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This paper provides a generalization of the main lemma and a correct proof of an extended version of Theorem 2, in [G. Vidossich, Differential inequalities for evolution equations, Nonlinear Anal. TMA 25 (1995) 1063–1069].

We prove a theorem for the existence of solutions to a variational problem, under assumptions that do not require the convexity of the integrand. © 2006 Elsevier Ltd. All rights reserved.

Our aim is to point out an error in the proof of Lemma 2 in the paper “Fixed point theory in symmetric spaces with applications to probabilistic spaces” by T.L. Hicks, B.E. Rhoades [Nonlinear Analysis 36 (1999) 331–344], and to indicate a way to repair it.

In this note, we will modify several gaps in Chidume and Ofoedu [C.E. Chidume, E.U. Ofoedu, A new iteration process for generalized Lipschitz pseudo-contractive and generalized Lipschitz accretive mappings, Nonlinear Anal. (2006), in press ( doi:10.1016/j.na.2006.05.012 )].

THROUGHOUT this note, X denotes a complete metric space with distance function d, and T is a map from X into itself. Powers of T are defined by Tax = x and T”+ lx = T(T”x), n 2 0. Occasionally, we use the notation xk = Tkx, in particular x0 = x, x1 = TX, for the sake of brevity. The set {x, TX, T'x, . . . , T”x,. . .} is called an orbit (starting at x) and denoted by O(x). The set 0(x, y) is the union of the two orbits starting at x and y, 0(x, y) = {xk, yk: k 3 O}. The diameter diam A of a nonempty subset A of X is the supremum of the numbers d(x, y), where x and y vary in A. The letter 4 denotes a contractive gauge function, i.e. a continuous, increasing function from R,, the non-negative reals, into R, which satisfies 4(s) 0.

This paper discusses a rearrangement minimization problem related to a boundary value problem where the differential operator is the p -Laplacian and the external force is indefinite. By applying the rearrangement theory established by G. R. Burton, we show that the minimization problem has a unique solution in the weak closure of the admissible set in an appropriate function space. What makes this paper different from many others in the literature is that the admissible set here is allowed to be indefinite (takes on both positive and negative values). We show that in this case the optimal solution will be a weak limit point of the admissible set and cannot be a rearrangement element itself. Optimality conditions are derived based on the mean value of the generator. In case the generator is a three valued function, we shall show that the optimality conditions lead to free boundary value problems that are of interest independently. We also discuss a minimization problem where the admissible set is not seemingly related to rearrangement classes but we shall prove otherwise, and apply the earlier results of the current paper to draw interesting conclusions regarding the optimal solution. We have also discussed the case where the domain of interest is radial, and shown that the optimal solutions are radial as well.

We consider the nonlinear boundary value problem consisting of the equation − u ″ = f ( u ) , on ( − 1 , 1 ) , where f : R → R is continuous, together with general Sturm–Liouville type, multi-point boundary conditions at ± 1 . We obtain Rabinowitz-type global bifurcation results, and then use these to obtain ‘nodal’ solutions of the problem. We conclude with a nonresonance result for an inhomogeneous form of the problem. These results rely on the spectral properties of the eigenvalue problem consisting of the equation − u ″ = λ u , on ( − 1 , 1 ) , together with the multi-point boundary conditions. In a previous paper it was shown that, under certain ‘optimal’ conditions, the basic spectral properties of this eigenvalue problem are similar to those of the standard Sturm–Liouville problem with single-point boundary conditions. In particular, for each integer k ⩾ 0 there exists a unique, simple eigenvalue λ k , whose eigenfunctions have ‘oscillation count’ equal to k , where the ‘oscillation count’ was defined in terms of a complicated Prufer angle construction. Unfortunately, it seems to be difficult to apply the Prufer angle construction to the nonlinear problem. Accordingly, in this paper we use alternative, non-optimal, oscillation counting methods to obtain the required spectral properties of the linear problem, and these are then applied to the nonlinear problem to yield the results mentioned above.

In this paper, we discuss a critical exponent with respect to the stability of solutions to an Henon type equation on the hyperbolic space. In Euclidean space, there exists a critical exponent on stable solutions of an Henon type equation. In particular, this exponent is called the Joseph–Lundgren exponent in case of the Lane–Emden equation. However, on the hyperbolic space, the Lane–Emden equation admits no such critical exponent on stability. We devote this paper to showing the existence of a critical exponent of Joseph–Lundgren type for a weighted Lane–Emden equation. More precisely, we prove non-existence of non-trivial stable solutions of the Henon type equation for the subcritical case. Moreover, we show that the Henon type equation has stable, positive, and radial solutions for the supercritical case.

