Symmetry

Research

13 pages, 1697 KiB

Open AccessArticle

Latent Infectious Capacities of Dengue Fever: Mathematical Modeling and Eco-Friendly Prevention Strategy

by Chung-Chien Hong, Wei-Shih Du and Yu-Hong Ge

Symmetry 2020, 12(2), 263; https://doi.org/10.3390/sym12020263 - 08 Feb 2020

Cited by 1 | Viewed by 1968

The main aim of this article is to propose a method for exploring the latent values about the capacities of spreading dengue for each potential site. First, a mathematical model connecting the observable public data and the capacities of spreading dengue is provided [...] Read more.

The main aim of this article is to propose a method for exploring the latent values about the capacities of spreading dengue for each potential site. First, a mathematical model connecting the observable public data and the capacities of spreading dengue is provided based on the split feasibility problem (SFP). Then, a proper iterative scheme for the SFP is presented to approach the values of infectious capacities (ICs) of potential sites—the capacities of spreading. The performance of our proposed method is demonstrated using public data from Kaohsiung City for 2014 and 2015. The results presented in this paper show that our proposed method is reliable and the sites with a high capacity of spreading are only a small portion of thousands of all potential sites and could be an alternative strategy for preventing the outbreak of dengue fever whilst also avoiding the damage of ecosystems caused by chemical insecticides. Full article

(This article belongs to the Special Issue Nonlinear, Convex, Nonsmooth, Functional Analysis in Symmetry)

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12 pages, 284 KiB

Open AccessArticle

w-b-Cone Distance and Its Related Results: A Survey

by Reza Babaei, Hamidreza Rahimi, Manuel De la Sen and Ghasem Soleimani Rad

Symmetry 2020, 12(1), 171; https://doi.org/10.3390/sym12010171 - 16 Jan 2020

Cited by 6 | Viewed by 1737

Abstract

In this work, we define the concept of a w-b-cone distance in

t v s

-cone b-metric spaces which differs from generalized c-distance in cone b-metric spaces, and we discuss its properties. Our results are significant, since [...] Read more.

In this work, we define the concept of a w-b-cone distance in

t v s

-cone b-metric spaces which differs from generalized c-distance in cone b-metric spaces, and we discuss its properties. Our results are significant, since all of the results in fixed point theory with respect to a generalized c-distance can be introduced in the version of w-b-cone distance. Moreover, using Minkowski functionals in topological vector spaces, we prove the equivalence between some fixed point results with respect to a

w t

-distance in general b-metric spaces and a w-b-cone distance in

t v s

-cone b-metric spaces. Full article

(This article belongs to the Special Issue Nonlinear, Convex, Nonsmooth, Functional Analysis in Symmetry)

20 pages, 1050 KiB

Open AccessArticle

A Modified Equation for Thickness of the Film Fabricated by Spin Coating

by Un Gi Lee, Woo-Byoung Kim, Do Hyung Han and Hyun Soo Chung

Symmetry 2019, 11(9), 1183; https://doi.org/10.3390/sym11091183 - 18 Sep 2019

Cited by 20 | Viewed by 8862

Abstract

According to the equation for Newtonian fluids, the film thickness after spin coating is determined by five parameters: angular velocity, spin coating time, viscosity, density of the coating material, and initial thickness of the material before spin coating. The spin coating process is [...] Read more.

According to the equation for Newtonian fluids, the film thickness after spin coating is determined by five parameters: angular velocity, spin coating time, viscosity, density of the coating material, and initial thickness of the material before spin coating. The spin coating process is commonly controlled by adjusting only the angular velocity parameter and the coating time in the Newtonian expression. However, the measured coating thickness obtained is then compared to the theoretical thickness calculated from the Newtonian fluid equation. The measured coating thickness usually varies somewhat from the theoretical thickness; further details are described in Section 1. Thus, the Newtonian fluid equation must be modified to better represent the actual film thickness. In this paper, we derive a new formula for the spin coating film thickness, which is based on the equation for Newtonian fluids, but modified to better represent film thicknesses obtained experimentally. The statistical analysis is performed to verify our modifications. Full article

(This article belongs to the Special Issue Nonlinear, Convex, Nonsmooth, Functional Analysis in Symmetry)

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12 pages, 291 KiB

Open AccessArticle

Pre-Schauder Bases in Topological Vector Spaces

by Francisco Javier García-Pacheco and Francisco Javier Pérez-Fernández

Symmetry 2019, 11(8), 1026; https://doi.org/10.3390/sym11081026 - 09 Aug 2019

Cited by 5 | Viewed by 2077

Abstract

A Schauder basis in a real or complex Banach space X is a sequence

{(e_{n})}_{n \in N}

in X such that for every

x \in X

there exists a unique sequence of scalars [...] Read more.

