Breast cancer seriously threatens women's physical and mental health. Mammography is one of the most effective methods for breast cancer diagnosis via artificial intelligence algorithms to identify diverse breast masses. The popular intelligent diagnosis methods require a large amount of breast images for training. However, collecting and labeling many breast images manually is extremely time consuming and inefficient. In this paper, we propose a distributed multi-latent code inversion enhanced Generative Adversarial Network (dm-GAN) for fast, accurate and automatic breast image generation. The proposed dm-GAN takes advantage of the generator and discriminator of the GAN framework to achieve automatic image generation. The new generator in dm-GAN adopts a multi-latent code inverse mapping method to simplify the data fitting process of GAN generation and improve the accuracy of image generation, while a multi-discriminator structure is used to enhance the discrimination accuracy. The experimental results show that the proposed dm-GAN can automatically generate breast images with higher accuracy, up to a higher 1.84 dB Peak Signal-to-Noise Ratio (PSNR) and lower 5.61% Fréchet Inception Distance (FID), as well as 1.38x faster generation than the state-of-the-art.
Citation: Jiajia Jiao, Xiao Xiao, Zhiyu Li. dm-GAN: Distributed multi-latent code inversion enhanced GAN for fast and accurate breast X-ray image automatic generation[J]. Mathematical Biosciences and Engineering, 2023, 20(11): 19485-19503. doi: 10.3934/mbe.2023863
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Breast cancer seriously threatens women's physical and mental health. Mammography is one of the most effective methods for breast cancer diagnosis via artificial intelligence algorithms to identify diverse breast masses. The popular intelligent diagnosis methods require a large amount of breast images for training. However, collecting and labeling many breast images manually is extremely time consuming and inefficient. In this paper, we propose a distributed multi-latent code inversion enhanced Generative Adversarial Network (dm-GAN) for fast, accurate and automatic breast image generation. The proposed dm-GAN takes advantage of the generator and discriminator of the GAN framework to achieve automatic image generation. The new generator in dm-GAN adopts a multi-latent code inverse mapping method to simplify the data fitting process of GAN generation and improve the accuracy of image generation, while a multi-discriminator structure is used to enhance the discrimination accuracy. The experimental results show that the proposed dm-GAN can automatically generate breast images with higher accuracy, up to a higher 1.84 dB Peak Signal-to-Noise Ratio (PSNR) and lower 5.61% Fréchet Inception Distance (FID), as well as 1.38x faster generation than the state-of-the-art.
The theory of fixed points takes an important place in the transition from classical analysis to modern analysis. One of the most remarkable works on fixed point of functions was done by Banach [9]. Various generalizations of Banach fixed point result were made by numerous mathematicians [1,2,3,6,12,13,14,17,19,20,25,26]. One of the generalizations of the metric space is the quasi metric space that was introduced by Wilson [30]. The commutativity condition does not hold in general in quasi metric spaces. Several authors used these concepts to prove some fixed point theorems, see [7,10,15,21]. On the other hand Bakhtin [8] and Czerwik [11] generalized the triangle inequality by multiplying the right hand side of triangle inequality in metric spaces by a parameter b≥1 and defined b-metric spaces, for more results, see [4,23,24]. Fixed point results for multivalued mappings generalizes the results for single-valued mappings. Many interesting results have been proved in the setting of multivalued mappings, for example, see [5,26,29]. Kamran et al. [12] introduced a new concept of generalized b metric spaces, named as extended b-metric spaces, see also [22]. They replaced the parameter b≥1 in triangle inequality by the control function θ:X×X→[1,∞). Mlaiki et al. [16] replaced the triangle inequality in b-metric spaces by using control function in a different style and introduced controlled metric type spaces. Abdeljawad et al. [1] generalized the idea of controlled metric type spaces and introduced double controlled metric type spaces. They replaced the control function ξ(x,y) in triangle inequality by two functions ξ(x,y) and ⊤(x,y), see also [27,28]. In this paper, the concept of double controlled quasi metric type spaces has been discussed. Fixed point results and several examples are established. First of all, we discuss the previous concepts that will be useful to understand the paper.
Now, we define double controlled metric type space.
