symmetry
S S
Article
On New Extensions of Darbo’s Fixed Point Theorem with Applications
Hüseyin I¸sık1 , Shahram Banaei2 , Farhan Golkarmanesh3, Vahid Parvaneh4,* and Choonkil Park5,* and Maryam Khorshidi6
1 Department of Mathematics, Faculty of Science and Arts, Mu¸s Alparslan University, Mu¸s 49250, Turkey;
2 Department of Mathematics, Bonab Branch, Islamic Azad University, Bonab, Iran;
3 Department of Mathematics, Sanandaj Branch, Islamic Azad University, Sanandaj, Iran;
4 Department of Mathematics, Gilan-E-Gharb Branch, Islamic Azad University, Gilan-E-Gharb, Iran
5 Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Korea
6 Department of Mathematics, Firouzabad Institute of Higher Education, Firouzabad, Fars, Iran;
* Correspondence: [email protected] (V.P.); [email protected] (C.P.)
Received: 11 December 2019; Accepted: 26 December 2019; Published: 6 March 2020 Abstract:In this paper, we extend Darbo’s fixed point theorem via weak JS-contractions in a Banach space. Our results generalize and extend several well-known comparable results in the literature.
The technique of measure of non-compactnessis the main tool in carrying out our proof. As an application, we study the existence of solutions for a system of integral equations. Finally, we present a concrete example to support the effectiveness of our results.
Keywords:fixed point; complete metric space; measure of non-compactness; integral equation
1. Introduction and Preliminaries
The notion of a measure of non-compactness (shortly, MNC) was introduced by Kuratowski [1]
in 1930. This concept is a very useful tool in functional analysis, such as in metric fixed point theory and operator equation theory in Banach spaces. This notion is also applied in studies of the existence of solutions of ODE and PDE problems, integral and integro-differential equations, etc. For more details, we refer the reader to [2–6]. In [7], the authors generalized the Darbo’s fixed point theorem via the concept of the class of operatorsO(f; .).
The aim of this paper is to generalize the Darbo’s fixed point theorem via weak JS-contractions in a Banach space and to study the existence of solutions for the following system of integral equations:
κ1(η) = f
η,κ1(α(η)),κ2(α(η)), Z β(η)
0 g(η,$,κ1(α($)),κ2(α($)))d$
κ2(η) = f
η,κ2(α(η)),κ1(α(η)), Z β(η)
0 g(η,$,κ2(α($)),κ1(α($)))d$
(1)
whereη∈[0,T].
We introduce some notations and definitions which are used throughout this paper. LetRdenotes the set of real numbers and letR+= [0,+∞). Let(E,k · k)be a real Banach space. Moreover,B(η,r)
Symmetry2020,12, 424; doi:10.3390/sym12030424 www.mdpi.com/journal/symmetry
denotes the closed ball centered atηwith radiusr. The symbolBrstands for the ballB(0,r). ForΛ, a nonempty subset ofE, we denote byΛandConvΛthe closure and the closed convex hull ofΛ, respectively. Furthermore, let us denote byME the family of nonempty bounded subsets ofE, and by NE, its subfamily consisting of all relatively compact subsets ofE.
Definition 1([8]). A mappingµ:ME −→R+is said to be a measure of non-compactness inE if it satisfies the following conditions:
1◦ The family kerµ={Λ∈ME :µ(Λ) =0}is nonempty and kerµ⊂NE; 2◦ Λ⊂Σ=⇒µ(Λ)≤µ(Σ);
3◦ µ(Λ) =µ(Λ); 4◦ µ(ConvΛ) =µ(Λ);
5◦ µ(λΛ+ (1−λ)Σ)≤λµ(Λ) + (1−λ)µ(Σ)for allλ∈[0, 1];
6◦ if {Λn} is a sequence of closed subsets from ME such that Λn+1 ⊂ Λn for n = 1, 2, . . . and if
n→lim∞µ(Λn) = 0, thenΛ∞=∩∞n=1Λn 6=∅.
