In the control of the self-driving vehicles, PID controllers are widely used due to their simple structure and good stability. However, in complex self-driving scenarios such as curvature curves, car following, overtaking, etc., it is necessary to ensure the stable control accuracy of the vehicles. Some researchers used fuzzy PID to dynamically change the parameters of PID to ensure that the vehicle control remains in a stable state. It is difficult to ensure the control effect of the fuzzy controller when the size of the domain is not selected properly. This paper designs a variable-domain fuzzy PID intelligent control method based on Q-Learning to make the system robust and adaptable, which is dynamically changed the size of the domain to further ensure the control effect of the vehicle. The variable-domain fuzzy PID algorithm based on Q-Learning takes the error and the error rate of change as input and uses the Q-Learning method to learn the scaling factor online so as to achieve online PID parameters adjustment. The proposed method is verified on the Panosim simulation platform.The experiment shows that the accuracy is improved by 15% compared with the traditional fuzzy PID, which reflects the effectiveness of the algorithm.
Citation: Yongqiang Yao, Nan Ma, Cheng Wang, Zhixuan Wu, Cheng Xu, Jin Zhang. Research and implementation of variable-domain fuzzy PID intelligent control method based on Q-Learning for self-driving in complex scenarios[J]. Mathematical Biosciences and Engineering, 2023, 20(3): 6016-6029. doi: 10.3934/mbe.2023260
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In the control of the self-driving vehicles, PID controllers are widely used due to their simple structure and good stability. However, in complex self-driving scenarios such as curvature curves, car following, overtaking, etc., it is necessary to ensure the stable control accuracy of the vehicles. Some researchers used fuzzy PID to dynamically change the parameters of PID to ensure that the vehicle control remains in a stable state. It is difficult to ensure the control effect of the fuzzy controller when the size of the domain is not selected properly. This paper designs a variable-domain fuzzy PID intelligent control method based on Q-Learning to make the system robust and adaptable, which is dynamically changed the size of the domain to further ensure the control effect of the vehicle. The variable-domain fuzzy PID algorithm based on Q-Learning takes the error and the error rate of change as input and uses the Q-Learning method to learn the scaling factor online so as to achieve online PID parameters adjustment. The proposed method is verified on the Panosim simulation platform.The experiment shows that the accuracy is improved by 15% compared with the traditional fuzzy PID, which reflects the effectiveness of the algorithm.
The new era of fixed point theory associated with metrics is now ineluctably associated with medical biological sciences, abstract terminology, space analysis and epidemiological data mining through engineering. This were often persisted by extending metric fixed point theory to a profusion of literature from computational engineering, fluid mechanics, and medical science. Fixed point theory has made a brief appearance as its own literature in the analysis of metric spaces, as keep referring to many other mathematical groups. Popular uses of metric fixed point theory involve defining and/or generalizing the various metric spaces and the notion of contractions. These extensions are also rendered with the intended consequence of a deeper comprehension of the geometric properties of Banach spaces, set theory, and non-expensive mappings.
A qualitative principle that concerns seeking conditions on the set M structure and choosing a mapping on M to get a fixed point is generally referred to as a fixed point theorem. The fixed point theory framework falls from the larger field of nonlinear functional analysis. Many of the natural sciences and engineering physical questions are usually developed in the form of numerical and analytical equations. Fixed-point assumptions find potential advantages in proving the existence of the solutions of some differential and integral equations which occur in the analysis of heat and mass transfer problems, chemical and electro-chemical processes, fluid dynamics, molecular physics and in many other fields. In 2006, Mustafa and Sims [17] introduced the concept of G-metric space as a generalization of metric space.
It is calculated that investigators get their fresh outputs from engineering mathematics and/or its applications from ∼60%. For example, non-linear integral equations: it has been commonly utilized both in engineering and technology streams of all kinds. These are also appealing to researchers because of the simplicity of using non-linear integral equations and/or their implementations for approximation/numerical/data analysis strategies. In bio-medical sciences, evolution, database technology and computational systems, the steady flow of non-linear integral equations will create fresh avenues in broad directions. Non-linear integral equations are gradually becoming methods for different aspects of hydrodynamics, cognitive science, respectively (see for example [20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36]). The impetus of this work is to prove coupled coincidence point theorems for two mappings via rational type contractions satisfying mixed g-monotone property which are the generalizations of theorems of Chouhan and Richa Sharma [5] and extensions of some other existed results.
The basic definitions and propositions which are used to derive our main results are given below and also note that G-metric and g-monotone property are denoted by ϑ−metric and B-monotone property respectively through out this paper.
Definition 1.1. ([17]) Let M be a set wihch is nonempty and ϑ:M×M×M→R such that
(ϑ1) ϑ(a,b,c)≥0 for all a,b,c∈M with ϑ(a,b,c)=0 if a=b=c;
(ϑ2) ϑ(a,a,b)>0 for all a,b∈M with a≠b;
(ϑ3) ϑ(a,a,b)≤ϑ(a,b,c) for all a,b,c∈M with c≠b;
(ϑ4) ϑ(a,b,c)=ϑ(a,c,b)=ϑ(b,a,c)=ϑ(c,a,b)=ϑ(b,c,a)=ϑ(c,b,a) for all a,b,c∈M;
(ϑ5) ϑ(a,b,c)≤ϑ(a,d,d)+ϑ(d,b,c) for all a,b,c,d∈M.
Then the pair (M,ϑ) is called a ϑ-metric space with ϑ-metric ϑ on M. Axioms (ϑ4) and (ϑ5) are referred to as the symmetry and the rectangle inequality (of ϑ) respectively.
If (M,≤) is partially ordered set in the definition of ϑ-metric space, then (M,ϑ,≤) is partially ordered ϑ-metric space.
Given a ϑ-metric space (M,ϑ), define
ρϑ(a,b)=ϑ(a,b,b)+ϑ(a,a,b) for all a,b,c∈M. | (1.1) |
Then it is seen in [17] that ϑ is a metric on M, and that the family of all ϑ-balls {Bϑ(a,r):a∈M,r>0} is the base topology, called the ϑ-metric topology τ(ϑ) on M, where Bϑ(a,r)={b∈M:ϑ(a,b,b)<r}. Further, it was shown that the ϑ-metric topology coincides with the metric topology induced by the metric ϑ, which allows us to readily transform many concepts from metric spaces into ϑ-metric space.
Definition 1.2. ([17]) If a sequence ⟨aκ⟩ ∞n=1 in (M,ϑ) converges to an element p∈M in the ϑ-metric topology τ(ϑ), then ⟨aκ⟩ ∞n=1 is called ϑ-convergent with limit p.
Proposition 1.1. ([17]) If (M,ϑ) is a ϑ-metric space, then following are equivalent.
(ⅰ) ⟨aκ⟩ ∞κ=1 is ϑ-convergent to a;
(ⅱ) ϑ(aκ,aκ,a)→0 as κ→∞;
(ⅲ) ϑ(aκ,a,a)→0 as κ→∞;
(ⅳ) ϑ(aκ,aη,a)→0 as κ,η→∞.
Definition 1.3. ([17]) A sequence ⟨aκ⟩ in (M,ϑ)is called ϑ-Cauchy if for every σ>0 there exists a positive integer N such that ϑ(aκ,aη,aζ)<σ for all κ,η,ζ≥N.
Proposition 1.2. ([18]) If (M,ϑ) is a ϑ-metric space, then ⟨aκ⟩ is ϑ-Cauchy if and only if for every σ>0, there exists a positive integer N such that ϑ(aκ,aη,aη)<σ for all η,κ≥N.
Proposition 1.3. ([17]) Every ϑ-convergent sequence in (M,ϑ) is ϑ-Cauchy.
Definition 1.4. ([17]) If every ϑ-Cauchy sequence in M converges in M, then (M,ϑ) is called ϑ-complete.
Proposition 1.4. ([17]) If (M,ϑ)is a ϑ-metric space and B is a self-map on M, then B is ϑ-continuous at a point a∈M iff the sequence⟨Baκ⟩ converges to Ba whenever ⟨aκ⟩ converges to a.
Proposition 1.5. ([17]) The ϑ-metric ϑ(a,b,c) is continuous jointly in all the variables a,b and c.
Proposition 1.6. ([17]) If (M,ϑ) is a ϑ-metric space, then
(ⅰ) if ϑ(a,b,c)=0 then a=b=c;
(ⅱ) ϑ(a,b,c)≤ϑ(a,a,b)+ϑ(a,a,c);
(ⅲ) ϑ(a,b,b)≤2ϑ(b,a,a);
(ⅳ) ϑ(a,b,c)≤ϑ(a,x,c)+ϑ(x,b,c) for all a,b,c,x∈M.