Definition 1.1.[1] Given non-comparable functions ξ,⊤:X×X→[1,∞). If △:X×X→[0,∞) satisfies:
(q1) △(ω,v)=0 if and only if ω=v,
(q2) △(ω,v)=△(v,ω),
(q3) △(ω,v)≤ξ(ω,e)△(ω,e)+⊤(e,v)△(e,v),
for all ω,v,e∈X. Then, △ is called a double controlled metric with the functions ξ, ⊤ and the pair (X,△) is called double controlled metric type space with the functions ξ,⊤.
The classical result to obtain fixed point of a mapping in double controlled metric type space is given below.
Theorem 1.2. [1] Let (X,△) be a complete double controlled metric type space with the functions ξ,⊤ :X×X →[1,∞) and let T:X→X be a given mapping. Suppose that there exists k∈(0,1) such that
△(T(x),T(y)≤k△(x,y), for all x,y∈X. |
For ω0∈X, choose ωg=Tgω0. Assume that
supm≥1limi→∞ξ(ωi+1,ωi+2)ξ(ωi,ωi+1)⊤(ωi+1,ωm)<1k. |
In the addition, assume that, for every ω∈X, we have
limg→∞ξ(ω,ωg), and limg→∞⊤(ω g,ω) exists and are finite. |
Then T has a unique fixed point ω∗∈X.
Now, we are introducing the concept of double controlled quasi metric type space and controlled quasi metric type space.
Definition 1.3. Given non-comparable functions ξ,⊤:X×X→[1,∞). If △:X×X→[0,∞) satisfies
(q1) △(ω,v)=0 if and only if ω=v,
(q2) △(ω,v)≤ ξ(ω,e)△(ω,e)+⊤(e,v)△(e,v),
for all ω,v,e∈X. Then, △ is called a double controlled quasi metric type with the functions ξ, ⊤ and (X,△) is called a double controlled quasi metric type space. If ⊤(e,v)=ξ(e,v) then (X,△) is called a controlled quasi metric type space.
Remark 1.4. Any quasi metric space or any double controlled metric type space is also a double controlled quasi metric type space but, the converse is not true in general, see examples (1.5, 2.4, 2.12 and 2.15).
Example 1.5. Let X={0,1,2}. Define △:X×X→[0, ∞) by △(0,1)=4, △(0,2)=1, △(1,0)=3=△(1,2), △(2,0)=0, △(2,1)=2, △(0,0)=△(1,1)=△(2,2)=0.
Define ξ,⊤:G×G→[1, ∞) as ξ(0,1)= ξ(1,0)= ξ(1,2)=1, ξ(0,2)=54, ξ(2,0)=109, ξ(2,1)=2019, ξ(0,0)= ξ(1,1)= ξ(2,2)=1, ⊤(0,1)=⊤(1,0)=⊤(0,2)=⊤(1,2)=1, ⊤(2,0)=32, ⊤(2,1)=118, ⊤(0,0)=⊤(1,1)=⊤(2,2)=1. It is clear that △ is double controlled quasi metric type with the functions ξ, ⊤. Let w=0, e=2, v=1, we have
△(0,1)=4>3=△(0,2)+△(2,1). |
So △ is not a quasi metric. Also, it is not a controlled quasi metric type. Indeed,
△(0,1)=4>134=ξ(0,2)△(0,2)+ξ(2,1)△(2,1). |
Moreover, it is not double controlled metric type space because, we have
△(0,1)=ξ(0,2)△(0,2)+⊤(2,1)△(2,1)=5516≠△(1,0). |
The convergence of a sequence in double controlled quasi metric type space is defined as:
Definition 1.6. Let (X,△) be a double controlled quasi metric type space with two functions. A sequence {ut} is convergent to some u in X if and only if limt→∞△(ut,u)=limt→∞△(u,ut)=0.
Now, we discuss left Cauchy, right Cauchy and dual Cauchy sequences in double controlled quasi metric type space.
Definition 1.7. Let (X,△) be a double controlled quasi metric type space with two functions.
(i) The sequence {ut} is a left Cauchy if and only if for every ε>0 such that △(um,ut)<ε, for all t>m>tε, where tε is some integer or limt,m→∞△(um,ut)=0.
(ii) The sequence {ut} is a right Cauchy if and only if for every ε>0 such that △(um,ut)<ε, for all m>t>tε, where tε is some integer.
