Citation: Fiammetta Iannuzzo, Silvia Crudo, Gianpaolo Antonio Basile, Fortunato Battaglia, Carmenrita Infortuna, Maria Rosaria Anna Muscatello, Antonio Bruno. Efficacy and safety of non-invasive brain stimulation techniques for the treatment of nicotine addiction: A systematic review of randomized controlled trials[J]. AIMS Neuroscience, 2024, 11(3): 212-225. doi: 10.3934/Neuroscience.2024014
[1] | Eric Chung, Yalchin Efendiev, Ke Shi, Shuai Ye . A multiscale model reduction method for nonlinear monotone elliptic equations in heterogeneous media. Networks and Heterogeneous Media, 2017, 12(4): 619-642. doi: 10.3934/nhm.2017025 |
[2] | Mario Ohlberger, Ben Schweizer, Maik Urban, Barbara Verfürth . Mathematical analysis of transmission properties of electromagnetic meta-materials. Networks and Heterogeneous Media, 2020, 15(1): 29-56. doi: 10.3934/nhm.2020002 |
[3] | Patrick Henning . Convergence of MsFEM approximations for elliptic, non-periodic homogenization problems. Networks and Heterogeneous Media, 2012, 7(3): 503-524. doi: 10.3934/nhm.2012.7.503 |
[4] | Antoine Gloria Cermics . A direct approach to numerical homogenization in finite elasticity. Networks and Heterogeneous Media, 2006, 1(1): 109-141. doi: 10.3934/nhm.2006.1.109 |
[5] | Patrick Henning, Mario Ohlberger . The heterogeneous multiscale finite element method for advection-diffusion problems with rapidly oscillating coefficients and large expected drift. Networks and Heterogeneous Media, 2010, 5(4): 711-744. doi: 10.3934/nhm.2010.5.711 |
[6] | Fabio Camilli, Claudio Marchi . On the convergence rate in multiscale homogenization of fully nonlinear elliptic problems. Networks and Heterogeneous Media, 2011, 6(1): 61-75. doi: 10.3934/nhm.2011.6.61 |
[7] | Thomas Abballe, Grégoire Allaire, Éli Laucoin, Philippe Montarnal . Application of a coupled FV/FE multiscale method to cement media. Networks and Heterogeneous Media, 2010, 5(3): 603-615. doi: 10.3934/nhm.2010.5.603 |
[8] | Nils Svanstedt . Multiscale stochastic homogenization of monotone operators. Networks and Heterogeneous Media, 2007, 2(1): 181-192. doi: 10.3934/nhm.2007.2.181 |
[9] | Frederike Kissling, Christian Rohde . The computation of nonclassical shock waves with a heterogeneous multiscale method. Networks and Heterogeneous Media, 2010, 5(3): 661-674. doi: 10.3934/nhm.2010.5.661 |
[10] | Liselott Flodén, Jens Persson . Homogenization of nonlinear dissipative hyperbolic problems exhibiting arbitrarily many spatial and temporal scales. Networks and Heterogeneous Media, 2016, 11(4): 627-653. doi: 10.3934/nhm.2016012 |
Fixed point theory plays a fundamental role in mathematics and applied science, such as optimization, mathematical models and economic theories. Also, this theory has been applied to show the existence and uniqueness of the solutions of differential equations, integral equations and many other branches of mathematics, see [1,2]. A prominent result in fixed point theory is the Banach contraction principle [3]. Since the appearance of this principle, there has been a lot of activity in this area. Bakhtin [4] in 1989 introduced the notion of a b-metric space (Bms). Shoaib et. al [5] proved certain fixed point results in rectangular metric spaces. Multivalued mappings in various types of metric spaces have been extensively studied by many researchers to establish fixed point results and their applications, see for instance [6,7,8,9,10,11,12].
In 1965, Zadeh [13] introduced the concept of a fuzzy set theory to deal with the unclear or inexplicit situations in daily life. Using this theory, Kramosil and Michálek [14] defined the concept of a fuzzy metric space (Fms). Grabiec [15] gave contractive mappings on a Fms and extended fixed point theorems of Banach and Edelstein in such a space. Successively, George and Veeramani [16] slightly modified the notion of a Fms introduced by Kramosil and Michálek [14] and then obtained a Hausdorff topology and a first countable topology on it. Many fixed point results have been established in a Fms. For instance, see [17,18,19,20,21,22,23,24,25] and the references therein. Recently, some coupled fuzzy fixed-point results on closed ball are established in fuzzy metric spaces [26]. The notion of generalized fuzzy metric spaces is studied in [27].
The notion of a fuzzy b-metric space (Fbms) was defined in [28]. The notion of a Hausdorff Fms is introduced in [29]. Fixed point theory for multivalued mapping in fuzzy metric spaces has been extended in many directions. For a multivalued mapping (Mvp) in a complete Fms, some fixed point results are establish in [30]. Some fixed point results for a Mvp in a Hausdorff fuzzy b-metric space (Hfbms) are proved in [31]. In this article, we prove some fixed point results for a Mvp using Geraghty type contractions in a Hfbms. Results in [31,32] and [30] turn out to be special cases of our results.
Throughout the article, ℧ refers to a non-empty set, N represents the set of natural numbers, R corresponds to the collection of real numbers, CB(℧) and ˆC0(℧) represent the collection of closed and bounded subsets and compact subsets of ℧, respectively.
Let us have a look at some core concepts that will be helpful for the proof of our main results.
Definition 1.1. [33] For a real number b≥1, the triplet (℧,Θfb,∗) is called a Fbms on ℧ if for all ψ1,ψ2,ψ3∈℧ and γ>0, the following axioms hold, where ∗ is a continuous t-norm and Θfb is a fuzzy set on ℧×℧×(0,∞):
[Fb1:] Θfb(ψ1,ψ2,γ)>0;
[Fb2:] Θfb(ψ1,ψ2,γ)=1 if and only if ψ1=ψ2;
[Fb3:] Θfb(ψ1,ψ2,γ)=Θfb(ψ2,ψ1,γ);
[Fb4:] Θfb(ψ1,ψ3,b(γ+β))≥Θfb(ψ1,ψ2,γ)∗Θfb(ψ2,ψ3,β) ∀γ,β≥0;
[Fb5:] Θfb(ψ1,ψ2,.):(0,∞)→[0,1] is left continuous.
The notion of a Fms in the sense of George and Veeramani [16] can be obtained by taking b=1 in the above definition.
Example 1.1. For a Bms (℧,Θb,∧), define a mapping Θfb:℧×℧×(0,∞)→[0,1] by
Θfb(ψ1,ψ2,γ)=γγ+db(ψ1,ψ1). |
Then (℧,Θfb,∧) is a Fbms.
Following Grabiec [15], the notions of G-Cauchyness and completeness are defined as follows:
Definition 1.2. [15]
(i) If for a sequence {ψn} in a Fbms (℧,Θfb,∗), there is ψ∈℧ such that
limn→∞Θfb(ψn,ψ,γ)=1,∀γ>0, |
then {ψn} is said to be convergent.
(ii) If for a sequence {ψn} in a Fbms (℧,Θfb,∗), limn→∞Θfb(ψn,ψn+q,γ)=1 then {ψn} is a G-Cauchy sequence for all γ>0 and positive integer q.
(iii) A Fbms is G-complete if every G-Cauchy sequence is convergent.
Definition 1.3. [30] Let B be any nonempty subset of a Fms (℧,Θfb,∗) and γ>0, then we define FΘfb(ϱ1,B,γ), the fuzzy distance between an element ϱ1∈℧ and the subset B, as follows:
FΘfb(ϱ1,B,γ)=sup{Θf(ϱ1,ϱ2,γ):ϱ2∈B}. |
Note that FΘfb(ϱ1,B,γ)=FΘfb(B,ϱ1,α).
Lemma 1.1. [31] Consider a Fbms (℧,Θfb,∗) and let CB(℧) be the collection of closed bounded subsets of ℧. If A∈CB(℧) then ψ∈A if and only if FΘfb(A,ψ,γ)=1∀γ>0.
