# Set-Valued SU-Type Fixed Point Theorems via Gauge Function with Applications

Authors: Amjad Ali; Monairah Alansari; Fahim Uddin; Muhammad Arshad; Awais Asif; Ghada Ali Basendwah
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this article, we have designed two existence of fixed point theorems which are regarding to set-valued SU-typeθη-contraction and Γα-contraction via gauge function in the setting of metric spaces. An extensive set of nontrivial example will be given to justify our claim. At the end, we will give an application to prove the existence behavior for the system of functional equation in dynamical system and integral inclusion.


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## Body

## 1. Introduction and Preliminaries

The most publicized famous result in nonlinear analysis is Banach contraction principle, which made clear a systematic rule to find the fixed point of a given mapping on a metric space. So far, numerous authors have studied this classical result to examine the existence and uniqueness of a solution for different forms of contractive structure.In 2014, Jleli and Samet [1] introduced the concept of a new contraction known as the θ-contraction, which generalizes the Banach contraction principle in a beautiful way.In 2015, Khojasteh et al. [2] introduced simulation function. Recently, many researchers have proved fixed point theorems for Suzuki-type (SU) mappings in metric space (see [3, 4]).LetX,ð be a metric space. For μ∈X and β1⊆X, let CLX and CBX denote the family of all nonempty closed subsets and the family of all nonempty closed bounded subsets of X. Design the Pompeiu–Hausdorff metric Hd induced by ð on CBX as(1)Hdβ1,β2=maxsupμ1∈β1ðμ1,β2,supμ2∈β2ðμ2,β1,for all β1,β2∈CLX and ðμ1,β1=infðμ1,μ2:μ2∈β1. A point μ∈X is said to be a fixed point of T:X⟶CBX, if μ∈Tμ. If, for μ0∈X, there exists a sequence μι in X such that μι∈Tμι−1, then OT,μ0=μ0,μ1,μ2,… is said to be an orbit of T:X⟶CBX. Mapping f:X⟶ℝ is said to be T-orbitally lower semicontinuous (o.l.s.c), if a sequence μι in OT,μ0 and μι⟶ϱ⇒f〉≤liminfιfμι.A multivalued mappingT:X⟶CBX is called a Nadler-contraction, if there exists γ∈0,1 such that(2)HdTμ1,Tμ2≤γðμ1,μ2for allμ1,μ2∈X.Nadler [5] obtained the multivalued version of the Banach contraction principle. Let X,ð be a complete metric space and T:X⟶CLX be a Nadler-contraction. Then, T has a fixed point. Recently, Vetro [6] proved the following result.to μ∗.Theorem 1.
LetX,ð be a complete metric space and T:X⟶CBX be a multivalued mapping. Suppose that there exist θ∈Ξ and k∈0,1 such that(3)μ1,μ2∈X,HdTμ1,Tμ2>0⇒θHdTμ1,Tμ2≤θðμ1,μ2k,where Ξ is the set of mapping θ:0,∞⟶1,∞ satisfying θi−θiii:(i)
θiθ is nondecreasing and right-continuous.(ii)
θii For each sι in 0,∞, limι⟶∞θsι=1⇔limι⟶∞sι=0.(iii)
θiii There exist r∈0,1 and μ∈0,+∞ such that lims⟶0+θs−1/sr=μ. Then, T has at least one fixed point.Remark 1.
LetX,ð be a metric space. If T:X⟶CBX is a multivalued mapping satisfying the above theorem, then(4)lnθHdTμ1,Tμ2≤klnθðμ1,μ2<lnθðμ1,μ2.
Sinceθ is nondecreasing, we obtain(5)HdTμ1,Tμ2<ðμ1,μ2,for allμ1,μ2∈X,Tμ1≠Tμ2.Example 1.
The functionsθ1,θ2:0,∞⟶1,∞, defined by θ1r=er and θ2r=1+r, are in Ξ.Lemma 1.
(see [6]). Let X,ð be a metric space and β1,β2∈CBX with Hdβ1,β2>0. Then, for all h>1 and μ∈β1, there exists ν=νμ∈β2 such that(6)ðμ,ν<hHdβ1,β2.Lemma 2.
(see [6]). Let X,ð be a metric space, β2∈CBX, and μ∈X. Then, for each ϵ>0, there exists ν∈β2 such that(7)ðμ,ν≤ðμ,β2+ε.Definition 1.
(see [2]). Mapping Γ:ℝ+×ℝ+⟶ℝ is called a simulation function such thatΓ1Γ0,0=0.Γ2Γμ1,μ2<μ2−μ1 for all μ1`,μ2>0.Γ3 If μ1ι,μ2ι∈0,∞ such that limι⟶∞μ1ι=limι⟶∞μ2ι>0, then(8)limsupι⟶∞Γμ1ι,μ2ι<0.
Due toΓ2, we have Γμ1,μ1<0 for all μ1>0. Here, the set ∇ satisfies the conditions (Γ1)–(Γ3).Example 2.
(see [2]). For j=1,2, let ϑj:ℝ+⟶ℝ+ be continuous functions such that ϑjμ1=0⇔μ1=0. Functions Γj:ℝ+×ℝ+⟶ℝ (j=1,2) are in ∇:(i)
Γ1μ1,μ2=ϑ1μ2−ϑ2μ1 for all μ1,μ2≥0, where ϑ1μ1≤μ1≤ϑ2μ1.(ii)
Γ2μ1,μ2=μ2−∫0μ1ςudu for allμ1,μ2≥0, where ς:ℝ+⟶ℝ+ is a function such that(9)∫0εςuduexists and∫0εςudu>εfor allε>0,Definition 2.
(see [1]). Let X,ð be a metric space and Λ be a nonempty subset of X, and T:Λ⟶CBX is known as α-admissible, if there exists a mapping α:Λ×Λ⟶0,∞ such that(10)αβ1,β2≥1⇒αμ,ν≥1,for all μ∈Tβ1∩Λ and ν∈Tβ2∩Λ.Lemma 3.
(see [7]). Suppose there is a point μ0∈Λ (Λ is a closed subset of X) that satisfies(11)ðμ0,Tμ0∈E^,and μι∈Λ for some ι≥0. Then, ðμι,Tμι∈E^ where E^ denotes an interval on ℝ+ containing 0.Definition 3.
(see [7]). (inclusion ball) Suppose μ0∈Λ and ðμ0,Tμ0∈E^. Then, for all iterates μιι≥0 which belongs to Λ, we define the closed-ball b¯μι,ρι with center μι and radius ρι=σðμι,Tμι, where σ:E^⟶R+ is defined by (13).Lemma 4.
(see [7]). Suppose there is a point μ0∈Λ that satisfies ðμ0,Tμ0∈E^ and b¯μι,ρι⊂Λ for some ι≥0; then, μι+1∈Λ and b¯μι+1,ρι+1⊂b¯μι,ρι.Definition 4.
(see [7]). Let j≥1, and η:E^⟶E^ is known as a gauge function of order j on E^, if it satisfies the following conditions:(i)
ηλμ<λjημ for all λ∈0,1 and μ∈E^.(ii)
ημ<μ for all μ∈E^−0.
Note that the first condition of Definition4 is equivalent to η0=0 and ημ/μi is nondecreasing on E^−0.Definition 5.
(see [7]). A gauge function η:E^⟶E^ is said to be a Bianchini–Grandolfi gauge function on E^ if(12)σμ=∑i=0∞ηiμ<∞,for allμ∈E^.
Note that a Bianchini–Grandolfi gauge function also satisfies the following functional equation:(13)σμ=σημ+μ.