(iii) The sequence {ut} is a dual Cauchy if and only if it is both left as well as right Cauchy.
Now, we define left complete, right complete and dual complete double controlled quasi metric type spaces.
Definition 1.8. Let (X,△) be a double controlled quasi metric type space. Then (X,△) is left complete, right complete and dual complete if and only if each left-Cauchy, right Cauchy and dual Cauchy sequence in X is convergent respectively.
Note that every dual complete double controlled quasi metric type space is left complete but the converse is not true in general, so it is better to prove results in left complete double controlled quasi metric type space instead of dual complete.
Best approximation in a set and proximinal set are defined as:
Definition 1.9. Let (ℑ,△) be a double controlled quasi metric type space. Let A be a non-empty set and l∈ℑ. An element y0∈A is called a best approximation in A if
△(l,A)=△(l,y0), where △(l,A)=infy∈A△(l,y)and △(A,l)=△(y0,l), where △(A,l)=infy∈A△(y,l). |
If each l∈ℑ has a best approximation in A, then A is know as proximinal set. P(ℑ) is equal to the set of all proximinal subsets of ℑ.
Double controlled Hausdorff quasi metric type space is defined as:
Definition 1.10. The function H△:P(E)×P(E)→[0,∞), defined by
H△(C,F)=max{supa∈C△(a,F), supb∈F△(C,b)} |
is called double controlled quasi Hausdorff metric type on P(E). Also (P(E),H△) is known as double controlled Hausdorff quasi metric type space.
The following lemma plays an important role in the proof of our main result.
Lemma 1.11. Let (X,△) be a double controlled quasi metric type space. Let (P(E),H△) be a double controlled Hausdorff quasi metric type space on P(E). Then, for all C,F∈P(E) and for each c∈C, there exists fc∈F, such that H△(C,F)≥△(c,fc) and H△(F,C)≥△(fc,c).
Let (X,△) be a double controlled quasi metric type space, u0∈X and T:X→P(X) be multifunctions on X. Let u1∈Tu0 be an element such that △(u0,Tu0)=△(u0,u1), △(Tu0,u0)=△(u1,u0). Let u2∈Tu1 be such that △(u1,Tu1)=△(u1,u2), △(Tu1,u1)=△(u2,u1). Let u3∈Tu2 be such that △(u2,Tu2)=△(u2,u3) and so on. Thus, we construct a sequence ut of points in X such that ut+1∈Tut with △(ut,Tut)=△(ut,ut+1), △(Tut,ut)=△(u t+1,ut), where t=0,1,2,⋯. We denote this iterative sequence by {XT(ut)}. We say that {XT(ut)} is a sequence in X generated by u0 under double controlled quasi metric △. If △ is quasi b-metric then, we say that {XT(ut)} is a sequence in X generated by u0 under quasi b -metric △. We can define {XT(ut)} in other metrics in a similar way.
Now, we define double controlled rational contracion which is a generalization of many other classical contractions.
Definition 2.1. Let (X,△) be a complete double controlled quasi-metric type space with the functions α,μ :X×X →[1,∞). A multivalued mapping T:X→P(X) is called a double controlled rational contracion if the following conditions are satisfied:
H△(Tx,Ty)≤k(Q(x,y)), | (2.1) |
for all x,y∈X, 0<k<1 and
Q(x,y)=max{△(x,y),△(x,Tx),△(x,Tx)△(x,Ty)+△(y,Ty)△(y,Tx)△(x,Ty)+△(y,Tx)}. |
Also, for the terms of the sequence {XT(ut)}, we have
supm≥1limi→∞α(ui+1,ui+2)α(ui,ui+1)μ(ui,um)<1k | (2.2) |
and for every u∈X
limt→∞α(u,ut) and limt→∞μ(ut,u) are finite. | (2.3) |
Now, we prove that an operator T satisfying certain rational contraction condition has a fixed point in double controlled quasi metric type space.
Theorem 2.2. Let (X,△) be a left complete double controlled quasi-metric type space with the functions α,μ:X×X→[1,∞) and T:X→P(X) be a double controlled rational contracion. Then, T has a fixed point u∗∈X.
