# Fixed Point Results for Rational Orbitally (Θ,δb)-Contractions with an Application

Authors: Zhenhua Ma; Jamshaid Ahmad; Abdullah Eqal Al-Mazrooei; Durdana Lateef
Journal: Journal of Function Spaces
 (2021)

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## Abstract

The purpose of this paper is to define a rational orbitally (Θ,δb)-contraction and prove some new results in the context of b-metric spaces. Our results extend, generalize, and unify some known results in the literature. As application of our main result, we investigate the solution of Fredholm integral inclusion. We also provide an example to substantiate the advantage and usefulness of obtained results.


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## Body

## 1. Introduction

The fixed point theory is a very essential tool for nonlinear analysis of solvability of nonlinear integral equations and others. A suitable selection of a generalized and extended metric space allows to get nontrivial conditions guaranteeing the existence of solutions for a considered equation. Therefore, it is necessary to flourish the fixed point theory in various generalization of metric spaces. One of the famous extensions of metric space is the notion ofb-metric space which has been given by Bakhtin [1] in 1989. It was properly defined by Czerwik [2] with the aspect of relaxing triangle inequality in metric spaces in 1993 and proved famous Banach Contraction Principle in this generalized metric space. Khamsi and Hussain [3] discussed the topology of b-metric space and established fixed point results for KKM mappings in metric type spaces. Van An et al. [4] proved the Stone-type theorem on b-metric spaces and obtained a sufficient condition for a b-metric space to be metrizable. On the other, Czerwik [5, 6] introduced set-valued mappings in b-metric spaces and generalized Nadler’s fixed point theorem. In 2012, Aydi et al. [7, 8] gave fixed point and common fixed point theorems for set-valued quasicontraction mappings and set-valued weak ϕ-contraction mappings in the setting of b-metric spaces, respectively. Many authors followed the concept of b-metric space and established impressive results [9–19].In 2012, Jleli and Samet [20] introduced a new type of contraction named as Θ-contraction and obtained a fixed point result to generalize the celebrated Banach Contraction Principle in Branciari metric spaces. Ali et al. [21] defined multivalued Suzuki-type θ-contractions and obtained some generalized fixed point results. Afterwards, Jleli et al. [22] established a new fixed point theorem for Θ-contraction in the setting of Branciari metric spaces and extended the main result of Jleli and Samet [20]. Recently, Alamri et al. [23] adapted Jleli’s approach to the b-metric space and obtained some generalized fixed point results. For more details in the direction of Θ-contractions, we refer the reader to [21–30].In this paper, we define the notion of the rational (Θ,δb)-contraction in b-metric spaces and explore the existence of solutions for certain integral problems of Fredholm type as applications of our main results. We obtain our results by using fixed point theorems for multivalued mappings, under new contractive conditions, in the setting of complete b-metric spaces. Evidently, the given results generalized some notable results of the literature to b-metric spaces.


