# Nonunique Fixed Point Results via KannanF-Contraction on Quasi-Partial b-Metric Space

Authors: Pragati Gautam; Santosh Kumar; Swapnil Verma; Gauri Gupta
Journal: Journal of Function Spaces
 (2021)

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## Abstract

This paper is aimed at acquainting with a new KannanF-expanding type mapping by the approach of Wardowski in the complete metric space. We establish some fixed point results for Kannan F-expanding type mapping and F-contractive type mappings which satisfy F-contraction conditions. Additionally, some new results are given which generalize several results present in the literature. Moreover, some applications and examples are provided to show the practicality of our results.


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## Body

## 1. Introduction and Preliminaries

In 1922, Banach [1] commenced one of the most essential and notable results called the Banach contraction principle, i.e., let P be a self-mapping on a nonempty set X and d be a complete metric, if there exists a constant k∈0,1 such that
(1)dPu,Pv≤kdu,v,for all u,v∈X. Then, it has a unique fixed point in X. Due to its significance, in 1968, Kannan [2] introduced a different intuition of the Banach contraction principle which removes the condition of continuity, i.e., for all u,v∈0,1/2, there exists a constant ρ∈0,1 such that
(2)dPu,Pv≤ρdu,Pu+dv,Pv.On the other hand, the notion of metric space has been generalized in several directions, and the abovementioned contraction principle has been enhanced in the new settings by considering the concept of convergence of functions. In 1989, Bakhtin [3] introduced the notion of b-metric space which was revaluated by Czerwik [4] in 1993.Definition 1.
Ab-metric space on a nonempty set X is a function d:X×X⟶0,∞ such that for all u,v,w∈X and for some real number s≥1, it satisfies the following:
(M1) If du,v=0, then u=v
(M2) du,v=dv,u
(M3) du,w≤sdu,v+dv,wThen, the pairX,d,s is called the b-metric space. Motivated by this, many researchers [5–8] generalized the concept of metric spaces and established on the existence of fixed points in the setting of b-metric space keeping in mind that, unlike standard metric, b-metric is not necessarily continuous due to the modified triangle inequality. In general, a b-metric does not induce a topology on X.Partial metric space is one of the attempts to generalize the notion of the metric space. In 1994, Matthews [9] introduced the notion of a partial metric space in which du,u are no longer necessarily zero.Definition 2.
A partial metric on a nonempty setX is a function p:X×X⟶0,∞ such that for all u,v,w∈X, it satisfies the following:
(PM1) If pu,u=pu,v=pv,v, then u=v
(PM2) pu,u≤pu,v
(PM3) pu,v=pv,u
(PM4) pu,v≤pu,w+pw,v−pw,wThen, the pairX,p is called the partial metric space.Definition 3.
LetX,p be a partial metric space. Then, several topological concepts for partial metric space can be easily defined as follows:(1)
A sequenceun in the partial metric space X,p converges to the limit u if pu,u=limn⟶∞pu,un(2)
It is said to be a Cauchy sequence iflimn⟶∞pun,um exists and is finite(3)
A partial metric spaceX,p is called complete if every Cauchy sequence un in X converges with respect to τp, to a point u∈X such that pu,u=limn⟶∞pun,umFor more details, see, for example, [10–12], and the related references therein. The following definition gives room for the lack of symmetry in the spaces under study. In 2013, Karapinar et al. [13] introduced quasi-partial metric space that satisfies the same axioms as metric spaces.Definition 4.
A quasi-partial metric on a nonempty setX is a function q:X×X⟶ℝ+ that satisfies the following:
(QPM1) If qu,u=qu,v=qv,v, then u=v
(QPM2) qu,u≤qu,v
(QPM3) qu,u≤qv,u
(QPM4) qu,v+qw,w≤qu,w+qw,vfor all u,v,w∈X. Then, the pair X,q is called quasi-partial metric space.Later on, Gupta and Gautam [14, 15] introduced quasi-partial b-metric space.Definition 5.
A quasi-partialb-metric on a nonempty set X is a function qpb:X×X⟶0,∞ such that for some real number ρ≥1, it satisfies the following:
(QPb1) If qpbu,u=qpbu,v=qpbv,v, then u=v (indistancy implies equality)
(QPb2) qpbu,u≤qpbu,v (small self-distances)
(QPb3) qpbu,u≤qpbv,u (small self-distances)
(QPb4) qpbu,v+qpbw,w≤ρqpbu,w+qpbw,v (triangularity)
for allu,v,w∈X. The infimum over all reals ρ≥1 satisfying QPb4 is called the coefficient of X,qpb and represented by RX,qpb.Lemma 6.
LetX,qpb be a quasi-partial b-metric space. Then, the following hold:(i)
Ifqpbu,v=0, then u=v(ii)
Ifu≠v, then qpbu,v>0 and qpbv,u>0Definition 7.
LetX,qpb be a quasi-partial b-metric. Then,
(i)
a sequenceun⊂X converges to u∈X if and only if(3)qpbu,u=limn⟶∞qpbu,un=limn⟶∞qpbun,u(ii)
a sequenceun⊂X is called a Cauchy sequence if and only if(4)limn,m⟶∞qpbun,um&limm,n⟶∞qpbum,unexist(iii)
the quasi-partialb-metric space X,qpb is said to be complete if every Cauchy sequence un⊂X converges with respect to τqpb to a point u∈X such that(5)qpbu,u=limn,m⟶∞qpbun,um=limm,n⟶∞qpbum,un(iv)
