# Hybrid Fixed Point Theorem with Applications to Forced Damped Oscillations and Infinite Systems of Fractional Order Differential Equations

Authors: Muhammad Shoaib; Muhammad Sarwar; Thabet Abdeljawad
Journal: Journal of Function Spaces
 (2020)

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## Abstract

In this manuscript, hybrid common fixed point results in the setting of ab-metric space are established. Our results generalized the results of Fisher, Khan, and Piri et al. for set-valued mapping in b-metric spaces. Applications to forced damped oscillations, infinite systems of fractional order differential equations, and system of functional equations are also studied. We construct an example to support our main result.


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## Body

## 1. Introduction and Preliminaries

The idea of a metric space was generalized by Czerwik [1] and Bakhtin [2]. They presented metric spaces called b-metric spaces. Several researchers took the idea of Czerwik and illustrated interesting results. For details, see [3–7]. For recent generalizations to b-metric spaces by employing control functions in the triangle inequality to replace the constant of the b-metric triangle inequality, we refer to [8–12] and the references therein.In 1973, the contraction was introduced by Geraghty [13] in which the contraction constant was quite changed by mapping owing to its interesting properties. After that, several papers for rational Geraghty contractive mappings have appeared (for details, see [14–18]). Khan [19] introduced one of the best works in this line, and Fisher [20] modified it. By rational expressions, Khan [19] and Fisher [20] results were lately extended by Piri et al. [21] by introducing a new general contractive condition. Fixed point results via F-Khan contractions were studied by Piri et al. [22] on complete metric spaces, and they discuss their application to integral equations. In [23], Ullah et al. established fixed point results and discuss the application to an infinite system of fractional order differential equations.Nadler [24] elaborated and extended the Banach contraction principle [25] to set-valued mapping by using the Hausdorff metric. After different generalizations of the Nadler contraction principle, Wardowski [26] introduced a contraction called F-contraction. In this way, Wardowski generalized the Banach contraction principle (BCP) in a different manner from the known results of literature. Following this direction, Sgroi and Vetro [27] studied set-valued F-contractions and discussed their application on certain functional and integral equations.Cosentino and Vetro [25] extended F-contraction in the setting of b-metric spaces and proved some fixed point results. Ali et al. [28] studied the fixed point, generalize the result of Cosentino et al. [25] for a new class of F-contractions in the setting of b-metric spaces, and apply the result to obtain existence results for Volterra-type integral inclusion in b-metric spaces. Several authors