# On Extended Branciarib-Distance Spaces and Applications to Fractional Differential Equations

Authors: Reena Jain; Hemant Kumar Nashine; Reny George; Zoran D. Mitrović
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this work, we define newα−λ-rational contractive conditions and establish fixed-points results based on aforesaid contractive conditions for a mapping in extended Branciari b-distance spaces. We furnish two examples to justify the work. Further, we discuss results on weak well-posed property, weak limit shadowing property, and generalized w-Ulam-Hyers stability in the underlying space. Finally, as an application of our main result, we obtain sufficient conditions for the existence of solutions of a nonlinear fractional differential equation with integral boundary conditions.


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## Body

## 1. Introduction and Preliminaries

The distance notion in the metric fixed-point theory is introduced and generalized in different ways by many authors [1–5]. Bakhtin [6] defined the notion of b-metric space which is further used by Czerwik in [7, 8]. In [9], Branciari extended the metric space and introduced the notion of the Branciari distance by changing the property of triangle inequality with quadrilateral one.Definition 1 [9].
LetΞ≠∅ be a set and let b:Ξ2⟶ℝ+ such that, for all ϑ,ν∈Ξ and all u,v∈Ξ\ϑ,ν(bd1)
bϑ,ν=0 if and only if ϑ=ν (self-distance/indistancy)(bd2)
bϑ,ν=bν,ϑ (symmetry)(bd3)
bϑ,ν≤bϑ,u+bu,v+bv,ν (quadrilateral inequality).The symbolΞ,b the denotes Branciari distance space and abbreviated as “BDS.”In [10], Kamran et al. introduced the notion of extended b-metric space as a generalization of b-metric space and proved the following result.Definition 2 [10].
LetΞ≠∅ be a set and w:Ξ2⟶ℝ+\0,1. We say that a function ρe:Ξ2⟶ℝ+ is an extended b-metric (ρe-metric, in short) if it satisfies
(eb1)
ρeϑ,ν=0 if and only if ϑ=ν(eb2)
ρeϑ,ν=ρeν,ϑ (symmetry)(eb3)
ρeϑ,ν≤wϑ,νρeϑ,υ+ρeυ,ν,
for allϑ,ν,υ∈Ξ. The symbol Ξ,ρe denotes a ρe-metric space.Theorem 3 [10].
LetΞ,ρe be a complete extended b-metric space such that ρe is a continuous functional. Let I:Ξ⟶Ξ satisfy ρeIϑ,Iν≤kρeϑ,ν for all ϑ,ν∈Ξ where k∈0,1 such that for each ϑ0∈Ξ, limn,m⟶∞wϑn,ϑm<1/k, here ϑn=Inϑ0, n=1,2,⋯. Then I has precisely one fixed-point ϑ. Moreover, for each ν∈Ξ, Inν⟶ϑ.In [3], Mitrović et al. extended Theorem 3 and proved the following:Theorem 4 [3].
LetΞ,ρe be a complete extended b-metric space such that ρe is a continuous functional. Let I:Ξ⟶Ξ satisfy
(1)ρeIϑ,Iν≤aρeϑ,ν+bρeϑ,Iϑ+cρeν,Iν,for all ϑ,ν∈Ξ where a,b,c are nonnegative real numbers with a+b+c<1. Then, I has a unique fixed-point ϑ. Moreover, there exists a sequence ϑnn∈ℕ in Ξ which converges to ϑ such that ϑn+1=Iϑn for every n∈ℕ.In [11], Abdeljawad et al. defined the notion of extended Branciari b-distance (EBbDS, in short) by combining the extended b-metric and Branciari distance.Definition 5 [11].
LetΞ≠∅ be a set and w:Ξ2⟶ℝ+\0,1. We say that a function eb:Ξ2⟶ℝ+ is an extended Branciari b-metric (eb-metric, in short) if it satisfies
(ebb1)
ebϑ,υ=0 if and only if ϑ=υ(ebb2)
ebϑ,υ=ebυ,ϑ(ebb3)
ebϑ,υ≤wϑ,υebϑ,ν+ebν,ρ+ebρ,υ,
for allϑ,υ∈Ξ, all distinct ν,ρ∈Ξ\ϑ,υ. The symbol Ξ,eb denotes the extended Branciari b-distance space. For wϑ,υ=1, Ξ,eb will be called a Branciari b-distance space (BbDS, in short).Example 1.
LetΞ=C0,1,ℝ and define eb:Ξ2⟶ℝ+ by ebP,Q=∫01Pt−Qt2dt with wP,Q=Pt+Qt+2. Note that ebP,Q≥0 for all P,Q∈Ξ, and ebP,Q=0 if and only if P=Q. Also, ebP,Q=ebQ,P. Hence, it is clear that Ξ,eb is an EBbDS, but it is neither an BDS nor metric space.Definition 6 [11].
LetΞ≠∅ be a set endowed with extended Branciari b-distance eb.
(a)
A sequenceϑn in Ξ converges to ϑ if for every ε>0 there exists N=Nε∈ℕ such that ebϑn,ϑ<ε for all n≥N. For this particular case, we write limn⟶∞ϑn=ϑ(b)
A sequenceϑn in Ξ is called Cauchy if for every ε>0 there exists N=Nε∈ℕ such that ebϑm,ϑn<ε for all m,n≥N(c)
Aneb-metric space Ξ,eb is complete if every Cauchy sequence in Ξ is convergent.
On the other hand, in [12], Samet et al. define the notion of α-admissible mappings which is further extended by Sintunavarat [13] and named as weakly α-admissible mapping.Definition 7.
For aΞ≠∅ set, let α:Ξ×Ξ⟶0,∞ and I:Ξ⟶Ξ be two mappings. Then I is called
(1)
[12] α-admissible if
(2)ϑ,ν∈Ξwithαx,ν≥1⇒αIϑ,Iν≥1(2)
[13] weakly α-admissible if
(3)ϑ∈Ξwithαϑ,Iϑ≥1⇒αIϑ,IIϑ≥1.For aΞ≠∅ set and a mapping α:Ξ×Ξ⟶0,∞, we use
(4)AΞ,α≔Thesetofallα−admissiblemappingsonΞ,WAΞ,α≔Thesetofallweaklyα−admissiblemappingsonΞ.It is noted that(5)AΞ,α⊂WAΞ,α.The notion of well-posedness of a fixed-point problem (fpp) has evoked much interest of several mathematicians, for example, Popa [14, 15] and others. In the paper [16], authors defined a weak well-posed (wwp) property in BbDS and in the papers [17, 18]; the authors have discussed limit shadowing property of fixed-point problems.The aim of this work is to introduceα−λ-rational contraction in an EBbDS and prove the existence of fixed points of such rational contraction in an EBbDS. We also discuss the weak well-posedness, limit shadowing property, and generalized weak-Ulam-Hyers stability of fixed-point problems in a EBbDS. As an application of our main result, we obtain sufficient conditions for the existence of solutions of a nonlinear fractional differential equation with integral boundary conditions. By doing these work, we generalize Theorems 3 and 4 in the sense that we use a more general contractive condition which depends on the variable (Lipschitz constants), function wx,y on the left-side of contractive condition, and proved results on the weakly α-admissible mapping on more general space structures. It is justifies the usefulness of these terms through illustrations, and the results are real generalization as the considered distances are neither metric space not Branciari distance space.