a mappingf:X⟶X is said to be continuous at u0∈X, if for every ε>0, there exists δ>0 such that fBu0,δ⊂Bfu0,εThe extensive application of the Banach contraction principle has motivated many researchers to study the possibility of its generalization. A great number of generalizations of this famous result have appeared in the literature. In 2012, Wardowski [16] established a new notion of F-contraction and proved the fixed point theorem which generalized the Banach contraction principle.Definition 8 (see [16]).
LetX,d be a metric space, and there exists a mapping F:0,∞⟶ℝ which satisfies the following condition:
(F1) F is strictly increasing
(F2) For any sequence xnn∈N, limn⟶∞xn=0 if and only if limn⟶∞Fxn=−∞
(F3) limx⟶0+xkFx=0 for some k∈0,1
Then, a mappingP:X⟶X is said to be Wardowski F-contraction if dPu,Pv>0 implies
(6)δ+FdPu,Pv≤Fdu,vfor all u,v∈XTheorem 9 (see [16]).
LetX,d be a complete metric space and T:X⟶X an F-contraction. Then, T has a unique fixed point x∗∈X, and for every x∈X, the sequence Tnxn∈N converges to x∗.In 2012, Samet et al. [17] established the class of α-admissible mappings as follows.Definition 10 (see [17]).
Letα:X×X⟶0,∞ be given mapping where X is a nonempty set. A self-mapping T is called α-admissible if for all x,y∈X, we have
(7)αx,y≥1⇒αTx,Ty≥1.Motivated by this, Aydi et al. [18] extended the notion of F-contraction and prove the following result.Theorem 11 (see [18]).
LetX,d be a metric space. A self-mapping T:X⟶X is said to be a modified F-contraction via α-admissible mappings. Suppose that
(i)
T is α-admissible(ii)
there existsx0∈X such that αx0,Tx0≥1(iii)
T is continuousThen,T has a fixed point. In 2015, Kumam et al. [19] generalized the contraction condition by adding four new values dT2x,x,dT2x,Tx,dT2x,y,dT2x,Ty and introduced F-Suzuki contraction mappings in complete metric space. The Suzuki-type generalization can be said to have many applications, as in computer science, game theory, biosciences, and in other areas of mathematical sciences such as in dynamic programming, integral equations, and data dependence. Recently, Wardowski [20] proposed the replacement of the positive constant δ in equation (6) by a function ϕ and relaxed the conditions on F.Definition 12 (see [20]).
LetX,d be a metric space, F:0,∞⟶ℝ and ϕ:0,∞⟶0,∞ satisfy the following:(1)
F is strictly increasing, i.e., x<y implies Fx<Fy for all x,y∈0,∞(2)
limα⟶0+Fα=−∞(3)
liminfα⟶s+ϕα>0 for all s>0
A mappingT:X⟶X is called an ϕ,F-contraction on X,d if
(8)ϕdx,y+FdTx,Ty≤Fdx,y,for all x,y∈X for which Tx≠Ty.Consider a functionFB:ℝ+⟶ℝ by FBu=lnu. Note that with F=FB, the F-contraction reduces to a Banach contraction. Therefore, the Banach contractions are a particular case of F-contractions. Meanwhile, there exist F-contractions which are not Banach contractions.The concept of anF-contraction has been generalized in many directions (see, e.g., [21–24]), and as an extension, engaging work was done by many authors [25–34], which enhanced this field. In 2015, Cosentino et al. [35] extended the concept of F-contraction in metric space to F-contraction in b-metric space by introducing the following condition with continuation of Definition 7.F4 For some δ>0 and any sequence xn, we have
(9)δ+Fsxn≤Fxn−1,δ+Fsnxn≤Fsn−1xn−1for all n∈N,s∈ℝIn 2017, Gornicki [36] established F-expanding type mappings.Definition 13.
LetX,d be a metric space. A mapping P:X⟶X is called F-expanding if for all u,v∈X and δ>0, we have
(10)du,v>0⇒FdPu,Pv≥Fdu,v+δ.The concept ofF-expanding type mappings was redefined as Kannan F-expanding type mappings by Goswami et al. [37].Definition 14 (see [37]).
A mappingP:X⟶X is said to be Kannan F-expanding type mapping if there exists Δ>0 such that du,Pudv,Pv≠0 implies
(11)Δ+Fsdu,v≤12Fdu,Pu+Fdv,Pv,and du,Pudv,Pv=0 implies
(12)Δ+Fsdu,v≤12Fdu,Pv+Fdv,Pufor all u,v∈X. Following this direction, we have established a new type of mapping, i.e., Kannan F-expanding type mapping, and proved some fixed point results for F-contractive type mappings as well as Kannan F-expanding type mappings in the setting of quasi-partial b-metric space without using the continuity of mapping. Also, we attain the nonunique fixed point in quasi-partial b-metric space which lacks symmetry property.The main motive behind this study is that today, this field of research has vast literature. The significance of the Kannan type mapping is that it characterizes completeness which the Banach contraction does not; also, it does not require continuous mapping. In this paper, some examples and applications for the solution of a certain integral equation and the existence of a bounded solution of the functional equation are also given to represent the practicality of the results obtained. The application shows the role of fixed point theorems in dynamic programming, which is used in computer programming and optimization.The future aspect of this study is to prove the existence of a unique fixed point in KannanF-expanding type mapping. Another field of research can be the existence of a common fixed point for the same. The notion of interpolative F-contraction as well as interpolation for Kannan F-expanding type mapping can also be future studies concerning the present manuscript.


