# 3D Dynamic Programming Approach to Functional Equations with Applications

Authors: Awais Asif; Sami Ullah Khan; Thabet Abdeljawad; Muahammad Arshad; Amjad Ali
Journal: Journal of Function Spaces
 (2020)

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## Abstract

This article demonstrates the graphical existence of a single fixed point while imposing the contractive condition of Chatterjea typeF-contraction on F-metric space (briefly as F-MS). We present two examples that verify the validity of the results given in the paper. The paper further explains the subsistence of the fixed point even if the contractive condition is valid only for a closed ball inside the space rather than imposing it on the whole F-MS. Moreover, the application of the mentioned results in finding a single solution of functional equations is described that is widely used in computer programming and optimization.


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## Body

## 1. Introduction and Preliminaries

After Banach presented his renowned Banach Contraction Principle, his idea was generalized by various authors into many interesting generalizations (see [1–8]). Wardowski [9] extended his idea to a more generalized form which he named as F-contraction. An additional strictly increasing function F with certain other restrictions was used to modify the Banach theorem. He investigated the fixed point of the contraction and explained the generality of his theorem with the help of concrete examples. This idea was furthered by Klim and Wardowski [10] into set-valued maps using a dynamic process instead of the ordinary Picard sequence. Later, Nazam et al. [11] extended Wardowski’s theorem into the form of Kannan’s theorem and hence proved the theorem for noncontinuous maps. He also described that a fixed point for such maps can be iterated even if the contractive inequality holds true only for a subset closed ball of the MS. The notion of F-contraction was extended by other authors as well (see [12–16]).This article relaxes the mapF by eliminating one of its restrictions, (F3) and hence iterates a fixed point for it. The investigation is carried out for single as well as set-valued maps. Our work is new which extends the preexisting theorems of Wardowski and their consequent results. This paper demonstrates the main idea of this research with the help of graphs which unifies work from the previous research carried out on the topic.Some basic definitions are given below which will be needed in a sequel.Definition 1 (see [17]).
AssumeGis a set of functions g:0,+∞⟶Rsatisfying the following conditions:
(F1)g is a nondecreasing function, i.e., 0<u<v⟹gu≤gv,
(F2) for any sequencetn⊂0,+∞, we have
(1)limn→∞tn=0⇔limn→∞gtn=−∞.Definition 2 (see [17]).
Assume a nonempty setEand d:E×E⟶0,∞is a map. Assume that there is some g,σ∈G×0,+∞ such that
(d1)a,x∈E×E,da,x=0⇔a=x,a,x∈E×E,da,x=dx,a,
(d2)a,x∈E×E,da,x=dx,a,
(d3) for everya,x∈E×E,for each N′∈N,N′≥2, and for every tii=1n⊂E with t1,tN′=a,x, we have a,x>0⟹gda,x≤g∑i=1N′−1dti,ti+1+σ.
Thend is called anF-metric on A, while E,d is named as F-MS.Example 3 (see [17]).
LetE=Nand d:E×E⟶0,∞be defined by
(2)da,x=a−x2,ifa,x∈0,3×0,3,a−x,ifa,x∉0,3×0,3,for all a,x∈E×E. Then, d is an F-MS.Example 4 (see [17]).
LetE=N and d:E×E⟶0,∞ is defined as
(3)da,x=0,ifa=x,ea−x,ifa≠x,for all a,x∈E×E. Then, d is an F-metric on E.Definition 5 (see [17]).
Letan∈E. If
(i)
limn→∞dan,a=0 for some a∈E. Then an is F-convergent to a(ii)
limn,m→∞dan,am=0 then the sequence an is F-Cauchy(iii)
For eachan⊂E implies an is F-convergent. Then, the space E,d is known as F-completeDefinition 6 (see [17]).
LetE,dbe an F-MS. A subset Oof E is said to be F-open if, for every a∈O, there is some r>0 such that Ba,r⊂O, where
(4)Ba,r=x∈E:da,x<r.
We say that a subsetC of E is F-closed if E\C is F-open.Definition 7 (see [17]).
LetB be a nonempty subset of E and d be an F-metric, then, the following statements are equivalent:
(i)
BisF-closed(ii)
For any sequencean⊂B, we have(5)limn→∞dan,a=0,a∈E⟹a∈B.Theorem 8 (see [17]).
AssumeE,dis an F-complete F-MS, and leth:E⟶E be a given map. Let there is some k∈0,1 such that
(6)dha,hx≤kda,x,a,x∈E×E.
Then,ha∗=a∗ for at most onea∗∈E. Moreover, for any a0∈E, the sequencean⊂E defined by an+1=han,n∈N is F-convergent to a∗.Theorem 9 (see [8]).
Assume thatEis a complete MS with metric d, and consider h:E⟶Ebe a function such that
(7)dha,hx≤αda,x+βda,hx+γda,hx,for all a,x∈E, where α,β,andγ are nonnegative numbers satisfying α+β+γ<1. Then, h has a unique fixed point.Lemma 10 (see [18]).
LetBW,·is a Banach space and d is a metric defined by
(8)dJ,h=∣∣J−h∣∣=maxa∈WJa−ha,J,h∈BW.
Then,BW,· is an F-MS.