Proof. By Lemma 1.11 and using inequality (2.1), we have
△(ut,ut+1)≤H△(Tut−1,Tut)≤k(Q(ut−1,ut)). |
Q(ut−1,ut)≤max{△(ut−1,ut),△(ut−1,ut),△(ut−1,ut)△(ut−1,Tut)+△(ut,ut+1)△(ut,Tut−1)△(ut−1,Tut)+△(ut,Tut−1)}=△(ut−1,ut). |
Therefore,
△(ut,ut+1)≤k△(ut−1,ut). | (2.4) |
Now,
△(ut−1,ut)≤H△(Tut−2,Tut−1)≤k(Q(ut−2,ut−1)). |
Q(ut−2,ut−1)=max{△(u t−2,ut−1),△(ut−2,ut−1), |
△(ut−2,ut−1)△(ut−2,Tut−1)+△(ut−1,ut)△(ut−1,Tut−2)△(ut−2,Tut−1)+△(ut−1,Tut−2)}. |
Therefore,
△(ut−1,ut)≤k△(ut−2,ut−1). | (2.5) |
Using (2.5) in (2.4), we have
△(ut,ut+1)≤k2△(ut−2,ut−1). |
Continuing in this way, we obtain
△(ut,ut+1)≤kt△(u0,u1). | (2.6) |
Now, by using (2.6) and by using the technique given in [1], it can easily be proved that {ut} is a left Cauchy sequence. So, for all natural numbers with t<m, we have
limt,m→∞△(ut,um)=0. | (2.7) |
Since (X,△) is a left complete double controlled quasi metric type space, there exists some u∗∈X such that
limt→∞△(ut,u∗)=limt→∞△(u∗,ut)=0. | (2.8) |
By using triangle inequality and then (2.1), we have
△(u∗,Tu∗)≤α(u∗,ut+1)△(u∗,ut+1)+μ(ut+1,Tu∗)△(ut+1,Tu∗) |
≤α(u∗,ut+1)△(u∗,ut+1)+μ(ut+1,Tu∗)max{△(ut,u∗),△(ut,ut+1),△(ut,ut+1)△(ut,Tu∗)+△(u∗,Tu∗)△(u∗,ut)△(ut,Tu∗)+△(u∗,ut)}. |
Using (2.3), (2.7) and (2.8), we get △(u∗,Tu∗)≤0. That is, u∗∈Tu∗. Thus u∗ is a fixed point of T.
If we take single-valued mapping instead of multivalued mapping, then we obtain the following result.
Theorem 2.3. Let (X,△) be a left complete double controlled quasi-metric type space with the functions α,μ:X×X→[1,∞) and T:X→X be a mapping such that:
△(Tx,Ty)≤k(Q(x,y)), |
for all x,y∈X, 0<k<1 and
Q(x,y)=max{△(x,y),△(x,Tx),△(x,Tx)△(x,Ty)+△(y,Ty)△(y,Tx)△(x,Ty)+△(y,Tx)}, |
Suppose that, for every u∈X and for the Picard sequence {ut}
limt→∞α(u,ut), limt→∞μ(ut,u) are finite and |
supm≥1limi→∞α(ui+1,ui+2)α(ui,ui+1)μ(ui,um)<1k. |
Then, T has a fixed point u∗∈X.
We present the following example to illustrate Theorem 2.3.
