Currently, interdigital capacitive (IDC) sensors are widely used in science, industry and technology. To measure the changes in capacitance in these sensors, many methods such as differentiation, phase delay between two signals, capacitor charging/discharging, oscillators and switching circuits have been proposed. These techniques often use high frequencies and high complexity to measure small capacitance changes of fF or aF with high sensitivity. An analog interface based on a capacitance multiplier for capacitive sensors is presented. This study includes analysis of the interface error factors, such as the error due to the components of the capacitance multiplier, parasitic capacitances, transient effects and non-ideal parameters of OpAmp. A design approach based on an IDC sensor to measure the quality of edible oils is presented and implemented. The quality relates to the total polar compounds (TPC) and consequently to relative electrical permittivity
Citation: Vasileios Delimaras, Kyriakos Tsiakmakis, Argyrios T. Hatzopoulos. Analog interface based on capacitance multiplier for capacitive sensors and application to evaluate the quality of oils[J]. AIMS Electronics and Electrical Engineering, 2023, 7(4): 243-270. doi: 10.3934/electreng.2023015
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Currently, interdigital capacitive (IDC) sensors are widely used in science, industry and technology. To measure the changes in capacitance in these sensors, many methods such as differentiation, phase delay between two signals, capacitor charging/discharging, oscillators and switching circuits have been proposed. These techniques often use high frequencies and high complexity to measure small capacitance changes of fF or aF with high sensitivity. An analog interface based on a capacitance multiplier for capacitive sensors is presented. This study includes analysis of the interface error factors, such as the error due to the components of the capacitance multiplier, parasitic capacitances, transient effects and non-ideal parameters of OpAmp. A design approach based on an IDC sensor to measure the quality of edible oils is presented and implemented. The quality relates to the total polar compounds (TPC) and consequently to relative electrical permittivity
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
\begin{align*} &\quad\max \lbrace \Theta_{fb}(\psi_{r}, \psi_{r+1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}) , \Theta_{fb}(\psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}) \rbrace \\ & = \Theta_{fb}\Bigl(\psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr), \end{align*} |
then (2.10) implies
\Theta_{fb}(\psi_{r}, \psi_{r+1}, t)\geq \Theta_{fb}\Bigl(\psi_{r}, \psi_{r+1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr) |
Then the proof follows by Lemma 1.2.
If
\begin{align*} &\quad\max \lbrace \Theta_{fb}(\psi_{r}, \psi_{r+1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}) , \Theta_{fb}(\psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}) \rbrace\\ & = \Theta_{fb}\Bigl(\psi_{r}, \psi_{r+1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr), \end{align*} |
then from (2.10) we have
\begin{align*} \Theta_{fb}(\psi_{r}, \psi_{r+1}, \gamma) &\geq \Theta_{fb}\Bigr( \psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr)\\ & \vdots\\ & \geqslant \Theta_{fb}\Bigl( \psi_{0}, \psi_{1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))\beta(\Theta_{fb}(\psi_{r-2}, \psi_{r-1}, \gamma))\ldots \beta(\Theta_{fb}(\psi_{0}, \psi_{1}, \gamma))}\Bigr). \end{align*} |
The remaining proof follows in the same way as in Theorem 2.1.
Theorem 2.6. Let (\mho, \Theta_{fb}, *) be a G -complete Fbms with b\geqslant 1 and (\hat{C_{0}} (\mho), \mathcal{H}_{F_{\Theta_{fb}}}, *) be a Hfbms. Let \Xi \colon \mho \rightarrow \hat{C}_{0}(\mho) be a Mvp satisfying
\begin{equation} \mathcal{H}_{F_{\Theta_{fb}}}(\Xi\psi, \Xi\varrho, \beta(\Theta_{fb}(\psi, \varrho, \gamma))\gamma)\geq \Gamma_{1}(\psi, \varrho, \gamma)* \Gamma_{2}(\psi, \varrho, \gamma), \end{equation} | (2.11) |
where,
\begin{align} \left. \begin{cases} \Gamma_{1}(\psi, \varrho, \gamma)& = \min \lbrace \mathcal{H}_{F_{\Theta_{fb}}}(\Xi\psi, \Xi\varrho, \gamma) , \mathcal{H}_{F_{\Theta_{fb}}}(\psi, \Xi\psi, \gamma), \mathcal{H}_{F_{\Theta_{fb}}}(\varrho, \Xi\varrho, \gamma), \Theta_{fb}(\psi, \varrho, \gamma) \rbrace\\ \Gamma_{2}(\psi, \varrho, \gamma)& = \max \lbrace \mathcal{H}_{F_{\Theta_{fb}}}(\psi, \Xi\varrho, \gamma) , \mathcal{H}_{F_{\Theta_{fb}}}(\Xi\psi, \varrho, \gamma) \rbrace \end{cases} \right\}, \end{align} | (2.12) |
for all \psi, \varrho \in \mho , and \beta \in \mathbb{F}_{fb}. Then \Xi has a fixed point.