## 2. Preliminaries

In this section, we give some fundamental notations, definitions, and lemmas which will be used throughout the paper. Throughout this paper, we denoteℕ the set of positive integers and ℝ+ the set of all nonnegative real numbers.Czerwik [2] gave the notion of the b-metric space as follows.Definition 1 (see [2]).
LetM be a nonempty set M and s≥1. A function σb: M×M⟶ℝ0+ is called a b-metric if these assertions hold: for all ρ,ϖ,ω∈M,
(B1)σbρ,ϖ=0⇔ρ=ϖ
(B2)σbρ,ϖ=σbϖ,ρ
(B3)σbρ,ω≤sσbρ,ϖ+σbϖ,ω
The tripleM,σb,s is called a b-metric space.Now, we give an elementary example of ab-metric space, but it is not a metric space as follows.Example 2 (see [2]).
LetM=ℝ and σb:M×M⟶ℝ be a mapping defined by
(1)σbρ,ϖ=ρ−ϖ2,∀ρ,ϖ∈M,s=2.
Then,M,σb,s is a b-metric space which is not a metric space.A brief but short history for multivalued mappings defined inM,σb,s is given in this way.LetPcbM be the family of all bounded and closed subsets of M. For any E1,E2,E3∈PbM, we define
(2)σbE1,E2=infσbρ,ϖ:ρ∈E1,ϖ∈E2,δbE1,E2=supσbρ,ϖ:ρ∈E1,ϖ∈E2,with
(3)σbρ,E3=σbρ,E3=infσbρ,ω:ρ∈E1,ω∈E3.Here, we provide some useful properties ofδb and σb (see [2, 5, 6]):
(1)
IfE1=ρ and E2=ϖ, then σbE1,E2=δbE1,E2=σbρ,ϖ(2)
σbE1,E2≤δbE1,E2(3)
σbρ,E2≤σbρ,ϖ for any ϖ∈E2(4)
δbE1,E3≤sδbE1,E2+δbE2,E3(5)
δbE1,E2=0⇔E1=E2=ρMoreover, we will always suppose that(6)
the functionσb is continuous in its variablesNow, we present the concepts of orbit and orbital continuity of a mapping in the setting ofM,σb,s which are generalization and extensions of the same notions for metric spaces given in [31, 32].Definition 3.
LetJ,J1,J2:M,σb,s⟶PbM.
(1)
AnorbitOρ0,J of J at ρ0 is any sequence ρn such that ρn∈Jρn−1 for each n∈ℕ(2)
If, for anyρ0∈M, there exists ρn in M such that ρ2n+1∈J2ρ2n and ρ2n+2∈J1ρ2n+1 for each n∈ℕ∪0, then Oρ0,J1,J2=ρn for each n∈ℕ is said to be an orbit of J1,J2 at ρ0(3)
M,σb,s is said to be J1,J2-orbitally complete if any Cauchy subsequence ρni of Oρ0,J1,J2 converges in M. Specifically, for J1=J2=J, M is said to be J-orbitally complete(4)
J is said to be orbitally continuous at ρ0∈M if, for any ρn⊂Oρ0,J for each n∈ℕ∪0 and ρ∗∈M,σbρn,ρ∗⟶0 as n⟶∞ gives that δbJρn,Jρ∗⟶0 as n⟶∞(5)
A graphGJ of J is constructed as follows:(4)GJ=ρ,ϖ:ρ∈M,ϖ∈JρWe need the following property ofGJ in our proof:(Gp) GJ is said to be J-orbitally closed if, for any sequence ρn in M, we get ρ∗,ρ∗∈GJ whenever ρn,ρn+1∈GJ and limn⟶∞ρn=ρ∗In 2012, Jleli and Samet [20] gave the notion of Θ-contractions and proved a contemporary fixed point theorem for these contractions in generalized metric spaces. Motivate by Jleli and Samet [20], Alamri et al. [23] present the following definition.Definition 4 (see [23]).
LetΩss≥1 denote the family of all mappings Θ:ℝ+⟶1,∞ which satisfy these conditions:
(Θ1)Θ is nondecreasing
(Θ2)For any ρn⊆ℝ+, limn⟶∞Θρn=1⇔limn⟶∞ρn=0
(Θ3)There exist h∈0,1 and l∈0,∞ such that limρ⟶0+Θρ−1/ρh=l
(Θ4)For ρn⊆ℝ+ such that Θsρn≤Θρn−1k for each n∈ℕ and some 0<k<1, then Θsnnρ≤Θsn−1ρn−1k for each n∈ℕThey provided the following example.Example 5.
LetΘ:ℝ+⟶1,∞ be a mapping given by θς=eςeς. Clearly, Θ satisfies the conditions (Θ1)-(Θ4). Here, we show only the condition (Θ4). Suppose that, for each n∈ℕ and some 0<k∈<1, we have θsρn≤θρn−1k. This implies that
(5)esρnesρn≤eρn−1eρn−1k,i.e.,
(6)sρnesρn≤kρn−1eρn−1.
This implies that(7)sρnesρn−ρn−1≤kρn−1.
It follows thatθsρn≤θρn−1k≤θρn−1 and θ is nondecreasing, and so, sρn≤ρn−1 and sρn−ρn−1≤0 implies esn−1sρn−ρn−1≤esρn−ρn−1. Therefore, (7) implies that
(8)sρnesn−1sρn−ρn−1≤kρn−1⇒sρnesnρnesn−1ρn−1≤kρn−1⇒sρnesnρn≤kρn−1esn−1ρn−1⇒snρnesnρn≤ksn−1ρn−1esn−1ρn−1⇒esnρnesnρn≤eksn−1ρn−1esn−1ρn−1⇒θsnρn≤θsn−1ρn−1k,and hence, the condition (Θ4) holds.


