geq 0 such that \varpi(\mathfrak{p}_{n-1}, \mathfrak{p}_{n}) = \rho. If \rho > 0, then
\begin{align*} \frac{\psi(\int_{0}^{\varpi(\mathfrak{p}_{n}, \mathfrak{p}_{n+1})}\varphi(\omega)\, d\omega)}{\psi(\int_{0}^{\varpi(\mathfrak{p}_{n-1}, \mathfrak{p}_{n})}\varphi(\omega)\, d\omega)} \leq\theta(\psi(\int_{0}^{\varpi(\mathfrak{p}_{n-1}, \mathfrak{p}_{n})}\varphi(\omega)\, d\omega)). \end{align*} |
Taking n\rightarrow +\infty , we obtain
\begin{align} 1\leq \lim\limits_{n\rightarrow +\infty}\theta(\psi(\int_{0}^{\varpi(\mathfrak{p}_{n-1}, \mathfrak{p}_{n})}\varphi(\omega)\, d\omega))\leq 1\nonumber \end{align} |
which implies \lim\limits_{n\rightarrow +\infty}\theta(\psi(\int_{0}^{\varpi(\mathfrak{p}_{n-1}, \mathfrak{p}_{n})}\varphi(\omega)\, d\omega)) = 1 . In view of the definition of \theta and \psi , one can deduce that
\begin{align} \lim\limits_{n\rightarrow +\infty}\int_0^{\varpi(\mathfrak{p}_{n-1}, \mathfrak{p}_{n})}\varphi(\omega)\, d\omega = 0\nonumber \end{align} |
i.e.,
\begin{align} \lim\limits_{n\rightarrow +\infty}\varpi(\mathfrak{p}_n, \mathfrak{p}_{n+1}) = 0, \nonumber \end{align} |
which is contradiction. It follows that \lim\limits_{n\rightarrow +\infty}\varpi(\mathfrak{p}_n, \mathfrak{p}_{n+1}) = 0.
Next, we want to show \{\mathfrak{p}_n\} is a Cauchy sequence. As in the proof of Theorem 3.3, we obtain that (3.13)–(3.17) hold. The transitivity property of \alpha implies that \alpha(\mathfrak{p}_{m(k)-1}, \mathfrak{p}_{n(k)-1})\geq s^p . Putting \mathfrak{u} = \mathfrak{p}_{m(k)-1} and \mathfrak{v} = \mathfrak{p}_{n(k)-1} into (3.20), we get
\begin{align} \psi(\int_0^{s^3\varepsilon}\varphi(\omega)\, d\omega) \leq&\liminf\limits_{k\rightarrow +\infty}\psi(\int_0^{s^p\varpi(\mathfrak{p}_{m(k)}, \mathfrak{p}_{n(k)})}\varphi(\omega)\, d\omega)\\ \leq&\liminf\limits_{k\rightarrow +\infty}\psi(\int_{0}^{\alpha(\mathfrak{p}_{m_k-1}, \mathfrak{p}_{n_k-1})\varpi(\mathbb{S} \mathfrak{p}_{m_k-1}, \mathbb{S}\mathfrak{p}_{n_k-1})}\varphi(\omega)\, d\omega)\\ \leq&\liminf\limits_{k\rightarrow +\infty}[\theta(\psi(\int_{0}^{\triangle(\mathfrak{p}_{m_k-1}, \mathfrak{p}_{n_k-1})}\varphi(\omega)\, d\omega)) \psi(\int_{0}^{\triangle(\mathfrak{p}_{m_k-1}, \mathfrak{p}_{n_k-1})}\varphi(\omega)\, d\omega)]\\ \leq&\limsup\limits_{k\rightarrow +\infty}\theta(\psi(\int_{0}^{\triangle(\mathfrak{p}_{m_k-1}, \mathfrak{p}_{n_k-1})}\varphi(\omega)\, d\omega)\cdot \liminf\limits_{k\rightarrow +\infty}\psi(\int_{0}^{\triangle(\mathfrak{p}_{m_k-1}, \mathfrak{p}_{n_k-1})}\varphi(\omega)\, d\omega)\\ = &\theta(\liminf\limits_{k\rightarrow +\infty}\psi(\int_{0}^{\triangle(\mathfrak{p}_{m_k-1}, \mathfrak{p}_{n_k-1})}\varphi(\omega)\, d\omega))\cdot \psi(\liminf\limits_{k\rightarrow +\infty}\int_{0}^{\triangle(\mathfrak{p}_{m_k-1}, \mathfrak{p}_{n_k-1})}\varphi(\omega)\, d\omega)\\ < &\psi(\int_{0}^{s^3\varepsilon}\varphi(\omega)\, d\omega)\nonumber \end{align} |
which is a contradiction. Hence, \{\mathfrak{p}_n\} is Cauchy. The completeness of \aleph ensures that there exists \mathfrak{p}^*\in\aleph such that \{\mathfrak{p}_n\}\rightarrow \mathfrak{p}^* as n\to +\infty .