Proof. In the same way as Theorem 2.1, we have
\begin{equation*} \Theta_{fb}(\psi_{1}, \psi_{2}, \gamma)\geqslant \mathcal{H}_{F_{\Theta_{fb}}}(\Xi \psi_{0}, \Xi \psi_{1}, \gamma), \quad \forall\; \gamma > 0.\\ \end{equation*} |
By induction we have \psi_{r+1} \in \Xi \psi_{r} satisfying
\begin{align} \mathcal{H}_{F_{\Theta_{fb}}}(\psi_{r}, \psi_{r+1}, \gamma)& = F_{\theta}(\Xi\psi_{r-1}, \Xi\psi_{r}, \gamma)\\ &\geq \Gamma_{1}\Bigl(\psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr)* \Gamma_{2}\Bigl(\psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr) \end{align} | (2.13) |
Now,
\begin{align} &\quad\Gamma_{1}\Bigl(\psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr)\\ & = \min \Biggl \lbrace \mathcal{H}_{F_{\Theta_{fb}}}\Bigl(\Xi\psi_{r-1}, \Xi\psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr), \mathcal{H}_{F_{\Theta_{fb}}}\Bigl(\psi_{r-1}, \Xi\psi_{r-1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr), \\ &\quad\quad\quad \mathcal{H}_{F_{\Theta_{fb}}}\Bigl(\psi_{r}, \Xi\psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr), \Theta_{fb}\Bigl(\psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr)\Biggr \rbrace \\ & = \min \Biggl \lbrace \Theta_{fb}\Bigl(\psi_{r}, \psi_{r+1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr), \Theta_{fb}\Bigl(\psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr), \\ & \quad\quad\quad \Theta_{fb}\Bigl(\psi_{r}, \psi_{r+1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr), \Theta_{fb}\Bigl(\psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr)\Biggr \rbrace. \\&\quad\Gamma_{1}(\psi_{r-1}, \psi_{r-1}, \frac{\gamma}{\beta(\Theta_{fb}\Bigl(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr) \\ & = \min \Biggl\lbrace \Theta_{fb}\Bigl(\psi_{r}, \psi_{r+1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr), \Theta_{fb}\Bigl(\psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr) \biggr\rbrace. \end{align} | (2.14) |
\begin{align} &\quad\Gamma_{2}\Bigl(\psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigl) \\ & = \max \Bigl\lbrace \mathcal{H}_{F_{\Theta_{fb}}}\Bigl(\psi_{r-1}, \Xi\psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr) , \mathcal{H}_{F_{\Theta_{fb}}}\Bigl(\Xi\psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr)\Bigr\rbrace\\ & = \max \Bigl\lbrace \Theta_{fb}\Bigl(\psi_{r-1}, \psi_{r+1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr) , \Theta_{fb}\Bigl(\psi_{r}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr) \Bigr\rbrace \\ & = \max \Bigl\lbrace \Theta_{fb}\Bigl(\psi_{r-1}, \psi_{r+1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr) , 1 \Bigr\rbrace. \notag \\ \\ & \ \ \ \ \ \ \ \ \ \Gamma_{2}\Bigl(\psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr) = 1. \end{align} | (2.15) |
Using (2.14) and (2.15) in (2.13) we have
\begin{align} \Theta_{fb}(\psi_{r}, \psi_{r+1}, \gamma)&\geq \min \Bigl\lbrace \Theta_{fb}\Bigl(\psi_{r}, \psi_{r+1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr), \Theta_{fb}\Bigl(\psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr) \Bigr\rbrace *1, \\ \Theta_{fb}(\psi_{r}, \psi_{r+1}, \gamma)&\geq \min \Bigl\lbrace \Theta_{fb}\Bigl(\psi_{r}, \psi_{r+1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr), \Theta_{fb}\Bigl(\psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr) \Bigr\rbrace. \end{align} | (2.16) |
If
\begin{align*} &\min \Bigg\lbrace \Theta_{fb}\Bigl(\psi_{r}, \psi_{r+1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr) , \Theta_{fb}\Bigl(\psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr) \Bigg\rbrace \\ & = \Theta_{fb}\Bigl(\psi_{r}, \psi_{r+1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigl), \end{align*} |
then (2.16) implies
\Theta_{fb}(\psi_{r}, \psi_{r+1}, \gamma)\geq \Theta_{fb}\Bigl(\psi_{r}, \psi_{r+1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigl) |
Then the proof follows by Lemma 1.2
If
\begin{align*} &\quad\min \Bigl\lbrace \Theta_{fb}\Bigl(\psi_{r}, \psi_{r+1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr) , \Theta_{fb}\Bigl(\psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr) \Bigr\rbrace\\ & = \Theta_{fb}\Bigl(\psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr), \end{align*} |
then from (2.16), we have
\begin{align*} \Theta_{fb}(\psi_{r}, \psi_{r+1}, \gamma) &\geq \Theta_{fb}\Bigl( \psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr)\\ & \vdots \\ &\geqslant \Theta_{fb}\Bigl( \psi_{0}, \psi_{1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))\beta(\Theta_{fb}(\psi_{r-2}, \psi_{r-1}, \gamma))\ldots \beta(\Theta_{fb}(\psi_{0}, \psi_{1}, \gamma))}\Big). \end{align*} |
The remaining proof is similar as in Theorem 2.1.
