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In this paper we will investigate the existence of multiple normalized solutions to the logarithmic Schrödinger equation given by - ϵ 2 Δ u + V (x) u = λ u + u log u 2 , in R N , ∫ R N | u | 2 d x = a 2 ϵ N , where N ≥ 1 , a , ϵ > 0 , λ ∈ R is an unknown parameter that appears as a Lagrange multiplier and V : R N → (- 1 , + ∞) is a continuous function. Our analysis demonstrates that the number of normalized solutions of the equation is associated with the topology of the set where the potential function V attains its minimum value. To prove the main result, we employ minimization techniques and use the Lusternik-Schnirelmann category. Additionally, we introduce a new function space where the energy functional associated with the problem is of class C 1 . [ABSTRACT FROM AUTHOR]

Let X be a space of homogeneous type. Assume that L is a non negative, selfadjoint operator on L 2 (X) satisfying the sub-Gaussian upper bounds. In this paper, we prove that ‖ (I + L) - n / 2 e i τ L f ‖ 1 , ∞ ≤ C (1 + | τ |) n / 2 ‖ f ‖ 1 , ∀ τ ∈ R. By interpolation, we obtain ‖ (I + L) - n | 1 / p - 1 / 2 | e i τ L f ‖ p ≤ C (1 + | τ |) n | 1 / p - 1 / 2 | ‖ f ‖ p , ∀ τ ∈ R , 1 < p < ∞. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in studying a generalized block space (denoted as B φ p , r ) on a space of homogeneous type. We show that this space is the predual of certain generalized Morrey–Lorentz space. By duality, we obtain the B φ p , r -bound of operators of Calderón–Zygmund type. In addition, we prove a weak Hardy factorization in terms of commutators of integral operator of Calderón–Zygmund type in block spaces. Thanks to the Hardy factorization result, we obtain a characterization of functions in BMO via the boundedness of commutators of homogeneous linear Calderón–Zygmund operators in the generalized block space (resp. the generalized Morrey–Lorentz space). Finally, we study a compactness characterization of commutators of Calderón–Zygmund type in generalized Morrey–Lorentz spaces. [ABSTRACT FROM AUTHOR]

In this paper, we give an estimate of the upper bound of the inner radius for a complete Finsler measure space (M , F , d μ) of weighted Ricci curvature bounded from below, in terms of the d μ -mean curvature bound of the boundary. We also prove the rigidity result that when the boundary is compact, the upper bound is achieved if and only if M is isometric to a forward Finsler geodesic ball of vanishing radical flag curvature. [ABSTRACT FROM AUTHOR]

In this paper, we deal with Serrin-type problems in Riemannian manifolds. First, we obtain a Heintze-Karcher inequality and a Soap Bubble result, with its respective rigidity, when the ambient space has a Ricci tensor bounded below. After, we approach a Serrin problem in bounded domains of manifolds endowed with a closed conformal vector field. Our primary tool, in this case, is a new Pohozaev identity, which depends on the scalar curvature of the manifold. Applications involve Einstein and constant scalar curvature spaces. [ABSTRACT FROM AUTHOR]

Multi-material design optimization problems can, after discretization, be solved by the iterative solution of simpler sub-problems which approximate the original problem at an expansion point to first order. In particular, models constructed from convex separable first order approximations have a long and successful tradition in the design optimization community and have led to powerful optimization tools like the prominently used method of moving asymptotes (MMA). In this paper, we introduce several new separable approximations to a model problem and examine them in terms of accuracy and fast evaluation. The models can, in general, be nonconvex and are based on the Sherman–Morrison–Woodbury matrix identity on the one hand, and on the mathematical concept of topological derivatives on the other hand. We show a surprising relation between two models originating from these two—at a first sight—very different concepts. Numerical experiments show a high level of accuracy for two of our proposed models while also their evaluation can be performed efficiently once enough data has been precomputed in an offline stage. Additionally it is demonstrated that suboptimal decisions can be avoided using our most accurate models. [ABSTRACT FROM AUTHOR]