## 2. Set-Valued SUθη-Contraction

The first main definition of this exposition is as follows.Definition 6.
LetX,ð be a metric space, Λ be a closed subset of X, and η be a Bianchini–Grandolfi gauge function on E^. A mapping T:Λ⟶CBX is known as set-valued SU θη-contraction, if there exists θ∈Ξ such that for Tμ∩Λ≠∅,(14)12minðμ,Tμ∩Λ,ðν,Tν∩Λ<ðμ,ν,which implies that(15)θHdTμ∩Λ,Tν∩Λ≤θηΩμ,νk,where(16)Ωμ,ν=maxðμ,ν,ðμ,Tμ,ðν,Tν,ðμ,Tν+ðν,Tμ2,for all μ∈Λ, ν∈Tμ∩Λ with ðμ,ν∈E^, where k∈0,1.Theorem 2.
LetX,ð be a complete meθηtric space and T:Λ⟶CBX be a multivalued SU -contraction. Suppose μ0∈Λ such that ðμ0,c∗∈E^ for some c∗∈Tμ0∩Λ. Then, there exists an orbit μι of T in Λ and ϑ∗∈Λ such that limι⟶∞μι=ϑ∗. In addition, ϑ∗ is a fixed point of T if and only if the function gμ:=ðμ,Tμ∩Λ is T-o.l.s.c at ϑ∗.Proof.
Chooseμ1=c∗∈Tμ0∩Λ. In the case that ðμ0,μ1=0, μ0 is a fixed point of T. Thus, we assume that ðμ0,μ1≠0. On the other hand, we have(17)12minðμ0,Tμ0∩Λ,ðμ1,Tμ1∩Λ<ðμ0,μ1.
Defineρ=σðμ0,μ1. From (13), we have σr≥r. Hence, ðμ0,μ1≤ρ, and so μ1∈b¯μ0,ρ. Since ðμ0,μ1∈E^, from (15) and (17), it follows that(18)θHdTμ0∩Λ,Tμ1∩Λ≤θηðμ0,μ1k<θΩμ0,μ1k.
By right continuity ofθ, there exists a real number h1>1 such that(19)θh1HdTμ0∩Λ,Tμ1∩Λ≤θΩμ0,μ1k.
From(20)ðμ1,Tμ1∩Λ≤HdTμ0∩Λ,Tμ1∩Λ<h1HdTμ0∩Λ,Tμ1∩Λ,by Lemma 1, there exists μ2∈Tμ1∩Λ such that ðμ1,μ2≤h1HdTμ0∩Λ,Tμ1∩Λ. Since θ is nondecreasing, by (19), this inequality gives that(21)θðμ1,μ2≤θh1HdTμ0∩Λ,Tμ1∩Λ≤θΩμ0,μ1k,where(22)Ωμ0,μ1=maxðμ0,μ1,ðμ0,Tμ0,ðμ1,Tμ1,ðμ0,Tμ1+ðμ1,Tμ02≤maxðμ0,μ1,ðμ1,Tμ1,ðμ0,Tμ12≤maxðμ0,μ1,ðμ1,Tμ1.
We claim that(23)θðμ1,μ2≤θh1HdTμ0∩Λ,Tμ1∩Λ≤θðμ0,μ1k.
LetΦ=maxðμ0,μ1,ðμ1,Tμ1. If Φ=ðμ1,Tμ1, we have μ2∈Tμ1∩Λ, so we obtain(24)θðμ1,μ2≤θh1HdTμ0∩Λ,Tμ1∩Λ≤θðμ1,μ2k,which is a contradiction. Thus, we conclude that Φ=ðμ0,μ1. We assume that ðμ1,μ2≠0; otherwise, μ1 is a fixed point of T. From Remark 1, we have ðμ1,μ2<ðμ0,μ1, and so ðμ1,μ2∈E^. Next, μ2∈b¯μ0,ρ because(25)ðμ0,μ2≤ðμ0,μ1+ðμ1,μ2≤ðμ0,μ1+ηðμ0,μ1≤ðμ0,μ1+σηðμ0,μ1=σðμ0,μ1=ρ.
Also, since(26)12minðμ1,Tμ1∩Λ,ðμ2,Tμ2∩Λ<ðμ1,μ2,from (15), we get(27)θHdTμ1∩Λ,Tμ2∩Λ≤θηðμ1,μ2k<θΩμ1,μ2k.
Sinceθ is right-continuous, there exists a real number h2>1 such that(28)θh2HdTμ1∩Λ,Tμ2∩Λ≤θΩμ1,μ2k.
Next, from(29)ðμ2,Tμ2∩Λ≤HdTμ1∩Λ,Tμ2∩Λ<h2HdTμ1∩Λ,Tμ2∩Λ,by Lemma 1, there exists μ3∈Tμ2∩Λ such that ðμ2,μ3≤h2HdTμ1∩Λ,Tμ2∩Λ. By (28), this inequality gives that(30)θðμ2,μ3≤θh2HdTμ1∩Λ,Tμ2∩Λ≤θΩμ1,μ2k≤θΩμ0,μ1k2.where(31)Ωμ1,μ2=maxðμ1,μ2,ðμ1,Tμ1,ðμ2,Tμ2,ðμ1,Tμ2+ðμ2,Tμ12≤maxðμ1,μ2,ðμ2,Tμ2,ðμ1,Tμ22≤maxðμ1,μ2,ðμ2,Tμ2.
We claim that(32)θðμ2,μ3≤θh2HdTμ1∩Λ,Tμ2∩Λ≤θðμ1,μ2k≤θðμ0,μ1k2
LetΦ=maxðμ1,μ2,ðμ2,Tμ2. If Φ=ðμ2,Tμ2, we have μ3∈Tμ2∩Λ, so we obtain(33)θðμ2,μ3≤θh2HdTμ1∩Λ,Tμ2∩Λ≤θðμ2,μ3k,which is a contradiction. Thus, we conclude that Φ=ðμ1,μ2. We assume that ðμ2,μ3≠0; otherwise, μ2 is a fixed point of T. From Remark 1, we have ðμ2,μ3<ðμ1,μ2, and so ðμ2,μ3∈E^. Also, we have μ3∈b¯μ0,ρ, since(34)ðμ0,μ3≤ðμ0,μ1+ðμ1,μ2+ðμ2,μ3≤ðμ0,μ1+ηðμ0,μ1+η2ðμ0,μ1≤∑i=0∞ηiðμ0,μ1=σðμ0,μ1=ρ.