Next, we prove the point \mathfrak{p}^* to be a fixed point of \mathbb{S} . Similar to the discussion related to Theorem 3.4, taking \mathfrak{u} = \mathfrak{p}_{n(k)-1} and \mathfrak{v} = \mathfrak{p}^* in (3.20), we get
\begin{align} \psi&(\int_{0}^{\alpha(\mathfrak{p}_{n(k)-1}, \mathfrak{p}^*)\varpi(\mathbb{S} \mathfrak{p}_{n(k)-1}, \mathbb{S} \mathfrak{p}^*)}\varphi(\omega)\, d\omega)\\ \leq&\theta(\psi(\int_{0}^{\triangle(\mathfrak{p}_{n(k)-1}, \mathfrak{p}^*)}\varphi(\omega)\, d\omega)) \psi(\int_{0}^{\triangle(\mathfrak{p}_{n(k)-1}, \mathfrak{p}^*)}\varphi(\omega)\, d\omega) \end{align} | (3.24) |
where
\begin{align} \triangle(\mathfrak{p}_{n(k)-1}, \mathfrak{p}^*) = \max&\{\varpi(\mathfrak{p}_{n(k)-1}, \mathfrak{p}^*), \varpi( \mathfrak{p}_{n(k)-1}, \mathbb{S} \mathfrak{p}_{n(k)-1}), \varpi( \mathfrak{p}^*, \mathbb{S} \mathfrak{p}^*), \frac{\varpi( \mathfrak{p}_{n(k)-1}, \mathbb{S} \mathfrak{p}^*)+\varpi( \mathfrak{p}^*, \mathbb{S} \mathfrak{p}_{n(k)-1})}{2s}\}\\ = \max&\{\varpi(\mathfrak{p}_{n(k)-1}, \mathfrak{p}^*), \varpi(\mathfrak{p}_{n(k)-1}, \mathfrak{p}_{n(k)}), \varpi( \mathfrak{p}^*, \mathbb{S}\mathfrak{p}^*), \frac{\varpi(\mathfrak{p}_{n(k)-1}, \mathbb{S} \mathfrak{p}^*)+\varpi(\mathfrak{p}^*, \mathfrak{p}_{n(k)})}{2s}\}.\\ \end{align} | (3.25) |
Taking the limit as n\rightarrow +\infty in (3.25), we get
\begin{align} \lim\limits_{n\rightarrow +\infty}\triangle(\mathfrak{p}_{n(k)-1}, \mathfrak{p}^*) = & \max\{0, 0, \varpi( \mathfrak{p}^*, \mathbb{S}\mathfrak{p}^*), \frac{\varpi( \mathfrak{p}^*, \mathbb{S}\mathfrak{p}^*)}{2}\} = \varpi( \mathfrak{p}^*, \mathbb{S}\mathfrak{p}^*). \end{align} | (3.26) |
According to (3.24), (3.26), and Lemma 2.5, we get
\begin{align} \psi(\int_0^{\varpi(\mathfrak{p}^*, \mathbb{S}\mathfrak{p}^*)}\varphi(t)\, dt) \le& \psi(\int_0^{s^3\cdot\frac{1}{s}\varpi(\mathfrak{p}^*, \mathbb{S} \mathfrak{p}^*)}\varphi(t)\, dt)\\ \le&\lim\limits_{n\rightarrow +\infty} \psi(\int_{0}^{\alpha(\mathfrak{p}_{n(k)-1}, \mathfrak{p}^*)\varpi(\mathbb{S} \mathfrak{p}_{n(k)-1}, \mathbb{S} \mathfrak{p}^*)}\varphi(\omega)\, d\omega)\\ \leq&\lim\limits_{n\rightarrow +\infty}\theta(\psi(\int_{0}^{\triangle(\mathfrak{p}_{n(k)-1}, \mathfrak{p}^*)}\varphi(\omega)\, d\omega)) \psi(\int_{0}^{\triangle(\mathfrak{p}_{n(k)-1}, \mathfrak{p}^*)}\varphi(\omega)\, d\omega)\\ = &\theta(\psi(\int_0^{\varpi( \mathfrak{p}^*, \mathbb{S}\mathfrak{p}^*)}\varphi(\omega)\, d\omega))\psi(\int_0^{\varpi( \mathfrak{p}^*, \mathbb{S}\mathfrak{p}^*)}\varphi(\omega)\, d\omega)\\ < &\psi(\int_0^{\varpi( \mathfrak{p}^*, \mathbb{S}\mathfrak{p}^*)}\varphi(\omega)\, d\omega)\nonumber \end{align} |
which is impossible. It follows that \mathbb{S} \mathfrak{p}^* = \mathfrak{p}^* .
At last, we show the uniqueness of the fixed point of \mathbb{S} . Suppose \mathfrak{q}^* is another fixed point of \mathbb{S} . It follows from the condition (ⅳ) that \alpha(\mathfrak{p}^*, \mathfrak{q}^*)\geq s^p . In light of (3.20), one can get
\begin{align} \psi(\int_{0}^{\alpha(\mathfrak{p}^*, \mathfrak{q}^*)\varpi(\mathbb{S} \mathfrak{p}, \mathbb{S} \mathfrak{q})}\varphi(\omega)\, d\omega) \leq&\theta(\psi(\int_{0}^{\triangle(\mathfrak{p}^*, \mathfrak{q}^*)}\varphi(\omega)\, d\omega)) \psi(\int_{0}^{\triangle(\mathfrak{p}^*, \mathfrak{q}^*)}\varphi(\omega)\, d\omega), \nonumber \end{align} |
\begin{align} \triangle( \mathfrak{p}^*, \mathfrak{q}^*) = \max&\{\varpi(\mathfrak{p}^*, \mathfrak{q}^*), \varpi( \mathfrak{p}^*, \mathbb{S} \mathfrak{p}^*), \varpi( \mathfrak{q}^*, \mathbb{S} \mathfrak{q}^*), \frac{\varpi(\mathfrak{p}^*, \mathbb{S} \mathfrak{q}^*)+\varpi( \mathfrak{q}^*, \mathbb{S} \mathfrak{p}^*)}{2s}\}\\ = \max&\{\varpi(\mathfrak{p}^*, \mathfrak{q}^*), 0, 0, \frac{\varpi(\mathfrak{p}^*, \mathfrak{q}^*)+\varpi(\mathfrak{q}^*, \mathfrak{p}^*)}{2s}\} = \varpi(\mathfrak{p}^*, \mathfrak{q}^*). \nonumber \end{align} |
Then
\begin{align} \psi(\int_{0}^{\varpi(\mathfrak{p}^*, \mathfrak{q}^*)}\varphi(\omega)\, d\omega) \leq&\psi(\int_{0}^{\alpha(\mathfrak{p}^*, \mathfrak{q}^*)\varpi(\mathbb{S} \mathfrak{p}, \mathbb{S} \mathfrak{q})}\varphi(\omega)\, d\omega)\\ \leq&\theta(\psi(\int_{0}^{\triangle(\mathfrak{p}^*, \mathfrak{q}^*)}\varphi(\omega)\, d\omega)) \psi(\int_{0}^{\triangle(\mathfrak{p}^*, \mathfrak{q}^*)}\varphi(\omega)\, d\omega)\\ < &\psi(\int_{0}^{\varpi(\mathfrak{p}^*, \mathfrak{q}^*)}\varphi(\omega)\, d\omega)\nonumber \end{align} |
a contradiction, which implies that \mathfrak{p}^* = \mathfrak{q}^* . This completes the proof.