The aim of this paper is to begin a systematic study of functional inequalities on symmetric spaces of noncompact type of higher rank. Our first main goal of this study is to establish the Stein–Weiss inequality, also known as a weighted Hardy–Littlewood–Sobolev inequality, for the Riesz potential on symmetric spaces of noncompact type. This is achieved by performing delicate estimates of ground spherical function with the use of polyhedral distance on symmetric spaces and by combining the integral Hardy inequality developed by Ruzhansky and Verma with the sharp Bessel-Green-Riesz kernel estimates on symmetric spaces of noncompact type obtained by Anker and Ji. As a consequence of the Stein–Weiss inequality, we deduce Hardy–Sobolev, Hardy–Littlewood–Sobolev, Gagliardo–Nirenberg and Caffarelli–Kohn–Nirenberg inequalities on symmetric spaces of noncompact type. The second main purpose of this paper is to show the applications of aforementioned inequalities for studying nonlinear PDEs on symmetric spaces. Specifically, we show that the Gagliardo-Nirenberg inequality can be used to establish small data global existence results for the semilinear wave equations with damping and mass terms for the Laplace–Beltrami operator on symmetric spaces. [ABSTRACT FROM AUTHOR]

In the recent paper "On a formula for sets of constant width in 2D, Comm. Pure Appl. Anal. 18 (2019), 2117–2131", we gave a constructive formula for all 2d sets of constant width. Based on this result we derive here a formula for the parametrization of the boundary of bodies of constant width in 3 dimensions, with the formula depending on one function defined on S 2 . Each such function gives a minimal value r 0 and for all r ≥ r 0 one finds a body of constant width 2r. Moreover, we show that all bodies of constant width in 3d have such a parametrization. The last result needs a tool that we describe as 'shadow domain' and which is explained in an appendix. The construction is explicit and offers a parametrization different from the one given by T. Bayen, T. Lachand-Robert and É. Oudet in "Analytic parametrization of three-dimensional bodies of constant width. Arch. Ration. Mech. Anal., 186 (2007), 225–249". [ABSTRACT FROM AUTHOR]

Fix λ > - 1 / 2 and λ ≠ 0 . Consider the Bessel operator (introduced by Muckenhoupt–Stein) ▵ λ : = - d 2 d x 2 - 2 λ x d dx on R + : = (0 , ∞) with d m λ (x) : = x 2 λ d x and dx the Lebesgue measure on R + . In this paper, we study the Muckenhoupt-type weights in this Bessel setting along the line of Muckenhoupt–Stein and Andersen–Kerman. Besides, exploiting more properties of the weights A p , λ introduced by Andersen–Kerman, we introduce a new class A ~ p , λ such that the Hardy–Littlewood maximal function is bounded on the weighted L w p space if and only if w is in A ~ p , λ . Moreover, along the line of Coifman–Rochberg–Weiss, we investigate the commutator [ b , R λ ] with R λ : = d dx (▵ λ) - 1 2 to be the Bessel Riesz transform. We show that for w ∈ A p , λ , the commutator [ b , R λ ] is bounded on weighted L w p if and only if b is in the BMO space associated with ▵ λ . [ABSTRACT FROM AUTHOR]

Let B g (1) ⊂ H 2 be the unit geodesic ball and W 0 1 , 2 be the standard Sobolev space. Let λ 1 (B g (1)) be the first Dirichlet eigenvalue in B g (1) associated to the Laplace–Beltrami operator - Δ g on H 2 . In this paper, we establish a sharp Moser–Trudinger inequality involving the Adimurthi–Duret term by a delicate blow-up analysis. Specifically, we prove that sup u ∈ W 0 1 , 2 (B g (1)) , ‖ ∇ g u ‖ 2 = 1 ∫ B g (1) e 4 π (1 + α ‖ u ‖ 2 2) u 2 d Vol g < ∞ if and only if α < λ 1 (B g (1)) . We also prove the existence of extremal functions for the above inequality when α is sufficiently small, which, to our knowledge, is the first result on the extremal function of Moser–Trudinger inequality involving Adimurthi–Druet term in Hyperbolic space. Besides, we extend the above results to the entire space H 2 partially, namely, we prove that sup u ∈ W 1 , 2 (H 2) , ‖ ∇ g u ‖ 2 = 1 ∫ H 2 e 4 π (1 + α ‖ u ‖ 2 2) u 2 - 1 d Vol g is finite if α < 1 / 4 , and is infinite if α > 1 / 4 . If α = 1 / 4 , we derive a different Adimurthi–Druet type result using an effective scaling argument, which is established by Chen et al. in [9] and [10]. [ABSTRACT FROM AUTHOR]

