A linear $(k+1)$ th-order discrete delayed equation $\Delta x\left(n\right)=-p\left(n\right)x(n-k)$ where $p\left(n\right)$ a positive sequence is considered for $n\to \infty$. This equation is known to have a positive solution if the sequence $p\left(n\right)$ satisfies an inequality. Our aim is to show that, in the case of the opposite inequality for $p\left(n\right)$ , all solutions of the equation considered are oscillating for $n\to \infty$ .

We consider a cooperating two-species Lotka-Volterra model of degenerate parabolic equations. We are interested in the coexistence of the species in a bounded domain. We establish the existence of global generalized solutions of the initial boundary value problem by means of parabolic regularization and also consider the existence of the nontrivial time-periodic solution for this system.

This contribution is devoted to the investigation of the asymptotic behavior of delayed difference equations with an integer delay. We prove that under appropriate conditions there exists at least one solution with its graph staying in a prescribed domain. This is achieved by the application of a more general theorem which deals with systems of first-order difference equations. In the proof of this theorem we show that a good way is to connect two techniques—the so-called retract-type technique and Liapunov-type approach. In the end, we study a special class of delayed discrete equations and we show that there exists a positive and vanishing solution of such equations.

We derive necessary and sufficient conditions for (some or all) positive solutions of the half-linear $q$-difference equation $D_q\left(\mathrm{\Phi }\right(D_{q}y\left(t\right)\left)\right)+p\left(t\right)\mathrm{\Phi }\left(y\right(\text{i}tqt\left)\right)=0$, $t\in \{q^k:k\in {\mathbb{N}}_0\}$ with , $\Phi \left(u\right)={\left|u\right|}^{\alpha -1}\mathrm{sgn}u$ with , to behave like $q$-regularly varying or $q$-rapidly varying or $q$-regularly bounded functions (that is, the functions $y$, for which a special limit behavior of $y\left(\text{i}tqt\right)/y\left(t\right)$ as $t\to \mathrm{\infty }$ is prescribed). A thorough discussion on such an asymptotic behavior of solutions is provided. Related Kneser type criteria are presented.

Singular differential equation $(p\left(t\right)u^{\prime }{)}^{\prime }=p\left(t\right)f\left(u\right)$ is investigated. Here $f$ is Lipschitz continuous on $ℝ$ and has at least two zeros 0 and . The function $p$ is continuous on $[0,\infty )$ and has a positive continuous derivative on $(0,\infty )$ and $p\left(0\right)=0$. An asymptotic formula for oscillatory solutions is derived.

The aim of this paper is to study properties of the third-order delay trinomial differential equation ${\left(\right(1/r\left(t\right)\left)y^{\prime \prime }\right(t\left)\right)}^{\prime }+p\left(t\right)y^{\prime }\left(t\right)+q\left(t\right)y\left(\sigma \right(t\left)\right)=0$, by transforming this equation onto the second-/third-order binomial differential equation. Using suitable comparison theorems, we establish new results on asymptotic behavior of solutions of the studied equations. Obtained criteria improve and generalize earlier ones.

A discrete equation $\Delta y\left(n\right)=\beta \left(n\right)[y(n-j)-y(n-k)]$ with two integer delays $k$ and is considered for $n\to \infty$. We assume $\beta :{\mathbb{Z}}_{n_0-k}^{\infty }\to (0,\infty )$, where ${\mathbb{Z}}_{n_0}^{\infty }=\{n_0,n_0+1,\dots \}$, $n_0\in \mathbb{N}$ and $n\in {\mathbb{Z}}_{n_0}^{\infty }$. Criteria for the existence of strictly monotone and asymptotically convergent solutions for $n\to \infty$ are presented in terms of inequalities for the function $\beta$. Results are sharp in the sense that the criteria are valid even for some functions $\beta$ with a behavior near the so-called critical value, defined by the constant ${(k-j)}^{-1}$. Among others, it is proved that, for the asymptotic convergence of all solutions, the existence of a strictly monotone and asymptotically convergent solution is sufficient.

