# A New Nonlinear Contraction Principle in Partial Metric Spaces

Authors: Yi Zhang; Jiang Zhu
Journal: Journal of Function Spaces
 (2016)

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## Abstract

We present a new nonlinear contraction principle on partial metric spaces and prove the existence of common fixed point. We also give some examples to show our results and apply our results to study the existence of common bounded solution of the system of functional equations.


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## Body

## 1. Introduction

There exist many generalizations of the well-known Banach contraction mapping principle in the literature. In particular, Matthews [1, 2] introduced the notion of a partial metric space as part of the study of denotational semantics of dataflow networks, showing that the Banach contraction mapping theorem can be generalized to the partial metric context for applications in program verification. After that, fixed point results in partial metric spaces have been studied by many authors. For example, Matthews [2] proved a Banach fixed point theorem for a self-map T of a partial metric spaces (X,p) under the following contractive condition:(1)pTx,Ty≤cpx,y,∀x,y∈X,where 0≤c<1. Altun and Erduran [3] improved (1) by the following contractive condition:(2)pTx,Ty≤φpx,y,∀x,y∈X,where φ:R+→R+ is a continuous nondecreasing function such that φ(t)<t for all t>0 and proved the existence of the unique fixed point. Altun et al. [4] and Altun and Sadarangani [5] improved (2) by the following contractive condition:(3)pTx,Ty≤φmax⁡px,y,px,Tx,py,Ty,12px,Ty+py,Tx,where φ:R+→R+ is continuous nondecreasing function such that φ(t)<t and the series ∑n≥1‍φn(t) converges for all t>0 and proved the existence of the unique fixed point. Romaguera [6] improved (3) by assuming that φ:R+→R+ is a right upper semicontinuous function such that φ(t)<t for all t>0 and proved the existence of the unique fixed point without assuming that the series ∑n≥1‍φn(t) converges for all t>0. Abdeljawad et al. [7] proved the existence of fixed point for self-maps T of (X,p) under the following:(4)pTx,Ty≤ψpx,y-ϕpx,y,∀x,y∈X,where ψ,ϕ:R+→R+ are continuous nondecreasing function such that φ(t)=ψ(t)=0 if and only if t=0. Abdeljawad et al. [8] proved the existence of common fixed point for self-maps S and T of (X,p) under the following:(5)pTx,Sy≤Nx,y-ϕNx,y,∀x,y∈X,where N(x,y)=max{p(x,y),p(Tx,x),⁡p(y,Sy),1/2[p(Tx,y)+p(x,Sy)]}, ϕ:R+→R+ is a continuous, nondecreasing function with ϕ(t)>0 for all t>0 and ϕ(0)=0.Recently, Haghi et al. made [9] a paper which stated that we should “be careful on partial metric fixed point results” along with giving some results. The authors noted that some fixed point theorems in partial metric spaces can be obtained from the corresponding results in metric spaces. Abdeljawad et al. [10] used the technique of Haghi et al. in [9] to partial b-metric spaces. Reminded of the warning of Haghi et al. [9], one can see that the main characteristic of a partial metric space is that the self-distance p(x,x) of a point x may be not zero. But the all above contraction conditions are not proper for the characteristics. For example, if (X,p) is a partial metric space and satisfies p0=infp(x,x):x∈X>0, then the all above contraction conditions do not hold at any fixed point in this partial metric space (X,p). This problem also shows that the abovementioned contractive conditions are only metric type contractive conditions but do not reflect the structure of a partial metric space. On the other hand, Ilić et al. [11] proved a more reasonable contraction principle in the partial metric space in which they used self-distance terms. They gave the following linear contractive condition allowing use of self-distance in the contractive condition so that completeness, rather than