Example 3.7. Let \aleph = [0, 1] and \varpi(\mathfrak{p}, \mathfrak{q}) = (\mathfrak{p}-\mathfrak{q})^2. Define mappings \mathbb{S}:\aleph\rightarrow \aleph by
\mathbb{S} \mathfrak{p} = \left\{\begin{aligned} &\frac{\mathfrak{p}}{32\sqrt[16]{e}}, \ \ \mathfrak{p}\in [0, \frac{1}{2}], \\ &\frac{1}{32\sqrt[16]{e}}, \quad \mathfrak{p}\in (\frac{1}{2}, 1] \end{aligned}\right.\\ |
and \alpha:\aleph \times \aleph\rightarrow [0, +\infty) by
\alpha(\mathfrak{p}, \mathfrak{q}) = 2^4, \mathfrak{p}, \mathfrak{q}\in [0, 1]. |
Define \theta:[0, +\infty)\rightarrow (0, 1) and \psi, \varphi:[0, +\infty)\rightarrow [0, +\infty) by
\theta(\omega) = e^{-4\omega}, \; \psi(\omega) = \omega \ \text{ and }\ \varphi(\omega) = 2\omega. |
One can deduce that \alpha(\mathfrak{u}, \mathfrak{v})\geq 2^4 , \int_0^{\varpi(\mathbb{S}\mathfrak{u}, \mathbb{S}\mathfrak{v})}\varphi(\omega)d\omega > 0 \Leftrightarrow \mathfrak{u}, \mathfrak{v} \in [0, 1] with \mathfrak{u}\neq\mathfrak{v} . It follows that we also consider two cases:
Case 1. \mathfrak{u}, \mathfrak{v} \in [0, \frac{1}{2}] , then
\begin{align*} \psi(\int_{0}^{\alpha(\mathfrak{u}, \mathfrak{v})\varpi(\mathbb{S}\mathfrak{u}, \mathbb{S}\mathfrak{v})}\varphi(\omega)\, d\omega) = &\int_{0}^{2^4(\frac{\mathfrak{u}}{32\sqrt[16]{e}}-\frac{\mathfrak{v}}{32\sqrt[16]{e}})^2} 2\omega\, d\omega\\ = &\frac{1}{64^2\times\sqrt[4]{e}}(\mathfrak{u}-\mathfrak{v})^4, \\ \end{align*} |
\begin{align*} \theta(\psi(\int_{0}^{\triangle(\mathfrak{u}, \mathfrak{v})}\varphi(\omega)\, d\omega))\psi(\int_{0}^{\triangle(\mathfrak{u}, \mathfrak{v})}\varphi(\omega)\, d\omega) = &e^{-4\int_{0}^{\triangle(\mathfrak{u}, \mathfrak{v})}2\omega\, d\omega}\cdot\int_{0}^{\triangle(\mathfrak{u}, \mathfrak{v})}2\omega\, d\omega\\ \geq&\frac{1}{\sqrt[4]{e}}(\mathfrak{u}-\mathfrak{v})^4.\\ \end{align*} |
Case 2. \mathfrak{u}\in [0, \frac{1}{2}], \mathfrak{v}\in (\frac{1}{2}, 1] . It is easy to obtain that
\begin{align*} \psi(\int_{0}^{\alpha(\mathfrak{u}, \mathfrak{v})\varpi(\mathbb{S}\mathfrak{u}, \mathbb{S}\mathfrak{v})}\varphi(\omega)\, d\omega) = &\int_{0}^{2^4(\frac{\mathfrak{u}}{32\sqrt[16]{e}}-\frac{1}{32\sqrt[16]{e}})^2} 2\omega\, d\omega\\ = &\frac{1}{64^2\times\sqrt[4]{e}}(\mathfrak{u}-1)^4\\ \leq &\frac{1}{64^2\times\sqrt[4]{e}}, \end{align*} |