We denote byΘthe set of all functionsθ:[0,∞)→[1,∞)satisfying the following conditions:
θ1. θis a continuous strictly increasing function;
θ2. for each sequence{tn} ⊆(0,∞), lim
n→∞θ(tn) =1 if and only if lim
n→∞tn=0.
LetΦbe the class of all functionsφ:[1,∞)→[0,∞)satisfying the following properties:
φ1. φis lower semi-continuous;
φ2. φ(1) =0.
φ3. for each sequence{tn} ⊆(1,∞), lim
n→∞φ(tn) =0 if and only if lim
n→∞tn=1.
Remark 1([9]). It is clear that f(t) =t−√n
t(n≥1)belongs toΦ. Other examples are f(t) =et−1−1and f(t) =lnt.
Definition 2([9]). Let (Λ,d)be a metric space and Υbe a self-mapping on Λ.We say that Υis a weak JS-contraction if, for allη,$∈Λwith d(Υη,Υ$)>0,we have
θ d(Υη,Υ$)≤θ d(η,$)−φ θ(d(η,$)) (2) whereφ∈Φandθ∈Θ.
Theorem 1 ([9]). Let (Λ,d) be a complete metric space. Let Υ : Λ → Λ be a self-mapping satisfying the following assertions:
(i) Υis a weak JS-contraction, (ii) Υis continuous.
Then,Υhas a unique fixed point.
Now we recall two important theorems playing a key role in the fixed point theory.
Theorem 2([10]). LetCbe a nonempty, bounded, closed, and convex subset of a Banach spaceE. Then, each continuous and compact mapΥ:C → Chas at least one fixed point in the setC.
Obviously, the above theorem is the well-known Schauder fixed point principle. Its generalization, called Darbo’s fixed point theorem, is arranged as follows.
Theorem 3([11]). LetC be a nonempty, bounded, closed, and convex subset of a Banach spaceE and let Υ:C → Cbe a continuous mapping. Assume that there exists a constant K∈[0, 1)such thatµ(ΥΛ)≤Kµ(Λ) for any nonempty subsetΛofC, whereµis an MNC defined inE. Then,Υhas at least one fixed point inC. 2. Main Results
Now, we state one of the main results in this article, which extends and generalizes Darbo’s fixed point theorem by using the concept of weak JS-contractions.
Theorem 4. LetCbe a nonempty, bounded, closed, and convex subset of a Banach spaceEand letΥ:C → C be a continuous operator such that
θ µ(Υ(Λ))≤θ µ(Λ)−φ θ(µ(Λ)) (3) for allΛ⊆ C, whereφ∈Φ,θ∈Θ, andµis an arbitrary MNC. Then,Υhas at least one fixed point inC. Proof. Define a sequence{Cn}such thatC0=CandCn+1=Conv(Υ(Cn))for alln∈N.
If there exists an integerN ∈ Nsuch thatµ(CN) =0, then CN is relatively compact, and the Schauder fixed point theorem implies thatΥhas a fixed point. So, we assume thatµ(CN)>0 for each n∈N.
It is clear that{Cn}n∈Nis a sequence of nonempty, bounded, closed and convex sets such that C0⊇ C1⊇ · · · ⊇ Cn ⊇ Cn+1.
We know that{µ(Cn)}n∈Nis a positive decreasing and bounded-below sequence of real numbers.
Thus,{µ(Cn)}n∈Nis a convergent sequence. Suppose that lim
n→∞µ(Cn) =r. Also, we have µ(Cn+1) =µ(Conv(Υ(Cn))) =µ(Υ(Cn)).
In view of condition (3), we have
θ(µ(Cn+1)) = θ(µ(Υ(Cn)))
≤ θ(µ(Cn))−φ(θ(µ(Cn))). (4) Taking the limsup in the above inequality, we have
lim sup
n→∞ θ(µ(Cn+1))≤lim sup
n→∞ θ(µ(Cn))−lim inf
n→∞ φ(θ(µ(Cn))). Therefore,
θ(r)≤θ(r)−φ(θ(r)).