generalized F-contraction by combining it with some existing contractive conditions (see [27, 29–32]).In the current work, we derive a hybrid (single and multivalued) common fixed point result for theF-Khan-type contraction in the b-metric space. Also, we shall provide an example and applications for the validity of the established result. Throughout this paper, CBΛ indicates the family of nonempty subsets of Λ, which is bounded and closed. ℝ+, ℕ0, and ℕ signify the set of any nonnegative real numbers, the set of nonnegative integers, and the set of positive integers. Now, we recall a few basic results and definitions.Definition 1 [1].
Consider a nonempty setΛ and let s≥1. Assume d:Λ×Λ⟶ℝ+ is a function satisfying the conditions:
(1)
dξ11,ξ22=0⇔ξ11=ξ22 for all ξ11,ξ22∈Λ(2)
dξ11,ξ22>0, ξ11≠ξ22, for all ξ11,ξ22∈Λ(3)
dξ11,ξ22=dξ22,ξ11, where ξ11,ξ22∈Λ(4)
dξ11,ξ22≤sdξ11,ξ33+dξ33,ξ22 for all ξ11,ξ22,ξ33∈Λ
Then, the tripleΛ,d,s is called the b-metric space.Definition 2 [1].
AssumeΛ,d,s is a b-metric space, where s≥1. Let σn be a sequence in Λ. Then, σ∈Λ is said to be the limit of the sequence σn if
(1)limn→∞dσn,σ=0,and the sequence σn is said to be convergent in Λ.Definition 3 [1].
If for eachϵ>0, there is a positive integer ℕ such that dσn,σm<ϵ for all n,m>ℕ, then a sequence σn is said to be a b-Cauchy sequence.Definition 4 [1].
Ab-metric space Λ,d,s is said to be complete (or a b-complete metric space) if every Cauchy sequence in Λ,d,s is convergent in Λ.Definition 5 [33].
Assumes≥1 is a real number and F represents the family of functions F:ℝ+⟶ℝ, with the below conditions:
(F1) For each γn⊂ℝ+ which is a positive term sequence, F is strictly increasing.
(F2) limn→∞γn=0⇔limn→∞Fγn=∞.
(F3) For each positive number sequence, i.e., γn⊂ℝ+, limn→∞γn=0, there exists k∈0,1 such that limn→∞γnkFγn=0.
(F4) For γn⊂ℝ+, such that λ+Fsγn≤Fγn−1∀n∈ℕ and some λ∈ℝ+, then λ+Fsnγn≤Fsn−1γn−1.
Now, we give some basics defined in ab-metric space Λ,d,s for set-valued mappings. Define the mapping ℍ:CBΛ×CBΛ⟶ℝ+ for Λ1,Λ2∈CBΛby
(2)ℍΛ1,Λ2=maxδΛ1,Λ2,δΛ2,Λ1,where
(3)δΛ1,Λ2=supda,Λ2,a∈Λ1,δΛ2,Λ1=supdb,Λ1,b∈Λ1,da,Λ=infda,σ1,σ1∈Λ.Lemma 6 [16, 34].
Assume ab-metric space Λ,d,s. Suppose Λ1,Λ2,Λ3∈CBΛ and σ1,σ2∈Λ; then, the following holds:
(1)
dσ1,Λ2≤dσ1,bforanyb∈Λ2(2)
δΛ1,Λ2≤ℍΛ1,Λ2(3)
dσ1,Λ2≤ℍΛ1,Λ2foranyσ1∈Λ1(4)ℍΛ1,Λ1=0,(4)
ℍΛ1,Λ2=ℍΛ2,Λ1(5)ℍΛ1,Λ3≤sℍΛ1,Λ2+ℍΛ2,Λ3,ℍσ1,Λ1≤sdσ1,σ2+dσ2,Λ1.Lemma 7 [20, 24].
LetΛ,d be a metric space. Further, Λ1,Λ2∈CBΛ; then, for δ≥1 and for each ζ1∈Λ1, there exists ζ2ζ1∈Λ2 such that dζ1,ζ2≤δℍΛ1,Λ2.Lemma 8 [16, 34].
Assume ab-metric space Λ,d,s and Λ1,Λ2∈CBΛ. Then, for δ>1,ζ1∈Λ1, there exists ζ2ζ1∈Λ2 such that dζ1,ζ2≤δℍΛ1,Λ2.Theorem 9 [24].
Assumeλ:Λ⟶CBΛ is a multivalued mapping on a complete metric space Λ,d such that
(6)ℍλσ1,λσ2≤μdσ1,σ2,∀σ1,σ2∈Λ,where μ∈0,1. Then, λ has a fixed point.Theorem 10 [25].
LetΛ,d,s be a b-metric space, where s≥1. For F∈F, a multivalued mapping λ:Λ⟶CBΛ is called a Nadler-type F-contraction such that for τ∈ℝ+,
(7)2τ+Fsℍλξ1,λξ2≤Fdξ1,ξ2.∀ξ1,ξ2∈Λλξ1≠λξ2; then, λ has a fixed point in Λ.