Definition 1.5. ([6]) Let (M,ϑ) be a ϑ-metric space and X:M×M→M be a mapping on M×M. Then X is called continuous if the sequence ⟨X(an,bn)⟩ converge to X(a,b) whenever the sequences ⟨an⟩ and ⟨bn⟩ are converge to a and b respectively.
Definition 1.6. ([11]) Let M be a set which is nonempty and X:M×M→M, B:M→M be two mappings. Then X is said to be commute with B if X(Ba,Bb)=B(X(a,b)).
Definition 1.7. ([11]) Let M be a set which is nonempty set and X:M×M→M, B:M→M be two mappings. Then X is said to have mixed B-monotone property if
a1,a2∈M,Ba1≤Ba2⇒X(a1,b)≤X(a2,b)andb1,b2∈M,Bb1≤Bb2⇒X(b1,a)≥X(b2,a)for all a,b∈M. |
If B is an identity mapping in the above deinition, then X has mixed monotone property.
Definition 1.8. ([2]) If M is a set which is nonempty and X:M×M→M is a mapping such thatX(a,b)=a and X(b,a)=b, then (a,b)∈M×M is called coupled fixed point of X.
Definition 1.9. ([17]) If M is a set which is nonempty set and X:M×M→M, B:M→M are two mappings such that X(a,b)=Ba and X(b,a)=Bb, then (a,b)∈M×M is called coupled coincidence point of X and B.
If B is an identity mapping in the above definition, then (a,b) is called coupled fixed point of a mapping X.
Theorem 2.1. Let (M,ϑ,≤) be a partially ordered complete ϑ-metric space and X:M×M→M, B:M→M be two continuous mappings such that X has mixed B-monotone property and
(i) there exist α,β,γ∈[0,1) and L≥0 with 8α+β+γ<1 such that
ϑ(X(x,y),X(u,v),X(w,z))≤αϑ(B,X(x,y),X(y,x))ϑ(Bu,X(u,v),X(u,v))ϑ(Bw,X(w,z),X(w,z))[ϑ(Bx,Bu,Bw)]2+βϑ(Bx,Bu,Bw),+γϑ(By,Bv,Bz)+Lmin{ϑ(Bx,X(u,v),X(w,z)),ϑ(Bu,X(x,y),X(w,z)), ϑ(Bw,X(x,y),X(u,v)),ϑ(Bx,X(x,y),X(x,y)), ϑ(Bu,X(u,v),X(u,v)),ϑ(Bw,X(w,z),X(w,z))}for allx,y,u,v,w,z∈MwithBx≥Bu≥Bw and By≤Bv≤Bz; | (2.1) |
(ii) X(M×M)⊆B(M);
(iii) B commutes with X.
If there exist x0,y0∈M such that Bx0≤X(x0,y0) and By0≥X(y0,x0), then X and B have a coupled coincidence point in M×M.
Proof. Let x0 and y0 be any two elements in M such that Bx0≤X(x0,y0) and By0≥X(y0,x0). Since X(M×M)⊆B(M), we construct two sequences ⟨xκ⟩∞κ=1 and ⟨yκ⟩∞κ=1 in M as follows:
Bxκ+1=X(xκ,yκ) and Byκ+1=X(yκ,xκ) for κ∈N. |
Since X has mixed B-monotone property, we have
Bxκ=X(xκ−1,yκ−1)≤X(xκ,yκ−1)≤X(xκ,yκ)=Bxκ+1 and Byκ+1=X(yκ,xκ)≤X(yκ−1,xκ)≤X(yκ−1,xκ−1)=Byκ. |
Now using (2.1) with x=xκ, y=yκ, u=w=xκ−1 and v=z=yκ−1, we get
ϑ(Bxκ+1,Bxκ,Bxκ)=ϑ(X(xκ,yκ),X(xκ−1,yκ−1),X(xκ−1,yκ−1))≤αϑ(Bxκ,X(xκ,yκ),X(xκ,yκ))ϑ(Bxκ−1,X(xκ−1,yκ−1),X(xκ−1,yκ−1))ϑ(Bxκ−1,X(xκ−1,yκ−1),X(xκ−1,yκ−1))[ϑ(Bxκ,Bxκ−1,Bxκ−1)]2+βϑ(Bxκ,Bxκ−1,Bxκ−1)+γϑ(Byκ,Byκ−1,Byκ−1) +Lmin{ϑ(Bxκ,X(xκ−1,yκ−1),X(xκ−1,yκ−1)),ϑ(Bxκ−1,X(xκ,yκ),X(xκ−1,yκ−1)),ϑ(Bxκ−1,X(xκ,yκ),X(xκ−1,yκ−1)),ϑ(Bxκ,X(xκ,yκ),X(yκ,xκ)),ϑ(Bxκ−1,X(xκ−1,yκ−1),X(xκ−1,yκ−1)),ϑ(Bxκ−1,X(xκ−1,yκ−1),X(xκ−1,yκ−1))}=αϑ(Bxκ,Bxκ+1,Bxκ+1)ϑ(Bxκ−1,Bxκ,Bxκ)ϑ(Bxκ−1,Bxκ,Bxκ)[ϑ(Bxκ,Bxκ−1,Bxκ−1)]2+βϑ(Bxκ,Bxκ−1,Bxκ−1)+γϑ(Byκ,Byκ−1,Byκ−1)+Lmin{ϑ(Bxκ,Bxκ,Bxκ),ϑ(Bxκ−1,Bxκ+1,Bxκ),ϑ(Bxκ−1,Bxκ+1,Bxκ)),ϑ(Bxκ,Bxκ+1,Bxκ+1),ϑ(Bxκ−1,Bxκ,Bxκ),ϑ(Bxκ−1,Bxκ,Bxκ)}≤8αϑ(Bxκ+1,Bxκ,,Bxκ)ϑ(Bxκ,Bxκ−1,Bxκ−1)ϑ(Bxκ,Bxκ−1,Bxκ−1)[ϑ(Bxκ,Bxκ−1,Bxκ−1)]2+βϑ(Bxκ,Bxκ−1,Bxκ−1)+γϑ(Byκ,Byκ−1,Byκ−1) |
so that
ϑ(Bxκ+1,Bxκ,Bxκ)≤1(1−8α)[βϑ(Bxκ,Bxκ−1,Bxκ−1)+γϑ(Byκ,Byκ−1,Byκ−1)] | (2.2) |
Similarly,
ϑ(Byκ+1,Byκ,Byκ)≤1(1−8α)[βϑ(Byκ,Byκ−1,Byκ−1)+γϑ(Bxκ,Bxκ−1,Bxκ−1)] | (2.3) |
Adding (2.2) and (2.3), we have
ϑ(Bxκ+1,Bxκ,Bxκ)+ϑ(Byκ+1,Byκ,Byκ)≤β+γ1−8α[ϑ(Bxκ,Bxκ−1,Bxκ−1)+ϑ(Byκ,Byκ−1,Byκ−1)]. | (2.4) |
Let Δκ=ϑ(Bxκ+1,Bxκ,Bxκ)+ϑ(Byκ+1,Byκ,Byκ) and c=β+γ1−8α
where 0≤c<1 inview of choice of α,β and γ.
Now the inequality (2.4) becomes as follows
Δκ≤c.Δκ−1 for κ∈N
Consequently Δκ≤cΔκ−1≤c2Δκ−2≤...≤cκΔ0
If Δ0=0, we have ϑ(Bx1,Bx0,Bx0)+ϑ(By1,By0,By0)=0
or ϑ(X(x0,y0),Bx0,Bx0)+ϑ(X(y0,x0),By0,By0)=0 which implies that X(x0,y0)=Bx0 and X(y0,x0)=By0.
That is, (x0,y0) is a coupled coincidence point of X and B.
Suppose that Δ0>0.
Now by applying rectangle inequality of ϑ-metric repeatedly and using inequality Δκ≤cκΔ0, we have
ϑ(Bxη,Bxκ,Bxκ)+ϑ(Byη,Byκ,Byκ)≤[ϑ(Bxκ+1,Bxκ,Bxκ)+ϑ(Bxη,Bxκ+1,Bxκ+1)]+[ϑ(Byκ+1,Byκ,Byκ)+ϑ(Byη,Byκ+1,Byκ+1)]≤[ϑ(Bxκ+1,Bxκ,Bxκ)+ϑ(Bxκ+2,Bxκ+1,Bxκ+1)+ϑ(Bxη,Bxκ+2,Bxκ+2)][ϑ(Byκ+1,Byκ,Byκ)+ϑ(Byκ+2,Byκ+1,Byκ+1)+ϑ(Byη,Byκ+2,Byκ+2)]⋮≤[ϑ(Bxκ+1,Bxκ,Bxκ)+ϑ(Bxκ+2,Bxκ+1,Bxκ+1)+⋯+ϑ(Bxη,Bxη−1,Bxη−1)][ϑ(Byκ+1,Byκ,Byκ)+ϑ(Byκ+2,Byκ+1,Byκ+1)+⋯+ϑ(Byη,Byη−1,Byη−1)]=Δκ+Δκ+1+Δκ+2+⋯+Δη−1≤[cκ+cκ+1+⋯+cη−1]Δ0≤cκ⋅11−cΔ0 for η>κ |
or
ϑ(Bxη,Bxκ,Bxκ)+ϑ(Byη,Byκ,Byκ)≤cκ⋅11−cΔ0 for η>κ. | (2.5) |
Since 0≤c<1, cκ→0 as κ→∞.