Hence, φ(θ(r)) must be 0, which means that θ(r) = 1. Consequently, r = 0. Therefore, limn→∞µ(Cn) = 0. According to axiom(6◦)of Definition1, we derive that the setC∞=T∞n=1Cnis a nonempty, closed and convex set, that it is invariant under the operatorΥ, and that it belongs toKerµ.
Then, in view of the Schauder theorem,Υhas a fixed point.
Remark 2. We obtain the Darbo’s fixed point theorem if we takeθ(t) =et,φ(t) =t−tλin Theorem4, where λ∈(0, 1).
In [12], Bhaskar and Lakshmikantham introduced the notion of coupled fixed point and proved some coupled fixed point theorems for some mappings, and discussed the existence and uniqueness of solutions for periodic boundary value problems.
Definition 3([12]). An element(η,$)∈ E × Eis called a coupled fixed point of the mappingΥ:E × E → E ifΥ(η,$) =ηandΥ($,η) =$.
Theorem 5([6]). Suppose thatµ1,µ2, . . . ,µnare measures of non-compactness in Banach spacesE1,E2, . . . ,En, respectively. Moreover, assume that the functionΓ:[0,∞)n−→[0,∞)is convex andΓ(η1, . . . ,ηn) =0if and only ifηi =0for i=1, 2, . . . ,n. Then
˜
µ(Λ) =Γ(µ1(Λ1),µ2(Λ2), . . . ,µn(Λn))
defines a measure of non-compactness inE1× E2×. . .× En, whereΛidenotes the natural projection ofΛinto Ei, for i=1, 2, . . . ,n.
From now on, we assume thatθis a subadditive mapping. For instance, a concave function f : [0,∞) → [0,∞)with f(0) ≥ 0 is a subadditive function. θ(t) = √
t+1, θ(t) = √
t+1 and θ(t) =1+ln(t+1)are some subadditive functions which belong toΘ.
Theorem 6. LetCbe a nonempty, bounded, closed and convex subset of a Banach spaceEand letΥ:C × C → C be a continuous function such that
θ
µ(Υ(Λ1×Λ2))≤ 1 2
θ(µ(Λ1) +µ(Λ2))−φ(θ(µ(Λ1) +µ(Λ2))) (5) for all subsetsΛ1,Λ2ofC, whereµis an arbitrary MNC andθ,φare as in Theorem4. In addition, we assume thatθis a subadditive mapping. Then,Υhas at least one coupled fixed point.
Proof. We define the mappingΥe :C2→ C2by
Υe(η,$) = (Υ(η,$),Υ($,η)),
whereC2 =C × C. It is clear thatΥeis continuous. We show thatΥesatisfies all of the conditions of Theorem4. LetΛ⊂ C2be a nonempty subset. We know thatµe(Λ) =µ(Λ1) +µ(Λ2)is an MNC (see, [8]), whereΛ1andΛ2denote the natural projections ofΛintoE. From (5), we have
θ[µe(Υe(Λ))]≤θ(µe(Υ(Λ1×Λ2)×Υ(Λ2×Λ1)))
=θ[µ(Υ(Λ1×Λ2)) +µ(Υ(Λ2×Λ1))]
≤ 1
2[θ(µ(Λ1) +µ(Λ2))−φ(θ(µ(Λ1) +µ(Λ2)))]
+1
2[θ(µ(Λ2) +µ(Λ1))−φ(θ(µ(Λ2) +µ(Λ1)))]
=θ(µe(Λ))−φ(θ(µe(Λ))).
Now, from Theorem4, we deduce thatΥehas at least one fixed point which implies thatΥhas at least one coupled fixed point.
Takingφ(t) =t−tλ(λ∈(0, 1)) in Theorem6, we have the following result.
Corollary 1. LetCbe a nonempty, bounded, closed and convex subset of a Banach spaceE and letΥ:C × C → Cbe a continuous function such that
θ[µ(Υ(Λ1×Λ2))]≤ 1
2(θ[µ(Λ1) +µ(Λ2)])λ (6) for all subsetsΛ1,Λ2ofC, whereµis an arbitrary MNC,λ∈ (0, 1)and the subadditive mappingθis as in Theorem4. Then,Υhas at least one coupled fixed point.