We investigate the existence of solutions for a class of second-order $q$-difference inclusions with nonseparated boundary conditions. By using suitable fixed-point theorems, we study the cases when the right-hand side of the inclusions has convex as well as nonconvex values.

Conditions are derived of the existence of solutions of nonlinear boundary-value problems for systems of $n$ ordinary differential equations with constant coefficients and single delay (in the linear part) and with a finite number of measurable delays of argument in nonlinearity: $\dot{z}\left(t\right)=Az(t-\tau )+g\left(t\right)+ϵZ(z\left(h_i\right(t),t,ϵ)$, $t\in [a,b]$, assuming that these solutions satisfy the initial and boundary conditions $z\left(s\right):=\psi \left(s\right)$ if $s\notin [a,b]$, $lz(\cdot )=\alpha \in {ℝ}^m$. The use of a delayed matrix exponential and a method of pseudoinverse by Moore-Penrose matrices led to an explicit and analytical form of sufficient conditions for the existence of solutions in a given space and, moreover, to the construction of an iterative process for finding the solutions of such problems in a general case when the number of boundary conditions (defined by a linear vector functional $𝓁$) does not coincide with the number of unknowns in the differential system with a single delay.

Some new integral inequalities are given, and bounds of solutions of the following integro-differential equation are determined: $x\text{'}\left(t\right)-ℱ(t,x(t),{{\displaystyle \int }}_0^{t}k(t,s,x\left(t\right),x\left(s\right))ds)=h\left(t\right)$, $x\left(0\right)=x_0$, where $h:R_{+}\to R$, $k:R_{+}^2\times R^2\to R$, $ℱ:R_{+}\times R^2\to R$ are continuous functions, $R_{+}=[0,\infty )$.

In the first part of this paper sufficient conditions for nonuniqueness of the classical Cauchy problem $\dot{x}=f(t,x)$, $x\left(t_0\right)=x_0$ are given. As the essential tool serves a method which estimates the “distance” between two solutions with an appropriate Lyapunov function and permits to show that under certain conditions the “distance” between two different solutions vanishes at the initial point. In the second part attention is paid to conditions that are obtained by a formal inversion of uniqueness theorems of Kamke-type but cannot guarantee nonuniqueness because they are incompatible.

We study oscillation properties of $2n$-order Sturm-Liouville difference equations. For these equations, we show a conjugacy criterion using the $p$-criticality (the existence of linear dependent recessive solutions at $\mathrm{\infty }$ and $-\mathrm{\infty }$). We also show the equivalent condition of $p$-criticality for one term $2n$-order equations.

This paper investigates some initial value problems in discrete fractional calculus. We introduce a linear difference equation of fractional order along with suitable initial conditions of fractional type and prove the existence and uniqueness of the solution. Then the structure of the solutions space is discussed, and, in a particular case, an explicit form of the general solution involving discrete analogues of Mittag-Leffler functions is presented. All our observations are performed on a special time scale which unifies and generalizes ordinary difference calculus and $q$-difference calculus. Some of our results are new also in these particular discrete settings.

A nonlinear stochastic differential-difference control system with delay of neutral type is considered. Sufficient conditions for the exponential stability are derived by using Lyapunov-Krasovskii functionals of quadratic form with exponential factors. Upper bound estimates for the exponential rate of decay are derived.

Solutions of the equation $\dot{y}\left(t\right)=-f(t,y_t)$ are considered for $t\to \infty$. The existence of two classes of positive solutions which are asymptotically different is proved using the retract method combined with Razumikhin's technique. With the aid of two auxiliary linear equations, which are constructed using upper and lower linear functional estimates of the right-hand side of the equation considered, inequalities for both types of positive solutions are given as well.