\begin{align*} \theta(\psi(\int_{0}^{\triangle(\mathfrak{u}, \mathfrak{v})}\varphi(\omega)\, d\omega))\psi(\int_{0}^{\triangle(\mathfrak{u}, \mathfrak{v})}\varphi(\omega)\, d\omega) = &e^{-4\int_{0}^{\triangle(\mathfrak{u}, \mathfrak{v})}2\omega\, d\omega}\cdot\int_{0}^{\triangle(\mathfrak{u}, \mathfrak{v})}2\omega\, d\omega\\ \geq&\frac{1}{e^4}\times \frac{1}{16}\times (1-\frac{1}{32\sqrt[16]{e}})^4\\ \geq &\frac{1}{64^2\times\sqrt[4]{e}}. \end{align*} |
That is,
\psi(\int_{0}^{\alpha(\mathfrak{u}, \mathfrak{v})\varpi(\mathbb{S} \mathfrak{u}, \mathbb{S} \mathfrak{v})}\varphi(\omega)\, d\omega) \leq\theta(\psi(\int_{0}^{\triangle(\mathfrak{u}, \mathfrak{v})}\varphi(\omega)\, d\omega)) \psi(\int_{0}^{\triangle(\mathfrak{u}, \mathfrak{v})}\varphi(\omega)\, d\omega). |
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 \mathbb{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 S\subseteq O and D\subseteq P be the state and decision spaces. B(S) denotes the Banach space of all bounded real-valued functions on S with norm
\begin{align} \parallel m \parallel = \sup\{|m(\xi)|:\xi \in S\} \text{ for any }\ m\in B(S). \end{align} | (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) = \inf\limits_{y\in D}\max\{r(x, y), s(x, y), f(b(x, y))\}, |
f(x) = \inf\limits_{y\in D}\max\{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) = opt_{y\in D}\{H(x, y, f(T(x, y)))\}, \forall x\in S |
where the opt represents \sup or \inf . Bhakta and Mitra [22] obtained the existence and uniqueness of solutions for the functional equations
f(x) = \sup\limits_{y\in D}\{p(x, y)+A(x, y, f(a(x, y))\} |
in a Banach space B(S) and
f(x) = \sup\limits_{y\in D}\{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), \varpi) is a complete b -metric space with
\begin{align*} \begin{aligned} \varpi(\xi, \eta) = \parallel \xi-\eta \parallel^2, \forall \xi, \eta\in B(S). \end{aligned} \end{align*} |
Consider the functional equations arising in dynamic programming:
\begin{align} \mathfrak{f}(x) = \inf\limits_{y\in D}\{u(x, y)+H(x, y, \mathfrak{f}(\mathrm{T}(x, y)))\}, \forall x\in S \end{align} | (4.2) |
where u:S\times D\rightarrow \mathbb{R} , \mathrm{T}:S\times D\rightarrow S and H:S\times D\times \mathbb{R}\rightarrow \mathbb{R} are mappings. Let
\begin{align} \mathbb{S}\mathfrak{f}(x) = \inf\limits_{y\in D}\{u(x, y)+H(x, y, \mathfrak{f}(\mathrm{T}(x, y)))\}, \forall (x, \mathfrak{f})\in S\times B(S). \end{align} | (4.3) |