The storage requirement for distributed tracing can be reduced significantly by sampling only the anomalous or interesting traces that occur rarely at runtime. In this paper, we introduce an unsupervised sampling pipeline for distributed tracing that ensures high sampling accuracy while reducing the storage requirement. The proposed method, SampleHST-X, extends our recent work SampleHST. It operates based on a budget which limits the percentage of traces to be sampled while adjusting the storage quota of normal and anomalous traces depending on the size of this budget. The sampling process relies on accurately defining clusters of normal and anomalous traces by leveraging the distribution of mass scores, which characterize the probability of observing different traces, obtained from a forest of Half Space Trees (HST). In our experiments, using traces from a cloud data center, SampleHST yields 2.3 × to 9.5 × better sampling performance. SampleHST-X further extends the SampleHST approach by incorporating a novel class of Half Space Trees, namely Approximate HST, that uses approximate counters to update the mass scores. These counters significantly reduces the space requirement for HST while the sampling performance remains similar. In addition to this extension, SampleHST-X includes a Family of Graph Spectral Distances (FGSD) based trace characterization component, which, in addition to point anomalies, enables it to sample traces with collective anomalies. For such traces, we observe that the SampleHST-X approach can yield 1.2 × to 19 × better sampling performance. [ABSTRACT FROM AUTHOR]

Virtual network embedding (VNE) is the process of allocating resources in a substrate (i.e. physical) network to support virtual networks optimally. The VNE problem is an NP-hard problem that has been studied for more than a decade in the continuous seek to maximize the revenue of physical infrastructures with more efficient VNE solutions. Metaheuristics have been widely used in online VNE as they incorporate mechanisms to avoid local optimum solutions, explore larger search spaces, and keep acceptable execution times. All metaheuristic optimization algorithms require initialization for which the vast majority of online VNE solutions implement random initialization. This paper proposes three novel initialization functions namely, Initialization Based on Node Selection (IFNS), Initialization Function Based on Community Detection (IFCD), and Initialization Function Based on Previous Solutions (IFPS), intending to enhance the performance of the online VNE process. Through simulation, our initialization functions have been proven to enhance the acceptance rate, revenue, and revenue-to-cost metrics of the VNE process. The enhancements achieved by our initialization functions are statistically significant and their implementation does not add computational overhead to the classic VNE approaches. [ABSTRACT FROM AUTHOR]

What happens at the end of a clinical trial for an investigational neural implant? It may be surprising to learn how difficult it is to answer this question. While new trials are initiated with increasing regularity, relatively little consensus exists on how best to conduct them, and even less on how to ethically end them. The landscape of recent neural implant trials demonstrates wide variability of what happens to research participants after an neural implant trial ends. Some former research participants continue to receive support for their devices (e.g., battery and component replacements, software updates, etc.). Others, when safe, have their neural implants removed through surgical explantation. Still others continue to live with a deactivated neural implant embedded in their body. In the United States, there are no uniform requirements to provide services, of any kind, after an neural implant study ends, and other nations are similarly facing this challenge. The existence of a post-trial gap in an expanding neural implant research ecosystem invites obvious questions: What is owed to neural implant research participants post-trial, and why has providing it been so difficult to accomplish in practice? To take a step forward on this difficult issue, we assembled one group of stakeholders – researchers funded for neuroethics grants by the National Institutes of Health – to explore possible starting points on one topic: ethical guidance for post-trial care of research participants in neural implant trials. Based on shared concerns discussed in the expert workshop the current paper is a call to action. It reports the key areas of convergence from the meeting and highlights important next steps towards developing much needed guidance. [ABSTRACT FROM AUTHOR]

