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In this paper we first extend from normed spaces to locally convex spaces some characterizations of denting points in convex sets. On the other hand, we also prove that in an infrabarreled locally convex space a point in a convex set is denting if and only if it is a point of continuity and an extreme point of the closure of such a convex set under the strong topology in the second dual. The version for normed spaces of the former equivalence is new and contains, as particular cases, some known and remarkable results. We also extend from normed spaces to locally convex spaces some known characterizations of the weak property (π) of cones. Besides, we provide some new results regarding the angle property of cones and related. We also state that the class of cones in normed spaces having a pointed completion is the largest one for which the vertex is a denting point if and only if it is a point of continuity. Finally we analyse and answer several problems in the literature concerning geometric properties of cones which are related with density problems into vector optimization. The authors Fernando García-Castaño and M. A. Melguizo Padial have been supported by MINECO and FEDER (MTM2017-86182-P).

In this paper we have given conditions on exponential polynomials $$P_{n}(s)$$ of Dirichlet type to be attained the equality between each of two pairs of bounds, called essential bounds, $$a_{P_{n}(s)}$$ , $$\rho _{N}$$ and $$b_{P_{n}(s)}$$ , $$\rho _{0}$$ associated with $$P_{n}(s)$$ . The reciprocal question has been also treated. The bounds $$a_{P_{n}(s)}$$ , $$b_{P_{n}(s)}$$ are defined as the end-points of the minimal closed and bounded real interval $$I= [ a_{P_{n}(s)},b_{P_{n}(s)} ] $$ such that all the zeros of $$P_{n}(s)$$ are contained in the strip $$I\times {\mathbb {R}}$$ of the complex plane $${\mathbb {C}}$$ . The bounds $$\rho _{N}$$ , $$\rho _{0}$$ are defined as the unique real solutions of Henry equations of $$P_{n}(s)$$ . Some applications to the partial sums of the Riemann zeta function have been also showed.

Based on Bohr’s equivalence relation for general Dirichlet series, in this paper we connect the families of equivalent exponential polynomials with a geometrical point of view related to lines in crystal-like structures. In particular we characterize this equivalence relation, and give an alternative proof of Bochner’s property referring to these functions, through this new geometrical perspective. The first author’s research was partially supported by PGC2018-097960-B-C22 (MCIU/AEI/ERDF, UE).

This paper presents a full generalization of Bohr’s equivalence theorem for the case of almost periodic functions, which improves a recent result that was uniquely formulated in the case of existence of an integral basis for the set of exponents of the associated Dirichlet series. J.M. Sepulcre research was partially supported by MICIU of Spain under Project Number PGC2018-097960-B-C22.

The performance of turbo codes at the error floor region is largely determined by the effective free distance, which corresponds to the minimum Hamming weight among all codeword sequences generated by input sequences of weight two. In this paper, we study turbo codes of dimension one obtained from the concatenation of two equal codes and present an upper bound on the effective free distance of a turbo code with these parameters defined over any finite field. We do that making use of the so-called (A, B, C, D) state-space representations of convolutional codes and restrict to the case where A is invertible. A particular construction, from a linear systems point of view, of a recursive systematic convolutional code of rate 1/n so that the effective free distance of the corresponding turbo code attains this upper bound is also presented. D. Napp was partially supported by the the Universitat d’Alacant (Grant No. VIGROB-287) and Generalitat Valenciana (Grant No. AICO/2017/128). V. Herranz and C. Perea were supported by the Ministerio de Economa, Industria y Competitividad within project TIN2016-80565-R.

In this paper it is shown that the complex functional equation $$F(z)+F(2z)=0$$ , $$z\in \varOmega :=\mathbb {C}\setminus ( -\infty ,0 ] $$ , is stable in the strong sense of Ulam. It means that given an analytic and bounded function f(z) on $$\varOmega $$ satisfying $$\left| f(z)+f(2z)\right| 0$$ , there exists an analytic solution F(z) of the above functional equation such that $$\left| f(z)-F(z)\right| 0$$ , not necessarily bounded on $$\varOmega $$ .