Continuing this setup, we have two sequencesμι⊂b¯μ0,ρ and hι⊂1,∞ such that μι+1∈Tμι∩Λ, μι≠μι+1 with ðμι,μι+1∈E^ and(35)1<θðμι,μι+1≤θhιHdTμι−1∩Λ,Tμι∩Λ≤θðμι−1,μιk,for all ι∈ℕ. Then,(36)1<θðμι,μι+1≤θðμ0,μ1kι,for allι∈ℕ,which gives that(37)limι⟶∞θðμι,μι+1=1,and by θii, we have(38)limι⟶∞ðμι,μι+1=0.
Next, we prove thatμι is a Cauchy sequence in X. Setting δι:=ðμι,μι+1, from (θiii), there exist r∈0,1 and μ∈0,∞ such that(39)limι⟶∞θδι−1διr=μ.
Takeλ∈0,μ. From the definition of limit, there exists ι0∈ℕ such that(40)διr≤λ−1θδι−1,for allι>ι0.
Using (36) and the above inequality,(41)ιδιr≤λ−1ιθδ0kι−1,for allι>ι0.
This implies that(42)limι⟶∞ιδιr=limι⟶∞ιdμι,μι+1r=0.
Hence, there existsι1∈ℕ such that(43)dμι,μι+1≤1ι1/r,for allι>ι1.
Letp>ι>ι1. Then, using the triangular inequality and (43), we get(44)ðμι,μp≤∑j=ιp−1ðμj,μj+1≤∑j=ιp−11j1/r<∑j=ι∞1j1/r.
Owing to the convergence of the series∑j=ι∞1/j1/r, μι is a Cauchy sequence in b¯μ0,ρ. Since b¯μ0,ρ is closed in X, there exists ϑ∗∈b¯μ0,ρ such that μι⟶ϑ∗. Note that ϑ∗∈Λ because μι+1∈Tμι∩Λ. Now, we claim that(45)12minðμι,Tμι∩Λ,ðϑ∗,Tϑ∗∩Λ<ðμι,ϑ∗,or(46)12minðϑ∗,Tϑ∗∩Λ,ðμι+1,Tμι+1∩Λ<ðμι+1,ϑ∗,for all ι∈N. Assume, on the contrary, that there exists ι′∈N such that(47)12minðμι′,Tμι′∩Λ,ðϑ∗,Tϑ∗∩Λ≥ðμι′,ϑ∗,and(48)12minðϑ∗,Tϑ∗∩Λ,ðμι′+1,Tμι′+1∩Λ≥ðμι′+1,ϑ∗.
By (47), we have(49)2ðμι′,ϑ∗≤minðμι′,Tμι′∩Λ,ðϑ∗,Tϑ∗∩Λ≤minðμι′,ϑ∗+ðϑ∗,Tμι′∩Λ,ðϑ∗,Tϑ∗∩Λ≤ðμι′,ϑ∗+ðϑ∗,Tμι′∩Λ<ðμι′,ϑ∗+ðϑ∗,Tμι′≤ðμι′,ϑ∗+ðϑ∗,μι′+1,which implies that(50)ðμι′,ϑ∗≤ðϑ∗,μι′+1,which together with (38) gives(51)ðμι′,ϑ∗≤ðϑ∗,μι′+1≤12minðϑ∗,Tϑ∗∩Λ,ðμι′+1,Tμι′+1∩Λ.
Since(52)12minðμι′,Tμι′∩Λ,ðμι′+1,Tμι′+1∩Λ<ðμι′,μι′+1,from contractive condition (15), we have(53)θðμι′+1,μι′+2≤θh2HdTμι′∩Λ,Tμι′+1∩Λ≤θΩμι′,μι′+1k,where(54)Ωμι′,μι′+1=maxðμι′,μι′+1,ðμι′,Tμι′,ðμι′+1,Tμι′+1,ðμι′,Tμι′+1+ðμι′+1,Tμι′2≤maxðμι′,μι′+1,ðμι′+1,μι′+2,ðμι′,μι′+22≤maxðμι′,μι′+1,ðμι′+1,μι′+2.
We claim that(55)θðμι′+1,μι′+2≤θh2HdTμι′∩Λ,Tμι′+1∩Λ≤θðμι′,μι′+1k.
LetΔ=maxðμι′,μι′+1,ðμι′+1,μι′+2. If Δ=ðμι′+1,μι′+2. Since μι′+2∈Tμι′+1∩Λ, we have(56)θðμι′+1,μι′+2≤θh2HdTμι′∩Λ,Tμι′+1∩Λ≤θðμι′+1,μι′+2k,which is a contradiction. Thus, we conclude that Δ=ðμι′,μι′+1. From Remark 1, we have(57)ðμι′+1,μι′+2<ðμι′,μι′+1.
From (38), (43), and (47), we obtain(58)ðμι′+1,μι′+2<ðμι′,μι′+1≤ðμι′,ϑ∗+ðϑ∗,μι′+1≤12minðϑ∗,Tϑ∗∩Λ,ðμι′+1,Tμι′+1∩Λ+12minðϑ∗,Tϑ∗∩Λ,ðμι′+1,Tμι′+1∩Λ≤minðϑ∗,Tϑ∗∩Λ,ðμι′+1,Tμι′+1∩Λ≤ðμι′+1,Tμι′+1∩Λ,which is a contradiction. Hence, (45) holds true:(59)12minðμι,Tμι∩Λ,ðϑ∗,Tϑ∗∩Λ<ðμι,ϑ∗for allι≥2.
