In this paper, we study the quasilinear elliptic system with Sobolev critical exponent involving both concave-convex and Hardy terms in bounded domains. By employing the technique introduced by Benci and Cerami (1991), we obtain at least $\mathrm{c}\mathrm{a}\mathrm{t}(\mathrm{\Omega })+\mathrm{1}$ distinct positive solutions.

In this paper, we derive explicit determinantal representation formulas of general, Hermitian, and skew-Hermitian solutions to the generalized Sylvester matrix equation involving $*$-Hermicity $\mathbf{A}\mathbf{X}{\mathbf{A}}^{*}+\mathbf{B}\mathbf{Y}{\mathbf{B}}^{*}=\mathbf{C}$ over the quaternion skew field within the framework of the theory of noncommutative column-row determinants.

In this paper, a mathematical model of pneumococcal pneumonia with time delays is proposed. The stability theory of delay differential equations is used to analyze the model. The results show that the disease-free equilibrium is asymptotically stable if the control reproduction ratio $R_{\mathrm{0}}$ is less than unity and unstable otherwise. The stability of equilibria with delays shows that the endemic equilibrium is locally stable without delays and stable if the delays are under conditions. The existence of Hopf-bifurcation is investigated and transversality conditions are proved. The model results suggest that, as the respective delays exceed some critical value past the endemic equilibrium, the system loses stability through the process of local birth or death of oscillations. Further, a decrease or an increase in the delays leads to asymptotic stability or instability of the endemic equilibrium, respectively. The analytical results are supported by numerical simulations.

We study convergence of solutions of a space and time inhomogeneous fractional wave equation on the quarter-plane to the stationary regime described by solutions of the Helmholtz equation.

Via the concentration compactness principle, delicate energy estimates, the strong maximum principle, and the Mountain Pass lemma, the existence of positive solutions for a nonlinear PDE with multi-singular inverse square potentials and critical Sobolev-Hardy exponent is proved. This result extends several recent results on the topic.

A general Ostrowski’s type inequality for double integrals is given. We utilize function whose partial derivative of order four exists and is bounded.

In this paper, we present some common fixed point theorems for a pair of self-mappings in fuzzy cone metric spaces under the generalized fuzzy cone contraction conditions. We extend and improve some recent results given in the literature.

In the present investigation, we introduce certain new subclasses of the class of biunivalent functions in the open unit disc defined by quasi-subordination. We obtained estimates on the initial coefficients $\left|a_{\mathrm{2}}\right|$ and $\left|a_{\mathrm{3}}\right|$ for the functions in these subclasses. The results present in this paper would generalize and improve those in related works of several earlier authors.

In this work, we want to detect the shape and the location of an inclusion $\omega$ via some boundary measurement on $\partial \mathrm{\Omega }$. In practice, the body $\omega$ is immersed in a fluid flowing in a greater domain $\mathrm{\Omega }$ and governed by the Stokes equations. We study the inverse problem of reconstructing $\omega$ using shape optimization methods by defining the Kohn-Vogelius cost functional. We aim to study the inverse problem with Neumann and mixed boundary conditions.

We investigate the optimal vaccination and screening strategies to minimize human papillomavirus (HPV) associated morbidity and the interventions cost. We propose a two-sex compartmental model of HPV-infection with time-dependent controls (vaccination of adolescents, adults, and screening) which can act simultaneously. We formulate optimal control problems complementing our model with two different objective functionals. The first functional corresponds to the protection of the vulnerable group and the control problem consists of minimizing the cumulative level of infected females over a fixed time interval. The second functional aims to eliminate the infection, and, thus, the control problem consists of minimizing the total prevalence at the end of the time interval. We prove the existence of solutions for the control problems, characterize the optimal controls, and carry out numerical simulations using various initial conditions. The results and properties and drawbacks of the model are discussed.

In this paper, by the use of the weight functions, and the idea of introducing parameters, a discrete Mulholland-type inequality with the general homogeneous kernel and the equivalent form are given. The equivalent statements of the best possible constant factor related to a few parameters are provided. As applications, the operator expressions and a few particular examples are considered.

We introduce the two-dimensional quaternion linear canonical transform (QLCT), which is a generalization of the classical linear canonical transform (LCT) in quaternion algebra setting. Based on the definition of quaternion convolution in the QLCT domain we derive the convolution theorem associated with the QLCT and obtain a few consequences.