The weakly perturbed linear nonhomogeneous impulsive systems in the form $\dot{x}=A\left(t\right)x\hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}\hspace{0.17em}\epsilon A_1\left(t\right)x\hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}\hspace{0.17em}f\left(t\right),\mathrm{\hspace{0.17em}\hspace{0.17em}}t\in \mathbb{R},\mathrm{\hspace{0.17em}\hspace{0.17em}}t\notin \mathcal{T}:=\{{\tau }_i{\}}_{\mathbb{Z}},\Delta x{|}_{t={\tau }_i}={\gamma }_i+\epsilon A_{1i}x\left({\tau }_i-\right),\mathrm{\hspace{0.17em}\hspace{0.17em}}{\tau }_i\in \mathcal{T}\subset \mathbb{R},\mathrm{\hspace{0.17em}\hspace{0.17em}}{\gamma }_i\in {\mathbb{R}}^n$, and $i\in \mathbb{Z}$ are considered. Under the assumption that the generating system (for $\epsilon =0$) does not have solutions bounded on the entire real axis for some nonhomogeneities and using the Vishik-Lyusternik method, we establish conditions for the existence of solutions of these systems bounded on the entire real axis in the form of a Laurent series in powers of small parameter $\epsilon$ with finitely many terms with negative powers of $\epsilon$, and we suggest an algorithm of construction of these solutions.

This paper contains some sufficient conditions for the existence of positive solutions which are bounded below and above by positive functions for the first-order nonlinear neutral differential equations. These equations can also support the existence of positive solutions approaching zero at infinity

Asymptotic properties of solutions of the singular differential equation ${\left(p\right(t\left)u^{\prime }\right(t\left)\right)}^{\prime }=p\left(t\right)f\left(u\right(t\left)\right)$ are described. Here, $f$ is Lipschitz continuous on $ℝ$ and has at least two zeros 0 and . The function p is continuous on $[0,\infty )$ and has a positive continuous derivative on $(0,\infty )$ and $p\left(0\right)=0$. Further conditions for f and p under which the equation has oscillatory solutions converging to 0 are given.

By applying a fixed point theorem for mappings that are decreasing with respect to a cone, this paper investigates the existence of positive solutions for the nonlinear fractional boundary value problem: $D_{0^{+}}^{\alpha }u\left(t\right)+f(t,u(t\left)\right)=0$, $0<t<1$, $u\left(0\right)=u\text{'}\left(0\right)=u\text{'}\left(1\right)=0$, where $2<\alpha <3$, $D_{0^{+}}^{\alpha }$ is the Riemann-Liouville fractional derivative.

We all-sidedly consider a three-point boundary value problem for $p$-Laplacian differential equation with nonlinear term involving derivative. Some new sufficient conditions are obtained for the existence of at least one, triple, or arbitrary odd positive pseudosymmetric solutions by using pseudosymmetric technique and fixed-point theory in cone. As an application, two examples are given to illustrate the main results.

This work is concerned with a system of nonlinear wave equations with nonlinear damping and source terms acting on both equations. We prove a global nonexistence theorem for certain solutions with positive initial energy.

The issue of $H_{\infty }$ estimation for a class of Lipschitz nonlinear discrete-time systems with time delay and disturbance input is addressed. First, through integrating the $H_{\infty }$ filtering performance index with the Lipschitz conditions of the nonlinearity, the design of robust estimator is formulated as a positive minimum problem of indefinite quadratic form. Then, by introducing the Krein space model and applying innovation analysis approach, the minimum of the indefinite quadratic form is obtained in terms of innovation sequence. Finally, through guaranteeing the positivity of the minimum, a sufficient condition for the existence of the $H_{\infty }$ estimator is proposed and the estimator is derived in terms of Riccati-like difference equations. The proposed algorithm is proved to be effective by a numerical example.

The present investigation deals with global instability of a general $n$-dimensional system of ordinary differential equations with quadratic right-hand sides. The global instability of the zero solution in a given cone is proved by Chetaev's method, assuming that the matrix of linear terms has a simple positive eigenvalue and the remaining eigenvalues have negative real parts. The sufficient conditions for global instability obtained are formulated by inequalities involving norms and eigenvalues of auxiliary matrices. In the proof, a result is used on the positivity of a general third-degree polynomial in two variables to estimate the sign of the full derivative of an appropriate function in a cone.