## 3. Main Results

In this way, we define the notion of rational (Θ,δb)-contraction.Definition 6.
LetM,σb be a b-metric space. A mapping J:M⟶PbM is called a rational (Θ,δb)-contraction if ∃Θ∈Ωs,0<k<1 and 0≤L such that
(9)ΘsδbJρ,Jϖ≤Θm1ρ,ϖ+Lm2ρ,ϖk,
∀ρ,ϖ∈M with minδbJρ,Jϖ,σbρ,ϖ>0, where
(10)m1ρ,ϖ=maxσbρ,ϖ,σbρ,Jρ,σbϖ,Jϖ,σbρ,Jϖ+σbϖ,Jρ2s,σbϖ,Jϖ1+σbρ,Jρs1+σbρ,ϖ,σbϖ,Jρ1+σbρ,Jρs1+σbρ,ϖ,m2ρ,ϖ=minσbρ,Jρ,σbϖ,Jϖ,σbρ,Jϖ,σbϖ,Jρ.If (9) holds for all ρ,ϖ∈Oρ0,J¯ for some ρ0∈M, then J is called a rational orbitally (Θ,δb)-contraction.Theorem 7.
LetM,σb be a b-metric space such that σb is a continuous functional and J:M⟶PbM be a rational orbitally (Θ,δb)-contraction. If the following conditions hold:
(a)
M,σb,s is J-orbitally complete for some ρ0∈M(b)
Θ is continuous and Jρ is closed, ∀ρ∈Oρ0,J¯ or the property (Gp) holds, then there exists ρ∗∈M such that ρ∗∈Jρ∗Proof.
For anyρ0∈M, we generate a sequence ρn in M as ρn+1∈Jρn for each n≥0.
If there existsn0∈ℕ∪0 for which ρn0=ρn0+1, then ρn0 is a fixed point of J, and so, the proof is completed. Thus, assume that, for each n∈ℕ∪0,ρn≠ρn+1. So, we have σbρn+1,ρn+2>0 and δbJρn,Jρn+1>0 for each n≥0. Then, it follows from (9) with ρ=ρn and ϖ=ρn+1 that
(11)Θsσbρn+1,ρn+2≤ΘsδbJρn,Jρn+1≤Θm1ρn,ρn+1+Lm2ρn,ρn+1k,where
(12)m1ρn,ρn+1=maxσbρn,ρn+1,σbρn,Jρn,σbρn+1,Jρn+1,σbρn,Jρn+1+σbρn+1,Jρn2s,σbρn+1,Jρn+11+σbρn,Jρns1+σbρn,ρn+1,σbρn+1,Jρn1+σbρn,Jρns1+σbρn,ρn+1≤maxσbρn,ρn+1,σbρn,ρn+1,σbρn+1,ρn+2,12sσbρn,ρn+2,1sσbρn+1,ρn+2=maxσbρn,ρn+1,σbρn+1,ρn+2,12sσbρn,ρn+2,m2ρn,ρn+1=minσbρn,Jρn,σbρn+1,Jρn+1,σbρn,Jρn+1,σbρn+1,Jρn=0.
Since(13)12sσbρn,ρn+2≤maxσbρn,ρn+1,σbρn+1,ρn+2,it follows from (11) that
(14)Θsσbρn+1,ρn+2≤Θmaxσbρn,ρn+1,σbρn+1,ρn+2k.
Assume thatσbρn,ρn+1≤σbρn+1,ρn+2 for some n∈ℕ. Then, from (14), we get
(15)Θsσbρn+1,ρn+2≤Θσbρn+1,ρn+2k,which is a contradiction with (Θ1). Hence, we have
(16)maxσbρn,ρn+1,σbρn+1,ρn+2=σbρn,ρn+1,and consequently,
(17)Θsσbρn+1,ρn+2≤Θσbρn,ρn+1k,∀n∈ℕ∪0.
It follows from (17) and (Θ4) that
(18)Θsnσbρn,ρn+1≤Θsn−1σbρn−1,ρnk,∀n∈ℕ∪0.
Let us representλn=σbρn,ρn+1 for n∈ℕ∪0. Then, λn>0 for all n∈ℕ∪0. Thus, we have
(19)Θsnλn≤Θsn−1λn−1k≤Θsn−2λn−2k2≤⋯≤Θλ0kn,∀n∈ℕ∪0.
Takingn⟶∞ in (19), we get
(20)limn⟶∞Θsnλn=1,which implies that
