Search

Abstract An analytical vibration response in the time domain for an axially translating and laterally vibrating string with mixed boundary conditions is considered in this paper. The domain of the string is a constant, dependent upon the general initial conditions. The translating tensioned strings possess different types of mixed boundary conditions, such as fixed_dashpot, fixed_spring-dashpot, fixed_mass-spring-dashpot. An analytical solution using a reflected wave superposition method is presented for a finite translating string. Firstly, the cycle of boundary reflection for strings is provided, which is dependent upon the string length. Each cycle is divided into three time intervals according to the travelling speed and direction of the string. Applying D'Alembert's principle and the reflection properties, expressions for the reflected waves under three different non-classical boundary conditions are derived. Then, the vibrational response of the axially translating string is solved for three time intervals by using a reflected wave superposition method. The accuracy and efficiency of the proposed method are confirmed numerically by comparison to simulations produced using a Newmark- β method solution. The energy expressions for a travelling string with a fixed_dashpot boundary condition is obtained and the time domain curves for the total energy and the change of energy at the boundaries are given. [ABSTRACT FROM AUTHOR]

Abstract This paper deals with the Neumann problem for a fully parabolic chemotaxis–haptotaxis model of cancer invasion given by { u t = Δ u − χ ∇ ⋅ (u ∇ v) − ξ ∇ ⋅ (u ∇ w) + u (a − μ u r − 1 − λ w) , x ∈ Ω , t > 0 , τ v t = Δ v − v + u , x ∈ Ω , t > 0 , w t = − v w , x ∈ Ω , t > 0. Here, Ω ⊂ R N (N ≥ 1) is a bounded domain with smooth boundary and τ > 0 , r > 1 , λ ≥ 0 , a ∈ R , μ , ξ and χ are positive constants. It is shown that the corresponding initial–boundary value problem possesses a unique global bounded classical solution in the cases r > 2 or r = 2 , with μ > μ ⁎ = (N − 2) + N (χ + C β) C N 2 + 1 1 N 2 + 1 for some positive constants C β and C N 2 + 1. Furthermore, the large time behavior of solutions to the problem is also investigated. Specially speaking, when a is appropriately large, the corresponding solution of the system exponentially decays to ((a μ) 1 r − 1 , (a μ) 1 r − 1 , 0) if μ is large enough. This result improves or extends previous results of several authors. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we consider an initial-boundary value problem of the p-Laplacian parabolic equation { u t (x , t) = div (| ∇ u (x , t) | p − 2 ∇ u (x , t)) + f (u (x , t)) , (x , t) ∈ Ω × (0 , + ∞) , u (x , t) = 0 , (x , t) ∈ ∂ Ω × [ 0 , + ∞) , u (x , 0) = u 0 ≥ 0 , x ∈ Ω ‾ , where p ≥ 2 and Ω is a bounded domain of R N (N ≥ 1) with smooth boundary ∂Ω. The main contribution of this work is to introduce a new condition (C p) α ∫ 0 u f (s) d s ≤ u f (u) + β u p + γ , u > 0 for some α , β , γ > 0 with 0 < β ≤ (α − p) λ 1 , p p , where λ 1 , p is the first eigenvalue of p-Laplacian Δ p , and we use the concavity method to obtain the blow-up solutions to the above equations. In fact, it will be seen that the condition (C p) improves the conditions ever known so far. [ABSTRACT FROM AUTHOR]

Abstract Large computation time is widely considered to be the most important issue in scientific research especially in solving structural evolution problems. Recent developments in this domain have shown that the use of non-incremental schemes through Model Order Reduction led to important results in saving time. Yet, the question arises here how to obtain more time-saving. This paper examines an approach based on a collection of significant modes given by Proper Generalized Decomposition (PGD) solution for different time scales in order to save more computation time. The dictionary of the significant modes allows to construct an accurate solution for different characteristic times and different boundary problems compared to the full solution with a relative error rate less than 5% and with a large time saving of order 50 compared to Finite Element Method (FEM). The ability of the approach with respect to cycle time is discussed. Highlights • A new approach is proposed to decrease computational time for cyclic problems involving different time scales. • The approach is based on the use of pre-computed space–time modes which are further collected in a dictionary. • For a physical time belonging to the range of the dictionary, the solution can be accurately predicted. • For a given cycle time, the dictionary leads to an accurate if it contains the modes associated with this specific time. [ABSTRACT FROM AUTHOR]

