The present article deals with the existence of weak solutions to a class of p(z)p\left(z)-Kirchhoff-type problems. To address these problems, we employ a variational approach in conjunction with the theory of variable exponent Sobolev spaces, while imposing suitable assumptions on the source term. Furthermore, we utilize the theory of Young measures.
We consider conformal iterated function systems (CIFS) that are induced by an infinite rooted tree TT, which we call TT-conformal iterated function systems (TT-CIFS). We focus on the parameterized families of TT-CIFS that satisfy the trasversality condition and give a dimension formula for the exiting measure for a.e. parameter. We also give a sufficient condition that the exiting measure is absolutely continuous for a.e. parameter.
In this article, we study weighted Stepanov-like pseudo-almost automorphic functions with infinite delay using measure theory. We present a new concept of weighted ergodic functions, which is more general than the classical one. Then, we establish many interesting results on the space of such functions. We also study the existence and uniqueness of (μ,ν)\left(\mu ,\nu )-Stepanov-like pseudo-almost automorphic solutions of infinite class in the α\alpha -norm for some partial functional differential equations in a Banach space with unbounded delay, using the spectral decomposition of the phase space developed by Adimy and his co-authors. An example is given to illustrate this work.
This article aims to exhibit new instability results on the effects of arbitrary delay on a class of second-order evolution equations involving in a Hilbert space. More precisely, we prove that arbitrary finite, large or small delay might remarkably destroy the stability of a well-behaved (stable) system, that is by driving the resulting system with delay to generate non-trivial periodic solutions with constant energy, solutions with exponential growth rate and solutions with blow-up energy. Also, an interesting new effect of large delay is constructively established. We further supply this study with applications and numerical simulations.
This article discusses the concept of quasi-semigroups of bounded linear operators within the framework of time scales, encompassing both continuous and discrete cases. It extends the classical theory of quasi-semigroups of operators, which serves as a generalization of strongly continuous semigroups of operators in time scales. In addition, the text mentions providing applications in abstract evolution equations, indicating practical implications of this theoretical framework.
This article is dedicated to establishing the existence and uniqueness of solutions for the following problem: Dαx(t)=F(t,x(t))x(0)=x0,\left\{\begin{array}{l}{D}^{\alpha }x\left(t)=F\left(t,x\left(t))\hspace{1.0em}\\ x\left(0)={x}_{0},\hspace{1.0em}\end{array}\right. where x0{x}_{0} is the singular generalized function and F satisfies L∞{L}^{\infty } logarithmic type, Dα{D}^{\alpha } is the Caputo derivative of order m−1<α
In this article, we consider a nonlinear elliptic unilateral equation whose model is − ∑ i = 1 N ∂ i σ i ( x , u , ∇ u ) + L ( x , u , ∇ u ) + N ( x , u , ∇ u ) = μ − div ϕ ( u ) in Ω . -\mathop{\sum }\limits_{i=1}^{N}{\partial }^{i}{\sigma }_{i}\left(x,u,\nabla u)+L\left(x,u,\nabla u)+N\left(x,u,\nabla u)=\mu -{\rm{div}}\phi \left(u)\hspace{1.0em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega . We prove the existence of entropy solutions for the aforementioned equation in the anisotropic Sobolev space, under the hypotheses, μ = f − div F \mu =f-{\rm{div}}F belongs to L 1 ( Ω ) + W − 1 , p ′ ( Ω ) {L}^{1}\left(\Omega )+{W}^{-1,{p}^{^{\prime} }}\left(\Omega ) . The nonlinear terms L ( x , s , ∇ u ) L\left(x,s,\nabla u) satisfy the sign and growth conditions, and N ( x , s , ∇ u ) N\left(x,s,\nabla u) verifies only the growth conditions.
In this study, an initial-value problem for a nonlinear Volterra functional integro-differential equation on a finite interval was considered. The nonlinear term in the equation contains multiple time delays. In addition to giving some new theorems on the existence and uniqueness of solutions to the equation, the authors also prove the Hyers-Ulam-Rassias stability and the Hyers-Ulam stability of the equation. The proofs use several different tools including Banach’s fixed point theorem, the construction of a Picard operator, and an application of Pachpatte’s inequality. An example is provided to illustrate the existence, uniqueness, and stability properties of solutions.
In this work, we investigate the existence of mild solutions in the α\alpha -norm for a class of nonlocal integrodifferential equations. We employ the theory of resolvent operators introduced by R. Grimmer. The Leray-Schauder alternative is the principal working tool for our analysis.
Abstract In this article, we investigate some generalized Sigmoid Beverton-Holt models in time scales, and we obtain the existence and uniqueness of a globally attractive almost periodic solution to the associated dynamic equations with or without survival rates under some suitable assumptions. An example is given to illustrate our abstract results.
This article establishes the existence of a weak solution for a class of p(x)p\left(x)-Kirchhoff-type problem under no-flux boundary conditions with a reaction term depending also on the gradient convection. The proof of the main result is constructed by utilizing the theory of topological degree and the theory of generalized Sobolev spaces.
In this paper, an inverse problem of determining a kernel in a one-dimensional integro-differential time-fractional diffusion equation with initial-boundary and overdetermination conditions is investigated. An auxiliary problem equivalent to the problem is introduced first. By Fourier method this auxilary problem is reduced to equivalent integral equations. Then, using estimates of the Mittag-Leffler function and successive aproximation method, an estimate for the solution of the direct problem is obtained in terms of the norm of the unknown kernel which will be used in study of inverse problem. The inverse problem is reduced to the equivalent integral equation. For solving this equation the contracted mapping principle is applied. The local existence and global uniqueness results are proven.