Remark 2.5. If we set \hat{C_{0}} (\mho) = \mho the map \Xi 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 (\mho, \Theta_{fb}, *) be a G -complete Fbms with b\geqslant 1 and (\hat{C_{0}} (\mho), \mathcal{H}_{F_{\Theta_{fb}}}, *) be a Hfbms. Let \Xi \colon \mho \rightarrow \hat{C}_{0}(\mho) be a Mvp satisfying
\begin{align} \mathcal{H}_{F_{\Theta_{fb}}}(\Xi\psi, \Xi\varrho, \beta(\Theta_{fb}(\psi, \varrho, \gamma))\gamma)\geq \dfrac{\Gamma_{1}(\psi, \varrho, \gamma)* \Gamma_{2}(\psi, \varrho, \gamma)}{\Gamma_{3}(\psi, \varrho, \gamma)}, \end{align} | (2.17) |
where
\begin{align} \left. \begin{cases} \Gamma_{1}(\psi, \varrho, \gamma)& = \min \lbrace \mathcal{H}_{F_{\Theta_{fb}}}(\Xi\psi, \Xi\varrho, \gamma). \Theta_{fb}(\psi, \varrho, \gamma) , \mathcal{H}_{F_{\Theta_{fb}}}(\psi, \Xi\psi, \gamma). \mathcal{H}_{F_{\Theta_{fb}}}(\varrho, \Xi\varrho, \gamma) \rbrace\\ \Gamma_{2}(\psi, \varrho, \gamma)& = \max \lbrace \mathcal{H}_{F_{\Theta_{fb}}}(\psi, \Xi\psi, \gamma). \mathcal{H}_{F_{\Theta_{fb}}}(\psi, \Xi\varrho, \gamma) , \mathcal{H}_{F_{\Theta_{fb}}}(\varrho, \Xi\psi, \gamma))^{2}\rbrace\\ \Gamma_{3}(\psi, \varrho, \gamma)& = \max \lbrace \mathcal{H}_{F_{\Theta_{fb}}}(\psi, \Xi\psi, \gamma), \mathcal{H}_{F_{\Theta_{fb}}}(\varrho, \Xi\varrho, \gamma)\rbrace \end{cases} \right\}, \end{align} | (2.18) |
for all \psi, \varrho \in \mho , and \beta \in \mathbb{F}_{fb}. Then \Xi has a fixed point.
Proof. In the same way as Theorem 2.1, we have
\begin{equation*} \Theta_{fb}(\psi_{1}, \psi_{2}, \gamma)\geqslant \mathcal{H}_{F_{\Theta_{fb}}}(\Xi \psi_{0}, \Xi \psi_{1}, \gamma), \quad \forall\; \gamma > 0. \end{equation*} |
By induction we have \psi_{r+1} \in \Xi \psi_{r} satisfying
\begin{align} \Theta_{fb}(\psi_{r}, \psi_{r+1}, \gamma)& = \mathcal{H}_{F_{\Theta_{fb}}}(\Xi\psi_{r-1}, \Xi\psi_{r}, \gamma)\\ &\geq \dfrac{\Gamma_{1}(\psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))})* \Gamma_{2}(\psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))})}{\Gamma_{3}(\psi, \varrho, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))})}. \end{align} | (2.19) |
\begin{align} &\quad\Gamma_{1}\Bigl(\psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr) \\ & = \min \Biggl \lbrace \mathcal{H}_{F_{\Theta_{fb}}}\Bigl(\Xi\psi_{r-1}, \Xi\psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr).F_{\theta}\Bigl(\psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr) , \\ &\quad\quad\quad \mathcal{H}_{F_{\Theta_{fb}}}(\psi_{r-1}, \Xi\psi_{r-1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}). \mathcal{H}_{F_{\Theta_{fb}}}(\psi_{r}, \Xi\psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}) \Biggr \rbrace \\ & = \min \Biggl \lbrace \Theta_{fb}(\psi_{r}, \psi_{r+1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}).\Theta_{fb}(\psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}) , \\ &\quad\quad\quad\Theta_{fb}\Bigl(\psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr). \Theta_{fb}\Bigl(\psi_{r}, \psi_{r+1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr) \Biggr \rbrace \\ & = \Theta_{fb}\Bigl(\psi_{r}, \psi_{r+1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr).\Theta_{fb}\Bigl(\psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr). \end{align} | (2.20) |
Similarly,
\begin{align} &\quad\Gamma_{2}\Bigl(\psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr)\\ & = \max \Biggl \lbrace \mathcal{H}_{F_{\Theta_{fb}}}\Bigl(\psi_{r-1}, \Xi\psi_{r-1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr), \mathcal{H}_{F_{\Theta_{fb}}}\Bigl(\psi_{r-1}, \Xi\psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr) , \\ &\quad\quad\quad \Bigl(\mathcal{H}_{F_{\Theta_{fb}}}\Bigl(\psi_{r}, \Xi\psi_{r-1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr)\Bigr)^{2} \Biggr \rbrace\\ & = \max \Biggl \lbrace \Theta_{fb}\Bigl(\psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr), \Theta_{fb}\Bigl(\psi_{r-1}, \psi_{r+1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr) , \\ &\quad\quad\quad \Bigl(\Theta_{fb}\Bigl(\psi_{r}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr)\Bigr)^{2} \Biggr \rbrace\\ & = \max \Biggl\lbrace \Theta_{fb}\Bigl(\psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr), \Theta_{fb}\Bigl(\psi_{r-1}, \psi_{r+1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr) , 1 \Biggr\rbrace. \end{align} |