A Schauder basis in a real or complex Banach space X is a sequence

{(e_{n})}_{n \in N}

in X such that for every

x \in X

there exists a unique sequence of scalars

{(λ_{n})}_{n \in N}

satisfying that

x = \sum_{n = 1}^{\infty} λ_{n} e_{n}

. Schauder bases were first introduced in the setting of real or complex Banach spaces but they have been transported to the scope of real or complex Hausdorff locally convex topological vector spaces. In this manuscript, we extend them to the setting of topological vector spaces over an absolutely valued division ring by redefining them as pre-Schauder bases. We first prove that, if a topological vector space admits a pre-Schauder basis, then the linear span of the basis is Hausdorff and the series linear span of the basis minus the linear span contains the intersection of all neighborhoods of 0. As a consequence, we conclude that the coefficient functionals are continuous if and only if the canonical projections are also continuous (this is a trivial fact in normed spaces but not in topological vector spaces). We also prove that, if a Hausdorff topological vector space admits a pre-Schauder basis and is

w^{*}

-strongly torsionless, then the biorthogonal system formed by the basis and its coefficient functionals is total. Finally, we focus on Schauder bases on Banach spaces proving that every Banach space with a normalized Schauder basis admits an equivalent norm closer to the original norm than the typical bimonotone renorming and that still makes the basis binormalized and monotone. We also construct an increasing family of left-comparable norms making the normalized Schauder basis binormalized and show that the limit of this family is a right-comparable norm that also makes the normalized Schauder basis binormalized. Full article

(This article belongs to the Special Issue Nonlinear, Convex, Nonsmooth, Functional Analysis in Symmetry)

16 pages, 311 KiB

Open AccessArticle

Nonlinear Rayleigh Quotients and Nonlinear Spectral Theory

by Raffaele Chiappinelli

Symmetry 2019, 11(7), 928; https://doi.org/10.3390/sym11070928 - 16 Jul 2019

Cited by 1 | Viewed by 2522

Abstract

We give a new and simplified definition of spectrumfor a nonlinear operator F acting in a real Banach space X, and study some of its features in terms of (qualitative and) quantitative properties of F such as the measure of noncompactness, [...] Read more.

We give a new and simplified definition of spectrumfor a nonlinear operator F acting in a real Banach space X, and study some of its features in terms of (qualitative and) quantitative properties of F such as the measure of noncompactness,

α (F)

, of F. Then, using as a main tool the Ekeland Variational Principle, we focus our attention on the spectral properties of F when F is a gradient operator in a real Hilbert space, and in particular on the role played by its Rayleigh quotient

R (F)

and by the best lower and upper bounds,

m (F)

and

M (F)

, of

R (F)

. Full article

(This article belongs to the Special Issue Nonlinear, Convex, Nonsmooth, Functional Analysis in Symmetry)

10 pages, 238 KiB

Open AccessEditor’s ChoiceArticle

Hyers-Ulam Stability for Linear Differences with Time Dependent and Periodic Coefficients

by Constantin Buşe, Donal O’Regan and Olivia Saierli

Symmetry 2019, 11(4), 512; https://doi.org/10.3390/sym11040512 - 09 Apr 2019

Cited by 11 | Viewed by 1942

Abstract

Let

q \geq 2

be a positive integer and let

(a_{j})

,

(b_{j})

, and

(c_{j})

(with j a non-negative integer) be three given

C

-valued and q-periodic sequences. Let [...] Read more.