Most philosophers think that phenomenal consciousness underlies, or at any rate makes a large contribution, to moral considerability. This paper argues that many such accounts invoke question-begging arguments. Moreover, they're unable to explain apparent differences in moral status across and within different species. In the light of these problems, I argue that we ought to take very seriously a view according to which moral considerability is grounded in functional properties. Phenomenal consciousness may be sufficient for having a moral value, but it may not be necessary, and it may contribute relatively little to our overall considerability. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new F-normed space, namely Orlicz–Lorentz spaces equipped with the Mazur–Orlicz F-norm. Some basic properties in Orlicz–Lorentz spaces equipped with the Mazur–Orlicz F-norm are given. We find a tool to study the geometry property of Orlicz–Lorentz function spaces, the necessary and sufficient conditions for strict monotonicity, lower local uniform monotonicity, upper local uniform monotonicity in Orlicz–Lorentz spaces endowed with the Mazur–Orlicz F-norm are obtained without any assumptions. The tool also can simplify the proof of the corresponding results of Orlicz–Lorentz spaces equipped with the Luxemburg norm without condition (+). [ABSTRACT FROM AUTHOR]

We introduce topological definitions of strong chain transitivity and we prove that topological average shadowing and topological pseudo-orbital specification each imply topological strong chain transitivity. Furthermore, we introduce the notion of the topological Lipschitz property and we show that it is a sufficient condition for the coincidence of the strong chain recurrent set of a dynamical system and its iterations. [ABSTRACT FROM AUTHOR]

In this work, we prove some new results concerning the class of Eberlein weakly almost periodic functions in Stepanov’s sense. We prove that if the forcing term of a partial functional differential equation with infinite delay is Eberlein-weakly almost periodic in Stepanov’s sense, then the solution is even Eberlein-weakly almost periodic. This shows that a less regular almost periodic behavior in the forcing term yields a more regular almost periodic behavior in the solution. The theoretical results are illustrated in the Lotka–Volterra model. [ABSTRACT FROM AUTHOR]

This research paper has been dedicated to the investigation of Coupled System of Fractional Stochastic Differential Equations (CSFSDEs), which is an extension of Fractional Stochastic Differential equations, an emerging field that is in a developmental stage and demands the focused attention of experts. Within this context, the research article is mainly focused on determining solutions for CSFSDEs featuring variable order derivatives. This article will lay the foundation for establishing the necessary conditions for the existence and uniqueness of the considered CSFSDEs. To achieve this, we have employed Picard’s iteration techniques, which have proven effective in this domain. Furthermore, we will set the groundwork for defining conditions of Ulam’s type stabilities specific to the proposed model. To conclude this work, we have presented an illustrative example that serves to explain the primary findings of our research. [ABSTRACT FROM AUTHOR]

This paper focuses on using piecewise derivatives to simulate the dynamic behavior and investigate the crossover effect within the coupled fractional system with delays by dividing the study interval into two subintervals. We establish and prove significant lemmas concerning piecewise derivatives. Furthermore, we extend and develop the necessary conditions for the existence and uniqueness of solutions, while also investigating the Hyers–Ulam stability results of the proposed system. The results are derived using the Banach contraction principle and the Leary–Schauder alternative fixed-point theorem. Additionally, we employ a numerical method based on Newton’s interpolation polynomials to compute approximate solutions for the considered system. Finally, we provide an illustrative example demonstrating our theoretical conclusions’ practical application. [ABSTRACT FROM AUTHOR]

In this paper, we consider the delay dynamic equation Based on Lyapunov’s method, we study global attractivity of the trivial solution of ( ∗ ). Our new result generalizes some well-known results in the theory of difference and differential equations to dynamic equations of the form ( ∗ ). We also present some examples on nonstandard time scales to illustrate the importance of the new result. [ABSTRACT FROM AUTHOR]