We shall consider a result of Fel’dman, where a sharp Baker-type lower bound is obtained for linear forms in the values of some E-functions. Fel’dman’s proof is based on an explicit construction of Padé approximations of the first kind for these functions. In the present paper, we introduce Padé approximations of the second kind for the same functions and use these to obtain a slightly improved version of Fel’dman’s result.

In this paper we study extensions between Cohen–Macaulay modules for algebras arising in the categorifications of Grassmannian cluster algebras. We prove that rank 1 modules are periodic, and we give explicit formulas for the computation of the period based solely on the rim of the rank 1 module in question. We determine \(\mathrm{Ext}^i(L_I, L_J)\) for arbitrary rank 1 modules \(L_I\) and \(L_J\). An explicit combinatorial algorithm is given for the computation of \(\mathrm{Ext}^i(L_I, L_J)\) when i is odd, and when i even, we show that \(\mathrm{Ext}^i(L_I, L_J)\) is cyclic over the centre, and we give an explicit formula for its computation. At the end of the paper we give a vanishing condition of \(\mathrm{Ext}^i(L_I, L_J)\) for any \(i>0\).

This paper is concerned with the distributions of zeros of solutions to first order delay dynamic equations on time scales. The results are obtained using iterative sequences.

Calvert and Gupta’s results concerning the perturbations on the ranges of m-accretive mappings have been employed widely in the discussion of the existence of solutions of nonlinear elliptic differential equation with Neumann boundary. In this paper, we shall focus our attention on certain hyperbolic differential equation with mixed boundaries. By defining some suitable nonlinear mappings, we shall demonstrate that Calvert and Gupta’s results can be applied to hyperbolic equations, in addition to its wide usage in elliptic equations. Due to the differences between hyperbolic and elliptic equations, some new techniques have been developed in this paper, which can be regarded as the complement and extension of the previous work.

In this paper, we are interested in the eigenvalues and its algebraic multiplicities of a fractional linear boundary value problem with mixed set of Neumann and Dirichlet boundary conditions. The research results are then applied to consider the sign-changing solutions of the corresponding nonlinear problem by fixed point index and Leray-Schauder degree. To date, no paper has appeared in the literature which discusses sign-changing solutions of fractional boundary value problems. This paper attempts to fill this gap in the literature.

In this paper, we are concerned with nonlinear elliptic equations of the $p(x)$ -Laplace type operators $$\textstyle\begin{cases} -\operatorname{div}(a(x,\nabla u))+\vert u\vert ^{p(x)-2}u=\lambda f(x,u) &\mbox{in } \Omega, \\ a(x,\nabla u)\frac{\partial u}{\partial n} = \lambda\theta g(x,u) &\mbox{on } \partial\Omega, \end{cases} $$ which are subject to nonlinear Neumann boundary conditions. Here the function $a(x,v)$ is of type $\vert v\vert ^{p(x)-2}v$ with a continuous function $p: \overline{\Omega} \to(1,\infty)$ and the functions $f, g$ satisfy a Caratheodory condition. The main purpose of this paper is to establish the existence of at least three weak solutions of the above problem by applying an abstract three critical points theorem which is inspired by the work of Ricceri (Nonlinear Anal. 74:7446-7454, 2011) Furthermore, we determine two intervals of λ’s precisely such that the first is where the given problem admits only the trivial solution, and the second is where the given problem has at least two nontrivial solutions as considering the positive principal eigenvalue for the $p(x)$ -Laplacian Neumann problems and an estimate of the Sobolev trace embedding’s constant.

Sufficient oscillation conditions involving $\limsup $ and $\liminf $ for first-order differential equations with several non-monotone deviating arguments and nonnegative coefficients are obtained. The results are based on the iterative application of the Gr\"{o}nwall inequality. Examples illustrating the significance of the results are also given., Comment: The paper was originally open access. The purpose of this publication is correct the original mistake that occurred in Theorems 6 and 10 (in particular, the variables under the integral in (2.11) and (2.27)) in the published paper

In this paper, we introduce a new class of multivalued contractions with respect to b-generalized pseudodistances and prove a best proximity point theorem for this class of non-self mappings. In this way, we improve and extend several existing results in the literature. Examples are given to support our main results. This work is a continuation of studies on the use of a new type of distances in the fixed point theory. The pioneering effort in direction of defining distance is inter alia paper of O. Kada, T. Suzuki and W. Takahashi.

