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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 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: Let X be a Banach space whose dual space is uniformly convex. We demonstrate that, for any demicontinuous, weakly Nagumo, k-pseudo-contractive mapping with closed domain, weakly generates a semigroup on . In this paper, we project the consequences of this result on fixed point theory. In particular, we show that if (id est, if T is strongly pseudo-contractive), then T has a unique fixed point. This implies that, if T is pseudo-contractive () and is closed, bounded, and convex, then T has at least one fixed point. Consequently, any demicontinuous pseudo-contractive mapping (for an appropriate C) has a fixed point, which has been an important open question in fixed point theory for quite some time. In a subsequent paper, we explore the consequences of the semigroup result on the existence of solutions to certain partial differential equations. The semigroup result directly implies the existence of unique global solutions to time evolution equations of the form where A is a combination of derivatives. The fixed point results from this paper imply the existence of solutions to partial differential equations of the form . [Copyright &y& Elsevier]

Abstract: In this paper, we study the growth of solutions of a first-order linear differential equation and that of a second-order linear differential equation. From this we obtain some uniqueness theorems of a nonconstant entire function and its first derivative having the same fixed points with the same multiplicities. The results in this paper also improve some known results. Some examples show that the results in this paper are best possible. [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: 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 paper we characterize the relation between two entire functions which are permutable and satisfy certain algebraic differential equations. [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: In this paper we provide a new approach for the derivation of parameterizations for the Eisenstein series. We demonstrate that a variety of classical formulas may be derived in an elementary way, without knowledge of the inversion formulae for the corresponding Schwarzian triangle functions. In particular, we provide a new derivation for the parametric representations of the Eisenstein series in terms of the complete elliptic integral of the first kind. The proof given here is distinguished from existing elementary proofs in that we do not employ the Jacobi–Ramanujan inversion formula relating theta functions and hypergeometric series. Our alternative approach is based on a Lie symmetry group for the differential equations satisfied by certain Eisenstein series. We employ similar arguments to obtain parameterizations from Ramanujan''s alternative signatures and those associated with the inversion formula for the modular J-function. Moreover, we show that these parameterizations represent the only possible signatures under a certain assumed form for the Lie group parameters. [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: In this paper we consider the stability of the inverse problem of determining a function in a wave equation in a bounded smooth domain in from boundary observations. This information is enclosed in the hyperbolic (dynamic) Dirichlet-to-Neumann map associated to the solutions to the wave equation. We prove in the case of that is uniquely determined by the range restricted to a subboundary of the Dirichlet-to-Neumann map whose stability is a type of double logarithm. [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: In this paper we prove an existence and uniqueness theorem for solving the operator equation , where F is a Gateaux differentiable continuous operator while the operator G satisfies a Lipschitz-condition on an open convex subset of a Banach space. As corollaries, a theorem of Tapia on a weak Newton''s method and the classical convergence theorem for modified Newton-iterates are deduced. An existence theorem for a generalized Euler–Lagrange equation in the setting of Sobolev space is obtained as a consequence of the main theorem. We also obtain a class of Gateaux differentiable operators which are nowhere Frechet differentiable. Illustrative examples are also provided. [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 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: 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: 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: Let X be a real Banach space, an m-accretive operator and a multi-function which is 2π-periodic with respect to its first argument, has nonempty, closed, convex and weakly compact values and is strongly–weakly upper semicontinuous. In this paper we prove the existence of at least one solution for the problem in the case in which F satisfies an appropriate “sign” condition. [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: The Tonelli existence theorem in the calculus of variations and its subsequent modifications were established for integrands f which satisfy convexity and growth conditions. In this paper we consider a large class of optimal control problems which is identified with a complete metric space of integrands without convexity assumptions and show that for a generic integrand the corresponding optimal control problem possesses a unique solution and this solution is Lipschitzian. [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: The paper is concerned with the delay differential equation satisfying for and , where denotes the Riemann–Stieltjes integral. By applying the fixed point theorem in cones, we show the relationship between the asymptotic behaviors of the quotient (at zero and infinity) and the open intervals (eigenvalue intervals) of the parameter λ such that the problem has zero, one and two positive solution(s). If , by using a property of the Riemann–Stieltjes integral, the above nonlocal boundary value problem educes a three-point boundary value problem with delay, for which some similar results are established. [Copyright &y& Elsevier]

Abstract: In this paper we study the existence of solutions for Lidstone boundary value problems on time scale. Firstly, by using Schauder fixed point theorem in a cone, we obtain the existence of solutions to a Lidstone boundary value problem (LBVP). Secondly, existence result for this problem is also given by the monotone method. Finally, by using Krasnosel''skii fixed point theorem, it is proved that the LBVP has a positive solution. [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: In this paper, variational conclusions of set-valued bifunctions on convex subsets of Banach spaces are investigated and several new results are obtained. The results are applied to the fixed point theory and variational inequalities. We obtain two fixed point theorems and existence theorems of solutions of variational inequalities. [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]