Definition 1.4. [31] Let (℧,Θfb,∗) be a Fbms. Define HFΘfb on ˆC0(℧)׈C0(℧)×(0,∞) by
HFΘfb(A,B,γ)=min{ infψ∈AFΘfb(ψ,B,γ),infϱ∈BFΘfb(A,ϱ,γ)}, |
for all A,B∈^C0(℧) and γ>0.
For Geraghty type contractions, follow [33] to define a class FΘb of all functions β:[0,∞)→[0,1b) for b≥1, as
FΘb={β:[0,∞)→[0,1b)|limn→∞β(γn)=1b⇒limn→∞γn=0}. | (1.1) |
Lemma 1.2. [31] Let (℧,Θfb,∗) be a G-complete Fbms. If ψ,ϱ∈℧ and for a function β∈FΘfb
Θfb(ψ,ϱ,β(Θfb(ψ,ϱ,γ))γ)≥Θfb(ψ,ϱ,γ), |
then ψ=ϱ.
Lemma 1.3. [31] Let (^C0(℧),HFΘfb,∗) be a Hfbms where (Θfb,∗) is a Fbm on ℧. If for all A,B∈^C0(℧), for each ψ∈A and for γ>0 there exists ϱψ∈B, satisfying FΘfb(ψ,B,γ)=Θfb(ψ,ϱψ,γ), then
HFΘfb(A,B,γ)≤Θfb(ψ,ϱψ,γ). |
In this section, we develop some fixed point results by using the idea of a Hfbms. Furthermore, an example is also presented for a deeper understanding of our results.
Recall that, given a multivalued mapping Ξ:℧→ˆC0(℧), a point ψ is said to be a fixed point of Ξ if ψ∈Ξψ.
Theorem 2.1. Let (℧,Θfb,∗) be a G-complete Fbms and (^C0(℧),HFΘfb,∗) be a Hfbms. Let Ξ:℧→^C0(℧) be a Mvp satisfying
HFΘfb(Ξψ,Ξϱ,β(Θfb(ψ,ϱ,γ))γ)≥Θfb(ψ,ϱ,γ), | (2.1) |
for all ψ,ϱ∈℧, where β∈FΘfb as defined in (1.1). Then Ξ has a fixed point.
Proof. Choose {ψn} for ψ0∈℧ as follows: Let ψ1∈℧ such that ψ1∈Ξψ0 by the application of Lemma 1.3, we can choose ψ2∈Ξψ1 such that for all γ>0,
Θfb(ψ1,ψ2,γ)⩾HFΘfb(Ξψ0,Ξψ1,γ). |
By induction, we have ψr+1∈Ξψr satisfying
Θfb(ψr,ψr+1,γ)⩾HFΘfb(Ξψr−1,Ξψr,γ)∀r∈N. |
By the application of (2.1) and Lemma 1.3, we have
Θfb(ψr,ψr+1,γ)≥HFΘfb(Ξψr−1,Ξψr,γ)≥Θfb(ψr−1,ψr,γβ(Θfb(ψr−1,ψr,γ)))≥HFΘfb(Ξψr−2,Ξψr−1,γβ(Θfb(ψr−1,ψr,γ)))≥Θfb(ψr−2,ψr−1,γβ(Θfb(ψr−1,ψr,γ))β(Θfb(ψr−2,ψr−1,γ)))⋮≥HFΘfb(Ξψ0,Ξψ1,γβ(Θfb(ψr−1,ψr,γ))β(Θfb(ψr−2,ψr−1,γ))…β(Θfb(ψ1,ψ2,γ)))≥Θfb(ψ0,ψ1,γβ(Θfb(ψr−1,ψr,γ))β(Θfb(ψr−2,ψr−1,γ))…β(Θfb(ψ0,ψ1,γ))). | (2.2) |
For any q∈N, writing q(γq)=γq+γq+…+γq and using [Fb4] repeatedly,
Θfb(ψr,ψr+q,γ)≥Θfb(ψr,ψr+1,γqb)∗Θfb(ψr+1,ψr+2,γqb2)∗Θfb(ψr+2,ψr+3,γqb3)∗…∗Θfb(ψr+q−1,ψr+q,γqbq). |
Using (2.2) and [Fb5], we get
Θfb(ψr,ψr+q,γ)≥Θfb(ψ0,ψ1,γqbβ(Θfb(ψr−1,ψr,γ))β(Θfb(ψr−2,ψr−1,γ))…β(Θfb(ψ0,ψ1,γ)))∗Θfb(ψ0,ψ1,γqb2β(Θfb(ψr,ψr+1,γ))β(Θfb(ψr−1,ψr,γ))…β(Θfb(ψ0,ψ1,γ)))∗Θfb(ψ0,ψ1,γqb3β(Θfb(ψr+1,ψr+2,γ))β(Θfb(ψr,ψr+1,γ))…β(Θfb(ψ0,ψ1,γ)))∗…∗Θfb(ψ0,ψ1,γqbqβ(Θfb(ψr+q−2,ψr+q−1,γ))β(Θfb(ψr+q−3,ψr+q−2,γ))…β(Θfb(ψ0,ψ1,γ))). |
That is,
Θfb(ψr,ψr+q,γ)≥Θfb(ψ0,ψ1,br−1γq)∗Θfb(ψ0,ψ1,br−1γq)∗Θfb(ψ0,ψ1,br−1γq)∗…∗Θfb(ψ0,ψ1,br−1γq). |
Taking limit as r→∞, we get
limn→∞Θfb(ψr,ψr+q,γ)=1∗1∗…∗1=1. |
Hence, {ψr} is G-Cauchy sequence. By the G-completeness of ℧, there exists ϕ∈℧ such that
Θfb(ϕ,Ξϕ,γ)≥Θfb(ϕ,ψr+1,γ2b)∗Θfb(ψr+1,Ξϕ,γ2b)≥Θfb(ϕ,ψr+1,γ2b)∗HFΘfb(Ξψr,Ξϕ,γ2b)≥Θfb(ϕ,ψr+1,γ2b)∗Θfb(ψr,ϕ,γ2bβ(Θfb(ψr,ϕ,γ)))⟶1asr→∞. |
By Lemma 1.1, it follows that ϕ∈Ξϕ. That is, ϕ is a fixed point for Ξ.
Remark 2.1.
(1) If we take β(Θfb(ψ,ϱ,γ))=k with bk<1, we get Theorem 3.1 of [31].
(2) By setting ^C0(℧)=℧ the mapping Ξ:℧→^C0(℧) becomes a single valued and we get Theorem 3.1 of [32]. Notice that when Ξ is a singlevalued map, Ξψ becomes a singleton set and the fact that HFΘfb(Ξψ,Ξϱ,γ)=Θfb(Ξψ,Ξϱ,γ) indicates that the fixed point will be unique as proved in [32].
(3) Set b=1 and ^C0(℧)=℧ and let k∈(0,1) be such that β(Θfb(ψ,ϱ,γ))=k then we get the main result of [15].
The next example illustrates Theorem 2.1.
Example 2.1. Let ℧=[0,1] and define a G-complete Fbms by
Θfb(ψ,ϱ,γ)=γγ+(ψ−ϱ)2, |
with b≥1. For β∈Fb, define a mapping Ξ:℧→^C0(℧) by
Ξψ={{0}if ψ=0,{0,√β(Θfb(ψ,ϱ,γ))ψ2}otherwise. |
For ψ=ϱ,
HFΘfb(Ξψ,Ξϱ,β(Θfb(ψ,ϱ,γ))γ)=1=Θfb(ψ,ϱ,γ). |
If ψ≠ϱ, then following cases arise.