## 2. Main Results

Definition 11.
LetΛ,d,s be a b-metric space. Let θ:Λ⟶Λ and Θ:Λ⟶CBΛ be two mappings. Then, Θ is said to be F-Khan contraction w.r.t θ if there exists 0<τ<∞ and F∈F such that for all ξ,ζ∈Λ, maxdξ,θζ,dθξ,ζ≠0; then, θξ≠θζ and
(8)τ+Fsℍθξ,θζ≤Fdξ,θξdξ,θζ+dζ,θζdζ,θξξdξ,θζ,dθξ,ζ,ifmaxdξ,θζ,dθξ,ζ≠0,0,ifmaxdξ,θζ,dθξ,ζ=0.Theorem 12.
AssumeΛ,d,s is a complete b-metric space. Let θ,Θ be the F-Khan contraction. Further, θ is continuous. Then, Cθ,Θ≠∅.
(i)
θ and Θ have a common fixed point if θθα=θα, and θ is occasionally Θ-weakly commuting at α. Then, θ and Θ have a common fixed pointProof.
Letζ0∈Λ and ζ1∈Θζ0. Let ζ2≔θζ1. By Lemma 8, there exists ζ3∈Θζ2 such that dζ3,ζ2≤ℍΘζ2,θζ1. Inductively, we let ζ2n≔θζ2n−1, and by Lemma 8, we choose ζ2n+1∈Θζ2n such that
(9)dζ2n−1,ζ2n≤ℍΘζ2n,θζ2n−1.Using equation (8), we have
(10)τ+FsℍΘζ2n,Θζ2n+1≤Fdζ2n,θζ2ndζ2n,θζ2n+1+dζ2n+1,θζ2n+1dζ2n+1,θζ2nmaxdθζ2n,ζ2n+1,dζ2n,θζ2n+1≤Fdζ2n,θζ2n,which implies
(11)τ+FsℍΘζ2n,Θζ2n+1≤Fdζ2n,θζ2n.We deduce that
(12)τ+Fsdζ2n+1,ζ2n+2≤Fdζ2n,ζ2n+1.LetQn=dζ2n+1,ζ2n+2>0, ∀n∈ℕ. It follows from (12) and axiom F4 that
(13)τ+Fsndζ2n+1,ζ2n+2≤Fsn−1dζ2n,ζ2n+1,∀n∈ℕ.Thus, by equation (13),
(14)FsnQn≤Fsn−1Qn−1−τ,Fsn−1Qn−1≤Fsn−2Qn−2−2τ,⋮FsnQn≤FsoQ0−nτ,which implies that(15)limn→∞FsnQn=−∞.By usingF2, we have
(16)limn→∞snQn=0.By theF3 property, there exists 0<k<1 such that
(17)limn→∞snQnkFsnQn=0.Equation (14) implies that
(18)Fsnpn≤Fp0−nτ.Multiplying (18) by snpnk, we have
(19)snpnkFsnpn≤snpnkFp0−nτsnpnk,which implies that
(20)snpnkFsnpn−snpnkFp0≤−nτsnpnk≤0.Applying limitn⟶1, we have
(21)limn→∞nsnpnk=0.From (21), there exists n1∈ℕ such that nsnQnk<1 such that
(22)snQn≤1nn1/k,∀n≥n1.To show thatζn is a b-Cauchy sequence, consider m,n∈ℕ such that m>n>n1, using triangular inequality, and using (18), we have
(23)dζ2n,ζ2m≤sdζ2n,ζ2n+1+s2dζ2n+1,ζ2n+2+⋯+sm−ndζ2m−1,ζ2m,=sQn−1+s2Qn+⋯+sm−nQm−2,=∑j=n−1m−2sj−n+2Qj≤∑j=n−1∞sj−n+2Qj≤∑j=n−1∞s2−n1j1/k.By taking limit, we getdζn,ζm⟶0. Hence, ζn is a b-Cauchy sequence, but a b-metric space Λ,d,s is a complete space so there exists ζ∈Λ such that ζn⟶ζ as n⟶∞.The next step is to show thatζ is a common fixed point of the mapping Θ and θ. We have
(24)dζ2n+2,Θζ≤ℍΘζ2n+1,Θζ≤sℍΘζ2n+1,Θζ,which implies that
(25)dζ2n+2,Θζ≤sℍΘζ2n+1,Θζ.SinceF is strictly increasing, therefore
(26)Fdζ2n+2,Θζ≤FsℍΘζ2n+1,Θζ.Addingτ to both sides and using equation (7), we have
(27)τ+Fdζ2n+2,Θζ≤τ+FsℍΘζ2n+1,Θζ≤Fdζ2n,θζ2ndζ2n,θζ+dζ,θζdζ,θζ2nmaxdζ2n+1,θζ,dθζ2n+1,ζ.Sinceτ∈ℝ+, we have