Also, we know thatðμι,μι+1∈E^ for all n. Thus, from (15), we have(60)θðμι+1,Tμι+1∩Λ≤θHdTμι∩Λ,Tμι+1∩Λ≤θηΩμι,μι+1k<θΩμι,μι+1k,where(61)Ωμι,μι+1=maxðμι,μι+1,ðμι,Tμι,ðμι+1,Tμι+1,ðμι,Tμι+1+ðμι+1,Tμι2≤maxðμι,μι+1,ðμι+1,μι+2,ðμι,μι+22≤maxðμι,μι+1,ðμι+1,μι+2.
We claim that(62)θðμι+1,Tμι+1∩Λ≤θHdTμι∩Λ,Tμι+1∩Λ<θðμι,μι+1k.
LetΦ=maxðμι,μι+1,ðμι+1,μι+2. If Φ=ðμι+1,μι+2, we have μι+2∈Tμι+1∩Λ, so we obtain(63)θðμι+1,Tμι+1∩Λ≤θHdTμι∩Λ,Tμι+1∩Λ<θðμι+1,μι+2k,which is a contradiction. Thus, we obtain Φ=ðμι,μι+1. From Remark 1, we deduce(64)ðμι+1,Tμι+1∩Λ<ðμι,μι+1.
Taking limitι⟶∞ in (64),(65)limι⟶∞ðμι+1,Tμι+1∩Λ=0.
Sincegμ=ðμ,Tμ∩Λ is T-o.l.s.c at ϑ∗, then(66)ðϑ∗,Tϑ∗∩Λ=gϑ∗≤liminfιgμι+1=liminfιðμι+1,Tμι+1∩Λ=0.
SinceTϑ∗ is closed, we have ϑ∗∈Tϑ∗. Conversely, if ϑ∗ is a fixed point of T, then gϑ∗=0≤liminfιgμι, since ϑ∗∈Λ.Corollary 1.
LetX,ð be a complete metric space, η be a Bianchini–Grandolfi gauge function on an interval E^, and T:Λ⟶CBX be a given set-valued mapping. If k∈0,1 and Tμ∩Λ≠∅ exist,(67)12minðμ,Tμ∩Λ,ðν,Tν∩Λ<ðμ,ν,which implies that(68)θHdTμ∩Λ,Tν∩Λ≤θηðμ,νk,for all μ∈Λ, ν∈Tμ∩Λ with ðμ,ν∈E^. Suppose μ0∈Λ such that ðμ0,c∗∈E^ for some c∗∈Tμ0∩Λ. Then, there exists an orbit μι of T in Λ and ϑ∗∈Λ such that limι⟶∞μι=ϑ∗. In addition, ϑ∗ is a fixed point of T if and only if the function gμ:=ðμ,Tμ∩Λ is T-o.l.s.c at point ϑ∗.Corollary 2.
LetX,ð be a complete metric space, η be a Bianchini–Grandolfi gauge function on an interval E^, and T:Λ⟶CBX be a given set-valued mapping. If k∈0,1 and for Tμ∩Λ≠∅ exist,(69)12minðμ,Tμ∩Λ,ðν,Tν∩Λ<ðμ,ν,implies that(70)HdTμ∩Λ,Tν∩Λ≤kηðμ,ν,for all μ∈Λ, ν∈Tμ∩Λ, and ðμ,ν∈E^. In addition, suppose μ0∈Λ such that ðμ0,c∗∈E^ for some c∗∈Tμ0∩Λ. Then, there exists an orbit μι of T in Λ, ϑ∗∈Λ such that limι⟶∞μι=ϑ∗ and ϑ∗ is a fixed point of T if and only if the function gμ:=ðμ,Tμ∩Λ is T-o.l.s.c at ϑ∗.Corollary 3.
LetX,ð be a complete metric space,η be a Bianchini–Grandolfi gauge function on E^, and T:Λ⟶CBX be a given set-valued mapping. If θ∈Ξ and k∈0,1 exist, then(71)12minðμ,Tμ∩Λ,ðν,Tν∩Λ<ðμ,ν,⇒θHdTμ,Tν≤θηðμ,νk,for all μ∈X, ν∈Tμ, and ðμ,ν∈E^. Suppose that μ0∈X such that ðμ0,c∗∈E^ for some c∗∈Tμ0. Then, there exists an orbit μι of T in X that converges to the fixed point ϑ∗∈ℱ=μ∈X:ðμ,ϑ∗∈E^ of T.Example 3.
LetX=−10,∞ be an usual metric ð and let E^=0,∞. Mapping T:Λ⟶CBX is defined as(72)Tμ=0,μ8,μ∈0,4,0,μ,μ∈−10,0∪4,∞.
Clearly,1/2minðμ,Tμ∩Λ,ðν,Tν∩Λ<ðμ,ν if and only if μ,ν∈0,4. Let μ0=4; then, we have c∗=1/2∈Tμ0 such that ðμ0,c∗∈E^. Firstly, we claim that T satisfies inequality (68) with setting θr=erer, ηr=r/2, and k=1/2. For μ∈0,4 and ν∈Tμ, we obtain(73)θHdTμ,Tν=θμ−ν8=eμ−ν/8eμ−ν/8≤e1/2μ−ν/2eμ−ν/2=e1/2ηd^υ,νeηd^υ,ν=θηðμ,νk.
Consequently, the requirements of Corollary1 are fulfilled and 0 is a fixed point of T. For μ=0 and ν=5,(74)θHdTμ,Tν=θHdT0,T5=θ5>θ5k=θðμ,νk,for all θ∈Ξ and k∈0,1. Then, Corollary 1 cannot be satisfied.