The paper addresses the asymptotic properties of Camassa-Holm equation on the half-line. That is, using the method of asymptotic density, under the assumption that it is unique, the paper proves that the positive momentum density of the Camassa-Holm equation is a combination of Dirac measures supported on the positive axis. This means that as time goes to infinity, the momentum density concentrates in small intervals moving right with different constant speeds.

The paper studies the equation $$ {u_{t}}= \operatorname{div} \bigl(a(x)\vert {\nabla u} \vert ^{p(x) - 2}\nabla u \bigr), $$ with the boundary degeneracy coming from $a(x)\vert_{x\in \partial \Omega }=0$ . The paper introduces a new kind of weak solutions of the equation. One can study the stability of the new kind of weak solutions without any boundary value condition.

We complete the stability results of the paper Bourles et al. (SIAM J Control Optim 53:2725---2761, 2015), and for this purpose use the linear time-varying (LTV) discrete-time behaviors and the exponential stability (e.s.) of this paper. In the main theorem we characterize the e.s. of an autonomous LTV system by standard spectral properties of a complex matrix connected with the system. We extend the theory of discrete-time LTV behaviors, developed in the quoted publication, from the coefficient field of rational functions to that of locally convergent Laurent series or even of Puiseux series. The stability test can and has to be applied in connection with the construction of stabilizing compensators.

This paper is concerned with a thermoelastic suspension bridge equations with memory effects. For the suspension bridge equations without memory, there are many classical results. However, the suspension bridge equations with both viscoelastic and thermal memories were not studied before. The object of the present paper is to provide a result on the global attractor to a thermoelastic suspension bridge equation with past history.

This paper is devoted to the study of the sign of the Green’s function related to a general linear nth-order operator, depending on a real parameter, $T_{n}[M]$ , coupled with the $(k,n-k)$ boundary value conditions. If the operator $T_{n}[\bar{M}]$ is disconjugate for a given M, we describe the interval of values on the real parameter M for which the Green’s function has constant sign. One of the extremes of the interval is given by the first eigenvalue of the operator $T_{n}[\bar{M}]$ satisfying $(k,n-k)$ conditions. The other extreme is related to the minimum (maximum) of the first eigenvalues of $(k-1,n-k+1)$ and $(k+1,n-k-1)$ problems. Moreover, if $n-k$ is even (odd) the Green’s function cannot be nonpositive (nonnegative). To illustrate the applicability of the obtained results, we calculate the parameter intervals of constant sign Green’s functions for particular operators. Our method avoids the necessity of calculating the expression of the Green’s function. We finalize the paper by presenting a particular equation in which it is shown that the disconjugation hypothesis on operator $T_{n}[\bar{M}]$ for a given M cannot be eliminated.

In this paper, we propose a novel stabilized mixed finite element method for the approximation of optimal control problems governed by reaction-diffusion equations. Compared with the classical mixed finite element methods, the main contributions of this paper are as follows. First, the novel method uses only one stabilization parameter which is not mesh-dependent, and, the new mixed bilinear formulation is coercive and continuous. Second, the novel method is easy to be implemented on a computer using the standard Lagrange finite element. Third, the solutions of the novel method to the optimal control problems require low regularities. Fourth, the Ladyzhenkaya-Babuska-Brezzi (LBB) or the inf-sup condition for the mixed element spaces is unnecessary. Based on the novel method, we derive both continuous and discrete optimality systems for the corresponding constrained optimal control problems, and then a priori error analysis in a weighted norm is discussed. Finally, numerical experiments are given to confirm the efficiency and reliability of the novel stabilized method.

