# Generalized Darbo-TypeF-Contraction and F-Expansion and Its Applications to a Nonlinear Fractional-Order Differential Equation

Authors: Mian Bahadur Zada; Muhammad Sarwar; Thabet Abdeljawad; Fahd Jarad
Journal: Journal of Function Spaces
 (2020)

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## Abstract

In this work, we introduce various Darbo-typeF£-contractions, and utilizing these contractions, we present some fixed point theorems. Moreover, we introduce a Darbo-type F£-expanding mapping and prove fixed point theorems under the Darbo-type F£-expanding mapping. Employing our results, we check the existence of a solution to the nonlinear fractional-order differential equation under the integral type boundary conditions. For its validity, an appropriate example is given.


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## Body

## 1. Introduction and Preliminaries

For the sake of completeness, we provide a brief introduction and recollect basic notions, definitions, and fundamental results. In the sequel, we symbolize byℝ the set of all real numbers, by ℕ the set of all positive integers, by M¯ the closure of M and by co¯M the convex hull closure of M. Additionally, Ξ denotes a Banach space BΞ=Λ≠0:ΛisboundedsubsetsofΞ, ker£=Λ∈BΞ:£Λ=0 the kernel of function £:BΞ⟶0,∞ and Ω=Λ:Λ≠0,convex,bounded,andclosedsubsetofΞ.Many researchers have been interested in the fixed point theory. This theory is branched into two notable areas. One deals with contraction mappings on metric spaces. In this area, the first important result is the Banach contraction principle. The second deals with continuous operators on convex and compact subsets of a Banach space. In this area, two important results are Brouwer’s fixed point theorem and its infinite dimensional form, Schauder’s fixed point theorem.Theorem 1.
Every continuous mapping from the unit ball ofℝn into itself has a fixed point.Theorem 2 (see [1]).
LetΛ∈Ω. Then, a compact continuous operator ϒ:Λ⟶Λ has a fixed point.In both theorems, compactness plays a crucial role. To overcome such hurdle, one of the techniques is to use the notion of a measure of noncompactness (in short MNC). The axiomatic definition of an MNC is as below.Definition 3 (see [2]).
A map £:BΞ⟶0,∞ is an MNC in Ξ if for all Λ,Λ1,Λ2∈BΞ it satisfies the following axioms:
(i)
ker£≠0 and relatively compact in Ξ(ii)
Λ1⊂Λ2⇒£Λ1≤£Λ2(iii)
£Λ¯=£Λ(iv)
£co¯Λ=£Λ(v)
£ηΛ1+1−ηΛ2≤η£Λ1+1−η£Λ2,∀η∈0,1(vi)
IfΛn is a sequence of closed sets in BΞsuch that Λn+1⊂Λn,∀n∈ℕ,lims→+∞£Λn=0, then Λ∞=∩n=1+∞Λn≠0.The Kuratowski MNC [3] is the function £:BΞ⟶0,∞ defined by
(1)£K=infε>0:K⊂∪i=1nSi,Si⊂Ξ,diamSi<ε,where diamS is the diameter of a set S. Using the notion of an MNC, Darbo [4] published a fixed point result, which determines the existence of a fixed point.Theorem 4 (see [4]).
LetΛ∈Ω and ϒ:Λ⟶Λ be a continuous function. If there exists k∈0,1 such that
(2)£ϒΛ0≤k£Λ0,where Λ0⊂Λ and £ is an MNC defined on Ξ. Then, ϒ has a fixed point in Λ.It generalizes the well-known Schauder fixed point result and includes the existence part of the Banach contraction principle. Many extensions and generalizations of the Darbo fixed point result can be noticed in the existing literature.The contraction on underlying mappings plays a central role for finding the fixed point. Inspired from this natural idea, the Banach contraction has been improved and extended by several researchers [5–9]. Wardowski [10] proposed a new contraction, called the ℱ-contraction and established fixed point theorems.Definition 5 (see [10]).
Letℱ:0,∞⟶ℝ be a map such that
(F1) ℱisnondecreasing
(F2) limn→+∞δn=0⇔limn→+∞ℱδn=−∞, for any sequence δn⊂0,∞
(F3) onecanfindk∈0,1suchthatlims→0+δkℱδ=0.Symbolized byF, the family of all maps ℱ:0,∞⟶ℝ which fulfill the axioms F1andF2, and by S, the family of functions τ:0,∞⟶ℝ such that limt→s+infτt>0,∀s∈0,∞.Using the specific form ofℱ, we deduce other known existing contractions. Many articles concerning ℱ-contractions and its generalizations have come into view (see, e.g., [11, 12] and the references cited therein). In particular, Jleli et al. [13] generalized ℱ-contraction such that ℱL-contraction is an established Darbo-type fixed point result.Definition 6 (see [13]).
LetΛ∈Ω. Then, the mapping ϒ:Λ⟶Λ is ℱ£-contraction if there exists τ∈S and F∈F such that
(3)τ£Λ0+ℱ£ϒΛ0≤ℱ£Λ0,where Λ0 is a subset of Λ,£Λ0,£TΛ0>0, and £ is an MNC defined in Ξ.Theorem 7 (see [13]).
LetΛ∈Ω. If the mapping ϒ:Λ⟶Λ is continuous and an ℱ£-contraction, then ϒ has a fixed point in Λ.Gillespie et al. [14] introduced the concept of expanding mapping. Gòrnicki [15] introduced the idea of ℱ-expanding mappings and presented some fixed point results. To find the fixed point of expanding mappings, one needs the following lemma.Lemma 8 (see [15]).
For surjective mapf:X⟶X, there exists a map f∗:X⟶X such that f∘f∗:X⟶X is an identity map.This manuscript has two aims. Firstly, we prove various fixed point theorems: theℱ£-weak contraction, ℱ£-weak Suzuki contraction, almost ℱ£-contraction, Hardy-Rogers-type ℱ£-contraction and Reich-type ℱ£-contraction. Secondly, we prove fixed point results under the Darbo-type ℱ£-expanding mapping. We also observe that several existing results can be concluded from our main results. Furthermore, we check the existence of a solution to the nonlinear fractional-order differential equation via integral type boundary conditions, and for its validity we construct an example.