## 2. Fixed Point forF-Contractive Type Mappings

In this section, the existence of a fixed point forF-contractive type mappings in a quasi-partial b-metric space is obtained.Definition 15.
For a quasi-partialb-metric space X,qpb, a mapping P:X⟶X is said to be an F-contractive type mapping if there exists δ>0 such that, if qpbu,Puqpbv,Pv≠0, then
(13)δ+FρqpbPu,Pv≤13Fqpbu,v+Fqpbu,Pv+Fqpbv,Pu−Fqpbw,Pw,and if qpbu,Puqpbv,Pv=0, then
(14)δ+FρqpbPu,Pv≤13Fqpbu,v+Fqpbu,Pu+Fqpbv,Pv−Fqpbw,Pwfor all u,v,w∈X and ρ≥1.Definition 16.
LetX,qpb be a quasi-partial b-metric space. A self-mapping P on X is called an F-contraction if there exist τ∈ℝ+ such that
(15)τ+FρqpbPu,Pv≤Fqpbu,v,for all u,v∈X with qpbPu,Pv>0.Example 17.
LetF:ℝ+⟶ℝ be given by Fu=logu. Here, F satisfies (F1)-(F3) for any k∈0,1. Each mapping P:X⟶X satisfying Definition 16 is an F-contraction such that
(16)qpbPu,Pv≤ρe−τqpbu,vfor all u,v∈X,Pu≠Pv.It is clear that foru,v∈X such that Pu=Pv, the previous inequality also holds, and hence, P is a contraction as shown in Figure 1.Example 18.
Consider a functionFu=−1/u,u>0 where F satisfies (F1)-(F3) for any k∈1/2,1. In this case, a mapping P:X⟶X satisfies
(17)ρqpbPu,Pv≤11+τqpbu,v2qpbu,vfor all u,v∈X,Pu≠Pv. Hence, P is a contraction as shown in Figure 2.Theorem 19.
LetX,qpb be a quasi-partial b-metric space and P:X⟶X be an F-contractive type mapping. Then, P has a unique fixed point u∗∈X, and for every u0∈X, a sequence Pnx0n∈ℕ is convergent to u∗.Proof.
Letu0 be an arbitrary and fixed point in X, and we assume a sequence unn∈N⊂X such that un+1=Pun,n=0,1,⋯. To prove P has a fixed point, we need to show that if un0+1=un0, then Pun0=un0 for all n0∈ℕ. Suppose that un+1≠un for every n∈ℕ, then qpbun+1,un>0, and using equation (6), we have
(18)Fqpbun+1,un≤ρFqpbun,un−1−δ≤ρFqpbun−1,un−2−2δ≤⋮≤ρFqpbu,u0−nδ,which implies
(19)limn⟶∞ρFqpbun+1,un=−∞.
UsingF2, we get
(20)limn⟶∞ρqpbun+1,un=0.
Also, usingF3, there exists k∈0,1 such that
(21)limn⟶∞qpbun+1,unkFqpbun+1,un=0.
Let us denoteqpbun+1,un by αn. From inequality (18), the following holds
(22)Fαn−αnkFα0≤ραnkFα0−nδ−αnkFα0=−ραnknδ≤0,which implies
(23)limn⟶∞nαnk=0.
Also, if there existsn1<n∈ℕ such that nαnk≤1, we have
(24)ραn≤n−1/k.
To proveunn∈ℕ is a Cauchy sequence, let us consider m,n∈ℕ such that m>n≥n1. From the definition of quasi-partial b-metric space and equation (24), we have
(25)qpbum,un≤ραm−1+αm−2+⋯+αn≤ρ∑i=n∞αi≤ρ∑i=n∞i−1/k.
Using the convergence of series, we get thatunn∈ℕ is a Cauchy sequence. Since X is complete, there exists u∗∈X such that limn⟶∞un=u∗, and the continuity of P implies
(26)qpbPu∗,u∗=ρlimn⟶∞qpbPun,un=ρqpbun+1,un=0.
Hence,P has a unique fixed point.☐Theorem 20.
For a quasi-partialb-metric space X,qpb, we say X is complete if for every closed subset Y of X, P:Y⟶Y is an F-contractive type mapping having a fixed point.Proof.
Suppose that there does not exist any Cauchy sequence inX which has a convergent subsequence and we have a sequence