## 2. Main Results

### 2.1.α−λ-Rational Contractive Mapping and Fixed Points

We start with introducing the notion ofα−λ-rational contraction in a EBbDS as follows.Definition 8.
LetΞ,eb be an EBbDS and α:Ξ2⟶ℝ+ and λ:Ξ⟶0,1. A mapping I:Ξ⟶Ξ is said to be an α−λ-rational contraction, if there exist
(6)ϑ,ν∈Ξ,with
(7)αϑ,ν≥1,ebϑ,ν>0,ebIϑ,Iν>0,which implies
(8)wϑ,νebIϑ,Iν≤λumaxebϑ,ν,ebϑ,Iϑ,ebν,Iν,ebν,Iν1+ebϑ,Iϑwϑ,ν1+ebϑ,ν,ebϑ,Iϑ.ebν,Iνwϑ,ν.ebϑ,ν.We denote byΛΞ,α the collection of all α−λ-rational contractive mappings on Ξ,eb.The set of all fixed points of a self-mappingI on a set Ξ≠∅ will be denoted by FixI.We are now in a position to state and prove the result.Theorem 9.
LetΞ,eb be a complete EBbDS and α:Ξ×Ξ⟶0,∞. Let I:Ξ⟶Ξ be a mapping satisfying the following:
(A1)
I∈ΛΞ,α∩WAΞ,α(A2)
There existsu0∈Ξ such that αu0,Iu0≥1(A3)
I is continuous.Then,FixI≠∅. Furthermore, for any u0∈Ξ, the sequence un satisfying un=Iun−1 is convergent.Proof.
By virtue of condition (A2), there existsu0∈Ξ such that αu0,Iu0≥1. Define the sequence un∈Ξ by un+1=Iun. If there exists n0∈ℕ such that un0=un0+1, then un0∈FixI, and we are complete. Therefore, we assume that un≠un+1 for all n∈ℕ.
It follows that(9)ebun,un+1>0,∀n∈ℕ.
It follows fromI∈WAΞ,α and αu0,Iu0≥1 that
(10)αu1,u2=αIu0,IIu0≥1.
Continuing this process, we obtain(11)αun,un+1≥1∀n∈ℕ.Step 1.
First, we prove that(12)limn⟶∞ebun,un+1=0.
It follows fromI∈ΛΞ,α that
(13)wun−1,unebIun−1,Iun≤λun−1maxebun−1,un,ebun−1,Iun−1,ebun,Iun,ebun,Iun1+ebun−1,Iun−1wun−1,un1+ebun−1,un,ebun−1,Iun−1.ebun,Iunwun−1,un.ebun−1,un=λun−1maxebun−1,un,ebun−1,un,ebun,un+1,ebun,un+11+ebun−1,unwun−1,un1+ebun−1,un,ebun−1,un.ebun,un+1wun−1,un.ebun−1,un,(14)≤λun−1maxebun−1,un,ebun−1,un,ebun,un+1,ebun,un+1,ebun,un+1≤λun−1maxebun−1,un,ebun,un+1.
Ifebun−1,un≤ebun,un+1 for some n∈ℕ, then from (13), we have wun,un+1ebun−1,un≤λun−1ebun,un+1, which is a contradiction since w≥1 and λ<1. Thus, ebun,un+1≤ebun−1,un for all n∈ℕ, and the sequence ebun,un+1 is a decreasing sequence of real numbers. Therefore, there exists ζ such that
(15)limn⟶∞ebun,un+1=ζ.
Again applying the limit in (13), we get
(16)limn⟶∞wun−1,unζ≤limn⟶∞λun−1ζ,which leads to ζ=0 as w≥1. Thus, we get
(17)limn⟶∞ebun,un+1=0.Step 2.
At this step, we will prove thatun is a Cauchy sequence, that is, for m>n, we prove
(18)limn,m⟶∞ebun,um=0.
Using (ebb3), we have(19)ebun,um≤wun,umebun,un+1+ebun+1,un+2+ebun+2,un+m≤wun,umebun,un+1+wun,umebun,un+1+wun,umebun+2,um≤wun,umebun,un+1+wun,umebun,un+1+wun,umwun+2,umebun+2,un+3+ebun+3,un+4+ebun+4,um≤wun,umebun,un+1+wun,umebun,un+1+wun,umwun+2,umebun,un+1+wun,umwun+2,umebun,un+1+wun,umwun+2,umebun+4,um⋮≤wun,umebun,un+1+wun,umebun,un+1+wun,umwun+2,umebun,un+1+wun,umwun+2,umebun,un+1+⋯+wun,umwun+2,um⋯wum−2,umebun,un+1+wun,umwun+2,um⋯wum−2,umebun,un+1≤wun,umebun,un+1+wun,umwun+1,umebun,un+1+wun,umwun+1,umwun+2,umebun,un+1+wun,umwun+1,umwun+2,umwun+3,umebun,un+1+⋯+wun,umwun+1,umwun+2,um⋯wum−2,umebun,un+1+wun,umwun+1,umwun+2,um⋯wum−2,umwum−1,umebun,un+1.
Applyingn,m⟶∞ and using (12), we get
(20)limn,m⟶∞ebun,um=0.
Hence,un is a Cauchy sequence. Since Ξ,eb is a complete EBbDS, then there exists a point u∗∈Ξ such that un⟶u∗ as n⟶+∞, that is
(21)limn⟶+∞ebun,u∗=0.
Next, we prove thatu∗∈FixI. Indeed, we write
(22)ebu∗,Iu∗≤wu∗,Iu∗ebu∗,un+ebun,un+1+ebun+1,Iu∗.
SinceI is continuous, on letting n⟶+∞, we obtain ebu∗,Iu∗=0, that is, Iu∗=u∗, and hence, u∗ is a fixed point of I.
To prove the uniqueness of fixed-pointu∗, we impose an additional requirement.
(A4)
For every pairu∗ and v∗ of fixed points of I, αu∗,v∗≥1.Theorem 10.
In addition of condition (A4) in Theorem9, FixI is a singleton set.Proof.
Following Theorem9, u∗∈FixI. To prove FixI is a singleton set, assume that there exist u∗,v∗∈FixI with u∗≠v∗, and by (A4), we have αu∗,v∗≥1. It follows from I∈ΛΞ,α that
(23)wu∗,v∗ebIu∗,Iv∗≤λu∗maxebu∗,v∗,ebu∗,Iu∗,ebv∗,Iv∗,ebv∗,Iv∗1+ebu∗,Iv∗wu∗,v∗1+ebu∗,v∗,ebu∗,Iu∗.ebv∗,Iv∗wu∗,v∗.ebu∗,v∗≤λu∗maxebu∗,v∗,0,0,0,0,which implies that
(24)wu∗,v∗ebu∗,v∗≤λu∗ebu∗,v∗,
a contradiction, and hence,u∗=v∗.