Now applying limit as κ→∞ with η>κ in the inequality (2.5), we have
ϑ(Bxη,Bxκ,Bxκ)+ϑ(Byη,Byκ,Byκ)≤0 which follows that ⟨xκ⟩∞κ=1 and ⟨yκ⟩∞κ=1 are cauchy sequences in M.
Since (M,ϑ) is a partially ordered complete ϑ-metric space, there exist p,q∈M such that xκ→p and yκ→q.
Now we prove that (p,q) is a coupled coincidence point of X and B.
Since X commutes with B, we have
X(Bxκ,Byκ)=B(X(xκ,yκ))=B(Bxκ+1) andX(Byκ,Bxκ)=B(X(yκ,xκ))=B(Byκ+1) |
Since X and B are continuous, we have
limκ→∞X(Bxκ,Byκ)=limκ→∞B(Bxκ+1)=Bp andlimκ→∞X(Byκ,Bxκ)=limκ→∞B(Byκ+1)=Bq |
Snice ϑ is continuous in all its variables, we have
ϑ(X(p,q),Bp,Bp)=ϑ(limκ→∞X(Bxκ,Byκ),Bp,Bp))=ϑ(Bp,Bp,Bp) |
so that
ϑ(X(p,q),Bp,Bp)=0 |
which implies that X(p,q)=Bp.
Similarly, it can be proved that X(q,p)=Bq.
Hence (p,q) is a coupled coincidence point of X and B.
Remark 2.1. (ⅰ.) If we assume B is an identity mapping and γ=0 in the Theorem 2.1, then we get Theorem 3.1 in the results of the Chouhan and Richa Sharma [5];
(ⅱ.) If we take, α=0 and β=α, γ=β in the Theorem 2.1, we get Theorem 3.1 in the results of Chandok et al. [3];
(ⅲ.) By taking γ=0 and L=0, we get Theorem 2.1 in the results of Chakrabarti [4].
That is Theorem 2.1 is generalization and extension of above three results.
The following is the example to illusrative Theorem 2.1.
Example 2.1. Let M=[0,1], with ϑ-metric ϑ(x,y,z)={0, x=y=z,max{x,y,z},otherwise.
Define partial order on X as x≥y for any x,y∈M. Then(M,ϑ,≤) is a partially ordered ϑ-complete.
Define X:M×M→M by X(x,y)={x3(y2+2), x≥y0 otherwise
and B:M→M by Bx=x4.
Then clearlyX, B are continuous and X satisfies mixed B-monotone property. We show that X satisfies the inequality (2.1) with α=β=γ=132 so that 0≤8α+β+γ<1 and for any L≥0.
Let x,y,z,u,v,w∈M be such that x≥u≥w and y≤v≤z.
We discuss four cases:
Case (ⅰ): If x≥y,u≥v and w≥z then we have X(x,y)=x3(y2+2), X(u,v)=u3(v2+2) and X(w,z)=w3(z2+2)
ϑ(X(x,y),X(u,v),X(w,z))=max{x3(y2+2),u3(v2+2),w3(z2+2)}=x3(y2+2)≤132[16wuxx2+x4+u4]=132ϑ(Bx,X(x,y),X(y,x))ϑ(Bu,X(u,v),X(u,v))ϑ(Bw,X(w,z),X(w,z))[ϑ(Bx,Bu,Bw)]2+132ϑ(Bx,Bu,Bw)+132ϑ(By,Bv,Bz)≤132ϑ(Bx,X(x,y),X(y,x))ϑ(Bu,X(u,v),X(u,v))ϑ(Bw,X(w,z),X(w,z))[ϑ(Bx,Bu,Bw)]2+132ϑ(Bx,Bu,Bw)+132ϑ(By,Bv,Bz)+Lmin{ϑ(Bx,X(u,v),X(w,z)),ϑ(Bu,X(x,y),X(w,z)), ϑ(Bw,X(x,y),X(u,v)),ϑ(Bx,X(x,y),X(x,y)), ϑ(Bu,X(u,v),X(u,v)),ϑ(Bw,X(w,z),X(w,z)} |
Case (ⅱ): If x≥y,u≥v and w<z then we have X(x,y)=x3(y2+2), X(u,v)=u3(v2+2) and X(w,z)=0
ϑ(X(x,y),X(u,v),X(w,z))=max{x3(y2+2),u3(v2+2),0}=x3(y2+2)≤132[16wuxx2+x4+u4]≤132ϑ(Bx,X(x,y),X(y,x))ϑ(Bu,X(u,v),X(u,v))ϑ(Bw,X(w,z),X(w,z))[ϑ(Bx,Bu,Bw)]2+132ϑ(Bx,Bu,Bw)+132ϑ(By,Bv,Bz)+Lmin{ϑ(Bx,X(u,v),X(w,z)),ϑ(Bu,X(x,y),X(w,z)), ϑ(Bw,X(x,y),X(u,v)),ϑ(Bx,X(x,y),X(x,y)), ϑ(Bu,X(u,v),X(u,v)),ϑ(Bw,X(w,z),X(w,z))} |
Case (ⅲ): If x≥y,u<v and w<z, then we have X(x,y)=x3(y2+2), X(u,v)=0 and X(w,z)=0
ϑ(X(x,y),X(u,v),X(w,z))=max{x3(y2+2),0,0}=x3(y2+2)≤132[16wuxx2+x4+u4]≤132ϑ(Bx,X(x,y),X(y,x))ϑ(Bu,X(u,v),X(u,v))ϑ(Bw,X(w,z),X(w,z))[ϑ(Bx,Bu,Bw)]2+132ϑ(Bx,Bu,Bw)+132ϑ(By,Bv,Bz)+Lmin{ϑ(Bx,X(u,v),X(w,z)),ϑ(Bu,X(x,y),X(w,z)), ϑ(Bw,X(x,y),X(u,v)),ϑ(Bx,X(x,y),X(x,y)), ϑ(Bu,X(u,v),X(u,v)),ϑ(Bw,X(w,z),X(w,z))} |
Case (iv): If x<y,u<v and w<z, then X(x,y)=x3(y2+2), X(u,v)=0 and X(w,z)=0, it follows that
ϑ(X(x,y),X(u,v),X(w,z))≤132ϑ(Bx,X(x,y),X(y,x))ϑ(Bu,X(u,v),X(u,v))ϑ(Bw,X(w,z),X(w,z))[ϑ(Bx,Bu,Bw)]2+132ϑ(Bx,Bu,Bw)+132ϑ(By,Bv,Bz)+Lmin{ϑ(Bx,X(u,v),X(w,z)),ϑ(Bu,X(x,y),X(w,z)), ϑ(Bw,X(x,y),X(u,v)),ϑ(Bx,X(x,y),X(x,y)), ϑ(Bu,X(u,v),X(u,v)),ϑ(Bw,X(w,z),X(w,z))} |
In similar manner, the cases x<y,u≥v,w≥z; x<y,u<v,w≥z and all others can be handled.
Thus X and B satify all the conditions of Theorem 2.1 and also note that (0,0) is a coupled coincidence point of X and B.
Theorem 2.2. Let (M,ϑ,≤) be a partially ordered complete ϑ-metric space and X:M×M→M, B:M→M be two continuous mappings such that X has mixed B-monotone property and
(i) there exist α,β,γ∈[0,1) and L≥0 with 2α+β+γ<1 such that
ϑ(X(x,y),X(u,v),X(w,z))≤αϑ(Bx,X(x,y),X(y,x))[1+ϑ(Bu,X(u,v),X(u,v))][1+ϑ(Bw,X(w,z),X(w,z))][1+2ϑ(Bx,Bu,Bw)]2+βϑ(Bx,Bu,Bw),+γϑ(By,Bv,Bz)+Lmin{ϑ(Bx,X(u,v),X(w,z)),ϑ(Bu,X(x,y),X(w,z)), ϑ(Bx,X(x,y),X(x,y),ϑ(Bu,X(u,v),X(u,v))}for allx,y,u,v,w,z∈MwithBx≥Bu≥Bw and By≤Bv≤Bz; | (2.6) |
(ii) X(M×M)⊆g(M);
(iii) B commutes with X.