## 2. Common Fixed Points Results of Reich TypeF-Contractions

This section of the paper investigates the fixed point of single-valuedF-contractions for two maps and single map in F-MS.Theorem 11.
AssumeF∈G,X,d is an F-complete F-MS and S,T:X⟶Xare self-mappings. Assume that for nonnegative functionsαand β with maxa,x∈Xαa,x+2βa,x<1, there is someτ>σ>0such that
(9)τ+FdSa,Tx≤Fαa,xda,x+βa,xdx,Sa+da,Tx,with mindSa,Tx,da,x>0, for all a,x∈X×X. Then, Sy=Ty=yfor some y in X.Proof.
Choose an arbitrary pointa0 and iterate a sequence an by
(10)Sa2c=a2c+1andTa2c+1=a2c+2;c=0,1,2,⋯
Using (9) and (10), we can write
(11)τ+Fda2c+1,a2c+2=τ+FdSa2c,Ta2c+1≤Fαa2c,a2c+1da2c,a2c+1+βa2c,a2c+1da2c+1,Sa2c+da2c,Ta2c+1=Fαa2c,a2c+1da2c,a2c+1+βa2c,a2c+1a2c,a2c+2≤Fαa2c,a2c+1da2c,a2c+1+βa2c,a2c+1a2c,a2c+1+a2c+1,a2c+2+σ.
Using (F1), we have(12)da2c+1,a2c+2<αa2c,a2c+1+βa2c,a2c+11−βa2c,a2c+1da2c,a2c+1=λ1da2c,a2c+1,sayαa2c,a2c+1+βa2c,a2c+11−βa2c,a2c+1=λ1.
Similarly,(13)da2c+2,a2c+3<αa2c+1,a2c+2+βa2c+1,a2c+21−βa2c+1,a2c+2da2c+1,a2c+2=λ2da2c+1,a2c+2.
Hence, forλ=maxλ1,λ2,⋯,λl, we have
(14)dan,an+1<λdan−1,anforalln∈N,which yields
(15)dan,an+1<λdan−1,an<λ2dan−2,an−1<⋯<λnda0,a1,n∈N.
Using (15), we can write
(16)∑k=nm−1dak,ak+1<λn1+λ+λ2+⋯+λm−n−1da0,a1≤λn1−λda0,a1,m>n.
Sincelimn→∞λn/1−λda0,a1=0, for any δ>0, ∃ some n′∈N such that
(17)0<λn1−λda0,a1<δ,n≥n′.
Further, letg,σ∈G×0,∞ satisfies (d3) andϵ>0is fixed. By (F2), there is someδ>0 such that
(18)0<t<δ⟹gt<gϵ−σ.
By (17) and (18), we write
(19)g∑k=nm−1dak,ak+1≤gλn1−λda0,a1<gϵ−σ,m>n≥n′.
Using the above equation and (d3), we have(20)dan,am>0,m>n>n′⟹gdan,am<gϵ.
This shows(21)dan,am<ϵ,m>n≥n′.
Hence, (an) isF-Cauchy in X. Since X,d is F-complete, ∃ y∈X such that an is F-convergent to y, i.e.,
(22)limn→∞dan,y=0.
Now, assume thatdSy,y>0. Then,