## 2. Generalization of Darbo-Type Results viaF-Contractions

In this section, we introduce various types ofℱ£-contractions of Darbo type, and then, we prove fixed point results for mappings satisfying such contractive condition in the Banach space endowed with an MNC. We first give the definition of the ℱ£-weak contraction.Definition 9.
LetΛ∈Ω. Then, the mapping ϒ:Λ⟶Λ is a weak contraction if there exists τ∈S and F∈F such that
(4)τ£Λ1+ℱ£ϒΛ1≤ℱΔΛ1,Λ2,where Λ1 and Λ2 are subsets of Λ,£Λ1,£ϒΛ1,£ϒΛ2>0 and £ is an MNC defined in Ξ and
(5)ΔΛ1,Λ2=max£Λ1,£ϒΛ1,£ϒΛ2,12£ϒΛ1∪ϒΛ2.In the light ofℱ£-weak contraction, we present the first result.Theorem 10.
LetΛ∈Ω. If the mapping ϒ:Λ⟶Λ is continuous and an ℱ£-weak contraction, then ϒ has a fixed point in Λ.Proof.
Define a sequenceΛnn=0∞ such that
(6)Λ0=ΛandΛn=co¯ϒΛn−1,∀n∈ℕ.We need to prove thatΛn+1⊂ΛnandϒΛn⊂Λn,∀n∈ℕ. For the first inclusion, we use induction. If n=1, then by (6), we get Λ0=Λ and Λ1=co¯ϒΛ0⊂Λ0. Next, for n>1, we assume that
(7)Λn⊂Λn−1.Then(8)co¯ϒΛn⊂co¯ϒΛn−1.Using (6), we get the first inclusion
(9)Λn+1⊂Λn.With the help of inclusion (9), we obtained the second inclusion as
(10)ϒΛn⊂co¯ϒΛn=Λn+1⊂Λn.Now, we discuss two cases subject to £. If we can find an integerm≥0 such that £Λm=0, then Λm is compact. But since ϒ:Λm⟶Λm, by Theorem 2, ϒ has a fixed point in Λm⊂Λ. Instead, if we take £Λn>0,∀n∈ℕ, then we have to testify that Λ∞⊂Λn∈Ω. First, we need to show that £Λn⟶0asn⟶+∞. From inclusion (9), we write £Λn+1<£Λn, that is £Λn is a decreasing sequence and hence converges to s∈ℝ with s≥0. Now, since £Λn∈0,∞ and s∈0,∞, so by assumption on τ,limt→s+infτt>0, we can find r>0 and n0∈ℕ such that τ£Λn≥r,∀n≥n0. Using contraction condition (4) with Λ1=Λn and Λ2=Λn+1, we write
(11)τ£Λn+ℱ£Λn+1=τ£Λn+ℱ£co¯ϒΛn=τ£Λn+ℱ£ϒΛn≤ℱΔΔn,Δn+1,where
(12)ΔΛn,Λn+1=max£Λn,£ϒΛn,£ϒΛn+1,12£ϒΛn∪ϒΛn+1≤max£Λn,£Λn,£Λn+1,12£Λn∪Λn+1=max£Λn,£Λn,£Λn+1,12£Λn=£Λn.Thus, from (11), we obtain
(13)τ£Λn+ℱ£Λn+1≤ℱ£Λn.From here, we write(14)ℱ£Λn+1≤ℱ£Λn−τ£Λn≤ℱ£Λn−r.Consequently(15)ℱ£Λn≤ℱ£Λn−1−r,∀n.Thus(16)ℱ£Λn≤ℱ£Λn−1−r≤ℱ£Λn−2−2r≤ℱ£Λn−3−3r⋮≤ℱ£Λn0−n−n0r,∀n>n0.Clearly,limn→+∞ℱ£Λn=−∞ and using property F2, we can writelimn→+∞£Λn=0. Thus, by Definition 3υi,Λ∞=∩n=1+∞≠0 and ϒΛ∞⊂Λ∞ as ϒΛn⊂Λn. Also, since Λ∞⊂Λn for all n∈ℕ, so by Definition 3(ii), £Λ∞≤£Λn,∀n∈ℕ. Thus, £Λ∞=0, that is Λ∞∈ker£, and hence, Λ∞ is bounded. But Λ∞ is closed such that Λ∞ is compact. Therefore, by Theorem 2, ϒ has a fixed point in Λ∞⊂Λ.For the support of Theorem10, we construct the following example.Example 1.