The singularity in the traditional spherical polar coordinate system at the poles is a major factor in the lack of scalability of atmospheric models on massively parallel machines. Overset grids such as the Yin-Yang grid introduced by Kageyama and Sato [1] offer a potential solution to this problem. In this paper a three-dimensional, compressible, non-hydrostatic atmospheric model is developed and tested on the Yin-Yang grid building on ideas previously developed by the authors on the solution of Elliptic boundary value problems and conservation on overset grids. Using several tests from the literature, it is shown that this model is highly stable (even with little off-centering), accurate, and highly efficient in terms of computational cost. The model also incorporates highly efficient and accurate approaches to achieve positivity, monotonicity and conservative transport, which are paramount requirements for any atmospheric model. The parallel scalability of this model, using in excess of 212 million unknowns and more than 6000 processors, is also discussed and shown to compare favourably with a highly optimised latitude–longitude model in terms of scalability and actual run times. [ABSTRACT FROM AUTHOR]

Abstract: The aim of the paper is to study necessary and sufficient conditions for the existence of the global solution of the one-dimensional semilinear equation appearing in the boundary value problems of gas dynamics. We investigate the Cauchy problem for such equation in the domain where the operator is weakly hyperbolic. We obtain the necessary condition for the existence of the self-similar solutions for the semilinear Gellerstedt-type equation. The approach used in the paper is based on the fundamental solution of the linear Gellerstedt operator and the estimates. [Copyright &y& Elsevier]

Abstract: Convoluted C-cosine functions and semigroups in a Banach space setting extending the classes of fractionally integrated C-cosine functions and semigroups are systematically analyzed. Structural properties of such operator families are obtained. Relations between convoluted C-cosine functions and analytic convoluted C-semigroups, introduced and investigated in this paper are given through the convoluted version of the abstract Weierstrass formula which is also proved in the paper. Ultradistribution and hyperfunction sines are connected with analytic convoluted semigroups and ultradistribution semigroups. Several examples of operators generating convoluted cosine functions, (analytic) convoluted semigroups as well as hyperfunction and ultradistribution sines illustrate the abstract approach of the authors. As an application, it is proved that the polyharmonic operator , , acting on with appropriate boundary conditions, generates an exponentially bounded -convoluted cosine function, and consequently, an exponentially bounded analytic -convoluted semigroup of angle , for suitable exponentially bounded kernels and . [Copyright &y& Elsevier]

Abstract: In this paper, we develop an efficient preconditioning method on the basis of the modified hierarchy basis for solving the singular boundary value problem by the Galerkin method. After applying the preconditioning method, we show that the condition number of the linear system arising from the Galerkin method is uniformly bounded. In particular, the condition number of the preconditioned system will be bounded by 2 for the case (see Eq. (1) in the paper). Numerical results are presented to confirm our theoretical results. [Copyright &y& Elsevier]

Abstract: This paper deals with a class of boundary value problem of singular differential equations on time scales. The conditions we used here differ from those in the majority of papers as we know. An existence theorem of positive solutions is established by using the Krasnosel''skii fixed point theorem and an example is given to illustrate it. [Copyright &y& Elsevier]

Abstract: In this paper we are interested in the global existence and large-time behavior of solutions to the initial-boundary value problem for subcritical Kuramoto–Sivashinsky-type equationwhere the nonlinear term depends on the unknown function u and its derivative and satisfy the estimatewith such that The aim of this paper is to prove the global existence of solutions to the initial-boundary value problem (0.1) in subcritical case, when the nonlinear term has a time decay rate less than that of the linear terms of Eq. (0.1). Also we find the main term of the asymptotic representation of solutions. [Copyright &y& Elsevier]