(23)τ+FdSy,a2c+2≤Fαy,a2c+1dy,a2c+1+βy,a2c+1da2c+1,Sy+dy,a2c+2.
By (F1) and lettingc⟶∞, we have
(24)1−βy,a2c+1dSy,y<0,which is a contradiction, as βa,x is nonnegative. Hence, dSy,y=0,i.e.,Sy=y.
Following the same steps, we getTy=y. Hence, Ty=Sy=y.Uniqueness: assume another common fixedzof the maps S and Tandy≠z.Then(25)τ+Fdy,z=FdSy,Tz≤Fαy,zdy,z+βy,zdz,Sy+dy,Tz=Fαy,zdy,z+βy,zdz,y+dy,z.Using (F1), we get (1−αy,z−2βy,z)dy,z<0, which is a contradiction. Hence, y=z.Example 12.
Assume thatX=Xc≔6c+2/3,c∈N,
(26)dXc,Xk=0,ifXc=Xk,eXc−Xk,ifXc≠Xk,FXc=lnXc,and S,T:X⟶X are defined by
(27)TXc=X1,ifc=1,2,Xc−1,ifc>2,and
(28)SXn=X1,ifc=1,X2,ifc=2,Xc−2,ifc>4.
On the other hand, we defineα,β:X×X⟶0,∞ by
(29)αXc,Xk=0ifXc=Xk,e−1/2ifXc≠Xk,βXc,Xk=e−2,ifXk=SXcandXc=TXk,0,otherwise.
One can verify thatF fulfill conditions (F1) and (F2) and that d is an F-metric. Assume c≠k, then
(30)FdSXc,TXk=lneXc−2−Xk−1=lne2c−k−2<lne−1/2.e2c−k=FαXc,XkdXc,Xk<FαXc,XkdXc,Xk+βXc,XkdXk,SXc+dXc,TXk,whenever mindSXc,TXk,dXc,Xk>0. One can verify that maxXc,XkαXc,Xk+2βXc,Xk<1. For τ∈σ,lne−1/2.e2c−k/e2c−k−2=σ,lne3, the inequality (9) is true. Furthermore, X1 is a unique point such that SX1=TX1=X1.Choosingαa,x=0 in the previous result, a result of Chatterjea type F-contraction is obtained.Corollary 13.
AssumeF∈G,X,d is an F-complete F-MS and S,T:X⟶Xare self-maps. Assume that for β:X×X⟶0,1, there is some τ>σsuch that
(31)τ+FdSa,Tx≤Fβa,x2dx,Sa+da,Tx,with mindSa,Tx,da,Sa,dx,Tx>0, for all a,x∈X×X. Then, Ty=Sy=y for a unique y in X.SubstitutingS with T in Corollary 13, we obtain the following result.Corollary 14.
Assume thatX,d is an F-complete F-MS,F∈G and T:X⟶Xis a self-map. Assume that for β:X×X⟶0,1, there is some τ>σsuch that
(32)τ+FdTa,Tx≤Fβa,x2da,Tx+dx,Ta,with mindTa,Tx,da,Ta+dx,Tx>0, for all a,x∈X×X. Then, Thas at most one fixed point in X.