Let

q \geq 2

be a positive integer and let

(a_{j})

,

(b_{j})

, and

(c_{j})

(with j a non-negative integer) be three given

C

-valued and q-periodic sequences. Let

A (q) : = A_{q - 1} \dots A_{0}

, where

A_{j}

is as is given below. Assuming that the “monodromy matrix”

A (q)

has at least one multiple eigenvalue, we prove that the linear scalar recurrence

x_{n + 3} = a_{n} x_{n + 2} + b_{n} x_{n + 1} + c_{n} x_{n}, n \in Z_{+}

is Hyers-Ulam stable if and only if the spectrum of

A (q)

does not intersect the unit circle

Γ : = {w \in C : | w | = 1}

. Connecting this result with a recently obtained one it follows that the above linear recurrence is Hyers-Ulam stable if and only if the spectrum of

A (q)

does not intersect the unit circle. Full article

(This article belongs to the Special Issue Nonlinear, Convex, Nonsmooth, Functional Analysis in Symmetry)

8 pages, 238 KiB

Open AccessArticle

Hyers-Ulam Stability for Linear Differences with Time Dependent and Periodic Coefficients: The Case When the Monodromy Matrix Has Simple Eigenvalues

by Constantin Buşe, Donal O’Regan and Olivia Saierli

Symmetry 2019, 11(3), 339; https://doi.org/10.3390/sym11030339 - 07 Mar 2019

Cited by 5 | Viewed by 2056

Abstract

Let

q \geq 2

be a positive integer and let

(a_{j}), (b_{j})

and

(c_{j})

(with j nonnegative integer) be three given

C

-valued and q-periodic sequences. Let [...] Read more.

Let

q \geq 2

be a positive integer and let

(a_{j}), (b_{j})

and

(c_{j})

(with j nonnegative integer) be three given

C

-valued and q-periodic sequences. Let

A (q) : = A_{q - 1} \dots A_{0},

where

A_{j}

is defined below. Assume that the eigenvalues

x, y, z

of the “monodromy matrix”

A (q)

verify the condition

(x - y) (y - z) (z - x) \neq 0 .

We prove that the linear recurrence in

C

x_{n + 3} = a_{n} x_{n + 2} + b_{n} x_{n + 1} + c_{n} x_{n}, n \in Z_{+}

is Hyers–Ulam stable if and only if

(| x | - 1) (| y | - 1) (| z | - 1) \neq 0,

i.e., the spectrum of

A (q)

does not intersect the unit circle

Γ : = {w \in C : | w | = 1} .

Full article

(This article belongs to the Special Issue Nonlinear, Convex, Nonsmooth, Functional Analysis in Symmetry)

15 pages, 313 KiB

Open AccessArticle

Nonoscillatory Solutions to Higher-Order Nonlinear Neutral Dynamic Equations

by Yang-Cong Qiu, Irena Jadlovská, Dhaou Lassoued and Tongxing Li

Symmetry 2019, 11(3), 302; https://doi.org/10.3390/sym11030302 - 28 Feb 2019

Cited by 4 | Viewed by 2114

Abstract

For a class of nonlinear higher-order neutral dynamic equations on a time scale, we analyze the existence and asymptotic behavior of nonoscillatory solutions on the basis of hypotheses that allow applications to equations with different integral convergence and divergence of the reciprocal of [...] Read more.

For a class of nonlinear higher-order neutral dynamic equations on a time scale, we analyze the existence and asymptotic behavior of nonoscillatory solutions on the basis of hypotheses that allow applications to equations with different integral convergence and divergence of the reciprocal of the coefficients. Two examples are presented to demonstrate the efficiency of new results. Full article

(This article belongs to the Special Issue Nonlinear, Convex, Nonsmooth, Functional Analysis in Symmetry)

18 pages, 283 KiB

Open AccessArticle

β–Hyers–Ulam–Rassias Stability of Semilinear Nonautonomous Impulsive System

by Xiaoming Wang, Muhammad Arif and Akbar Zada

Symmetry 2019, 11(2), 231; https://doi.org/10.3390/sym11020231 - 15 Feb 2019

Cited by 25 | Viewed by 3045

Abstract

In this paper, we study a system governed by impulsive semilinear nonautonomous differential equations. We present the

β

–Ulam stability,

β

–Hyers–Ulam stability and

β

–Hyers–Ulam–Rassias stability for the said system on a compact interval and then extended it to an unbounded interval. [...] Read more.