LetΛ=−8,9 be a subset of a Banach space ℝ. Then, clearly, Λ∈Ω. Define ϒ:Λ⟶Λ,F:0,∞⟶ℝ,τ:0,∞⟶ℝ by ϒx=1−x,τx+1/2 and Fx=lnx, respectively. One can easily check that ϒ is continuous, τ∈S and F∈F. Also, define an MNC, £:BΞ→0,∞ by £Λ=diamΛ=supx,y∈Λx−y.
Now, letΛ1=0,1 and Λ2=2,7 be two subsets of Λ. Then, £Λ1=£ϒΛ1=1,£ϒΛ1=6, £ϒΛ1∪Λ1=7, and hence
(17)ΔΛ1,Λ2=max£Λ1,£ϒΛ1,£ϒΛ2,12£ϒΛ1∪ϒΛ2=max1,1,6,72=6.
Thus, from (4), we write
(18)τ£Λ1+ℱ£ϒΛ1=τ1+ℱ1=1.5<In6=ℱΔΛ1.Λ2.
That isϒ is an ℱ£-weak contraction. Hence, by Theorem 10, ϒ has a fixed point 1/2∈Λ.From Theorem10, we can deduce several pivotal results. We demonstrate some preferable corollaries that cover and extend several well-known results in the existing literature. The special case, if we take £ϒΛn>0, we deduce the following corollary.Corollary 11.
LetΛ∈Ω. If the mapping ϒ:Λ⟶Λ is continuous such that
(19)£ϒΛ1≤e−τ£Λ1ΔΛ1.Λ2,where Λ1 and Λ2 are subsets of Λ, then ϒ has a fixed point in Λ.If we takeℱx=lnx+x,x>0, in Theorem 10, we deduce the following corollary.Corollary 12.
LetΛ∈Ω. If the mapping ϒ:Λ⟶Λ is continuous such that
(20)£ϒΛ1≤eΔΛ1,Λ2−τ£Λ1−£Λ1ΔΛ1,Λ2,where Λ1 and Λ2 are subsets of Λ, then ϒ has a fixed point in Λ.If we takeℱx=lnx2+x,x>0, in Theorem 10, we deduce the following corollary.Corollary 14.
LetΛ∈Ω. If the mapping ϒ:Λ⟶Λis continuous such that
(21)£ϒΛ12+£ϒΛ1≤e−τ£Λ1ΔΛ1,Λ22+ΔΛ1,Λ2,where Λ1 and Λ2 are subsets of Λ, then ϒ has a fixed point in Λ.Definition 13.
LetΛ∈Ω. Then, the mapping ϒ:Λ⟶Λ is an ℱ£-weak Suzuki contraction if there exist τ∈S and F∈F such that
(22)τ£Λ1+ℱ£ϒΛ1≤ℱΔΛ1,Λ2,where Λ1 and Λ2 are subsets of Λ,ϒΛ1<2£Λ1,£Λ1,£ϒΛ1,£ϒΛ2>0, and £ is an MNC defined on Ξ and
(23)ΔΛ1,Λ2=max£Λ1,£ϒΛ1,£ϒΛ2,12£ϒΛ1∪ϒΛ2.In the light of theℱ£-weak Suzuki contraction, we provide the following result. Since the proof is very easy, we omit it.Theorem 14.
LetΛ∈Ω. If the map ϒ:Λ⟶Λ is continuous and an ℱ£-weak Suzuki contraction, then ϒ has a fixed point in Λ.Definition 15.
LetΛ∈Ω. Then, the mapping ϒ:Λ⟶Λis almost an ℱ£-contraction if there exist τ∈S and F∈F such that
(24)£ϒN>0⇒τ£N+ℱ£ϒN≤ℱ£N+L£ϒN.In the light of an almostℱ-contraction, we present the following result.Theorem 16.
LetΛ∈Ω. If the functionϒ:Λ⟶Λis continuous and an almost ℱ£-contraction, then ϒ has a fixed point in Λ.Proof.