(28)Fdζ2n+2,Θζ≤Fdζ2n,θζ2ndζ2n,θζ+dζ,θζdζ,θζ2nmaxdζ2n+1,θζ,dθζ2n+1,ζ.SinceF is strictly increasing, therefore
(29)dζ2n+2,Θζ≤dζ2n,θζ2ndζ2n,θζ+dζ,θζdζ,θζ2nmaxdζ2n+1,θζ,dθζ2n+1,ζ.Applying limitn⟶∞, we get
(30)dζ,Θζ≤0,which implies ζ∈Θζ.But(31)dζ2n+2,Θζ=dθζ2n+1,Θζ.By taking limit and using continuity ofθ, we have
(32)dζ,Θζ=limn→∞dθζ2n+1,Θζ=dθζ,Θζ,which implies θζ∈Θζ. Since θθζ=θζ and θζ∈Θθζ, therefore γ=θγ∈Θγ.By puttingθ=Θ in Theorem 12, we get the following.Corollary 13.
AssumeΛ,d,s is a complete b-metric space. Let Θ be an F-Khan contraction. Then, Θ has a fixed point.Consider the following class of function.Definition 14.
Assumes≥1 is a real number and F↑ represents the family of functions F:ℝ+⟶ℝ, which is strictly increasing.Definition 15.
LetΛ,d,s be a b-metric space. Let θ:Λ⟶Λ and Θ:Λ⟶CBΛ be two mapping. Then, Θ is said to be the F-Khan contraction w.r.t θ if there exists 0<τ<∞ and F∈F↑ such that for all ξ,ζ∈Λ, maxdξ,θζ,dθξ,ζ≠0; then, θξ≠θζ and
(33)τ+FsℍΘξ,Θζ≤Fdξ,θξdξ,θζ+dζ,θζdζ,θξξdξ,θζ,dθξ,ζ,ifmaxdθξ,ζ,dθξ,ζ≠0,0,ifmaxdξ,θζ,dθξ,ζ=0.Theorem 16.
AssumeΛ,d,s is a complete b-metric space. Let θ,Θ be an F-Khan contraction. Further, if θ is continuous, then Cθ,Θ≠∅.
(i)
θ and Θ have a common fixed point if θθα=θα, and θ is occasionally Θ-weakly commuting at α. Then, θ and Θ have a common fixed point
Remarks. Our result extended the results of
(i)
Fisher [20] for set-valued mapping in the setting of b-metric spaces(ii)
Khan [19] for set-valued mapping in b-metric spaces(iii)
Piri et al. [21, 22] for set-valued mapping in b-metric spacesExample 17.
Consider the sequenceSq:q∈1,2⋯100 as follows:
(34)S1=1.2,S2=2.3,⋯,Sq=qq+1.LetΛ=Sq:q∈1,2,⋯,100 and d:Λ×Λ⟶0,∞be defined by
(35)dξ,ζ=maxξ,ζ2,ifξ≠ζdξ,ζ=0,ifξ=ζ.Then,Λ,d,s is a complete b-metric space.Define the mappingΘ:Λ⟶CBΛ by
(36)ΘS1=S1,ΘSq=Sq−1,and θ:Λ⟶Λ by
(37)θS1=S1,θSq=Sq−1.Let us consider the following calculation. First, observe thatmaxdxq,θxp,dθxq,xp≠0⇔p>2∧q=1∨p>q>1.For eachp∈ℕ, p>2, we have
(38)ℍΘSp,ΘS1dSp,θSpdSp,θS1+dS1,θS1dS1,θSp/maxdSp,θS1dS1,θSp=Sp−12Sp2⋅S12+S12⋅Sp−12/Sp2≤Sp−12Sp2=p−12p+12<e−0.02.For eachp,q∈N, p>q>1, we have
(39)ℍΘSp,ΘSqdSp,θSpdSp,θSq+dSq,θSqdSq,θSp/maxdSp,θSqdSq,θSp=Sp2Sp−12Sp2Sp−12+Sq2Sp2≤Sp2Sq2+Sp2=pp−12qq+12+pp+12<e−0.02.Multiplying 1.01 on both sides and taking log to the basee on both sides, we get inequality (7) and also find that τ=0.004365⋯.Therefore, θ and Θ have a fixed point.