### 2.2. Illustrations

Example 2.
LetΞ=0.2,0.25,0.3,0.5,1. Define eb:Ξ2⟶ℝ+ so that ebζ,ξ=ebξ,ζ for all ζ,ξ∈Ξ, and
(25)eb0.5,0.3=0.07,eb0.5,0.25=0.015,eb0.25,0.2=0.02,eb0.3,0.25=0.02,eb0.3,0.2=0.02,
ebζ,ξ=ζ−ξ2, otherwise. Then Ξ,eb is a EBbDS with wζ,ξ=ζ+ξ+2 but neither a BDS Ξ,b nor a metric space Ξ,d. For instance
(26)eb0.5,0.3=0.07≤0.035=eb0.5,0.25+eb0.25,0.3,eb0.5,0.3=0.07≤0.055=eb0.5,0.25+eb0.25,0.2+eb0.2,0.3,but
(27)eb0.5,0.3=0.07≤0.154=wζ,νeb0.5,0.25+eb0.25,0.2+eb0.2,0.3.
Consider the self-mappingI on Ξ, α:Ξ2⟶ℝ+ and λ:Ξ⟶0,1(28)I:0.20.250.30.510.30.50.20.50.25,αζ,ν=1,ζ,ν∈0.2,1∪0.5,1∪1,0.2∪1,0.50,otherwise,and λζ=2ζ/3 for all ζ∈Ξ.It is easy to see thatI∈WAΞ,α. We will check that I satisfies (8) for ζ≠ξ with Iζ≠Iξ and αζ,ξ>1. We demonstrate by three nontrivial possible cases. Here, wζ,ξ∈2.4,6.Case 1.
ζ=0.5, ξ=1 (or vice versa if ζ, ξ change places). Then, ebIζ,Iξ=0.015, wζ,ξ=3.5, λζ=0.333 and
(29)Mwζ,ξ=max0.25,0,0.56250.56251+03.51+0.25,00.56253.50.25=0.5625.
Therefore, (8) implies that 0.0525<0.1873, and (8) holds true.Case 2.
ζ=0.2, ξ=1 (or vice versa if ζ, ξ change places). Then, ebIζ,Iξ=0.02, wζ,ξ=3.2, λζ=0.1333 and
(30)Mwζ,ξ=max0.64,0.02,0.5625,0.56251+0.023.21+0.64,0.20.56253.20.64=0.64,and it is easily seen that (8) is fulfilled.Thus, all the conditions are fulfilled, andI has a unique fixed point (which is ζ∗=0.5).Note that in this example the use of weaklyα-admissibility and λζ was crucial because, e.g., if we take ζ=0.2, ξ=0.5, we get ebIζ,Iξ=0.07, wζ,ξ=2.7 and
(31)Mwζ,ξ=max0.09,0.02,0,01+0.022.71+0.09,0.0202.70.09=0.09,and no contractive condition for any λζ<1 can be chosen which would holds for these points.Example 3.
ConsiderΞ=0,1 and define eb:Ξ2⟶ℝ+ by ebζ,ξ=ζ−ξ2. Then, Ξ,eb is a EBbDS with wζ,ξ=ζ+ξ+2.5 but neither a BDS Ξ,b not a metric space Ξ,d. For instance
(32)eb0,1=1≤0.5=eb0,0.5+eb0.5,1,eb0,1=1≤0.4902=eb0,0.5+eb0.5,0.99+eb0.99,1,but
(33)ebζ,ξ=ζ−ξ2=ζ−μ+μ−υ+υ−ξ2≤ζ−μ2+μ−υ2+υ−ξ2+2ζ−μμ−υ+2μ−υυ−ξ+2υ−ξζ−μ≤ζ+ξ+52ζ−μ2+μ−υ2+υ−ξ2=wζ,ξebζ,μ+ebμ,υ+ebυ,ξ,for all ζ,ξ,μ,υ∈Ξ.
Consider the self-mappingI on Ξ given by Iζ=ζ2/2. Taking α:Ξ2⟶ℝ+ and λ:Ξ⟶0,1 such that λζ=8.95+ζ/10 for all ζ∈Ξ, and αζ,ξ=1 for ζ,ξ∈Ξ, it is obvious to see I∈WAΞ,α. Here, wζ,ξ∈2,4.
Then equation (8) for ζ≠ξ would be of the form
(34)ζ+ξ+2.5ζ22−ξ222≤8.95+ζ10maxζ−ξ2,ζ−ζ222,ξ−ξ222,ξ−ξ2/221+ζ−ζ2/22ζ+ξ+2.51+∣ζ−ξ2,ζ−ζ2/22.ξ−ξ2/22ζ+ξ+2.5.ζ−ξ2holds whenever ebIζ,Iξ>0 and αζ,ξ≥1.For example, we demonstrate (34) is true for two cases:Case 1.
ζ=0, ξ=1 (or vice versa if ζ, ξ change places). Then, (34) will be
(35)3.50.25=0.875≤8.9510max1,0,14,1/41+03.51+1,0=0.895,which is true.Case 2.
ζ=1, ξ=0.9 (or vice versa if ζ, ξ change places). Then, (34) will be
(36)4.40.192=0.15884≤9.9510max0.01,0.444,0.3969,0.39691+0.4444.41+0.01,0.4440.39694.40.01=0.44178,which holds true.Similarly, it can be verified for anyζ≠ξ∈Ξ with αζ,ξ≥1. Thus, all the conditions are fulfilled, and the FixI=0 is a singleton set.