Abstract: Necessary and sufficient conditions are obtained for the existence of symmetric positive solutions to the boundary value problem Applications of our results to the special case where f is a power function of u and its derivatives are also discussed. Moreover, similar conclusions for a more general higher order boundary value problem are established. Our results extend some recent work in the literature on boundary value problems for ordinary differential equations. The analysis of this paper mainly relies on fixed point theorems on cones. [Copyright &y& Elsevier]

Abstract: In this paper we study the initial boundary value problem for fourth order wave equations with nonlinear strain and source terms. First we introduce a family of potential wells and prove the invariance of some sets and vacuum isolating of solutions. Then we obtain a threshold result of global existence and nonexistence. Finally we discuss the global existence of solutions for the problem with critical initial condition , . So the Esquivel-Avila''s results are generalized and improved. [Copyright &y& Elsevier]

Abstract: Asymptotic stability by Lyapunov of the steady-state solution to the linear initial-boundary value problem which is formulated in the first part of this paper [A.M. Blokhin, D.L. Tkachev, L.O. Baldan, Well-posedness of a modified initial-boundary value problem on stability of shock waves in a viscous gas. Part I, J. Math. Anal. Appl. 331 (1) (2007) 408–423 (this issue)] proved under two conditions. The main of these conditions is that zeros of Lopatinsky determinant excluding , lie in the left half-plane. The problem arises while providing grounds for replacing of thin transitional zones of strong gradients of basic flow parameters for viscous heat-conducting gas with surfaces of strong discontinuity. [Copyright &y& Elsevier]

Abstract: This paper studies the existence of extremal solutions for a class of singular boundary value problems of second order impulsive differential equations. By using the method of upper and lower solutions and the monotone iterative technique, criteria of the existence of extremal solutions are established. [Copyright &y& Elsevier]

Abstract: In this paper, the global solutions of initial value problems for nonlinear second-order integro-differential equations of mixed type in Banach spaces are investigated. The existence and uniqueness of solutions and their iterative approximation are obtained by using Mönch fixed point theorem and new comparison results. The results presented here essentially improve, generalize and unify many well-known results. [Copyright &y& Elsevier]

Abstract: The Ginzburg–Landau equation has been used as a simplified mathematical model for various pattern formation systems in mechanics, physics and chemistry. In this paper, we study a generalized complex Ginzburg–Landau equation in two spatial dimension with a fifth order nonlinear term and cubic terms containing spatial derivatives. We prove a global existence and uniqueness theorem. [Copyright &y& Elsevier]

Abstract: In this paper, we are concerned with the existence of positive solutions to a n-point nonhomogeneous boundary value problem. By using the Krasnoselskii''s fixed point theorem in Banach spaces, some sufficient conditions guaranteeing the existence of positive solution is established for the n-point nonhomogeneous boundary value problem. [Copyright &y& Elsevier]

Abstract: This work is a continuation of our previous work [Z.-Q. Shao, D.-X. Kong, Y.-C. Li, Shock reflection for general quasilinear hyperbolic systems of conservation laws, Nonlinear Anal. TMA 66 (1) (2007) 93–124]. In this paper, we study the global structure instability of the Riemann solution containing shocks, at least one rarefaction wave for general quasilinear hyperbolic systems of conservation laws in the presence of a boundary. We prove the nonexistence of global piecewise solution to a class of the mixed initial-boundary value problem for general quasilinear hyperbolic systems of conservation laws on the quarter plane. Our result indicates that this kind of Riemann solution mentioned above for general quasilinear hyperbolic systems of conservation laws in the presence of a boundary is globally structurally unstable. Some applications to quasilinear hyperbolic systems of conservation laws arising from physics and mechanics are also given. [Copyright &y& Elsevier]

Abstract: In this paper, the three-point boundary value problem and its conjugate boundary value problem are studied under some conditions concerning the same first eigenvalue of the both linear problems. By applying the fixed point index theory, the coexistence of single and multiple positive solutions of the above mentioned problems is verified. As an application, some examples are given to illustrate our results. [Copyright &y& Elsevier]

Abstract: In this paper, the second-order four-point boundary value problem is studied, where , , and , are nonnegative continuous functions. By the use of the property of the corresponding Green''s function, fixed-point index theory, Leray–Schauder degree and upper and lower solution method, some existence, nonexistence, and multiplicity results of positive solutions are acquired. [Copyright &y& Elsevier]