Corollary 2. LetCbe a nonempty, bounded, closed and convex subset of a Banach spaceE and letΥ:C × C → Cbe a continuous function such that
1+ln[1+µ(Υ(Λ1×Λ2))]≤ [1+ln(1+µ(Λ1) +µ(Λ2))]λ
2 (7)
for all subsetsΛ1,Λ2ofC, whereµis an arbitrary MNC andλ∈(0, 1). Then,Υhas at least one coupled fixed point.
Theorem 7. LetCbe a nonempty, bounded, closed and convex subset of a Banach spaceEand letΥ:C × C → C be a continuous function such that
θ[µ(Υ(Λ1×Λ2))]≤θ(max{µ(Λ1),µ(Λ2)})−φ(θ(max{µ(Λ1),µ(Λ2)})) (8) for all subsetsΛ1,Λ2ofC, whereµis an arbitrary MNC andθ,φare as in Theorem4. In addition, we assume thatθis a subadditive mapping. Then,Υhas at least one coupled fixed point.
Proof. We define the mappingΥe :C2→ C2by
Υe(η,$) = (Υ(η,$),Υ($,η)).
It is clear thateΥis continuous. We show thatΥesatisfies all of the conditions of Theorem4. We know thatµe(Λ) =max{µ(Λ1),µ(Λ2)}is an MNC (see, [8]), whereΛ1andΛ2denote the natural projections ofΛintoE. LetΛ⊂ C2be a nonempty subset. From (8), we have
θ[µe(Υe(Λ))]≤θ(µe(Υ(Λ1×Λ2)×Υ(Λ2×Λ1)))
=θ[max{µ(Υ(Λ1×Λ2)),µ(Υ(Λ2×Λ1))}]
≤max{θ[µ(Υ(Λ1×Λ2))],θ[µ(Υ(Λ2×Λ1))]}
≤max{θ(max{µ(Λ1),µ(Λ2)})−φ(θ(max{µ(Λ1),µ(Λ2)})), θ(max{µ(Λ2),µ(Λ1)})−φ(θ(max{µ(Λ2),µ(Λ1)}))}
=θ(max{µ(Λ1),µ(Λ2)})−φ(θ(max{µ(Λ1),µ(Λ2)}))
=θ(µe(Λ))−φ(θ(µe(Λ))).
Now, from Theorem4, we deduce thatΥehas at least one fixed point which implies thatΥhas at least one coupled fixed point.
Corollary 3. LetCbe a nonempty, bounded, closed and convex subset of a Banach spaceE and letΥ:C × C → Cbe a continuous function such that
θ[µ(Υ(Λ1×Λ2))]≤(θ[max{µ(Λ1),µ(Λ2)}])λ (9) for all subsetsΛ1,Λ2ofC, whereµis an arbitrary MNC,λ∈ (0, 1)and the subadditive mappingθis as in Theorem4. Then,Υhas at least one coupled fixed point.
3. Application
In this section, as an application of Corollary3, we study the existence of solutions for the system of integral Equation (1).
Let the spaceΞ:=C(J,R)(J = [0,T])consists of all real valued functions which are bounded and continuous onJ equipped with the standard norm
kηk=sup{|η(t)|:t∈ J }.
Recall that the modulus of continuity of a functionη∈Ξis defined by ω(η,e) =sup{|η(t)−η(s)|:t,s∈ J, |t−s| ≤e}.
Sinceηis uniformly continuous onJ, thenω(η,e)→0 ase→0, and the Hausdorff measure of non-compactness for all bounded setsΩofΞis
χ(Ω) =lim
e→0
nsup
η∈Ω
ω(η,e)o. (See [6], for more details).