Example 2.4. Let X={0,1,2,3}. Define △:X×X→[0, ∞) by: △(0,1)=1, △(0,2)=4, △(0,3)=5, △(1,0)=0, △(1,2)=10, △(1,3)=1, △(2,0)=7, △(2,1)=3, △(2,3)=5, △(3,0)=3, △(3,1)=6, △(3,2)=2, △(0,0)=△(1,1)=△(2,2)=△(3,3)=0. Define α,μ:X×X→[1, ∞) as: α(0,1)=2, α(1,2)= α(0,2)= α(1,0)=α(2,0)=α(3,1)=α(2,3)=α(0,3)=1, α(1,3)=2, α(2,1)=73, α(3,0)=43, α(3,2)=32, α(0,0)=α(1,1)=α(2,2)=α(3,3)=1, μ(1,2)=μ(2,1)=μ(2,0)=μ(3,0)=μ(0,3)=1, μ(1,0)=32, μ(0,1)=2, μ(1,3)=3, μ(3,1)=1, μ(3,2)=4, μ(2,3)=1, μ(0,2)=52, μ(0,0)=μ(1,1)=μ(2,2)=μ(3,3)=1. Clearly (X,△) is a double controlled quasi metric type space, but it is not a controlled quasi metric type space. Indeed,
△(1,2)=10>4=α(1,0)△(1,0)+α(0,2)△(0,2). |
Also, it is not a double controlled metric type space. Take T0=T1={0}, T2=T3={1} and k=13. We observe that
△(Tx,Ty)≤k(Q(x,y)), for all x,y∈X. |
Let u0=2, we have u1=Tu0=T2=1, u2=Tu1=0, u3=Tu2=0,⋯⋅
supm≥1limi→∞α(ui+1,ui+2)α(ui,ui+1)μ(ui,um)=2<3=1k. |
Also, for every u∈X, we have
limt→∞α(u,ut)<∞ and limt→∞μ(ut,u)<∞. |
All hypotheses of Theorem 2.3 are satisfied and u∗=0 is a fixed point.
As every quasi b-metric space is double controlled quasi metric type space but the converse is not true in general, so we obtain a new result in quasi b-metric space as a corollary of Theorem 2.3.
Theorem 2.5. Let (X,△) be a left complete quasi b -metric space and T:X→X be a mapping. Suppose that there exists k∈(0,1) such that
△(Tx,Ty)≤k(Q(x,y)) |
whenever,
Q(x,y)=max{△(x,y),△(x,Tx),△(x,Tx)△(x,Ty)+△(y,Ty)△(y,Tx)△(x,Ty)+△(y,Tx)}, |
for all x,y∈X. Assume that 0<bk<1. Then, T has a fixed point u∗∈X.
Remark 2.6. In the Example 2.3, note that △ is quasi b -metric with b=103, but we can not apply Theorem 2.5 for any b=103 and k=13, because bk≮1.
Quasi metric version of Theorem 2.2 is given below:
Theorem 2.7. Let (X,△) be a left complete quasi metric space and T:X→P(X) be a multivalued mapping. Suppose that there exists 0<k<1 such that
H△(Tx,Ty)≤k(max{△(x,y),△(x,Tx),△(x,Tx)△(x,Ty)+△(y,Ty)△(y,Tx)△(x,Ty)+△(y,Tx)}) |
for all x,y∈X. Then T has a fixed point u∗∈X.
Now, we extend the sequence {XT(ut)} for two mappings. Let (X,△) be a double controlled quasi metric type space, u0∈X and S,T:X→P(X) be the multivalued mappings on X. Let u1∈Su0 such that △(u0,Su0)=△(u0,u1) and △(Su0,u0)=△(u1,u0). Now, for u1∈X, there exist u2∈Tu1 such that △(u1,Tu1)=△(u1,u2) and △(Tu1,u1)=△(u2,u1). Continuing this process, we construct a sequence ut of points in X such that u2t+1∈Su2t, and u2t+2∈Tu2t+1 with △(u2t,Su2t)=△(u2t,u2t +1), △(Su2t,u2t)=△(u2t+1,u2t) and △(u2t+1,Tu2t+1)=△(u2t+1,u2t+2), △(Tu2t+1,u2t+1)=△(u2t+2,u2t+1). We denote this iterative sequence by {TS(ut)} and say that {TS(ut)} is a sequence in X generated by u0.
Now, we introduce double controlled Reich type contraction.
Definition 2.8. Let X be a non empty set, (X,△) be a left complete double controlled quasi-metric type space with the functions α,μ:X×X→[1,∞) and S,T:X→P(X) be a multivalued mappings. Suppose that the following conditions are satisfied:
H△(Sx,Ty)≤c(△(x,y))+k(△(x,Sx)+△(y,Ty)), | (2.9) |
H△(Tx,Sy)≤c(△(x,y))+k(△(x,Tx)+△(y,Sy)), | (2.10) |
for each x,y∈X, 0<c+2k<1. For u0∈X, choose ut∈{TS(ut)}, we have
supm≥1limi→∞α(ui+1,ui+2)α(ui,ui+1)μ(ui+1,um)<1−kc+k. | (2.11) |
Then the pair (S,T) is called a double controlled Reich type contraction.