## 3. Application to Force Damping Oscillation

Throughout in this section,maxdh,θk,dθh,k≠0.Assume that an object of massm moves to and fro on the x-axis around an equilibrium position x=0 (see Figure 1). The object has position xt at time t. It undergoes a force due to a spring:
(40)Fs=−kx,Figure 1
Forced oscillations.Furthermore, a damping force that resists the movement of the object is shown:(41)Fd=bdxdt.Now, by the second law of motion,(42)Fnet=md2xdt2,where m, k, and b are all positive constants. Up to that feature, the system is simply the damped harmonic oscillator.Now, suppose the additional time-dependent forceft is applied to the object. Then, by Newton’s second law,
(43)−kx−bdxdt+ft=md2xdt2,md2xdt2+bdxdt+kx=ft,x0=0,x′0=0.The problem (43) can be written in the form of the Fredholm integral equation:
(44)vt1=∫01Gβ,sΘ1t1,vs1ds1,t1∈0,1.Here, Green’s function for critically damping oscillation is defined as(45)Gβ,s1=−s1eτβ−t,0≤s1≤β≤1,−teτt−β,0≤β≤s1≤1,where τ can be found in terms of m, b, and k.LetΛ=C0,1 be the set of all continuous functions defined on 0,1. For u∈C0,1, define the supremum norm as
(46)v=supt1∈0,tvt1e−τ.LetC0,1,ℝ be endowed with the b-metric:
(47)dv,ζ=supt1∈0,1vt1−ζt1e−τ.With these contexts,C0,1,R,. becomes the Banach space. We give the following theorem.Theorem 18.
Assume the assumptions given below hold.(A1) There exists a continuous function Θ1:0,1×0,1⟶0,∞ and for t1,s1∈0,t such that
(48)Θ1t1,s1,u−Θ1t1,s1,v≤e−τsMsh,k,Msh,k=e−τsdh,θhdh,θk+dk,θkdk,θhmaxdh,,dθh,k,where θ:C0,1⟶C0,1 is any function.(A2) ∫01Gβ,s1ds1≤1, for some τ>0.Then, the problem (43) has a solution in C0,1.Proof.
DefineΘ:C0,1⟶C0,1 by
(49)Θvt1=∫01Gβ,s1Θ1t1,vs1ds1,t1∈0,t.We have(50)Θv−Θζ=∫01Gβ,s1Θ1t1,s1,vs1ds1−Θ1t1,s1,ζs1ds1,Θv−Θζ≤1s∫01Gβ,s1e−τMsh,kds1≤1sGβ,s1e−τMsh,kds1≤1e−2τse−τMsh,k∫01β,s1ds1≤1se−τMsh,k.Squaring both sides, we get(51)Θv−Θζ2≤12e−τdh,θhdh,θk+dk,θkdk,θhmaxdh,θk,dθh,k.Multiplying on both sidese−2τ and taking sup, we get
(52)sdΘv,Θζ≤e−2τdh,θhdh,θk+dk,θkdk,θhmaxdh,θk,dθh,k.By takingFν=lnν from Theorem 12, the integral equation (44) has a common solution.Similarly, we can apply our theorem for the existence of underdamping oscillation and overdamping oscillation.