Sufficient conditions for the existence and asymptotic stability of the invariant sets of an impulsive system of differential equations defined in the direct product of a torus and an Euclidean space are obtained.

New explicit conditions of asymptotic and exponential stability are obtained for the scalar nonautonomous linear delay differential equation $\dot{x}\left(t\right)+{{\displaystyle \sum }}_{k=1}^m{a}_k\left(t\right)x\left(h_k\right(t\left)\right)=0$ with measurable delays and coefficients. These results are compared to known stability tests.

Existence of nonoscillatory solutions for the second-order dynamic equation ${\left(A_0{x}^{\Delta }\right)}^{\Delta }\left(t\right)+{{\displaystyle \sum }}_{i\in {[1,n]}_{\mathbb{N}}}{A}_i\left(t\right)x\left({\alpha }_i\right(t\left)\right)=0$ for $t\in {[t_0,\infty )}_{𝕋}$ is investigated in this paper. The results involve nonoscillation criteria in terms of relevant dynamic and generalized characteristic inequalities, comparison theorems, and explicit nonoscillation and oscillation conditions. This allows to obtain most known nonoscillation results for second-order delay differential equations in the case $A_0\left(t\right)\equiv 1$ for $t\in [t_0,\infty ){}_{ℝ}$ and for second-order nondelay difference equations (${\alpha }_i\left(t\right)=t+1$ for $t\in [t_0,\infty ){}_{\mathbb{N}}$). Moreover, the general results imply new nonoscillation tests for delay differential equations with arbitrary $A_0$ and for second-order delay difference equations. Known nonoscillation results for quantum scales can also be deduced.

We consider equation $x^{\prime \prime }+g\left(x\right)=0$, where $g\left(x\right)$ is a polynomial, allowing the equation to have multiple period annuli. We detect the maximal number of possible period annuli for polynomials of odd degree and show how the respective optimal polynomials can be constructed.

Existence of positive solutions for advanced equations with several terms $\dot{x}\left(t\right)+{{\displaystyle \sum }}_{k=1}^m{a}_k\left(t\right)x\left(h_k\right(t\left)\right)=0$, $h_k\left(t\right)\ge t$ is investigated in the following three cases: (a) all coefficients $a_k$ are positive; (b) all coefficients $a_k$ are negative; (c) there is an equal number of positive and negative coefficients. Results on asymptotics of nonoscillatory solutions are also presented.

For a system of linear functional differential equations, we consider a three-point problem with nonseparated boundary conditions determined by singular matrices. We show that, to investigate such a problem, it is often useful to reduce it to a parametric family of two-point boundary value problems for a suitably perturbed differential system. The auxiliary parametrised two-point problems are then studied by a method based upon a special kind of successive approximations constructed explicitly, whereas the values of the parameters that correspond to solutions of the original problem are found from certain numerical determining equations. We prove the uniform convergence of the approximations and establish some properties of the limit and determining functions.

The stability of the zero solution of a system of first-order linear functional differential equations with nonconstant delay is considered. Sufficient conditions for stability, uniform stability, asymptotic stability, and uniform asymptotic stability are established.

We consider the following real two-dimensional nonlinear analytic quasi-periodic Hamiltonian system $\dot{x}=J{\nabla }_{x}H$, where $H(x,t,ϵ)=(1/2)\beta (x_1^2+x_2^2)+F(x,t,ϵ)$ with $\beta \ne 0,{\partial }_{x}F(0,t,ϵ)=O\left(ϵ\right)$ and ${\partial }_{xx}F(0,t,ϵ)=O\left(ϵ\right)$ as $ϵ\to 0$. Without any nondegeneracy condition with respect to ε, we prove that for most of the sufficiently small ε, by a quasi-periodic symplectic transformation, it can be reduced to a quasi-periodic Hamiltonian system with an equilibrium.