If there exist x0,y0∈M such that Bx0≤X(x0,y0) and By0≥X(y0,x0), then X and B have a coupled coincidence point in M×M.
Proof. Let x0 and y0 be any two elements in M such that Bx0≤X(x0,y0) and By0≥X(y0,x0). Since X(M×M)⊆B(M), we constuct two sequences ⟨xκ⟩∞κ=1 and ⟨yκ⟩∞κ=1 in M as follows:
Bxκ+1=X(xκ,yκ) and Byκ+1=X(yκ,xκ) for κ∈N. |
Since X has mixed B-monotone property, we have
Bxκ=X(xκ−1,yκ−1)≤X(xκ,yκ−1)≤X(xκ,yκ)=Bxκ+1 and Byκ+1=X(yκ,xκ)≤X(yκ−1,xκ)≤X(yκ−1,xκ−1)=Byκ. |
Now using (2.1) with x=xκ, y=yκ, u=w=xκ−1 and v=z=yκ−1, we get
ϑ(Bxκ+1,Bxκ,Bxκ)=ϑ(X(xκ,yκ),X(xκ−1,yκ−1),X(xκ−1,yκ−1))≤αϑ(Bxκ,X(xκ,yκ),X(xκ,yκ))[1+ϑ(Bxκ−1,X(xκ−1,yκ−1),X(xκ−1,yκ−1))][1+ϑ(Bxκ−1,X(xκ−1,yκ−1),X(xκ−1,yκ−1))][1+2ϑ(Bxκ,Bxκ−1,Bxκ−1)]2+βϑ(Bxκ,Bxκ−1,Bxκ−1)+γϑ(Byκ,Byκ−1,Byκ−1) +Lmin{ϑ(Bxκ,X(xκ−1,yκ−1),X(xκ−1,yκ−1)),ϑ(Bxκ−1,X(xκ,yκ),X(xκ−1,yκ−1)),ϑ(Bxκ,X(xκ,yκ),X(yκ,xκ)),ϑ(Bxκ−1,X(xκ−1,yκ−1),X(xκ−1,yκ−1))}=αϑ(Bxκ,Bxκ+1,Bxκ+1)[1+ϑ(Bxκ−1,Bxκ,Bxκ)][1+ϑ(Bxκ−1,Bxκ,Bxκ)][1+2ϑ(Bxκ,Bxκ−1,Bxκ−1)]2+βϑ(Bxκ,Bxκ−1,Bxκ−1)+γϑ(Byκ,Byκ−1,Byκ−1)+Lmin{ϑ(Bxκ,Bxκ,Bxκ),ϑ(Bxκ−1,Bxκ+1,Bxκ),ϑ(Bxκ,Bxκ+1,Bxκ+1),ϑ(Bxκ−1,Bxκ,Bxκ)}≤α2ϑ(Bxκ+1,Bxκ,Bxκ)[1+2ϑ(Bxκ,Bxκ−1,Bxκ−1)][1+2ϑ(Bxκ,Bxκ−1,Bxκ−1)][1+2ϑ(Bxκ,Bxκ−1,Bxκ−1)]2+βϑ(Bxκ,Bxκ−1,Bxκ−1)+γϑ(Byκ,Byκ−1,Byκ−1) |
so that
ϑ(Bxκ+1,Bxκ,Bxκ)≤1(1−2α)[βϑ(Bxκ,Bxκ−1,Bxκ−1)+γϑ(Byκ,Byκ−1,Byκ−1)] | (2.7) |
Similarly,
ϑ(Byκ+1,Byκ,Byκ)≤1(1−2α)[βϑ(Byκ,Byκ−1,Byκ−1)+γϑ(Bxκ,Bxκ−1,Bxκ−1)] | (2.8) |
Adding (2.7) and (2.8), we have
ϑ(Bxκ+1,Bxκ,Bxκ)+ϑ(Byκ+1,Byκ,Byκ)≤β+γ1−2α[ϑ(Bxκ,Bxκ−1,Bxκ−1)+ϑ(Byκ,Byκ−1,Byκ−1)]. | (2.9) |
Let Δκ=ϑ(Bxκ+1,Bxκ,Bxκ)+ϑ(Byκ+1,Byκ,Byκ) and c=β+γ1−2α where 0≤c<1 inview of choice of α,β and γ.
Now the inequality (2.9) becomes as follows
Δκ≤c.Δκ−1 for n∈N.
Consequently Δκ≤cΔκ−1≤c2Δκ−2≤...≤cκΔ0.
If Δ0=0, we have ϑ(Bx1,Bx0,Bx0)+ϑ(By1,By0,By0)=0
or ϑ(X(x0,y0),Bx0,Bx0)+ϑ(X(y0,x0),By0,By0)=0 which implies that X(x0,y0)=Bx0 and X(y0,x0)=By0.
That is, (x0,y0) is a coupled coincidence point of X and B.
Suppose that Δ0>0.
Now using repeated application of rectangle inequality of ϑ-metric and inequality Δκ≤cκΔ0, we have
ϑ(Bxη,Bxκ,Bxκ)+ϑ(Byη,Byκ,Byκ)≤[ϑ(Bxκ+1,Bxκ,Bxκ)+ϑ(Bxη,Bxκ+1,Bxκ+1)]+[ϑ(Byκ+1,Byκ,Byκ)+ϑ(Byη,Byκ+1,Byκ+1)]≤[ϑ(Bxκ+1,Bxκ,Bxκ)+ϑ(Bxκ+2,Bxκ+1,Bxκ+1)+ϑ(Bxη,Bxκ+2,Bxκ+2)][ϑ(Byκ+1,Byκ,Byκ)+ϑ(Byκ+2,Byκ+1,Byκ+1)+ϑ(Byη,Byκ+2,Byκ+2)]⋮≤[ϑ(Bxκ+1,Bxκ,Bxκ)+ϑ(Bxκ+2,Bxκ+1,Bxκ+1)+⋯+ϑ(Bxη,Bxη−1,Bxη−1)][ϑ(Byκ+1,Byκ,Byκ)+ϑ(Byκ+2,Byκ+1,Byκ+1)+⋯+ϑ(Byη,Byη−1,Byη−1)]=Δκ+Δκ+1+Δκ+2+⋯+Δη−1≤[cκ+cκ+1+⋯+cη−1]Δ0≤cκ⋅11−cΔ0 for η>κ |
or
ϑ(Bxη,Bxκ,Bxκ)+ϑ(Byη,Byκ,Byκ)≤cκ⋅11−cΔ0 for η>κ. | (2.10) |
Since 0≤c<1, cκ→0 as κ→∞.
Now applying limit as κ→∞ with η>κ in the inequality (2.10), we have
ϑ(Bxη,Bxκ,Bxκ)+ϑ(Byη,Byκ,Byκ)≤0 which follows that ⟨xκ⟩∞κ=1 and ⟨yκ⟩∞κ=1 are ϑ−cauchy sequences in M.
Since (M,ϑ) is a partially ordered ϑ-complete, there exist a,b∈M such that xκ→a and yκ→b.
Now we prove that (a,b) is a coupled coincidence point of X and B.
Since X commutes with B, we have
X(Bxκ,Byκ)=B(X(xκ,yκ))=B(Bxκ+1) andX(Byκ,Bxκ)=B(X(yκ,xκ))=B(Byκ+1) |
Since X and B are continuous, we have
limκ→∞X(Bxκ,Byκ)=limκ→∞B(Bxκ+1)=Ba andlimκ→∞X(Byκ,Bxκ)=limκ→∞B(Byκ+1)=Bb |
Since ϑ is continuous in all its variables, we have
ϑ(X(a,b),Ba,Bb)=ϑ(limκ→∞X(Bxκ,Byκ),Ba,Ba)=ϑ(Ba,Ba,Ba) |
so that
ϑ(X(a,b),Bb,Bb)=0 |
which implies that X(a,b)=Ba.
Similarly, it can be verified that X(b,a)=Bb.
Thus(a,b) is a coupled coincidence point of X and B in M×M.
Remark 2.2. If we take B is an identity mapping and γ=0 in the Theorem 2.2, we get Theorem 3.1 in the results of the Chouhan and Richa Sharma[5].