Theorem 8. Suppose that the following assumptions are satisfied:
(i) α,β:J −→ J are continuous functions;
(ii) the function f :J ×R3−→Ris continuous and
θ(|f(η,κ1,κ2,$)− f(η,ν1,ν2,ζ)|)≤(θ(max{|κ1−ν1|,|κ2−ν2|}+|$−ζ|))λ, whereλ∈(0, 1)andθis a subadditive mapping such thatθ∈Θ;
(iii) N:=sup{|f(η, 0, 0, 0)|:η∈ J }; (iv) g:J × J ×R2−→Ris continuous and
G:=supn
Z β(η)
0 g(η,$,κ1(α($)),κ2(α($)))d$
:η,$∈ J, κ1,κ2∈Ξo
;
(iv) there exists a positive solution r0to the inequality
θ(r+G)λ+θ(N)≤r.
Then, the system of the integral Equation(1)has at least one solution in the spaceΞ2. Proof. Let us consider the operator
Υ:Ξ×Ξ−→Ξ with the formula
Υ(κ1,κ2)(η) = f(η,κ1(α(η)),κ2(α(η)), (10) Z β(η)
0 g(η,$,κ1(α($)),κ2(α($)))d$).
We observe that for anyη∈ J, the functionΥis continuous. For arbitrary fixedη∈ J, by applying the assumptions(i)–(iv), we have
θ(|Υ(κ1,κ2)(η)|)≤θ
|f(η,κ1(α(η)),κ2(α(η)), Z β(η)
0 g(η,$,κ1(α($)),κ2(α($)))d$
− f(η, 0, 0, 0)|+|f(η, 0, 0, 0)|
≤θ
max{|κ1(α(η))|,|κ2(α(η))|}
+| Z β(η)
0 g(η,$,κ1(α($)),κ2(α($)))d$|λ +θ(|f(η, 0, 0, 0)|)
≤θ(max{kκ1k,kκ2k}+G)λ+θ(N). Therefore,
θ(kΥ(κ1,κ2)k)≤θ(max{kκ1k,kκ2k}+G)λ+θ(N). (11) Due to the inequality (11) and using(iv), the functionΥ maps(B¯r0)2 into(B¯r0). Now, we prove that the operator Υ is a continuous operator on (B¯r0)2. Let us arbitrarily fix ε > 0 and take (κ1,κ2),(ν1,ν2)∈ (B¯r0)2such that max{kκ1−ν1k,kκ2−ν2k}<ε. Then, for allη∈ J, we have
θ(|Υ(κ1,κ2)(η)−Υ(ν1,ν2)(η)|)
≤θ
|f(η,κ1(α(η)),κ2(α(η)), Z β(η)
0 g(η,$,κ1(α($)),κ2(α($)))d$)
−f(η,ν1(α(η)),ν2(α(η)), Z β(η)
0 g(η,$,ν1(α($)),ν2(α($))d$)|
≤θ
max{|κ1(α(η))−ν1(α(η))|,|κ2(α(η))−ν2(α(η))|}
+| Z β(η)
0 [g(η,$,κ1(α($)),κ2(α($)))−g(η,$,ν1(α($)),ν2(α($)))]d$|λ
≤θ(ε+β(T)ωT(g,ε))λ, where
ωT(g,ε) =sup{|g(η,$,κ1,κ2)−g(η,$,ν1,ν2)|:η∈ J,$∈[0,β(T)], κ1,κ2,ν1,ν2∈[−r0,r0], max{kκ1−ν1k,kκ2−ν2k}<ε} and
β(T) =sup{β(η):η∈ J }.
Applying the continuity of gon J ×[0,β(T)]×[−r0,r0]2, we haveωT(g,ε) → 0 asε → 0, hence θ(ε+β(T)ωT(g,ε))λ→1 asε→0, which implies thatΥis a continuous function on(B¯r0)2.