This paper deals with the vectorial Schrodinger operators with δ-interactions generated by $L_{X,A,Q}:=-\frac{d^{2}}{dx^{2}} +Q(x)+\sum_{k=1}^{\infty}A_{k}\delta(x-x_{k})$ , $x\in[ 0,+\infty)$ . First, we obtain an embedding inequality. Then using standard form methods, we prove that the operator $\mathbf{H}_{X,A,Q}$ given in this paper is self-adjoint. Finally, a sufficient condition and a necessary condition are given for the spectrum of the operator $\mathbf {H}_{X,A,Q}$ to be discrete. By giving additional restrictions on the symmetric potential matrix $Q(x)$ and $A_{k}$ , we also give a necessary and sufficient condition for a special case. The conditions are analogous to Molchanov’s discreteness criteria.

This paper is devoted to the analysis of an approximately controllable system perturbed by a certain Lipschitz-continuous nonlinearity. By assuming that the intensity of the nonlinear influence is ‘weak’, we prove that the perturbed system is still approximately controllable. Using this perturbation theory, we prove that a third-order semilinear dispersion equation on a finite interval is approximately controllable whenever the nonlinearity effect is ‘weak’. The result in the paper is a complement of the results obtained in (Zhou in SIAM J. Control Optim. 21:551-565, 1983), in which the control operator (resp. infinitesimal generator of the dynamics) of the unperturbed linear system is assumed to be bounded (resp. generated a differentiable semigroup on the state space).

In this paper, we consider the existence of periodic solutions for a class of nonautonomous second-order discrete Hamiltonian systems in case the sum on the time variable of potential is periodic. The tools used in our paper are the direct variational minimizing method and Rabinowitz’s saddle point theorem.

We study the third-order linear difference equation with quasi-differences and its adjoint equation. The main results of the paper describe relationships between the oscillatory and nonoscillatory solutions of both equations.

In this paper, we investigate the singular Sturm-Liouville problem , , , where is a nonnegative parameter, , , and . We discuss the existence of multiple positive solutions and show that for certain values of , there also exist solutions that vanish on a subinterval , the so-called dead core solutions. The theoretical findings are illustrated by computational experiments for and for some model problems from the class of singular differential equations discussed in Agarwal et al. (2007). For the numerical simulation, the collocation method implemented in our MATLAB code bvpsuite has been applied.

Given a simple digraph $D$ on $n$ vertices (with $n\ge2$), there is a natural construction of a semigroup $\langle D\rangle$ associated with $D$. For any edge $(a,b)$ of $D$, let $a\to b$ be the idempotent of defect $1$ mapping $a$ to $b$ and fixing all vertices other than $a$; then define $\langle D\rangle$ to be the semigroup $\langle a\to b:(a,b)\in E(D)\rangle$. For $\alpha \in \langle D \rangle$, let $\ell(D,\alpha)$ be the minimal length of a word in $E(D)$ expressing $\alpha$. When $D=K_n$ is the complete undirected graph, Howie and Iwahori, independently, obtained a formula to calculate $\ell(K_n,\alpha)$, for any $\alpha \in \langle K_n \rangle = \text{Sing}_n$; however, no analogous nontrivial results are known when $D \neq K_n$. In this paper, we characterise all simple digraphs $D$ such that either $\ell(D,\alpha)$ is equal to Howie-Iwahori's formula for all $\alpha \in \langle D \rangle$, or $\ell(D,\alpha) = n - \text{fix}(\alpha)$ for all $\alpha \in \langle D \rangle$, or $\ell(D,\alpha) = n - \text{rk}(\alpha)$ for all $\alpha \in \langle D \rangle$. When $D$ is an acyclic digraph and $\alpha \in \langle D \rangle$, we find a tight upper bound for $\ell(D,\alpha)$. Finally, we study the case when $D$ is a strong tournament (which corresponds to a smallest generating set of idempotents of defect $1$ of $\text{Sing}_n$), and we propose some conjectures., Comment: 17 pages