Theorem 2.3. Let (M,ϑ,≤) be a partially ordered complete ϑ-metric space and X:M×M→M, B:M→M be two continuous mappings such that X has mixed B-monotone property and
(i) there exist non negative real numbers Ψ1,Ψ2,Ψ3,Ψ4,Ψ5,Ψ6,Ψ7,Ψ8 and Ψ9 with 0≤Ψ1+Ψ2+Ψ3+Ψ4+Ψ5+Ψ6+Ψ7+Ψ8+Ψ9<1 such that
ϑ(X(x,y),X(u,v),X(w,z))≤Ψ1ϑ(Bx,Bu,Bw)+ϑ(By,Bv,Bz)2+Ψ2ϑ(X(x,y),X(u,v),X(w,z))⋅ϑ(Bx,Bu,Bw)1+ϑ(Bx,Bu,Bw)+ϑ(By,Bv,Bz),+Ψ3ϑ(X(x,y),X(u,v),X(w,z))⋅ϑ(By,Bv,Bz)1+ϑ(Bx,Bu,Bw)+ϑ(By,Bv,Bz)+Ψ4ϑ(Bx,Bx,X(x,y))⋅ϑ(Bx,Bu,Bw)1+ϑ(Bx,Bu,Bw)+ϑ(By,Bv,Bz),+Ψ5ϑ(Bx,Bx,X(x,y))⋅ϑ(By,Bv,Bz)1+ϑ(Bx,Bu,Bw)+ϑ(By,Bv,Bz)+Ψ6ϑ(Bu,Bu,X(u,v))⋅ϑ(Bx,Bu,Bw)1+ϑ(Bx,Bu,Bw)+ϑ(By,Bv,Bz),+Ψ7ϑ(Bu,Bu,X(u,v))⋅ϑ(By,Bv,Bz)1+ϑ(Bx,Bu,Bw)+ϑ(By,Bv,Bz)+Ψ8ϑ(Bw,Bw,X(w,z))⋅ϑ(Bx,Bu,Bw)1+ϑ(Bx,Bu,Bw)+ϑ(By,Bv,Bz),+Ψ9ϑ(Bw,Bw,X(w,z))⋅ϑ(By,Bv,Bz)1+ϑ(Bx,Bu,Bw)+ϑ(By,Bv,Bz)for allx,y,u,v,w,z∈MwithBx≥Bu≥Bw and By≤Bv≤Bz; | (2.11) |
(ii) X(M×M)⊆B(M);
(iii) B commutes with X.
If there exist x0,y0∈M such that Bx0≤X(x0,y0) and By0≥X(y0,x0), then X and B have a coupled coincidence point in M×M.
Proof. Let x0 and y0 be any two elements in M such that Bx0≤X(x0,y0) and By0≤X(y0,x0). Since X(M×M)⊆g(M), we constuct two sequences ⟨xκ⟩∞κ=1 and ⟨yκ⟩∞κ=1 in M as follows:
Bxκ+1=X(xκ,yκ) and Byκ+1=X(yκ,xκ) for κ∈N. |
Since X has mixed B-monotone property, we have
Bxκ=X(xκ−1,yκ−1)≤X(xκ,yκ−1)≤X(xκ,yκ)=Bxκ+1 and Byκ+1=X(yκ,xκ)≤X(yκ−1,xκ)≤X(yκ−1,xκ−1)=Byκ. |
Now using (2.11) with x=xκ, y=yκ, u=w=xκ−1 and v=z=yκ−1, we get
ϑ(Bxκ+1,Bxκ,Bxκ)=ϑ(X(xκ,yκ),X(xκ−1,yκ−1),X(xκ−1,yκ−1))≤Ψ1ϑ(Bxκ,Bxκ−1,Bxκ−1)+ϑ(Byκ,Byκ−1,Byκ−1)2+Ψ2ϑ(X(xκ,yκ),X(xκ−1,yκ−1),X(xκ−1,yκ−1))⋅ϑ(Bxκ,Bxκ−1,Bxκ−1)1+ϑ(Bxκ,Bxκ−1,Bxκ−1)+ϑ(Byκ,Byκ−1,Byκ−1),+Ψ3ϑ(X(xκ,yκ),X(xκ−1,yκ−1),X(xκ−1,yκ−1))⋅ϑ(Byκ,Byκ−1,Byκ−1)1+ϑ(Bxκ,Bxκ−1,Bxκ−1)+ϑ(Byκ,Byκ−1,Byκ−1)+Ψ4ϑ(Bxκ,Bxκ,X(xκ,yκ))⋅ϑ(Bxκ,Bxκ−1,Bxκ−1)1+ϑ(Bxκ,Bxκ−1,Bxκ−1)+ϑ(Byκ,Byκ−1,Byκ−1),+Ψ5ϑ(Bxκ,Bxκ,X(xκ,yκ))⋅ϑ(Byκ,Byκ−1,Byκ−1)1+ϑ(Bxκ,Bxκ−1,Bxκ−1)+ϑ(Byκ,Byκ−1,Byκ−1)+Ψ6ϑ(Bxκ−1,Bxκ−1,X(xκ−1,yκ−1))⋅ϑ(Bxκ,Bxκ−1,Bxκ−1)1+ϑ(Bxκ,Bxκ−1,Bxκ−1)+ϑ(Byκ,Byκ−1,Byκ−1),+Ψ7ϑ(Bxκ−1,Bxκ−1,X(xκ−1,yκ−1))⋅ϑ(Byκ,Byκ−1,Byκ−1)1+ϑ(Bxκ,Bxκ−1,Bxκ−1)+ϑ(Byκ,Byκ−1,Byκ−1)+Ψ8ϑ(Bxκ−1,Bxκ−1,X(xκ−1,yκ−1))⋅ϑ(Bxκ,Bxκ−1,Bxκ−1)1+ϑ(Bxκ,Bxκ−1,Bxκ−1)+ϑ(Byκ,Byκ−1,Byκ−1),+Ψ9ϑ(Bxκ−1,Bxκ−1,X(xκ−1,yκ−1))⋅ϑ(Byκ,Byκ−1,Byκ−1)1+ϑ(Bxκ,Bxκ−1,Bxκ−1)+ϑ(Byκ,Byκ−1,Byκ−1) |
=Ψ1ϑ(Bxκ,Bxκ−1,Bxκ−1)+ϑ(Byκ,Byκ−1,Byκ−1)2+Ψ2ϑ(Bxκ+1,Bxκ,Bxκ)⋅ϑ(Bxκ,Bxκ−1,Bxκ−1)1+ϑ(Bxκ,Bxκ−1,Bxκ−1)+ϑ(Byκ,Byκ−1,Byn−1),+Ψ3ϑ(Bxκ+1,Bxκ,Bxκ)⋅ϑ(Byκ,Byκ−1,Byκ−1)1+ϑ(Bxκ,Bxκ−1,Bxκ−1)+ϑ(Byκ,Byκ−1,Byκ−1)+Ψ4ϑ(Bxκ,Bxκ,Bxκ+1)⋅ϑ(Bxκ,Bxκ−1,Bxκ−1)1+ϑ(Bxκ,Bxκ−1,Bxκ−1)+ϑ(Byκ,Byκ−1,Byκ−1),+Ψ5ϑ(Bxκ,Bxκ,Bxκ+1)⋅ϑ(Byκ,Byκ−1,Byκ−1)1+ϑ(Bxκ,Bxκ−1,Bxκ−1)+ϑ(Byκ,Byκ−1,Byκ−1)+Ψ6ϑ(Bxκ−1,Bxκ−1,Bxκ)⋅ϑ(Bxκ,Bxκ−1,Bxκ−1)1+ϑ(Bxκ,Bxκ−1,Bxκ−1)+ϑ(Byκ,Byκ−1,Byκ−1),+Ψ7ϑ(Bxκ−1,Bxκ−1,Bxκ)⋅ϑ(Byκ,Byκ−1,Byκ−1)1+ϑ(Bxκ,Bxκ−1,Bxκ−1)+ϑ(Byκ,Byκ−1,Byκ−1)+Ψ8ϑ(Bxκ−1,Bxκ−1,Bxκ)⋅ϑ(Bxκ,Bxκ−1,Bxκ−1)1+ϑ(Bxκ,Bxκ−1,Bxκ−1)+ϑ(Byκ,Byκ−1,Byκ−1),+Ψ9ϑ(Bxκ−1,Bxκ−1,Bxκ)⋅ϑ(Byκ,Byκ−1,Byκ−1)1+ϑ(Bxκ,Bxκ−1,Bxκ−1)+ϑ(Byκ,Byκ−1,Byκ−1) |
≤Ψ1ϑ(Bxκ,Bxκ−1,Bxκ−1)+ϑ(Byκ,Byκ−1,Byκ−1)2+Ψ2ϑ(Bxκ+1,Bxκ,Bxκ)+Ψ3ϑ(Bxκ+1,Bxκ,Bxκ)+Ψ4ϑ(Bxκ+1,Bxκ,Bxκ)+Ψ5ϑ(Bxκ+1,Bxκ,Bxκ)+Ψ6ϑ(Bxκ−1,Bxκ−1,Bxκ)+Ψ7ϑ(Bxκ−1,Bxκ−1,Bxκ)+Ψ8ϑ(Bxκ−1,Bxκ−1,Bxκ)+Ψ9ϑ(Bxκ−1,Bxκ−1,Bxκ) |
so that
ϑ(Bxκ+1,Bxκ,Bxκ)≤Ψ12+Ψ6+Ψ7+Ψ8+Ψ9[1−(Ψ2+Ψ3+Ψ4+Ψ5)]ϑ(Bxκ,Bxκ−1,Bxκ−1)+Ψ12[1−(Ψ2+Ψ3+Ψ4+Ψ5)]ϑ(Byκ,Byκ−1,Byκ−1) | (2.12) |
Similarly,
ϑ(Byκ+1,Byκ,Byκ)≤Ψ12+Ψ6+Ψ7+Ψ8+Ψ9[1−(Ψ2+Ψ3+Ψ4+Ψ5)]ϑ(Byκ,Byκ−1,Byκ−1)+Ψ12[1−(Ψ2+Ψ3+Ψ4+Ψ5)]ϑ(Bxκ,Bxκ−1,Bxκ−1) | (2.13) |
Adding (2.12) and (2.13), we have
ϑ(Bxκ+1,Bxκ,Bxκ)+ϑ(Byκ+1,Byκ,Byκ)≤Ψ1+Ψ6+Ψ7+Ψ8+Ψ91−(Ψ2+Ψ3+Ψ4+Ψ5)[ϑ(Bxκ,Bxκ−1,Bxκ−1)+ϑ(Byκ,Byκ−1,Byκ−1)]. | (2.14) |
Let Δκ=ϑ(Bxκ+1,Bxκ,Bxκ)+ϑ(Byκ+1,Byκ,Byκ) and c=Ψ1+Ψ6+Ψ7+Ψ8+Ψ91−(Ψ2+Ψ3+Ψ4+Ψ5) where 0≤c<1 inview of choice of Ψ1,Ψ2,Ψ3,Ψ4,Ψ5,Ψ6,Ψ7,Ψ8 and Ψ9.