Abstract: The paper deals with canard solutions at very general turning points of smooth singular perturbation problems in two dimensions. We follow a geometric approach based on the use of -normal forms, centre manifolds and (family) blow up, as we did in (Trans. Amer. Math. Soc., to appear). In (Trans. Amer. Math. Soc., to appear) we considered the existence of manifolds of canard solutions for given appropriate boundary conditions. These manifolds need not be smooth at the turning point. In this paper we essentially study the transition time along such manifolds, as well as the divergence integral, providing a structure theorem for these integrals. As a consequence we get a nice structure theorem for the transition equation, governing the canard solutions. It permits to compare different control manifolds and to obtain a precise description of the entry–exit relation of different canard solutions. Attention is also given to the special case in which the canard manifolds are smooth, i.e. when “formal” canard solutions exist. [Copyright &y& Elsevier]

Abstract: In this work a general class of nonlinear abstract equations satisfying a generalized strong maximum principle is considered in order to study the behavior of the bounded components of positive solutions bifurcating from the curve of trivial states at a nonlinear eigenvalue with geometric multiplicity one. Since the unilateral theorems of Rabinowitz (J. Funct. Anal. 7 (1971) 487, Theorems 1.27 and 1.40) are not true as originally stated (cf. the very recent counterexample of Dancer, Bull. London Math. Soc. 34 (2002) 533), in order to get our main results the unilateral theorem of López-Gómez (Spectral Theory and Nonlinear Functional Analysis, Research Notes in Mathematics, vol. 426, CRC Press, Boca Raton, FL, 2001, Theorem 6.4.3) is required. Our analysis fills some serious gaps existing is some published papers that were provoked by a direct use of Rabinowitz''s unilateral theory. Actually, the abstract theory developed in this paper cannot be covered with the pioneering results of Rabinowitz (1971), since in Rabinowitz''s context any component of positive solutions must be unbounded, by a celebrated result attributable to Dancer (Arch. Rational Mech. Anal. 52 (1973) 181). [Copyright &y& Elsevier]

Abstract: In this first part of a series of two papers, we present in considerable detail a collision-driven molecular dynamics algorithm for a system of non-spherical particles, within a parallelepiped simulation domain, under both periodic or hard-wall boundary conditions. The algorithm extends previous event-driven molecular dynamics algorithms for spheres, and is most efficient when applied to systems of particles with relatively small aspect ratios and with small variations in size. We present a novel partial-update near-neighbor list (NNL) algorithm that is superior to previous algorithms at high densities, without compromising the correctness of the algorithm. This efficiency of the algorithm is further increased for systems of very aspherical particles by using bounding sphere complexes (BSC). These techniques will be useful in any particle-based simulation, including Monte Carlo and time-driven molecular dynamics. Additionally, we allow for a non-vanishing rate of deformation of the boundary, which can be used to model macroscopic strain and also alleviate boundary effects for small systems. In the second part of this series of papers we specialize the algorithm to systems of ellipses and ellipsoids and present performance results for our implementation, demonstrating the practical utility of the algorithm. [Copyright &y& Elsevier]

In computations, it is now common to surround artificial boundaries of a computational domain with a perfectly matched layer (PML) of finite thickness in order to prevent artificially reflected waves from contaminating a numerical simulation. Unfortunately, the PML does not give us an indication about appropriate boundary conditions needed to close the edges of the PML, or how those boundary conditions should be enforced in a numerical setting. Terminating the PML with an inappropriate boundary condition or an unstable numerical boundary procedure can lead to exponential growth in the PML which will eventually destroy the accuracy of a numerical simulation everywhere. In this paper, we analyze the stability and the well-posedness of boundary conditions terminating the PML for the elastic wave equation in first order form. First, we consider a vertical modal PML truncating a two space dimensional computational domain in the horizontal direction. We freeze all coefficients and consider a left half-plane problem with linear boundary conditions terminating the PML. The normal mode analysis is used to study the stability and well-posedness of the resulting initial boundary value problem (IBVP). The result is that any linear well-posed boundary condition yielding an energy estimate for the elastic wave equation, without the PML, will also lead to a well-posed IBVP for the PML. Second, we extend the analysis to the PML corner region where both a horizontal and vertical PML are simultaneously active. The challenge lies in constructing accurate and stable numerical approximations for the PML and the boundary conditions. Third, we develop a high order accurate finite difference approximation of the PML subject to the boundary conditions. To enable accurate and stable numerical boundary treatments for the PML we construct continuous energy estimates in the Laplace space for a one space dimensional problem and two space dimensional PML corner problem. We use summation-by-parts finite difference operators to approximate the spatial derivatives and impose boundary conditions weakly using penalties. In order to ensure numerical stability of the discrete PML, it is necessary to extend the numerical boundary procedure to the auxiliary differential equations. This is crucial for deriving discrete energy estimates analogous to the continuous energy estimates. Numerical experiments are presented corroborating the theoretical results. Moreover, in order to ensure longtime numerical stability, the boundary condition closing the PML, or its corresponding discrete implementation, must be dissipative. Furthermore, the numerical experiments demonstrate the stable and robust treatment of PML corners. [ABSTRACT FROM AUTHOR]