The paper deals with the class of jump control systems with semi-Markov coefficients. The control system is described as the system of linear differential equations. Every jump of the random process implies the random transformation of solutions of the considered system. Relations determining the optimal control to minimize the functional are derived using Lyapunov functions. Necessary conditions of optimization which enables the synthesis of the optimal control are established as well.

Some oscillation criteria are established for the second-order nonlinear neutral differential equations of mixed type ${\left[{\left(x\right(t)+p_{1}x(t-{\tau }_1)+p_{2}x(t+{\tau }_2\left)\right)}^{\gamma }\right]}^{\prime \prime }=q_1\left(t\right)x^{\gamma }(t-{\sigma }_1)+q_2\left(t\right)x^{\gamma }(t+{\sigma }_2)$, $t\ge t_0$, where $\gamma \ge 1$ is a quotient of odd positive integers. Our results generalize the results given in the literature.

We establish some new oscillation criteria for the second-order neutral delay dynamic equations of Emden-Fowler type, ${\left[a\left(t\right)\right(x\left(t\right)+r\left(t\right)x{\left(\tau \right(t\left)\right))}^{\Delta }]}^{\Delta }+p\left(t\right)x^{\gamma }\left(\delta \right(t\left)\right)=0$, on a time scale unbounded above. Here is a quotient of odd positive integers with a and p being real-valued positive functions defined on $𝕋$. Our results in this paper not only extend and improve the results in the literature but also correct an error in one of the references.

Some oscillation criteria are established for the second-order superlinear neutral differential equations $\left(r\right(t\left)\right|z\text{'}\left(t\right){|}^{\alpha -1}z\text{'}\left(t\right))\text{'}+f(t,x(\sigma \left(t\right)\left)\right)=0$, $t\ge t_0$, where $z\left(t\right)=x\left(t\right)+p\left(t\right)x\left(\tau \right(t\left)\right)$, $\tau \left(t\right)\ge t$, $\sigma \left(t\right)\ge t$, $p\in C\left(\right[t_0,\infty ),[0,p_0])$, and $\alpha \ge 1$. Our results are based on the cases ${{\displaystyle \int }}_{t_0}^{\infty }1/r^{1/\alpha }\left(t\right)\text{d}t=\infty$ or ${{\displaystyle \int }}_{t_0}^{\infty }1/r^{1/\alpha }\left(t\right)\text{d}t<\infty$. Two examples are also provided to illustrate these results.

We study the following second-order neutral functional differential equation with mixed nonlinearities ${\left(r\right(t\left)\right|\left(u\right(t)+p(t\left)u\right(t-\sigma \left)\right)\text{'}{|}^{\alpha -1}{\left(u\right(t)+p(t\left)u\right(t-\sigma \left)\right)}^{\prime })}^{\prime }+q_0\left(t\right)|u\left({\tau }_0\right(t\left)\right){|}^{\alpha -1}u\left({\tau }_0\right(t\left)\right)+q_1\left(t\right)|u\left({\tau }_1\right(t\left)\right){|}^{\beta -1}u\left({\tau }_1\right(t\left)\right)+q_2\left(t\right)|u\left({\tau }_2\right(t\left)\right){|}^{\gamma -1}u\left({\tau }_2\right(t\left)\right)=0$, where , ${{\displaystyle \int }}_{t_0}^{\infty }(1/r^{1/\alpha }(t\left)\right)\text{d}t<\infty$. Oscillation results for the equation are established which improve the results obtained by Sun and Meng (2006), Xu and Meng (2006), Sun and Meng (2009), and Han et al. (2010).

Oscillation criteria obtained by Kusano and Onose (1973) and by Belohorec (1969) are extended to second-order sublinear impulsive differential equations of Emden-Fowler type: $x\left(t\right)+p\left(t\right)|x\left(\tau \right(t\left)\right){|}^{\alpha -1}x\left(\tau \right(t\left)\right)=0$ , $t\ne {\theta }_k$; $\Delta x\text{'}\left(t\right){|}_{t={\theta }_k}+q_k|x\left(\tau \right({\theta }_k\left)\right){|}^{\alpha -1}x\left(\tau \right({\theta }_k\left)\right)=0$; $\Delta x\left(t\right){|}_{t={\theta }_k}=0$, $(0<\alpha <1)$ by considering the cases $\tau \left(t\right)\le t$ and $\tau \left(t\right)=t$, respectively. Examples are inserted to show how impulsive perturbations greatly affect the oscillation behavior of the solutions.