For ψ=0 and ϱ∈(0,1], we have
HFΘfb(Ξ0,Ξϱ,β(Θfb(ψ,ϱ,γ))γ)=min{ infa∈Ξ0FΘfb(a,Ξϱ,β(Θfb(ψ,ϱ,γ))γ),infb∈ΞϱFΘfb(Ξ0,b,β(Θfb(ψ,ϱ,γ))γ))}=min{infa∈Ξ0FΘfb(a,{0,√β(Θfb(ψ,ϱ,γ))ϱ2},β(Θfb(ψ,ϱ,γ))γ),infb∈ΞϱFΘfb({0},b,β(Θfb(ψ,ϱ,γ))γ)}=min{inf{FΘfb(0,{0,√β(Θfb(ψ,ϱ,γ))ϱ2},β(Θfb(ψ,ϱ,γ))γ)},inf{FΘfb({0},0,β(Θfb(ψ,ϱ,γ))γ),FΘfb({0},√β(Θfb(ψ,ϱ,γ))ϱ2,β(Θfb(ψ,ϱ,γ))γ)}}=min{inf{sup{FΘfb(0,0,β(Θfb(ψ,ϱ,γ))γ),FΘfb(0,√β(Θfb(ψ,ϱ,γ))ϱ2,β(Θfb(ψ,ϱ,γ))γ)}},inf{FΘfb(0,0,β(Θfb(ψ,ϱ,γ))γ),FΘfb(0,√β(Θfb(ψ,ϱ,γ))y2,β(Θfb(ψ,ϱ,γ))γ)}}=min{inf{sup{1,γγ+ϱ24}},inf{1,γγ+ϱ24}}=min{inf{1},γγ+ϱ24}=min{1,γγ+ϱ24}=γγ+ϱ24. |
It follows that
HFΘfb(Ξ0,Ξϱ,β(Θfb(ψ,ϱ,γ))γ)>Θfb(0,ϱ,γ)=γγ+ϱ2. |
For ψ and ϱ∈(0,1], after simplification we have
HFΘfb(S(ψ),Ξϱ,β(Θfb(ψ,ϱ,γ))γ)=min{sup{γγ+ψ24,γγ+(ψ−ϱ)24},sup{γγ+ϱ24,γγ+(ψ−ϱ)24}}≥γγ+(ψ−ϱ)24>γγ+(ψ−ϱ)2=Θfb(ψ,ϱ,γ). |
Thus, for all cases, we have
HFΘfb(Ξψ,Ξϱ,β(Θfb(ψ,ϱ,γ))γ)≥Θfb(ψ,ϱ,γ). |
Since all conditions of Theorem 2.1 are satisfied and 0 is a fixed point of Ξ.
Theorem 2.2. Let (℧,Θfb,∗) be a G-complete Fbms with b⩾1 and (^C0(℧),HFΘfb,∗) be a Hfbms. Let Ξ:℧→^C0(℧) be a Mvp satisfying
HFΘfb(Ξψ,Ξϱ,β(Θfb(ψ,ϱ,γ))γ)≥min{FΘfb(ϱ,Ξϱ,γ)[1+FΘfb(ψ,Ξψ,γ)]1+Θfb(ψ,ϱ,γ),Θfb(ψ,ϱ,γ)}, | (2.3) |
for all ψ,ϱ∈℧, where β∈FΘfb as given in (1.1). Then Ξ has a fixed point.
Proof. Choose {ψn} for ψ0∈℧ as follows: Let ψ1∈℧ such that ψ1∈Ξψ0. By the application of Lemma 1.3 we can choose ψ2∈Ξψ1 such that
Θfb(ψ1,ψ2,γ)⩾HFΘfb(Ξψ0,Ξψ1,γ),∀γ>0. |
By induction, we have ψr+1∈Ξψr satisfying
Θfb(ψr,ψr+1,γ)⩾HFΘfb(Ξψr−1,Ξψr,γ),∀r∈N. |
By the application of (2.3) and Lemma 1.3 we have
Θfb(ψr,ψr+1,γ)≥HFΘfb(Ξψr−1,Ξψr,γ)≥min{FΘfb(ψr,Ξψr,γβ(Θfb(ψr−1,ψr,γ)))[1+FΘfb(ψr−1,Ξψr−1,γβ(Θfb(ψr−1,ψr,γ)))]1+Θfb(ψr−1,ψr,γβ(Θfb(ψr−1,ψr,γ))),Θfb(ψr−1,ψr,γβ(Θfb(ψr−1,ψr,γ)))}≥min{Θfb(ψr,ψr+1,γβ(Θfb(ψr−1,ψr,γ)))[1+Θfb(ψr−1,ψr,γβ(Θfb(ψr−1,ψr,γ)))]1+Θfb(ψr−1,ψr,γβ(Θfb(ψr−1,ψr,γ))),Θfb(ψr−1,ψr,γβ(Θfb(ψr−1,ψr,γ)))},Θfb(ψr,ψr+1,γ)≥min{Θfb(ψr,ψr+1,γβ(Θfb(ψr−1,ψr,γ))),Θfb(ψr−1,ψr,γβ(Θfb(ψr−1,ψr,γ)))}. | (2.4) |
If
min{Θfb(ψr,ψr+1,γβ(Θfb(ψr−1,ψr,γ))),Θfb(ψr−1,ψr,γβ(Θfb(ψr−1,ψr,γ)))}=Θfb(ψr,ψr+1,γβ(Θfb(ψr−1,ψr,γ))), |
then (2.4) implies
Θfb(ψr,ψr+1,γ)≥Θfb(ψr,ψr+1,γβ(Θfb(ψr−1,ψr,γ))). |
The result is obvious by Lemma 1.2.
If
min{Θfb(ψr,ψr+1,γβ(Θfb(ψr−1,ψr,γ))),Θfb(ψr−1,ψr,γβ(Θfb(ψr−1,ψr,γ)))}=Θfb(ψr−1,ψr,γβ(Θfb(ψr−1,ψr,γ))), |
then from (2.4) we have
Θfb(ψr,ψr+1,γ)≥Θfb(ψr−1,ψr,γβ(Θfb(ψr−1,ψr,γ)))≥Θfb(ψr−2,ψr−1,γβ(Θfb(ψr−1,ψr,γ))β(Θfb(ψr−2,ψr−1,γ)))⋮⩾Θfb(ψ0,ψ1,γβ(Θfb(ψr−1,ψr,γ))β(Θfb(ψr−2,ψr−1,γ))…β(Θfb(ψ0,ψ1,γ)). |
The rest of the proof can be done by proceeding same as in Theorem 2.1.
Remark 2.2.
(1) If we take β(Θfb(ψ,ϱ,γ))=k with bk<1, we get Theorem 3.2 of [31].
(2) By taking b=1 and for some 0<k<1 setting β(Θfb(ψ,ϱ,γ))=k in Theorem 2.2, we get the main result of [30].
Theorem 2.3. Let (℧,Θfb,∗) be a G-complete Fbms with b⩾1 and (^C0(℧),HFΘfb,∗) be a Hfbms. Let Ξ:℧→ˆC0(℧) be a Mvp satisfying
HFΘfb(Ξψ,Ξϱ,β(Θfb(ψ,ϱ,γ))γ)≥min{FΘfb(ϱ,Ξϱ,γ)[1+FΘfb(ψ,Ξψ,γ)+FΘfb(ϱ,Ξψ,γ)]2+Θfb(ψ,ϱ,γ),Θfb(ψ,ϱ,γ)} | (2.5) |
for all ψ,ϱ∈℧, where β∈FΘfb, the class of functions defined in (1.1). Then Ξ has a fixed point.