## 3. Investigation of Fixed Points ofF-Contractions on F-Closed Balls

This section of the paper investigates a fixed point of single-valuedF-contractions for two maps and single map imposed only on a F-closed subset of F-MS.Definition 15.
Assume anF-MS X,d which is F-complete,F∈G, and S,T:X⟶Xare self-maps, let α,β be nonnegative functions with maxa,x∈Xαa,x+2βa,x<1. Then, T is named as Reich type F-contraction on Ba0,r⊆X if there is some τ>σsatisfying
(33)τ+FdSa,Tx≤Fαa,xda,x+βa,xdx,Sa+da,Tx,∀a,x∈Ba0,r.Theorem 16.
Assumeg,σ∈G×0,∞, an F-MS X,d which is F-complete and Tis a Reich type F-contraction on Ba0,r. Assume thatfora0∈X and r>0, the conditions given below are fulfilled:
(a)
Ba0,r is F-closed(b)
da0,a1≤1−λr, for a1∈X and λ=αa,x+βa,x/1−βa,x(c)
∃0<ϵ<r such as g1−λk+1r≤gϵ−σ, where k∈NThen,Ty∗=Sy=y for some y in Ba0,r.Proof.
Choose and arbitrary pointa0and iterate a sequence an by
(34)Ta2c=a2c+1andSa2c+1=a2c+2;c=0,1,2,⋯.
Using mathematical induction, we show thatan is in Ba0,r for all n∈N. By hypothesis
(35)da0,a1<r.
Therefore,a1∈Ba0,r. Assume a2,⋯,ak∈Ba0,r for some k∈N. Now, if a2c+1≤ak, then by (33), we can write
(36)τ+Fda2c,a2c+1=τ+FdSa2c−1,Ta2c≤Fαa2c−1,a2cda2c−1,a2c+βa2c−1,a2cda2c,Sa2c−1+da2c−1,Ta2c=Fαa2c−1,a2cda2c−1,a2c+βa2c−1,a2cda2c−1,a2c+1≤Fαa2c−1,a2cda2c−1,a2c+βa2c−1,a2cda2c−1,a2c+da2c,a2c+1+σ.
From (F1), we have(37)da2c,a2c+1<αa2c−1,a2c+βa2c−1,a2c1−βa2c−1,a2cda2c−1,a2c=λ1da2c−1,a2c.
On the other hand, ifa2c≤ak(38)da2c−1,a2c<αa2c−1,a2c+βa2c−1,a2c1−βa2c−1,a2cda2c−2,a2c−1=λ2da2c−2,a2c−1.
Continuity this way, forλ=maxλ1,λ2,⋯,λl, we deduce from inequality (37) and (38) that
(39)da2c,a2c+1<λda2c−1,a2c<⋯<λ2cda0,a1,(40)da2c−1,a2c<λda2c−2,a2c−1<⋯<λ2c−1da0,a1.
From (39) and (40), we write
(41)dak,ak+1≤λkda0,a1forsomek∈N.
Now, using (41), we have
(42)gda0,ak+1≤g∑i=1k+1dai−1,ai+σ=gda0,a1+⋯+dak,ak+1+σ≤g1+λ+λ2+⋯+λkda0,a1+σ=g1−λk+11−λda0,a1+σ.
From (b) and (c), we obtain(43)gda0,ak+1≤g1−λk+1r+σ≤gϵ<gr.
Hence by (F1), we deduce that(44)ak+1∈Ba0,r.
Therefore,an∈Ba0,r for all n∈N. Now for p∈N,we have by (33)
(45)τ+Fda2p+1,a2p+2=τ+FdSa2p,Ta2p+1≤Fαa2p,a2p+1da2p,a2p+1+βa2p,a2p+1da2p+1,Sa2p+da2p,Ta2p+1=Fαa2p,a2p+1da2p,a2p+1+βa2p,a2p+1da2p,a2p+2≤Fαa2p,a2p+1da2p,a2p+1+βa2p,a2p+1da2p,a2p+1+da2p+1,a2p+2+σ.
Using (a) and repeating the steps done in heorem11, we get to the conclusion that an is F-Cauchy to a point y in Ba0,r. Proceeding in a similar way as in heorem 11, we obtain that y=Sy=Ty
SubstitutingS by T in the previous theorem, the following result is obtained.Corollary 17.
Assumeg,σ∈G×0,∞, F,τ∈G×0,∞,X,d is an F-complete F-MS and T:X⟶Xis a self-map. Let α,β:X×X⟶0,∞ such that maxa,x∈aαa,x+2βa,x<1. Assume thatfora0∈X and r>0, the below conditions are fulfilled:
(a)
