Consider $(M, g)$ as an $n$-dimensional compact Riemannian manifold. Our main aim in this paper is to study the first eigenvalue of $(p, q)$-Laplacian system on $C$-totally real submanifold in Sasakian space of form $\overline{M}^{2m+1} (\kappa).$ Also in the case of $p, q > n$ we show that for $λ_{1,p,q}$ arbitrary large there exists a Riemannian metric of volume one conformal to the standard metric of $S^n.$

In this paper, a compartment model is formulated to study the transmission dynamics of avian influenza virus among birds, poultry and human population. Due to the strong coupling of the system caused by the transmission route, there are some mathematical challenges to obtain the existence of the endemic equilibrium and its global asymptotic stability. Firstly, we give the well-posedness of the model and then discuss the threshold dynamics for three sub-models. Next, we give the threshold dynamics for the whole system: the disease-free equilibrium is locally asymptotically stable with $R_0<1,$ and the other conditions are required for the global asymptotic stability of the disease-free equilibrium. The endemic-equilibrium is globally attractive when $R_0>1.$ Furthermore, the sensitivity analysis and an optimal control problem are discussed. Finally, some numerical simulations are carried out to illustrate our theoretical results and visualize the impact of various parameters on model dynamics, which suggest that decreasing the recruitment rate and increasing the death rate of poultry, can only control the disease by simultaneously cutting off the transmission from birds to poultry and humans even if the ultimate scale of the disease can be effectively controlled. In addition, enhancing public awareness of prevention to reduce the transmission from birds and poultry to humans is also effective in controlling the final scale of disease.

Based on Lie symmetry theory, the exact solutions of the hyperbolic Monge-Ampère equation are studied. Firstly, the invariance of the Lie symmetry group is applied to obtain the six-dimensional Lie algebras, then the commutator table and the adjoint representation of the equation are obtained, based on which the optimal system is found. Finally, the exact solutions are obtained by symmetry reduction which transforms the PDEs into easily solvable ODEs.

This paper introduces an equality that is valid for tempered fractional integrals. By using this equality, we establish some Euler-Maclaurin-type inequalities for the case of differentiable convex functions involving tempered fractional integrals. Furthermore, our results are provided by using special cases of obtained theorems.

Multiple fractional derivatives enrich the dynamic properties of fractional differential equations. This paper concerns with neutral fractional functional differential equations with two Caputo fractional derivatives. By using the fixed point methods with the fractional integral inequalities, the existence results and controllability of the equations are considered in the cases of finite delay and infinite delay, respectively. An example is given to illustrate the main results.

In this article, exact solutions of the Lonngren-wave equation are investigated. Firstly, the equation is transformed into an ordinary differential equation by traveling wave transformation. Based on the homogeneous balance method, bright-solitons and singular periodic wave solutions of the equation are derived by applying the simple function expansion method and the Riccati equation method. Applying the $Exp(−φ(ς))$ expansion method, we construct dark-solitons and kink wave solutions of the equation. Moreover, the 3-D, 2-D and density plots are drawn by choosing the appropriate parameters so that the properties of the solutions can be better studied. According to the Figures, the analysis of the dynamical behavior of the solutions is provided. This article enriches the diversity of the solutions of the equation.

This paper focuses on constructing the exact solutions of a (4+1)- dimensional BLMP equation via the two variables $(G'/G,1/G)$-expansion method and extended generalized Riccati equation mapping method. Firstly, the main ideas of the methods are described. Then, the methods are applied to the equation to derive the exact solutions including singular, kink (or anti-kink) and periodic solutions. Finally, the 3D plots of some exact solutions are observed graphically and intuitively by assigning the values of unknown parameters. The results prove that the methods are powerful, enriching the diversity of forms of exact solutions.

The geometrical singular perturbation theory has been successfully applied in studying a large range of mathematical biological models with different time scales. In this paper, we use the geometrical singular perturbation theory to investigate a slow-fast Leslie-Gower prey-predator model with Monod-Haldane function response and get some interesting dynamical phenomena such as singular Hopf bifurcation, canard explosion phenomenon, relaxation oscillation cycle, heteroclinic and homoclinic orbits and the coexistence of canard cycle and relaxation oscillation cycle.

In this paper, we study the positive solutions of the semilinear elliptic equation $$\begin{cases} Lu+g(x,u)u=0 \ \ &{\rm in}& \Omega, \\ Bu=0 \ \ &{\rm on}& ∂Ω, \end{cases}$$where $\Omega ⊂\mathbb{R}^N$ is a bounded smooth domain, $L$ is an elliptic operator, $B$ is a general boundary operator and $g(·, ·)$ is a continuous function. This is a general problem proposed by Amann [Arch. Rational Mech. Anal. 44 (1972)], Cac [J. London Math. Soc. 25 (1982)] and Hess [Math. Z. 154 (1977)]. We obtain various uniqueness results when the nonlinearity function $g$ satisfies some additional conditions.