Proof. Choose {ψn} for ψ0∈℧ as follows: Let ψ1∈℧ such that ψ1∈Ξψ0. by the application of Lemma 1.3 we can choose ψ2∈Ξψ1 such that
Θfb(ψ1,ψ2,γ)⩾HFΘfb(Ξψ0,Ξψ1,γ),∀γ>0. |
By induction, we have ψr+1∈Ξψr satisfying
Θfb(ψr,ψr+1,γ)⩾HFΘfb(Ξψr−1,Ξψr,γ),∀r∈N. |
By the application of (2.5) and Lemma 1.3, we have
Θfb(ψr,ψr+1,γ)≥HFΘfb(Ξψr−1,Ξψr,γ)≥min{FΘfb(ψr,Ξψr,γβ(Θfb(ψr−1,ψr,γ)))[1+FΘfb(ψr−1,Ξψr−1,γβ(Θfb(ψr−1,ψr,γ)))+FΘfb(ψr,Ξψr−1,γβ(Θfb(ψr−1,ψr,γ)))]2+Θfb(ψr−1,ψr,γβ(Θfb(ψr−1,ψr,γ))),Θfb(ψr−1,ψr,γβ(Θfb(ψr−1,ψr,γ)))}≥min{Θfb(ψr,ψr+1,γβ(Θfb(ψr−1,ψr,γ)))[1+Θfb(ψr−1,ψr,γβ(Θfb(ψr−1,ψr,γ)))+Θfb(ψr,ψr,γβ(Θfb(ψr−1,ψr,γ)))]2+Θfb(ψr−1,ψr,γβ(Θfb(ψr−1,ψr,γ))),Θfb(ψr−1,ψr,γβ(Θfb(ψr−1,ψr,γ)))}≥min{Θfb(ψr,ψr+1,γβ(Θfb(ψr−1,ψr,γ)))[1+Θfb(ψr−1,ψr,γβ(Θfb(ψr−1,ψr,γ)))+1]2+Θfb(ψr−1,ψr,γβ(Θfb(ψr−1,ψr,γ))),Θfb(ψr−1,ψr,γβ(Θfb(ψr−1,ψr,γ)))}≥min{Θfb(ψr,ψr+1,γβ(Θfb(ψr−1,ψr,γ)))[2+Θfb(ψr−1,ψr,γβ(Θfb(ψr−1,ψr,γ)))]2+Θfb(ψr−1,ψr,γβ(Θfb(ψr−1,ψr,γ))),Θfb(ψr−1,ψr,γβ(Θfb(ψr−1,ψr,γ)))}≥min{Θfb(ψr,ψr+1,γβ(Θfb(ψr−1,ψr,γ))),Θfb(ψr−1,ψr,γβ(Θfb(ψr−1,ψr,γ)))}. | (2.6) |
If
min{Θfb(ψr,ψr+1,γβ(Θfb(ψr−1,ψr,γ))),Θfb(ψr−1,ψr,γβ(Θfb(ψr−1,ψr,γ)))}=Θfb(ψr,ψr+1,γβ(Θfb(ψr−1,ψr,γ))), |
then (2.6) implies
Θfb(ψr,ψr+1,γ)≥Θfb(ψr,ψr+1,γβ(Θfb(ψr−1,ψr,γ))), |
and the proof follows by Lemma 1.2.
If
min{Θfb(ψr,ψr+1,γβ(Θfb(ψr−1,ψr,γ))),Θfb(ψr−1,ψr,γβ(Θfb(ψr−1,ψr,γ)))}=Θfb(ψr−1,ψr,γβ(Θfb(ψr−1,ψr,γ))). |
Then from (2.6) we have
Θfb(ψr,ψr+1,γ)≥Θfb(ψr−1,ψr,γβ(Θfb(ψr−1,ψr,γ)))⩾…⩾Θfb(ψr−1,ψr,γβ(Θfb(ψr−1,ψr,γ))β(Θfb(ψr−2,ψr−1,γ))…β(Θfb(ψ0,ψ1,γ))). |
The rest of the proof is same as in Theorem 2.1.
Remark 2.3. Theorem 3.3 of [31] becomes a special csae of the above theorem by setting β(Θfb(ψ,ϱ,γ))=k where k is chosen such that bk<1.
Theorem 2.4. Let (℧,Θfb,∗) be a G-complete Fbms with b⩾1 and (^C0(℧),HFΘfb,∗) be a Hfbms. Let Ξ:℧→ˆC0(℧) be a multivalued mapping satisfying
HFΘfb(Ξψ,Ξϱ,β(Θfb(ψ,ϱ,γ))γ)≥min{FΘfb(ψ,Ξψ,γ)[1+FΘfb(ϱ,Ξϱ,γ)]1+FΘfb(Ξψ,Ξϱ,γ),FΘfb(ψ,Ξϱ,γ)[1+FΘfb(ψ,Ξψ,γ)]1+Θfb(ψ,ϱ,γ),FΘfb(ψ,Ξψ,γ)[2+FΘfb(ψ,Ξϱ,γ)]1+Θfb(ψ,Ξϱ,γ)+FΘfb(ϱ,Ξψ,γ),Θfb(ψ,ϱ,γ)}, | (2.7) |
for all ψ,ϱ∈℧, where β∈Ffb, the class of functions defined in (1.1). Then Ξ has a fixed point.
Proof. In the same way as Theorem 2.1, we have
Θfb(ψ1,ψ2,γ)⩾HFΘfb(Ξψ0,Ξψ1,γ),∀γ>0. |
By induction, we obtain ψr+1∈Ξψr satisfying
Θfb(ψr,ψr+1,γ)⩾HFΘfb(Ξψr−1,Ξψr,γ),∀n∈N. |
Now, by (2.7) together with Lemma 1.3, we have
Θfb(ψr,ψr+1,γ)≥HFΘfb(Ξψr−1,Ξψr,γ)≥min{FΘfb(ψr−1,Ξψr−1,γβ(Θfb(ψr−1,ψr,γ)))[1+FΘfb(ψr,Sψr,γβ(Θfb(ψr−1,ψr,γ)))]1+FΘfb(Ξψr−1,Ξψr,γβ(Θfb(ψr−1,ψr,γ))),FΘfb(ψr,Ξψr,γβ(Θfb(ψr−1,ψr,γ)))[1+FΘfb(ψr−1,Ξψr−1,γβ(Θfb(ψr−1,ψr,γ)))]1+Θfb(ψr−1,ψr,γβ(Θfb(ψr−1,ψr,γ))),FΘfb(ψr−1,Ξψr−1,γβ(Θfb(ψr−1,ψr,γ)))[2+FΘfb(ψr−1,Ξψr,γβ(Θfb(ψr−1,ψr,γ)))]1+FΘfb(ψr−1,Ξψr,γβ(Θfb(ψr−1,ψr,γ)))+FΘfb(ψr,Ξψr−1,γβ(Θfb(ψr−1,ψr,γ))),Θfb(ψr−1,ψr,γβ(Θfb(ψr−1,ψr,γ)))}≥min{Θfb(ψr−1,ψr,γβ(Θfb(ψr−1,ψr,γ)))[1+Θfb(ψr,ψr+1,γβ(Θfb(ψr−1,ψr,γ)))]1+Θfb(ψr,ψr+1,γβ(Θfb(ψr−1,ψr,γ))),Θfb(ψr,ψr+1,γβ(Θfb(ψr−1,ψr,γ)))[1+Θfb(ψr−1,ψr,γβ(Θfb(ψr−1,ψr,γ)))]1+Θfb(ψr−1,ψr,γβ(Θfb(ψr−1,ψr,γ))),Θfb(ψr−1,ψr,γβ(Θfb(ψr−1,ψr,γ)))[2+Θfb(ψr−1,ψr+1,γβ(Θfb(ψr−1,ψr,γ)))]1+Θfb(ψr−1,ψr+1,γβ(Θfb(ψr−1,ψr,γ)))+Θfb(ψr,ψr,γβ(Θfb(ψr−1,ψr,γ))),Θfb(ψr−1,ψr,γβ(Θfb(ψr−1,ψr,γ)))},Θfb(ψr,ψr+1,γ)≥min{Θfb(ψr,ψr+1,γβ(Θfb(ψr−1,ψr,γ))),Θfb(ψr−1,ψr,γβ(Θfb(ψr−1,ψr,γ)))}. | (2.8) |
If
min{Θfb(ψr,ψr+1,γβ(Θfb(ψr−1,ψr,γ))),Θfb(ψr−1,ψr,γβ(Θfb(ψr−1,ψr,γ)))}=Θfb(ψr,ψr+1,γβ(Θfb(ψr−1,ψr,γ))), |
then (2.8) implies
Θfb(ψr,ψr+1,γ)≥Θfb(ψr,ψr+1,γβ(Θfb(ψr−1,ψr,γ))). |
Then the proof follows by Lemma 1.2.
If
min{Θfb(ψr,ψr+1,γβ(Θfb(ψr−1,ψr,γ))),Θfb(ψr−1,ψr,γβ(Θfb(ψr−1,ψr,γ)))}=Θfb(ψr−1,ψr,γβ(Θfb(ψr−1,ψr,γ))), |
then from (2.6) we have
Θfb(ψr,ψr+1,γ)≥Θfb(ψr−1,ψr,γβ(Θfb(ψr−1,ψr,γ)))⋮⩾Θfb(ψ0,ψ1,γβ(Θfb(ψr−1,ψr,γ))β(Θfb(ψr−2,ψr−1,γ))…β(Θfb(ψ0,ψ1,γ))). |
The rest of the proof is similar as in Theorem 2.1.