The symmetrical properties of dynamic and/or differential equations are kind of oscillation properties that allow us to conclude the character of solutions for dynamic equations. In this paper, we obtain some symmetrical properties of solutions to an even-order nonlinear dynamic equations with superlinear and sublinear neutral terms on time scales. Our approach is based on linearizing the considered equation in the sense that we would deduce the properties of the considered equation from that of the linear form and provide new oscillation results via comparing with first order as well as n th order non-neutral delay dynamic inequalities. The new obtained results outfit a general podium that enables to analyse the oscillatory behaviour for many types of even-order nonlinear dynamic equations. An example is provided to demonstrate the validity of the theoretical outcomes. [ABSTRACT FROM AUTHOR]

We study the dynamics of a spatial four-body problem where the bodies maintain a rhombus-shape configuration at all times: two of the bodies of equal mass move in the horizontal plane symmetrically with respect to the origin while another pair of bodies of equal mass move symmetrically opposed along the vertical axis. The bodies interact via the Jacobi potential, an attractive binary potential of the form - 1 / x 2 , where x is the distance between the particles. We use appropriate transformations to blow up total collision into a manifold pasted onto the phase space for all levels of energy. We find that the topology of the total collision manifold changes as the angular momentum varies, and also that the dynamics on the collision manifold changes as a function of angular momentum and the mass ratio. We also give a qualitative description of the global flow of the problem for negative energy. This description utilizes knowledge concerning the flow on and near the collision manifold, the presence of an additional integral of motion, and takes advantage of the time-reversing symmetry inherent in the system of equations. [ABSTRACT FROM AUTHOR]

This article explores the topological entropy and topological sequence entropy of hom tree-shifts on unexpandable trees. Regarding topological entropy, we establish that the entropy, denoted as h (T X) on an unexpandable tree, equals the entropy h(X) of the base shift X when X is a subshift satisfying the almost specification property. Additionally, we derive some fundamental properties such as entropy approximation and the denseness of entropy for subsystems. Concerning topological sequence entropy, we show that the set of sequence entropies of hom tree-shifts with a base shift is generated by an irreducible matrix A, forming a subset of log N . Precisely, these entropies correspond to the logarithms of the largest cardinalities of the periodic classes of A. [ABSTRACT FROM AUTHOR]

In this paper, we characterize and study dynamical properties of cubic vector fields on the sphere S 2 = { (x , y , z) ∈ R 3 | x 2 + y 2 + z 2 = 1 } . We start by classifying all degree three polynomial vector fields on S 2 and determine which of them form Kolmogorov systems. Then, we show that there exist completely integrable cubic vector fields on S 2 and also study the maximum number of various types of invariant great circles for homogeneous cubic vector fields on S 2 . We find a tight bound in each case. Further, we also discuss phase portraits of certain cubic Kolmogorov vector fields on S 2 . [ABSTRACT FROM AUTHOR]

The primary focus of this paper is threefold: first, to investigate the existence of mild solutions; second, to analyze the topological and geometrical structure of the solution sets; and third, to determine the continuous dependence of the solution for second-order semilinear integro-differential inclusion. In this study, we employ Bohnenblust–Karlin’s fixed point theorem in conjunction with the theory of resolvent operators, as presented by Grimmer. An illustrative example is employed to showcase the achieved outcomes. [ABSTRACT FROM AUTHOR]

A compact space Y is called homeo-product-minimal if given any minimal system (X, T), it admits a homeomorphism S : Y → Y , such that the product system (X × Y , T × S) is minimal. We show that a large class of cofrontiers is homeo-product-minimal. This class contains Bing’s pseudo-circle, answering a question of Dirbák et al. (Trans Am Math Soc 375:6453–6506, 2022). [ABSTRACT FROM AUTHOR]

For family x ′ = (a 0 + a 1 cos t + a 2 sin t) | x | + b 0 + b 1 cos t + b 2 sin t , we solve three basic problems related with its dynamics. First, we characterize when it has a center (Poincaré center focus problem). Second, we show that each equation has a finite number of limit cycles (finiteness problem), and finally we give a uniform upper bound for the number of limit cycles (Hilbert problem 16). [ABSTRACT FROM AUTHOR]