The following results extend the results of Reich [18].
Theorem 2.9. Let S,T:X→P(X) be the multivalued mappings, (X,△) be a left complete double controlled quasi metric type space and (S,T) be a pair of double controlled Reich type contraction. Suppose that, for all u∈X
limt→∞α(u,ut) is finite and limt→∞μ(ut,u)<1k. | (2.12) |
Then, S and T have a common fixed point ˙z in X.
Proof. Consider the sequence {TS(ut)}. Now, by Lemma 1.11, we have
△(u2t,u2t+1)≤H△(Tu2t−1,Su2t) | (2.13) |
By using the condition (2.10), we get
△(u2t,u2t+1)≤c(△(u2t−1,u2t)+k(△(u2t−1,Tu2t−1)+△(u2t,Su2t)≤c(△(u2t−1,u2t)+k(△(u2t−1,u2t)+△(u2t,u2t+1)) |
△(u2t,u2t+1)≤η(△(u2t−1,u2t)), | (2.14) |
where η=c+k1−k. Now, by Lemma 1.11, we have
△(u2t−1,u2t)≤H△(Su2t−2,Tu2t−1). |
So, by using the condition (2.9), we have
△(u2t−1,u2t)≤c△(u2t−2,u2t−1)+k(△(u2t−2,Su2t−2)+△(u2t−1,Tu2t−1))≤c△(u2t−2,u2t−1)+k(△(u2t−2,u2t−1)+△(u2t−1,u2 t)) |
△(u2t−1,u2t)≤c+k1−k(△(u2t−2,u2t−1))=η(△(u2t−2,u2t−1)). | (2.15) |
Using (2.14) in (2.15), we have
△(u2t,u2t+1)≤η2△(u2t−2,u2t−1). | (2.16) |
Now, by Lemma 1.11 we have
△(u2t−2,u2t−1)≤H△(Tu2t−3,Su2t−2). |
Using the condition (2.10), we have
△(u2t−2,u2t−1)≤c△(u2t −3,u2t−2)+k(△(u2t−3,u2t−2)+△(u2t−2,u2t−1)) |
implies
△(u2t−2,u2t−1)≤η(△(u2t−3,u2t−2)). | (2.17) |
From (2.16) and (2.17), we have
η2(△(u2t−2,u2t−1))≤η3(△(u2t−3,u2t−2)). | (2.18) |
Using (2.18) in (2.14), we have
△(u2t,u2t+1)≤η3(△(u2t−3,u2t−2)). |
Continuing in this way, we get
△(u2t,u2t+1)≤η2t(△(u0,u1)). | (2.19) |
Similarly, by Lemma 1.11, we have
△(u2t−1,u2t)≤η2t−1(△(u0,u1)). |
Now, we can write inequality (2.19) as
△(ut,ut+1)≤ηt(△(u0,u1)). | (2.20) |
Now, by using (2.20) and by using the technique given in [1], it can easily be proved that {ut} is a left Cauchy sequence. So, for all natural numbers with t<m, we have
limt,m→∞△(ut,um)=0. | (2.21) |
Since (X,△) is a left complete double controlled quasi metric type space. So {ut}→˙z∈X, that is
limt→∞△(ut,˙z)=limt→∞△(˙z,ut)=0. | (2.22) |
Now, we show that ˙z is a common fixed point. We claim that △(˙z,T˙z)=0. On contrary suppose △(˙z,T˙z)>0. Now by Lemma 1.11, we have
△(u2t+1,T˙z)≤H△(Su2t,T˙z). |
△(u2t+1,T˙z)≤c(△(u2t,˙z))+k[△(u2t,u2t+1)+△(˙z,T˙z))] | (2.23) |
Taking limt→∞ on both sides of inequality (2.23), we get
limt→∞△(u2t+1,T˙z)≤climt→∞△(u2t,˙z)+klimt→∞[△(u2t,u2t+1)+△(˙z,T˙z))] |
By using inequalities (2.21) and (2.22), we get
limt→∞△(u2t+1,T˙z)≤k(△(˙z,T˙z)) | (2.24) |
Now,
△(˙z,T˙z)≤α(˙z,u2t+1)△(˙z,u2t+1)+μ(u2t+1,T˙z)△(u2t+1,T˙z) |
Taking limt→∞ and by using inequalities (2.12), (2.22) and (2.24), we get
△(˙z,T˙z)<△(˙z,T˙z). |
It is a contradiction, therefore
△(˙z,T˙z)=0. |
Thus, ˙z∈T˙z. Now, suppose △(˙z,S˙z)>0. By Lemma 1.11, we have
△(u2t,S˙z)≤H△(Tu2t−1,S˙z). |
By inequality (2.10), we get
△(u2t,S˙z)≤c(△(u2t−1,˙z))+k[△(u2t−1,u2t)+△(˙z,S˙z)]. |
Taking limt→∞ on both sides of above inequality, we get
limt→∞△(u2t,S˙z)≤k(△(˙z,S˙z)). | (2.25) |
Now,
△(˙z,S˙z)≤α(˙z,u2t)△(˙z,u2t)+μ(u2t,S˙z)△(u2t,S˙z). |
Taking limt→∞ and by using inequality (2.12), (2.22) and (2.25), we get
△(˙z,S˙z)<△(˙z,S˙z). |
It is a contradiction. Hence, ˙z∈S˙z. Thus, ˙z is a common fixed point for S and T.