We study the existence of oscillatory periodic solutions for two nonautonomous differential-difference equations which arise in a variety of applications with the following forms: $\dot{x}\left(t\right)=-f(t,x(t-r\left)\right)$ and $\dot{x}\left(t\right)=-f(t,x(t-s\left)\right)-f(t,x(t-2s\left)\right)$, where $f\in C(ℝ\times ℝ,ℝ)$ is odd with respect to $\mathrm{x}$, and are two given constants. By using a symplectic transformation constructed by Cheng (2010) and a result in Hamiltonian systems, the existence of oscillatory periodic solutions of the above-mentioned equations is established.

The necessary and sufficient conditions for solvability of the family of difference equations with periodic boundary condition were obtained using the notion of relative spectrum of linear bounded operator in the Banach space and the ergodic theorem. It is shown that when the condition of existence is satisfied, then such periodic solutions are built using the formula for the generalized inverse operator to the linear limited one.

We study the existence of positive solutions for the boundary value problem of nonlinear fractional differential equations $D_{0^{+}}^{\alpha }u\left(t\right)+\lambda f\left(u\right(t\left)\right)=0$, $0<t<1$, $u\left(0\right)=u\left(1\right)=u\text{'}\left(0\right)=0$, where $2<\alpha \le 3$ is a real number, $D_{0^{+}}^{\alpha }$ is the Riemann-Liouville fractional derivative, $\lambda$ is a positive parameter, and $f:(0,+\infty )\to (0,+\infty )$ is continuous. By the properties of the Green function and Guo-Krasnosel'skii fixed point theorem on cones, the eigenvalue intervals of the nonlinear fractional differential equation boundary value problem are considered, some sufficient conditions for the nonexistence and existence of at least one or two positive solutions for the boundary value problem are established. As an application, some examples are presented to illustrate the main results.

The aim of this paper is to offer sufficient conditions for property (B) and/or the oscillation of the third-order nonlinear functional differential equation with mixed arguments ${\left[a\left(t\right){\left[x^{\prime \prime }\left(t\right)\right]}^{\gamma }\right]}^{\prime }=q\left(t\right)f\left(x\left[\tau \right(t\left)\right]\right)+p\left(t\right)h\left(x\left[\sigma \right(t\left)\right]\right)$ . Both cases ${{\displaystyle \int }}^{\infty }{a}^{-1/\gamma }\left(s\right)\text{d}s=\infty$ and ${{\displaystyle \int }}^{\infty }{a}^{-1/\gamma }\left(s\right)\text{d}s<\infty$ are considered. We deduce properties of the studied equations via new comparison theorems. The results obtained essentially improve and complement earlier ones.

A Lie group acting on finite-dimensional space is generated by its infinitesimal transformations and conversely, any Lie algebra of vector fields in finite dimension generates a Lie group (the first fundamental theorem). This classical result is adjusted for the infinite-dimensional case. We prove that the (local, $C^{\infty }$ smooth) action of a Lie group on infinite-dimensional space (a manifold modelled on ${ℝ}^{\infty }$) may be regarded as a limit of finite-dimensional approximations and the corresponding Lie algebra of vector fields may be characterized by certain finiteness requirements. The result is applied to the theory of generalized (or higher-order) infinitesimal symmetries of differential equations.

Using the Kato theorem for abstract differential equations, the local well-posedness of the solution for a nonlinear dissipative Camassa-Holm equation is established in space $C\left(\right[0,T),H^s\left(R\right))\cap C^1\left(\right[0,T),H^{s-1}\left(R\right))$ with . In addition, a sufficient condition for the existence of weak solutions of the equation in lower order Sobolev space $H^s\left(R\right)$ with $1\le s\le 3/2$ is developed.