Now the inequality (2.14) becomes as follows
Δκ≤c.Δκ−1 for n∈N,
Consequently Δκ≤cΔκ−1≤c2Δκ−2≤...≤cκΔ0.
If Δ0=0, we have ϑ(Bx1,Bx0,Bx0)+ϑ(By1,By0,By0)=0,
or ϑ(X(x0,y0),Bx0,Bx0)+ϑ(X(y0,x0),By0,By0)=0 which implies that X(x0,y0)=Bx0 and X(y0,x0)=By0.
That is, (x0,y0) is a coupled coincidence point of X and B.
Suppose that Δ0>0.
Now using rectangle inequality of ϑ-metric repeatedly and inequality Δκ≤cκΔ0, we have
ϑ(Bxη,Bxκ,Bxκ)+ϑ(Byη,Byκ,Byκ)≤[ϑ(Bxκ+1,Bxκ,Bxκ)+ϑ(Bxη,Bxκ+1,Bxκ+1)]+[ϑ(Byκ+1,Byκ,Byκ)+ϑ(Byη,Byκ+1,Byκ+1)]≤[ϑ(Bxκ+1,Bxκ,Bxκ)+ϑ(Bxκ+2,Bxκ+1,Bxκ+1)+ϑ(Bxη,Bxκ+2,Bxκ+2)][ϑ(Byκ+1,Byκ,Byκ)+ϑ(Byκ+2,Byκ+1,Byκ+1)+ϑ(Byη,Byκ+2,Byκ+2)]⋮≤[ϑ(Bxκ+1,Bxκ,Bxκ)+ϑ(Bxκ+2,Bxκ+1,Bxκ+1)+⋯+ϑ(Bxη,Bxη−1,Bxη−1)][ϑ(Byκ+1,Byκ,Byκ)+ϑ(Byκ+2,Byκ+1,Byκ+1)+⋯+ϑ(Byη,Byη−1,Byη−1)]=Δκ+Δκ+1+Δκ+2+⋯+Δη−1≤[cκ+cκ+1+⋯+cη−1]Δ0≤cκ⋅11−cΔ0 for η>κ |
or
ϑ(Bxη,Bxκ,Bxκ)+ϑ(Byη,Byκ,Byκ)≤cκ⋅11−cΔ0 for η>κ. | (2.15) |
Since 0≤c<1, cκ→0 as κ→∞.
Now applying limit as κ→∞ with η>κ in the inequality (2.15), we have
ϑ(Bxη,Bxκ,Bxκ)+ϑ(Byη,Byκ,Byκ)≤0 which follows that ⟨xκ⟩∞κ=1 and ⟨yκ⟩∞κ=1 are ϑ−cauchy sequences in M.
Since (M,ϑ) is a partially ordered complete ϑ-metric space, there exist r,s∈M such that xκ→r and yκ→s.
Now we prove that (r,s) is a coupled coincidence point of X and B.
Since X commutes with B, we have
X(Bxκ,Byκ)=B(X(xκ,yκ))=B(Bxκ+1) andX(Byκ,Bxκ)=B(X(yκ,xκ))=B(Byκ+1) |
Since X and B are continuous, we have
limκ→∞X(Bxκ,Byκ)=limκ→∞B(Bxκ+1)=Br andlimκ→∞X(Byκ,Bxκ)=limκ→∞B(Byκ+1)=Bs |
Since ϑ is continuous in all its variables, we have
ϑ(X(r,s),Br,Bs)=ϑ(limκ→∞X(Bxκ,Byκ),Br,Br)=ϑ(Br,Br,Br), |
so that
ϑ(X(r,s),Br,Bs)=0 |
which implies that X(r,s)=Br.
Similarly, it can be proved that X(s,r)=Bs.
Hence (s,r) is a coupled coincidence point of X and B.
Taking α=0, L=0 and B is an identity mapping in Theorem 2.1, we get
Corollary 2.1. Let (M,ϑ,≤) be a partially ordered complete ϑ-metric space and X:M×M→M be a continuous mapping such that X has mixed monotone property and there exist β,γ∈[0,1) with β+γ<1 such that
ϑ(X(x,y),X(u,v),X(w,z))≤βϑ(x,u,w)+γϑ(y,v,z). | (2.16) |
If there exist x0,y0∈M such that x0≤X(x0,y0) and y0≥X(y0,x0), then X has a coupled fixed point in M×M.
Consider the following system of nonlinear integral equations:
f(s)=q(s)+∫a0λ(s,t)[X1(t,f(t))+X2(t,g(t))]dt,g(s)=q(s)+∫a0λ(s,t)[X1(t,g(t))+X2(t,f(t))]dt,s∈[0,L],L>0. | (3.1) |
Let M=C([0,L],R) be the class of all real valued continuous functions on [0,L].