Theorem 2.10 presents a result for single-valued mapping which is a consequence of the previous result.
Theorem 2.10. Let (X,△) be a left complete double controlled quasi-metric type space with the functions α,μ:X×X→[1,∞) and S,T:X→X be the mappings such that:
△(Sx,Ty)≤c(△(x,y))+k(△(x,Sx)+△(y,Ty)) |
and
△(Tx,Sy)≤c(△(x,y))+k(△(x,Tx)+△(y,Sy)), |
for each x,y∈X, 0<c+2k<1. Suppose that, for every u∈X and for the Picard sequence {ut}
limt→∞α(u,ut) is finite, limt→∞μ(ut,u)<1k and |
supm≥1limi→∞α(ui+1,ui+2)α(ui,ui+1)μ(ui,um)<1−kc+k. |
Then S and T have a common fixed point u∗∈X.
Quasi b-metric version of Theorem 2.10 is given below:
Theorem 2.11. Let (X,△) be a left complete quasi b-metric type space with the functions α,μ:X×X→[1,∞) and S,T:X→X be the mappings such that:
△(Sx,Ty)≤c(△(x,y))+k(△(x,Sx)+△(y,Ty)) |
and
△(Tx,Sy)≤c(△(x,y))+k(△(x,Tx)+△(y,Sy)), |
for each x,y∈X, 0<c+2k<1 and b<1−kc+k. Then S and T have a common fixed point u∗∈X.
The following example shows that how double controlled quasi metric type spaces can be used where the quasi b-metric spaces cannot.
Example 2.12. Let X={0,12,14,1}. Define △:X×X→[0, ∞) by △(0,12)=1, △(0,14)=13, △(14,0)=15, △(12,0)=1, △(14,12)=3, △(14,1)=12, △(1,14)=13 and △(x,y)=|x−y|, if otherwise. Define α,μ:X×X→[1, ∞) as follows α(12,14)=165, α(0,14)=32, α(14,1)=3, α(1,14)=125 and α(x,y)=1, if otherwise. μ(0,12)=145, μ(1,12)=3 and μ(x,y)=1, if otherwise. Clearly △ is double controlled quasi metric type for all x,y,z∈X. Let, T0={0}, T12={14}, T14={0}, T1={14}, S0=S14={0}, S12={14}, S1={0} and c=25, k=14. Now, if we take the case x=12, y=14, we have
H△(S12,T14)=H△({14},{0})=△(14,0)=15≤1780=c(△(x,y))+k(△(x,Sx)+△(y,Ty)). Also, satisfied for all cases x,y∈X. That are inequalities (2.9) and (2.10) hold. Take u0=1, then u1=Su0=0, u2=Tu1=0, u3=Su2=0⋯.
supm≥1limi→∞α(ui+1,ui+2)α(ui,ui+1)μ(ui+1,um)=1<1513=1−kc+k. |
which shows that inequality (2.11) holds. Thus the pair (S,T) is double controlled Reich type contraction. Finally, for every u∈X, we obtain
limt→∞α(u,ut) is finite, limt→∞μ(ut,u)≤1k. |
All hypotheses of Theorem 2.9 are satisfied and ˙z=0 is a common fixed point.