Abstract: In this paper there are established some analogues of the Hilbert formulas on the unit sphere for the theory of time-harmonic (monochromatic) relativistic Dirac bispinors. The formulas relate a pair of the components of the limit value of a time-harmonic Dirac bispinor in the unit ball to the other pair of components. The obtained results are based on the intimate connection between time-harmonic solutions of the relativistic Dirac equation and the three-dimensional α-hyperholomorphic function theory. Hilbert formulas for α-hyperholomorphic function theory for α being a complex quaternionic number are also presented, such formulas relate a pair of components of the boundary value of an α-hyperholomorphic function in the unit ball to the other pair of components, in an analogy with what is known for the case of the theory of functions of one complex variable. [Copyright &y& Elsevier]

Abstract: In this paper, we present analytical characterizations of Mazurʼs intersection property (MIP), the CIP and the MIP⁎ via a specific class of convex functions and their conjugates. More precisely, let X be a Banach space and be its dual. Then X has the MIP if and only if for every extended real-valued lower semi-continuous convex function f defined on X with bounded domain, f is the supremum of all functions of the form: for some and , . And X has the CIP if and only if for every extended real-valued lower semi-continuous convex function on X with relatively compact domain, is the infimum of all functions which are of the form: [Copyright &y& Elsevier]

Abstract: In this paper we consider the solutions of micropolar fluid equations in space dimension two with periodic boundary condition. We show that the strong solutions are analytic in time with values in an appropriate Gevrey class of function, provided that external forces and moments are time-independent and are in a Gevrey class. [Copyright &y& Elsevier]

Abstract: Based on the complex variable function method, a new approach for solving the scattering of plane P wave by circular-arc alluvial valley in poroelastic half-space is developed in the paper. In this analysis, the poroelastic half-space and the circular-arc valley are modeled as poroelastic medium based on Biot''s dynamic theory. By introducing three potentials, the governing equations for Biot''s theory are reduced to three Helmholtz equations. The series solutions of the Helmholtz equations are obtained by the wave function expansion method. Here, the large circle assumption is applied to simulate the boundary conditions at the half-space boundary. The stresses and pore pressures are obtained by using the boundary conditions and continuous conditions of the poroelastic half-space and the circular-arc alluvial valley. Numerical results show that the dynamic stresses concentration and pore pressures concentration are mainly relative to the wave shape of incidence, angle of incidence, dimensionless frequency of incident wave, stiffness and pore ratio of the poroelastic half-space and valley. [Copyright &y& Elsevier]

Abstract: We address the analysis of a nonlinear and degenerating PDE system, proposed by M. Frémond for modelling phase transitions in viscoelastic materials subject to thermal effects. The system features an internal energy balance equation, governing the evolution of the absolute temperature ϑ, an evolution equation for the phase change parameter χ, and a stress–strain relation for the displacement variable u. The main novelty of the model is that the equations for χ and u are coupled in such a way as to take into account the fact that the properties of the viscous and of the elastic parts influence the phase transition phenomenon in different ways. However, this brings about an elliptic degeneracy in the equation for u which needs to be carefully handled. In this paper, we first prove a local (in time) well-posedness result for (a suitable initial–boundary value problem for) the above mentioned PDE system, in the (spatially) three-dimensional setting. Secondly, we restrict to the one-dimensional case, in which, for the same initial–boundary value problem, we indeed obtain a global well-posedness theorem. [Copyright &y& Elsevier]