the 0-completeness, of the partial metric space is needed:(6)pTx,Ty≤max⁡αpx,y,px,x,py,y,∀x,y∈X,where α∈[0,1). After that, Chi et al. [12] gave the following linear generalization of (6): (7)pTx,Ty≤max⁡apx,y,bpx,Tx,cpy,Ty,dpxTy+py,Tx,px,x,py,y,∀x,y∈X, where a,b,c∈[0,1) and d∈[0,1/2). Abdeljawad and Zaidan [13] gave the following linear generalization of (6): (8)pTx,Ty≤max⁡φpx,y,px,x,py,y, where φ:[0,∞)→[0,∞) is an increasing function such that f(t)=t-φ(t) is increasing with f-1 being right continuous at 0. Also assume limn→∞φn(t)=0 for all t≥0 (and hence φ(0)=0 and φ(t)<t for t>0). The method of self-distance term is also used in [14–16]. In [17], the authors formulated the concept of M-metric spaces as a generalization of partial metric spaces, as well as a generalization of metric spaces. It is worth mentioning that the technique of Haghi et al. in [9] also is not applicable in M-metric spaces and the contraction condition presented in this paper is also not reflecting the structure of a M-metric space. After that, very recently, Abodayeh et al. [18] succeeded in studying M-metric and partial metric spaces topologically and proved a Caristi type fixed point theorem there by using self-distance term in their contractive conditions. For more general fixed point theorems, one can refer to [19, 20]. In [19], the authors proved some common fixed point theorems for four mappings in partial metric spaces by using weakly compatible concept. In [20], the authors proved some fixed point theorems in more general ordered partial b-metric spaces. But these results still have similar gap between the contractive conditions and the structure of the spaces.Inspired by the above works, in the present manuscript, we will focus on presenting a new nonlinear contraction principle on partial metric spaces for two mappings by using self-distance term in contractive conditions and proving the existence of common fixed point. Our result generalize and improve the results in [2–8, 11–14]. We also give some examples to show our results and apply our results to study the existence of common bounded solution of the system of functional equations arising in dynamic programing.


## 2. Preliminaries

A partial metric space (see, e.g., [1]) is a pair (X,p:X×X→R+) (where R+ denotes the set of all nonnegative real numbers) such that for all x,y,z∈X(PM1)
p(x,y)=p(y,x);(PM2)
x=y if and only if p(x,y)=p(x,x)=p(y,y);(PM3)
p(x,x)≤p(x,y);(PM4)
p(x,z)≤p(x,y)+p(y,z)-p(y,y).Each partial metricp on X generates a T0 topology τp on X with a base of the family of open p-balls {Bp(x,ε):x∈X,ε>0}, where Bp(x,ε)={y∈X:p(x,y)<p(x,x)+ε} for all x∈X and ε>0.In a partial metric space, the concepts of convergence, the Cauchy sequence, completeness, and continuity are defined as follows.Definition 1 (see [1, 2]).
(i) A sequence{xn} in a partial metric space (X,p) converges to x∈X if and only if p(x,x)=limn→∞p(x,xn).
(ii) A sequence{xn} in a partial metric space (X,p) is called Cauchy if limn,m→∞p(xn,xm) exists and is finite.
(iii) A partial metric space(X,p) is said to be complete if every Cauchy sequence {xn} in X converges, with respect to τp, to a point x∈X such that p(x,x)=limn,m→∞p(xn,xm).Example 2.
LetX=R+ and define p(x,y)=max{x,y}, for all x,y∈X. Then (X,p) is a complete partial metric space. It is clear that p is not a (usual) metric.


## 3. Main Results

Let(X,p) be a partial metric space, A, B be self-maps of a set X, and (9)Mx,y=max⁡px,y,px,Ax,py,By,px,By+py,Ax2,for any x,y∈X. For any x0∈X, the sequence {xn} defined by x2n-1=Ax2n-2 and x2n=Bx2n-1, for n∈N, is called an A,B-iteration sequence of x0 We first present the following contraction condition.(C1)p(Ax,By)≤maxϕ(M(x,y)),p(x,x),p(y,y), for any x,y∈X, where ϕ:[0,∞)→[0,∞) is upper-semicontinuous (i.e., limsupt→t0ϕ(t)≤ϕ(t0),∀t0∈[0,∞)), such that ϕ(t)<t for any t>0.Theorem 3.