Due to the importance of some physical systems, in this paper, we aim to investigate a generalized impulsive ρ -Caputo differential equation equipped with a p-Laplacian operator. In fact, our problem is a generalization of fractional differential equations equipped with the integral boundary conditions, impulsive forms and p-Laplacian operators under the Nemytskii operators. In this direction, we prove some theorems on the existence property along with the uniqueness of solutions under the Nemytskii operator. More precisely, we use the Schauder’s and Schaefer’s fixed point theorems, along with the Banach contraction principle. In the sequel, two examples are provided to show the validity of the obtained results in practical. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the non-degeneracy of a Liénard equation with a linear difference operator by means of the generalized Wirtinger inequality. In addition, by the non-degenerate results and an extension of Mawhin’s continuation theorem, the existence and uniqueness of periodic solutions for a Liénard equation with a linear difference operator are proved. [ABSTRACT FROM AUTHOR]

The main concern of this paper is to study the boundary stabilization problem of the disk-beam system. To do so, we assume that the boundary control is either of a force type control or a moment type control and is subject to the presence of a constant time-delay. First, we show that in both cases, the system is well-posed in an appropriate functional space. Next, the exponential stability property is established. Finally, the obtained outcomes are ascertained through numerical simulations. [ABSTRACT FROM AUTHOR]

This paper is concerned with the existence and nonexistence of traveling wave solutions for a discrete diffusive mosquito-borne epidemic model with general incidence rate and constant recruitment. It is observed that whether the traveling wave solutions exist or not depend on the so-called basic reproduction ratio R 0 of the corresponding kinetic system and the critical wave speed c ∗ . More precisely, when R 0 > 1 and c ≥ c ∗ , the system admits a nontrivial traveling wave solution by constructing an invariant cone in a bounded domain with initial functions being defined on, and employing the method of upper and lower solution, Schauder’s fixed point theorem and a limiting approach. Moreover, the asymptotic behavior of traveling wave solutions at positive infinity is obtained by constructing a suitable Lyapunov functional. When 0 < c < c ∗ or R 0 ≤ 1 , the system has no nontrivial traveling wave solution by using a contradictory approach and two-sided Laplace transforms. [ABSTRACT FROM AUTHOR]

In this paper, the reaction-diffusion equation in space 1D with negative power nonlinearity is presented as an example the stationary solutions of which are characterized by the bifurcation at infinity. The all dynamics, including to infinity, of the ordinary differential equation concerning with the stationary problem of the reaction-diffusion equation are obtained by the Poincaré-type compactification. The results of this paper give relationship between the bifurcation at infinity of the stationary solutions for the reaction-diffusion equation and the connecting orbits of the ordinary differential equations, especially, that connect the equilibria at infinity. Then, for the bifurcation at infinity, the question of what kind of bifurcation structures are present in this ODEs not only before the solutions go to infinity but also after they go to infinity is answered by a precise analysis. Moreover, this result also answers the question of what kind of solutions appear (or disappear) associated with the bifurcation. In addition, from the bifurcation theory view point, to clarify the bifurcation structures of the equilibria in the ordinary differential equation, the dynamics of it through the one-point compactification called the Bendixson compactification is also shown. [ABSTRACT FROM AUTHOR]

This paper studies the bifurcations of limit cycles from the system x ˙ = y , y ˙ = - x with the switching curve y = x 3 / 3 - x under the perturbations of arbitrary polynomials of x and y with degree n. We obtain the lower bound and upper bound of the maximum number of limit cycles bifurcating from h ∈ (0 , 3 / 2) if the first order Melnikov function is not identically 0. When the degree of perturbing terms is low, we obtain a precise result on the number of zeros of the first order Melnikov function. We also give an example to illustrate our result. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence of solutions for a new coupled system of fractional differential equations that involves ψ -Caputo fractional derivatives equipped with coupled integro multistrip–multipoint boundary conditions. The uniqueness result for the given problem is obtained by utilizing the Banach contraction principle, while the existence results are established with the help of Schaefer’s fixed point theorem under specific assumptions. We also discuss the Ulam–Hyers stability for the problem at hand. Numerical examples are constructed for the illustration of the abstract results. [ABSTRACT FROM AUTHOR]