## 4. Application to Infinite Systems of Fractional Order Differential Equations

Now, we have to derive sufficient conditions for the solutions in spacec to the following nonlinear infinite systems of fractional order differential equations:
(53)ϑαi=e−τbtbitϑi+θiϑ,η1,η2,⋯,αi∈0,1,ϑβj=e−τbtbjtϑj+θjϑ,ϑ1,ϑ2,⋯,βj∈0,1,with the initial condition ϑ0=ϑi0, where t∈J, i,j=1,2⋯, and τ is the positive real number. J is any fixed interval on the real line. Let Λ=c be the space of all real sequences whose limit is finite.
(54)η=supηie−τ.Letc,ℝ be endowed with the b-metric:
(55)dη,ϑ=supηi−ϑie−τ.Theorem 19.
Assume the assumptions given below hold.(A1) ϑ0=ϑi0∈c.θ=θ1,θ2,⋯:J×c⟶c.(A2) bit represents the continuous function on J.(A3)
(56)θit,η−θit,ϑ≤e−τ2sdη,θjηdη,θjϑ+dϑ,θjϑdϑ,θjηηdη,θjϑ,dθjη,ϑ1/2,where η=η1,η2,η3⋯ and ϑ=ϑ1,ϑ2,ϑ3⋯.(A4)
(57)bitηi−bjtϑi≤bt2sdη,θjηdη,θjϑ+dϑ,θjϑdϑ,θjηηdη,θjϑ,dθjη,ϑ1/2,then, the infinite systems of fractional order differential (53) have a common solution in c.Proof.
Definehit,η,hjt,η:J×c⟶c by
(58)hit,η=e−τbtbitηi+θiη,η1,η2,⋯,hjt,ϑ=e−τbtbjtηj+θjϑ,ϑ1,ϑ2,⋯.
We have(59)hit,η−hit,ϑ=e−τbtbitηi+θiη,η1,η2,⋯−e−τbtbjtϑ1−θiϑ,ϑ1,ϑ2,⋯,≤e−τbtbitηi−e−τbtbjtϑj+θiη,η1,η2,⋯−θjϑ,ϑ1,ϑ2,⋯,≤e−τ2sdη,θjηdη,θjϑ+dϑ,θjϑdϑ,θjηmaxdη,θjϑ,dθjη,ϑ1/2+e−τ2sdη,θjηdη,θjϑ+dϑ,θjϑdϑ,θjηηdη,θjϑ,dθjη,ϑ1/2.
Squaring both sides, we get(60)hit,η−hit,ϑ2e−τ≤e−2τs2dη,θjηdη,θjϑ+dϑ,θjϑdϑ,θjηmaxdη,θjϑ,dθjη,ϑ.
Taking the supremum, we get(61)s2dhit,η−hit,ϑ≤e−τdη,θjηdη,θjϑ+dϑ,θjϑdϑ,θjηmaxdη,θjϑ,dθjη,ϑ.
By takingFν=lnν from Theorem 12, the infinite systems of fractional order differential equation (53) have a common solution.


