Volume 24, Issue 1

The strict stability properties are generalized to nonlinear Caputo fractional differential equations in the case when both initial points and initial times are changeable. Using Lyapunov functions, some criteria for strict stability, eventually strict stability and strict practical stability are obtained. A brief overview of different types of derivatives in the literature related to the application of Lyapunov functions to Caputo fractional equations are given, and their advantages and disadvantages are discussed with several examples. The Caputo fractional Dini derivative with respect to to initial time difference is used to obtain some sufficient conditions.

The existence of unbounded solutions with different asymptotic behavior for a second order nonlinear differential equation with p -Laplacian is considered. The oscillation of all solutions is investigated. Some discrepancies and similarities between equations of Emden–Fowler-type and equations with mixed nonlinearities are pointed out.

The periodic boundary value problem for linear second order functional differential equations is considered. Sharp sufficient conditions for the positiveness of solutions are obtained.

In this paper, we are interested in the study of the existence of solutions of a class of nonlinear fractional differential equations coupled with Dirichlet boundary conditions. We make an exhaustive study of the sign of Green’s function related to the linear problem and obtain the exact values for which it has the constant sign on its whole square of definition. This constant sign is equivalent to the validity of anti-maximum principles. The existence of solutions is deduced from a recent fixed point theorem valid for operators defined on suitable cones of Banach spaces. Moreover, the method of lower and upper solutions is developed for the non-homogeneous Dirichlet problem.

The authors establish the existence of at least three solutions to a quasilinear elliptic problem subject to Dirichlet boundary conditions in a bounded domain in ℝN${\mathbb{R}^{N}}$. A critical point result for differentiable functionals is used to prove the results.

The structure and the asymptotic behavior of positive increasing solutions of functional differential systems of the form x′⁢(t)=p⁢(t)⁢φα⁢(y⁢(k⁢(t))),y′⁢(t)=q⁢(t)⁢φβ⁢(x⁢(l⁢(t)))$x^{\prime}(t)=p(t)\varphi_{\alpha}\bigl{(}y(k(t))\bigr{)},\quad y^{\prime}(t)=% q(t)\varphi_{\beta}\bigl{(}x(l(t))\bigr{)}$ are investigated in detail, where α and β are positive constants, p⁢(t)${p(t)}$ and q⁢(t)${q(t)}$ are positive continuous functions on [0,∞)${[0,\infty)}$, k⁢(t)${k(t)}$ and l⁢(t)${l(t)}$ are positive continuous functions on [0,∞)${[0,\infty)}$ tending to ∞${\infty}$ with t , and φγ⁢(u)=|u|γ⁢sgn⁡u${\varphi_{\gamma}(u)=\lvert u\rvert^{\gamma}\operatorname{sgn}u}$, γ>0${\gamma>0}$, u∈ℝ${u\in\mathbb{R}}$. An extreme class of positive increasing solutions, called rapidly increasing solutions , of the system above is analyzed by means of regularly varying functions. The results obtained find applications to systems of the form x′⁢(g⁢(t))=p⁢(t)⁢φα⁢(y⁢(k⁢(t))),y′⁢(h⁢(t))=q⁢(t)⁢φβ⁢(x⁢(l⁢(t))),$x^{\prime}(g(t))=p(t)\varphi_{\alpha}\bigl{(}y(k(t))\bigr{)},\quad y^{\prime}(% h(t))=q(t)\varphi_{\beta}\bigl{(}x(l(t))\bigr{)},$ and to scalar equations of the type (p⁢(t)⁢φα⁢(x′⁢(g⁢(t))))′=p⁢(t)⁢φβ⁢(x⁢(l⁢(t))).$\Bigl{(}p(t)\varphi_{\alpha}\bigl{(}x^{\prime}(g(t))\bigr{)}\Bigr{)}^{\prime}=% p(t)\varphi_{\beta}\bigl{(}x(l(t))\bigr{)}.$

The Liénard–Mathieu equation with a small parameter ε is examined. The existence of periodic and quasiperiodic solutions is proved using the averaging method, the method of complexification and the phase space analysis of an associated autonomous equation. The results extend and generalize those of Kalas and Kadeřábek [6].