(21)limn⟶∞snλn=0,by (Θ2). By (Θ3), ∃h∈0,1 and τ∈0,∞ such that
(22)limn⟶∞Θsnλn−1snλnh=τ.
Assume thatτ<∞. For this case, let q2=τ/2>0. So, ∃n1∈ℕ such that
(23)Θsnλn−1snλnh−τ≤q2,∀n>n1,which implies that
(24)Θsnλn−1snλnh≥τ−q2=τ2=q2,∀n>n1.
Then, we have(25)nsnλnh≤q1nΘsnλn−1,∀n>n1,where q1=1/q2.
Now, assume thatτ=∞. Let q2>0. So, ∃n1∈ℕ such that
(26)q2≤Θsnλn−1snλnh,∀n>n1,which implies that
(27)nsnλnh≤q1nΘsnλn−1,∀n>n1,where q1=1/q2. Thus, in all cases, there exist q1>0 and n1∈ℕ such that
(28)nsnλnh≤q1nΘsnλn−1,∀n>n1.
Hence, by (19) and (28), we get
(29)nsnλnh≤q1nΘλ0kn−1.
Takingn⟶∞ in (29), we have
(30)limn⟶∞nsnλnh=0,and hence, limn⟶∞n1/hsnλn=0, which yields that ∑n=1∞snλn is convergent. Thus, ρn is a Cauchy sequence in Oρ0,J¯. Since M is J-orbitally complete, there exists ρ∗∈M such that
(31)ρn⟶ρ∗asn⟶∞.
Suppose thatJρ∗ is closed. We notice that if ∃nk⊂ℕ such that ρnk∈Jρ∗ for each k∈ℕ. Since Jρ∗ is closed and limk⟶∞ρnk=ρ∗, we conclude that ρ∗∈Jρ∗, and so, the proof is finished. Hence, we suppose that there exists n0∈ℕ so that ρn∈Jρ∗ for each n∈ℕ with n≥n0. This implies that δbJρn,Jρ∗>0 for each n≥n0. Then, it follows from (9) with ρ=ρn and ϖ=ρ∗ that
(32)Θsσbρn+1,Jρ∗=ΘsδbJρn,Jρ∗≤Θm1ρn,ρ∗+Lm2ρn,ρ∗k,where
(33)m1ρn,ρ∗=maxσbρn,ρ∗,σbρn,Jρn,σbρ∗,Jρ∗,σbρn,Jρ∗+σbρ∗,Jρn2s,σbρ∗,Jρ∗1+σbρn,Jρns1+σbρn,ρ∗,σbρ∗,Jρn1+σbρn,Jρns1+σbρn,ρ∗≤maxσbρn,ρ∗,σbρn,ρn+1,σbρ∗,Jρ∗,σbρn,Jρ∗+σbρ∗,ρn+12s,σbρ∗,Jρ∗1+σbρn,ρn+1s1+σbρn,ρ∗,σbρ∗,ρn+11+σbρn,ρn+1s1+σbρn,ρ∗⟶σbρ∗,Jρ∗asn⟶∞,m2ρn,ρ∗=minσbρn,Jρn,σbρ∗,Jρ∗,σbρn,Jρ∗,σbρ∗,Jρn≤minσbρn,ρn+1,σbρ∗,Jρ∗,σbρn,Jρ∗,σbρ∗,ρn+1⟶0asn⟶∞.
Using the continuity ofΘ and σb, so applying the limit of (32) as n⟶∞, we get
(34)Θsσbρ∗,Jρ∗≤Θσbρ∗,Jρ∗k,which is impossible from 0<k∈<1 and s≥1. By the condition (Θ1), we get σbρ∗,Jρ∗=0. Since Jρ∗ is closed, thus we get ρ∗∈Jρ∗. Assume that GJ is J-orbitally closed. Since ρn,ρn+1∈GJ and limn⟶∞ρn=ρ∗∀n∈ℕ∪0, we get ρ∗,ρ∗∈GJ. Hence, ρ∗∈Jρ∗. This completes the proof.☐IfΘς=eς for any ς>0 in Theorem 7, we get following result.Corollary 8.
LetM,σb be a b-metric space such that σb is a continuous functional and J:M,σb⟶PbM be a mapping satisfying the following condition: for some 0<k<1,ρ0∈M and L≥0,