## 5. Application to Functional Equations

In this section, we study the solvability of functional equations using the established fixed point theorem.LetΛ1 and Λ2 be Banach spaces, Θ1⊂Λ1, Θ2⊂Λ2, and ℝ be the field of real numbers. Suppose Λ=BΘ1 represents the set of all functions defined on Θ1 which is bounded and real valued. Define d:Λ×Λ⟶ℝ+ by
(62)dν,ζ=ν−ζ2.Then,Λ,d,s is called the b-metric space. Consider the system of functional equations given as
(63)pν=supζ∈Θ2gν,ζ+Φ1ν,ζ,Qτν,ζ,ν∈Θ1,qν=supζ∈Θ2gν,ζ+Φ2ν,ζ,Qτν,ζ,ν∈Θ1,Here,g:Θ1×Θ2⟶ℝ and Φ1,Φ2:Θ1×Θ2×ℝ⟶ℝ are functions which are bounded. Θ1 and Θ2 are the state and decision spaces, respectively. τ:Θ1×Θ2⟶Θ1 shows the transformation of the process, and pν and qν signify sup return functions with initial state Λ. Let Θ,θ:BΘ1⟶BΘ1 be defined by
(64)Θh1ν=supζ∈Θ2gν,ζ+Φ1ν,ζ,Pτν,ζ,ν∈Θ1,θh1ν=supζ∈Θ2gν,ζ+Φ2ν,ζ,Pτν,ζ,ν∈Θ1,Theorem 20.
LetΦ1,Φ2:Θ1×Θ2×ℝ⟶ℝ and g:Θ1×Θ2⟶ℝ be continuous and bounded and satisfy the following assumption.
(65)Φ1ν,ζ,hν−Φ1ν,ζ,kν≤e−τsdh,θhdh,θk+dk,θkdk,θhmaxdh,θk,dθh,k,for all ν∈Θ1 and ζ∈Θ2. Then, the system of functional equation (63) has a common solution.Proof.
Letδ be an arbitrary positive number, ν∈Θ1, and h1,h2∈BΛ; then, there exist ζ1,ζ2∈Θ2 such that
(66)Θh1ν<gν,ζ1+Φ1ν,ζ1,h1τ1+δ,(67)Θh2ν<gν,ζ2+Φ1ν,ζ2,h2τ2+δ,(68)Θh1ν≥gν,ζ2+Φ1ν,ζ2,h1τ2,(69)Θh2ν≥gν,ζ1+Φ1ν,ζ1,h2τ1.Using equations (66) and (69),
(70)Θh1ν−Θh2ν≤Φ1ν,ζ1,h1τν,ζ1−Φ1ν,ζ1,h2τν,ζ1+δ≤∣Φ1ν,ζ1,h2τν,ζ1−Φ1ν,ζ1,h1τν,ζ1∣+δ≤e−τsdh,θhdh,θk+dk,θkdk,θhmaxdh,θk,dθh,k+δ.It follows that(71)Θh1ν−Θh2ν≤e−τsdh,θhdh,θk+dk,θkdk,θhmaxdh,θk,dθh,k+δ.Similarly, using equations (67) and (68), we have
(72)Θh2ν−Θh1ν≤e−τsdh,θhdh,θk+dk,θkdk,θhmaxdh,θk,dθh,k+δ.Using equations (71) and (72), we get
(73)Θh1ν−Θh2ν≤e−τsdh,θhdh,θk+dk,θkdk,θhmaxdh,θk,dθh,k+δ,for allν∈Θ1, and δ>0 is arbitrary; therefore,
(74)sΘh1ν−Θh2ν2≤e−2τdh,θhdh,θk+dk,θkdk,θhmaxdh,θk,dθh,k2.Taking logarithms, we have(75)lnsΘh1ν−Θh2ν2≤lne−2τdh,θhdh,θk+dk,θkdk,θhmaxdh,θk,dθh,k2.After simple calculation, we get(76)2τ+lnsdΘh1ν,Θh2ν≤lndh,θhdh,θk+dk,θkdk,θhmaxdh,θk,dθh,k.By takingFν=lnν from Theorem 12, the system of functional equation (63) has a common solution.



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*Source: 4843908-2020-12-08.xml*