For a class of functionals with nonlocal integrant, necessary conditions of local extremum under general linear constraints are obtained. An analog of Euler’s equation in an integral form is derived. An example of application to a variational problem for a functional with deviating argument and two-point boundary conditions is presented.

We consider the system of difference equations Δ(Δ⁢un-11-|Δ⁢un-1|2)=∇Vn(un)+hn,un=un+T (n∈ℤ),$\Delta\bigg{(}\frac{\Delta u_{n-1}}{\sqrt{1-|\Delta u_{n-1}|^{2}}}\bigg{)}=% \nabla V_{n}(u_{n})+h_{n},\quad u_{n}=u_{n+T}\quad(n\in\mathbb{Z}),$ with Δ⁢un=un+1-un∈ℝN${\Delta u_{n}=u_{n+1}-u_{n}\in{\mathbb{R}}^{N}}$, Vn=Vn⁢(x)∈C2⁢(ℝN,ℝ)${V_{n}=V_{n}(x)\in C^{2}({\mathbb{R}}^{N},\mathbb{R})}$, Vn+T=Vn${V_{n+T}=V_{n}}$, hn+T=hn${h_{n+T}=h_{n}}$ for all n∈ℤ${n\in\mathbb{Z}}$ and some positive integer T , Vn⁢(x)${V_{n}(x)}$ is ωi${\omega_{i}}$-periodic (ωi>0${\omega_{i}>0}$) with respect to each xi${x_{i}}$ (i=1,…,N${i=1,\ldots,N}$) and ∑j=1Thj=0${\sum_{j=1}^{T}h_{j}=0}$. Applying a modification argument to the corresponding problem with a left-hand member of p -Laplacian type, and using Morse theory, we prove that if all its solutions are non-degenerate, then the difference system above has at least 2N${2^{N}}$ geometrically distinct T -periodic solutions.

We discuss existence, multiplicity, localisation and stability properties of solutions of the Dirichlet problem associated with the gradient dependent prescribed mean curvature equation in the Lorentz–Minkowski space {-div⁡(∇⁡u1-|∇⁡u|2)=f⁢(x,u,∇⁡u)in ⁢Ω,u=0on ⁢∂⁡Ω.$\left\{\begin{aligned} \displaystyle{-}\operatorname{div}\biggl{(}\frac{\nabla u% }{\sqrt{1-|\nabla u|^{2}}}\biggr{)}&\displaystyle=f(x,u,\nabla u)&&% \displaystyle\phantom{}\text{in }\Omega,\\ \displaystyle u&\displaystyle=0&&\displaystyle\phantom{}\text{on }\partial% \Omega.\end{aligned}\right.$ The obtained results display various peculiarities, which are due to the special features of the involved differential operator and have no counterpart for elliptic problems driven by other quasilinear differential operators. This research is also motivated by some recent achievements in the study of prescribed mean curvature graphs in certain Friedmann–Lemaître–Robertson–Walker, as well as Schwarzschild–Reissner–Nordström, spacetimes.

We study piecewise synergetic systems originating from Biochemical Systems Theory. In the first part of the paper, the emphasis is put on practical calculations with such systems. We consider four examples: calculation of trajectories and steady states, solution of an optimization problem and a method of estimation of parameters (kinetic orders), all examples being biologically motivated. In the second part of the paper, we study convergence of solutions, in particularly, steady states, of a sequence of piecewise synergetic systems approximating an arbitrary compartment model. This convergence analysis is then applied to the optimization problem and the method of estimating sensitivities (kinetic orders) in a generic compartment model. In this paper we put forward arguments for the importance of the theoretical and numerical analysis of piecewise synergetic systems.

We obtain new conditions sufficient for the unique solvability of the Darboux problem for linear partial functional differential equations of hyperbolic type. The main results are applied to the hyperbolic differential equations with argument deviations.