(27)θun=infqpbun,um:m>n>0for all n∈ℕ where θun≤θum for m≥n. Also, we consider a subsequence unk such that
(28)ρqpbui,uj<aθunk−1,for any a with 0<a<1 and for all i,j≥nk. Then, Y=unk:k∈ℕ is a closed subset of X. Define P:X⟶X by
(29)Punk=unk+1for all k∈ℕ, which implies P has no fixed point. Now,
(30)ρqpbPu,Pv=ρqpbPunk,Punk+1=ρqpbunk+1,Punk+i+1<aθunk.
By definition,(31)θunk≤ρqpbunk,unk+i=ρqpbu,v≤ρqpbunk,unk+1=ρqpbu,Pu≤θunk+i≤ρqpBunk+i,unk+i+1=ρqpbv,Pv,which implies
(32)δ+FρqpbPu,Pv≤13Fqpbu,v+Fqpbu,Pu+Fqpbv,Pv−qpbw,Pwfor some δ>0. Hence, it proves that P is an F-contractive type mapping on a closed subset of X which has no fixed point. Thus, this is a contradiction and X is complete.☐Theorem 21.
LetX,qpb be a quasi-partial b-metric space and P:X×CX be a closed F-contraction. Then, P has a fixed point.Proof.
Letu0∈X be an arbitrary point of X, and we have u1∈Pu0. If u1=u0, then u1 is a fixed point of P, and hence, the proof is completed. Now, assume that u1≠u0. Since P is a F-contraction, there exists u2∈Pu1 such that
(33)τ+Fρqpbu1,u2≤FMu0,u1,where
(34)Mu0,u1=maxqpbu0,u1,qpbu0,Pu0,qpbu1,Pu1,12qpbu0,Pu1+qpbu1,Pu0,and u2≠u1. Also, there exists u3∈Pu2 such that
(35)τ+Fρqpbu2,u3≤FMu1,u2,and u3≠u2. With the recurrence of the same process, we get
(36)τ+Fρqpbun,un+1≤FMun−1,unfor all n∈ℕ. It implies
(37)τ+Fρqpbun,un+1≤FMun−1,un.
Assume thatqpbn=qpbxn,xn+1>0 for all n∈ℕ∪0. By equation (37), we have
(38)Fρqpbn≤Fρqpbn−1≤⋯<≤Fρqpb0−nτfor all n∈ℕ. Letting n⟶∞, property F2 implies
(39)limn⟶∞Fρqpbn=−∞.
Letk∈0,1 such that
(40)limn⟶∞qpbnkFρqpbn=0.
By equation (38), the following holds
(41)qpbnkFρqpbn−qpbnkFρqpb0≤qpbnkFρqpb0−nτ−qpbnkFρqpb0=−nτqpbnk≤0for all n∈ℕ. Letting n⟶∞, we get
(42)limn⟶∞nqpbnk=0.
This implieslimn⟶∞n1/kqpbn=0 and ∑n=1+∞qpbn is convergent. Hence, un is Cauchy sequence. Since X is complete, there exists x∈X such that un=x as n⟶+∞. Since P is closed, un,un+1⟶x,x, we get x∈Px, and hence, x is the fixed point of P.☐Corollary 22.
LetX,qpb be a quasi-partial b-metric space and P:X⟶CX be an upper semicontinuous F-contraction. Then, P has a fixed point.Example 23.
Consider the quasi-partialb-metric space X,qpb where X=0,2,4,⋯ and qpb:X×X⟶0,∞ is given by
(43)qpbu,v=u+v,u≠v,0,u=v,which is also shown in Figure 3, and P:X⟶CX is defined by
(44)Pu=0,u∈0,1,0,2,⋯,2u−2,u≥4.
Now, we show thatP satisfies Definition 16, where ρ=2,τ=2 and Fu=logu+u for each u∈ℝ+. Let for all u,v∈X with v∈Pu, we have w=0∈Pv. Here, qpbv,w>0 iff x≥4 and w>0. If it is true, then
(45)qpbv,w=v<u+v=qpbu,v.
This implies(46)qpbv,w−Mu,v≤qpbv,w−qpbu,v≤−4,qpbv,wMu,veqpbu,v−Mu,v≤e−2.
Hence,(47)2+Fρqpbv,w≤FMu,vfor all u,v∈X and qpbv,w>0. Then, by Theorem 21, P has a fixed point.Figure 1
The graphical surface represents a 3-D view ofF-contractive function Fu=logu where u∈0,5,v∈0,5 and w∈ℝ+.Figure 2
The graphical surface represents a 3-D view ofF-contractive function Fu=−1/u where u∈0,10,v∈0,10 and w∈ℝ+.Figure 3
3-D view. The plane in yellow represents the quasi-partialb-metric space defined by qpb:X×X⟶0,∞ and qpbu,v=u+v when u≠v for all u,v∈X.