It follows that
\begin{equation} \Gamma_{2}\Bigl(\psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr) = 1. \end{equation} | (2.21) |
\begin{align} &\quad\Gamma_{3}\Bigl(\psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr)\\ & = \max\Big\lbrace \mathcal{H}_{F_{\Theta_{fb}}}\Bigl(\psi_{r-1}, \Xi\psi_{r-1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Big), \mathcal{H}_{F_{\Theta_{fb}}}\Bigl(\psi_{r}, \Xi\psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr) \Big\rbrace \\ & = \max \Big\lbrace \Theta_{fb}\Bigl(\psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr), \Theta_{fb}\Bigl(\psi_{r}, \psi_{r+1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Big) \Big\rbrace. \end{align} | (2.22) |
Using (2.20), (2.21) and (2.22) in (2.19), we have
\begin{align} \Theta_{fb}(\psi_{r}, \psi_{r+1}, t)\geq \dfrac{ \Theta_{fb}(\psi_{r}, \psi_{r+1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}).\Theta_{fb}(\psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))})}{\max \lbrace \Theta_{fb}(\psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}), \Theta_{fb}(\psi_{r}, \psi_{r+1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}) \rbrace}. \end{align} | (2.23) |
If
\begin{align*} &\quad\max \Bigg \lbrace \Theta_{fb}\Bigl(\psi_{r}, \psi_{r+1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr) , \Theta_{fb}\Bigl(\psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr) \Bigg\rbrace \\ & = \Theta_{fb}\Bigl(\psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr), \end{align*} |
then (2.23) implies
\Theta_{fb}(\psi_{r}, \psi_{r+1}, \gamma)\geq \Theta_{fb}\Bigl(\psi_{r}, \psi_{r+1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr). |
It is obvious by Lemma 1.2.
If
\begin{align*} &\quad\max \Bigg\lbrace \Theta_{fb}\Bigl(\psi_{r}, \psi_{r+1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr) , \Theta_{fb}\Bigl(\psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr) \Bigg\rbrace \\ & = \Theta_{fb}\Bigl(\psi_{r}, \psi_{r+1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr), \end{align*} |
then from (2.23), we have
\begin{equation*} \Theta_{fb}(\psi_{r}, \psi_{r+1}, \gamma)\geq \Theta_{fb}\Bigl(\psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr). \end{equation*} |
Continuing in this way, we will get
\begin{align*} \Theta_{fb}(\psi_{r}, \psi_{r+1}, t)&\geq \Theta_{fb}\Bigl(\psi_{r-1}, \psi_{r}, \frac{\gamma}{\beta(F_{\theta}(\psi_{r-1}, \psi_{r}, \gamma))}\Bigr)\\ & \vdots\\ & \geq \Theta_{fb}\Bigl(\psi_{0}, \psi_{1}, \frac{\gamma}{\beta(\Theta_{fb}(\psi_{r-1}, \psi_{r}, \gamma)).\beta(\Theta_{fb}(\psi_{n-2}, \psi_{r-1}, \gamma))\ldots\beta(\Theta_{fb}(\psi_{0}, \psi_{1}, \gamma))}\Bigr). \end{align*} |
The rest of the proof follows in the same way as in Theorem 2.1.
Remark 2.6. By setting \hat{C_{_0}} (\mho) = \mho , the mapping \Xi \colon \mho \rightarrow \hat{C_{_0}} (\mho) 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], \mathbb{R}) . Now set \mho = C([0, 1], \mathbb{R}) and define the G -complete Fbm on \mho by
\Theta_{fb} (\psi, \varrho, \gamma) = e^{-\dfrac{ {\sup\limits_{u \in[0, 1]}}\vert \psi(u)-\varrho(u) \vert^{2}} {\gamma}}, \quad \forall \;\; \gamma > 0 \;\; \text{and} \; \; \psi, \varrho \in \mho. |
Consider
\begin{equation} \psi(u)\in \int_0^u G(u, v, \psi(v))dv + h(u) \quad \text{for all} \;\; u, v \in[0, 1], \quad \text{whereas}\quad h, \psi\in C([0, 1], \mathbb{R}). \end{equation} | (3.1) |
Here G \colon [0, 1]\times[0, 1] \times \mathbb{R}\rightarrow P_{cv}(\mathbb{R}) is multivalued function and P_{cv}(\mathbb{R}) represents the collections of convex and compact subsets of \mathbb{R} . Moreover, for each \psi in C([0, 1], \mathbb{R}) the operator G(\cdot, \cdot, \psi) is lower semi-continuous.