Since first introduced by John von Neumann, the notion of cellular automaton has grown into a key concept in computer science, physics and theoretical biology. In its classical setting, a cellular automaton is a transformation of the set of all configurations of a regular grid such that the image of any particular cell of the grid is determined by a fixed local function that only depends on a fixed finite neighbourhood. In recent years, with the introduction of a generalised definition in terms of transformations of the form $\tau : A^G \to A^G$ (where $G$ is any group and $A$ is any set), the theory of cellular automata has been greatly enriched by its connections with group theory and topology. In this paper, we begin the finite semigroup theoretic study of cellular automata by investigating the rank (i.e. the cardinality of a smallest generating set) of the semigroup $\text{CA}(\mathbb{Z}_n; A)$ consisting of all cellular automata over the cyclic group $\mathbb{Z}_n$ and a finite set $A$. In particular, we determine this rank when $n$ is equal to $p$, $2^k$ or $2^kp$, for any odd prime $p$ and $k \geq 1$, and we give upper and lower bounds for the general case., Comment: 12 pages

In this paper, we study a predator-prey-mutualist system with digestion delay. First, we calculate the threshold value of delay and prove that the positive equilibrium is locally asymptotically stable when the delay is less than the threshold value and the system undergoes a Hopf bifurcation at the positive equilibrium when the delay is equal to the threshold value. Second, by applying the normal form method and center manifold theorem, we investigate the properties of Hopf bifurcation, such as the direction and stability. Finally, some numerical simulations are carried out to verify the main theoretical conclusions.

The purpose of this paper is to discuss boundary value problems of fractional differential equations with $p(t)$ -Laplacian operator. Some new existence, uniqueness, and multiplicity results were acquired by employing some fixed point theorems. Moreover, some examples are supplied to verify our main results.

In this paper we consider the existence of nonoscillatory solutions for a system of higher-order neutral differential equations with distributed coefficients and delays. We use the $Banach$ contraction principle to obtain new sufficient conditions for the existence of nonoscillatory solutions.

In this paper, we study the existence of multiple positive solutions for fourth-order impulsive differential equation with integral boundary conditions and deviating argument. The main tool is based on the Avery and Peterson fixed point theorem. Meanwhile, an example to demonstrate the main results is given.

In this paper we study the long-time dynamics of the semilinear viscoelastic equation $$u_{tt} - \Delta u_{tt} - \Delta u + \int_{0}^{\infty} \mu(s) \Delta u(t-s) \,ds + f(u) = h , $$ defined in a bounded domain of $\mathbb{R}^{3}$ with Dirichlet boundary condition. The functions $f=f(u)$ and $h=h(x)$ represent forcing terms and the kernel function $\mu\ge0$ is assumed to decay exponentially. Then, by exploring only the dissipation given by the memory term, we establish the existence of a global attractor to the corresponding dynamical system.

In this paper, we establish two sufficient conditions for the oscillation of forced fractional difference equations with damping term of the form $$\bigl(1+p(t)\bigr)\Delta\bigl(\Delta^{\alpha}x(t)\bigr)+p(t) \Delta^{\alpha}x(t)+f\bigl(t,x(t)\bigr)=g(t),\quad t\in\mathbb{N}_{0}, $$ with initial condition $\Delta^{\alpha-1}x(t)|_{t=0}=x_{0}$ , where $0

In this paper, the nonlocal boundary value problems for a class of nonlinear functional convection reaction-diffusion equations with the singular reaction function are studied by using the method of upper and lower solutions and monotone iterative technique. Some of sufficient results on the existence and uniqueness of positive global solutions or positive solutions for the boundary value problems are presented, which are a generalization of some recent results in the area.

In this paper, we study the existence of the positive solutions for a class of high-order differential equations with multi-point boundary value conditions involving the Caputo fractional derivative on infinite interval. Moreover, we develop two computable explicit monotone iterative sequences for approximating the two minimal and maximal positive solutions. An example is given to show the applicability of our main results.

This paper concerns the dynamics of a stochastic SIVR epidemic model with imperfect vaccine where, differently from the epidemic model with perfect vaccine, the vaccinated is perturbed by the noise. This difference is the main difficulty to be conquered to give the threshold $R_{0}^{S}$ . Firstly, $R_{0}^{S}>1$ is proved to be sufficient for persistence in mean of the system. Then, three conditions for the disease to die out are given, which improve the previously known results on extinction of the disease. In case that the disease goes extinct, we show that the disease-free equilibrium is almost surely stable by using the nonnegative semimartingale convergence theorem.