## 3. Fixed Point for KannanF-Expanding Type Mapping

In this section, we prove the fixed point results for KannanF-expanding type mappings in a quasi-partial b-metric space.Definition 24.
Let us consider a mappingP:X⟶X; it is said to be Kannan F-expanding type mapping if there exists Δ>0 such that qpbu,Puqpbv,Pv≠0 implies
(48)Δ+Fρqpbu,v≤12Fqpbu,Pu+Fqpbv,Pv−FqpbPw,Pw,and qpbu,Puqpbv,Pv=0 implies
(49)Δ+Fρqpbu,v≤12Fqpbu,Pv+FqpbPv,Pu−FqpbPw,Pwfor all u,v,w∈X.Lemma 25.
LetX,qpb be a quasi-partial b-metric space and P:X⟶X be surjective. Then, there exists a mapping P∗:X⟶X such that P∘P∗ is the identity map on X.Proof.
For any pointu∈X, let vu∈X be any point such that Pvu=u. Let P∗u=vu for all u∈X. Then, P∘P∗u=PP∗u=Pvu=u for all u∈X.☐Theorem 26.
LetX,qpb be a quasi-partial b-metric space and P:X⟶X be surjective and a Kannan F-expanding type mapping. Then, P has a unique fixed point γ∈X.Proof.
Assume that there exists a mappingP∗:X⟶X such that P∘P∗ is the identity map on X. Let u,v be arbitrary points of X such that u≠v and x=P∗u,y=P∗v which also implies that x≠y. Applying equation (48) on x,y, we have
(50)Δ+Fρqpbx,y≤12Fqpbx,Px+Fqpby,Py−Fqpbz,Pzfor qpbx,Pxqpby,Py≠0 and
(51)Δ+Fρqpbx,y≤12Fqpbx,Py+Fqpby,Px−Fqpbz,Pzfor qpbx,Pxqpby,Py=0. Since Px=PP∗u=u and Py=PP∗v=v, we get
(52)Δ+FρqpbP∗u,P∗v≤12Fqpbu,P∗u+Fqpbv,P∗v−Fqpbw,P∗wfor qpbu,Puqpbv,Pv≠0 and
(53)Δ+FρqpbP∗u,P∗v≤12Fqpbu,P∗v+Fqpbv,P∗u−Fqpbw,P∗wfor qpbu,Puqpbv,Pv=0, which implies P∗ is Kannan F-contractive type mapping. Also, we know that P∗ has a unique fixed point γ∈X, and for every u0∈X, the sequence P∗nu0 converges to γ. In particular, γ is also a fixed point of P since P∗γ=γ implies that
(54)Pγ=PP∗γ=γ.
Finally, ifγ0=Pγ0 is another fixed point, then from equation (49),
(55)Δ+FρqpBγ,γ0≤12Fqpbγ,Pγ0+Fqpbγ0,Pγ−Fqpbv,Pv,which is not possible, and hence, P has a unique fixed point.☐


## 4. Applications ofF-Contraction

In this section, we discuss the applications of the results obtained to prove the existence of the solution of an integral equation and a functional equation.