Remark 2.4. Again by taking β(Θfb(ψ,ϱ,γ))=k with kb<1, we get Theorem 3.4 of [31].
Theorem 2.5. Let (℧,Θfb,∗) be a G-complete Fbms with b⩾1 and (^C0(℧),HFΘfb,∗) be a Hfbms. Let Ξ:℧→ˆC0(℧) be a Mvp satisfying
HFΘfb(Ξψ,Ξϱ,β(Θfb(ψ,ϱ,γ))γ)≥min{HFΘfb(Ξψ,Ξϱ,γ).Θfb(ψ,ϱ,γ),HFΘfb(ψ,Ξψ,γ).HFΘfb(ϱ,Ξϱ,γ)})max{HFΘfb(ψ,Ξψ,γ),HFΘfb(ϱ,Ξϱ,γ)}, | (2.9) |
for all ψ,ϱ∈℧, where β∈Ffb. Then Ξ has a fixed point.
Proof. In the same way as Theorem 2.1, we have
Θfb(ψ1,ψ2,γ)⩾HFΘfb(Ξψ0,Ξψ1,γ),∀γ>0. |
By induction we have ψr+1∈Ξψr satisfying
Θfb(ψr,ψr+1,γ)⩾HFΘfb(Ξψr−1,Ξψr,γ),∀n∈N. |
Now by (2.7) together with Lemma 1.3 and some obvious simplification step, we have
Θfb(ψr,ψr+1,γ)≥HFΘfb(Ξψr−1,Ξψr,γ)≥min{Θfb(ψr,ψr+1,γβ(Θfb(ψr−1,ψr,γ))).Θfb(ψr−1,ψr,γβ(Θfb(ψr−1,ψr,γ))),Θfb(ψr−1,ψr,γβ(Θfb(ψr−1,ψr,γ))).Θfb(ψr,ψr+1,γβ(Θfb(ψr−1,ψr,γ)))}max{Θfb(ψr−1,ψr,γβ(Θfb(ψr−1,ψr,γ))),Θfb(ψr,ψr+1,γβ(Θfb(ψr−1,ψr,γ)))}≥Θfb(ψr−1,ψr,γβ(Θfb(ψr−1,ψr,γ))).Θfb(ψr,ψr+1,γβ(Θfb(ψr−1,ψr,γ)))max{Θfb(ψr−1,ψr,γβ(Θfb(ψr−1,ψr,γ))),Θfb(ψr,ψr+1,γβ(Θfb(ψr−1,ψr,γ)))} | (2.10) |
If
max{Θfb(ψr,ψr+1,γβ(Θfb(ψr−1,ψr,γ))),Θfb(ψr−1,ψr,γβ(Θfb(ψr−1,ψr,γ)))}=Θfb(ψr−1,ψr,γβ(Θfb(ψr−1,ψr,γ))), |
then (2.10) implies
Θfb(ψr,ψr+1,t)≥Θfb(ψr,ψr+1,γβ(Θfb(ψr−1,ψr,γ))) |
Then the proof follows by Lemma 1.2.
If
max{Θfb(ψr,ψr+1,γβ(Θfb(ψr−1,ψr,γ))),Θfb(ψr−1,ψr,γβ(Θfb(ψr−1,ψr,γ)))}=Θfb(ψr,ψr+1,γβ(Θfb(ψr−1,ψr,γ))), |
then from (2.10) we have
Θfb(ψr,ψr+1,γ)≥Θfb(ψr−1,ψr,γβ(Θfb(ψr−1,ψr,γ)))⋮⩾Θfb(ψ0,ψ1,γβ(Θfb(ψr−1,ψr,γ))β(Θfb(ψr−2,ψr−1,γ))…β(Θfb(ψ0,ψ1,γ))). |
The remaining proof follows in the same way as in Theorem 2.1.
Theorem 2.6. Let (℧,Θfb,∗) be a G-complete Fbms with b⩾1 and (^C0(℧),HFΘfb,∗) be a Hfbms. Let Ξ:℧→ˆC0(℧) be a Mvp satisfying
HFΘfb(Ξψ,Ξϱ,β(Θfb(ψ,ϱ,γ))γ)≥Γ1(ψ,ϱ,γ)∗Γ2(ψ,ϱ,γ), | (2.11) |
where,
{Γ1(ψ,ϱ,γ)=min{HFΘfb(Ξψ,Ξϱ,γ),HFΘfb(ψ,Ξψ,γ),HFΘfb(ϱ,Ξϱ,γ),Θfb(ψ,ϱ,γ)}Γ2(ψ,ϱ,γ)=max{HFΘfb(ψ,Ξϱ,γ),HFΘfb(Ξψ,ϱ,γ)}}, | (2.12) |
for all ψ,ϱ∈℧, and β∈Ffb. Then Ξ has a fixed point.
Proof. In the same way as Theorem 2.1, we have
Θfb(ψ1,ψ2,γ)⩾HFΘfb(Ξψ0,Ξψ1,γ),∀γ>0. |
By induction we have ψr+1∈Ξψr satisfying
HFΘfb(ψr,ψr+1,γ)=Fθ(Ξψr−1,Ξψr,γ)≥Γ1(ψr−1,ψr,γβ(Θfb(ψr−1,ψr,γ)))∗Γ2(ψr−1,ψr,γβ(Θfb(ψr−1,ψr,γ))) | (2.13) |
Now,
Γ1(ψr−1,ψr,γβ(Θfb(ψr−1,ψr,γ)))=min{HFΘfb(Ξψr−1,Ξψr,γβ(Θfb(ψr−1,ψr,γ))),HFΘfb(ψr−1,Ξψr−1,γβ(Θfb(ψr−1,ψr,γ))),HFΘfb(ψr,Ξψr,γβ(Θfb(ψr−1,ψr,γ))),Θfb(ψr−1,ψr,γβ(Θfb(ψr−1,ψr,γ)))}=min{Θfb(ψr,ψr+1,γβ(Θfb(ψr−1,ψr,γ))),Θfb(ψr−1,ψr,γβ(Θfb(ψr−1,ψr,γ))),Θfb(ψr,ψr+1,γβ(Θfb(ψr−1,ψr,γ))),Θfb(ψr−1,ψr,γβ(Θfb(ψr−1,ψr,γ)))}.Γ1(ψr−1,ψr−1,γβ(Θfb(ψr−1,ψr,γ)))=min{Θfb(ψr,ψr+1,γβ(Θfb(ψr−1,ψr,γ))),Θfb(ψr−1,ψr,γβ(Θfb(ψr−1,ψr,γ)))}. | (2.14) |
Γ2(ψr−1,ψr,γβ(Θfb(ψr−1,ψr,γ)))=max{HFΘfb(ψr−1,Ξψr,γβ(Θfb(ψr−1,ψr,γ))),HFΘfb(Ξψr−1,ψr,γβ(Θfb(ψr−1,ψr,γ)))}=max{Θfb(ψr−1,ψr+1,γβ(Θfb(ψr−1,ψr,γ))),Θfb(ψr,ψr,γβ(Θfb(ψr−1,ψr,γ)))}=max{Θfb(ψr−1,ψr+1,γβ(Θfb(ψr−1,ψr,γ))),1}. Γ2(ψr−1,ψr,γβ(Θfb(ψr−1,ψr,γ)))=1. | (2.15) |
Using (2.14) and (2.15) in (2.13) we have
Θfb(ψr,ψr+1,γ)≥min{Θfb(ψr,ψr+1,γβ(Θfb(ψr−1,ψr,γ))),Θfb(ψr−1,ψr,γβ(Θfb(ψr−1,ψr,γ)))}∗1,Θfb(ψr,ψr+1,γ)≥min{Θfb(ψr,ψr+1,γβ(Θfb(ψr−1,ψr,γ))),Θfb(ψr−1,ψr,γβ(Θfb(ψr−1,ψr,γ)))}. | (2.16) |
If
min{Θfb(ψr,ψr+1,γβ(Θfb(ψr−1,ψr,γ))),Θfb(ψr−1,ψr,γβ(Θfb(ψr−1,ψr,γ)))}=Θfb(ψr,ψr+1,γβ(Θfb(ψr−1,ψr,γ))), |
then (2.16) implies
Θfb(ψr,ψr+1,γ)≥Θfb(ψr,ψr+1,γβ(Θfb(ψr−1,ψr,γ))) |
Then the proof follows by Lemma 1.2
If
min{Θfb(ψr,ψr+1,γβ(Θfb(ψr−1,ψr,γ))),Θfb(ψr−1,ψr,γβ(Θfb(ψr−1,ψr,γ)))}=Θfb(ψr−1,ψr,γβ(Θfb(ψr−1,ψr,γ))), |
then from (2.16), we have
Θfb(ψr,ψr+1,γ)≥Θfb(ψr−1,ψr,γβ(Θfb(ψr−1,ψr,γ)))⋮⩾Θfb(ψ0,ψ1,γβ(Θfb(ψr−1,ψr,γ))β(Θfb(ψr−2,ψr−1,γ))…β(Θfb(ψ0,ψ1,γ))). |
The remaining proof is similar as in Theorem 2.1.