Note that △ is quasi b-metric with b=3, but Theorem 2.11 can not be applied because b≮1−kc+k, for all b=3. Therefore, this example shows that generalization from a quasi b-metric spaces to a double controlled quasi metric type spaces is real.
Taking c=0 in Theorem 2.9, we get the following result of Kannan-type.
Theorem 2.13. Let (X,△) be a left complete double controlled quasi-metric type space with the functions α,μ:X×X→[1,∞) and S,T:X→P(X) be the multivalued mappings such that:
H△(Sx,Ty)≤k(△(x,Sx)+△(y,Ty)) |
and
H△(Tx,Sy)≤k(△(x,Tx)+△(y,Sy)), |
for each x,y∈X, 0<c+2k<1. Suppose that, for every u∈X and for the sequence {TS(ut)}, we have
limt→∞α(u,ut) is finite, limt→∞μ(ut,u)<1k and |
supm≥1limi→∞α(ui+1,ui+2)α(ui,ui+1)μ(ui,um)<1−kk. |
Then S and T have a common fixed point u∗∈X. Then, S and T have a common fixed point ˙z in X.
Taking c=0 and S=T in Theorem 2.9, we get the following result.
Theorem 2.14. Let (X,△) be a left complete double controlled quasi-metric type space with the functions α,μ:X×X→[1,∞) and T:X→P(X) be a multivalued mapping such that:
H△(Tx,Ty)≤k(△(x,Tx)+△(y,Ty)) |
for each x,y∈X, 0<c+2k<1. Suppose that, for every u∈X and for the sequence {T(ut)}, we have
limt→∞α(u,ut) is finite, limt→∞μ(ut,u)<1k and |
supm≥1limi→∞α(ui+1,ui+2)α(ui,ui+1)μ(ui,um)<1−kk. |
Then T has a fixed point.
Now, the following example illustrates Theorem 2.14.
Example 2.15. Let X=[0,3). Define △:X×X→[0, ∞) as
△(x,y)={0, if x=y,(x−y)2+x, if x≠y. |
with
α(x,y)={2, if x,y≥1,x+22, otherwise., μ(x,y)={1, if x,y≥1, y+22, otherwise. |
Clearly (X,△) is double controlled quasi metric type space. Choose Tx={x4} and k=25. It is clear that T is Kannan type double controlled contraction. Also, for each u ∈X, we have
limt→∞α(u,ut)<∞, limt→∞μ(ut,u)<1k. |
Thus, all hypotheses of Theorem 2.14 are satisfied and ˙z=0 is the fixed point.
Quasi b-metric version of Theorem 2.14 is given below:
Theorem 2.16. Let (X,△) be a left complete quasi b -metric space and T:X→P(X) be a mapping such that:
H△(Tx,Ty)≤k[△(x,Tx)+△(y,Ty)], |
for all x,y∈X, k∈[0,12) and
b<1−kk. |
Then T has a fixed point u∗∈X.
The following remark compare, distinguish and relate the quasi b-metric with the double controlled quasi metric type spaces and illustrate the importance of double controlled quasi metric type spaces.
Remark 2.17. In the Example 2.15, △(x,y)=(x−y)2+x is a quasi b-metric with b≥2, but we can not apply Theorem 2.16 because T is not Kannan type b-contraction. Indeed b≮1−kk, for all b≥2.
It has been shown that double controlled quasi metric is general and better than other metrics, like controlled quasi metric, controlled metric, extended quasi-b-metric, extended b-metric, quasi-b-metric and quasi metric. Also, left, right and dual completeness has been discussed. Results in dual complete spaces can be obtained as corollaries. These results may be extended to obtain results for other contractions. Double controlled quasi metric like ordered spaces can be introduced to establish new results. These results may be applied to find applications to random impulsive differential equations, dynamical systems, graph theory and integral equations.
The authors would also like to thank the editor and the reviewers for their helpful comments and suggestions. This research was funded by the Center of Excellence in Theoretical and Computational Science (TaCS-CoE), KMUTT.
The authors declare that they do not have any competing interests.
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