Construct a sequenceΛnn=0∞ such that
(25)Λ0=ΛandΛn=co¯ϒΛn−1,foralln∈ℕ.
Then,Λn+1⊂Λn and ϒΛn⊂Λn,∀n∈N.
If we take an integerm≥0 such that £Λm=0, then Λm is a compact, and by Theorem 2, we can find a fixed point of ϒ in Λm⊂Λ. Let us take £Λn>0,∀n∈ℕ. Then, {£ (Λn)} is a decreasing sequence and hence converges to s∈ℝ with s≥0. Now, since £Λn∈0,∞ and s∈0,∞, so by assumption on τ,limt→s+infτt>0, we can find r>0 and n0∈ℕ such that τ£Λn≥r,∀n≥n0. Now, assume that
(26)£ϒΛn>0.
Then, settingN=Λn in (24), we have
(27)τ£Λn+ℱ£Λn+1=τ£Λn+ℱ£co¯ϒΛn+1=τ£Λn+ℱ£ϒΛn≤ℱ£Λn+L£ϒΛn.
From here, we write(28)ℱ£Λn+1≤ℱ£Λn+L£ϒΛn−τ£Λn<ℱ£Λn+L£Λn−r,foralln.
Consequently(29)ℱ£Λn<ℱ1+L£Λn−1−r,foralln.
Thus(30)ℱ£Λn≤ℱ1+L£Λn−1−r≤ℱ1+L£Λn−2−2r≤ℱ1+L£Λn−3−3r≤ℱ1+L£Λn0−n−n0r,∀n>n0.
Clearlylimn→+∞F£Λn=−∞, and using the property F2, we can write limn→+∞F£Λn=0.
Following the same steps as in Theorem10, we can easily show that ϒ has a fixed point in Λ∞⊂Λ.Definition 17.
LetΛ∈Ω. Then, the mapping ϒ:Λ⟶Λ is the Hardy-Rogers F£-contraction, if we can find τ∈S,F∈F, and A0⊂Ξ such that
(31)£ϒA>0⇒τ£A+ℱ£ϒA≤ℱλ1£A+λ2£A0+λ3£ϒA+λ4£ϒA0.for all A⊂Λ, where λ1+λ2+λ3+λ4≤1 with λ1,λ2,λ3,λ4≥0.Remark 18.
(1)
Ifλ2=0, then (31) is a Reich-type F£-contraction(2)
Ifλ2=λ3=λ4=0 andλ1=1, then the contraction (31) becomes a Darbo-type F£-contraction.Theorem 19.
LetΛ∈Ω. If the function ϒ:Λ⟶Λ is continuous and a Hardy-Rogers-type F£v-contraction, then ϒ has a fixed point in Λ.Proof.
Construct a sequence such that(32)Λ0=ΛandΛn=co¯ϒΛn−1,foralln∈ℕ.
Then,Λn+1⊂Λn and ϒΛn⊂Λn∀n∈ℕ.
Now, we discuss two cases subject to £. If we considerm as a nonnegative integer with £Λm=0, then Λm is a compact. But ϒ:Λm⟶Λm, so by Theorem 2, ϒ has a fixed point in Λm⊂Λ. Instead, let us take £Λn>0,∀n∈ℕ. From (9), we write £Λn+1<£Λn, that is £Λn is a decreasing sequence and hence converges s∈ℝ with s≥0. Now since £Λn∈0,∞ and s∈0,∞, so by assumption on τ,limt→s+infτ£Λn>0, we can find r>0 and n0∈ℕ such that τ£Λn≥r∀n≥n0. Using (31) with A=Λn and A0=Λn+1, we have
(33)τ£Λn+ℱ£Λn+1=τ£Λn+ℱ£c¯o¯ϒΛn=τ£Λn+ℱ£ϒΛn+1≤ℱλ1£Λn+λ2£Λn+1+λ3££ϒΛn+λ4£ϒΛn+1<ℱλ1£Λn+λ2£Λn+1+λ3£,but λ1+λ2+λ3+λ4≤1 and ℱ is nondecreasing; thus, we have
(34)T£Λn+ℱ£Λn+1≤ℱ£Λn.
The rest of the proof is analogous to that of Theorem10.We provide the following result, the proof is easy, so we omit it.Theorem 20.
LetΛ∈Ω. If the function ϒ:Λ⟶Λ is continuous and a Reich F£-contraction then ϒ has a fixed point in Λ.