### 2.3. Weak Well-Posedness, Weak Limit Shadowing, and Generalizedw-Ulam-Hyers Stability

The notion of well-posedness of an fpp has evoked much interest of several mathematicians, for example, Popa [14, 15] and others. In the paper [16], the authors defined a weak well-posed (wwp) property in BbDS. In what follows, we extend this notion to EBbDS.Definition 11.
LetΞ,eb be a complete EBbDS and I:Ξ⟶Ξ be a mapping. The fpp of I is said to be weak well-posed if it satisfies the following:
(1)
u∗∈FixI is a singleton set in Ξ(2)
For any sequenceup in Ξ with limp⟶∞ebup,Iup=0 and(37)limp,q⟶∞ebIup,Iuq=0,onehaslimp⟶∞ebup,u∗=0.Theorem 12.
LetΞ,eb be a complete EBbDS and I:Ξ⟶Ξ be a mapping satisfying all the conditions of Theorem 9 and a sequence un in Ξ such that limn⟶∞ebun,Iun=0, limn,m⟶∞ebIun,Ium=0, and u∗∈FixI. Then, the fpp of I is wwp.Proof.
Letun be a sequence in Ξ such that limn⟶∞ebun,Iun=0 and limn,m⟶∞ebIun,Ium=0, for m>n; we obtain from (ebb3) that
(38)ebun,u∗≤wun,u∗ebun,Ium+ebIum,Iun+ebIun,u∗.
Taking limitn⟶∞(39)limn⟶∞ebun,u∗≤limn⟶∞wun,u∗ebun,Ium+ebIun,u∗.
WLOG, we can assume that there exists a distinct subsequenceIunk of Iun. Otherwise, there exists u0∈Ξ and n1∈ℕ such that Iun=u0 for n≥n1. Since limn⟶∞ebun,Iun=0, we get limn⟶∞ebun,u0=0. If u0≠u∗, then u0≠Iu0 due to uniqueness of the fixed point of I. For n≥n1, we obtain u0=Iun≠Iu0. So, we have
(40)ebu0,Iu0=ebIun,Iu0≤wun,u0ebIun,Iu0.
Forαu0,Iu0≥1 and I∈ΛΞ,α, we have
(41)wun,u0ebIun,Iu0≤λunmaxebun,u0,ebun,Iun,ebu0,Iu0,ebu0,Iu01+ebun,Iunwun,u01+ebun,u0,ebun,Iun.ebu0,Iu0wun,u0.ebun,u0≤λunmaxebun,u0,ebun,u0,ebu0,Iu0,0,ebu0,Iu0=λunmaxebun,u0,ebu0,Iu0.
Therefore, sincelimn⟶∞ebun,u0=0, we get
(42)limn⟶∞wun,u0ebu0,Iu0≤limn⟶∞λunebu0,Iu0.
Soebu0,Iu0=0, i.e., u0=Iu0, a contradiction. Hence, there exist m,q,n>n0m>q>n such that Ium≠Iuq≠Iun≠un. Then
(43)ebun,Ium≤wun,Iumebun,Iun+ebIun,Iuq+ebIuq,Ium,which ⟶0 as n⟶∞. On replacing the value in (39), we get
(44)limn⟶∞ebun,u∗≤limn⟶∞wun,u∗ebIun,u∗.
Again, sinceαun,u∗≥1 and I∈ΛΞ,α, we have
(45)wun,u∗ebIun,Iu∗≤λunmaxebun,u∗,ebun,Iun,ebu∗,Iu∗,ebun,Iun1+ebu∗,Iu∗wun,u∗1+ebun,u∗,ebu∗,Iu∗.ebun,Iunwun,u∗.ebun,u∗≤λunmaxebun,u∗,ebun,Iun,0,ebun,Iunwun,u∗1+ebun,u∗,0,which implies
(46)limn⟶∞wun,u∗ebIun,Iu∗≤limn⟶∞λunebun,u∗.
On placing in (39), we get
(47)limn⟶∞ebun,u∗≤limn⟶∞wun,u∗ebIun,Iu∗≤limn⟶∞λunebun,u∗.
Therefore,limn⟶∞ebun,u∗=0.The limit shadowing property of fpps has been discussed in the papers [17, 18]. We define weak limit shadowing property (wlsp) in EBbDS.Definition 13.
LetΞ,eb be a complete EBbDS and I:Ξ⟶Ξ be a mapping. The fpp of I is said to have wlsp in Ξ if assuming that un in Ξ satisfies ebun,Iun⟶0 as n⟶∞ and ebIun,Ium⟶0, it follows that there exists u∈Ξ such that ebun,Inu⟶0 as n⟶∞.Theorem 14.
LetΞ,eb be a complete EBbDS and I:Ξ⟶Ξ be an α−λ-contractive mapping for α:Ξ2⟶ℝ+ and λ:Ξ⟶0,1 with un in Ξ such that limn⟶∞ebun,Iun=0, limn,m⟶∞ebIun,Ium=0 and u∗∈FixI. Then, I has the wlsp.Proof.
Sinceu∗ is a fixed point of I, we have ebu∗,Iu∗=0, and let un in Ξ such that limn⟶∞ebun,Iun=0, limn,m⟶∞ebIun,Ium=0; then, by virtue of Theorem 12, we have limn⟶∞ebun,u∗=0, and therefore, we can write limn⟶∞ebun,Inu∗=0.In the following, we define the generalizedw-Ulam-Hyers stability (Gw-UHS) of fixed-point problem (fpp) in EBbDS as an extension of b-metric space case discussed in [19, 20] (see also [21]).Definition 15.
LetΞ,eb be a complete EBbDS and I:Ξ⟶Ξ be a mapping. The fixed-point equation (FPE)
(48)u=Iu,u∈Ξis called the generalized weak-Ulam-Hyers stable (Gw-UHS in short) in the setting of EBbDS if there exists an increasing function ϕ:ℝ+⟶ℝ+, continuous at 0, with ϕ0=0, such that for each ε>0 and an ε-solution υ∈Ξ, that is
(49)ebυ,Iυ≤ε,there exists a solution u∗∈Ξ of (48) such that
(50)ebυ,u∗≤ϕwu∗,υε.Ifϕξ=αξ for all ξ∈ℝ+, where α>0, then FPE (48) is said to be w-UHS in the setting of EBbDS.Theorem 16.
LetΞ,eb be a complete EBbDS and I:Ξ⟶Ξ be an α−λ-contractive mapping for α:Ξ2⟶ℝ+ and λ:Ξ⟶0,1 and also that the function φ:ℝ+⟶ℝ+ is strictly increasing and onto. Then the FPE (48) is Gw-UHS.Proof.
Following Theorem14, we have Iu∗=u∗, that is, u∗∈Ξ is a solution of the FPE (48) with ebu∗,u∗=0. Let ε>0 and υ∗∈Ξ be an ε-solution of FPE (48), that is
(51)e4bυ∗,Iυ∗≤ε.
Sinceebu∗,Iu∗=ebu∗,u∗=0≤ε, u∗ and υ∗ are ε-solutions. Since we have αu∗,υ∗≥1, so
(52)ebu∗,υ∗≤wu∗,υ∗ebu∗,Iu∗+ebIu∗,Iυ∗+ebIυ∗,υ∗≤wu∗,υ∗ebIu∗,Iυ∗+εwu∗,υ∗≤λu∗maxebu∗,υ∗,ebu∗,Iu∗,ebυ∗,Iυ∗,ebυ∗,Iυ∗1+ebu∗,Iu∗wu∗,υ∗1+ebu∗,υ∗,ebu∗,Iu∗.ebυ∗,Iυ∗wu∗,υ∗.ebu∗,υ∗+εwu∗,υ∗≤λu∗maxebu∗,υ∗,0,ε,0,0+εwu∗,υ∗.Let us discuss the two possible cases.Case 1.
Ifebu∗,υ∗>ε, then we get
(53)ebu∗,υ∗≤λu∗ebu∗,υ∗+wu∗,υ∗ε,that is
(54)ebu∗,υ∗1−λu∗≤wu∗,υ∗ε,which implies that
(55)ebu∗,υ∗≤11−λu∗wu∗,υ∗ε=ϕwu∗,υ∗ε.Case 2.
Ifebu∗,υ∗<ε, then (12) gives
(56)ebu∗,υ∗≤λu∗ε+wu∗,υ∗ε≤λu∗wu∗,υ∗ε+wu∗,υ∗ε=λu∗+1wu∗,υ∗ε=ϕwu∗,υ∗ε.
It shows that the inequality (50) is true for all cases, and thus the FPE (48) is Gw-UHS.