For the integral inclusion given in (3.1), define a multivalued operator S: \mho\rightarrow \hat{C_{0}} (\mho) by
S\psi(u) = \Biggl\lbrace w \in \mho : w \in \int_0^u G(u, v, \psi(v))dv + h(u), \quad u \in [0, 1]\Biggr\rbrace. |
Now for arbitrary \psi \in (C([0, 1], \mathbb{R}) , denote G_\psi(u, v) = G(u, v, \psi(v)) where u, v\in [0, 1] . For the multivalued map G_\psi : [0, 1]\times [0, 1]\rightarrow P_{cv}(\mathbb{R}) , by Michael selection theorem [34], there exists a continuous selection g_\psi :[0, 1] \times [0, 1]\rightarrow \mathbb{R} such that g_\psi(u, v)\in G_\psi(u, v) for each u, v\in [0, 1] . It follows that
\int_0^u g_\psi(u, v)dv + h(u) \in S\psi(u). |
Since g_\psi is continuous on [0, 1]\times [0, 1] and h is continuous on [0, 1] , therefore both g_\psi and h are bounded realvalued functions. It follows that, the operator S\psi is nonempty and S\psi \in \hat{C_{0}} (\mho) .
With the above setting, the upcoming outcome shows the existence of a solution of the integral inclusion (3.1).
Theorem 3.1. Let \mho = C([0, 1], \mathbb{R}) and define the multivalued operator S: \mho \rightarrow \hat{C_{0}} (\mho) by
S\psi(u) = \Biggl\lbrace w \in \mho : w \in \int_0^u G(u, v, \psi(v))dv + h(u), \quad u \in [0, 1]\Biggr\rbrace, |
where h:[0, 1]\rightarrow \mathbb{R} is continuous and the map G:[0, 1]\times [0, 1]\times \mathbb{R}\rightarrow P_{cv}(\mathbb{R}) is defined in such a way that for every \psi\in C([0, 1], \mathbb{R}) , the operator G(\cdot, \cdot, \psi) is lower semi-continuous. Assume further that the given terms are satisfied:
(i) There exists a continuous mapping f \colon [0, 1]\times [0, 1]\rightarrow [0, \infty) such that
\mathcal{H}_{F_{\Theta_{fb}}}( G(u, v, \psi(v))-G(u, v, \varrho(v))\leq f^{2}(u, v)\vert \psi(v)-\varrho(v)\vert^{2}, |
for each \psi, \varrho \in \mho and u, v\in [0, 1] .
(ii) There exists \beta \in \mathbb{F}_{\Theta 2} , such that
{ \sup\limits_{u \in[0, 1]}}\int_0^u f^{2}(u, v)dv \leq \beta({\Theta_{fb}}(\psi, \varrho, \gamma)) . |
Then (3.1) has a solution in \mho .
Proof. We will show that the operator S satisfies the conditions of Theorem 2.1. In particular we prove (2.1) as follows:
Let \psi, \varrho \in \mho be such that q\in S\psi . As stated earlier, by selection theorem there is g_\psi(u, v)\in G_\psi(u, v) = G(u, v, \psi(v)) for u, v \in [0, 1] such that
q(u) = \int_0^u g_\psi(u, v)dv+h(u), \quad u\in [0, 1]. |
Further, the condition (ⅰ) ensures that there is some g(u, v)\in G_\varrho(u, v) such that
\vert g_\psi(u, v)-g(u, v)\le f^{2}(u, v)\vert \psi(v)-\varrho(v)\vert^{2}, \quad\forall\, u, v\in [0, 1]. |
Now consider the multivalued operator T defined as follows:
T(u, v) = G_\varrho(u, v)\cap \left\lbrace w\in \mathbb{R} : \left\vert g_\psi(u, v)-w\right\vert\le f^{2}(u, v)\vert \psi(v)-\varrho(v)\vert^{2}\right\rbrace. |
Since, by construction, T is lower semi-continuous, it follows again by the selection theorem that there is continuous function g_\varrho(u, v):[0, 1]\times [0, 1]\rightarrow \mathbb{R} such that for each u, v \in [0, 1] , g_\varrho(u, v) \in T(u, v) .
Thus, we have
r(u) = \int_0^u g_\varrho(u, v)dv+h(u)\, \in \int_0^u G(u, v, \varrho(v))dv+h(u), \quad u\in [0, 1]. |
Therefore, for each u\in [0, 1] we get
\begin{align*} e^{-\dfrac{ {\sup\limits_{t \in[0, 1]}}\vert q(u)- r(u)) \vert^{2}}{\beta({\Theta_{fb}}(\psi, \varrho, \gamma))\gamma}}& \geq e^{-\dfrac{ {\sup\limits_{u \in[0, 1]}}\int_0^u \vert g_{\psi}(u, v)-g_{\varrho}(u, v) \vert^{2} dv}{\beta({\Theta_{fb}}(\psi, \varrho, \gamma))\gamma}}\\ & \geq e^{-\dfrac{ {\sup\limits_{u \in[0, 1]}}\int_0^u f^{2}(u, v) \vert \psi(v)-\varrho(v) \vert^{2} dv}{\beta({\Theta_{fb}}(\psi, \varrho, \gamma))\gamma}}\\ & \geq e^{-\dfrac{\vert \psi(v)-\varrho(v) \vert^{2} {\sup\limits_{u \in[0, 1]}}\int_0^u f^{2}(u, v) dv}{\beta({\Theta_{fb}}(\psi, \varrho, \gamma))\gamma}}\\ & \geq e^{-\dfrac{\beta({\Theta_{fb}}(\psi, \varrho, \gamma))\vert \psi(v)-\varrho(v) \vert^{2}}{\beta({\Theta_{fb}}(\psi, \varrho, \gamma))\gamma}}\\ & = e^{-\dfrac{\vert \psi(v)-\varrho(v) \vert^{2}}{\gamma}}\\ & \geq e^{-\dfrac{ {\sup\limits_{v \in[0, 1]}}\vert \psi(v)-\varrho(v) \vert^{2}}{\gamma}}\\ & = \Theta_{fb}(\psi, \varrho, \gamma). \end{align*} |