## 3. Darbo-Type Result viaF-Expansion

In this section, we introduce the Darbo-typeF-expanding mapping and establish some fixed point results.Definition 21.
LetΛ∈Ω. Then the mapping ϒ:Λ⟶Λ is F£-expanding if there exist τ∈S and F∈F such that
(35)£Λ0>0⇒ℱ£ϒΛ0≥ℱ£Λ0+τ£Λ0,where Λ0⊂Λ.In the light ofF£-expanding mapping, we present the following result.Theorem 22.
LetΛ∈Ω. If the mapping ϒ:Λ⟶Λ is continuous, surjective, and F£-expanding, then ϒ has a fixed point in Λ.Proof.
Sinceϒ:Λ⟶Λ is surjective, so by Lemma 8, we can find a function ϒ∗:Λ⟶Λ such that ϒ∘ϒ∗ is the identity function on Λ. Let Λ1 and Λ2 be any subsets of Λ such that Λ2=ϒ∗Λ1. Assume that £Λ2>0; then on using (35), we can write
(36)ℱ£ϒΛ2≥ℱ£Λ2+τ£Λ2.
SinceϒΛ2=ϒϒ∗Λ1=ϒ∘ϒ∗Λ1=Λ1, then (36) becomes
(37)ℱ£Λ1≥ℱ£ϒ∗Λ1+τ£ϒ∗Λ1.
Now, ifu is fixed point of ϒ∗, then ϒu=ϒϒ∗u=u. Thus, to show that ϒ has a fixed point, it is sufficient to show that ϒ∗ has a fixed point. To do this, construct a sequence such that
(38)Λ0=ΛandΛn=co¯ϒ∗Λn−1,foralln∈ℕ.
Then, we can easily show thatΛn+1⊂Λn and ϒ∗Λn⊂Λn.
Next, if we take an integerm≥0 with £Λm=0, then Λmis compact. So, by Theorem 2, ϒ∗ has a fixed point in Λm⊂Λ. Now, let us take £Λn>0, for all n∈N. We have to prove that Λ∞⊂Λn is a nonempty, bounded, closed, and convex subset of Ξ. For this, since the sequence £Λn is decreasing, it converges to s≥0. Now, assume that £Λn+1>0. Then
(39)0<£Λn+1=£co¯ϒ∗Λn=£ϒ∗Λn,that is, £ϒ∗Λn∈0,∞ and s∈0,∞, so by assumption on τ, limt→s+inf τ£ϒ∗Λn>0, we can find r>0 and n0∈N such that τ£ϒ∗Λn≥r for all n≥n0. Using (37) with Λ=Λn, we write
(40)τ£ϒ∗Λn+ℱ£Λn+1=τ£ϒ∗Λn+ℱ£co¯ϒ∗Λn=τ£ϒ∗Λn+ℱ£ϒ∗Λn≤ℱ£Λn.
From here, we write(41)ℱ£Λn+1≤ℱ£Λn−τ£ϒ∗Λn≤ℱ£Λn−r.
Consequently,(42)ℱ£Λn≤ℱ£Λn−1−r.
By routine calculation, one can easily obtain(43)ℱ£Λn≤ℱ£Λn0−n−n0r,∀n>n0.
Clearly,limn→+∞ℱ£Λn=−∞ and using the property F2, we can write limn→+∞£Λn=0. Thus, by Definition 3(vi), Λ∞=∩n=1+∞ is nonempty and ϒ∗Λ∞⊂Λ∞ as ϒ∗Λn⊂Λn. Also, since Λ∞⊂Λn, for all n∈ℕ, so by Definition 3(ii), £Λ∞≤£Λn,foralln∈ℕ. Thus, £Λ∞=0, and hence, Λ∞∈ker£, that is Λ∞ is bounded. But Λ∞is closed such that Λ∞ is compact. Therefore, by Theorem 2, ϒ∗ has a fixed point in Λ∞⊂Λ. Consequently, ϒ has a fixed point in Λ∞⊂Λ.From Theorem22, we can deduce several pivotal results. We demonstrate some preferable corollaries that cover and extend several known theorems in the literature. The special case, if we take Fx=lnx,x>0, in Theorem 22, we deduce the following corollary.Corollary 23.
LetΛ∈Ωand ϒ:Λ⟶Λ be a surjective and continuous mapping such that
(44)£ϒΛ≥£Λeτ£Λ,forallΛ⊂Λ.lim
Thenϒ has a fixed point in Λ.If we takeFx=lnx+x,x>0, in Theorem 22, we get the following corollary.Corollary 24.
LetΛ∈Ω andϒ:Λ⟶Λ be a surjective and continuous mapping such that
(45)£ϒΛ≥£Λe2τ£Λ−£ϒΛ,forallΛ⊂Λ.
Then,ϒ has a fixed point in Λ.If we takeℱx=lnx2+x,x>0, in Theorem 22, we deduce the following corollary.Corollary 25.
LetΛ∈Ω and ϒ:Λ⟶Λbe a surjective and continuous mapping such that
(46)£ϒΛ£ϒΛ+1≥£Λ£Λ+1eτ£Λ,forallΛ⊂Λ.
Thenϒ has a fixed point in the set Λ.If we takeℱx=arctan−1/x with x>0, in Theorem 22, we deduce the following corollary.Corollary 26.
LetΛ∈Ω and ϒ:Λ⟶Λ be a surjective and continuous mapping such that
(47)£ϒΛ≥£Λ+tanτ£Λ1−tanτ£Λ.£Λ,forallΛ⊂Λ.
Then,ϒ has a fixed point in Λ.