## 2.1.α−λ-Rational Contractive Mapping and Fixed Points

We start with introducing the notion ofα−λ-rational contraction in a EBbDS as follows.Definition 8.
LetΞ,eb be an EBbDS and α:Ξ2⟶ℝ+ and λ:Ξ⟶0,1. A mapping I:Ξ⟶Ξ is said to be an α−λ-rational contraction, if there exist
(6)ϑ,ν∈Ξ,with
(7)αϑ,ν≥1,ebϑ,ν>0,ebIϑ,Iν>0,which implies
(8)wϑ,νebIϑ,Iν≤λumaxebϑ,ν,ebϑ,Iϑ,ebν,Iν,ebν,Iν1+ebϑ,Iϑwϑ,ν1+ebϑ,ν,ebϑ,Iϑ.ebν,Iνwϑ,ν.ebϑ,ν.We denote byΛΞ,α the collection of all α−λ-rational contractive mappings on Ξ,eb.The set of all fixed points of a self-mappingI on a set Ξ≠∅ will be denoted by FixI.We are now in a position to state and prove the result.Theorem 9.
LetΞ,eb be a complete EBbDS and α:Ξ×Ξ⟶0,∞. Let I:Ξ⟶Ξ be a mapping satisfying the following:
(A1)
I∈ΛΞ,α∩WAΞ,α(A2)
There existsu0∈Ξ such that αu0,Iu0≥1(A3)
I is continuous.Then,FixI≠∅. Furthermore, for any u0∈Ξ, the sequence un satisfying un=Iun−1 is convergent.Proof.
By virtue of condition (A2), there existsu0∈Ξ such that αu0,Iu0≥1. Define the sequence un∈Ξ by un+1=Iun. If there exists n0∈ℕ such that un0=un0+1, then un0∈FixI, and we are complete. Therefore, we assume that un≠un+1 for all n∈ℕ.
It follows that(9)ebun,un+1>0,∀n∈ℕ.
It follows fromI∈WAΞ,α and αu0,Iu0≥1 that
(10)αu1,u2=αIu0,IIu0≥1.
Continuing this process, we obtain(11)αun,un+1≥1∀n∈ℕ.Step 1.
First, we prove that(12)limn⟶∞ebun,un+1=0.
It follows fromI∈ΛΞ,α that
(13)wun−1,unebIun−1,Iun≤λun−1maxebun−1,un,ebun−1,Iun−1,ebun,Iun,ebun,Iun1+ebun−1,Iun−1wun−1,un1+ebun−1,un,ebun−1,Iun−1.ebun,Iunwun−1,un.ebun−1,un=λun−1maxebun−1,un,ebun−1,un,ebun,un+1,ebun,un+11+ebun−1,unwun−1,un1+ebun−1,un,ebun−1,un.ebun,un+1wun−1,un.ebun−1,un,(14)≤λun−1maxebun−1,un,ebun−1,un,ebun,un+1,ebun,un+1,ebun,un+1≤λun−1maxebun−1,un,ebun,un+1.
Ifebun−1,un≤ebun,un+1 for some n∈ℕ, then from (13), we have wun,un+1ebun−1,un≤λun−1ebun,un+1, which is a contradiction since w≥1 and λ<1. Thus, ebun,un+1≤ebun−1,un for all n∈ℕ, and the sequence ebun,un+1 is a decreasing sequence of real numbers. Therefore, there exists ζ such that
(15)limn⟶∞ebun,un+1=ζ.
Again applying the limit in (13), we get
(16)limn⟶∞wun−1,unζ≤limn⟶∞λun−1ζ,which leads to ζ=0 as w≥1. Thus, we get
(17)limn⟶∞ebun,un+1=0.Step 2.
At this step, we will prove thatun is a Cauchy sequence, that is, for m>n, we prove
(18)limn,m⟶∞ebun,um=0.
Using (ebb3), we have(19)ebun,um≤wun,umebun,un+1+ebun+1,un+2+ebun+2,un+m≤wun,umebun,un+1+wun,umebun,un+1+wun,umebun+2,um≤wun,umebun,un+1+wun,umebun,un+1+wun,umwun+2,umebun+2,un+3+ebun+3,un+4+ebun+4,um≤wun,umebun,un+1+wun,umebun,un+1+wun,umwun+2,umebun,un+1+wun,umwun+2,umebun,un+1+wun,umwun+2,umebun+4,um⋮≤wun,umebun,un+1+wun,umebun,un+1+wun,umwun+2,umebun,un+1+wun,umwun+2,umebun,un+1+⋯+wun,umwun+2,um⋯wum−2,umebun,un+1+wun,umwun+2,um⋯wum−2,umebun,un+1≤wun,umebun,un+1+wun,umwun+1,umebun,un+1+wun,umwun+1,umwun+2,umebun,un+1+wun,umwun+1,umwun+2,umwun+3,umebun,un+1+⋯+wun,umwun+1,umwun+2,um⋯wum−2,umebun,un+1+wun,umwun+1,umwun+2,um⋯wum−2,umwum−1,umebun,un+1.
Applyingn,m⟶∞ and using (12), we get
(20)limn,m⟶∞ebun,um=0.
Hence,un is a Cauchy sequence. Since Ξ,eb is a complete EBbDS, then there exists a point u∗∈Ξ such that un⟶u∗ as n⟶+∞, that is
(21)limn⟶+∞ebun,u∗=0.
Next, we prove thatu∗∈FixI. Indeed, we write
(22)ebu∗,Iu∗≤wu∗,Iu∗ebu∗,un+ebun,un+1+ebun+1,Iu∗.
SinceI is continuous, on letting n⟶+∞, we obtain ebu∗,Iu∗=0, that is, Iu∗=u∗, and hence, u∗ is a fixed point of I.
To prove the uniqueness of fixed-pointu∗, we impose an additional requirement.
(A4)
For every pairu∗ and v∗ of fixed points of I, αu∗,v∗≥1.Theorem 10.
In addition of condition (A4) in Theorem9, FixI is a singleton set.Proof.
Following Theorem9, u∗∈FixI. To prove FixI is a singleton set, assume that there exist u∗,v∗∈FixI with u∗≠v∗, and by (A4), we have αu∗,v∗≥1. It follows from I∈ΛΞ,α that
(23)wu∗,v∗ebIu∗,Iv∗≤λu∗maxebu∗,v∗,ebu∗,Iu∗,ebv∗,Iv∗,ebv∗,Iv∗1+ebu∗,Iv∗wu∗,v∗1+ebu∗,v∗,ebu∗,Iu∗.ebv∗,Iv∗wu∗,v∗.ebu∗,v∗≤λu∗maxebu∗,v∗,0,0,0,0,which implies that
(24)wu∗,v∗ebu∗,v∗≤λu∗ebu∗,v∗,
a contradiction, and hence,u∗=v∗.