LetA, B be self-mappings of a complete partial metric space (X,p) such that (C1) holds; then A, B have a common fixed point z∈X.Proof.
Fixx∈X. Let {xn} be A,B-iteration sequence of x0=x. For n∈N define pn=p(xn,xn+1).
Step one: we prove that{pn} is a decreasing sequence; that is, for any k∈N,(10)pk+1≤pk.
(1) Assume thatk=2n. Then p2n+1=p(x2n+1,x2n+2)=p(Ax2n,Bx2n+1), and(11)Mx2n,x2n+1=max⁡px2n,x2n+1,px2n,Ax2n,px2n+1,Bx2n+1,12px2n,Bx2n+1+px2n+1,Ax2n=max⁡px2n,x2n+1,px2n,x2n+1,px2n+1,x2n+2,12px2n,x2n+2+px2n+1,x2n+1=max⁡px2n,x2n+1,px2n+1,x2n+2=max⁡p2n,p2n+1.It follows from (C1) that (12)p2n+1=px2n+1,x2n+2=pAx2n,Bx2n+1≤max⁡ϕMx2n,x2n+1,px2n,x2n,px2n+1,x2n+1=max⁡ϕmaxp2n,p2n+1,px2n,x2n,px2n+1,x2n+1.If ϕ(max{p2n,p2n+1})>maxp(x2n,x2n),p(x2n+1,x2n+1), we have that (13)p2n+1≤max⁡ϕmax⁡p2n,p2n+1,px2n,x2n,px2n+1,x2n+1=ϕmax⁡p2n,p2n+1.This implies that p2n+1<p2n. If ϕ(max{p2n,p2n+1})≤maxp(x2n,x2n),p(x2n+1,x2n+1), we have that (14)p2n+1≤max⁡px2n,x2n,px2n+1,x2n+1≤p2n.Thus, we have proved that p2n+1≤p2n.
(2) Assume thatk=2n+1; using a similar argument as in (1), one can verify that p2n+2≤p2n+1.
Combining the results of (1) and (2), we may conclude that (10) holds. Hence there exists rx≥0, such that limn→∞pn=rx.
Step two: we prove thatlimn→∞p(xn,xn)=rx. Assume that limn→∞¯p(xn,xn)=px. Note that p(xn,xn)≤pn+1, and we can obtain that (15)limn→∞-⁡pxn,xn=px≤limn→∞⁡pn=rx.Since pn is decreasing, by the proof of Step one, we have that (16)pn+1≤max⁡ϕpn,pxn,xn,pxn+1,xn+1.If there are infinite number of n, such that (17)max⁡ϕpn,pxn,xn,pxn+1,xn+1=ϕpn,then we have that pn+1≤ϕ(pn). Thus, by letting n→∞ we can obtain that rx≤ϕ(rx). This implies that rx=0, so we have that px=0. Thus, (18)limn→∞⁡pxn,xn=rx=px=0.Therefore, if rx>0, then there exists K>0, such that, for any n>K, (19)ϕpn<max⁡pxn,xn,pxn+1,xn+1,so we have that (20)pn+1≤max⁡pxn,xn,pxn+1,xn+1.This has two cases. Case one: there exists K1>K, such that pn+1≤p(xn,xn), for any n>K1, or pn+1≤p(xn+1,xn+1), for any n>K1. In this case we have that (21)limn→∞⁡pn+1=rx≤lim_n→∞⁡pxn,xn≤limn→∞-⁡pxn,xn≤limn→∞pn=rx, and, thus, limn→∞p(xn,xn)=rx. Case two: if case one does not hold, then there are subsequences {nk},{nk′}, such that {n}={nk}∪{nk′}, n>K, and (22)pnk+1≤pxnk,xnk,pnk′+1≤pxnk′+1,xnk′+1. In this case we also have that(23)limk→∞⁡pxnk,xnk=rx,limk→∞⁡pxnk′+1,xnk′+1=rx.From this, we obtain that limn→∞p(xn,xn)=rx. By the conclusion of Step one, we know that pn+1≥rx for any n∈N. Then, by case one and case two, we obtain that there exists positive integer L>0, such that p(xn,xn)≥rx for any n>L.