Abstract: In this paper, we establish the global attractivity of the positive steady state of the diffusive Nicholson''s equation with homogeneous Neumann boundary value under a condition that makes the equation a non-monotone dynamical system. To achieve this, we develop a novel method: combining a dynamical systems argument with maximum principle and some subtle inequalities. [Copyright &y& Elsevier]

Abstract: This paper presents and compares various numerical techniques for the long-wave short-wave interaction equations. In addition to the standard explicit, implicit schemes and the spectral methods, a novel scheme SRK which is based on a time-splitting approach combined with the Runge–Kutta method is presented. We demonstrate that not only the SRK scheme is efficient compared to the split step spectral methods, but it can apply directly to problems with general boundary conditions. The conservation properties of the numerical schemes are discussed. Numerical simulations are reported for case studies with different types of initial data. The present study enhances our understanding of the behavior of nonlinear dispersive waves in the semi-classical limit. [Copyright &y& Elsevier]

Abstract: In this paper, we apply trigonometric wavelet method to investigate the numerical solution of elliptic boundary value problem in an exterior unit disk. The simple computation formulae of the entries in the stiffness matrix are obtained. It shows that we only need to compute elements of one stiffness matrix. Moreover, the error estimates of the approximation solutions are given and some test examples are presented. [Copyright &y& Elsevier]

Abstract: This paper deals with correctness of initial boundary value problems for general dispersive equations of finite odd orders. For the Kawahara and KdV equations we prove existence, uniqueness and stability of strong global solutions in a bounded domain for different signs of a coefficient of the highest derivative as well as their asymptotics when the coefficient of the higher-order derivative in the Kawahara equation approaches zero. [Copyright &y& Elsevier]

Abstract: In this paper, we consider the following p-Laplacian multipoint boundary value problem on time scales:where , , , . By using fixed point index, we provide some sufficient conditions for the existence of multiple positive solutions to the above boundary value problem. Especially, the nonlinear term is allowed to change sign. As an application, an example is given to demonstrate our result. [Copyright &y& Elsevier]

Abstract: The method of fundamental solutions (MFS) is a popular tool to solve Laplace and Helmholtz boundary value problems. Its main drawback is that it often leads to ill-conditioned systems of equations. In this paper, we investigate for the interior Helmholtz problem on analytic domains how the singularities (charge points) of the MFS basis functions have to be chosen such that approximate solutions can be represented by the MFS basis in a numerically stable way. For Helmholtz problems on the unit disc we give a full analysis which includes the high frequency (short wavelength) limit. For more difficult and nonconvex domains such as crescents we demonstrate how the right choice of charge points is connected to how far into the complex plane the solution of the boundary value problem can be analytically continued, which in turn depends on both domain shape and boundary data. Using this we develop a recipe for locating charge points which allows us to reach error norms of typically on a wide variety of analytic domains. At high frequencies of order only 3 points per wavelength are needed, which compares very favorably to boundary integral methods. [Copyright &y& Elsevier]

Abstract: Let Ω be a bounded domain in , , with the boundary . We consider the following singularly perturbed nonlinear elliptic problem on Ω where ν is an exterior normal to ∂Ω and a nonlinearity f of subcritical growth. Under rather strong conditions on f, it has been known that for small , there exists a solution of the above problem which exhibits a spike layer near local maximum points of the mean curvature H on ∂Ω as . In this paper, we obtain the same result under some conditions on f (Berestycki–Lions conditions), which we believe to be almost optimal. [Copyright &y& Elsevier]

Abstract: In this paper we give a simple proof of well-posedness of multidimensional scalar conservations laws with a strong boundary condition. The proof is based on a result of strong trace for solutions of scalar conservation laws and kinetic formulation. [Copyright &y& Elsevier]