(35)sδbJρ,Jϖ≤km1ρ,ϖ+Lm2ρ,ϖfor all ρ,ϖ∈M with minδbJρ,Jϖ,σbρ,ϖ>0, where
(36)m1ρ,ϖ=maxσbρ,ϖ,σbρ,Jρ,σbϖ,Jϖ,σbρ,Jϖ+σbϖ,Jρ2s,σbϖ,Jϖ1+σbρ,Jρs1+σbρ,ϖ,σbϖ,Jρ1+σbρ,Jρs1+σbρ,ϖ,m2ρ,ϖ=minσbρ,Jρ,σbϖ,Jϖ,σbρ,Jϖ,σbϖ,Jρfor all ρ,ϖ∈Oρ0,J¯. Assume that M,σb,s is J-orbitally complete for some ρ0∈M. If Jρ is closed, for all ρ∈Oρ0,J¯ or the property (Gp) holds, then ∃ρ∗∈M such that ρ∗∈Jρ∗.Example 9.
LetM=0,1. The b-metric is defined by
(37)σbρ,ϖ=ρ−ϖ2,with coefficient s=2. Define J:M⟶PcbM given by
(38)Jρ=13,0≤ρ<1,0,14,ρ=1.
Ifρ,ϖ∈0,1, then δbJρ,Jϖ=0. Let ρ∈0,1 and ϖ=1. Then, Jρ=1/3,Jϖ=0,1/4, and δbJρ,Jϖ=1/12,
(39)m1ρ,ϖ=max1−ρ2,13−ρ2,342,14σbρ,Jϖ+232,3/421+1/3−ρ221+1−ρ2,2/321+1/3−ρ221+1−ρ2≥916,m2ρ,ϖ=min13−ρ2,342,σbρ,Jϖ,232≥0.
Takek=2/9, Θς=eς and 0≤L. Then,
(40)ΘsδbJρ,Jϖ=Θ2·112=e1/6<e2/3=e8/9·3/4=Θm1ρ,ϖ+Lm2ρ,ϖk.Thus, all conditions of Theorem7 are satisfied and ρ∗=1/3 is the required point.The familyΩs contains a wide set of functions; that is, if we take
(41)Θρ=2−2πarctan1ρβ,where ρ>0 and 0<β<1, then we can obtain the following corollary from Theorem 7.Corollary 10.
LetM,σb be a b-metric space such that σb is a continuous functional. Assume that J:M⟶PbM be a mapping such that for some 0<k,β<1,ρ0∈Mand0≤L,
(42)2−2πarctan1sδbJρ,Jϖβ≤2−2πarctan1m1ρ,ϖ+Lm2ρ,ϖβk,
∀ρ,ϖ∈M with minδbJρ,Jϖ,σbρ,ϖ>0, where
(43)m1ρ,ϖ=maxσbρ,ϖ,σbρ,Jρ,σbϖ,Jϖ,σbρ,Jϖ+σbϖ,Jρ2s,σbϖ,Jϖ1+σbρ,Jρs1+σbρ,ϖ,σbϖ,Jρ1+σbρ,Jρs1+σbρ,ϖ,m2ρ,ϖ=minσbρ,Jρ,σbϖ,Jϖ,σbρ,Jϖ,σbϖ,Jρfor all ρ,ϖ∈Oρ0,J¯. Suppose that M,σb,s is J-orbitally complete for some ρ0∈M. If Jρ is closed for all ρ∈Oρ0,J¯ or the property (Gp) holds, then there exists ρ∗∈M such that ρ∗∈Jρ∗.If we replace self-mappingJ:M⟶M on the place of multivalued mapping in Theorem 7, we can get following results as consequences.Corollary 11.
LetM,σb be a b-metric space such that σb is a continuous functional and J:M⟶M be a mapping such that M is J-orbitally complete at some ρ0. Assume that there exist Θ∈Ωs, k∈0,1 and L≥0 such that
(44)ΘsσbJρ,Jϖ≤Θm1/ρ,ϖ+Lm2/ρ,ϖkfor all ρ,ϖ∈M with minσbJρ,Jϖ,σbρ,ϖ>0, where
(45)m1/ρ,ϖ=maxσbρ,ϖ,σbρ,Jρ,σbϖ,Jϖ,σbρ,Jϖ+σbϖ,Jρ2s,σbϖ,Jϖ1+σbρ,Jρs1+σbρ,ϖ,σbϖ,Jρ1+σbρ,Jρs1+σbρ,ϖ,m2/ρ,ϖ=minσbρ,Jρ,σbϖ,Jϖ,σbρ,Jϖ,σbϖ,Jρ.
IfΘ is continuous, then there exists ρ∗∈M such that ρ∗=Jρ∗.