Ba0,r⊆X is F-closed(b)
da0,a1≤1−λr, for a1∈X and λ=αa,x+βa,x/1−βa,x(c)
∃0<ϵ<rsuchasg1−λk+1r≤gϵ−σ,wherek∈N(d)
τ+FdTa,Tx≤Fαa,xda,x+βa,xdx,Ta+da,Tx, ∀a,x∈Ba0,r
withmindTa,Tx,da,x>0.Then there is a unique y in Ba0,r such that Ty=y.Example 18.
LetX=R0+ andFa=ga=lna. Define T:X⟶X by
(46)Ta=a3,ifa∈0,1,a2,ifa∈1,∞.See Figure1. Define dby(47)da,x=a−x2,ifa,x∈0,1×0,1,a−x,ifa,x∉0,1×0,1.Figure 1One can verify thatF fulfill conditions (F1) and (F2) and that d is and F-metric.On the other hand, defineα,:X×X⟶0,∞ by
(48)αa,x=0,ifa=x,e−3/7,ifa≠x,βa,x=e−2,ifx=Taαnda=Tx,0,otherwise.Fixa0=r=1/2, then Ba0,r=0,1. Clearly, Ba0,r is F-closed hence is satisfied. Now, since a0≠a1, therefore, αa,x=e−3/7 and βa,x=0, which implies that λ=αa,x and
(49)da0,a1=da0,Ta0=12−162<1−e−3/712=1−λr.Therefore, condition (b) is obeyed. Moreover, assumek=1, then g1−λk+1r=ln1−e−3/721/2=ln2/5−ln2/7=gε−σ is satisfied. i.e., ε=2/5≤1/2=r and σ=ln2/7. In a similar way, for each k∈N, ∃ some 0<ϵ<r and σ satisfying condition (c). Now checking for condition (d), we have two cases:Case 19.
Ifa,x∈Ba0,r×Ba0,r, then
(50)FdTa,Tx=lna3−x32<lne−3/7a−x2=Fαa,xda,x+βa,xda,Ta+dx,Tx.Figures2–4 illustrate this inequality, where
(51)τ∈27,lne−3/7a−x2−lna3−x32=27,lne−3/71/9=27,ln9e−3/737.Figure 2Figure 3Figure 4Therefore, for alla,x∈Ba0,r×Ba0,r, condition (d) is also satisfied.Case 20.
Ifa,x∉Ba0,r×Ba0,r, e.g., a=2 and x=3, then
(52)FdTa,Tx=ln22−32>lne−3/72−3=Fαa,xda,x=Fαda,x+βda,Ta+γdx,Tx.FdTa,Tx=ln22−32>Ine−3/72−3=Fαa,xda,x=Fαda,x+βda,Ta+γdx,Tx.Hence, condition (b) holds only forBa0,r and not on X×X. Moreover, 0∈Ba0,r is the fixed point of T. Given below Figure 1 shows two maps fa=a/3 and ga=lnaFigures2 and 3 are 3D graphs of the functions z=dfa,fx=a/3−x/32 and z=da,x=a−x2, respectively.Multiplyingαa,x=e−3/7 to da,x of the contractive inequality and combining Figures 2 and 3, we get the below 3D graph which clearly demonstrate that the graph of αa,xda,x is dominating the graph of dfa,fx.
(53)αa,xda,x+βa,xda,fa+dx,fx.dfa,fx.As we see in Figure1, lnx is an increasing function. Therefore, it will not change the inequality, i.e., the right side of the inequality will still be dominant. Note that z-axis represents the values of the function g, and it can be observed that for every value of a and x, Fdfa,fx<Fαa,xda,x and hence satisfy the inequality of the above example.Corollary 21.
Assumeg,σ∈G×0,∞,F,τ∈G×0,∞such that σ<τ andX,d is an F-complete F-MS. Let S.T:X⟶Xare self-maps andk:X×X⟶0,1. Assume that for a0∈X and r>0, the below conditions are fulfilled:
(a)
Ba0,r⊆X is F-closed(b)
τ+FdSa,Tx≤Fka,x/2da,Sa+dx,Tx, for all a,x∈Ba0,r(c)
da0,a1≤1−λr, for a1∈X and λ=ka,x/2−ka,x(d)
∃0<ϵ<r such as g1−λn+1r≤gϵ−σ, where n∈NThenSy=Ty=yfor a unique y in Ba0,r.


