The methods of lower and upper solutions and monotone iterative technique are employed to the study of integral boundary value problems for a class of first-order impulsive functional differential equations. Sufficient conditions are obtained for the existence of extreme solutions.

We study the qualitative behavior of a class of predator-prey models with Beddington-DeAngelis-type functional response, primarily from the viewpoint of permanence (uniform persistence). The Beddington-DeAngelis functional response is similar to the Holling type-II functional response but contains a term describing mutual interference by predators. We establish criteria under which we have boundedness of solutions, existence of an attracting set, and global stability of the coexisting interior equilibrium via Lyapunov function.

This work is focused on the doubly nonlinear equation ${\partial }_{tt}u+{\partial }_{xxxx}u+(p-\parallel {\partial }_{x}u{\parallel }_{L^2\left(\mathrm{0,1}\right)}^2){\partial }_{xx}u+{\partial }_{t}u+k^2{u}^{+}=f$, whose solutions represent the bending motion of an extensible, elastic bridge suspended by continuously distributed cables which are flexible and elastic with stiffness $k^2$. When the ends are pinned, long-term dynamics is scrutinized for arbitrary values of axial load $p$ and stiffness $k^2$. For a general external source $f$, we prove the existence of bounded absorbing sets. When $f$ is time-independent, the related semigroup of solutions is shown to possess the global attractor of optimal regularity and its characterization is given in terms of the steady states of the problem.

This paper outlines a detailed study of the coupling of He's polynomials with correction functional of variational iteration method (VIM) for solving various initial and boundary value problems. The elegant coupling gives rise to the modified versions of VIM which is very efficient in solving nonlinear problems of diversified nature. It is observed that the variational iteration method using He's polynomials (VIMHP) is very efficient, easier to implements, and more user friendly. Several examples are given to reconfirm the efficiency of the proposed VIMHP.

Using the theory of uniform global attractors of multivalued semiprocesses, we prove the existence of a uniform global attractor for a nonautonomous semilinear degenerate parabolic equation in which the conditions imposed on the nonlinearity provide the global existence of a weak solution, but not uniqueness. The Kneser property of solutions is also studied, and as a result we obtain the connectedness of the uniform global attractor.

We consider one-phase nonclassical unidimensional Stefan problems for a source function $F$ which depends on the heat flux, or the temperature on the fixed face $x=0$. In the first case, we assume a temperature boundary condition, and in the second case we assume a heat flux boundary condition or a convective boundary condition at the fixed face. Exact solutions of a similarity type are obtained in all cases.

The paper is devoted to the study of initial-boundary value problems for quasilinear second-order systems. Existence and uniqueness of the solution in the space $H^s\left(\overline{\Omega }\times \left[0,T\right]\right)$, with $s>d/2+3$, is proved in the case where $\Omega$ is a half-space of ${\Re }^d$. The proof of the main theorem relies on two preliminary results: existence of the solution to mixed problems for linear second-order systems with smooth coefficients, and existence of the solution to initial-boundary value problems for linear second-order operators whose coefficients depend on the variables $x$ and $t$ through a function $v\in H^s\left({\Re }^{d+1}\right)$. By means of the results proved for linear operators, the well posedness of the mixed problem for the quasi-linear system is established by studying the convergence of a suitable iteration scheme.

We present some new oscillation criteria for second-order neutral partial functional differential equations of the form $(\partial /\partial t)\left\{p\right(t\left)\right(\partial /\partial t\left)\right[u(x,t)+{\sum }_{i=1}^l{\lambda }_i\left(t\right)u(x,t-{\tau }_i)\left]\right\}=a\left(t\right)\Delta u(x,t)+{\sum }_{k=1}^s{a}_k\left(t\right)\Delta u(x,t-{\rho }_k(t\left)\right)-q(x,t)f\left(u\right(x,t\left)\right)-{\sum }_{j=1}^m{q}_j(x,t)f_j\left(u\right(x,t-{\sigma }_j\left)\right)$, $(x,t)\in \Omega \times R^{+}\equiv G$, where $\Omega$ is a bounded domain in the Euclidean $N$-space $R^N$ with a piecewise smooth boundary $\partial \Omega$ and $\Delta$ is the Laplacian in $R^N$. Our results improve some known results and show that the oscillation of some second-order linear ordinary differential equations implies the oscillation of relevant nonlinear neutral partial functional differential equations.

This paper is concerned with the existence of bounded positive solution for the semilinear elliptic problem $\Delta u=\lambda p\left(x\right)f\left(u\right)$ in $\Omega$ subject to some Dirichlet conditions, where $\Omega$ is a regular domain in ${ℝ}^n$ $\left(n\ge 3\right)$ with compact boundary. The nonlinearity $f$ is nonnegative continuous and the potential $p$ belongs to some Kato class $K\left(\Omega \right)$. So we prove the existence of a positive continuous solution depending on $\lambda$ by the use of a potential theory approach.

We study the boundary value problem for a linear first-order partial differential system with characteristic boundary of constant multiplicity. We assume the problem to be “weakly” well posed, in the sense that a unique $L^2$-solution exists, for sufficiently smooth data, and obeys an a priori energy estimate with a finite loss of tangential/conormal regularity. This is the case of problems that do not satisfy the uniform Kreiss-Lopatinskiĭ condition in the hyperbolic region of the frequency domain. Provided that the data are sufficiently smooth, we obtain the regularity of solutions, in the natural framework of weighted conormal Sobolev spaces.

A generalized Fisher's equation is solved by using the modified Adomian decomposition method (MADM), variational iteration method (VIM), homotopy analysis method (HAM), and modified homotopy perturbation method (MHPM). The approximation solution of this equation is calculated in the form of series whose components are computed easily. The existence, uniqueness, and convergence of the proposed methods are proved. Numerical example is studied to demonstrate the accuracy of the present methods.

Certain nonlinear autonomous ordinary differential equations of the second order are reduced to Abel equations of the first kind ((Ab-1) equations). Based on the results of a previous work, concerning a closed-form solution of a general (Ab-1) equation, and introducing an arbitrary function, exact one-parameter families of solutions are derived for the original autonomous equations, for the most of which only first integrals (in closed or parametric form) have been obtained so far. Two-dimensional autonomous systems of differential equations of the first order, equivalent to the considered herein autonomous forms, are constructed and solved by means of the developed analysis.