## 4. Applications

In this section, we solve the Fredholm integral inclusion:(46)ρς∈gς+∫abKς,ζ,ρζσζ,∀ς∈a,b,given in the start of this paper.Letσb on Ca,b be given by
(47)σbρ,ϖ=maxς∈a,b∣ρς−ϖς∣p=maxς∈a,bρς−ϖςp,withp≥1.Then,Ca,b,σb,2p−1 is a complete b-metric space.Assume that forς,ζ∈a,b, these conditions hold:
(a)
∀ρ∈Ca,b,Kρς,ζ=Kς,ζ,ρζ is continuous(b)
∃Y:a,b×a,b⟶0,+∞ such that(48)kφς,ζ−kψς,ζp≤Yς,ζmaxσbφζ,ψζ,σbφζ,Kς,ζ,φζ,σbψζ,Kς,ζ,ψζ,σbφζ,Kς,ζ,ψζ+σbψζ,Kς,ζ,φζ2p,σbψζ,Kς,ζ,ψζ1+σbφζ,Kς,ζ,φζ2p−11+σbφζ,ψζ,σbψζ,Kς,ζ,φζ1+σbφζ,Kς,ζ,φζ2p−11+σbφζ,ψζ+λminσbφζ,Kς,ζ,φζ,σbψζ,Kς,ζ,ψζ,σbφζ,Kς,ζ,ψζ,σbψζ,Kς,ζ,φζ∀ς,ζ∈a,b,φ,ψ∈Ca,b and kφς,ζ∈Kφς,ζ,kψς,ζ∈Kψς,ζ, where p>1 and λ≥0(c)
∃0<k<1 such that(49)supς∈a,b∫abYς,ζσζ≤k2p−1Theorem 12.
Under the conditions (a)–(c), the integral inclusion given in (46) has a solution in Ca,b.Proof.
LetM=Ca,b. Define b-metric σb as in (47) and a multivalued mapping J:M⟶PcbM by
(50)Jρ=ϖ∈M:ϖς∈gς+∫abKς,ζ,ρζσζ,ς∈a,b.
Letρ∈M. For a mapping Kρς,ζ:a,b×a,b⟶Pcbℝ, there exists kρς,ζ:a,b×a,b⟶ℝ such that
(51)kρς,ζ∈Kρς,ζ,
∀ς,ζ∈a,b. Then,
(52)gς+∫abkρς,ζσζ∈Jρ.
Hence,Jρ≠∅. Since Kρς,ζ is continuous and g∈Ca,b, the ranges of both functions are bounded. This shows that Jρ is also bounded.
Now, we show that (9) holds for any J on M with some 0<k<1, 0≤L and Θ∈Ωζ, i.e.,
(53)ΘζδbJρ1,Jρ2≤Θmaxσbρ1,ρ2,σbρ1,Jρ1,σbρ2,Jρ2,σbρ1,Jρ2+σbρ2,Jρ12p,σbρ2,Jρ21+σbρ1,Jρ12p−11+σbρ1,ρ2,σbρ2,Jρ11+σbρ1,Jρ12p−11+σbρ1,ρ2+Lminσbρ1,Jρ1,σbρ2,Jρ2,σbρ1,Jρ2,σbρ2,Jρ1kfor all ρ1,ρ2∈M. Let ϖ1∈Jρ1 be such that