Step three: we prove that(24)limn,m→∞⁡pxn,xm=rx.
By the conclusion of Step two, we get thatp(xn,xm)≥p(xn,xn)≥rx for any n>L. Thus, if (24) does not hold, then there exist ε0>0 and increasing sequence of integers {m(k)} and {n(k)} such that, for all integers k, (25)nk>mk>k+2,pxnk,xmk≥rx+ε0.Since limn→∞p(xn,xn)=limn→∞p(xn+1,xn)=rx, we have that (26)pxmk,xmk<rx+ε0,pxmk+1,xmk<rx+ε0,when m(k) is large enough. So we can assume that n(k) is the minimum positive integer such that (25) holds. Then we have that(27)pxnk-1,xmk<rx+ε0.Using (25), (27), and the triangle inequality for partial metric we have (28)rx+ε0≤pxnk,xmk≤pxnk,xnk-1+pxnk-1,xmk-pxnk-1,xnk-1≤pxnk,xnk-1+rx+ε0-pxnk-1,xnk-1.Thus,(29)limk→∞⁡pxnk,xmk=rx+ε0,limk→∞⁡pxnk-1,xmk=rx+ε0.Again, the triangle inequality gives us (30)pxnk,xmk-1≤pxnk,xmk+pxmk,xmk-1-pxmk,xmk,pxnk,xmk≤pxnk,xmk-1+pxmk-1,xmk-pxmk-1,xmk-1.Thus,(31)pxnk,xmk-pxmk-1,xmk+pxmk-1,xmk-1≤pxnk,xmk-1≤pxnk,xmk+pxmk,xmk-1-pxmk,xmk.Letting k→∞, we get(32)limk→∞⁡pxnk,xmk-1=rx+ε0.Similarly, we can prove that(33)limk→∞⁡pxnk,xmk+1=limk→∞⁡pxnk-1,xmk+1=limk→∞⁡pxnk-1,xmk-1=rx+ε0,limk→∞⁡pxnk+1,xmk+1=limk→∞⁡pxnk+1,xmk=limk→∞⁡pxnk+1,xmk-1=rx+ε0,putting(34)xnk∗,xmk∗=xnk-1,xmk-1,nkisodd,mkiseven,xnk-1,xmk,nkisodd,mkisodd,xnk,xmk-1,nkiseven,mkiseven,xnk,xmk,nkiseven,mkisodd.By use of (29), (32), and (33), we have (35)limk→∞⁡Mxnk∗,xmk∗=limk→∞⁡max⁡pxnk∗,xmk∗,pxnk∗,Axnk∗,pxmk∗,Bxmk∗,pxnk∗,Bxmk∗+pAxnk∗,xmk∗2=rx+ε0.And limk→∞p(Axn(k)∗,Bxm(k)∗)=rx+ε0. But, by using (C1), we know that (36)pAxnk∗,Bxmk∗≤max⁡ϕMxnk∗,xmk∗,pxnk∗,xnk∗,pxmk∗,xmk∗.Thus, by letting k→∞, we get (37)rx+ε0≤max⁡ϕrx+ε0,rx,rx<rx+ε0.This yields a contradiction. Therefore, {xn} is a Cauchy sequence. By the completeness of (X,p),{xn} is a convergent sequence. Then there exists x~∈X such that (38)limn→∞⁡pxn,x~=limn,m→∞⁡pxn,xm=limn→∞⁡pxn,xn=rx=px~,x~.