We consider the Sturm-Liouville boundary value problem $$\left \{ \textstyle\begin{array}{@{}l} y^{(m)}(t)+ F (t,y(t),y'(t),\ldots,y^{(q)}(t) )=0, \quad t\in[0,1],\\ y^{(k)}(0)=0,\quad 0\leq k\leq m-3, \\ \zeta y^{(m-2)}(0)-\theta y^{(m-1)}(0)=0,\qquad \rho y^{(m-2)}(1)+\delta y^{(m-1)}(1)=0, \end{array}\displaystyle \right . $$ where $m\geq3$ and $1\leq q\leq m-2$ . We note that the nonlinear term F involves derivatives. This makes the problem challenging, and such cases are seldom investigated in the literature. In this paper we develop a new technique to obtain existence criteria for one or multiple positive solutions of the boundary value problem. Several examples with known positive solutions are presented to dwell upon the usefulness of the results obtained.

In this paper, a genetic oscillator model with time delay is discretized by the Euler method. The discrete oscillator model is discussed by using Neimark-Sacker bifurcation theory. The direction and the stability of the Neimark-Sacker bifurcation has been studied using the center manifold theorem and normal form theory. Numerical simulations illustrate the theoretical results.

The purpose of this paper is to study solvability of the higher order nonlinear neutral delay differential equation $$\begin{aligned}& \frac{d^{n}}{dt^{n}}\bigl[x(t)+c(t)x(t-\tau)\bigr]+(-1)^{n+1}f\bigl(t,x \bigl(\sigma _{1}(t)\bigr),x\bigl(\sigma_{2}(t)\bigr), \ldots,x\bigl(\sigma_{k}(t)\bigr)\bigr) \\ & \quad =g(t),\quad t\geq t_{0}, \end{aligned}$$ where n and k are positive integers, $\tau>0 $ , $t_{0}\in{\mathbb{R}}$ , $f\in C ([t_{0},+\infty)\times {\mathbb{R}}^{k},{\mathbb{R}} ) $ , $c,g,\sigma_{i}\in C([t_{0},+\infty),{\mathbb{R}})$ and $\lim_{t\rightarrow +\infty}\sigma_{i}(t)=+ \infty$ for $i \in\{1,2,\ldots,k\}$ . Under suitable conditions, several existence results of uncountably many nonoscillatory solutions and convergence of Mann iterative approximations for the above equation are shown. Three nontrivial examples are given to demonstrate the advantage of our results over the existing ones in the literature.

This paper studies a Caputo type anti-periodic boundary value problem of impulsive fractional q-difference equations involving a q-shifting operator of the form ${}_{a}\Phi_{q}(m) = qm + (1-q)a$ . Concerning the existence of solutions for the given problem, two theorems are proved via Schauder’s fixed point theorem and the Leray-Schauder nonlinear alternative, while the uniqueness of solutions is established by means of Banach’s contraction mapping principle. Finally, we discuss some examples illustrating the main results.

In this paper, we formulate a two-dimensional autonomous predator-prey model with state-dependent impulsive effects and square root response function. The square root response function indicates that the prey population gather together for self-defense purposes. Firstly, we prove that the system has semitrivial periodic solution under some conditions. Furthermore, by the Poincare map and the theory of impulsive differential equations, the existence and stability of positive order-1 or order-2 periodic solution of the system are investigated in detail. The validity of all results is illustrated by numerical simulations.