Define
ϑ(a,b,c)=sup{|a(s)−b(s)|/s∈[0,L]}x+sup{|b(s)−c(s)|/s∈[0,L]}+sup{|c(s)−a(s)|/s∈[0,L]} |
and the partial ordered relation on M as
a≤b⇔a(s)≤b(s) for all a,b∈M and s∈[0,L]. | (3.2) |
Then (M,ϑ,≤) is a partially ordered complete ϑ-metric space. We make the following assumptions:
(a) The mappings X1:[0,L]×R→R, X2:[0,L]×R→R, q:[0,L]→R and λ:[0,L]×R→[0,∞) are continuous;
(b) there exist c>0 and β,γ∈[0,1) with β+γ<1 such that
0≤X1(s,b)−X1(s,a)≤cβ(b−a)0≤X2(s,a)−X2(s,b)≤cγ(b−a) |
for all a,b∈R with b≥a and s∈[0,L];
(c) csup{∫L0λ(s,t)dt:s∈[0,L]}<1;
(d) there exists u0 and v0 in M such that
u0(s)≥q(s)+∫L0λ(s,t)[X1(t,u0(t))+X2(t,v0(t))]dt,v0(s)≤q(s)+∫L0λ(s,t)[X1(t,v0(t))+X2(t,u0(t))]dt. |
Then the system (3.1) has a solution in M×M where M=C([0,L],R). To achieve this, define X:M×M→M as
X(f,g)(s)=q(s)+∫L0λ(s,t)[X1(t,f(t))+X2(t,g(t))]dt for all f,g∈M and s∈[0,L]. | (3.3) |
Using condition (b), it can be shown that X has mixed monotone property. Now for x,y,u,v,w,z∈M with x≥u≥w, y≤v≤z,
ϑ(X(x,y),X(u,v),X(w,z))=sup{|X(x,y)(s)−X(u,v)(s)|/s∈[0,L]}+sup{|X(u,v)(s)−X(w,z)(s)|/s∈[0,L]}+sup{|X(w,z)(s)−X(x,y)(s)|/s∈[0,L]}=sup{|∫L0λ(s,t)[X1(t,x(t))+X2(t,y(t))]dt−∫L0λ(s,t)[X1(t,u(t))+X2(t,v(t))]dt|/s∈[0,L]}+sup{|∫L0λ(s,t)[X1(t,u(t))+X2(t,v(t))]dt−∫L0λ(s,t)[X1(t,w(t))+X2(t,z(t))]dt|/s∈[0,L]}+sup{|∫L0λ(s,t)[X1(t,w(t))+X2(t,z(t))]dt−∫L0λ(s,t)[X1(t,x(t))+X2(t,y(t))]dt|/s∈[0,L]} |
≤sup{|∫L0[X1(t,x(t))−X1(t,u(t))]||λ(s,t)|dt/s∈[0,L]}+sup|∫L0[X2(t,y(t))−X2(t,v(t))]||λ(s,t)|dt/s∈[0,L]}+sup{|∫L0[X1(t,u(t))−X1(t,w(t))]||λ(s,t)|dt/s∈[0,L]}+sup|∫L0[X2(t,v(t))−X2(t,z(t))]||λ(s,t)|dt/s∈[0,L]}+sup{|∫L0[X1(t,w(t))−X1(t,x(t))]||λ(s,t)|dt/s∈[0,L]}+sup|∫L0[X2(t,z(t))−X2(t,y(t))]||λ(s,t)|dt/s∈[0,L]}≤cβsup{∫L0|x(t)−u(t)||λ(s,t)|dt/s∈[0,L]}+cγsup{∫L0|y(t)−v(t)||λ(s,t)|dt/s∈[0,L]}+cβsup{∫L0|u(t)−w(t)||λ(s,t)|dt/s∈[0,L]}+cγsup{∫L0|v(t)−z(t)||λ(s,t)|dt/s∈[0,L]}+cβsup{∫L0|w(t)−x(t)||λ(s,t)|dt/s∈[0,L]}+cγsup{∫L0|z(t)−y(t)||λ(s,t)|dt/s∈[0,L]} |
≤β[sup{|x(s)−u(s)|/s∈[0,L]}+sup{|u(s)−w(s)|/s∈[0,L]}+sup{|w(s)−x(s)|/s∈[0,L]}]⋅c sup{∫L0|λ(s,t)|dt/s∈[0,L]}+γ[sup{|y(s)−v(s)|/s∈[0,L]}+sup{|v(s)−z(s)|/s∈[0,L]}+sup{|z(s)−y(s)|/s∈[0,L]}]⋅c sup{∫L0|λ(s,t)|dt/s∈[0,L]}≤β[sup{|x(s)−u(s)|/s∈[0,L]}+sup{|u(s)−w(s)|/s∈[0,L]}+sup{|w(s)−x(s)|/s∈[0,L]]+γsup{|y(s)−v(s)|/s∈[0,L]}+sup{|v(s)−z(s)|/s∈[0,L]}+sup{|z(s)−y(s)|/s∈[0,L]]=βϑ(x,u,w)+γϑ(y,v,z) |
So that
ϑ(X(x,y),X(u,v),X(w,z))≤βϑ(x,u,w)+γϑ(y,v,z) |
Hence all the conditions of Corollary 2.1 are satisfied. Therefore, X has a coupled fixed point in M×M. In other words, the system (3.1) of nonlinear integral equations has a solution in M×M.
The aforesaid application is illustrated by the following example:
Example 3.1. Let M=C([0,1],R), Now consider the integral equation in M as
X(f,g)(s)=s3+74+∫10t224(s+3)[f(t)+2g(t)+3]dt. | (3.4) |
Then clearly the above equation is in the form of following equation:
X(f,g)(s)=q(s)+∫L0λ(s,t)[X1(t,f(t))+X2(t,g(t))]dt for all f,g∈M and s∈[0,L], |
where q(s)=s3+74, λ(s,t)=t224(s+3), X1(t,s)=s, X2(t,s)=2s+3 and L=1.
That is, (3.4) is a special case of (3.3) in Theorem 3.1.
Here it is easy to verify that the functions q(s),λ(s,t),X1(t,s) and X2(t,s) are continuous. Moreover, there exist c=9, β=13 and γ=12 with β+γ<1 such that
0≤X1(s,b)−X1(s,a)≤cβ(b−a)0≤X2(s,a)−X2(s,b)≤cγ(b−a) |
for all a, b\in\mathbb{R} with b\geq a and s\in [0, 1] .
and
\begin{align*} c\sup\{\int_{0}^{L}\lambda(s,t)dt:s\in [0,L]\} = &9\sup\{\int_{0}^{1}\frac{t^2}{24(s+3)}dt:s\in [0,1]\}.\\ = &9\sup\{\frac{1}{72(s+3)}:s\in [0,1]\} \lt 1.\\ \end{align*} |
Thus the conditions (a), (b) and (c) of Theorem 3.1 are satisfied.
Now consider u_0(s) = 1 and v_0(s) = 1. Then we have
\begin{align*} q(s)+\int_{0}^{1}\lambda(s,t)[\mathscr{X}_1(t,v_0(t))+\mathscr{X}_2(t,u_0(t))]dt& = \frac{s^3+7}{4}+\int_{0}^{1}\frac{t^2}{24(s+3)}[1+\frac{2}{4}]dt\\ & = \frac{s^3+7}{4}+\frac{1}{48(s+3)}\geq 1\\ \end{align*} |
That is, v_0 \leq\mathscr{X}(v_0, u_0) . Similarly, it can be shown that u_0 \geq\mathscr{X}(u_0, v_0) .
Thus all the conditions of Theorem 3.1 are satisfied. It follows that the integral Eq (3.4) has a solution in \mathscr{M}\times \mathscr{M} with \mathscr{M} = C([0, 1], \mathbb R) .
Some coupled coincidence point theorems for two mappings established using rational type contractions in the setting of partially ordered \mathscr{G}- metric spaces. By considering \mathscr{G}- metric space, we propose a fairly simple solution for a system of nonlinear integral equations by using fixed point technique. Moreover, supporting example (exact solution) is provided to strengthen our obtained results.
The third and fourth authors would like to thank Prince Sultan University for funding this work through research group Nonlinear Analysis Methods in Applied Mathematics (NAMAM), Group Number RG-DES-2017-01-17.
The authors declare that they have no competing interests.