Remark 2.5. If we set ^C0(℧)=℧ the map Ξ becomes a singlevalued and we get Theorem 3.11 of [32]. Again as stated in Remark 2.1, the corresponding fixed point will be unique.
Theorem 2.7. Let (℧,Θfb,∗) be a G-complete Fbms with b⩾1 and (^C0(℧),HFΘfb,∗) be a Hfbms. Let Ξ:℧→ˆC0(℧) be a Mvp satisfying
HFΘfb(Ξψ,Ξϱ,β(Θfb(ψ,ϱ,γ))γ)≥Γ1(ψ,ϱ,γ)∗Γ2(ψ,ϱ,γ)Γ3(ψ,ϱ,γ), | (2.17) |
where
{Γ1(ψ,ϱ,γ)=min{HFΘfb(Ξψ,Ξϱ,γ).Θfb(ψ,ϱ,γ),HFΘfb(ψ,Ξψ,γ).HFΘfb(ϱ,Ξϱ,γ)}Γ2(ψ,ϱ,γ)=max{HFΘfb(ψ,Ξψ,γ).HFΘfb(ψ,Ξϱ,γ),HFΘfb(ϱ,Ξψ,γ))2}Γ3(ψ,ϱ,γ)=max{HFΘfb(ψ,Ξψ,γ),HFΘfb(ϱ,Ξϱ,γ)}}, | (2.18) |
for all ψ,ϱ∈℧, and β∈Ffb. Then Ξ has a fixed point.
Proof. In the same way as Theorem 2.1, we have
Θfb(ψ1,ψ2,γ)⩾HFΘfb(Ξψ0,Ξψ1,γ),∀γ>0. |
By induction we have ψr+1∈Ξψr satisfying
Θfb(ψr,ψr+1,γ)=HFΘfb(Ξψr−1,Ξψr,γ)≥Γ1(ψr−1,ψr,γβ(Θfb(ψr−1,ψr,γ)))∗Γ2(ψr−1,ψr,γβ(Θfb(ψr−1,ψr,γ)))Γ3(ψ,ϱ,γβ(Θfb(ψr−1,ψr,γ))). | (2.19) |
Γ1(ψr−1,ψr,γβ(Θfb(ψr−1,ψr,γ)))=min{HFΘfb(Ξψr−1,Ξψr,γβ(Θfb(ψr−1,ψr,γ))).Fθ(ψr−1,ψr,γβ(Θfb(ψr−1,ψr,γ))),HFΘfb(ψr−1,Ξψr−1,γβ(Θfb(ψr−1,ψr,γ))).HFΘfb(ψr,Ξψr,γβ(Θfb(ψr−1,ψr,γ)))}=min{Θfb(ψr,ψr+1,γβ(Θfb(ψr−1,ψr,γ))).Θfb(ψr−1,ψr,γβ(Θfb(ψr−1,ψr,γ))),Θfb(ψr−1,ψr,γβ(Θfb(ψr−1,ψr,γ))).Θfb(ψr,ψr+1,γβ(Θfb(ψr−1,ψr,γ)))}=Θfb(ψr,ψr+1,γβ(Θfb(ψr−1,ψr,γ))).Θfb(ψr−1,ψr,γβ(Θfb(ψr−1,ψr,γ))). | (2.20) |
Similarly,
Γ2(ψr−1,ψr,γβ(Θfb(ψr−1,ψr,γ)))=max{HFΘfb(ψr−1,Ξψr−1,γβ(Θfb(ψr−1,ψr,γ))),HFΘfb(ψr−1,Ξψr,γβ(Θfb(ψr−1,ψr,γ))),(HFΘfb(ψr,Ξψr−1,γβ(Θfb(ψr−1,ψr,γ))))2}=max{Θfb(ψr−1,ψr,γβ(Θfb(ψr−1,ψr,γ))),Θfb(ψr−1,ψr+1,γβ(Θfb(ψr−1,ψr,γ))),(Θfb(ψr,ψr,γβ(Θfb(ψr−1,ψr,γ))))2}=max{Θfb(ψr−1,ψr,γβ(Θfb(ψr−1,ψr,γ))),Θfb(ψr−1,ψr+1,γβ(Θfb(ψr−1,ψr,γ))),1}. |
It follows that
Γ2(ψr−1,ψr,γβ(Θfb(ψr−1,ψr,γ)))=1. | (2.21) |
Γ3(ψr−1,ψr,γβ(Θfb(ψr−1,ψr,γ)))=max{HFΘfb(ψr−1,Ξψr−1,γβ(Θfb(ψr−1,ψr,γ))),HFΘfb(ψr,Ξψr,γβ(Θfb(ψr−1,ψr,γ)))}=max{Θfb(ψr−1,ψr,γβ(Θfb(ψr−1,ψr,γ))),Θfb(ψr,ψr+1,γβ(Θfb(ψr−1,ψr,γ)))}. | (2.22) |
Using (2.20), (2.21) and (2.22) in (2.19), we have
Θfb(ψr,ψr+1,t)≥Θfb(ψr,ψr+1,γβ(Θfb(ψr−1,ψr,γ))).Θfb(ψr−1,ψr,γβ(Θfb(ψr−1,ψr,γ)))max{Θfb(ψr−1,ψr,γβ(Θfb(ψr−1,ψr,γ))),Θfb(ψr,ψr+1,γβ(Θfb(ψr−1,ψr,γ)))}. | (2.23) |
If
max{Θfb(ψr,ψr+1,γβ(Θfb(ψr−1,ψr,γ))),Θfb(ψr−1,ψr,γβ(Θfb(ψr−1,ψr,γ)))}=Θfb(ψr−1,ψr,γβ(Θfb(ψr−1,ψr,γ))), |
then (2.23) implies
Θfb(ψr,ψr+1,γ)≥Θfb(ψr,ψr+1,γβ(Θfb(ψr−1,ψr,γ))). |
It is obvious by Lemma 1.2.
If
max{Θfb(ψr,ψr+1,γβ(Θfb(ψr−1,ψr,γ))),Θfb(ψr−1,ψr,γβ(Θfb(ψr−1,ψr,γ)))}=Θfb(ψr,ψr+1,γβ(Θfb(ψr−1,ψr,γ))), |
then from (2.23), we have
Θfb(ψr,ψr+1,γ)≥Θfb(ψr−1,ψr,γβ(Θfb(ψr−1,ψr,γ))). |
Continuing in this way, we will get
Θfb(ψr,ψr+1,t)≥Θfb(ψr−1,ψr,γβ(Fθ(ψr−1,ψr,γ)))⋮≥Θfb(ψ0,ψ1,γβ(Θfb(ψr−1,ψr,γ)).β(Θfb(ψn−2,ψr−1,γ))…β(Θfb(ψ0,ψ1,γ))). |
The rest of the proof follows in the same way as in Theorem 2.1.
Remark 2.6. By setting ^C0(℧)=℧, the mapping Ξ:℧→^C0(℧) becomes a self (singlevalued) mapping and we get Theorem 3.13 of [32].
