Open Journal of Mathematical Analysis (OMA)

Here we study the existence of solutions of a nonlocal two-point, with parameters, boundary value problem of a first order nonlinear differential equation. The maximal and minimal solutions will be proved. The continuous dependence of the unique solution on the parameters of the nonlocal condition will be proved. The anti-periodic boundary value problem will be considered as an application.

This paper presents an existence theorem of the solutions for a boundary value problem of fractional order differential equations with integral boundary conditions, by using measure of noncompactness combined with Mönch fixed point theorem. An example is furnished to illustrate the validity of our outcomes.

Recently, asymptotic estimates for the unimodular Fourier multipliers \(e^{i\mu(D)}\) have been studied for the function \(\alpha\)-modulation space. In this paper, using the almost orthogonality of projections and some techniques on oscillating integrals, we obtain asymptotic estimates for the unimodular Fourier multiplier \(e^{it(I-\Delta)^{\frac{\beta}{2}}}\) on the \(\alpha\)-modulation space. For an application, we give the asymptotic estimate of the solution for the Klein-Gordon equation with initial data in a \(\alpha\)-modulation space. We also obtain a quantitative form about the solution to the nonlinear Klein-Gordon equation.

In this work, we consider a plate equation with nonlinear source and partially hinged boundary conditions. Our goal is to show analytically that the solution blows up in finite time. The background of the problem comes from the modeling of the downward displacement of suspension bridge using a thin rectangular plate. The result in the article shows that in the present of fractional damping and a nonlinear source such as the earthquake shocks, the suspension bridge is bound to collapse in finite time.

Using the Krasnoselskii’s fixed point theorem and the contraction mapping principle we give sufficient conditions for the existence and uniqueness of solutions for initial value problems for delay fractional differential equations with the mixed Riemann-Liouville and Caputo fractional derivatives. At the end, an example is given to illustrate our main results.

Ation frical contact problem between a piezoelectric body and a deformable conductive foundation is numerically studied in this paper. The process is quasistatic and the material’s behavior is modelled with an electro-viscoelastic constitutive law. Contact is described with the normal compliance condition, a version of Coulomb’s law of dry friction, and a regularized electrical conductivity condition. A fully discrete scheme is introduced to solve the problem. Under certain solution regularity assumptions, we derive an optimal order error estimate. Some numerical simulations are included to show the performance of the method.

Using the technique of differential subordination, we here, obtain certain sufficient conditions for starlike and convex functions. In most of the results obtained here, the region of variability of the differential operators implying starlikeness and convexity of analytic functions has been extended. The extended regions of the operators have been shown pictorially.

In this paper, we investigate the nonlinear dynamics for an attraction-repulsion chemotaxis Keller-Segel model with logistic source term
\(u_{1t}=d_{1}\Delta{u_{1}}-\chi \nabla (u_{1}\nabla{u_{2}})+ \xi{ \nabla (u_{1}\nabla{u_{3}})}+\mathbf g(u),{\mathbf x}\in\mathbb{T}^{d}, t>0,\)
\( u_{2t}=d_{2}\Delta{u_{2}}+\alpha u_{1}-\beta u_{2},{\mathbf x}\in\mathbb{T}^{d}, t>0,\)
\(u_{3t}=d_{3}\Delta{u_{3}}+\gamma u_{1}- \eta u_{3},{\mathbf x}\in\mathbb{T}^{d}, t>0,\)
\( \frac{\partial{u_{1}}}{\partial{x_{i}}}=\frac{\partial{u_{2}}}{\partial{x_{i}}}=\frac{\partial{u_{3}}}{\partial{x_{i}}}=0,x_{i}=0,\pi, 1\leq i\leq d,\)
\( u_{1}(x,0)=u_{10}(x), u_{2}(x,0)=u_{20}(x), u_{3}(x,0)=u_{30}(x), {\mathbf x}\in\mathbb{T}^{d} (d=1,2,3).\)
Under the assumptions of the unequal diffusion coefficients, the conditions of chemotaxis-driven instability are given in a \(d\)-dimensional box \(\mathbb{T}^{d}=(0,\pi)^{d} (d=1,2,3)\). It is proved that in the condition of the unique positive constant equilibrium point \({\mathbf w_{c}}=(u_{1c},u_{2c},u_{3c})\) of above model is nonlinearly unstable. Moreover, our results provide a quantitative characterization for the early-stage pattern formation in the model.

This paper studies the movement of a molecule in two types of cell complexes: the square tiling and the hexagonal one. This movement from a cell \(i\) to a cell \(j\) is referred to as an homogeneous Markov chain. States with the same stochastic behavior are grouped together using symmetries of states deduced from groups acting on the cellular complexes. This technique of lumpability is effective in forming new chains from the old ones without losing the primitive properties and simplifying tedious calculations. Numerical simulations are performed using R software to determine the impact of the shape of the tiling and other parameters on the achievement of the equilibrium. We start from small square tiling to small hexagonal tiling before comparing the results obtained for each of them. In this paper, only continuous Markov chains are considered. In each tiling, the molecule is supposed to leave the central cell and move into the surrounding cells.

In this work, we are concerned with a problem for a viscoelastic wave equation with strong damping, nonlinear source and distributed delay terms. We show the exponential growth of solution with \(L_{p}\)-norm, i.e., \(\lim\limits_{t\rightarrow \infty}\Vert u\Vert_p^p \rightarrow \infty\).