## 3. Set-Valued SU-TypeΓα-Contraction

In this section, we prove the existence of fixed point in the class of metric space with respect to a simulation function.Definition 7.
LetX,ð be a metric space, Λ be a closed subset of X, and η be a Bianchini–Grandolfi gauge function on E^. Mapping T:Λ⟶CBX is known as set-valued SU-type Γα-contraction, if there exists Γ∈∇ such that for Tμ∩Λ≠∅,(75)12minðμ,Tμ∩Λ,ðν,Tν∩Λ<ðμ,ν,which implies that(76)Γαμ,νHdTμ∩Λ,Tν∩Λ,ηΩμ,ν≥0,where(77)Ωμ,ν=maxðμ,ν,ðμ,Tμ,ðν,Tν,ðμ,Tν+ðν,Tμ2,for all μ∈Λ, ν∈Tμ∩Λ with ðμ,ν∈E^.Theorem 3.
LetX,ð be a complete metric space and T:Λ⟶CBX be a set-valued SU-type Γα-contraction such that the following conditions are satisfied:(a)
T is α-admissible.(b)
There existsμ0∈Λ with ðμ0,μ1∈E^ for some μ1∈Tμ0∩Λ such that αμ0,μ1≥1. Then, there exists an orbit μι of T in Λ and ϑ∗∈Λ such that limι⟶∞μι=ϑ∗. In addition, ϑ∗ is a fixed point of T if and only if the function gμ:=ðμ,Tμ∩Λ is T-o.l.s.c at ϑ∗.Proof.
By the hypothesis, there existsμ0∈Λ with ðμ0,μ1∈E^ for some μ1∈Tμ0∩Λ such that αμ0,μ1≥1. On the other hand, we have(78)12minðμ0,Tμ0∩Λ,ðμ1,Tμ1∩Λ<ðμ0,μ1.
In the case thatðμ0,μ1=0, μ0 is a fixed point of T. Thus, we assume that ðμ0,μ1≠0. Define ρ=σðμ0,μ1. From (13), we have σr≥r. Hence, ðμ0,μ1≤ρ, and so μ1∈b¯μ0,ρ. Since αμ0,μ1≥1 and ðμ0,μ1∈E^, from (76) and (78), it follows that(79)0≤Γαμ0,μ1HdTμ0∩Λ,Tμ1∩Λ,ηðμ0,μ1<ηΩμ0,μ1−αμ0,μ1HdTμ0∩Λ,Tμ1∩Λ,which implies that(80)αμ0,μ1HdTμ0∩Λ,Tμ1∩Λ<ηΩμ0,μ1.
We can chooseε1>0 such that(81)αμ0,μ1HdTμ0∩Λ,Tμ1∩Λ+ε1≤ηΩμ0,μ1.
Thus,(82)ðμ1,Tμ1∩Λ+ε1≤HdTμ0∩Λ,Tμ1∩Λ+ε1≤αμ0,μ1HdTμ0∩Λ,Tμ1∩Λ+ε1≤ηΩμ0,μ1.
It follows from Lemma2 that there exists μ2∈Tμ1∩Λ such that(83)ðμ1,μ2≤ðμ1,Tμ1∩Λ+ε1.
Fromfd82(82) and (83), we infer that(84)ðμ1,μ2≤ηΩμ0,μ1,where(85)Ωμ0,μ1=maxðμ0,μ1,ðμ0,Tμ0,ðμ1,Tμ1,ðμ0,Tμ1+ðμ1,Tμ02≤maxðμ0,μ1,ðμ1,Tμ1,ðμ0,Tμ12≤maxðμ0,μ1,ðμ1,Tμ1.
We claim that(86)ðμ1,μ2≤ηðμ0,μ1.
LetΦ=maxðμ0,μ1,ðμ1,Tμ1. If Φ=ðμ1,Tμ1, we have μ2∈Tμ1∩Λ, so we obtain(87)ðμ1,μ2≤ηðμ1,μ2,which is a contradiction. Thus, we obtain Φ=ðμ0,μ1. We assume that ðμ1,μ2≠0; otherwise, μ1 is a fixed point of T. Since ðμ1,μ2≤ηðμ0,μ1<ðμ0,μ1, we deduce that ðμ1,μ2∈E^. Next, μ2∈b¯μ0,ρ because(88)ðμ0,μ2≤ðμ0,μ1+ðμ1,μ2≤ðμ0,μ1+ηðμ0,μ1≤ðμ0,μ1+σηðμ0,μ1=σðμ0,μ1=ρ.
BecauseT is α-admissible, αμ1,μ2≥1. Also, since(89)12minðμ1,Tμ1∩Λ,ðμ2,Tμ2∩Λ<ðμ1,μ2,from (76), we get(90)0≤Γαμ1,μ2HdTμ1∩Λ,Tμ2∩Λ,ηΩμ1,μ2<ηΩμ1,μ2−αμ1,μ2HdTμ1∩Λ,Tμ2∩Λ.
This implies that(91)αμ1,μ2HdTμ1∩Λ,Tμ2∩Λ<ηΩμ1,μ2.
Now chooseε2>0 such that(92)αμ1,μ2HdTμ1∩Λ,Tμ2∩Λ+ε2≤ηΩμ1,μ2.
Thus,(93)ðμ2,Tμ2∩Λ+ε2≤HdTμ1∩Λ,Tμ2∩Λ+ε2≤αμ1,μ2HdTμ1∩Λ,Tμ2∩Λ+ε2≤ηΩμ1,μ2.
It follows from Lemma2 that there exists μ3∈T, μ2∩Λ such that(94)ðμ2,μ3≤ðμ2,Tμ2∩Λ+ε2.
From (93) and (94),(95)ðμ2,μ3≤η2Ωμ0,μ1,where(96)Ωμ1,μ2=maxðμ1,μ2,ðμ1,Tμ1,ðμ2,Tμ2,ðμ1,Tμ2+ðμ2,Tμ12≤maxðμ1,μ2,ðμ2,Tμ2,ðμ1,Tμ22≤maxðμ1,μ2,ðμ2,Tμ2.
We claim that(97)ðμ2,μ3≤ηðμ1,μ2.