In this paper, we study the existence of infinitely many weak solutions for nonlocal elliptic equations with critical exponent driven by the fractional $p$-Laplacian of order $s$. We show the above result when $\lambda >\mathrm{0}$ is small enough. We achieve our goal by making use of variational methods, more specifically, the Nehari Manifold and Lusternik-Schnirelmann theory.

We consider an abstract coupled evolution system of second order in time. For any positive value of the initial energy, in particular for high energies, we give sufficient conditions on the initial data to conclude nonexistence of global solutions. We compare our results with those in the literature and show how we improve them.

We establish fractional integral and derivative formulas by using Marichev-Saigo-Maeda operators involving the S-function. The results are expressed in terms of the generalized Gauss hypergeometric functions. Corresponding assertions in terms of Saigo, Erdélyi-Kober, Riemann-Liouville, and Weyl type of fractional integrals and derivatives are presented. Also we develop their composition formula by applying the Beta and Laplace transforms. Further, we point out also their relevance.

The Cauchy initial value problem of the modified coupled Hirota equation is studied in the framework of Riemann-Hilbert approach. The N-soliton solutions are given in a compact form as a ratio of $(N+\mathrm{1})\times (N+\mathrm{1})$ determinant and $N\times N$ determinant, and the dynamical behaviors of the single-soliton solution are displayed graphically.

In this paper, we consider the following two-point boundary value problems of fuzzy linear fractional differential equations: $({{}^{c}D}_{\mathrm{1,1}}^{\alpha }y)(t)\oplus b(t)\otimes ({{}^{c}D}_{\mathrm{1,1}}^{\beta }y)(t)\oplus c(t)\otimes y(t)=f(t)$, $t\in (\mathrm{0,1})$, $y(\mathrm{0})=y_{\mathrm{0}}$ and $y(\mathrm{1})=y_{\mathrm{1}}$, where $b,c\in C(I)$, $b(t),c(t)\ge \mathrm{0}$, $y,f\in C(I,{\mathbf{R}}_{\mathrm{F}})$, $I=[\mathrm{0,1}]$, $y_{\mathrm{0}},y_{\mathrm{1}}\in {\mathbf{R}}_{\mathrm{F}}$ and . Our existence result is based on Banach fixed point theorem and the approximate solution of our problem is obtained by applying the Haar wavelet operational matrix.

This paper deals with solving numerically partial integrodifferential equations appearing in biological dynamics models when nonlocal interaction phenomenon is considered. An explicit finite difference scheme is proposed to get a numerical solution preserving qualitative properties of the solution. Gauss quadrature rules are used for the computation of the integral part of the equation taking advantage of its accuracy and low computational cost. Numerical analysis including consistency, stability, and positivity is included as well as numerical examples illustrating the efficiency of the proposed method.

We constitute some necessary and sufficient conditions for the system $A_{\mathrm{1}}{X}_{\mathrm{1}}=C_{\mathrm{1}}$, $X_{\mathrm{1}}{B}_{\mathrm{1}}=C_{\mathrm{2}}$, $A_{\mathrm{2}}{X}_{\mathrm{2}}=C_{\mathrm{3}}$, $X_{\mathrm{2}}{B}_{\mathrm{2}}=C_{\mathrm{4}}$, $A_{\mathrm{3}}{X}_{\mathrm{1}}{B}_{\mathrm{3}}+A_{\mathrm{4}}{X}_{\mathrm{2}}{B}_{\mathrm{4}}=C_c$, to have a solution over the quaternion skew field in this paper. A novel expression of general solution to this system is also established when it has a solution. The least norm of the solution to this system is also researched in this article. Some former consequences can be regarded as particular cases of this article. Finally, we give determinantal representations (analogs of Cramer’s rule) of the least norm solution to the system using row-column noncommutative determinants. An algorithm and numerical examples are given to elaborate our results.

We define and study $C^1$-solutions of the Aronsson equation (AE), a second order quasi linear equation. We show that such super/subsolutions make the Hamiltonian monotone on the trajectories of the closed loop Hamiltonian dynamics. We give a short, general proof that $C^1$-solutions are absolutely minimizing functions. We discuss how $C^1$-supersolutions of (AE) become special Lyapunov functions of symmetric control systems, and allow to find continuous feedbacks driving the system to a target in finite time, except on a singular manifold. A consequence is a simple proof that the corresponding minimum time function is locally Lipschitz continuous away from the singular manifold, despite classical results showing that it should only be Hölder continuous unless appropriate conditions hold. We provide two examples for Hörmander and Grushin families of vector fields where we construct $C^1$-solutions (even classical) explicitly.