Theorem 4.1. Let u:S\times D\rightarrow \mathbb{R} , \mathrm{T}:S\times D\rightarrow S , H:S\times D\times \mathbb{R}\rightarrow \mathbb{R} , \mathbb{S}:B(S)\to B(S) , \alpha:B(S)\times B(S)\to \mathbb{R} . If
(ⅰ) u and H are bounded,
(ⅱ) \mathbb{S} is \alpha_{s^p} -admissible,
(ⅲ) there is \mathfrak{p}_0\in B(S) satisfying \alpha(\mathfrak{p}_0, \mathbb{S} \mathfrak{p}_0)\geq s^p ,
(ⅳ) \alpha satisfies transitive property, i.e., for \xi, \eta, \zeta\in B(S) such that
\alpha(\xi, \eta)\geq s^p \text{ and } \alpha(\eta, \zeta)\geq s^p\Rightarrow \alpha(\xi, \zeta)\geq s^p, |
(ⅴ) if \{\mathfrak{p}_n\} is a sequence in B(S) satisfying \mathfrak{p}_n\rightarrow \mathfrak{p} as n\rightarrow +\infty , then there is a subsequence \{\mathfrak{p}_{n(k)}\} of \{\mathfrak{p}_{n}\} with \alpha(\mathfrak{p}_{n(k)}, \mathfrak{p})\geq s^p ,
(ⅵ) for \mathfrak{p}, \mathfrak{q}\in Fix(\mathbb{S}) , one can get the condition of \alpha(\mathfrak{p}, \mathfrak{q})\geq s^p and \alpha(\mathfrak{q}, \mathfrak{p})\geq s^p , where Fix(\mathbb{S} ) represents the collection of all fixed points of \mathbb{S} ,
(ⅶ) if there exists \ell\in (0, 1) , \varphi \in \Im such that
\begin{align} &\alpha(\mathfrak{u}, \mathfrak{v})\geq s^p, \int_{0}^{\parallel\mathbb{S}\mathfrak{u}-\mathbb{S}\mathfrak{v}\parallel^2}\varphi(\omega)\, d\omega > 0\\ \Rightarrow &\\ &\exp(\int_{0}^{2\alpha(\mathfrak{u}, \mathfrak{v})|H(\mathfrak{u}, \mathfrak{v}, \mathfrak{g}(\mathrm{T}(\mathfrak{u}, \mathfrak{v}))) -H(\mathfrak{u}, \mathfrak{v}, \mathfrak{h}(\mathrm{T}(\mathfrak{u}, \mathfrak{v})))|^2 }\varphi(\omega)\, d\omega)\leq[\exp(\ \int_{0}^{\triangle(\mathfrak{u}, \mathfrak{v})}\varphi(\omega)\, d\omega)]^\ell, \end{align} | (4.4) |
then the functional Eq (4.2) has a unique solution \mathfrak{p^*}\in B(S) .
Proof. It follows from (i) that there exists M > 0 satisfying
\sup\{|u(x, y)|, |H(x, y, t)|:(x, y, t)\in S\times D\times \mathbb{R}\}\leq M. |
It is easy to see that \mathbb{S} is a self-mapping in B(S) . Define \alpha:B(S)\times B(S)\rightarrow [0, \infty) by
\alpha(\mathfrak{u}, \mathfrak{v}) = \left\{\begin{aligned} &s^{p}, \ \ \varpi(\mathbb{S}\mathfrak{u}, \mathbb{S}\mathfrak{v}) > 0, \\ &0, \ \text{ otherwise}. \end{aligned}\right.\\ |
By (i) and \varphi\in \Im , we have for each \varepsilon > 0 , there exists \delta > 0 such that
\begin{align} \int_{C}\varphi(t)dt < \varepsilon, \forall C\subset [0, 2M] \text{ with } m(C)\leq \delta, \end{align} | (4.5) |
where m(C) denotes the Lebesgue measure of C .