### 4.1. Existence of Solution of Integral Equation

Now, we study the existence of solution of the following Volterra type integral equation(56)ux=∫0xfx,y,uydy+gx,x∈0,σ where σ>0. Let C0,σ,ℝ denote space of all continuous functions on 0,σ, and for an arbitrary u∈C0,σ,ℝ, we define
(57)uτ=supx∈0,σuxe−τx,where τ>0 is taken arbitrary. Clearly, C0,σ,ℝ,·τ is endowed with quasi-partial b-metric defined by
(58)qpbu,v=supx∈0,σ∣ux−vx∣e−τxfor all u,v∈C0,σ,ℝ is a Banach space and
(59)u≤v⇔ux≤vxfor all x∈0,σ.Theorem 27.
Let us consider that for the integral equation (56), the following conditions are satisfied:
(i)
f and g are continuous where f:0,σ×0,σ×ℝ⟶ℝ and g:0,σ⟶ℝ(ii)
fx,y,.:ℝ⟶ℝ is increasing(iii)
u0x≤∫0xfx,y,u0ydy+gx for some u0∈C0,σ,ℝ(iv)
There existsτ∈1,∞ such that(60)fx,y,u−fx,y,v≤τe−2τρu−vfor all x,y∈0,σ and u,v∈ℝ. Then, integral equation (56) has a solution.Proof.
Here,C0,σ,ℝ,qpb is a quasi-partial b-metric space, where qpb is the quasi-partial b-metric given by equation (58). Let us define P:C0,σ,ℝ⟶C0,σ,ℝ by
(61)Pux=∫0xfx,y,uydy+gx,x∈0,σ.
From (iv) we have,(62)∣Pux−Pvx∣≤∫0x∣fx,y,uy−fx,y,vy∣dy≤∫0xτe−2τρ∣uy−vy∣dy=∫0xτe−2τρ∣uy−vy∣e−τyeτydy≤∫0xτe−2τeτyρ∣uy−vy∣e−τydy≤τe−2τρu−vτ∫0xeτydy≤τe−2τρ1τu−vτeτx.
It implies(63)∣Pux−Pvx∣e−τx≤e−2τρu−vτor
(64)qpbPu,Pv≤e−2τρqpbu,v.
Taking logarithm in both sides, we get(65)lnqpbPu,Pv≤lne−2τρqpbu,v,which on solving reduces to
(66)2τ+lnqpbPu,Pv≤lnρqpbu,v.
Now, we observe that the functionF:ℝ+⟶ℝ defined by Fu=logu for each u∈C0,σ,ℝ is F-contraction. Clearly, from (iii), we have
(67)u0≤Pu0,and hence, Theorem 19 applies to P, which has a fixed point u∗∈C0,σ,ℝ. Hence, u∗ is a solution of integral equation (56).☐