In this article, we consider a harmonic analysis of locally compact groups and introduce a generalization of the classical cross-Wigner distribution defined on $G × \hat{G}$ by $$W_{\mathfrak{F}}(\psi,\varphi)(g,\xi)=\int_{G}\overline{\xi(h)}\psi(\tau_1(g,h))\overline{\varphi(\tau_2(g,h))}d\mu(h).$$We construct the so-called Weyl-Heisenberg frame on a locally compact communicative Hausdorff group and establish its properties. Thus, we show that assume $\Lambda$ and $\Gamma$ are closed cocompact subgroups of $G$ and $\hat{G}$, respectively, then, for a given window $\phi∈ L^2(G),$ either both systems $\{m_{\gamma}\tau_{\lambda}\phi\}_{\lambda\in\Lambda,\gamma\in\Gamma}$ and $\{m_{\kappa}\tau_v\phi\}_{\kappa\in\Lambda^{⊥},v\in\Gamma^{⊥}}$ are Gabor systems in $L^2 (G),$ simultaneously, with the same upper bound, or neither $\{m_{\gamma}\tau_{\lambda}\phi\}_{\lambda\in\Lambda,\gamma\in\Gamma}$ nor $\{m_{\kappa}\tau_v\phi\}_{\kappa\in\Lambda^{⊥},v\in\Gamma^{⊥}}$ comprises a Gabor system. Finally, pseudo-differential operators on locally compact groups are studied, we establish that assuming a pseudo-differential operator $A_a$ corresponds to the symbol $a\in W^{\infty,1}_{\tau,1oι^{-1}}(G\times\hat{G})$ then $A_a$ is bounded operator $W^{p,q}_{\tau}(G)\rightarrow W^{p,q}_{\tau}(G)$.

The global dynamics of a SIR model characterized by both vaccination and treatment are considered in the present paper. Global stability ensures convergence to an equilibrium solution irrespective of the initial state of infection. Various independent sets of sufficient conditions on parameters and functional relations are obtained through Lyapunov functionals for stability. It is also established how a disease-free environment can be provided by a proper combination of treatment and vaccination, which is a unique feature as far as SIR models are concerned, as many of the studies have ignored the influence of treatment. Results are illustrated with numerical examples and simulations are provided to visualize the illustrations.

In this paper, we consider a Leslie-Gower predator-prey model with a square root functional response while prey forms a herd as a form of group defense. We show that the solution of the system is non-negative and bounded. By applying the blow-up technique, it can be deduced that the origin displays instability. Moreover, employing the proof-by-contradiction approach, we demonstrate that the unique equilibrium point can be globally asymptotically stable under certain conditions. The sufficient conditions for the occurrence, stability, and direction of Hopf bifurcation are obtained. We further explore the conditions for the existence and uniqueness of the limit cycle. Theoretical results are validated through numerical simulations. Thus, our findings reveal that herd behavior has an important impact on the Leslie-Gower prey-predator system.

Based on the properties of Mittag-Leffler function, the behavior of eigenvalue and the eigenfunction for a class of fractional differential equations with integral boundary value condition is discussed, and the asymptotic expression of the eigenfunction is given.

A new stochastic SIR epidemic model with vaccination is established and its dynamical behavior is analyzed. Considering the random effects of vaccination rates and mortality in this model, it is demonstrated that the extinction and persistence of the virus is only correlated with the threshold $R^s_0.$ If $R^s_0<1,$ the disease dies out with probability one. And if $R^s_0>1,$ the disease is stochastic persistent in the means with probability one. In addition, the existence and uniqueness of a smooth distribution are proven using the Itôs formula, and the sufficiency criterion is obtained using the Lyapunov function. Finally, the accuracy and efficiency of the stochastic SIR epidemic model with vaccination in predicting disease transmission trends were verified through simulation. Unlike the singularity of stochastic perturbations in existing infectious disease models, the innovation of this paper is in the addition of multiple stochastic perturbations, especially distinguishing the stochastic perturbations of mortality under vaccination, which are used to study the dynamics of the model.

This paper delves into an examination of the existence, uniqueness and controllability results concerning impulsive functional fuzzy nonlinear neutral integro-differential equations with non-local conditions. Additionally, we explore the fuzzy solution within the context of normal, convex, upper semi-continuous and compactly supported interval fuzzy numbers. Our findings are derived through the application of the Banach fixed point theorem. Also, we illustrate the result with an example.

The purpose of this paper is to investigate a frictional contact problem between a thermo-piezoelectric body and an obstacle (such as a foundation). The thermo-piezoelectric constitutive law is assumed to be nonlinear. Modified Signorini’s contact conditions are used to describe the contact, and these are adjusted to account for temperature-dependent unilateral conditions, which are associated with a nonlocal Coulomb friction law. The problem is formulated as a coupled system of displacement field, electric potential, and temperature, which is solved using a variational approach. The existence of a weak solution is established through the utilization of elliptic quasi-variational inequalities, strongly monotone operators, and the fixed point method. Finally, an iterative method is suggested to solve the coupled system, and a convergence analysis is established under appropriate conditions.