## 4. Applications

This section deals with some practicing of our fixed point results. LetE,. be a Banach space having the zero element 0. Let Ba,r be the closed ball with center a and radius r and Br be the ball B0,r. Our aim is to illustrate sufficient conditions for the existence of a solution of a nonlinear fractional-order differential equation:
(48)Dc℘ut=ψt,ut,t∈0,Δ,Δ≥1,under the integral type boundary conditions:
(49)p1u0+q1uΔ=I℘ϕ1Δ,uΔ,(50)p2u′0+q2u′Δ=I℘ϕ2Δ,uΔ,where ℘∈1,2, cD is the Caputo fractional derivative, ψ,φ1,φ2:0,△×ℝ⟶ℝ are continuous functions, and pk and qk (k=1,2) are positive real numbers.Lemma 27 (see [16]).
Forpj∈ℝ,j=0,1,2,⋯,r−1, we have
(51)I℘Dc℘ht=ht+p0+p1t+p2t2+..⋯+pr−1tr−1.To proceed further, we convert the nonlinear fractional-order differential equation (48) to an integral equation. For this, we prove the following lemma.Lemma 28.
A solution of the fractional-order boundary value problem (48) is
(52)ut=I℘ψt,ut−q1p1+q1I℘ψΔ,uΔ+q2p2+q2q1Δp1+q1−tI℘−1ψΔ,uΔ+1p1+q1I℘ψΔ,uΔ+1p2+q2t−q1Δp1+q1I℘φ2Δ,uΔ.Proof.
First of all, apply the Riemma-Liouville fractional integrable operatorI℘ of order ℘ to equation (48), and using Lemma 27, we can easily deduce that
(53)ut=C0+C1+I℘ψt,ut,and by differentiating (53), we get
(54)u′t=C0+C1t+I℘−1ψt,ut.
Butu0=C0,u′0=C1, uΔ=C0+C1Δ+I℘ψΔ,uΔ, and u′Δ=C1+I℘−1ψΔ,uΔ.
Substituting the values ofu0 and u∆ in (49), we get
(55)C0=−q1Δp1+q1C1−q1p1+q1I℘ψΔ,uΔ+1p1+q1I℘φ1Δ,uΔ.
Similarly, substituting the values ofu′0 and u′Δ in (50), we get
(56)C1=−q2p2+q2I℘−1ψΔ,uΔ+1p2+q2I℘ϕ2Δ,uΔ.
Putting the value ofC1 in (55), we deduce
(57)C0=−q1q2Δp1+q1p2+q2I℘−1ψΔ,uΔ−q1Δp1+q1p2+q2I℘ϕ2Δ,uΔ−q1p1+q1I℘ψΔ,uΔ+1p1+q1I℘ϕ2Δ,uΔ.
Thus, by switching the values ofC0 and C1 in (53) and by routine calculations, we get equation (52).Notice that Lemma (49) indicates that the solution of differential equation (48) is equivalent to the solution of integral equation (52). Now, we are in a position to present the existence result.Theorem 29.
Letu,v∈Br and ψ,φ1,φ2:0,∆×ℝ⟶ℝ are continuous mapping such that