We present a new perspective concerning the study of the asymptotic behavior of variational equations by employing function spaces techniques. We give a complete description of the dichotomous behaviors of the most general case of skew-product flows, without any assumption concerning the flow, the cocycle or the splitting of the state space, our study being based only on the solvability of some associated control systems between certain function spaces. The main results do not only point out new necessary and sufficient conditions for the existence of uniform and exponential dichotomy of skew-product flows, but also provide a clear chart of the connections between the classes of translation invariant function spaces that play the role of the input or output classes with respect to certain control systems. Finally, we emphasize the significance of each underlying hypothesis by illustrative examples and present several interesting applications.

We study perturbations of the nonoscillatory half-linear differential equation $\left(r\right(t\left)\Phi \right(x\text{'}\left)\right)\text{'}+c\left(t\right)\Phi \left(x\right)=0$, $\Phi \left(x\right):=|x{|}^{p-2}x$, . We find explicit formulas for the functions $\dot{r}$, $\dot{c}$ such that the equation $\left[\left(r\right(t)+\lambda \dot{r}(t\left)\right)\Phi (x\text{'})\right]\text{'}+[c\left(t\right)+\mu \dot{c}\left(t\right)]\Phi \left(x\right)=0$ is conditionally oscillatory, that is, there exists a constant $\gamma$such that the previous equation is oscillatory if and nonoscillatory if $\mu -\lambda <\gamma$. The obtained results extend the previous results concerning two-parametric perturbations of the half-linear Euler differential equation.

The purpose of this paper is to investigate the existence and uniqueness of positive solutions for the following fourth-order boundary value problem: $y^{\left(4\right)}\left(t\right)=f(t,y(t\left)\right)$, $t\in [0,1]$, $y\left(0\right)=y\left(1\right)=y^{\prime }\left(0\right)=y^{\prime }\left(1\right)=0$. Moreover, under certain assumptions, we will prove that the above boundary value problem has a unique symmetric positive solution. Finally, we present some examples and we compare our results with the ones obtained in recent papers. Our analysis relies on a fixed point theorem in partially ordered metric spaces.

A linear Volterra difference equation of the form $x(n+1)=a\left(n\right)+b\left(n\right)x\left(n\right)+{{\displaystyle \sum }}_{i=0}^{n}K(n,i)x\left(i\right),$ where $x:{\mathbb{N}}_0\to ℝ$, $a:{\mathbb{N}}_0\to ℝ$, $K:{\mathbb{N}}_0\times {\mathbb{N}}_0\to ℝ$ and $b:{\mathbb{N}}_0\to ℝ\setminus \left\{0\right\}$ is $\omega$-periodic, is considered. Sufficient conditions for the existence of weighted asymptotically periodic solutions of this equation are obtained. Unlike previous investigations, no restriction on ${{\displaystyle \prod }}_{j=0}^{\omega -1}b\left(j\right)$ is assumed. The results generalize some of the recent results.

We develop the Weyl-Titchmarsh theory for time scale symplectic systems. We introduce the $M\left(\lambda \right)$ -function, study its properties, construct the corresponding Weyl disk and Weyl circle, and establish their geometric structure including the formulas for their center and matrix radii. Similar properties are then derived for the limiting Weyl disk. We discuss the notions of the system being in the limit point or limit circle case and prove several characterizations of the system in the limit point case and one condition for the limit circle case. We also define the Green function for the associated nonhomogeneous system and use its properties for deriving further results for the original system in the limit point or limit circle case. Our work directly generalizes the corresponding discrete time theory obtained recently by S. Clark and P. Zemánek (2010). It also unifies the results in many other papers on the Weyl-Titchmarsh theory for linear Hamiltonian differential, difference, and dynamic systems when the spectral parameter appears in the second equation. Some of our results are new even in the case of the second-order Sturm-Liouville equations on time scales.