LetΦ=maxðμ1,μ2,ðμ2,Tμ2. If Φ=ðμ2,Tμ2. Since μ3∈Tμ2∩Λ, we have(98)ðμ2,μ3≤ηðμ2,μ3,which is a contradiction. Thus, we have Φ=ðμ1,μ2. We assume that ðμ2,μ3≠0; otherwise, μ2 is a fixed point of T. From (95), we have ðμ2,μ3<ðμ1,μ2, and so ðμ2,μ3∈E^. Also, we have μ3∈b¯μ0,ρ, since(99)ðμ0,μ3≤ðμ0,μ1+ðμ1,μ2+ðμ2,μ3≤ðμ0,μ1+ηðμ0,μ1+η2ðμ0,μ1≤∑i=0∞ηiðμ0,μ1=σðμ0,μ1=ρ.
Continuing this setup, we obtain a sequenceμι⊂b¯μ0,ρ such that μι+1∈Tμι∩Λ, μι≠μι+1 with αμι,μι+1≥1 and ðμι,μι+1∈E^ and(100)ðμι,μι+1≤ηιðμ0,μ1,for allι∈ℕ.
Forι,m∈ℕ with m>ι, by using the triangular inequality and (100), we get(101)ðμι,μm≤ðμι,μι+1+ðμι+1,μι+2+⋯+ðμm−1,μm≤ηιðμ0,μ1+ηι+1ðμ0,μ1+⋯+ηm−1ðμ0,μ1≤∑j=ι∞ηjðμ0,μ1<∞.
To shows thatμι is a Cauchy sequence in b¯μ0,ρ. Since b¯μ0,ρ is closed in X, there exists an ϑ∗∈b¯μ0,ρ such that μι⟶ϑ∗. Note that ϑ∗∈Λ because μι+1∈Tμι∩Λ. By same argument of Theorem 2, we have(102)12sminðμι,Tμι∩Λ,ðμι+1,Tμι+1∩Λ<ðμι,μι+1.
Also, we know thatαμι,μι+1≥1 and ðμι,μι+1∈E^ for all n. Thus, from (76), we have(103)0≤Γαμι,μι+1HdTμι∩Λ,Tμι+1∩Λ,ηΩμι,μι+1<ηΩμι,μι+1−αμι,μι+1HdTμι∩Λ,Tμι+1∩Λ,which gives that(104)αμι,μι+1HdTμι∩Λ,Tμι+1∩Λ<ηΩμι,μι+1.
Sinceμι+1∈Tμι∩Λ, from (100), we get(105)ðμι+1,Tμι+1∩Λ≤αμι,μι+1HdTμι∩Λ,Tμι+1∩Λ≤ηΩμι,μι+1<Ωμ0,μ1,where(106)Ωμι,μι+1=maxðμι,μι+1,ðμι,Tμι,ðμι+1,Tμι+1,ðμι,Tμι+1+ðμι+1,Tμι2≤maxðμι,μι+1,ðμι+1,μι+2,ðμι,μι+22≤maxðμι,μι+1,ðμι+1,μι+2.
We claim that(107)ðμι+1,Tμι+1∩Λ≤αμι,μι+1HdTμι∩Λ,Tμι+1∩Λ<ηðμι,μι+1.
LetΦ=maxðμι,μι+1,ðμι+1,μι+2. If Φ=ðμι+1,μι+2. Since μι+2∈Tμι+1∩Λ, we have(108)ðμι+1,Tμι+1∩Λ≤αμι,μι+1HdTμι∩Λ,Tμι+1∩Λ<ηðμι+1,μι+2,which is a contradiction. Thus, we have Φ=ðμι,μι+1. Taking limit ι⟶∞ in (105), we obtain(109)limι⟶∞ðμι+1,Tμι+1∩Λ=0.
Sincegμ=ðμ,Tμ∩Λ is T-o.l.s.c at ϑ∗, then(110)ðϑ∗,Tϑ∗∩Λ=gϑ∗≤liminfιgμι+1=liminfιðμι+1,Tμι+1∩Λ=0.
SinceTϑ∗ is closed, ϑ∗∈Tϑ∗. Conversely, if ϑ∗ is a fixed point of T, then gϑ∗=0≤liminfιgμι, since ϑ∗∈Λ.TakingΓr,s=s−∫0rςtdt for all r,s≥0, in Theorem 3, we obtain the following theorem.Corollary 4.
LetX,ð be a complete metric space, η be a Bianchini–Grandolfi gauge function on an interval E^, and T:Λ⟶CBX be a given set-valued mapping. If Tμ∩Λ≠∅, then(111)12minðμ,Tμ∩Λ,ðν,Tν∩Λ<ðμ,ν,which implies that(112)∫0αμ,νHdTμ∩Λ,Tν∩Λςtdt≤ηðμ,ν,for all μ∈Λ, ν∈Tμ∩Λ, and ðμ,ν∈E^, where ς:ℝ+⟶ℝ+ is a function such that ∫0εςtdt exists and ∫0εςtdt>ε for all ε>0 such that the following holds:(a)
T is α-admissible.(b)
There existsμ0∈Λ with ðμ0,μ1∈E^ for some μ1∈Tμ0∩Λ such that αμ0,μ1≥1. Then, there exists an orbit μι of T in Λ and ϑ∗∈Λ such that limι⟶∞μι=ϑ∗. In addition, ϑ∗ is a fixed point of T if and only if the function gμ:=ðμ,Tμ∩Λ is T-o.l.s.c at ϑ∗.Corollary 5.
LetX,ð be a complete metric space, η be a Bianchini–Grandolfi gauge function on an interval E^, and T:X⟶CBX be a given set-valued mapping. If Γ∈∇ exists, then(113)12minðμ,Tμ∩Λ,ðν,Tν∩Λ<ðμ,ν⇒Γαμ,νHdTμ,Tν,ηðμ,ν≥0,for all μ∈X, ν∈Tμ, and ðμ,ν∈E^ such that the following holds:(a)
T is α-admissible.(b)
There existsμ0∈X with ðμ0,μ1∈E^ for some μ1∈Tμ0 such that αμ0,μ1≥1. Then, there exists an orbit μι of T in X that converges to the fixed point ϑ∗∈ℱ=μ∈X:ðμ,ϑ∗∈E^ of T.