(58)ψt,ut−ψt,ut≤e−2r+1/k3Δ℘℘Δ+2u−v,(59)ϕ1t,ut−ϕ1t,ut≤e−2r+1/k3Δ℘u−v,(60)ϕ2t,ut−ϕ2t,ut≤e−2r+1/k3Δ℘+1u−v,(61)2+1r℘M1+M2+M3Δ℘+1≤r,where ℘∈1,2,k>1,Δ≥1,sups∈0,Δψs,us=M1<∞,sups∈0,Δϕ1s,us=M2<∞, and sups∈0,Δϕ1s,us=M3<∞. Then, there exists a solution of the fractional-order integral equation (52) in Br. Accordingly, there exists a solution of the nonlinear fractional-order differential equation (48) in Br.Proof.
LetBr=u∈C0,T,ℝ:u≤r. Then, Br is a nonempty, closed, bounded, and convex subset of E,.. Define τ,F:0,∞⟶ℝbyτx=x+1/k with k >1 and Fx=lnx and the operator ϒ:Br→Br by
(62)ϒut=I℘ψt,ut−q1p1+q1I℘ψΔ,uΔ−q2p2+q2q1Δp1+q1−tI℘−1ψΔ,uΔ=1p1+q1I℘ϕ1Δ,uΔ+1p2+q2t−q1Δp1+q1I℘ϕ2Δ,uΔ.
First, we will show thatϒ:Br⟶Br is well defined. Let u∈Br, for some r. Then, for all x∈0,4, we have
(63)ϒut≤1Γ℘∫0tt−s℘−1ψs,usds+q1p1+q11Γ℘∫0ΔΔ−s℘−1ψs,usds+q2p2+q2q1Δp1+q1−t1Γ℘−1∫0ΔΔ−s℘−2ψs,usds+1p1+q11Γ℘∫0ΔΔ−s℘−1φ1s,usds+1p2+q2t−q1Δp1+q11Γ℘∫0ΔΔ−s℘℘−1φ1s,usds≤ψs,usΓ℘∫0tt−s℘−1ds+ψs,usΓ℘∫0ΔΔ−s℘−1ds+ψs,usΔΓ℘−1∫0ΔΔ−s℘−2ds+φ1s,usΓ℘∫0ΔΔ−s℘−1ds+φ2s,usΔΓ℘∫0ΔΔ−s℘−1ds≤M1Γ℘t℘℘+M1Γ℘Δ℘℘+M1ΔΓ℘−1Δ℘−1℘−1+M2Γ℘Δ℘℘+M3ΔΓ℘Δ℘℘≤2+1Γ℘M1+M2+M3Δ℘+1≤r.
That isϒu≤r for all u∈Br, which implies that ϒu∈Br, and hence, ϒ:Br⟶Br is well defined.
Now, we have to show thatϒ:Br⟶Br is continuous. For this, consider
(64)ϒut−ϒvt≤1Γ℘∫0tt−s℘−1ψs,us−ψs,vsds+q1p1+q1Γ℘∫0ΔΔ−s℘−1ψs,us−ψs,vsds+q2p2+q2q1p1+q1−t∫0ΔΔ−s℘−2ψs,us−ψs,vsΓ℘−1ds+1p1+q1Γ℘∫0ΔΔ−s℘−1ϕ1s,us−ϕ1s,vsds+1p2+q2t−q1Δp1+q1∫0ΔΔ−s℘−2ϕ2s,us−ϕ2s,vsΓ℘ds≤ψs,us−ψs,vsΔ℘Γ℘+1+ψs,us−ψs,vsΔ℘Γ℘+1+ψs,us−ψs,vsΔ℘+1Γ℘+ϕ1s,us−ϕ1s,vsΔ℘Γ℘+1+φ2s,us−ϕ2s,vsΔ℘+1Γ℘+1≤Δ℘+1φ2s,us−ϕ2s,vs.
Sinceψ,φ1, and φ2 are continuous, so for ∈1=1/Δ℘℘Δ+2,∈2=1/Δ℘,∈3=1/Δ℘+1>0, there exist δ1,δ2,δ3>0 such that
(65)ψs,us−ψs,vs<∈13,wheneveru−v<δ1,ϕ1s,us−ϕ1s,vs<∈23,wheneveru−v<δ2,ϕ2s,us−ϕ2s,vs<∈33,wheneveru−v<δ3.
If∈=min∈1,∈2,∈3 and δ=minδ1,δ2,δ3, then we conclude that ϒvtϒυt<∈, whenever u−v<δ.
Hence,ϒ:Br⟶Br is continuous. Next, we have to show that ϒ:Br⟶Br is an F£-weak contraction. Let Λ1 and Λ2 be any two subsets of Br and u,v∈Λ1. Then, by conditions (58), (59), and (60), we have