## 2.2. Illustrations

Example 2.
LetΞ=0.2,0.25,0.3,0.5,1. Define eb:Ξ2⟶ℝ+ so that ebζ,ξ=ebξ,ζ for all ζ,ξ∈Ξ, and
(25)eb0.5,0.3=0.07,eb0.5,0.25=0.015,eb0.25,0.2=0.02,eb0.3,0.25=0.02,eb0.3,0.2=0.02,
ebζ,ξ=ζ−ξ2, otherwise. Then Ξ,eb is a EBbDS with wζ,ξ=ζ+ξ+2 but neither a BDS Ξ,b nor a metric space Ξ,d. For instance
(26)eb0.5,0.3=0.07≤0.035=eb0.5,0.25+eb0.25,0.3,eb0.5,0.3=0.07≤0.055=eb0.5,0.25+eb0.25,0.2+eb0.2,0.3,but
(27)eb0.5,0.3=0.07≤0.154=wζ,νeb0.5,0.25+eb0.25,0.2+eb0.2,0.3.
Consider the self-mappingI on Ξ, α:Ξ2⟶ℝ+ and λ:Ξ⟶0,1(28)I:0.20.250.30.510.30.50.20.50.25,αζ,ν=1,ζ,ν∈0.2,1∪0.5,1∪1,0.2∪1,0.50,otherwise,and λζ=2ζ/3 for all ζ∈Ξ.It is easy to see thatI∈WAΞ,α. We will check that I satisfies (8) for ζ≠ξ with Iζ≠Iξ and αζ,ξ>1. We demonstrate by three nontrivial possible cases. Here, wζ,ξ∈2.4,6.Case 1.
ζ=0.5, ξ=1 (or vice versa if ζ, ξ change places). Then, ebIζ,Iξ=0.015, wζ,ξ=3.5, λζ=0.333 and
(29)Mwζ,ξ=max0.25,0,0.56250.56251+03.51+0.25,00.56253.50.25=0.5625.
Therefore, (8) implies that 0.0525<0.1873, and (8) holds true.Case 2.
ζ=0.2, ξ=1 (or vice versa if ζ, ξ change places). Then, ebIζ,Iξ=0.02, wζ,ξ=3.2, λζ=0.1333 and
(30)Mwζ,ξ=max0.64,0.02,0.5625,0.56251+0.023.21+0.64,0.20.56253.20.64=0.64,and it is easily seen that (8) is fulfilled.Thus, all the conditions are fulfilled, andI has a unique fixed point (which is ζ∗=0.5).Note that in this example the use of weaklyα-admissibility and λζ was crucial because, e.g., if we take ζ=0.2, ξ=0.5, we get ebIζ,Iξ=0.07, wζ,ξ=2.7 and
(31)Mwζ,ξ=max0.09,0.02,0,01+0.022.71+0.09,0.0202.70.09=0.09,and no contractive condition for any λζ<1 can be chosen which would holds for these points.Example 3.
ConsiderΞ=0,1 and define eb:Ξ2⟶ℝ+ by ebζ,ξ=ζ−ξ2. Then, Ξ,eb is a EBbDS with wζ,ξ=ζ+ξ+2.5 but neither a BDS Ξ,b not a metric space Ξ,d. For instance
(32)eb0,1=1≤0.5=eb0,0.5+eb0.5,1,eb0,1=1≤0.4902=eb0,0.5+eb0.5,0.99+eb0.99,1,but
(33)ebζ,ξ=ζ−ξ2=ζ−μ+μ−υ+υ−ξ2≤ζ−μ2+μ−υ2+υ−ξ2+2ζ−μμ−υ+2μ−υυ−ξ+2υ−ξζ−μ≤ζ+ξ+52ζ−μ2+μ−υ2+υ−ξ2=wζ,ξebζ,μ+ebμ,υ+ebυ,ξ,for all ζ,ξ,μ,υ∈Ξ.
Consider the self-mappingI on Ξ given by Iζ=ζ2/2. Taking α:Ξ2⟶ℝ+ and λ:Ξ⟶0,1 such that λζ=8.95+ζ/10 for all ζ∈Ξ, and αζ,ξ=1 for ζ,ξ∈Ξ, it is obvious to see I∈WAΞ,α. Here, wζ,ξ∈2,4.
Then equation (8) for ζ≠ξ would be of the form
(34)ζ+ξ+2.5ζ22−ξ222≤8.95+ζ10maxζ−ξ2,ζ−ζ222,ξ−ξ222,ξ−ξ2/221+ζ−ζ2/22ζ+ξ+2.51+∣ζ−ξ2,ζ−ζ2/22.ξ−ξ2/22ζ+ξ+2.5.ζ−ξ2holds whenever ebIζ,Iξ>0 and αζ,ξ≥1.For example, we demonstrate (34) is true for two cases:Case 1.
ζ=0, ξ=1 (or vice versa if ζ, ξ change places). Then, (34) will be
(35)3.50.25=0.875≤8.9510max1,0,14,1/41+03.51+1,0=0.895,which is true.Case 2.
ζ=1, ξ=0.9 (or vice versa if ζ, ξ change places). Then, (34) will be
(36)4.40.192=0.15884≤9.9510max0.01,0.444,0.3969,0.39691+0.4444.41+0.01,0.4440.39694.40.01=0.44178,which holds true.Similarly, it can be verified for anyζ≠ξ∈Ξ with αζ,ξ≥1. Thus, all the conditions are fulfilled, and the FixI=0 is a singleton set.