## 4. An Application to Dynamical System

Dynamical system is connected to a multistage operation reduced for solving the following functional equation:(114)Tμ1=supμ2∈Hhμ1,μ2+ðμ1,μ2,Tlμ1μ2forμ1∈β˜,where(115)l:β˜×H⟶β˜,h,h′:β˜×H⟶−∞,∞,D,D′:β˜×H×−∞,∞⟶−∞,∞.Assume thatG˜1 and G˜2 are Banach spaces, β˜⊂G˜1 is a state space, and H⊂G˜2 is a decision space. For more details, see [3]. Let Bβ˜ signify the set of all bounded real-valued functions on β˜. Choose an arbitrary point σ∈Bβ˜ defined as σ=supr∈β˜σr. Bβ˜,. endowed with the metric given by(116)ðμ1,μ2=supr∈β˜μ1r−μ2r,for all μ1,μ2∈Bβ˜, are BS. Define g:Bβ˜⟶Bβ˜ by(117)gϖr=maxt∈HVr,t,ϖlr,t+fr,t,for all ϖ∈Bβ˜ and r∈β˜. Also,(118)Hdgϖ1r,gϖ2r=HdVxr,t,ϖ1lr,t+fxr,t,Vyr,t,ϖ2lr,t+fyr,t≤HdVxr,t,ϖ1lr,t,Vyr,t,ϖ2lr,t.Considerϕ:Bβ˜⟶Bβ˜ such that(119)12minðϖ1,ϕϖ1∩Λ,ðϖ2,ϕϖ1∩Λ<ðϖ1,ϖ2,ϖ1,ϖ2∈Bβ˜,and we have(120)Vr,t,ϖ1r−Vr,t,ϖ2r≤1+Tϖ1r−ϖ2rα−12,for all ϖ1,ϖ2∈Bβ˜, where r∈β˜, t∈V and 0≤α<1.Theorem 4.
Letϕ:Bβ˜⟶Bβ˜ be a l.s.c mapping as defined in (117) such that the following conditions are satisfied:V and f are continuous and bounded.There exists an orbitϖι∈Λ of ϕ and c∗∈Λ such that limι⟶∞ϖι=c∗.c∗ is a fixed point iff ϕϖ1=ðϖ1,Tϖ1∩Λ is ϕ-o.l.s.c at c∗.
Then, functional (114) possesses a bounded solution.Proof.
Note thatBβ˜,ð is a complete metric, where ðμ1,μ2 is the metric, as defined by (82). There exist r∈β˜, t1,t2∈V and ψ:Bβ˜×Bβ˜⟶0,∞ such that(121)ψðϖ1,ϕϖ2∩Λ,ðϖ1,ϖ2<0,ϖ1,ϖ2∈Bβ˜,and we have(122)Hdϕ1ϖ1r,ϕ1ϖ2r≤HdVxr,t,ϖ1lr,t,Vyr,t,ϖ2lr,t≤maxr∈β˜maxt∈β˜Vr,t,ϖ1lr,t−Vr,t,ϖ2lr,t≤maxr∈β˜maxt∈β˜1+ηϖ1lr,t−ϖ2lr,tα−12≤maxr∈β˜1+ηϖ1−ϖ2α−12≤1+ηðϖ1,ϖ2α−12.
It implies that(123)Hdϕ1ϖ1r,ϕ1ϖ2r≤1+ηðϖ1,ϖ2α−12.
Owing to (123),(124)1+ηðϖ1,ϖ2≤1+ηðϖ1,ϖ2α.
Byθ∈Ξ and θz=1+z with (124), we obtain(125)θHdðϕ1ϖ1,ϕ1ϖ2≤θηðϖ1,ϖ2αfor allϖ1,ϖ2∈Bβ˜.
Furthermore,1−3 are equivalent to a−b of Theorem 3. So, there exists a fixed point c∗∈Λ in Bβ˜, which is a bounded solution of functional (117).


### 4.1. An Application to Integral Inclusion

In this section, we consider the following set-valued integral inclusion:(126)ςr∈κ+U∫r0rVt,ςtVt,where κ∈−∞,∞, U is a bounded compact subset of −∞,∞, and Vt,ςt is l.s.c. Let X=CI be the space of all continuous real-valued function and CI is complete w.r.t the metric ð, which defined by(127)ðμ1,μ2=supr∈Iμ1r−μ2r.Assume that there existsϕ:Bβ˜⟶Bβ˜ and V:−∞,∞×−∞,∞⟶−∞,∞ is continuous on(128)R=r,ς:r−r0≤1α11/2andς−κ≤12α2,such that for(129)12minðμ1,ϕμ1∩Λ,ðμ2,ϕμ2∩Λ<ðϖ1,ϖ2,μ1,μ2≥0,we have(130)Vr,ς1r−Vr,ς2r≤eα1/α2ς1r−ς2rα,where α2=maxu¯∈Uu¯, 0<α1≤α2, and 0≤α<1.(131)Vt,ς<12α11α11/2.Moreover, letCˇ=ς∈CI:ðς,κ≤1/2α2 be a closed subspace of CI and the operator ϕ be defined by(132)ϕςr∈κ+U∫r0rVt,ςtdt.SetVxr=∫r0rVt,ςtdt. Note that(133)Hdϕς1r,ϕς2r=Hdκ+UVxr,κ+UVyr≤HdUVxr,UVyr=maxmaxa¯∈UVxrða¯,UVyr,maxb¯∈UVyrðb¯,UVxr.Consider(134)maxa¯∈UVxrða¯,UVyr=maxa¯∈UVxrminb¯∈UVyrða¯,b¯=maxu¯∈Uminv¯∈Uðu¯Vr,ς1r,v¯Vr,ς2r=maxu¯∈Uminv¯∈Usupr∈Iu¯Vr,ς1r−v¯Vr,ς2r≤maxu¯∈Uminv¯∈Usupr∈Iu¯Vr,ς2r−v¯Vr,ς2r+u¯Vr,ς2r−u¯Vr,ς1r≤maxu¯∈Uminv¯∈Uu¯supr∈IVr,ς2r−Vr,ς1r+u¯−v¯supr∈IVr,ς2r=maxu¯∈Uu¯supr∈IVr,ς2r−Vr,ς1r=α2supr∈IVr,ς2r−Vr,ς1r.This implies that(135)maxa¯∈UVxrVa¯,UVyr≤α2supr∈IVr,ς2r−Vr,ς1r.Theorem 5.
LetX=CI and ϕ:Cˇ,d⟶DCˇ,Hd be a l.s.c mapping. Suppose that the following assumptions hold:(i)
ϕ is defined for all ς∈Cˇ.(ii)
ϕςr is a CS of Cˇ for all ς∈Cˇ.
Then, owing to (127)–(135), integral (126) has a solution on I.Proof.
Letϰ∈I. Then, ϰ−r0≤1/α11/2. Hence, we have ςϰ−κ≤1/2α1. If ϰ,ςϰ∈−∞,∞, the integral equation in (132) exists. Since κ∈−∞,∞ is continuous, ϰ is defined for all ϰ∈Cˇ. Next, let ϑr∈ϕςr. Then, ϑr=κ+u¯Vxr for u¯∈U:(136)ϑr−κ=u¯Vxr=u¯Vxr≤α2∫r0rVt,ςtdt≤α2∫r0rVt,ςtdt<α2r−r012α21α11/2≤12α1.
Thus,ϑr−κ≤1/2α1 for each ϑr∈ϕςr. So, ϕςr is a subset of Cˇ. Now, let ςi⊂ϕςr; then, ς=κ+ui¯Vxr for ui¯∈U. Since U is compact, there exists subsequence ui∗^∈ui^ such that ui∗^ is convergent to u¯∈U. Let u^=κ+u^Vxr; then,(137)dui∗^,u^=supr∈Iui∗^−u^Vxr≤ui∗^−u^supr∈IVxr⟶0, asi∗⟶∞.
Hence,ϕςr is a CS of Cˇ for all ς∈Cˇ. Next,(138)Vr,ς1r−Vr,ς2r≤∫r0rVt,ς1t−Vt,ς2tdt≤e∫r0rα1/α2ς1t−ς2tαdt=eα1/α2∫r0rς1t−ς2tαdt≤eα1/α2supr∈Iς1t−ς2tα1/2∫r0rdt=eα1/α2r−r0supr∈Iς1t−ς2tα1/2≤eα1/α21/α21/2ðς1t−ς2tα1/2=e1/α2ðς1t−ς2tα.
It yields that(139)Vr,ς1r−Vr,ς2r≤eα1/α21/α21/2ðς1t−ς2tα1/2.
Byθ∈Ξ, θz=ez and ηr=1/α2r with (139), we get(140)θHdðϕϖ1,ϕϖ2≤θηðϖ1,ϖ2α.
Furthermore,i-ii are equivalent to a−b of Theorem 3. So, there exists a fixed point c∗∈Λ in Cˇ, which is a bounded solution of (126).