This implies that,
\Theta_{fb}(q, r, \beta({\Theta_{fb}}(\psi, \varrho, \gamma))\gamma)\geq \Theta_{fb}(\psi, \varrho, \gamma). |
Interchanging the roles of \psi and \varrho , we get
\mathcal{H}_{F_{\Theta_{fb}}}(S\psi, S\varrho, \beta({\Theta_{fb}}(\psi, \varrho, \gamma))\gamma) \geq \Theta_{fb}( \psi, \varrho, \gamma). |
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] |
Khan AU, Islam T, George B, Rehman M (2019) An Efficient Interface Circuit for Lossy Capacitive Sensors. IEEE Trans Instrum Meas 68: 829–836. https://doi.org/10.1109/TIM.2018.2853219 doi: 10.1109/TIM.2018.2853219
![]() |
[2] |
Wang L, Veselinovic M, Yang L, Geiss BJ, Dandy DS, Chen T (2017) A sensitive DNA capacitive biosensor using interdigitated electrodes. Biosens Bioelectron 87: 646–653. https://doi.org/10.1016/j.bios.2016.09.006 doi: 10.1016/j.bios.2016.09.006
![]() |
[3] |
Mahalingam D, Gurbuz Y, Qureshi A, Niazi JH (2015) Design, fabrication and performance evaluation of interdigital capacitive sensor for detection of Cardiac Troponin-Ⅰ and Human Epidermal Growth Factor Receptor 2. 2015 IEEE SENSORS - Proc 1–4. https://doi.org/10.1109/ICSENS.2015.7370209 doi: 10.1109/ICSENS.2015.7370209
![]() |
[4] |
Couniot N, Afzalian A, Van Overstraeten-Schlogel N, Francis LA, Flandre D (2016) Capacitive Biosensing of Bacterial Cells: Sensitivity Optimization. IEEE Sens J 16: 586–595. https://doi.org/10.1109/JSEN.2015.2485120 doi: 10.1109/JSEN.2015.2485120
![]() |
[5] |
Schaur S, Jakoby B (2011) A numerically efficient method of modeling interdigitated electrodes for capacitive film sensing. Procedia Eng 25: 431–434. https://doi.org/10.1016/j.proeng.2011.12.107 doi: 10.1016/j.proeng.2011.12.107
![]() |
[6] |
Senevirathna BP, Lu S, Dandin MP, Basile J, Smela E, Abshire PA (2018) Real-Time Measurements of Cell Proliferation Using a Lab-on-CMOS Capacitance Sensor Array. IEEE Trans Biomed Circuits Syst 12: 510–520. https://doi.org/10.1109/TBCAS.2018.2821060 doi: 10.1109/TBCAS.2018.2821060
![]() |
[7] |
Claudel J, Ngo TT, Kourtiche D, Nadi M (2020) Interdigitated Sensor Optimization for Blood Sample Analysis. Biosensors 10. https://doi.org/10.3390/bios10120208 doi: 10.3390/bios10120208
![]() |
[8] |
Meng J, Huang J, Oueslati R, Jiang Y, Chen J, Li S, et al. (2021) A single-step DNAzyme sensor for ultra-sensitive and rapid detection of Pb2+ ions. Electrochim Acta 368: 137551. https://doi.org/10.1016/j.electacta.2020.137551 doi: 10.1016/j.electacta.2020.137551
![]() |
[9] | Ramanathan P, Ramasamy S, Jain P, Nagrecha H, Paul S, Arulmozhivarman P, et al. (2013) Low Value Capacitance Measurements for Capacitive Sensors – A Review. Sensors & Transducers 148: 1–10. |
[10] |
Ferlito U, Grasso AD, Pennisi S, Vaiana M, Bruno G (2020) Sub-Femto-Farad Resolution Electronic Interfaces for Integrated Capacitive Sensors: A Review. IEEE Access 8: 153969–153980. https://doi.org/10.1109/ACCESS.2020.3018130 doi: 10.1109/ACCESS.2020.3018130
![]() |
[11] |
Kanoun O, Kallel AY, Fendri A (2022) Measurement Methods for Capacitances in the Range of 1 pF–1 nF: A review. Measurement 195: 111067. https://doi.org/10.1016/j.measurement.2022.111067 doi: 10.1016/j.measurement.2022.111067
![]() |
[12] |
Forouhi S, Dehghani R, Ghafar-Zadeh E (2019) CMOS based capacitive sensors for life science applications: A review. Sensors Actuators, A Phys 297: 111531. https://doi.org/10.1016/j.sna.2019.111531 doi: 10.1016/j.sna.2019.111531
![]() |
[13] |
Preethichandra DMG, Shida K (2001) A simple interface circuit to measure very small capacitance changes in capacitive sensors. IEEE Trans Instrum Meas 50: 1583–1586. https://doi.org/10.1109/19.982949 doi: 10.1109/19.982949
![]() |
[14] |
Haider MR, Mahfouz MR, Islam SK, Eliza SA, Qu W, Pritchard E (2008) A low-power capacitance measurement circuit with high resolution and high degree of linearity. Midwest Symp Circuits Syst 261–264. https://doi.org/10.1109/MWSCAS.2008.4616786 doi: 10.1109/MWSCAS.2008.4616786
