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20 pages, 350 KiB

Open AccessArticle

Analysis of a SEIR-KS Mathematical Model For Computer Virus Propagation in a Periodic Environment

by Aníbal Coronel, Fernando Huancas, Ian Hess, Esperanza Lozada and Francisco Novoa-Muñoz

Mathematics 2020, 8(5), 761; https://doi.org/10.3390/math8050761 - 11 May 2020

Cited by 4 | Viewed by 2582

Abstract

In this work we develop a study of positive periodic solutions for a mathematical model of the dynamics of computer virus propagation. We propose a generalized compartment model of SEIR-KS type, since we consider that the population is partitioned in five classes: susceptible [...] Read more.

In this work we develop a study of positive periodic solutions for a mathematical model of the dynamics of computer virus propagation. We propose a generalized compartment model of SEIR-KS type, since we consider that the population is partitioned in five classes: susceptible (S); exposed (E); infected (I); recovered (R); and kill signals (K), and assume that the rates of virus propagation are time dependent functions. Then, we introduce a sufficient condition for the existence of positive periodic solutions of the generalized SEIR-KS model. The proof of the main results are based on a priori estimates of the SEIR-KS system solutions and the application of coincidence degree theory. Moreover, we present an example of a generalized system satisfying the sufficient condition. Full article

(This article belongs to the Special Issue Nonlinear Functional Analysis and Its Applications)

13 pages, 269 KiB

Open AccessArticle

A Sub-Supersolution Approach for Robin Boundary Value Problems with Full Gradient Dependence

by Dumitru Motreanu, Angela Sciammetta and Elisabetta Tornatore

Mathematics 2020, 8(5), 658; https://doi.org/10.3390/math8050658 - 27 Apr 2020

Cited by 4 | Viewed by 1700

Abstract

The paper investigates a nonlinear elliptic problem with a Robin boundary condition, which exhibits a convection term with full dependence on the solution and its gradient. A sub- supersolution approach is developed for this type of problems. The main result establishes the existence [...] Read more.

The paper investigates a nonlinear elliptic problem with a Robin boundary condition, which exhibits a convection term with full dependence on the solution and its gradient. A sub- supersolution approach is developed for this type of problems. The main result establishes the existence of a solution enclosed in the ordered interval formed by a sub-supersolution. The result is applied to find positive solutions. Full article

(This article belongs to the Special Issue Nonlinear Functional Analysis and Its Applications)

8 pages, 227 KiB

Open AccessArticle

A Class of Equations with Three Solutions

by Biagio Ricceri

Mathematics 2020, 8(4), 478; https://doi.org/10.3390/math8040478 - 1 Apr 2020

Cited by 4 | Viewed by 2225 | Correction

Abstract

Here is one of the results obtained in this paper: Let

Ω \subset R^{n}

be a smooth bounded domain, let

q > 1

, with

q < \frac{n + 2}{n - 2}

if

n \geq 3

and let

λ_{1}

be [...] Read more.

Here is one of the results obtained in this paper: Let

Ω \subset R^{n}

be a smooth bounded domain, let

q > 1

, with

q < \frac{n + 2}{n - 2}

if

n \geq 3

and let

λ_{1}

be the first eigenvalue of the problem

- Δ u = λ u

in

Ω

,

u = 0

on

\partial Ω

. Then, for every

λ > λ_{1}

and for every convex set

S \subseteq L^{\infty} (Ω)

dense in

L^{2} (Ω)

, there exists

α \in S

such that the problem

- Δ u = λ (u^{+} - {(u^{+})}^{q}) + α (x)

in

Ω

,

u = 0

on

\partial Ω

, has at least three weak solutions, two of which are global minima in

H_{0}^{1} (Ω)

of the functional

u \to \frac{1}{2} \int_{Ω} {| \nabla u (x) |}^{2} d x - λ \int_{Ω} (\frac{1}{2} | u^{+} {(x) |}^{2} - \frac{1}{q + 1} {| u^{+} (x) |}^{q + 1}) d x - \int_{Ω} α (x) u (x) d x

where

u^{+} = max {u, 0}

. Full article

(This article belongs to the Special Issue Nonlinear Functional Analysis and Its Applications)

15 pages, 309 KiB

Open AccessArticle

New Generalized Mizoguchi-Takahashi’s Fixed Point Theorems for Essential Distances and e⁰-Metrics

by Binghua Jiang, Huaping Huang and Wei-Shih Du

Mathematics 2019, 7(12), 1224; https://doi.org/10.3390/math7121224 - 11 Dec 2019

Cited by 2 | Viewed by 1957

Abstract

In this paper, we present some new generalizations of Mizoguchi-Takahashi’s fixed point theorem which also improve and extend Du-Hung’s fixed point theorem. Some new examples illustrating our results are also given. By applying our new results, some new fixed point theorems for essential [...] Read more.

In this paper, we present some new generalizations of Mizoguchi-Takahashi’s fixed point theorem which also improve and extend Du-Hung’s fixed point theorem. Some new examples illustrating our results are also given. By applying our new results, some new fixed point theorems for essential distances and e⁰-metrics were established. Full article

(This article belongs to the Special Issue Nonlinear Functional Analysis and Its Applications)

20 pages, 2710 KiB

Open AccessArticle

Design of Robust Trackers and Unknown Nonlinear Perturbation Estimators for a Class of Nonlinear Systems: HTRDNA Algorithm for Tracker Optimization

by Jiunn-Shiou Fang, Jason Sheng-Hong Tsai, Jun-Juh Yan, Chang-He Tzou and Shu-Mei Guo

Mathematics 2019, 7(12), 1141; https://doi.org/10.3390/math7121141 - 22 Nov 2019

Cited by 5 | Viewed by 2428

Abstract

A robust linear quadratic analog tracker (LQAT) consisting of proportional-integral-derivative (PID) controller, sliding mode control (SMC), and perturbation estimator is proposed for a class of nonlinear systems with unknown nonlinear perturbation and direct feed-through term. Since the derivative type (D-type) controller is very [...] Read more.