Let \mathfrak{u}\in S, \mathfrak{h}, \mathfrak{g}\in B(S) . By (4.3), there exists \mathfrak{v}, \mathfrak{w}\in D satisfying
\mathbb{S}\mathfrak{g}(\mathfrak{u}) > u(\mathfrak{u}, \mathfrak{v})+H(\mathfrak{u}, \mathfrak{v}, \mathfrak{g}(\mathrm{T}(\mathfrak{u}, \mathfrak{v})))-\frac{\sqrt{2\delta}}{2}, |
\mathbb{S}\mathfrak{h}(\mathfrak{u}) > u(\mathfrak{u}, \mathfrak{w})+H(\mathfrak{u}, \mathfrak{w}, \mathfrak{h}(\mathrm{T}(\mathfrak{u}, \mathfrak{w})))-\frac{\sqrt{2\delta}}{2}, |
\mathbb{S}\mathfrak{g}(\mathfrak{u})\leq u(\mathfrak{u}, \mathfrak{w})+H(\mathfrak{u}, \mathfrak{w}, \mathfrak{g}(\mathrm{T}(\mathfrak{u}, \mathfrak{w}))), |
\mathbb{S}\mathfrak{h}(\mathfrak{u})\leq u(\mathfrak{u}, \mathfrak{v})+H(\mathfrak{u}, \mathfrak{v}, \mathfrak{h}(\mathrm{T}(\mathfrak{u}, \mathfrak{v}))).\\ |
Thus,
\begin{align*} \mathbb{S}\mathfrak{g}(\mathfrak{u})-\mathbb{S}\mathfrak{h}(\mathfrak{u}) < &H(\mathfrak{u}, \mathfrak{w}, \mathfrak{g}(\mathrm{T}(\mathfrak{u}, \mathfrak{w})))- H(\mathfrak{u}, \mathfrak{w}, \mathfrak{h}(\mathrm{T}(\mathfrak{u}, \mathfrak{w})))+\frac{\sqrt{2\delta}}{2}\\ \leq&|H(\mathfrak{u}, \mathfrak{w}, \mathfrak{g}(\mathrm{T}(\mathfrak{u}, \mathfrak{w})))- H(\mathfrak{u}, \mathfrak{w}, \mathfrak{h}(\mathrm{T}(\mathfrak{u}, \mathfrak{w})))|+\frac{\sqrt{2\delta}}{2}, \end{align*} |
\begin{align*} \mathbb{S}\mathfrak{h}(\mathfrak{u})-\mathbb{S}\mathfrak{g}(\mathfrak{u}) < &H(\mathfrak{u}, \mathfrak{v}, \mathfrak{h}(\mathrm{T}(\mathfrak{u}, \mathfrak{v})))- H(\mathfrak{u}, \mathfrak{v}, \mathfrak{g}(\mathrm{T}(\mathfrak{u}, \mathfrak{v})))+\frac{\sqrt{2\delta}}{2}\\ \leq&|H(\mathfrak{u}, \mathfrak{v}, \mathfrak{h}(\mathrm{T}(\mathfrak{u}, \mathfrak{v})))- H(\mathfrak{u}, \mathfrak{v}, \mathfrak{g}(\mathrm{T}(\mathfrak{u}, \mathfrak{v})))|+\frac{\sqrt{2\delta}}{2}. \end{align*} |
It follows that
\begin{align} ||\mathbb{S}\mathfrak{g}-\mathbb{S}\mathfrak{h}|| = \sup\limits_{\mathfrak{u}\in S}|\mathbb{S}\mathfrak{g}(\mathfrak{u})-\mathbb{S}\mathfrak{h}(\mathfrak{u})|\le\max\{\mathrm{T_{1}}, \mathrm{T_{2}}\}+\frac{\sqrt{2\delta}}{2}, \end{align} | (4.6) |
where
\begin{align*} \begin{aligned} \mathrm{T}_{1} = |H(\mathfrak{u}, \mathfrak{w}, \mathfrak{g}(\mathrm{T}(\mathfrak{u}, \mathfrak{w})))- H(\mathfrak{u}, \mathfrak{w}, \mathfrak{h}(\mathrm{T}(\mathfrak{u}, \mathfrak{w})))|, \end{aligned} \end{align*} |