Abstract: The aim of this paper consists in to give sufficient conditions to ensure the existence and location of the solutions of a nonlinear fully fourth-order equation with functional boundary conditions. The arguments make use of the upper and lower solutions method, a ϕ-Laplacian operator and a fixed point theorem. An application of the beam theory to a nonlinear continuous model of the human spine allows to estimate its deformation under some loading forces. [Copyright &y& Elsevier]

Abstract: Let be near to 1 and be an RD-space, which includes any Carnot–Carathéodory space with a doubling measure. In this paper, the authors prove that a sublinear operator T extends to a bounded sublinear operator from Hardy spaces to some quasi-Banach space if and only if T maps all -atoms into uniformly bounded elements of . [Copyright &y& Elsevier]

Abstract: In this paper, the existence of positive solutions for a nonlinear general discrete boundary value problem is established. Such results extend and improve some known facts for the two-point and three-point boundary value problems. Particularly, the boundary value conditions can be nonlinear and the method is new. For explaining the main results, some numerical examples are also given. [Copyright &y& Elsevier]

Abstract: In this paper we establish existence-uniqueness of solution of a class of singular boundary value problem for and , , where satisfies (i) in , (ii) , and for some , (iii) is analytic in and satisfies (i) in , (ii) and for some , (iii) is analytic in with quite general conditions on . Region for multiple solutions have also been determined. [Copyright &y& Elsevier]

Abstract: In this paper, first we show that an analogy of the Christoffel–Darboux formula holds for the zonal spherical functions for the Gelfand pair . Next, making use of it, we deal with the problem on point-wise convergence of Fourier expansion by means of the zonal spherical functions for . [Copyright &y& Elsevier]

Abstract: In this paper, a method of separating variables is effectively implemented for solving a time-fractional telegraph equation (TFTE). We discuss and derive the analytical solution of the TFTE with three kinds of nonhomogeneous boundary conditions, namely, Dirichlet, Neumann and Robin boundary conditions. [Copyright &y& Elsevier]

Abstract: In this paper operator-valued multiplier theorems in Banach-valued weighted spaces are studied. Also weighted Sobolev–Lions type spaces are discussed when , E are two Banach spaces and is continuously and densely embedded on E. Several conditions are found that ensure the continuity of the embedding operators that are optimally regular in these spaces in terms of interpolations of . These results permit us to show the separability of the anisotropic differential operators in an E-valued weighted space. By using these results the maximal regularity of infinite systems of quasi elliptic partial differential equations are established. [Copyright &y& Elsevier]

Abstract: In this paper, exact number of solutions are obtained for the one-dimensional p-Laplacian in a class of two-point boundary value problems. The interesting point is that the nonlinearity f is general form: . Meanwhile, some properties of the solutions are given in details. The arguments are based upon a quadrature method. [Copyright &y& Elsevier]

Abstract: In this paper we obtain local , , gradient estimates for weak solutions of elliptic equations of p-Laplacian type with small BMO coefficients. [Copyright &y& Elsevier]

Abstract: This paper deals with the uniqueness of positive radial solutions to Dirichlet problems on annular domains in , . As an application we can obtain the results to equation , where , , and . [Copyright &y& Elsevier]

Abstract: This paper studies the operator acting on q-forms on an unbounded domain with smooth boundary, where d is the exterior derivative and is the adjoint of d calculated using the Sobolev space topology. The domain of is determined and an expression for is obtained. The operator gives rise to a boundary value problem. Global regularity is obtained using weighted norms and global existence is obtained by using the theory of compact operators. [Copyright &y& Elsevier]

Abstract: In [C. Amrouche, V. Girault, J. Giroire, Dirichlet and Neumann exterior problems for the n-dimensional Laplace operator. An approach in weighted Sobolev spaces, J. Math. Pures Appl. 76 (1997) 55–81], authors study Dirichlet and Neumann problems for the Laplace operator in exterior domains of . This paper extends this study to the resolution of a mixed exterior Laplace''s problem. Here, we give existence, unicity and regularity results in ''s theory with , in weighted Sobolev spaces. [Copyright &y& Elsevier]

