We introduce impulsive cellular neural network models with piecewise alternately advanced and retarded argument (in short IDEPCA). The model with the advanced argument is system with strong anticipation. Some sufficient conditions are established for the existence and global exponential stability of a unique equilibrium. The approaches are based on employing Banach’s fixed point theorem and a new IDEPCA integral inequality of Gronwall type. The criteria given are easily verifiable, possess many adjustable parameters, and depend on impulses and piecewise constant argument deviations, which provides exibility for the design and analysis of cellular neural network models. Several numerical examples and simulations are also given to show the feasibility and effectiveness of our results.

Newton-Kantorovich and Smale uniform type of convergence theorem of a deformed Newton method having the third-order convergence is established in a Banach space for solving nonlinear equations. The error estimate is determined to demonstrate the efficiency of our approach. The obtained results are illustrated with three examples.

In this paper, we propose the study of an integral equation, with deviating arguments, of the type $y(t)=\mathrm{\omega }(t)-{\int }_{\mathrm{0}}^{\mathrm{\infty }}\mathrm{‍}f(t,s,\mathrm{y}({\gamma }_{\mathrm{1}}(\mathrm{s})),\dots ,\mathrm{y}({\gamma }_N(\mathrm{s})))\mathrm{}ds,\mathrm{}\mathrm{}\mathrm{}\mathrm{}t\ge \mathrm{0},$ in the context of Banach spaces, with the intention of giving sufficient conditions that ensure the existence of solutions with the same asymptotic behavior at $\mathrm{\infty }$ as $\mathrm{\omega }(t)$. A similar equation, but requiring a little less restrictive hypotheses, is $y(t)=\mathrm{\omega }(t)-{\int }_{\mathrm{0}}^{\mathrm{\infty }}\mathrm{‍}q(t,s)\mathrm{}F(s,\mathrm{y}({\gamma }_{\mathrm{1}}(s)),\dots ,\mathrm{y}({\gamma }_N(s)))\mathrm{}ds,\mathrm{}\mathrm{}\mathrm{}\mathrm{}t\ge \mathrm{0}.$ In the case of $q(t,s)=(t-s{)}_{+}$, its solutions with asymptotic behavior given by $\mathrm{\omega }(t)$ yield solutions of the second order nonlinear abstract differential equation $y\text{'}\text{'}(t)-\omega \text{'}\text{'}(t)+F(t,\mathrm{y}({\gamma }_{\mathrm{1}}(t)),\dots ,\mathrm{y}({\gamma }_N(t)))=\mathrm{0},$ with the same asymptotic behavior at $\mathrm{\infty }$ as $\mathrm{\omega }(t)$.

A ratio-dependent predator-prey model with two time delays is studied. By means of an iteration technique, sufficient conditions are obtained for the global attractiveness of the positive equilibrium. By comparison arguments, the global stability of the semitrivial equilibrium is addressed. By using the theory of functional equation and Hopf bifurcation, the conditions on which positive equilibrium exists and the quality of Hopf bifurcation are given. Using a global Hopf bifurcation result of Wu (1998) for functional differential equations, the global existence of the periodic solutions is obtained. Finally, an example for numerical simulations is also included.

In this paper we study the following second-order periodic system: $x^{\prime \prime }+V^{\prime }(x)+p(x,t)=\mathrm{0,}$ where $V(x)$ has a singularity. Under some assumptions on the $V(x)$ and $p(x,t)$ by Ortega’ small twist theorem, we obtain the existence of quasi-periodic solutions and boundedness of all the solutions.

The paper develops a mathematical model of foreign currency exchange market in the form of a stochastic linear differential equation with coefficients depending on a semi-Markov process. The boundaries of the domain of its instability is determined by using moment equations.

The paper is devoted to the study of the solvability of a singular initial value problem for systems of ordinary differential equations. The main results give sufficient conditions for the existence of solutions in the right-hand neighbourhood of a singular point. In addition, the dimension of the set of initial data generating such solutions is estimated. An asymptotic behavior of solutions is determined as well and relevant asymptotic formulas are derived. The method of functions defined implicitly and the topological method (Ważewski's method) are used in the proofs. The results generalize some previous ones on singular initial value problems for differential equations.

The asymptotic phase property and reduction principle for stability of a trivial solution is generalized to the case of the noninvertible impulsive differential equations in Banach spaces whose linear parts split into two parts and satisfy the condition of separation.