Now, we prove thatΥsatisfies all conditions of Corollary3. To do this, letΛ1andΛ2be nonempty and bounded subsets of ¯Br0, and assume thatT>0 andε>0 are arbitrary constants. Letη1,η2∈ J, with|η2−η1| ≤εand(κ1,κ2)∈Λ1×Λ2. Then, we have
θ(|Υ(κ1,κ2)(η2)−Υ(κ1,κ2)(η1)|)
≤θ
|f(η2,κ1(α(η2)),κ2(α(η2)), Z β(η2)
0 g(η2,$,κ1(α($)),κ2(α($))d$)
−f(η2,κ1(α(η1)),κ2(α(η1)), Z β(η2)
0 g(η2,$,κ1(α($)),κ2(α($))d$)|
+|f(η2,κ1(α(η1)),κ2(α(η1)), Z β(η2)
0 g(η2,$,κ1(α($)),κ2(α($))d$)
−f(η1,κ1(α(η1)),κ2(α(η1)), Z β(η2)
0 g(η2,$,κ1(α($)),κ2(α($))d$)|
+|f(η1,κ1(α(η1)),κ2(α(η1)), Z β(η2)
0 g(η2,$,κ1(α($)),κ2(α($))d$)
−f(η1,κ1(α(η1)),κ2(α(η1)), Z β(η2)
0 g(η1,$,κ1(α($)),κ2(α($))d$)|
+|f(η1,κ1(α(η1)),κ2(α(η1)), Z β(η2)
0 g(η1,$,κ1(α($)),κ2(α($))d$)
−f(η1,κ1(α(η1)),κ2(α(η1)), Z β(η1)
0 g(η1,$,κ1(α($)),κ2(α($))d$)|
≤θ
max{|κ1(α(η2))−κ1(α(η1))|,|κ2(α(η2))−κ2(α(η1))|}+ωr0,G(f,ε) +
Z β(η2) 0
|g(η2,$,κ1(α($)),κ2(α($)))−g(η1,$,κ1(α($)),κ2(α($)))|d$
+| Z β(η2)
0 g(η1,$,κ1(α($)),κ2(α($)))d$− Z β(η1)
0 g(η1,$,κ1(α($)),κ2(α($)))d$|λ
≤θ
max{ω(κ1,ω(α,ε)),ω(κ2,ω(α,ε))}+ωr0,G(f,ε) +β(T)ωr0(g,ε) +Ur0ω(β,ε)λ (12) where
ω(α,ε) =sup{|α(η2)−α(η1)|:η1,η2∈ J, |η2−η1| ≤ε},
ω(κ1,ω(α,ε)) =sup{|κ1(η2)−κ1(η1)|:η1,η2∈ J, |η2−η1| ≤ω(α,ε)},
β(T) =sup{β(η):η∈ J },
Ur0 =sup{|g(η,$,κ1,κ2)|:η∈ J, $∈[0,β(T)], κ1,κ2∈[−r0,r0]},
G=β(T)sup{|g(η,$,κ1,κ2)|:η∈ J, $∈[0,βΛ], κ1,κ2∈[−r0,r0]},
ωr0,G(f,ε) =sup{|f(η2,κ1,κ2,z)−f(η1,κ1,κ2,z)|:η1,η2∈ J,
|η2−η1| ≤ε, κ1,κ2∈[−r0,r0], z∈[−G,G]},
ωr0(g,ε) =sup{|g(η2,$,κ1,κ2)−g(η1,$,κ1,κ2)|:η1,η2∈ J,
|η2−η1| ≤ε, κ1,κ2∈[−r0,r0], $∈[0,β(T)]}.
Since(κ1,κ2)is an arbitrary element ofΛ1×Λ2in (12), we have θ(ω(Υ(Λ1×Λ2),ε))≤θ
ωr0,G(f,ε) +max{ω(Λ1,ω(α,ε)),ω(Λ2,ω(α,ε))}
+β(T)ωr0(g,ε) +Ur0ω(β,ε)λ, and by the uniform continuity off andgon the compact sets
[0,T]×[−r0,r0]×[−r0,r0]×[−G,G] and
[0,T]×[0,β(T)]×[−r0,r0]×[−r0,r0]
respectively, we infer thatωr0,G(f,ε) −→0,ωr0(g,ε) −→0 andω(β,ε)−→0 asε−→0. Therefore, we obtain
θ(µ[Υ(Λ1×Λ2)])≤(θ(max{µ(Λ1),µ(Λ2)}))λ.