![]() |
[15] |
Dean RN, Rane A (2010) An improved capacitance measurement technique based of RC phase delay. 2010 IEEE Int Instrum Meas Technol Conf I2MTC 2010 - Proc 367–370. https://doi.org/10.1109/IMTC.2010.5488213 doi: 10.1109/IMTC.2010.5488213
![]() |
[16] |
Reverter F, Gasulla M, Pallàs-Areny R (2004) A low-cost microcontroller interface for low-value capacitive sensors. Conf Rec - IEEE Instrum Meas Technol Conf 3: 1771–1775. https://doi.org/10.1109/IMTC.2004.1351425 doi: 10.1109/IMTC.2004.1351425
![]() |
[17] |
Czaja Z (2020) A measurement method for capacitive sensors based on a versatile direct sensor-to-microcontroller interface circuit. Measurement 155: 107547. https://doi.org/10.1016/j.measurement.2020.107547 doi: 10.1016/j.measurement.2020.107547
![]() |
[18] |
Van Der Goes FML, Meijer GCM (1996) A novel low-cost capacitive-sensor interface. IEEE Trans Instrum Meas 45: 536–540. https://doi.org/10.1109/19.492782 doi: 10.1109/19.492782
![]() |
[19] |
Mohammad K, Thomson DJ (2017) Differential Ring Oscillator Based Capacitance Sensor for Microfluidic Applications. IEEE Trans Biomed Circuits Syst 11: 392–399. https://doi.org/10.1109/TBCAS.2016.2616346 doi: 10.1109/TBCAS.2016.2616346
![]() |
[20] |
Ashrafi A, Golnabi H (1999) A high precision method for measuring very small capacitance changes. Rev Sci Instrum 70: 3483–3487. https://doi.org/10.1063/1.1149941 doi: 10.1063/1.1149941
![]() |
[21] | Chatzandroulis S, Tsoukalas D (2001) Capacitance to frequency converter suitable for sensor applications using telemetry, ICECS'99. Proceedings of ICECS '99. 6th IEEE International Conference on Electronics, Circuits and Systems (Cat. No.99EX357), IEEE, 1791–1794. https://doi.org/10.1109/ICECS.1999.814557 |
[22] |
Ramfos I, Chatzandroulis S (2012) A 16-channel capacitance-to-period converter for capacitive sensor applications. Analog Integr Circuits Signal Process 71: 383–389. https://doi.org/10.1007/s10470-011-9738-y doi: 10.1007/s10470-011-9738-y
![]() |
[23] |
Bruschi P, Nizza N, Dei M (2008) A low-power capacitance to pulse width converter for MEMS interfacing. ESSCIRC 2008 - Proc 34th Eur Solid-State Circuits Conf 1: 446–449. https://doi.org/10.1109/ESSCIRC.2008.4681888 doi: 10.1109/ESSCIRC.2008.4681888
![]() |
[24] |
Bruschi P, Nizza N, Piotto M (2007) A current-mode, dual slope, integrated capacitance-to-pulse duration converter. IEEE J Solid-State Circuits 42: 1884–1891. https://doi.org/10.1109/JSSC.2007.903102 doi: 10.1109/JSSC.2007.903102
![]() |
[25] |
Nizza N, Dei M, Butti F, Bruschi P (2013) A low-power interface for capacitive sensors with PWM output and intrinsic low pass characteristic. IEEE Trans Circuits Syst Ⅰ Regul Pap 60: 1419–1431. https://doi.org/10.1109/TCSI.2012.2220461 doi: 10.1109/TCSI.2012.2220461
![]() |
[26] |
Arefin MS, Redouté JM, Yuce MR (2016) A Low-Power and Wide-Range MEMS Capacitive Sensors Interface IC Using Pulse-Width Modulation for Biomedical Applications. IEEE Sens J 16: 6745–6754. https://doi.org/10.1109/JSEN.2016.2587668 doi: 10.1109/JSEN.2016.2587668
![]() |
[27] |
Lu JHL, Inerowicz M, Joo S, Kwon JK, Jung B (2011) A low-power, wide-dynamic-range semi-digital universal sensor readout circuit using pulsewidth modulation. IEEE Sens J 11: 1134–1144. https://doi.org/10.1109/JSEN.2010.2085430 doi: 10.1109/JSEN.2010.2085430
![]() |
[28] |
Brookhuis RA, Lammerink TSJ, Wiegerink RJ (2015) Differential capacitive sensing circuit for a multi-electrode capacitive force sensor. Sensors Actuators, A Phys 234: 168–179. https://doi.org/10.1016/j.sna.2015.08.020 doi: 10.1016/j.sna.2015.08.020
![]() |
[29] |
Tan Z, Shalmany SH, Meijer GC, Pertijs MA (2012) An energy-efficient 15-bit capacitive-sensor interface based on period modulation. IEEE J Solid-State Circuits 47: 1703–1711. https://doi.org/10.1109/JSSC.2012.2191212 doi: 10.1109/JSSC.2012.2191212
![]() |
[30] |