An application of Theorem 2.1 is presented here. Recall that the space of all continuous realvalued functions on [0,1] is denoted by C([0,1],R). Now set ℧=C([0,1],R) and define the G-complete Fbm on ℧ by
Θfb(ψ,ϱ,γ)=e−supu∈[0,1]|ψ(u)−ϱ(u)|2γ,∀γ>0andψ,ϱ∈℧. |
Consider
ψ(u)∈∫u0G(u,v,ψ(v))dv+h(u)for allu,v∈[0,1],whereash,ψ∈C([0,1],R). | (3.1) |
Here G:[0,1]×[0,1]×R→Pcv(R) is multivalued function and Pcv(R) represents the collections of convex and compact subsets of R. Moreover, for each ψ in C([0,1],R) the operator G(⋅,⋅,ψ) is lower semi-continuous.
For the integral inclusion given in (3.1), define a multivalued operator S:℧→^C0(℧) by
Sψ(u)={w∈℧:w∈∫u0G(u,v,ψ(v))dv+h(u),u∈[0,1]}. |
Now for arbitrary ψ∈(C([0,1],R), denote Gψ(u,v)=G(u,v,ψ(v)) where u,v∈[0,1]. For the multivalued map Gψ:[0,1]×[0,1]→Pcv(R), by Michael selection theorem [34], there exists a continuous selection gψ:[0,1]×[0,1]→R such that gψ(u,v)∈Gψ(u,v) for each u,v∈[0,1]. It follows that
∫u0gψ(u,v)dv+h(u)∈Sψ(u). |
Since gψ is continuous on [0,1]×[0,1] and h is continuous on [0,1], therefore both gψ and h are bounded realvalued functions. It follows that, the operator Sψ is nonempty and Sψ∈^C0(℧).
With the above setting, the upcoming outcome shows the existence of a solution of the integral inclusion (3.1).
Theorem 3.1. Let ℧=C([0,1],R) and define the multivalued operator S:℧→^C0(℧) by
Sψ(u)={w∈℧:w∈∫u0G(u,v,ψ(v))dv+h(u),u∈[0,1]}, |
where h:[0,1]→R is continuous and the map G:[0,1]×[0,1]×R→Pcv(R) is defined in such a way that for every ψ∈C([0,1],R), the operator G(⋅,⋅,ψ) is lower semi-continuous. Assume further that the given terms are satisfied:
(i) There exists a continuous mapping f:[0,1]×[0,1]→[0,∞) such that
HFΘfb(G(u,v,ψ(v))−G(u,v,ϱ(v))≤f2(u,v)|ψ(v)−ϱ(v)|2, |
for each ψ,ϱ∈℧ and u,v∈[0,1].
(ii) There exists β∈FΘ2, such that
supu∈[0,1]∫u0f2(u,v)dv≤β(Θfb(ψ,ϱ,γ)). |
Then (3.1) has a solution in ℧.
Proof. We will show that the operator S satisfies the conditions of Theorem 2.1. In particular we prove (2.1) as follows:
Let ψ,ϱ∈℧ be such that q∈Sψ. As stated earlier, by selection theorem there is gψ(u,v)∈Gψ(u,v)=G(u,v,ψ(v)) for u,v∈[0,1] such that
q(u)=∫u0gψ(u,v)dv+h(u),u∈[0,1]. |
Further, the condition (ⅰ) ensures that there is some g(u,v)∈Gϱ(u,v) such that
|gψ(u,v)−g(u,v)≤f2(u,v)|ψ(v)−ϱ(v)|2,∀u,v∈[0,1]. |
Now consider the multivalued operator T defined as follows:
T(u,v)=Gϱ(u,v)∩{w∈R:|gψ(u,v)−w|≤f2(u,v)|ψ(v)−ϱ(v)|2}. |
Since, by construction, T is lower semi-continuous, it follows again by the selection theorem that there is continuous function gϱ(u,v):[0,1]×[0,1]→R such that for each u,v∈[0,1], gϱ(u,v)∈T(u,v).
Thus, we have
r(u)=∫u0gϱ(u,v)dv+h(u)∈∫u0G(u,v,ϱ(v))dv+h(u),u∈[0,1]. |
Therefore, for each u∈[0,1] we get
e−supt∈[0,1]|q(u)−r(u))|2β(Θfb(ψ,ϱ,γ))γ≥e−supu∈[0,1]∫u0|gψ(u,v)−gϱ(u,v)|2dvβ(Θfb(ψ,ϱ,γ))γ≥e−supu∈[0,1]∫u0f2(u,v)|ψ(v)−ϱ(v)|2dvβ(Θfb(ψ,ϱ,γ))γ≥e−|ψ(v)−ϱ(v)|2supu∈[0,1]∫u0f2(u,v)dvβ(Θfb(ψ,ϱ,γ))γ≥e−β(Θfb(ψ,ϱ,γ))|ψ(v)−ϱ(v)|2β(Θfb(ψ,ϱ,γ))γ=e−|ψ(v)−ϱ(v)|2γ≥e−supv∈[0,1]|ψ(v)−ϱ(v)|2γ=Θfb(ψ,ϱ,γ). |
This implies that,
Θfb(q,r,β(Θfb(ψ,ϱ,γ))γ)≥Θfb(ψ,ϱ,γ). |
Interchanging the roles of ψ and ϱ, we get
HFΘfb(Sψ,Sϱ,β(Θfb(ψ,ϱ,γ))γ)≥Θfb(ψ,ϱ,γ). |
Hence, by Theorem 2.1, the operator S has a fixed point which in turn proves the existence of a solution of integral inclusion (3.1).
In the present work, in the setting of a Hausdorff Fbms, some fixed fixed point results for multivalued mappings are established. The main result, that is, Theorem 2.1 shows that a multivalued mapping satisfying Geraghty type contractions on G-complete Hfbms has a fixed point. Example 2.1 illustrates the main result. Some other interesting fixed point theorems are also proved for the multivalued mappings satisfying certain contraction condition on G-complete Hfbms. The results proved in [30,31,32] turn out to be special cases of the results established in this work. For the significance of our results, an application is presented to prove the existence of solution of an integral inclusion.
The authors would like to thank the Deanship of Scientific Research at Umm Al-Qura University for supporting this work grant code: 22UQU4331214DSR02
The authors declare that they have no conflict of interest.