In this manuscript, we employ integrated resolvent operator to derive the variation of constants formula for the solution of the impulsive integro-differential system, in nonlocal domain with finite delay function. Using the Banach fixed point theorem and integrated resolvent operator, we explore the existence of mild solution of the aforementioned system. Additionally, we establish the Hyers–Ulam stability of the system. Finally, the main result is illustrated with the help of an example. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a system of two differential equations of fractional order for the fear effect in prey-predator interactions, in which the density of predators controls the mortality pace of the prey population. The non-integer order differential equation is interpreted in terms of the Caputo derivative, and the development of the non-integer order scheme is described in terms of the influence of memory on population increase. The primary goal of existing research is to explore how the changing aspects of the current scheme are impacted by various types of parameters, including time delay, fear effect, and fractional order. The solutions’ positivity, existence-uniqueness, and boundedness are established with precise mathematical conclusions. The requirements necessary for the local asymptotic stability of different equilibrium points and the global stability of coexistence equilibrium are established. Hopf bifurcation occurs in the system at various delay times. The model’s fractional-order derivatives enhance the model behaviours and provide stability findings for the solutions. We have observed that fractional order plays an important role in population dynamics. Also, Hopf bifurcation for the proposed system have been observed for certain values of order of derivatives. Thus, the stability conditions of the equilibrium points may be changed by changing the order of the derivatives without changing other parametric values. Finally, a numerical simulation is run to verify our conclusions. [ABSTRACT FROM AUTHOR]

In this paper we use topological degree to study the solvability for a first-order impulsive integral boundary value problem on time scales. We first construct a linear operator, and then obtain the existence of nontrivial solutions under some conditions concerning the spectral radius of this linear operator. Our method improves and generalizes some results in the literature. [ABSTRACT FROM AUTHOR]

This article studies the spreading properties in an asymptotic autonomous non-cooperative system that can be regarded as an epidemic model or a predator–prey system. In particular, due to the effect of nonlocal delay, it is possible that the system does not satisfy the comparison principle for mixed quasimonotone systems. When the initial value has proper decaying behavior, we obtain a constant spreading speed of solutions. Finally, the main result is applied to a stage-structured susceptible-latent-infected-recovered-matured epidemic model to show its spreading feature by presenting the spreading speed about the susceptible and the infected. [ABSTRACT FROM AUTHOR]

In this paper, the dynamical system bifurcation theory approach are employed to investigate the phase diagrams of the magnet-optic wave guides in Kuralay. With the use of the complete discrimination system, we obtain some new traveling wave solutions, including kink solitary, convex-periodic, Jacobian elliptic function solutions, dark-soliton and implicit analytical solutions. More details about the physical dynamical representation of the presented solutions are illustrated with profile pictures. We use Mathematica and Maple to plot three-dimensional diagrams, contour plots and two-dimensional diagrams to obtain complete configurations. This paper show that the fully discriminative system approach is simple and efficient method to reach the various type of the soliton solutions, provide a more powerful mathematical tool to solve many other nonlinear partial differential equations with the help of symbolic computation and computers. [ABSTRACT FROM AUTHOR]

In this work, the Hopf and Bogdanov–Takens bifurcations of a delayed Bazykin predator–prey model with predator intraspecific interactions and ratio-dependent functional response are studied. Sufficient conditions for the existence of Hopf bifurcation are established. In the Bogdanov–Takens bifurcation, the dynamics near the nonhyperbolic equilibrium can be reduced to the study of the dynamics of the corresponding normal form restricted to the associated two-dimensional center manifold. Some numerical simulations, such as the distribution of eigenvalues, the bifurcation diagrams of Hopf and Bogdanov–Takens bifurcations and phase portraits, are given to illustrate the theoretical criteria. The theoretical and numerical simulation results illustrate that there is supercritical Hopf bifurcation and subcritical Bogdanov–Takens bifurcation in this model. We show that, the dynamics of prey and predator are very sensitive to parameters and delay perturbations which can play a great role in controlling and regulating the number of biological populations. [ABSTRACT FROM AUTHOR]