Thus, from Corollary3, we deduce that the operatorΥhas a coupled fixed point. Therefore, the system of the functional integral Equation(1)has at least one solution inΞ2.
Example 1. Consider the following system of integral equations:
η(t) = 12e−t2+arctanη(t)+sinh8π+t8 −1$(t) +1
8 Z t
0
s2|sinη(s)|+ q
es(1+η2(s))(1+sin2$(s)) et(1+η2(s))(1+sin2$(s)) ds,
$(t) = 12e−t2+arctan$(t)+sinh8π+t8 −1η(t) +1
8 Z t
0
s2|sin$(s)|+ q
es(1+y2(s))(1+sin2η(s)) et(1+$2(s))(1+sin2η(s)) ds.
(13)
We observe that the system of integral Equation(13)is a special case of(1)withΞ=C([0, 1],R)and α(t) =β(t) =t, t∈[0, 1],
f(t,η,$,p) = 1
2e−t2+arctanη+sinh−1$ 8π+t8 + p
8,
g(t,s,η,$) = s
2|sinη|+ q
es(1+η2)(1+sin2$) et(1+η2)(1+sin2$) . To solve this system, we need to verify the conditions (i)–(iv) of Theorem8.
Letθ(t) =√
t+1andλ= 12. Condition (i) is clearly evident. On the other hand,
θ(|f(t,η,$,m)−f(t,κ,ν,n)|)
≤ s
|arctanη−arctanκ|+|sinh−1$−sinh−1ν|
8π+t8 + |m−n|
8 +1
≤ s
arctan|η−κ|
8π +sinh
−1|$−ν|
8π +|m−n|
8 +1
≤
r|η−κ|
8π +|$−ν|
8π +|m−n|
8 +1
≤qθ(max{|η−κ|,|$−ν|}+|m−n|). So, we find that f satisfies condition (ii) of Theorem8. In addition,
N=sup{|f(t, 0, 0, 0)|:t∈[0, 1]}=sup{1
2e−t2 :t∈[0, 1]}=0.5.
Hence, condition (iii) of Theorem8is valid. Moreover, g is continuous on[0, 1]×[0, 1]×R2and
G=supn
Z t 0
t2|sinη(s)|+ q
es(1+η2(s))(1+sin2$(s)) et(1+η2(s))(1+sin2$(s)) ds
:t,s∈[0, 1], η,$∈Ξo
<supt2
et '0.4.
Furthermore, it is easy to see that r≥2.65satisfies the inequality in condition (iv), i.e., θ(r+G)12 +θ(N) =
q√
r+0.4+1+√ 1.5≤r.
Consequently, all of the conditions of Theorem8are satisfied. Hence, the system of integral Equation(13)has at least one solution which belongs to the space(C([0, 1],R))2.
4. Conclusions
There are many generalizations of Darbo’s fixed point theorem. Some authors have made generalizations via certain contraction conditions. On the other hand, many authors have generalized Darbo’s fixed point theorem by changing the domain of mappings which possess a fixed point. In this paper, we used the notion of weak JS-contractions to verify that a mapping defined on a nonempty, bounded, closed, and convex subset of a given Banach space has at least one fixed point.
We applied our results to prove the existence of solutions for a system of functional integral equations.
Author Contributions:H.I. analyzed and prepared/edited the manuscript, S.B. analyzed and prepared/edited the manuscript, F.G. analyzed and prepared the manuscript, V.P. analyzed and prepared the manuscript, C.P.
analyzed and prepared the manuscript, M.K. analyzed and prepared the manuscript. All authors have read and agreed to the published version of the manuscript.
Funding:This research received no external funding.
Acknowledgments:The authors are thankful to the anonymous reviewers for their useful and critical remarks on the paper.
Conflicts of Interest:The authors declare no conflict of interest.
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