In this paper, we propose a multi-stage HIV infection model with general incidence rate to describe the influence of raltegravir intensification on viral dynamics. The basic reproduction number $R_0$ is established. The infection-free equilibrium $E_0$ is locally asymptotically stable if $R_0<1.$ The infection equilibrium $E^∗$ is locally asymptotically stable if $R^0 > 1.$

In Hilbert spaces, $K$-$g$-frames are an advanced version of $g$-frames that enable the reconstruction of objects from the range of a bounded linear operator $K.$ This research investigates $K$-$g$-frames in Hilbert space. Firstly, using the $g$-preframe operators, we characterize the dual $K$-$g$-Bessel sequence of a $K$-$g$ frame. We provide additional requirements that must be met for the sum of a given $K$-$g$-frame and its dual $K$-$g$-Bessel sequence to be a $K$-$g$-frame. At the end of this paper, we present the concept of $K$-$g$-orthonormal bases and explain their link to $g$-orthonormal bases in Hilbert space. We also provide an alternative definition of $K$-$g$-Riesz bases using $K$-$g$-orthonormal bases. This gives a better understanding of the concept.

The present paper introduces the Aboodh transform technique as a method for obtaining solutions to a class of non-homogeneous fractional integral equations. The emphasis is placed on equations characterized by expressions involving Riemann-Liouville fractional integrals of orders $1,$ $\frac{1}{2},$ and $\frac{1}{3}.$ The paper includes illustrative examples that demonstrate the application of the Aboodh transform technique. These examples elucidate how the technique can effectively yield solutions for specific instances of the mentioned equations. The obtained solutions are presented in the form of Mellin-Ross functions.

In this work, we discuss the existence and continuous dependence on initial data of solutions for non-local random impulsive neutral stochastic integrodifferential delayed equations. First, we prove the existence of mild solutions to the equations by using Krasnoselskii's-Schaefer type fixed point theorem. Next, we prove the continuous dependence on initial data results under the Lipschitz condition on a bounded and closed interval. Finally, we propose an example to validate the obtained results.

In this paper, we present a novel class of relations and investigate the connection between the ${\rm R}$-projective curvature tensor and other tensors of Finsler space $F_n.$ This space is characterized by the property for Cartan’s the third curvature tensor $R^i_{jkh}$ which satisfies the certain relationship with given covariant vectors field, as follows:

where $R^i_{jkh}\neq 0$ and $\mathcal{B}_n\mathcal{B}_m\mathcal{B}_l$ is the Berwald's third order covariant derivative with respect to $x^l,$ $x^m$ and $x^n$ respectively. The quantities $a_{lmn}= \mathcal{B}_nu_{lm} + u_{lm}\ λ_n,$ $b_{lmn} = \mathcal{B}_nv_{lm}+u_{lm} \ \mu_n,$ $c_{lm} = v_{lm},$ and $d_{ln} =\mathcal{B}_n\mu_l$ are non-zero covariant vector fields. We define this space a generalized $\mathcal{BR}-3rd$ recurrent space and denote it briefly by $\mathcal{GBR}-3RF_n.$ This paper aims to derive the third-order Berwald covariant derivatives of the torsion tensor $H^i_{kh}$ and the deviation tensor $H^i_h.$ Additionally, it demonstrates that the curvature vector $K_j,$ the curvature vector $H_k,$ and the curvature scalar $H$ are all non-vanishing within the considered space. We have some relations between Cartan’s third curvature tensor $R^i_{jkh}$ and some tensors that exhibit self-similarity under specific conditions. Furthermore, we have established the necessary and sufficient conditions for certain tensors in this space to have equal third-order Berwald covariant derivatives with their lower-order counterparts.

In this paper, we are concerned with the well-posedness and large time behavior of Cauchy problem for viscous Cahn-Hilliard-Oono system with chemotaxis in 3D whole space. By using the pure energy method, standard continuity arguments together with negative Sobolev norm estimates, one proves the global well-posedness and time decay estimates.

In this paper, the exact solution of one-dimensional isentropic Euler equations is studied. When the exponent of the state equation satisfies $\gamma = 2,$ we get an exact solution which is linear with respect to the spatial variable $x.$ For this end, we solve some ordinary differential equations with time dependent variable coefficients.

Fractional delay differential equations constitute a powerful mathematical framework for modeling complex dynamical phenomena exhibiting memory and delay effects. In this study, we investigate a class of fractional delay differential equations incorporating Caputo and Riemann-Liouville fractional derivatives with a delay term. Unlike previous approaches, we establish the existence and uniqueness of the analytical solution under relaxed Lipschitz conditions on the nonlinear terms, without requiring contraction assumptions. Utilizing Picard iteration techniques, we demonstrate convergence of the numerical method under these Lipschitz conditions, thereby broadening the applicability of our model to a wider range of real-world scenarios. Additionally, numerical tests are conducted to validate the effectiveness and accuracy of the proposed method, further highlighting its utility in practical applications. Our findings offer new insights into the modeling and analysis of complex dynamical systems, with implications for various scientific and engineering disciplines.