In this paper, we study a system governed by impulsive semilinear nonautonomous differential equations. We present the

β

–Ulam stability,

β

–Hyers–Ulam stability and

β

–Hyers–Ulam–Rassias stability for the said system on a compact interval and then extended it to an unbounded interval. We use Grönwall type inequality and evolution family as a basic tool for our results. We present an example to demonstrate the application of the main result. Full article

(This article belongs to the Special Issue Nonlinear, Convex, Nonsmooth, Functional Analysis in Symmetry)

20 pages, 325 KiB

Open AccessArticle

A New Approach to the Solution of Non-Linear Integral Equations via Various F_Be-Contractions

by Sumati Kumari Panda, Asifa Tassaddiq and Ravi P Agarwal

Symmetry 2019, 11(2), 206; https://doi.org/10.3390/sym11020206 - 12 Feb 2019

Cited by 27 | Viewed by 2795

Abstract

In this article, we introduce and establish various approaches related to the F-contraction using new sorts of contractions, namely the extended

F_{B_{e}}

-contraction, the extended

F_{B_{e}}

-expanding contraction, and the extended generalized

F_{B_{e}}

-contraction. Thereafter, we [...] Read more.

In this article, we introduce and establish various approaches related to the F-contraction using new sorts of contractions, namely the extended

F_{B_{e}}

-contraction, the extended

F_{B_{e}}

-expanding contraction, and the extended generalized

F_{B_{e}}

-contraction. Thereafter, we propose a simple and efficient solution for non-linear integral equations using the fixed point technique in the setting of a

B_{e}

-metric space. Moreover, to address conceptual depth within this approach, we supply illustrative examples where necessary. Full article

(This article belongs to the Special Issue Nonlinear, Convex, Nonsmooth, Functional Analysis in Symmetry)

12 pages, 285 KiB

Open AccessArticle

Optimal Control of Nonsmooth Production Systems with Deteriorating Items, Stock-Dependent Demand, with or without Backorders

by Messaoud Bounkhel, Lotfi Tadj, Yacine Benhadid and Ramdane Hedjar

Symmetry 2019, 11(2), 183; https://doi.org/10.3390/sym11020183 - 04 Feb 2019

Cited by 2 | Viewed by 2159

Abstract

We propose a nonsmooth dynamic system integrating production and inventory where the items may deteriorate and the demand is stock-dependent. We aim to derive the optimal production rate. In our first model, backorders are not allowed, while in the second model they are. [...] Read more.

We propose a nonsmooth dynamic system integrating production and inventory where the items may deteriorate and the demand is stock-dependent. We aim to derive the optimal production rate. In our first model, backorders are not allowed, while in the second model they are. Using optimal control, necessary optimality conditions are obtained for general forms of the cost, demand, and deterioration rates and closed form solutions are derived for specific forms of these rates. Numerical simulations are presented and sensitivity of the solutions are examined. Full article

(This article belongs to the Special Issue Nonlinear, Convex, Nonsmooth, Functional Analysis in Symmetry)

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14 pages, 298 KiB

Open AccessArticle

Existence Results for Second Order Nonconvex Sweeping Processes in q-Uniformly Convex and 2-Uniformly Smooth Separable Banach Spaces

by Djalel Bounekhel, Messaoud Bounkhel and Mostafa Bachar

Symmetry 2019, 11(1), 28; https://doi.org/10.3390/sym11010028 - 30 Dec 2018

Viewed by 1943

Abstract

We prove an existence result, in the separable Banach spaces setting, for second order differential inclusions of type sweeping process. This type of differential inclusion is defined in terms of normal cones and it covers many dynamic quasi-variational inequalities. In the present paper, [...] Read more.

We prove an existence result, in the separable Banach spaces setting, for second order differential inclusions of type sweeping process. This type of differential inclusion is defined in terms of normal cones and it covers many dynamic quasi-variational inequalities. In the present paper, we prove in the nonconvex case an existence result of this type of differential inclusions when the separable Banach space is assumed to be q-uniformly convex and 2-uniformly smooth. In our proofs we use recent results on uniformly generalized prox-regular sets. Part of the novelty of the paper is the use of the usual Lipschitz continuity of the set-valued mapping which is very easy to verify contrarily to the ones used in the previous works. An example is stated at the end of the paper, showing the application of our existence result. Full article

(This article belongs to the Special Issue Nonlinear, Convex, Nonsmooth, Functional Analysis in Symmetry)

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