## 4.1. An Application to Integral Inclusion

In this section, we consider the following set-valued integral inclusion:(126)ςr∈κ+U∫r0rVt,ςtVt,where κ∈−∞,∞, U is a bounded compact subset of −∞,∞, and Vt,ςt is l.s.c. Let X=CI be the space of all continuous real-valued function and CI is complete w.r.t the metric ð, which defined by(127)ðμ1,μ2=supr∈Iμ1r−μ2r.Assume that there existsϕ:Bβ˜⟶Bβ˜ and V:−∞,∞×−∞,∞⟶−∞,∞ is continuous on(128)R=r,ς:r−r0≤1α11/2andς−κ≤12α2,such that for(129)12minðμ1,ϕμ1∩Λ,ðμ2,ϕμ2∩Λ<ðϖ1,ϖ2,μ1,μ2≥0,we have(130)Vr,ς1r−Vr,ς2r≤eα1/α2ς1r−ς2rα,where α2=maxu¯∈Uu¯, 0<α1≤α2, and 0≤α<1.(131)Vt,ς<12α11α11/2.Moreover, letCˇ=ς∈CI:ðς,κ≤1/2α2 be a closed subspace of CI and the operator ϕ be defined by(132)ϕςr∈κ+U∫r0rVt,ςtdt.SetVxr=∫r0rVt,ςtdt. Note that(133)Hdϕς1r,ϕς2r=Hdκ+UVxr,κ+UVyr≤HdUVxr,UVyr=maxmaxa¯∈UVxrða¯,UVyr,maxb¯∈UVyrðb¯,UVxr.Consider(134)maxa¯∈UVxrða¯,UVyr=maxa¯∈UVxrminb¯∈UVyrða¯,b¯=maxu¯∈Uminv¯∈Uðu¯Vr,ς1r,v¯Vr,ς2r=maxu¯∈Uminv¯∈Usupr∈Iu¯Vr,ς1r−v¯Vr,ς2r≤maxu¯∈Uminv¯∈Usupr∈Iu¯Vr,ς2r−v¯Vr,ς2r+u¯Vr,ς2r−u¯Vr,ς1r≤maxu¯∈Uminv¯∈Uu¯supr∈IVr,ς2r−Vr,ς1r+u¯−v¯supr∈IVr,ς2r=maxu¯∈Uu¯supr∈IVr,ς2r−Vr,ς1r=α2supr∈IVr,ς2r−Vr,ς1r.This implies that(135)maxa¯∈UVxrVa¯,UVyr≤α2supr∈IVr,ς2r−Vr,ς1r.Theorem 5.
LetX=CI and ϕ:Cˇ,d⟶DCˇ,Hd be a l.s.c mapping. Suppose that the following assumptions hold:(i)
ϕ is defined for all ς∈Cˇ.(ii)
ϕςr is a CS of Cˇ for all ς∈Cˇ.
Then, owing to (127)–(135), integral (126) has a solution on I.Proof.
Letϰ∈I. Then, ϰ−r0≤1/α11/2. Hence, we have ςϰ−κ≤1/2α1. If ϰ,ςϰ∈−∞,∞, the integral equation in (132) exists. Since κ∈−∞,∞ is continuous, ϰ is defined for all ϰ∈Cˇ. Next, let ϑr∈ϕςr. Then, ϑr=κ+u¯Vxr for u¯∈U:(136)ϑr−κ=u¯Vxr=u¯Vxr≤α2∫r0rVt,ςtdt≤α2∫r0rVt,ςtdt<α2r−r012α21α11/2≤12α1.
Thus,ϑr−κ≤1/2α1 for each ϑr∈ϕςr. So, ϕςr is a subset of Cˇ. Now, let ςi⊂ϕςr; then, ς=κ+ui¯Vxr for ui¯∈U. Since U is compact, there exists subsequence ui∗^∈ui^ such that ui∗^ is convergent to u¯∈U. Let u^=κ+u^Vxr; then,(137)dui∗^,u^=supr∈Iui∗^−u^Vxr≤ui∗^−u^supr∈IVxr⟶0, asi∗⟶∞.
Hence,ϕςr is a CS of Cˇ for all ς∈Cˇ. Next,(138)Vr,ς1r−Vr,ς2r≤∫r0rVt,ς1t−Vt,ς2tdt≤e∫r0rα1/α2ς1t−ς2tαdt=eα1/α2∫r0rς1t−ς2tαdt≤eα1/α2supr∈Iς1t−ς2tα1/2∫r0rdt=eα1/α2r−r0supr∈Iς1t−ς2tα1/2≤eα1/α21/α21/2ðς1t−ς2tα1/2=e1/α2ðς1t−ς2tα.
It yields that(139)Vr,ς1r−Vr,ς2r≤eα1/α21/α21/2ðς1t−ς2tα1/2.
Byθ∈Ξ, θz=ez and ηr=1/α2r with (139), we get(140)θHdðϕϖ1,ϕϖ2≤θηðϖ1,ϖ2α.
Furthermore,i-ii are equivalent to a−b of Theorem 3. So, there exists a fixed point c∗∈Λ in Cˇ, which is a bounded solution of (126).


## 5. Conclusion

The paper deals with the set-valued fixed point theorems satisfying SU-type contraction via Bianchini–Grandolfi gauge function in the context of metric spaces. Within this framework, we introduced two related fixed point results in metric space. An extensive set of nontrivial example is given to justify our claim. Also, we have proven the existence theorem for the system of functional equation and integral inclusion.



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