Li X, Meijer GCM (2002) An accurate interface for capacitive sensors. IEEE Trans Instrum Meas 51: 935–939. https://doi.org/10.1109/TIM.2002.807793 doi: 10.1109/TIM.2002.807793
![]() |
[31] |
Gasulla M, Li X, Meijer GCM (2005) The noise performance of a high-speed capacitive-sensor interface based on a relaxation oscillator and a fast counter. IEEE Trans Instrum Meas 54: 1934–1940. https://doi.org/10.1109/TIM.2005.853684 doi: 10.1109/TIM.2005.853684
![]() |
[32] |
Yurish SY (2009) Universal Capacitive Sensors and Transducers Interface. Procedia Chem 1: 441–444. https://doi.org/10.1016/j.proche.2009.07.110 doi: 10.1016/j.proche.2009.07.110
![]() |
[33] |
Heidary A, Meijer GCM (2009) An integrated interface circuit with a capacitance-to-voltage converter as front-end for grounded capacitive sensors. Meas Sci Technol 20. https://doi.org/10.1088/0957-0233/20/1/015202 doi: 10.1088/0957-0233/20/1/015202
![]() |
[34] |
Liu Y, Chen S, Nakayama M, Watanabe K (2000) Limitations of a relaxation oscillator in capacitance measurements. IEEE Trans Instrum Meas 49: 980–983. https://doi.org/10.1109/19.872917 doi: 10.1109/19.872917
![]() |
[35] |
Czaja Z (2023) A New Approach to Capacitive Sensor Measurements Based on a Microcontroller and a Three-Gate Stable RC Oscillator. IEEE Trans Instrum Meas 72: 1–9. https://doi.org/10.1109/TIM.2023.3244851 doi: 10.1109/TIM.2023.3244851
![]() |
[36] |
De Marcellis A, Reig C, Cubells-Beltrán M-D (2019) A Capacitance-to-Time Converter-Based Electronic Interface for Differential Capacitive Sensors. Electronics 8: 80. https://doi.org/10.3390/electronics8010080 doi: 10.3390/electronics8010080
![]() |
[37] |
Ulla Khan A, Islam T, Akhtar J (2016) An Oscillator-Based Active Bridge Circuit for Interfacing Capacitive Sensors with Microcontroller Compatibility. IEEE Trans Instrum Meas 65: 2560–2568. https://doi.org/10.1109/TIM.2016.2581519 doi: 10.1109/TIM.2016.2581519
![]() |
[38] |
Rahman O, Islam T, Khera N, Khan SA (2021) A Novel Application of the Cross-Capacitive Sensor in Real-Time Condition Monitoring of Transformer Oil. IEEE Trans Instrum Meas 70. https://doi.org/10.1109/TIM.2021.3111979 doi: 10.1109/TIM.2021.3111979
![]() |
[39] |
Khaled AY, Aziz SA, Ismail WIW, Rokhani FZ (2016) Capacitive sensing system for frying oil assessment during heating. Proceeding - 2015 IEEE Int Circuits Syst Symp ICSyS 2015 137–141. https://doi.org/10.1109/CircuitsAndSystems.2015.7394081 doi: 10.1109/CircuitsAndSystems.2015.7394081
![]() |
[40] | Khamil KN, Mood MAUC (2017) Dielectric sensing (capacitive) on cooking oil's TPC level. J Telecommun Electron Comput Eng 9: 27–32. |
[41] |
Liu M, Qin X, Chen Z, Tang L, Borom B, Cao N, et al. (2019) Frying Oil Evaluation by a Portable Sensor Based on Dielectric Constant Measurement. Sensors: 1–11. https://doi.org/10.3390/s19245375 doi: 10.3390/s19245375
![]() |
[42] |
Kumar D, Singh A, Tarsikka PS (2013) Interrelationship between viscosity and electrical properties for edible oils. J Food Sci Technol 50: 549–554. https://doi.org/10.1007/s13197-011-0346-8 doi: 10.1007/s13197-011-0346-8
![]() |
[43] |
Pérez AT, Hadfield M (2011) Low-cost oil quality sensor based on changes in complex permittivity. Sensors 11: 10675–10690. https://doi.org/10.3390/s111110675 doi: 10.3390/s111110675
![]() |
[44] |
Behzadi G, Fekri L (2013) Electrical Parameter and Permittivity Measurement of Water Samples Using the Capacitive Sensor. Int J Water Resour Environ Sci 2: 66–75. https://doi.org/10.5829/idosi.ijwres.2013.2.3.2938 doi: 10.5829/idosi.ijwres.2013.2.3.2938
![]() |
[45] |
Golnabi H, Sharifian M (2013) Investigation of water electrical parameters as a function of measurement frequency using cylindrical capacitive sensors. Meas J Int Meas Confed 46: 305–314. https://doi.org/10.1016/j.measurement.2012.07.002 doi: 10.1016/j.measurement.2012.07.002
![]() |
[46] | Niranatlumpong P, Allen MA (2022) A 555 Timer IC Chaotic Circuit: Chaos in a Piecewise Linear System With Stable but No Unstable Equilibria. IEEE Trans Circuits Syst Ⅰ Regul Pap 69: 798–810. |
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