In this paper, we study the existence, nonexistence, and multiplicity of solutions to the following fractional $p\&q$ -Laplacian equation: 0.1 $$\begin{aligned} &(-\Delta)^{s}_{p}u + a(x)\vert u\vert ^{p-2}u +(-\Delta)^{s}_{q}u + b(x)\vert u\vert ^{q-2}u +\mu(x)\vert u\vert ^{r-2}u \\ &\quad= \lambda h(x)\vert u \vert ^{m-2}u,\quad x\in {\mathbb {R}}^{N}, \end{aligned}$$ where λ is a real parameter, $(-\Delta)_{p}^{s} $ and $(-\Delta )_{q}^{s} $ are the fractional $p\&q$ -Laplacian operators with $0 1$ and $sp< N$ , and the functions $a(x), b(x),\mu(x)$ , and $h(x)$ are nonnegative in ${\mathbb {R}}^{N}$ . Three cases on $p,q,r,m$ are considered: $p< m< r< p_{s}^{*}$ , $\max\{p,r\}< m< p_{s}^{*}$ , and $1< m< q< r< p_{s}^{*}$ . Using variational methods, we prove the existence, nonexistence, and multiplicity of solutions to Eq. (0.1) depending on $\lambda, p,q,r,m$ and the integrability properties of the ratio $h^{r-p}/\mu^{m-p}$ . Our results extend the previous work in Bartolo et al. (J. Math. Anal. Appl. 438:29-41, 2016) and Chaves et al. (Nonlinear Anal. 114:133-141, 2015) to the fractional $p\&q$ -Laplacian equation (0.1).

In this paper, we investigate the growth of meromorphic solutions of the differential equations $$f^{(k)}+A_{k-1}(z)f^{(k-1)}+\cdots+A_{1}(z)f'+A_{0}(z)f=0 $$ and $$f^{(k)}+A_{k-1}(z)f^{(k-1)}+\cdots+A_{1}(z)f'+A_{0}(z)f=F(z), $$ where $A_{0}(z)\not\equiv0, A_{1}(z), \ldots, A_{k-1}(z)$ and $F(z)\not \equiv0$ are meromorphic functions. A precise estimation of the hyper-order of meromorphic solutions of the above equations is given provided that there exists one dominant coefficient, which improves and extends previous results given by Belaidi, Chen, etc.

The fractional Black-Scholes (B-S) equation is an important mathematical model in finance engineering, and the study of its numerical methods has very significant practical applications. This paper constructs a new kind of universal difference method to solve the time-space fractional B-S equation. The universal difference method is analyzed to be stable, convergent, and uniquely solvable. Furthermore, it is proved that with numerical experiments the universal difference method is valid and efficient for solving the time-space fractional B-S equation. At the same time, numerical experiments indicate that the time-space fractional B-S equation is more consistent with the actual financial market.

In this paper, the problem of existence of periodic solution is studied for p-Laplacian Lienard equations with singular at $x=0$ and $x=+\infty$ . By using the topological degree theory, some new results are obtained, and an example is given to illustrate the effectiveness of our results. Our research enriches the contents of second order differential equations with singularity.

This paper is concerned with a discrete Nicholson’s blowflies model, which involves a linear harvesting term. In the case where the coefficients are pseudo-almost periodic functions, we establish the existence and local exponential stability of pseudo-almost periodic solutions for the addressed Nicholson’s blowflies model by using the contraction mapping theorem and a Lyapunov functional. An example is given to illustrate our results.

In this paper, we consider the 2nth-order p-Laplacian differential equation with singularity $$ \bigl(\varphi_{p} \bigl(x(t) \bigr)^{(n)} \bigr)^{(n)}+f \bigl(x(t) \bigr)x'(t)+g \bigl(t,x(t-\sigma) \bigr)=e(t). $$ By applications of coincidence degree theory and some analysis techniques, sufficient conditions for the existence of positive periodic solutions are established.

This paper investigates the existence of nonnegative multiple solutions for nonlinear fractional differential equations of Hadamard type, with nonlocal fractional integral boundary conditions on an unbounded domain by means of Leggett-Williams and Guo-Krasnoselskii’s fixed point theorems. Two examples are discussed for illustration of the main work.

In this paper, we discuss the fractional boundary value problem containing left and right fractional derivative operators and p-Laplacian. By using critical point theory we obtain some results on the existence of weak solutions of such a fractional boundary value problem.

In this paper, we give a reduction formula for a specific q-integral. Our formula is expressed as a three term recurrence relations for basic hypergeometric ${_{3}\phi_{2}}$ series. This is a q-analog of work by Watson and by Bailey of 1953.