(54)ϖ1ς∈gς+∫abKς,ζ,ρ1ζσζ,∀ς∈a,b.
It follows that∃kρ1ς,ζ∈Kρ1ς,ζ=Kς,ζ,ρ1ζ such that
(55)ϖ1ς=gς+∫abkρ1ς,ζσζ,∀ς,ζ∈a,b.
Now, for allρ1,ρ2∈M, it follows from (b) that
(56)kρ1ς,ζ−kρ2ς,ζp≤Yς,ζmaxσbρ1ζ,ρ2ζ,σbρ1ζ,Kς,ζ,ρ1ζ,σbρ2ζ,Kς,ζ,ρ2ζ,σbρ1ζ,Kς,ζ,ρ2ζ+σbρ2ζ,Kς,ζ,ρ1ζ2pσbρ2ζ,Kς,ζ,ρ2ζ1+σbρ1ζ,Kς,ζ,ρ1ζ2p−11+σbρ1ζ,ρ2ζ,σbρ2ζ,Kς,ζ,ρ1ζ1+σbρ1ζ,Kς,ζ,ρ1ζ2p−11+σbρ1ζ,ρ2ζ+Lminσbρ1ζ,Kς,ζ,ρ1ζ,σbρ2ζ,Kς,ζ,ρ2ζ,σbρ1ζ,Kς,ζ,ρ2ζ,σbρ2ρ,Kς,ζ,ρ1ζ
It follows that∃ως,ζ∈Kρ2ς,ζ such that
(57)kρ1ς,ζ−ως,ζp≤Yς,ζmaxσbρ1ζ,ρ2ζ,σbρ1ζ,Kς,ζ,ρ1ζ,σbρ2ζ,ως,ζ,σbρ1ζ,ως,ζ+σbρ2ζ,Kς,ζ,ρ1ζ2p,σbρ2ζ,ως,ζ1+σbρ1ζ,Kς,ζ,ρ1ζ2p−11+σbρ1ζ,ρ2ζ,σbρ2ζ,Kς,ζ,ρ1ζ1+σbρ1ζ,Kς,ζ,ρ1ζ2p−11+σbρ1ζ,ρ2ζ+Lminσbρ1ζ,Kς,ζ,ρ1ζ,σbρ2ζ,ως,ζ,σbρ1ζ,ως,ζ,σbρ2ρ,Kς,ζ,ρ1ζ=Rς,ζ,∀ς,ζ∈a,b.
Now, we show the functionUς,ζ:a,b×a,b⟶Pcbℝ by
(58)Uς,ζ=Kρ2ς,ζ⋂φ∈ℝ:σbkρ1ς,ζ,φ≤Rς,ζ.
Thus, by (a),U is lower semicontinuous; this implies that there exists kρ2ς,ζ:a,b×a,b⟶ℝ such that kρ2ς,ζ∈Uς,ζ,∀ς,ζ∈a,b. Then, ϖ2ς=gς+∫abkρ1ς,ζσζ satisfies
(59)ϖ1ς∈gς+∫abKς,ζ,ρ1ζσζ,∀ς∈a,b,which implies that ϖ2∈Jρ2 and
(60)σbϖ1,ϖ2≤maxς∈a,b∫abkρ1ς,ζ−kρ2ς,ζpσζ≤maxς∈a,b∫abYς,ζmaxσbρ1ζ,ρ2ζ,σbρ1ζ,Kς,ζ,ρ1ζ,σbρ2ζ,Kς,ζ,ρ2ζ,σbρ1ζ,Kς,ζ,ρ2ζ+σbρ2ζ,Kς,ζ,ρ1ζ2p,σbρ2ζ,Kς,ζ,ρ2ζ1+σbρ1ζ,Kς,ζ,ρ1ζ2p−11+σbρ1ζ,ρ2ζ,σbρ2ζ,Kς,ζ,ρ1ζ1+σbρ1ζ,Kς,ζ,ρ1ζ2p−11+σbρ1ζ,ρ2ζ+Lminσbρ1ζ,Kς,ζ,ρ1ζ,σbρ2ζ,Kς,ζ,ρ2ζ,σbρ1ζ,Kς,ζ,ρ2ζ,σbρ2ρ,Kς,ζ,ρ1ζσζ≤k2p−1maxσbρ1,ρ2,σbρ1,Jρ1,σbρ2,Jρ2,σbρ1,Jρ2+σbρ2,Jρ12p,σbρ2,Jρ21+σbρ1,Jρ1ζ1+σbρ1,ρ2,σbρ2,Jρ11+σbρ1,Jρ1ζ1+σbρ1,ρ2+Lminσbρ1,Jρ1,σbρ2,Jρ2,σbρ1,Jρ2,σbρ2Jρ1,∀ς,ζ∈a,b. Thus, we get