(66)diamϒΛ1=ϒut−ϒvt≤Δ℘℘Δ+2ψs,us−ψs,vs+Δ℘φ1s,us−φ1s,vs+Δ℘+1φ2s,us−φ2s,vs≤e−2r+1/ku−v=e−diamΛ1+1kdiamΛ1≤e−diamΛ1+1/kmaxdiamΛ1,diamϒΛ1,diamϒΛ2,diamϒΛ1∪ϒΛ2.
From here, we write(67)£ϒΛ1=e−£Λ1+1/kmax£Λ1,£ϒΛ1,£ϒΛ2,12£ϒΛ1∪ϒΛ2.
Consequently,(68)In£ϒΛ1≤−£Λ1+1k+InΔΛ1,Λ2,which implies that
(69)τ£Λ1+ℱ£ϒΛ1≤ℱΔΛ1,Λ2.
That isϒ:Br⟶Br is an F£-weak contraction. Thus, by Theorem 10, ϒ has a fixed point in Br. Consequently, equation (52) has a solution in Br.To illustrate the existence result (Theorem29), we present an example.Example 2.
Consider the nonlinear fractional-order differential equation(70)Dc3/4ut=e−t−2sinutt+73,t∈0,3,2u0+3u3=∫033−s1/2cosusΓ3/2s+44ds,13u′0+12u′3=∫033−s1/2e−us+6Γ3/2ds,where cD is the Caputo fractional-order derivative and ψ,φ1,φ2:0,3×ℝ⟶ℝ are continuous functions defined as
(71)ψt,ut=e−t−2sinutt+73,ϕ1t,ut=cosutt+44,andϕ2t,ute−ut+6.Now, we have to verify the conditions of Theorem29. Let Λ1⊂B1 and u,v∈Λ1; then, fork=10,t∈0,3,Δ=3, and ℘=3/2, we first show condition (58) of Theorem 29. To do this, we have
(72)ψt,ut−ψt,vt=e−t−2t+73sinut−sinvt≤e−2−1/103×33/23/2×3+2u−v=e−2r+1/k3Δ℘℘Δ+2u−v.Next, to show condition (59) of Theorem 29, we have
(73)ϕ1t,ut−ϕ2t,vt=cosutt+44−cosvtt+44≤1t+44u−v≤e−2−1/103×33/2u−v=e−2r+1/k3Δ℘u−v.Also, to show condition (60) of Theorem 29, we have
(74)ϕ2t,ut−ϕ2t,vt=e−ut+6−e−vt+6≤e−6u‐v≤e−2−1/103×35/2u‐v=e−2r+1/k3Δ℘+1u‐v.Finally, we have to verify condition (75) of Theorem 29. For this, since M1<0.0003945,M2<0.003906,andM3≈0.0024787, we get
(75)2+1Γ℘M1+M2+M3Δ℘+1<0.117977<1=r.Thus, all the conditions of Theorem29 are satisfied. Hence the nonlinear fractional-order differential equation (70) has a solution in B1.


## 5. Conclusion

Through measure of noncampactness, various newF-contraction and F-expanding mappings have been presented. In the Banach spaces, fixed point results were established, from which several existing results can be extracted. For the accuracy of our results, we have checked the existence of a solution to the nonlinear fractional-order differential equation under the integral-type boundary conditions with an example.



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