In this paper, the Hyers–Ulam stability of linear quaternion-valued differential equations with two-sided constant coefficients are studied. Utilizing the complex representation of quaternions, we transform a nth-order quaternion-valued differential equation with two-sided constant coefficients into four nth-order complex differential equations. Then we derive the Hyers–Ulam stability by means of the reduced order method and the relationship between the quaternion-valued differential equations and the system of complex-valued differential equations. Two examples are presented to illustrate the validity of the theoretical results. [ABSTRACT FROM AUTHOR]

This article is concerned with existence of mild solutions for jerk-type fractional differential equations in the sense of Hadamard and Caputo fractional derivatives with separated boundary conditions. For the uniqueness of mild solutions in both cases, Banach contraction principle are followed. Moreover, at least one mild solution of jerk-type Caputo–Hadamard and Hadamard–Caputo fractional differential equations can be analyzed using Krasnoselskii’s and Leray–Schauder fixed point theorems. Hyers–Ulam stability and its generalized case for both type of mentioned jerk-type problems can be find out with the help of some conditions and definitions. For the illustration of main results, an example is provided. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new class of metric spaces called the Menger probabilistic bipolar metric space and define some other notions related to this space. Moreover, we prove some uniqueness fixed point theorems for two cases including the covariant and contravariant maps. These fixed point theorems are new versions of the Banach contraction principle, Kannan theorem, and Reich-type theorem in the context of the Menger probabilistic bipolar metric space. Throughout the paper, we provide some examples to understand the definitions in the better manner. Finally, two applications are given for proving the uniqueness result in the form of an integral equation and a fractional boundary value problem. [ABSTRACT FROM AUTHOR]

This inquire about ponder is committed to investigating a few properties in connection to behaviors of solutions to an extended fractional structure of the standard jerk equation. Here, we define the scheme of the general fractional jerk problem using the generalized G operators. The existence result of such a new model is derived and analyzed based on some inequalities and fixed point tools. Furthermore, analysis of its Ulam–Hyers–Rassias type stability is performed and finally, we give numerical simulations for the existing parameters of the mentioned fractional G-jerk system in the Katugampola, Caputo–Hadamard and Caputo settings under different arbitrary orders. [ABSTRACT FROM AUTHOR]

In this paper, we prove that if the geodesic flow of a complete manifold without conjugate points with sectional curvatures bounded below by - c 2 is of Anosov type, then the constant of contraction of the flow is ≥ e - c . Moreover, if M has a finite volume, the equality holds if and only if the sectional curvature is constant. We also apply this result to get a certain rigidity for bi-Lipschitz, and consequently, for C 1 -conjugacy between two geodesic flows. [ABSTRACT FROM AUTHOR]

In this paper, we study the bifurcation of limit cycles, centers, and isochronous centers for a class of three-dimensional Kukles systems of degree 3. Through calculating the singular point quantities, we obtain a necessary condition for the origin to be a center, then the Darboux integrability theory is used to prove that the necessary condition is also sufficient. Then, we demonstrate that the origin is an isochronous center under the obtained center condition. Finally, we determine that the highest order of the origin to be a weak focus is ten, while the maximum number of small-amplitude limit cycles bifurcating from this weak focus is nine. Moreover, we show that this maximum number of 9 can be realized. It is worthwhile to say that this number is also a new lower bound on the number of limit cycles bifurcating from the single weak focus for three-dimensional cubic polynomial systems. [ABSTRACT FROM AUTHOR]

The paper deals with existence, localization and multiplicity of radial positive solutions in the annulus or the ball, for the Neumann problem involving a general ϕ -Laplace operator. Our results apply in particular to the classical Laplacian and to the mean curvature operators in the Euclidean and Minkowski spaces. Numerical experiments with the MATLAB object-oriented package Chebfun are performed to obtain numerical solutions for some concrete equations. [ABSTRACT FROM AUTHOR]