By the critical point theory, infinitely many 4σ-periodic solutions are obtained for the system of delay differential equations $ẋ\left(t\right)=-f(x(t-\sigma ))$, where $\sigma \in (0,+\infty )$ and $f\in C({ℝ}^n,{ℝ}^n)$. It is shown that all the periodic solutions derived here are brought about by the time delay.

It is proved that any first-order globally periodic linear inhomogeneous autonomous difference equation defined by a linear operator with closed range in a Banach space has an equilibrium. This result is extended for higher order linear inhomogeneous system in a real or complex Euclidean space. The work was highly motivated by the early works of Smith (1934, 1941) and the papers of Kister (1961) and Bas (2011).

A class of even order damped differential equations with distributed deviating arguments are investigated. Several new criteria that ensure the oscillation of solutions are obtained. To demonstrate the validity of the results obtained, two examples are given.

We give an existence and uniqueness theorem for solutions of inhomogeneous impulsive boundary value problem (BVP) for planar Hamiltonian systems. Green's function that is needed for representing the solutions is obtained and its properties are listed. The uniqueness of solutions is connected to a Lyapunov type inequality for the corresponding homogeneous BVP.

We study the half-linear delay differential equation $(r(t)\mathrm{\Phi }(x\mathrm{\text{'}}(t)))\mathrm{\text{'}}+c(t)\mathrm{\Phi }(x(\tau (t)))=0$, $\mathrm{\Phi }(x):=|x{|}^{p-2}x$, $p>1.$ We establish a new a priori bound for the nonoscillatory solution of this equation and utilize this bound to derive new oscillation criteria for this equation in terms of oscillation criteria for an ordinary half-linear differential equation. The presented results extend and improve previous results of other authors. An extension to neutral equations is also provided.

Some new nonlinear weakly singular difference inequalities are discussed, which generalize some known weakly singular inequalities and can be used in the analysis of nonlinear Volterra-type difference equations with weakly singular kernel. An application to the upper bound of solutions of a nonlinear difference equation is also presented.

The mathematical model for the incompressible two-dimensional/axisymmetric non-Newtonian fluid flows and heat transfer analysis in the region of stagnation point over a stretching/shrinking sheet and axisymmetric shrinking sheet is presented. The governing equations are transformed into dimensionless nonlinear ordinary differential equations by similarity transformation. Analytical technique, namely, the homotopy perturbation method (HPM) with general form of linear operator is used to solve dimensionless nonlinear ordinary differential equations. The series solution is obtained without using the diagonal Padé approximants to handle the boundary condition at infinity which can be considered as a clear advantage of homotopy perturbation technique over the decomposition method. The effects of the pertinent parameters on the velocity and temperature field are discussed through graphs. To the best of authors’ knowledge, HPM solution with general form of linear operator for two-dimensional/axisymmetric non-Newtonian fluid flows and heat transfer analysis in the region of stagnation point is presented for the first time in the literature.

We investigate the asymptotic behavior of solutions to a linear Volterra integrodifferential system $x_i^{\prime }(t)=a_i(t)+b_i(t)x_i(t)+{\sum }_{j=1}^n\mathrm{‍}{\int }_0^t\mathrm{‍}{K}_{ij}(t,s)x_j(s)ds,t\in {ℝ}^{+}$, $i=\mathrm{1,2},\dots ,n.$ We show that under some suitable conditions, there exists a solution for the above integrodifferential system, which is asymptotically equivalent to some given functions. Two examples are given to illustrate our theorem.

The current paper is devoted to the regularity of the mild solution for a stochastic fractional delayed reaction-diffusion equation driven by Lévy space-time white noise. By the Banach fixed point theorem, the existence and uniqueness of the mild solution are proved in the proper working function space which is affected by the delays. Furthermore, the time regularity and space regularity of the mild solution are established respectively. The main results show that both time regularity and space regularity of the mild solution depend on the regularity of initial value and the order of fractional operator. In particular, the time regularity is affected by the regularity of initial value with delays.

An inverse problem for Dirac differential operators with discontinuity conditions and discontinuous coefficient is studied. It is shown by Hochstadt and Lieberman's method that if the potential function $p(x)$ in $\Omega (x)$ is prescribed over the interval $(\pi /2,\pi )$, then a single spectrum suffices to determine $p(x)$ on the interval $(0,\pi )$ and it is also shown here that $\rho (x)$ is uniquely determined by a spectrum.

We analyze the dynamics of the forced singularly perturbed differential equations of Duffing’s type with a potential that is bounded from above. We explain the appearance of the large frequency nonlinear oscillations of the solutions. It is shown that the frequency can be controlled by a small parameter at the highest derivative.