### 4.2. Existence of Bounded Solutions of Functional Equations

Fixed point theory is widely used in the field of dynamic programming which is the most commonly used tool for mathematical optimization. With this approach, the problem of the dynamic programming process reduces to solving the functional equations.Let us consider thatU and V are Banach spaces, W⊂U is a state space, i.e., the set of the initial state of process, and D⊂V is a decision space, i.e., the set of possible actions that are allowed for the process.Here, we will prove the existence of the bounded solution of the following functional equation:(68)ϕu=supx∈Dfu,v+gu,v+ϕτu,v,where τ:W×D,f:W×D⟶ℝ,g:W×D×ℝ⟶ℝ. Let BW denote the set of all bounded real valued functions W and for an arbitrary α∈BW, define α=supx∈W∣αx∣. Clearly, Bw,· endowed with quasi-partial b-metric defined by
(69)qpbα,β=supx∈W∣αx−βx∣for all α,β∈BW is a Banach space. Thus, if we consider a Cauchy sequence αn in BW, then αn converges uniformly to a function, let α∗ that is bounded and so α∗∈Bw. Also, we have P:BW×BW defined by
(70)Pαx=supy∈Dfx,y+gx,y,ατx,yfor allα∈BW and x∈W. Hence, P is well defined if f and g are bounded.Theorem 28.
LetP:BW⟶BW be an upper semicontinuous operator defined by (70), and we assume that the following conditions are satisfied:
(i)
f and g are bounded and continuous where f:W×D⟶ℝ and g:W×D×ℝ⟶ℝ(ii)
There existsτ∈ℝ+ such that(71)∣gx,y,αx−gx,y,βx∣≤ρ∣α−β∣1+τ∣α−β∣2for all α,β∈BW,x∈W,y∈D,ρ≥1. Then, the functional equation (68) has a bounded solution.Proof.
Clearly,BW,qpb is a quasi-partial b-metric given by equation (69). Let σ be an arbitrary positive number, x∈W, α1,α2∈BW, then there exist y1,y2∈D such that
(72)Pα1x<fx,y1+gx,y1,α1τx,y1+σ,(73)Pα2x<fx,y2+gx,y2,α2τx,y2+σ,(74)Pα1x<fx,y2+gx,y2,α1τx,y2+σ,(75)Pα2x<fx,y1+gx,y1,α2τx,y1+σ.
From equations (72) and (75),
(76)Pα1x−Pα2x<gx,y1,α1τx,y1−gx,y1,α2τx,y1+σ≤∣gx,y1,α1τx,y1−gx,y1,α2τx,y1∣+σ≤ρ∣α1−α2∣1+τ∣α1−α2∣2+σ.
It implies,(77)Pα1x−Pα2x≤ρ∣α1−α2∣1+τ∣α1−α2∣2+σ.
Similarly, from equations (73) and (74),
(78)Pα2x−Pα1x≤ρ∣α1−α2∣1+τ∣α1−α2∣2+σ.
From equations (77) and (78), we get
(79)Pα1x−Pα2x≤ρα1−α21+τα1−α22+σ,i.e.,
(80)qpbPα1,Pα2≤ρ∣α1−α2∣1+τ∣α1−α2∣2+σ.
Hence, we conclude that(81)qpbPα1,Pα2≤ρ∣α1−α2∣1+τ∣α1−α2∣2.
Now, we observe that the functionF:ℝ+⟶ℝ defined by Fα=−1/α for each α∈W is F-contractive function, and hence, operator P is F-contractive.
Since any upper semicontinuousF-contractive function has a fixed point α∗∈BW, it implies that there exists a bounded solution of functional equation (68).☐


## 4.1. Existence of Solution of Integral Equation

Now, we study the existence of solution of the following Volterra type integral equation(56)ux=∫0xfx,y,uydy+gx,x∈0,σ where σ>0. Let C0,σ,ℝ denote space of all continuous functions on 0,σ, and for an arbitrary u∈C0,σ,ℝ, we define
(57)uτ=supx∈0,σuxe−τx,where τ>0 is taken arbitrary. Clearly, C0,σ,ℝ,·τ is endowed with quasi-partial b-metric defined by
(58)qpbu,v=supx∈0,σ∣ux−vx∣e−τxfor all u,v∈C0,σ,ℝ is a Banach space and
(59)u≤v⇔ux≤vxfor all x∈0,σ.Theorem 27.
Let us consider that for the integral equation (56), the following conditions are satisfied:
(i)
f and g are continuous where f:0,σ×0,σ×ℝ⟶ℝ and g:0,σ⟶ℝ(ii)
fx,y,.:ℝ⟶ℝ is increasing(iii)
u0x≤∫0xfx,y,u0ydy+gx for some u0∈C0,σ,ℝ(iv)
There existsτ∈1,∞ such that(60)fx,y,u−fx,y,v≤τe−2τρu−vfor all x,y∈0,σ and u,v∈ℝ. Then, integral equation (56) has a solution.Proof.
Here,C0,σ,ℝ,qpb is a quasi-partial b-metric space, where qpb is the quasi-partial b-metric given by equation (58). Let us define P:C0,σ,ℝ⟶C0,σ,ℝ by
(61)Pux=∫0xfx,y,uydy+gx,x∈0,σ.
From (iv) we have,(62)∣Pux−Pvx∣≤∫0x∣fx,y,uy−fx,y,vy∣dy≤∫0xτe−2τρ∣uy−vy∣dy=∫0xτe−2τρ∣uy−vy∣e−τyeτydy≤∫0xτe−2τeτyρ∣uy−vy∣e−τydy≤τe−2τρu−vτ∫0xeτydy≤τe−2τρ1τu−vτeτx.
It implies(63)∣Pux−Pvx∣e−τx≤e−2τρu−vτor
(64)qpbPu,Pv≤e−2τρqpbu,v.
Taking logarithm in both sides, we get(65)lnqpbPu,Pv≤lne−2τρqpbu,v,which on solving reduces to
(66)2τ+lnqpbPu,Pv≤lnρqpbu,v.
Now, we observe that the functionF:ℝ+⟶ℝ defined by Fu=logu for each u∈C0,σ,ℝ is F-contraction. Clearly, from (iii), we have
(67)u0≤Pu0,and hence, Theorem 19 applies to P, which has a fixed point u∗∈C0,σ,ℝ. Hence, u∗ is a solution of integral equation (56).☐