A robust linear quadratic analog tracker (LQAT) consisting of proportional-integral-derivative (PID) controller, sliding mode control (SMC), and perturbation estimator is proposed for a class of nonlinear systems with unknown nonlinear perturbation and direct feed-through term. Since the derivative type (D-type) controller is very sensitive to the state varying, a new D-type controller design algorithm is developed to avoid an unreasonable large value of the controller gain. Moreover, the boundary of D-type controller is discussed. To cope with the unknown perturbation effect, SMC is utilized. Based on the fast response of SMC controlled systems, the proposed perturbation estimator can estimate unknown nonlinear perturbation and improve the tracking performance. Furthermore, in order to tune the PID controller gains in the designed tracker, the nonlinear perturbation is eliminated by the SMC-based perturbation estimator first, then a hybrid Taguchi real coded DNA (HTRDNA) algorithm is newly proposed for the PID controller optimization. Compared with traditional DNA, a new HTRDNA is developed to improve the convergence performance and effectiveness. Numerical simulations are given to demonstrate the performance of the proposed method. Full article

(This article belongs to the Special Issue Nonlinear Functional Analysis and Its Applications)

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6 pages, 234 KiB

Open AccessArticle

The Topological Transversality Theorem for Multivalued Maps with Continuous Selections

by Donal O’Regan

Mathematics 2019, 7(11), 1113; https://doi.org/10.3390/math7111113 - 15 Nov 2019

Cited by 1 | Viewed by 1640

Abstract

This paper considers a topological transversality theorem for multivalued maps with continuous, compact selections. Basically, this says, if we have two maps F and G with continuous compact selections and

F ≅ G

, then one map being essential guarantees the essentiality of [...] Read more.

This paper considers a topological transversality theorem for multivalued maps with continuous, compact selections. Basically, this says, if we have two maps F and G with continuous compact selections and

F ≅ G

, then one map being essential guarantees the essentiality of the other map. Full article

(This article belongs to the Special Issue Nonlinear Functional Analysis and Its Applications)

17 pages, 310 KiB

Open AccessArticle

Existence of Solutions for Nonhomogeneous Choquard Equations Involving p-Laplacian

by Xiaoyan Shi, Yulin Zhao and Haibo Chen

Mathematics 2019, 7(9), 871; https://doi.org/10.3390/math7090871 - 19 Sep 2019

Cited by 1 | Viewed by 2238

Abstract

This paper is devoted to investigating a class of nonhomogeneous Choquard equations with perturbation involving p-Laplacian. Under suitable hypotheses about the perturbation term, the existence of at least two nontrivial solutions for the given problems is obtained using Nehari manifold and minimax methods. [...] Read more.

This paper is devoted to investigating a class of nonhomogeneous Choquard equations with perturbation involving p-Laplacian. Under suitable hypotheses about the perturbation term, the existence of at least two nontrivial solutions for the given problems is obtained using Nehari manifold and minimax methods. Full article

(This article belongs to the Special Issue Nonlinear Functional Analysis and Its Applications)

15 pages, 287 KiB

Open AccessArticle

Positive Solutions for Discontinuous Systems via a Multivalued Vector Version of Krasnosel’skiĭ’s Fixed Point Theorem in Cones

by Rodrigo López Pouso, Radu Precup and Jorge Rodríguez-López

Mathematics 2019, 7(5), 451; https://doi.org/10.3390/math7050451 - 20 May 2019

Cited by 1 | Viewed by 1639

Abstract

We establish the existence of positive solutions for systems of second–order differential equations with discontinuous nonlinear terms. To this aim, we give a multivalued vector version of Krasnosel’skiĭ’s fixed point theorem in cones which we apply to a regularization of the discontinuous integral [...] Read more.

We establish the existence of positive solutions for systems of second–order differential equations with discontinuous nonlinear terms. To this aim, we give a multivalued vector version of Krasnosel’skiĭ’s fixed point theorem in cones which we apply to a regularization of the discontinuous integral operator associated to the differential system. We include several examples to illustrate our theory. Full article

(This article belongs to the Special Issue Nonlinear Functional Analysis and Its Applications)

Review

Jump to: Research, Other

14 pages, 803 KiB

Open AccessFeature PaperReview

Variations on the Brouwer Fixed Point Theorem: A Survey

by Jean Mawhin

Mathematics 2020, 8(4), 501; https://doi.org/10.3390/math8040501 - 2 Apr 2020

Cited by 3 | Viewed by 3160

Abstract

This paper surveys some recent simple proofs of various fixed point and existence theorems for continuous mappings in

R^{n}

. The main tools are basic facts of the exterior calculus and the use of retractions. The special case of holomorphic functions is [...] Read more.

This paper surveys some recent simple proofs of various fixed point and existence theorems for continuous mappings in

R^{n}

. The main tools are basic facts of the exterior calculus and the use of retractions. The special case of holomorphic functions is considered, based only on the Cauchy integral theorem. Full article

(This article belongs to the Special Issue Nonlinear Functional Analysis and Its Applications)