Step four: we prove that(39)px~,Ax~=px~,Bx~=px~,x~.For any u∈X, we have that (40)pu,x~≤pu,xn+pxn,x~-pxn,xn,pu,xn≤pu,x~+px~,xn-px~,x~.This implies that(41)limn→∞⁡pu,xn=pu,x~.Again note that(42)limn→∞⁡x2n=limn→∞⁡Bx2n-1=x~,and we obtain that (43)px~,Ax~=limn→∞⁡pAx~,Bx2n-1≤limn→∞⁡max⁡ϕMx~,x2n-1,px~,x~,px2n-1,x2n-1=max⁡ϕpx~,Ax~,px~,x~.This implies that px~,Ax~=px~,x~. Similarly, one can prove that px~,Bx~=px~,x~.
Step five: let(44)X~=x~∈X:∃x∈X,theA,B-iterative  sequencexnofxconverges  tox~,and lp=inf⁡px~,x~:x~∈X~, we prove that there exists y~∈X, such that p(y~,y~)=lp.
For anyk∈N, we take xk∈X~, such that (45)pxk,xk<lp+1k.Assume that xnk is the A,B-iterative sequence of xk and converges to xk~. By using the conclusions of Step one and Step four, we obtain that (46)pxnk,xnk≤pxnk,xn+1k≤⋯≤px1k,x2k=pAxk,BAxk≤max⁡ϕmax⁡pxk,Axk,pxk,Axk,pAxk,BAxk,12pxk,BAxk+pAxk,Axk,pxk,xk,pAxk,Axk=max⁡ϕmax⁡pxk,xk,pAxk,BAxk,pxk,xk≤pxk,xk.This shows that, by using the conclusion of Step two,(47)pxk~,xk~=limn→∞⁡pxnk,xnk≤pxk,xk.Thus, we have that (48)lp≤pxk~,xk~≤pxk,xk<lp+1k.This implies that (49)limk→∞⁡pxk~,xk~=lp.Since(50)lp≤pxk~,xk~≤pxk~,xl~=limn→∞⁡pxnk,xnl=limn→∞⁡pAx2nk,Bx2n-1l,pAx2nk,Bx2n-1l≤max⁡ϕMx2nk,x2n-1l,px2nk,x2nk,px2n-1l,x2n-1l=max⁡ϕmax⁡px2nk,x2n-1l,px2nk,Ax2nk,px2n-1l,Bx2n-1l,12px2nk,Bx2n-1l+px2n-1l,Ax2nk,px2nk,x2nk,px2n-1l,x2n-1l,we obtain that (51)lp≤pxk~,xl~=limn→∞⁡pxnk,xnl=limn→∞⁡pAx2nk,Bx2n-1l≤limn→∞¯⁡max⁡ϕMx2nk,x2n-1l,px2nk,x2nk,px2n-1l,x2n-1l≤max⁡ϕpxk~,xl~,pxk~,xk~,pxl~,xl~=max⁡pxk~,xk~,pxl~,xl~≤max⁡pxk,xk,pxl,xl<lp+1k+1l.This shows that limk,l→∞pxk~,xl~=lp; that is, xn~ is a Cauchy sequence. Since the partial metric space X,p is completed, there is y∈X, such that(52)py,y=limn→∞⁡pxn~,y=limn,m→∞⁡pxn~,xm~=lp.