In this article, we provide sufficient conditions for the existence of periodic solutions for the polynomial differential system of the form x˙=−y+εP1(x,y,z,u,v)+h1(t),y˙=x+εP2(x,y,z,u,v)+h2(t),z˙=−u+εP3(x,y,z,u,v)+h3(t),u˙=z+εP4(x,y,z,u,v)+h4(t),v˙=λv+εP5(x,y,z,u,v)+h5(t),\begin{array}{r}\dot{x}=-y+\varepsilon {P}_{1}\left(x,y,z,u,v)+{h}_{1}\left(t),\\ \dot{y}=x+\varepsilon {P}_{2}\left(x,y,z,u,v)+{h}_{2}\left(t),\\ \dot{z}=-u+\varepsilon {P}_{3}\left(x,y,z,u,v)+{h}_{3}\left(t),\\ \dot{u}=z+\varepsilon {P}_{4}\left(x,y,z,u,v)+{h}_{4}\left(t),\\ \dot{v}=\lambda v+\varepsilon {P}_{5}\left(x,y,z,u,v)+{h}_{5}\left(t),\end{array} where P1,P2,P3,P4{P}_{1},{P}_{2},{P}_{3},{P}_{4}, and P5{P}_{5} are polynomials in the variables x,y,z,u,vx,y,z,u,v of degree nn, hi(t){h}_{i}\left(t) are 2π2\pi -periodic functions with i=1,5¯i=\overline{1,5}, λ\lambda is a real number, and ε\varepsilon is a small parameter.
This article deals with a class of generalized backward doubly stochastic differential equations driven by fractional Brownian motion with the Hurst parameter HH greater than 1/2. The existence and uniqueness of solutions to our equation as well as comparison theorems are obtained.
In this article, we study the following noncoercive quasilinear parabolic problem ∂u∂t−diva(x,t,u,∇u)+ν∣u∣s−1u=λ∣u∣p−2u∣x∣p+finQT,u=0onΣT,u(x,0)=u0inΩ,\left\{\begin{array}{ll}\frac{\partial u}{\partial t}-\hspace{0.1em}\text{div}\hspace{0.1em}a\left(x,t,u,\nabla u)+\nu {| u| }^{s-1}u=\lambda \frac{{| u| }^{p-2}u}{{| x| }^{p}}+f& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{Q}_{T},\\ u=0& \hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}{\Sigma }_{T},\\ u\left(x,0)={u}_{0}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,\end{array}\right. with f∈L1(QT)f\in {L}^{1}\left({Q}_{T}) and u0∈L1(Ω){u}_{0}\in {L}^{1}\left(\Omega ) and show the existence of entropy solutions for this noncoercive parabolic problem with Hardy potential and L1-data.
In this article, we consider an epidemiological model in which we take into account the effects of direct and indirect transmissions. The first mode occurs through direct contact between infectious and susceptible individuals, and the second one will take place through the shedding of virus particles by infectious individuals and their acquisition by susceptible ones. We also study the effect of latency period and time needed for a susceptible person to become infected by indirect transmission mode. By considering the direct and indirect basic reproduction numbers, we define the basic reproduction number R0{R}_{0} of the model, which helps us to analyze the stability of equilibria and bifurcation and determine the most sensitive parameters. In conclusion, some numerical simulations are given to confirm the analytical analysis.
We are interested in the existence of mild solutions for a class of partial integrodifferential inclusions in infinite dimensional Banach spaces. First, we show the existence of mild solutions with the help of a scale of Banach spaces, the theory of resolvent operators, and the fixed point theory for the measure of non-compactness. Moreover, we examine the existence of asymptotically almost periodic solutions for our problem. Finally, an example of the abstract results is provided.
Social distancing plays an essential role in controlling the spread of an epidemic, but changing the behavior of individuals regarding social distancing is costly. In order to make a rational decision, individuals must compare the cost of social distancing and the cost of infection. People are typically more likely to change their behavior if they are aware that the government is willing to incur additional cost to shorten the duration of an epidemic. I extend an optimal control problem of social distancing by integrating with the SIR model which describes the disease process. I present an optimal control problem to consider the behavior of susceptible individuals and the government in investment as control strategies and compute the equilibrium strategies under the potency of investment, using relative risk functions according to the investment that is made by susceptible individuals and the government. The equilibrium of this problem represents the optimal control strategies for minimizing the cost and duration of controlling an epidemic. Additionally, the model is evaluated using COVID-19 data from Egypt, Japan, Italy, Belgium, Nigeria, and Germany. The findings extracted from this model could be valuable in developing public health policy in the event of an epidemic.
We show the existence of unbounded solutions to difference equations of the form {xn+1=c′nxnBnyn,yn+1=bnxn+cnynAn+Cnyn for n=0,1,…,\left\{ {\matrix{{{x_{n + 1}} = {{{{c'}_n}{x_n}} \over {{B_n}{y_n}}},} \hfill \cr {{y_{n + 1}} = {{{b_n}{x_n} + {c_n}{y_n}} \over {{A_n} + {C_n}{y_n}}}} \hfill \cr } \,\,\,\,\,for} \right.\,\,\,n = 0,1, \ldots , where {c′n}n=0∞\left\{ {{{c'}_n}} \right\}_{n = 0}^\infty, {B′n}n=0∞\left\{ {{{B'}_n}} \right\}_{n = 0}^\infty, {bn}n=0∞\left\{ {{b_n}} \right\}_{n = 0}^\infty, {cn}n=0∞\left\{ {{c_n}} \right\}_{n = 0}^\infty, and {An}n=0∞\left\{ {{A_n}} \right\}_{n = 0}^\infty are all bounded above and below by positive constants, and {Cn}n=0∞\left\{ {{C_n}} \right\}_{n = 0}^\infty is either bounded above and below by positive constants or is identically zero. In the latter case, we give an example which can be reduced to a system of the form