## 4.2. Existence of Bounded Solutions of Functional Equations

Fixed point theory is widely used in the field of dynamic programming which is the most commonly used tool for mathematical optimization. With this approach, the problem of the dynamic programming process reduces to solving the functional equations.Let us consider thatU and V are Banach spaces, W⊂U is a state space, i.e., the set of the initial state of process, and D⊂V is a decision space, i.e., the set of possible actions that are allowed for the process.Here, we will prove the existence of the bounded solution of the following functional equation:(68)ϕu=supx∈Dfu,v+gu,v+ϕτu,v,where τ:W×D,f:W×D⟶ℝ,g:W×D×ℝ⟶ℝ. Let BW denote the set of all bounded real valued functions W and for an arbitrary α∈BW, define α=supx∈W∣αx∣. Clearly, Bw,· endowed with quasi-partial b-metric defined by
(69)qpbα,β=supx∈W∣αx−βx∣for all α,β∈BW is a Banach space. Thus, if we consider a Cauchy sequence αn in BW, then αn converges uniformly to a function, let α∗ that is bounded and so α∗∈Bw. Also, we have P:BW×BW defined by
(70)Pαx=supy∈Dfx,y+gx,y,ατx,yfor allα∈BW and x∈W. Hence, P is well defined if f and g are bounded.Theorem 28.
LetP:BW⟶BW be an upper semicontinuous operator defined by (70), and we assume that the following conditions are satisfied:
(i)
f and g are bounded and continuous where f:W×D⟶ℝ and g:W×D×ℝ⟶ℝ(ii)
There existsτ∈ℝ+ such that(71)∣gx,y,αx−gx,y,βx∣≤ρ∣α−β∣1+τ∣α−β∣2for all α,β∈BW,x∈W,y∈D,ρ≥1. Then, the functional equation (68) has a bounded solution.Proof.
Clearly,BW,qpb is a quasi-partial b-metric given by equation (69). Let σ be an arbitrary positive number, x∈W, α1,α2∈BW, then there exist y1,y2∈D such that
(72)Pα1x<fx,y1+gx,y1,α1τx,y1+σ,(73)Pα2x<fx,y2+gx,y2,α2τx,y2+σ,(74)Pα1x<fx,y2+gx,y2,α1τx,y2+σ,(75)Pα2x<fx,y1+gx,y1,α2τx,y1+σ.
From equations (72) and (75),
(76)Pα1x−Pα2x<gx,y1,α1τx,y1−gx,y1,α2τx,y1+σ≤∣gx,y1,α1τx,y1−gx,y1,α2τx,y1∣+σ≤ρ∣α1−α2∣1+τ∣α1−α2∣2+σ.
It implies,(77)Pα1x−Pα2x≤ρ∣α1−α2∣1+τ∣α1−α2∣2+σ.
Similarly, from equations (73) and (74),
(78)Pα2x−Pα1x≤ρ∣α1−α2∣1+τ∣α1−α2∣2+σ.
From equations (77) and (78), we get
(79)Pα1x−Pα2x≤ρα1−α21+τα1−α22+σ,i.e.,
(80)qpbPα1,Pα2≤ρ∣α1−α2∣1+τ∣α1−α2∣2+σ.
Hence, we conclude that(81)qpbPα1,Pα2≤ρ∣α1−α2∣1+τ∣α1−α2∣2.
Now, we observe that the functionF:ℝ+⟶ℝ defined by Fα=−1/α for each α∈W is F-contractive function, and hence, operator P is F-contractive.
Since any upper semicontinuousF-contractive function has a fixed point α∗∈BW, it implies that there exists a bounded solution of functional equation (68).☐


## 5. Conclusion

In this manuscript, we established a new type of mappings that is KannanF-expanding mappings and obtained fixed point theorems for contractive mappings in the framework of quasi-partial b-metric spaces. Moreover, we provided examples that demonstrate the usability of our results. As an application of our result, we also studied a system of integral and functional inclusions. It would be more engaging to work on the obtained results to prove the uniqueness of the fixed point in the future.



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*Source: 2163108-2021-11-02.xml*