It follows from(53)pxn~,Ay≤pxn~,y+py,Ay-py,y,py,Ay≤py,xn~+pxn~,Ay-pxn~,xn~that(54)limn→∞⁡pxn~,Ay=py,Ay.On the other hand, we know that (55)pxn~,Ay=limm→∞⁡pAy,x2mn=limm→∞⁡pAy,Bx2m-1n≤limm→∞¯⁡max⁡ϕMy,x2m-1n,py,y,px2m-1n,x2m-1n=max⁡ϕmax⁡py,xn~,py,Ay,pxn~,xn~,12py,xn~+pxn~,Ay,py,y,pxn~,xn~.Then by letting n→∞ we get that (56)py,Ay=limn→∞⁡pxn~,Ay≤max⁡ϕmax⁡py,y,py,Ay,py,y.This implies that p(y,Ay)≤p(y,y), so we have that(57)py,Ay=py,y.
Similarly, one can prove that(58)limn→∞⁡pxn~,By=py,By,py,By=py,y.Assume that yn is the A,B-iterative sequence of y and converges to y~. By using the conclusions of Step one and Step two, we get that(59)lp≤py~,y~≤pyn,yn+1≤⋯≤py,Ay=py,y=lp,and, thus, p(y~,y~)=lp.
Step six: we prove thatA, B have unique common fixed point u∈X with p(u,u)=lp.
By using the conclusions of Step one, Step two, and Step five, we know that there isK>0, such that p(y~,y~)≤p(yn,yn) for any n>K and (60)lp=py~,y~≤pyn,yn≤pyn,yn+1≤⋯≤py,Ay=py,y=lp.Thus, we have that(61)lp=py~,y~=pyn,yn=pyn,yn+1=⋯=py,y,n>K.
It follows from (61) that for n>K(62)lp=py~,y~=py2n,y2n=py2n,y2n+1=py2n+1,y2n+1,=py2n+1,y2n+2=py2n+2,y2n+2,and this shows that (63)y2n=y2n+1=y2n+2;that is (64)y2n=y2n+1=Ay2n=y2n+2=By2n+1=By2n.Then we have proved that y2n is a common fixed point of A, B. Assume that u,v∈X are two common fixed points of A, B, with the properties of p(u,u)=p(v,v)=lp. Clearly, u,v∈X~, since the A,B-iterative sequence of a fixed point is a constant sequence. By using (C1) we have that (65)pu,v=pAu,Bv≤max⁡ϕMu,v,pu,u,pv,v=max⁡ϕpu,v,lp,and this implies that p(u,v)=p(u,u)=p(v,v)=lp. Thus, u=v. The proof is completed.Remark 4.
If(X,p) is a metric space, then contraction condition (C1) reduces to the corresponding contraction conditions on metric spaces. Thus, the corresponding results in the set of a metric space is a special case of our results. But, in a partial metric space, since the partial metric space topology is only T0 topology, the limit of a sequence {xn} in a partial metric space (X,p) is not unique (refer to Remark  7 in [18]), and this induced that the fixed point may not be unique. In Theorem 3, we can only get that A, B have unique common fixed point u∈y∈X:p(y,y)=lp.Remark 5.
In Theorem3, if A=B, then the condition “ϕ:[0,∞)→[0,∞) is upper-semicontinuous” can be replaced by “ϕ:[0,∞)→[0,∞) is upper-semicontinuous from the right.” It is only needed to modify the proof of Theorem 3.Remark 6.
IfA=B, the conclusion of Theorem 3 is a generalization and improvement for the results in [2–6, 11–13].Remark 7.
In Theorem3, if ϕ is replaced by Φ=ψ-ϕ, where ψ,ϕ:R+→R+ are continuous function such that ϕ(t)=ψ(t)=0 if and only if t=0, then the conclusion of Theorem 3 in this case is a generalization and improvement for the results in [7, 8].Remark 8.
Let(X,Sp) be a complete partial S-metric space and T be a self-mapping on X (see [14]), if we define p(x,y)=Sp(x,x,y), ∀x,y∈X, then (X,p) is a complete partial metric space, and the contraction condition(66)SpTx,Tx,Ty≤max⁡kSpx,x,y,Spx,x,x,Spy,y,yof Theorem  2.1 in [14] will become (67)pTx,Ty≤maxkpx,y,px,x,py,y. The contraction condition (68)SpTx,Tx,Ty≤max⁡kSpx,x,y,Spx,x,x+Spy,y,y2 of Theorem  2.2 in [14] will become (69)pTx,Ty≤maxkpx,y,px,x+py,y2.Therefore, our Theorem 3 is a generalization and improvement for the results in [14].Remark 9.