(61)δbJρ1,Jρ2≤k2p−1maxσbρ1,ρ2,σbρ1,Jρ1,σbρ2,Jρ2,σbρ1,Jρ2+σbρ2,Jρ12p,σbρ2,Jρ21+σbρ1,Jρ12p−11+σbρ1,ρ2,σbρ2,Jρ11+σbρ1,Jρ12p−11+σbρ1,ρ2+Lminσbρ1,Jρ1,σbρ2,Jρ2,σbρ1,Jρ2,σbρ2,Jρ1,ζδbJρ1,Jρ2≤kmaxσbρ1,ρ2,σbρ1,Jρ1,σbρ2,Jρ2,σbρ1,Jρ2+σbρ2,Jρ12p,σbρ2,Jρ21+σbρ1,Jρ12p−11+σbρ1,ρ2,σbρ2,Jρ11+σbρ1,Jρ12p−11+σbρ1,ρ2+Lminσbρ1,Jρ1,σbρ2,Jρ2,σbρ1,Jρ2,σbρ2,Jρ1.
Taking the exponential on both sides, we have(62)eζδbJρ1,Jρ2δbJρ1,Jρ2≤ekmaxσbρ1,ρ2,σbρ1,Jρ1,σbρ2,Jρ2,σbρ1,Jρ2+σbρ2,Jρ12p,σbρ2,Jρ21+σbρ1,Jρ12p−11+σbρ1,ρ2,σbρ2,Jρ11+σbρ1,Jρ12p−11+σbρ1,ρ2+Lminσbρ1,Jρ1,σbρ2,Jρ2,σbρ1,Jρ2,σbρ2,Jρ1.
Taking the functionΘ∈Ωs, it follows that (18) is fulfilled. Therefore, by Theorem 7, we show that the integral inclusion (46) has a solution. This completes the proof.☐


## 5. Conclusion

In this paper, we have defined a rational (Θ,δb)-contraction and established some fixed point results in b-metric spaces. In this way, we generalized some known results of literature. We also discussed the solution of Fredholm integral inclusion as application of our obtained result. We expect that the results given in this article will make new directions for those who are working in the theory of fixed points.In this direction, our future work will pivot on studying the fixed points of fuzzy mappings andL-fuzzy mappings for Θ-contractions in b-metric spaces, with fractional differential inclusion problems as applications.



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