## 4. Application to Functional Equations

This section discusses the application of our results in finding a common solution of functional equations that are used in dynamic programming.The study of dynamic programming splits into two parts. A state space is a set of parameters of various states, i.e., initial states, transitional states, and action states. On the other hand, a decision space is a series of actions taking place for finding the possible solution to the indicated problem. The problem of dynamic program is transformed into functional equations:(54)gx=maxy∈XHx,y+Jx,y,gηx,yforx∈A,(55)fx=maxy∈XHx,y+Kx,y,gηx,yforx∈A,where U and V are Banach spaces such thatA⊆U and X⊆V and
(56)η:A×X⟶A,H:A×X⟶R,J,K:A×X×R⟶R.AssumeA and X are state space and decision spaces, respectively. AssumeWAdenotes a set of all-bounded real-valued maps on A. LethЄWAand say∣∣h∣∣=maxx∈Ahx. Then, WA,‖.‖ is a Banach space and d is the metric defined as
(57)dh,k=maxx∈Ahx−kx.Suppose the following conditions are satisfied:(C1): H,J, andK are bounded.(C2): For x∈A and h∈WA, define P,Q:WA⟶WA by
(58)Phx=maxy∈XHx,y+Jx,y,hηx,yforx∈A,Qhx=maxy∈XHx,y+Kx,y,hηx,yforx∈A.Observe that the functionsH,J, and Kare bounded hence P and Qare well-defined.(C3): For σ<τ:R+⟶R+, x,y∈A×X, h,k∈WA and t∈A, we have
(59)Jx,y,ht−Kx,y,kt≤e−τMh,k,where
(60)Mh,k=αdh,k+βdk,Ph+γdh,Qkfor α,β,γ∈0,∞such thatα+β+γ<1, where mindPh,Qk,dh,k>0.Based on the above hypothesis, we present the below theorem.Theorem 22.
LetC1−C3 are satisfied, then at most one bounded common solution exists for Equations (54) and (55).Proof.
We know by Lemma10 that WA,d is anF-complete F-MS, d is stated by (57) and (C1) say that P and Q are self-maps on WA. Choose any positive numberωand h1,h2∈WA. Takex∈Aand y1,y2∈X such that
(61)Phc<Hx,yc+Jx,yc,hcηx,yc+ω,(62)Qhc<Hx,yc+Kx,yc,hcηx,yc+ω,(63)Ph1≥Hx,y2+Jx,y2,h1ηx,y2,(64)Qh2≥Hx,y1+Kx,y1,h2ηx,y1.Then using (61) and (64), we get
(65)Ph1x−Qh2x<Jx,y1,h1ηx,y1−Jx,y1,h2ηx,y1+ω≤Kx,y1,h1ηx,y1−Kx,y1,h2ηx,y1+ω≤e−τMh1x,h2x+ω.Similarly, by (62) and (63), we get
(66)Qh2x−Ph1x<e−τMh1x,h2x+ω.Combining the above two inequalities, we get(67)Ph1x−Qh2x<e−τMh1x,h2x+ω,for all ω>0. Hence,
(68)dPh1x,Qh2x≤e−τMh1x,h2x,that is,
(69)dPh1,Qh2≤e−τMh1,h2,for eachx∈A. Applying logarithms on both sides, we get
(70)lndPh1,Qh2≤lne−τMh1,h2.This shows thatF:R+⟶R defined asFx=lnx is a member ofG, and
(71)τ+FdPh1,Qh2≤FMh1,h2.As every condition in Theorem11 is fulfilled, therefore using heorem 11P and T have a unique common and bounded solution of the Equations (54) and (55).


## 5. Conclusion

This article instigated the establishment of fixed point result of Reich typeF-contractions, while imposing the contractive conditions on both the whole F-MS as well as only on a subset (F-closed ball) of the F-MS. However, the constants in the inequality of Reich type contractive conditions are replaced by real-valued functions. The validity of the inequality is verified graphically, making the results clearer and more certain. At last, the use of our results in assuring the existence of a solution to the functional equation is described.



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