Some nice results for more general fixed point theorems were proved in [19, 20]. In [19], the authors proved some common fixed point theorems for four mappings in partial metric spaces by using weakly compatible concept and some control functions, but these control functions are different from upper-semicontinuous ones. In [20], the authors proved some fixed point theorems in more general ordered partial b-metric spaces. However, there is still the problem mentioned in the Introducion. That is, the contractive conditions presented in these papers do not reflect the structure of the partial metric space or partrial Gpb metric space. Thus, our results can not be deduced from the results in [19, 20]. But the results in [19, 20] also show that there is still more greater research space to prove some contraction principles with the contractive conditions includeing self-distance terms.The following simple example is an illustration of our extension.Example 10.
Suppose thatX=R+ and p(x,y)=emax{x,y}. Then (X,p) is a partial metric space. Suppose that A:X→X such that (70)Ax=x21+x,x≠1,1,x=1,for all x∈X and ϕ:[0,∞)→[0,∞) and ϕ(t)=t/1+t2, for any x,y∈R+. Then, for all x,y∈X with x≥y, we have that (71)pAx,Ay=emaxx2/1+x,y2/1+y=ex2/1+x,Mx,y=max⁡ex,ex,ey,12ex+emaxy,x2/1+x=ex,pAx,Ay=ex2/1+x≤max⁡ϕex,ex,ey.
Therefore, the conditions of Theorem3 are satisfied. Indeed x=0 and x=1 are the fixed point. But (72)pAx,Ay=ex2/1+x>ϕMx,y=ϕex=ex1+e2x.Thus, we cannot apply the results of [2–8] to this example.Example 11.
Suppose thatX=R+ and p(x,y)=max{x,y}. Then (X,p) is a partial metric space. Define A,B:X→X by Ax=x/2(1+x) and Bx=x-x2/2(1+x) for all x∈X and define ϕ:[0,∞)→[0,∞) by ϕ(t)=ln(1+t), for all t∈[0,1]. Then, for all x,y∈X, we have (73)pAx,By=max⁡x21+x,y-y221+y,Mx,y=max⁡px,y,px,Ax,py,By,px,By+py,Ax2=max⁡x,y.If x≥y, then p(Ax,By)=x/2(1+x)≤ϕ(x)=ϕ(M(x,y)). If y>x and x/2(1+x)≤y-y2/2(1+y), we have that (74)pAx,By=y-y221+y>ln⁡1+y=ϕy=ϕMx,y.Thus, we cannot apply the results of [2–8] to this example. But in both cases x≥y and y>x, and we always have that (75)pAx,By≤max⁡ϕMx,y,px,x,py,yholds. Thus, the conditions of Theorem 3 are satisfied and A, B have a common fixed point x=0.


