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Akgöl, Sibel D.; Zafer, Agacik
doi: 10.1002/mma.8812pmid: N/A
We present Wong‐type oscillation criteria for nonlinear impulsive differential equations having discontinuous solutions and involving both negative and positive coefficients. We use a technique that involves the use of a nonprincipal solution of the associated linear homogeneous equation. The existence of principal and nonpricipal solutions was recently obtained by the present authors. As in special cases, we have superlinear and sublinear Emden–Fowler equations under impulse effects. It is shown that the oscillatory behavior may change due to impulses. An example is also given to illustrate the importance of the results.
Luo, Xiankang; Zhu, Quanxin; Zhang, Ying; Chen, Peimin
doi: 10.1002/mma.8813pmid: N/A
In this paper, a hybrid strategy with optimal proportional reinsurance, dividends, and capital injections is discussed by a diffusion model with impulse process of discrete dividend events and discrete capital injections. Due to the presence of fixed transaction costs, the problem is formulated as a more complicated impulse stochastic control problem. By using methodologies from impulse control theory and the techniques of translation transformation, we obtain the explicit solutions of the value function and the corresponding optimal strategy. Firstly, the original problem is transformed into two types of suboptimal impulse control problems, one of which is “the case without capital injections” and the other one of which is “the case with compulsory capital injections.” Then, by verification theorem and the expressions of value functions of these two suboptimal problems, we verify that the optimal solution of the original problem is either “keeping bankruptcy without injections” or “never bankruptcy with injections.” It depends on the parameters of the model.
doi: 10.1002/mma.8814pmid: N/A
This paper is concerned with the dynamics of a reaction–diffusion–advection mutualist model with four different free boundaries in one‐dimensional space. These free boundaries are used to describe the spreading of two mutualistic species along two sides in the line, and two ones in the same direction may intersect each other as time evolves. The existence, uniqueness, and long time behaviors of global solutions are established. The sharp criteria for spreading and vanishing are provided.
doi: 10.1002/mma.8815pmid: N/A
The convexity of the free boundaries for axially symmetric Réthy flows will be established. As byproducts, we will set up the deflection angle estimate of the Réthy flow in the far‐field, the positive axial velocity, and the optimal regularity of velocity up to one separation point.
Shao, Minna; Zhang, Qimin; Zhao, Hongyong
doi: 10.1002/mma.8817pmid: N/A
Near‐optimal controls are as meaningful as optimal controls given both theory and applications; however, they are usually much easier to be obtained than optimal ones, which are simply impossible to be obtained in many complicated systems. Therefore, this paper focuses on the near‐optimal control problems of a stochastic West Nile virus system with spatial diffusion describing the virus transmission among the bird, mosquito, and human populations. First, we introduce two control variables, namely, antimosquito and the treatment of the infected humans, into the system and prove the existence and uniqueness of the global positive solution of the system. Second, the necessary and sufficient conditions for two controls to be near‐optimal are acquired in terms of a Hamiltonian function and a small parameter ε$$ \varepsilon $$ based on the Ekeland principle, the adjoint equation, and some prior estimates. Finally, we use numerical simulations to illustrate the theoretical results and conclude that mosquito control is the most critical factor preventing West Nile virus.
Jose, Sayooj Aby; Ramachandran, Raja; Baleanu, Dumitru; Panigoro, Hasan S.; Alzabut, Jehad; Balas, Valentina E.
doi: 10.1002/mma.8818pmid: N/A
In this paper, the ABC fractional derivative is used to provide a mathematical model for the dynamic systems of substance addiction. The basic reproduction number is investigated, as well as the equilibrium points' stability. Using fixed point theory and nonlinear analytic techniques, we verify the theoretical results of solution existence and uniqueness for the proposed model. A numerical technique for getting the approximate solution of the suggested model is established by using the Adams type predictor‐corrector rule for the ABC‐fractional integral operator. There are several numerical graphs that correspond to different fractional orders. Furthermore, we present a numerical simulation for the transmission of substance addiction in two scenarios with fundamental reproduction numbers greater than and fewer than one.
doi: 10.1002/mma.8819pmid: N/A
The paper deals with a new type of inverse spectral problems for second‐order quadratic differential pencils when one of the boundary conditions involves arbitrary entire functions of the spectral parameter. Although various aspects of the inverse spectral theory for the pencils have been of a special interest during the last decades, such settings were considered before only in the particular case of a Sturm–Liouville equation. We develop an approach covering also the quadratic dependence on the spectral parameter in the differential equation, which is based on the completeness and basisness of certain functional systems. By this approach, we obtain a uniqueness theorem and an algorithm for solving the inverse problem along with sufficient properties of the mentioned systems. The presented results give a universal tool for studying a number of important specific situations, including various Hochstadt–Lieberman‐type inverse problems both on an interval and on geometrical graphs, which is illustrated as well.
Goyal, Prashant; Patel, Dhiraj; Sivananthan, S.
doi: 10.1002/mma.8821pmid: N/A
In this article, we consider the random sampling in the image space V$$ V $$ of an idempotent integral operator on mixed Lebesgue space Lp,qℝn+1$$ {L}^{p,q}\left({\mathbb{R}}^{n+1}\right) $$. We assume some decay and regularity conditions on the integral kernel and show that the bounded functions in V$$ V $$ can be approximated by an element in a finite‐dimensional subspace of V$$ V $$ on CR,S=−R2,R2n×−S2,S2$$ {C}_{R,S}={\left[-\frac{R}{2},\frac{R}{2}\right]}^n\times \left[-\frac{S}{2},\frac{S}{2}\right] $$. Consequently, we show that the set of bounded functions concentrated on CR,S$$ {C}_{R,S} $$ is totally bounded and prove with an overwhelming probability that the random sample set uniformly distributed over CR,S$$ {C}_{R,S} $$ is a stable set of sampling for the set of concentrated functions on CR,S$$ {C}_{R,S} $$. Further, we propose an iterative scheme to reconstruct the concentrated functions from their random measurements.
doi: 10.1002/mma.8822pmid: N/A
A malaria transmission disease model with host selectivity and insecticide‐treated bed nets (ITNs), as an intervention for controlling the disease, is formulated. Since the vector is an insect, the vector time scale is faster than the host time scale. This leads to a singularly perturbed model with two distinctive intrinsic time scales, two‐slow for the host and one‐fast for the vector. The basic reproduction number ℜo$$ {\mathfrak{\Re}}_{\mathfrak{o}} $$ is calculated, and the local stability analysis is performed at equilibria of the model when the perturbation parameter ϵ>0$$ \epsilon >0 $$. The model is analyzed when ϵ→0$$ \epsilon \to 0 $$ using asymptotic expansions technique. The results show that if over 30% of humans use ITNs, then ℜo$$ {\mathfrak{\Re}}_{\mathfrak{o}} $$ can be reduced below 1, and hence, malaria disease can be eliminated. In addition, the dynamics on the slow surface indicate that the infected vectors decay fast when ϵ=0.001$$ \epsilon =0.001 $$ according to the numerical simulations.
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