## 2.3. Weak Well-Posedness, Weak Limit Shadowing, and Generalizedw-Ulam-Hyers Stability

The notion of well-posedness of an fpp has evoked much interest of several mathematicians, for example, Popa [14, 15] and others. In the paper [16], the authors defined a weak well-posed (wwp) property in BbDS. In what follows, we extend this notion to EBbDS.Definition 11.
LetΞ,eb be a complete EBbDS and I:Ξ⟶Ξ be a mapping. The fpp of I is said to be weak well-posed if it satisfies the following:
(1)
u∗∈FixI is a singleton set in Ξ(2)
For any sequenceup in Ξ with limp⟶∞ebup,Iup=0 and(37)limp,q⟶∞ebIup,Iuq=0,onehaslimp⟶∞ebup,u∗=0.Theorem 12.
LetΞ,eb be a complete EBbDS and I:Ξ⟶Ξ be a mapping satisfying all the conditions of Theorem 9 and a sequence un in Ξ such that limn⟶∞ebun,Iun=0, limn,m⟶∞ebIun,Ium=0, and u∗∈FixI. Then, the fpp of I is wwp.Proof.
Letun be a sequence in Ξ such that limn⟶∞ebun,Iun=0 and limn,m⟶∞ebIun,Ium=0, for m>n; we obtain from (ebb3) that
(38)ebun,u∗≤wun,u∗ebun,Ium+ebIum,Iun+ebIun,u∗.
Taking limitn⟶∞(39)limn⟶∞ebun,u∗≤limn⟶∞wun,u∗ebun,Ium+ebIun,u∗.
WLOG, we can assume that there exists a distinct subsequenceIunk of Iun. Otherwise, there exists u0∈Ξ and n1∈ℕ such that Iun=u0 for n≥n1. Since limn⟶∞ebun,Iun=0, we get limn⟶∞ebun,u0=0. If u0≠u∗, then u0≠Iu0 due to uniqueness of the fixed point of I. For n≥n1, we obtain u0=Iun≠Iu0. So, we have
(40)ebu0,Iu0=ebIun,Iu0≤wun,u0ebIun,Iu0.
Forαu0,Iu0≥1 and I∈ΛΞ,α, we have
(41)wun,u0ebIun,Iu0≤λunmaxebun,u0,ebun,Iun,ebu0,Iu0,ebu0,Iu01+ebun,Iunwun,u01+ebun,u0,ebun,Iun.ebu0,Iu0wun,u0.ebun,u0≤λunmaxebun,u0,ebun,u0,ebu0,Iu0,0,ebu0,Iu0=λunmaxebun,u0,ebu0,Iu0.
Therefore, sincelimn⟶∞ebun,u0=0, we get
(42)limn⟶∞wun,u0ebu0,Iu0≤limn⟶∞λunebu0,Iu0.
Soebu0,Iu0=0, i.e., u0=Iu0, a contradiction. Hence, there exist m,q,n>n0m>q>n such that Ium≠Iuq≠Iun≠un. Then
(43)ebun,Ium≤wun,Iumebun,Iun+ebIun,Iuq+ebIuq,Ium,which ⟶0 as n⟶∞. On replacing the value in (39), we get
(44)limn⟶∞ebun,u∗≤limn⟶∞wun,u∗ebIun,u∗.
Again, sinceαun,u∗≥1 and I∈ΛΞ,α, we have
(45)wun,u∗ebIun,Iu∗≤λunmaxebun,u∗,ebun,Iun,ebu∗,Iu∗,ebun,Iun1+ebu∗,Iu∗wun,u∗1+ebun,u∗,ebu∗,Iu∗.ebun,Iunwun,u∗.ebun,u∗≤λunmaxebun,u∗,ebun,Iun,0,ebun,Iunwun,u∗1+ebun,u∗,0,which implies
(46)limn⟶∞wun,u∗ebIun,Iu∗≤limn⟶∞λunebun,u∗.
On placing in (39), we get
(47)limn⟶∞ebun,u∗≤limn⟶∞wun,u∗ebIun,Iu∗≤limn⟶∞λunebun,u∗.
Therefore,limn⟶∞ebun,u∗=0.The limit shadowing property of fpps has been discussed in the papers [17, 18]. We define weak limit shadowing property (wlsp) in EBbDS.Definition 13.
LetΞ,eb be a complete EBbDS and I:Ξ⟶Ξ be a mapping. The fpp of I is said to have wlsp in Ξ if assuming that un in Ξ satisfies ebun,Iun⟶0 as n⟶∞ and ebIun,Ium⟶0, it follows that there exists u∈Ξ such that ebun,Inu⟶0 as n⟶∞.Theorem 14.
LetΞ,eb be a complete EBbDS and I:Ξ⟶Ξ be an α−λ-contractive mapping for α:Ξ2⟶ℝ+ and λ:Ξ⟶0,1 with un in Ξ such that limn⟶∞ebun,Iun=0, limn,m⟶∞ebIun,Ium=0 and u∗∈FixI. Then, I has the wlsp.Proof.
Sinceu∗ is a fixed point of I, we have ebu∗,Iu∗=0, and let un in Ξ such that limn⟶∞ebun,Iun=0, limn,m⟶∞ebIun,Ium=0; then, by virtue of Theorem 12, we have limn⟶∞ebun,u∗=0, and therefore, we can write limn⟶∞ebun,Inu∗=0.In the following, we define the generalizedw-Ulam-Hyers stability (Gw-UHS) of fixed-point problem (fpp) in EBbDS as an extension of b-metric space case discussed in [19, 20] (see also [21]).Definition 15.
LetΞ,eb be a complete EBbDS and I:Ξ⟶Ξ be a mapping. The fixed-point equation (FPE)
(48)u=Iu,u∈Ξis called the generalized weak-Ulam-Hyers stable (Gw-UHS in short) in the setting of EBbDS if there exists an increasing function ϕ:ℝ+⟶ℝ+, continuous at 0, with ϕ0=0, such that for each ε>0 and an ε-solution υ∈Ξ, that is
(49)ebυ,Iυ≤ε,there exists a solution u∗∈Ξ of (48) such that
(50)ebυ,u∗≤ϕwu∗,υε.Ifϕξ=αξ for all ξ∈ℝ+, where α>0, then FPE (48) is said to be w-UHS in the setting of EBbDS.Theorem 16.
LetΞ,eb be a complete EBbDS and I:Ξ⟶Ξ be an α−λ-contractive mapping for α:Ξ2⟶ℝ+ and λ:Ξ⟶0,1 and also that the function φ:ℝ+⟶ℝ+ is strictly increasing and onto. Then the FPE (48) is Gw-UHS.Proof.
Following Theorem14, we have Iu∗=u∗, that is, u∗∈Ξ is a solution of the FPE (48) with ebu∗,u∗=0. Let ε>0 and υ∗∈Ξ be an ε-solution of FPE (48), that is
(51)e4bυ∗,Iυ∗≤ε.
Sinceebu∗,Iu∗=ebu∗,u∗=0≤ε, u∗ and υ∗ are ε-solutions. Since we have αu∗,υ∗≥1, so
(52)ebu∗,υ∗≤wu∗,υ∗ebu∗,Iu∗+ebIu∗,Iυ∗+ebIυ∗,υ∗≤wu∗,υ∗ebIu∗,Iυ∗+εwu∗,υ∗≤λu∗maxebu∗,υ∗,ebu∗,Iu∗,ebυ∗,Iυ∗,ebυ∗,Iυ∗1+ebu∗,Iu∗wu∗,υ∗1+ebu∗,υ∗,ebu∗,Iu∗.ebυ∗,Iυ∗wu∗,υ∗.ebu∗,υ∗+εwu∗,υ∗≤λu∗maxebu∗,υ∗,0,ε,0,0+εwu∗,υ∗.Let us discuss the two possible cases.Case 1.
Ifebu∗,υ∗>ε, then we get
(53)ebu∗,υ∗≤λu∗ebu∗,υ∗+wu∗,υ∗ε,that is
(54)ebu∗,υ∗1−λu∗≤wu∗,υ∗ε,which implies that
(55)ebu∗,υ∗≤11−λu∗wu∗,υ∗ε=ϕwu∗,υ∗ε.Case 2.
Ifebu∗,υ∗<ε, then (12) gives
(56)ebu∗,υ∗≤λu∗ε+wu∗,υ∗ε≤λu∗wu∗,υ∗ε+wu∗,υ∗ε=λu∗+1wu∗,υ∗ε=ϕwu∗,υ∗ε.
It shows that the inequality (50) is true for all cases, and thus the FPE (48) is Gw-UHS.