[1] |
R. K. Khadanga, A. Kumar, S. Panda, Frequency control in hybrid distributed power systems via type-2 fuzzy pid controller, IET Renewable Power Gener., 15 (2021), 1706–1723. https://doi.org/10.1049/rpg2.12140 doi: 10.1049/rpg2.12140
![]() |
[2] | M. K. Diab, H. H. Ammar, R. E. Shalaby, Self-driving car lane-keeping assist using pid and pure pursuit control, in 2020 International Conference on Innovation and Intelligence for Informatics, Computing and Technologies (3ICT), IEEE, (2020), 1–6. https://doi.org/10.1109/3ICT51146.2020.9311987 |
[3] |
H. Maghfiroh, M. Ahmad, A. Ramelan, F. Adriyanto, Fuzzy-pid in bldc motor speed control using matlab/simulink, J. Rob. Control (JRC), 3 (2022), 8–13. https://doi.org/10.18196/jrc.v3i1.10964 doi: 10.18196/jrc.v3i1.10964
![]() |
[4] |
J. R. Nayak, B. Shaw, B. K. Sahu, K. A. Naidu, Application of optimized adaptive crow search algorithm based two degree of freedom optimal fuzzy pid controller for agc system, Eng. Sci. Technol. Int. J., 32 (2022), 101061. https://doi.org/10.1016/j.jestch.2021.09.007 doi: 10.1016/j.jestch.2021.09.007
![]() |
[5] |
N. Ma, D. Li, W. He, Y. Deng, J. Li, Y. Gao, et al., Future vehicles: interactive wheeled robots, Sci. China Inf. Sci., 64 (2021), 1–3. https://doi.org/10.1007/s11432-020-3171-4 doi: 10.1007/s11432-020-3171-4
![]() |
[6] | N. Ma, Y. Gao, J. Li, D. Li, Interactive cognition in self-driving, Chin. Sci.: Inf. Sci., 48 (2018), 1083–1096. |
[7] |
D. Li, N. Ma, Y. Gao, Future vehicles: learnable wheeled robots, Sci. China Inf. Sci., 63 (2020), 1–8. https://doi.org/10.1007/s11432-019-2787-2 doi: 10.1007/s11432-019-2787-2
![]() |
[8] |
T. Yang, N. Sun, Y. Fang, Adaptive fuzzy control for a class of mimo underactuated systems with plant uncertainties and actuator deadzones: Design and experiments, IEEE Trans. Cybern., 52 (2022), 8213–8226. https://doi.org/10.1109/TCYB.2021.3050475 doi: 10.1109/TCYB.2021.3050475
![]() |
[9] |
S. H. Park, K. W. Kim, W. H. Choi, M. S. Jie, Y. Kim, The autonomous performance improvement of mobile robot using type-2 fuzzy self-tuning PID controller, Adv. Sci. Technol. Lett., 138 (2016), 182–187. https://doi.org/10.14257/astl.2016.138.37 doi: 10.14257/astl.2016.138.37
![]() |
[10] |
P. Parikh, S. Sheth, R. Vasani, J. K. Gohil, Implementing fuzzy logic controller and pid controller to a dc encoder motor–-"a case of an automated guided vehicle", Procedia Manuf., 20 (2018), 219–226. https://doi.org/10.1016/j.promfg.2018.02.032 doi: 10.1016/j.promfg.2018.02.032
![]() |
[11] |
Q. Bu, J. Cai, Y. Liu, M. Cao, L. Dong, R. Ruan, et al., The effect of fuzzy pid temperature control on thermal behavior analysis and kinetics study of biomass microwave pyrolysis, J. Anal. Appl. Pyrolysis, 158 (2021), 105176. https://doi.org/10.1016/j.jaap.2021.105176 doi: 10.1016/j.jaap.2021.105176
![]() |
[12] | M. S. Jie, W. H. Choi, Type-2 fuzzy pid controller design for mobile robot, Int. J. Control Autom., 9 (2016), 203–214. |
[13] | N. Kumar, M. Takács, Z. Vámossy, Robot navigation in unknown environment using fuzzy logic, in 2017 IEEE 15th International Symposium on Applied Machine Intelligence and Informatics (SAMI), IEEE, (2017), 279–284. https://doi.org/10.1109/SAMI.2017.7880317 |
[14] | T. Muhammad, Y. Guo, Y. Wu, W. Yao, A. Zeeshan, Ccd camera-based ball balancer system with fuzzy pd control in varying light conditions, in 2019 IEEE 16th International Conference on Networking, Sensing and Control (ICNSC), IEEE, (2019), 305–310. https://doi.org/10.1109/ICNSC.2019.8743305 |
[15] |
A. Wong, T. Back, A. V. Kononova, A. Plaat, Deep multiagent reinforcement learning: Challenges and directions, Artif. Intell. Rev., 2022 (2022). https://doi.org/10.1007/s10462-022-10299-x doi: 10.1007/s10462-022-10299-x
![]() |
[16] |
Z. Cao, S. Xu, H. Peng, D. Yang, R. Zidek, Confidence-aware reinforcement learning for self-driving cars, IEEE Trans. Intell. Transp. Syst., 23 (2022), 7419–7430. https://doi.org/10.1109/TITS.2021.3069497 doi: 10.1109/TITS.2021.3069497
![]() |
[17] | T. Ribeiro, F. Gonçalves, I. Garcia, G. Lopes, A. F. Ribeiro, Q-learning for autonomous mobile robot obstacle avoidance, in 2019 IEEE International Conference on Autonomous Robot Systems and Competitions (ICARSC), IEEE, (2019), 1–7. https://doi.org/10.1109/ICARSC.2019.8733621 |
[18] | S. Danthala, S. Rao, K. Mannepalli, D. Shilpa, Robotic manipulator control by using machine learning algorithms: A review, Int. J. Mech. Prod. Eng. Res. Dev., 8 (2018), 305–310. |
[19] |
X. Lei, Z. Zhang, P. Dong, Dynamic path planning of unknown environment based on deep reinforcement learning, J. Rob., 2018 (2018). https://doi.org/10.1155/2018/5781591 doi: 10.1155/2018/5781591
![]() |
[20] |
Y. Shan, B. Zheng, L. Chen, L. Chen, D. Chen, A reinforcement learning-based adaptive path tracking approach for autonomous driving, IEEE Trans. Veh. Technol., 69 (2020), 10581–10595. https://doi.org/10.1109/TVT.2020.3014628 doi: 10.1109/TVT.2020.3014628
![]() |
[21] | T. P. Lillicrap, J. J. Hunt, A. Pritzel, N. Heess, T. Erez, Y. Tassa, et al., Continuous control with deep reinforcement learning, preprint, arXiv: 1509.02971. https://doi.org/10.48550/arXiv.1509.02971 |
[22] |
P. Ramanathan, K. K. Mangla, S. Satpathy, Smart controller for conical tank system using reinforcement learning algorithm, Measurement, 116 (2018), 422–428. https://doi.org/10.1016/j.measurement.2017.11.007 doi: 10.1016/j.measurement.2017.11.007
![]() |
[23] |
L. Brunke, M. Greeff, A. W. Hall, Z. Yuan, S. Zhou, J. Panerati, et al., Safe learning in robotics: From learning-based control to safe reinforcement learning, Annu. Rev. Control Rob. Auton. Syst., 5 (2022), 411–444. https://doi.org/10.1146/annurev-control-042920-020211 doi: 10.1146/annurev-control-042920-020211
![]() |
[24] | A. I. Lakhani, M. A. Chowdhury, Q. Lu, Stability-preserving automatic tuning of PID control with reinforcement learning, preprint, arXiv: 2112.15187. https://doi.org/10.20517/ces.2021.15 |
[25] |
O. Dogru, K. Velswamy, F. Ibrahim, Y. Wu, A. S. Sundaramoorthy, B. Huang, et al., Reinforcement learning approach to autonomous pid tuning, Comput. Chem. Eng., 161 (2022), 107760. https://doi.org/10.1016/j.compchemeng.2022.107760 doi: 10.1016/j.compchemeng.2022.107760
![]() |
[26] |
X. Yu, Y. Fan, S. Xu, L. Ou, A self-adaptive sac-pid control approach based on reinforcement learning for mobile robots, Int. J. Robust Nonlinear Control, 32 (2022), 9625–9643. https://doi.org/10.1002/rnc.5662 doi: 10.1002/rnc.5662
![]() |
[27] |
B. Guo, Z. Zhuang, J. S. Pan, S. C. Chu, Optimal design and simulation for pid controller using fractional-order fish migration optimization algorithm, IEEE Access, 9 (2021), 8808–8819. https://doi.org/10.1109/ACCESS.2021.3049421 doi: 10.1109/ACCESS.2021.3049421
![]() |
[28] | M. Praharaj, D. Sain, B. Mohan, Development, experimental validation, and comparison of interval type-2 mamdani fuzzy pid controllers with different footprints of uncertainty, Inf. Sci., 601 (2022), 374–402. |
[29] |
Y. Jia, R. Zhang, X. Lv, T. Zhang, Z. Fan, Research on temperature control of fuel-cell cooling system based on variable domain fuzzy pid, Processes, 10 (2022), 534. https://doi.org/10.3390/pr10030534 doi: 10.3390/pr10030534
![]() |
[30] |
J. Wei, L. Gang, W. Tao, G. Kai, Variable universe fuzzy pid control based on adaptive contracting-expanding factors, Eng. Mech., 38 (2021), 23–32. https://doi.org/10.6052/j.issn.1000-4750.2020.11.0786 doi: 10.6052/j.issn.1000-4750.2020.11.0786
![]() |
[31] | R. S. Sutton, A. G. Barto, Reinforcement Learning: An Introduction, MIT press, 2018. |
[32] |
P. R. Montague, Reinforcement learning: an introduction, by Sutton, RS and Barto, AG, Trends Cognit. Sci., 3 (1999), 360. https://doi.org/10.1016/S1364-6613(99)01331-5 doi: 10.1016/S1364-6613(99)01331-5
![]() |
[33] | D. Wang, R. Walters, X. Zhu, R. Platt, Equivariant q learning in spatial action spaces, in Conference on Robot Learning, PMLR, (2022), 1713–1723. |
[34] |
E. Anderlini, D. I. Forehand, P. Stansell, Q. Xiao, M. Abusara, Control of a point absorber using reinforcement learning, IEEE Trans. Sustainable Energy, 7 (2016), 1681–1690. https://doi.org/10.1109/TSTE.2016.2568754 doi: 10.1109/TSTE.2016.2568754
![]() |
1. | Donal O’Regan, Reza Saadati, Chenkuan Li, Fahd Jarad, The Hausdorff–Pompeiu Distance in Gn-Menger Fractal Spaces, 2022, 10, 2227-7390, 2958, 10.3390/math10162958 |