\begin{align*} \begin{aligned} \mathrm{T}_{2} = |H(\mathfrak{u}, \mathfrak{v}, \mathfrak{h}(\mathrm{T}(\mathfrak{u}, \mathfrak{v})))- H(\mathfrak{u}, \mathfrak{v}, \mathfrak{g}(\mathrm{T}(\mathfrak{u}, \mathfrak{v})))|. \end{aligned} \end{align*} |
It is easy to get that ||\mathbb{S}\mathfrak{g}-\mathbb{S}\mathfrak{h}||^2\leq \max\{2\mathrm{T_{1}}^2, 2\mathrm{T_{2}}^2\}+\delta . Under (4.4) and (4.6), we have
\begin{align*} \begin{aligned} &\exp(\int_{0}^{s^{p}||\mathbb{S}\mathfrak{g}(\mathfrak{u})-\mathbb{S}\mathfrak{h}(\mathfrak{u})||^2}\varphi(\omega)\, d\omega)\\ \leq&\exp(\int_{0}^{s^{p}\max\{2\mathrm{T}_{1}^2, 2\mathrm{T}_{2}^2\}+\delta}\varphi(\omega)\, d\omega)\\ = &\max\{\exp(\int_{0}^{2s^{p}\mathrm{T}_{1}^2+\delta}\varphi(\omega)\, d\omega), \exp(\int_{0}^{2s^{p}\mathrm{T}_{2}^2+\delta}\varphi(\omega)\, d\omega)\}\\ = &\max\{\exp(\int_{0}^{2s^{p}\mathrm{T}_{1}^2}\varphi(\omega)\, d\omega)\cdot\exp(\int_{2s^{p}\mathrm{T}_{1}^2}^{2s^{p}\mathrm{T}_{1}^2+\delta}\varphi(\omega)\, d\omega), \\ &\quad\exp(\int_{0}^{2s^{p}\mathrm{T}_{2}^2}\varphi(\omega)\, d\omega)\cdot\exp(\int_{2s^{p}\mathrm{T}_{2}^2}^{2s^{p}\mathrm{T}_{2}^2+\delta}\varphi(\omega)\, d\omega)\}\\ \leq&\max\{\exp(\int_{0}^{2s^{p}\mathrm{T}_{1}^2}\varphi(\omega)\, d\omega), \exp(\int_{0}^{2s^{p}\mathrm{T}_{2}^2}\varphi(\omega)\, d\omega)\}\\ &\quad\cdot \max\{\exp(\int_{2s^{p}\mathrm{T}_{1}^2}^{2s^{p}\mathrm{T}_{1}^2+\delta}\varphi(\omega)\, d\omega), \exp(\int_{2s^{p}\mathrm{T}_{2}^2}^{2s^{p}\mathrm{T}_{2}^2+\delta}\varphi(\omega)\, d\omega)\}\\ \leq&[\exp(\ \int_{0}^{\triangle(\mathfrak{u}, \mathfrak{v})}\varphi(\omega)\, d\omega)]^\ell\cdot\exp(\varepsilon).\\ \end{aligned} \end{align*} |
Letting \varepsilon\rightarrow0^{+} in the above inequality, we get
\begin{align*} \begin{aligned} \exp(\int_{0}^{\alpha(\mathfrak{u}, \mathfrak{v})\parallel \mathbb{S}\mathfrak{g}-\mathbb{S}\mathfrak{h}\parallel^2 }\varphi(\omega)\, d\omega)\leq[\exp(\ \int_{0}^{\triangle(\mathfrak{u}, \mathfrak{v})}\varphi(\omega)\, d\omega)]^\ell. \end{aligned} \end{align*} |
Thus, the conditions of Theorem 3.3 are satisfied by taking \theta(\omega) = \exp(\omega) , so the functional Eq (4.2) has a unique fixed sloution \mathfrak{p^*} \in B(S) . This completes the proof.
In this manuscript, we first defined two new types of weak contractions named \theta -weak contraction and \theta - \psi -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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