Abstract: This paper deals with the solvability and uniqueness of the second-order three-point boundary value problems at resonance on a half-line and where , are continuous and . By using the coincidence degree theory, we establish some existence and uniqueness criteria. [Copyright &y& Elsevier]

Abstract: In this paper, we consider the existence of positive solutions for the singular fourth-order p-Laplacian equation with the four-point boundary conditions where , , , may be singular at and/or 1 and . By using the upper and lower solution method and fixed-point theorems, the existence of positive solutions to the above the boundary value problem is obtained. [Copyright &y& Elsevier]

Abstract: In this paper, we discuss the limit behaviour of optimal control problems governed by elliptic boundary value problems with equivalued surface when the equivalued surface boundary shrinks to a fixed point on the outer boundary of a bounded domain. [Copyright &y& Elsevier]

Abstract: This paper deals with the second order three-point boundary value problem The existence and multiplicity of positive solutions are investigated by means of the fixed point theorem in cones. [Copyright &y& Elsevier]

Abstract: In this paper, we provide sufficient conditions for the existence of at least three solutions to a three-point boundary value problem for higher-order ordinary differential equations. The nonlinear term f in the differential equation under consideration may depend on higher-order derivatives of arbitrary order and this is where the main novelty of this work lies. By applying the two pairs of upper and lower solutions method of Henderson and Thompson, as well as degree theory, the existence of at least three solutions of the problem is given. [Copyright &y& Elsevier]

Abstract: This paper concerns hyperbolic systems of two linear first-order PDEs in one space dimension with periodicity conditions in time and reflection boundary conditions in space. The coefficients of the PDEs are supposed to be time independent, but allowed to be discontinuous with respect to the space variable. We construct two scales of Banach spaces (for the solutions and for the right-hand sides of the equations, respectively) such that the problem can be modeled by means of Fredholm operators of index zero between corresponding spaces of the two scales. [Copyright &y& Elsevier]

Abstract: In this paper we consider two different initial–boundary value problems in temperature dependent viscous flow when the temperature equation has a nonlinear heat source term. When blow-up occurs we derive lower bounds for the blow-up time in each case. [Copyright &y& Elsevier]

Abstract: In this paper, a food chain model with ratio-dependent functional response is studied under homogeneous Neumann boundary conditions. The large time behavior of all non-negative equilibria in the time-dependent system is investigated, i.e., conditions for the stability at equilibria are found. Moreover, non-constant positive steady-states are studied in terms of diffusion effects, namely, Turing patterns arising from diffusion-driven instability (Turing instability) are demonstrated. The employed methods are comparison principle for parabolic problems and Leray–Schauder Theorem. [Copyright &y& Elsevier]

Abstract: In this paper we consider the one-dimensional p-Laplacian boundary value problem on time scales where is p-Laplacian operator, i.e., , . Some new results are obtained for the existence of at least single, twin or triple positive solutions of the above problem by using Krasnosel''skii''s fixed point theorem, new fixed point theorem due to Avery and Henderson and Leggett–Williams fixed point theorem. This is probably the first time the existence of positive solutions of one-dimensional p-Laplacian boundary value problems on time scales has been studied. [Copyright &y& Elsevier]

Abstract: The present paper is devoted to the study of a boundary value problem for abstract first order linear differential equation with integral boundary conditions. We obtain necessary and sufficient conditions for the unique solvability and well-posedness. We also study the Fredholm solvability. Finally, we obtain a result of the stability of solution with respect to small perturbation. [Copyright &y& Elsevier]

Abstract: Let ϕ and θ be two increasing homeomorphisms from onto with , . Let be a function satisfying Carathéodory''s conditions, and for each i, , let , be a continuous function, with , , . In this paper we first prove a suitable continuation lemma of Leray–Schauder type which we use to obtain several existence results for the m-point boundary value problem: We note that this problem is at resonance, in the sense that the associated m-point boundary value problem has the non-trivial solution , where is an arbitrary constant vector, in view of the assumption . [Copyright &y& Elsevier]