## 4. An Application to a Dynamical Process

Generally, a dynamical process consists of a state space and a decision space. The state space is the set of the initial state, actions, and transition model of the process; the decision space is the set of possible actions that are allowed for the process.In this section we assume thatX and Y are Banach spaces; X1⊂X is a state space and Y1⊂Y is a decision space. It is well-known that the dynamic programing provides useful tools for mathematical optimization and computer programing as well. In particular, the problem of dynamic programing related to multistage process reduces to the problem of solving the system of functional equations(76)q1x=supy∈Y1⁡fx,y+Gx,y,q1τ1x,y,x∈X1,(77)q2x=supy∈Y1⁡fx,y+Hx,y,q2τ2x,y,x∈X1,where x and y signify the state and decision vectors, f:X1×Y1→R, G,H:X1×Y1×R→R, and τi:X1×Y1→X1 (i=1,2) represent the transformations of the process, and q1, q2 denote the optimal return functions with the initial state x. However, for the detailed background of the problem, the reader can refer to [21, 22]. Here, we study the existence of the common bounded solution of the system of functional equations (76).LetZ denote the set of all bounded real-valued functions on X1 and, for an arbitrary z∈Z, define z=supx∈X1zx. Clearly, (Z,·) endowed with the metric d defined by (78)dz1,z2=supx∈X1⁡z1x-z2x,z1,z2∈Zis a Banach space. Indeed, the convergence in the space Z with respect to · is uniform. Thus, if we consider a Cauchy sequence {zn} in Z, then {zn} converges uniformly to a function, for example, z∗, that is bounded and so z∗∈Z. Define p(z1,z2)=ed(z1,z2)/1+Kd(z1,z2) for all z1,z2∈Z and a fixed positive number K. Then (Z,p) is a complete partial metric space.We defineA,B:Z→Z by(79)Az1x=supy∈Y1⁡fx,y+Gx,y,z1τ1x,y,Bz2x=supy∈Y1⁡fx,y+Hx,y,z2τ2x,yfor all z1,z2∈Z and x∈X1. Obviously, if the functions f and G, H are bounded then A, B are well-defined.We will prove the following theorems.Theorem 12.
Let(80)Mz1,z2=max⁡pz1,z2,pz1,Az1,pz2,Bz2,pz1,Bz2+pz2,Az12,where p(z1,z2)=ed(z1,z2)/1+Kd(z1,z2), K>0. Suppose the following conditions are satisfied:(H1)
for all(x,y,z1,z2)∈X1×Y1×Z×Z, (81)Gx,y,z1τ1x,y-Hx,y,z2τ2x,y≤ln⁡max⁡ϕMz1,z2,pz1,z1,pz2,z2,for all x∈X1 and y∈Y1, where ϕ:[0,∞)→[0,∞) is upper-semicontinuous, such that ϕ(t)<t for any t>0. Then system (76) has a common bounded solution.Proof.
Letε be an arbitrary positive number, x∈X1, and z1,z2∈Z. From (79), there exist y1,y2∈Y1 such that (82)Az1x<fx,y1+Gx,y1,z1τ1x,y1+ε,(83)Bz2x<fx,y2+Hx,y2,z2τ2x,y2+ε,(84)Az1x≥fx,y2+Gx,y2,z1τ1x,y2,(85)Bz2x≥fx,y1+Gx,y1,z2τ2x,y1,Then, from (82) and (85), it follows easily that (86)Az1x-Bz2x<Gx,y1,z1τ1x,y1+ε-Hx,y1,z2τ2x,y1≤Gx,y1,z1τx,y1-Gx,y1,z2τx,y1+ε≤ln⁡max⁡ϕMz1,z2,pz1,z1,pz2,z2+ε,and, similarly, from (83) and (84) we have (87)Bz2x-Az1x<ln⁡max⁡ϕMz1,z2,pz1,z1,pz2,z2+ε.
Therefore, we have(88)Az1x-Bz2x<ln⁡max⁡ϕMz1,z2,pz1,z1,pz2,z2+ε.We have(89)pAz1,Bz2=edAz1,Bz2/1+KdAz1,Bz2≤edAz1,Bz2≤max⁡ϕMz1,z2,pz1,z1,pz2,z2eε.
Since the above inequality does not depend onx∈X1 and ε>0 is taken arbitrary, then we conclude immediately that (90)pAz1,Bz2≤max⁡ϕMz1,z2,pz1,z1,pz2,z2.
Then all the hypotheses of Theorem3 are satisfied with A, B. Thus we deduce that A, B have a common fixed point z∈Z. That is, the system of functional equations (76) has a common bounded solution.



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*Source: 8404529-2016-12-07.xml*