## 3. Application

In this section, we discuss the existence of solutions of a nonlinear fractional differential equation (FDE) [22] as an application of Theorem 9. Some other FDE-related work can be seen in [23–25].The Caputo fractional derivative of orderβ is defined as
(57)cDβpρ=1Γn−β∫0ρρ−σn−β−1pnσdsn−1<β<n,n=β+1,where p:0,∞⟶ℝ is a continuous function, β denotes the integer part of the positive real number β, and Γ is the gamma function.Consider the nonlinear FDE(58)cDβϑρ=ℏρ,ϑρ0<ρ<1,1<β≤2,with the integral boundary conditions
(59)ϑ0=0,ϑ1=∫0ηϑσds0<η<1,where J=0,1, ϑ∈CJ,ℝ, and ℏ:J×ℝ⟶ℝ are a continuous function.LetΞ=CJ,ℝ be endowed with the EBbDS function
(60)ebϑ,ν=maxρ∈Jϑρ−νρ2,and wϑ,ν=ϑρ+νρ+2.Theorem 17.
LetI:Ξ⟶Ξ be the operator defined by
(61)Iϑρ=1Γβ∫0ρρ−σβ−1ℏσ,ϑσdσ−2ρ2−η2Γβ∫011−σβ−1ℏσ,ϑσdσ+2ρ2−η2Γβ∫0η∫0σσ−ςβ−1ℏz,ϑzdςdσ,for ϑ∈Ξ, ρ∈J. Also, let ζ:ℝ×ℝ⟶ℝ be a given function. Assume the following:
(F1)
ℏ:J×ℝ⟶ℝ is a continuous function, non-decreasing in the second variable(F2)
There existsϑ0∈Ξ such that ζϑ0ρ,Iϑ0ρ≥0 for all ρ∈J(F3)
ϑ,ν∈Ξ2 and ζϑρ,Iϑρ≥0 for all ρ∈J imply that ζIϑρ,IIϑρ≥0 for all ρ∈J(F4)
There existsλ:Ξ⟶0,1 such that for ϑ,ν∈Ξ with ζϑ,ν≥0, and ρ∈J, we have
(62)ℏρ,ϑρ−ℏρ,νρ2≤λϑρΘϑ,νρθ×maxρ∈Jϑρ+νρ+2,where
(63)Θϑ,νρ=maxϑρ−νρ2,ϑρ−Iϑρ2,νρ−Iνρ2,νρ−Iνρ21+ϑρ−Iϑρ2maxρ∈Jϑρ+νρ+21+maxρ∈Jϑρ−νρ2,ϑρ−Iϑρ2νρ−Iνρ2maxρ∈Jϑρ+νρ+2maxρ∈Jϑρ−νρ2,and θ=2β−1ΓβΓβ+1/25β+2. Then, the problems (58) and (59) have at least one solution ϑ∗∈Ξ.Proof.
Define a functionα:Ξ2⟶0,∞ by
(64)αϑ,ν=1,ifforζϑρ,νρ≥0,forallρ∈Jγ,otherwise,where γ∈0,1. It is obvious to check that the assumption (F2) implies the condition (A2) of Theorem 9. Assumption (F3) clearly implies that I∈WAΞ,α.
Letϑ,ν∈Ξ be αϑ,ν≥1, i.e., ζϑρ,νρ≥0 for all ρ∈J. For each ρ∈J, by the definition (61) of operator I, we have (using Cauchy-Schwartz inequality)
(65)Iϑρ−Iνρ2=1Γβ∫0ρρ−σβ−1ℏσ,ϑσdσ−2ρ2−η2Γβ∫011−σβ−1ℏσ,ϑσdσ+2ρ2−η2Γβ∫0η∫0σσ−ςβ−1ℏς,ϑzdςdσ−1Γβ∫0ρρ−σβ−1ℏσ,νσdσ−2ρ2−η2Γβ∫011−σβ−1ℏσ,νσdσ+2ρ2−η2Γβ∫0η∫0σσ−ςβ−1ℏς,νςdςdσ2≤2Γ2β∫0ρρ−σβ−1ℏσ,ϑσ−ℏσ,νσ2dσ+8ρ22−η22Γ2β∫011−σβ−1ℏs,ϑσ−ℏσ,νσ2dσ+8ρ22−η22Γ2β∫0η∫0σσ−ςβ−1ℏς,ϑς−ℏς,νς2dςdσ,that is
(66)Iϑρ−Iνρ2≤2Γ2β∫0ρρ−σ2β−2dσ∫0ρℏσ,ϑσ−ℏσ,νσ2dσ+8ρ22−η22Γ2β∫011−σ2β−2dσ∫01ℏσ,ϑσ−ℏσ,νσ2dσ+8ρ22−η22Γ2β∫0η∫0σσ−ς2β−2dςdσ×∫0η∫0σℏς,ϑς−ℏς,νς2dςdσ.
Applying (F4) and small calculations, we get(67)Iϑρ−Iνρ2≤λϑρΘIϑ,νρϑρ+νρ+2.
This implies that(68)wϑ,νebIϑ,Iν=wϑ,νmaxt∈IIϑρ−Iνρ2≤λϑΘIϑ,ν,for all ϑ,ν∈Ξ with ebIϑ,Iν>0 where
(69)ΘIϑ,ν=maxebϑ,ν,ebϑ,Iϑ,ebν,Iν,ebν,Iν1+ebϑ,Iϑwϑ,ν1+ebϑ,ν,ebϑ,Iϑ.ebν,Iνwϑ,ν.ebϑ,ν.
Thus,I∈ΛΞ,α. Therefore, all the requirements of Theorem 9 are fulfilled, and we conclude that there is a fixed-point ϑ∗∈Ξ of the operator I. It is well known (see, e.g., [22], Theorem 17) that in this case ϑ∗ is also a solution of the integral equation (61) and the FDE (58) with the condition (59).



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*Source: 9949147-2021-05-15.xml*