[1] | Reitsma MB, Kendrick PJ, Ababneh E, et al. (2021) Spatial, temporal, and demographic patterns in prevalence of smoking tobacco use and attributable disease burden in 204 countries and territories, 1990–2019: a systematic analysis from the Global Burden of Disease Study 2019. Lancet 397: 2337-2360. https://doi.org/10.1016/S0140-6736(21)01169-7 |
[2] | Rezk-Hanna M, Benowitz NL (2019) Cardiovascular Effects of Hookah Smoking: Potential Implications for Cardiovascular Risk. Nicotine Tob Res 21: 1151-1161. https://doi.org/10.1093/ntr/nty065 |
[3] | Bade BC, Dela Cruz CS (2020) Lung Cancer 2020. Clin Chest Med 41: 1-24. https://doi.org/10.1016/j.ccm.2019.10.001 |
[4] | Duffy SP, Criner GJ (2019) Chronic Obstructive Pulmonary Disease. Med Clin N Am 103: 453-461. https://doi.org/10.1016/j.mcna.2018.12.005 |
[5] | Le Foll B, Piper ME, Fowler CD, et al. (2022) Tobacco and nicotine use. Nat Rev Dis Primers 8: 19. https://doi.org/10.1038/s41572-022-00346-w |
[6] | Potvin S, Tikàsz A, Dinh-Williams LL-A, et al. (2015) Cigarette Cravings, Impulsivity, and the Brain. Front Psychiatry 6. https://doi.org/10.3389/fpsyt.2015.00125 |
[7] | Goldstein RZ, Volkow ND (2011) Dysfunction of the prefrontal cortex in addiction: neuroimaging findings and clinical implications. Nat Rev Neurosci 12: 652-669. https://doi.org/10.1038/nrn3119 |
[8] | Basile GA, Bertino S, Bramanti A, et al. (2021) Striatal topographical organization: Bridging the gap between molecules, connectivity and behavior. Eur J Histochem 65. https://doi.org/10.4081/ejh.2021.3284 |
[9] | Pipe AL, Evans W, Papadakis S (2022) Smoking cessation: health system challenges and opportunities. Tob Control 31: 340-347. https://doi.org/10.1136/tobaccocontrol-2021-056575 |
[10] | Rachid F (2016) Neurostimulation techniques in the treatment of nicotine dependence: A review. Am J Addict 25: 436-451. https://doi.org/10.1111/ajad.12405 |
[11] | Chase HW, Boudewyn MA, Carter CS, et al. (2020) Transcranial direct current stimulation: a roadmap for research, from mechanism of action to clinical implementation. Mol Psychiatry 25: 397-407. https://doi.org/10.1038/s41380-019-0499-9 |
[12] | Page MJ, McKenzie JE, Bossuyt PM, et al. (2021) The PRISMA 2020 statement: an updated guideline for reporting systematic reviews. BMJ : n71. https://doi.org/10.1136/bmj.n71 |
[13] | Abdelrahman AA, Noaman M, Fawzy M, et al. (2021) A double-blind randomized clinical trial of high frequency rTMS over the DLPFC on nicotine dependence, anxiety and depression. Sci Rep 11: 1640. https://doi.org/10.1038/s41598-020-80927-5 |
[14] | Amiaz R, Levy D, Vainiger D, et al. (2009) Repeated high-frequency transcranial magnetic stimulation over the dorsolateral prefrontal cortex reduces cigarette craving and consumption. Addiction 104: 653-660. https://doi.org/10.1111/j.1360-0443.2008.02448.x |
[15] | Li X, Hartwell KJ, Henderson S, et al. (2020) Two weeks of image-guided left dorsolateral prefrontal cortex repetitive transcranial magnetic stimulation improves smoking cessation: A double-blind, sham-controlled, randomized clinical trial. Brain Stimul 13: 1271-1279. https://doi.org/10.1016/j.brs.2020.06.007 |
[16] | Sheffer CE, Bickel WK, Brandon TH, et al. (2018) Preventing relapse to smoking with transcranial magnetic stimulation: Feasibility and potential efficacy. Drug Alcohol Depend 182: 8-18. https://doi.org/10.1016/j.drugalcdep.2017.09.037 |
[17] | Trojak B, Meille V, Achab S, et al. (2015) Transcranial Magnetic Stimulation Combined With Nicotine Replacement Therapy for Smoking Cessation: A Randomized Controlled Trial. Brain Stimul 8: 1168-1174. https://doi.org/10.1016/j.brs.2015.06.004 |
[18] | Dieler AC, Dresler T, Joachim K, et al. (2014) Can Intermittent Theta Burst Stimulation as Add-On to Psychotherapy Improve Nicotine Abstinence? Results from a Pilot Study. Eur Addict Res 20: 248-253. https://doi.org/10.1159/000357941 |
[19] | Dinur-Klein L, Dannon P, Hadar A, et al. (2014) Smoking Cessation Induced by Deep Repetitive Transcranial Magnetic Stimulation of the Prefrontal and Insular Cortices: A Prospective, Randomized Controlled Trial. Biol Psychiatry 76: 742-749. https://doi.org/10.1016/j.biopsych.2014.05.020 |
[20] | Zangen A, Moshe H, Martinez D, et al. (2021) Repetitive transcranial magnetic stimulation for smoking cessation: a pivotal multicenter double-blind randomized controlled trial. World Psychiatry 20: 397-404. https://doi.org/10.1002/wps.20905 |
[21] | Ibrahim C, Tang VM, Blumberger DM, et al. (2023) Efficacy of insula deep repetitive transcranial magnetic stimulation combined with varenicline for smoking cessation: A randomized, double-blind, sham controlled trial. Brain Stimul 16: 1501-1509. https://doi.org/10.1016/j.brs.2023.10.002 |
[22] | Ghorbani Behnam S, Mousavi SA, Emamian MH (2019) The effects of transcranial direct current stimulation compared to standard bupropion for the treatment of tobacco dependence: A randomized sham-controlled trial. Eur Psychiat 60: 41-48. https://doi.org/10.1016/j.eurpsy.2019.04.010 |
[23] | Tseng P, Jeng J, Zeng B, et al. (2022) Efficacy of non-invasive brain stimulation interventions in reducing smoking frequency in patients with nicotine dependence: a systematic review and network meta-analysis of randomized controlled trials. Addiction 117: 1830-1842. https://doi.org/10.1111/add.15624 |
[24] | Petit B, Dornier A, Meille V, et al. (2022) Non-invasive brain stimulation for smoking cessation: a systematic review and meta-analysis. Addiction 117: 2768-2779. https://doi.org/10.1111/add.15889 |
[25] | Lefaucheur J-P, Aleman A, Baeken C, et al. (2020) Evidence-based guidelines on the therapeutic use of repetitive transcranial magnetic stimulation (rTMS): An update (2014–2018). Clin Neurophysiol 131: 474-528. https://doi.org/10.1016/j.clinph.2019.11.002 |
[26] | Lefaucheur J-P, André-Obadia N, Antal A, et al. (2014) Evidence-based guidelines on the therapeutic use of repetitive transcranial magnetic stimulation (rTMS). Clin Neurophysiol 125: 2150-2206. https://doi.org/10.1016/j.clinph.2014.05.021 |
[27] | Mahoney JJ, Hanlon CA, Marshalek PJ, et al. (2020) Transcranial magnetic stimulation, deep brain stimulation, and other forms of neuromodulation for substance use disorders: Review of modalities and implications for treatment. J Neurol Sci 418: 117149. https://doi.org/10.1016/j.jns.2020.117149 |
[28] | Liu Q, Yuan T (2021) Noninvasive brain stimulation of addiction: one target for all?. Psychoradiology 1: 172-184. https://doi.org/10.1093/psyrad/kkab016 |
[29] | Kang N, Kim RK, Kim HJ (2019) Effects of transcranial direct current stimulation on symptoms of nicotine dependence: A systematic review and meta-analysis. Addict Behav 96: 133-139. https://doi.org/10.1016/j.addbeh.2019.05.006 |
[30] | Watson NL, Carpenter MJ, Saladin ME, et al. (2010) Evidence for greater cue reactivity among low-dependent vs. high-dependent smokers. Addict Behav 35: 673-677. https://doi.org/10.1016/j.addbeh.2010.02.010 |
[31] | Kim W-S, Paik N-J (2021) Safety Review for Clinical Application of Repetitive Transcranial Magnetic Stimulation. Brain Neurorehabilitation 14. https://doi.org/10.12786/bn.2021.14.e6 |
[32] | Mattioli F, Maglianella V, D'Antonio S, et al. (2024) Non-invasive brain stimulation for patients and healthy subjects: Current challenges and future perspectives. J Neurol Sci 456: 122825. https://doi.org/10.1016/j.jns.2023.122825 |
[33] | Wessel MJ, Egger P, Hummel FC (2021) Predictive models for response to non-invasive brain stimulation in stroke: A critical review of opportunities and pitfalls. Brain Stimul 14: 1456-1466. https://doi.org/10.1016/j.brs.2021.09.006 |
1. | Wee Chin Tan, Viet Ha Hoang, High dimensional finite element method for multiscale nonlinear monotone parabolic equations, 2019, 345, 03770427, 471, 10.1016/j.cam.2018.04.002 | |
2. | Shubin Fu, Eric Chung, Tina Mai, Generalized multiscale finite element method for a strain-limiting nonlinear elasticity model, 2019, 359, 03770427, 153, 10.1016/j.cam.2019.03.047 | |
3. | Xinliang Liu, Eric Chung, Lei Zhang, Iterated Numerical Homogenization for MultiScale Elliptic Equations with Monotone Nonlinearity, 2021, 19, 1540-3459, 1601, 10.1137/21M1389900 | |
4. | Barbara Verfürth, Numerical homogenization for nonlinear strongly monotone problems, 2022, 42, 0272-4979, 1313, 10.1093/imanum/drab004 | |
5. | Minam Moon, Generalized multiscale hybridizable discontinuous Galerkin (GMsHDG) method for flows in nonlinear porous media, 2022, 415, 03770427, 114441, 10.1016/j.cam.2022.114441 | |
6. | Weifeng Qiu, Ke Shi, Analysis on an HDG Method for the p-Laplacian Equations, 2019, 80, 0885-7474, 1019, 10.1007/s10915-019-00967-6 |