In this survey paper, by using variational methods, we are concerned with the qualitative analysis of solutions to nonlinear elliptic problems of the type { − div A ( x , ∇ u ) = λ | u | q ( x ) − 2 u in Ω u = 0 on ∂ Ω , where Ω is a bounded or an exterior domain of R N and q is a continuous positive function. The results presented in this paper extend several contributions concerning the Lane–Emden equation and we focus on new phenomena which are due to the presence of variable exponents.

Let X ⊂ R n be a bounded Lipschitz domain and consider the σ 2 -energy functional F σ 2 [ u ; X ] : = ∫ X | ∇ u ∧ ∇ u | 2 d x , over the space of admissible maps A ( X ) = { u ∈ W 1 , 4 ( X , R n ) : u | ∂ X = x , det ∇ u = 1 for L n -a.e. in X } . A good measure of how much a map u stretches areas (of 2 -dimensional sub-manifolds of the domain X ) is the norm of ∇ u ∧ ∇ u : ∧ 2 T X → ∧ 2 T X , analogously to | ∇ u | 2 (the Dirichlet energy density) that is a measure of length’s stretching. These kinds of functionals also were arisen as a physical model describing the strong interactions of quantum fields which was introduced by T. Skyrme in 1961. In this paper we introduce a class of maps referred to as generalised twists and examine them in connection with the Euler–Lagrange equations associated with F σ 2 [ ⋅ ; X ] over A ( X ) . In particular we present a novel characterisation of all twist solutions and this points at a surprising discrepancy between even and odd dimensions which follows very closely to the ideas that have been introduced by the second author in his recent paper Shahrokhi-Dehkordi and Taheri (2009) [17] . Indeed we show that in even dimensions the latter system of equations admits infinitely many smooth solutions, modulo isometries, amongst such maps. In odd dimensions this number reduces to one. The result relies on a careful analysis of the full versus the restricted Euler–Lagrange equations. We investigate various qualitative properties of these solutions in view of a remarkably interesting previously unknown explicit formula.

In this paper, we present some new results of homoclinic solutions for second-order Hamiltonian systems u − λ L ( t ) u + W u ( t , u ) = 0 ; here λ > 0 is a parameter, L ∈ C ( R , R N × N ) and W ∈ C 1 ( R × R N , R ) . Unlike most other papers on this problem, we require that L ( t ) is a positive semi-definite symmetric matrix for all t ∈ R , that is, L ( t ) ≡ 0 is allowed to occur in some finite interval T of R . Under some suitable assumptions on W , we prove the existence of two different homoclinic solutions u λ ( 1 ) , u λ ( 2 ) , which both vanish on R ∖ T as λ → ∞ , and converge to u 0 ( 1 ) , u 0 ( 2 ) in H 1 ( R ) , respectively; here u 0 ( 1 ) ≠ u 0 ( 2 ) ∈ H 0 1 ( T ) are two nontrivial solutions of the Dirichlet BVP for Hamiltonian systems on the finite interval T .

In this paper we consider the second-order nonlinear differential equation ( ∗ ) ( t α − 1 Φ ( x ′ ) ) ′ + t α − 1 − p f ( x ) = 0 , Φ ( x ) = | x | p − 2 x , p > 1 , α ∈ R , with f satisfying x f ( x ) > 0 , x ≠ 0 . We analyze the difference between the cases α p , α > p , and α = p . In each case we give a condition on the function f which guarantees that solutions of Eq. ( ∗ ) are (non)oscillatory. The principal methods used in this paper are the Riccati technique and its modifications. The results of our paper complement and extend several previously obtained results on the subject.

In this paper, we study the pointwise convergence of the function e i t Δ α 2 f ( x ) whenever f belongs to some Sobolev space. When α = 2 , e i t Δ f ( x ) is the solution to the standard Schrodinger equation and this problem has been widely studied. Here we focus on the case 0 α ≠ 1 . The same convergence problem for the nonelliptic Schrodinger equation is also discussed. The proof here follows the idea of Carleson’s first paper on this subject. But we do reprove some known theorems by this straightforward method and get some new results as well.

In the previous papers Nakauchi and Takenaka (2011) and Kawai and Nakauchi (2011), the first author introduced a stationary map for the functional of L 2 -norm of pullbacks of metrics. We call it a symphonic map. In this paper using the argument of Xu and Yang (1993) (see also Smith (1975) and Ding (1988)), we give a notion of the symphonic join of two symphonic maps between the standard spheres.

In this paper, we investigate the long-time behavior and solvability of the reaction-diffusion equation, which has a polynomial growth nonlinearity of arbitrary order, with Robin boundary condition. We begin this paper with the existence and uniqueness of the solution to the problem. For the long-time behavior, we firstly prove the existence of an absorbing set in two different spaces. Secondly, for the autonomous case, the existence of a global attractor is obtained in W-2(1) (Omega) boolean AND L rho+2 (Omega). (C) 2014 Elsevier Ltd. All rights reserved.

In this paper we consider extended stationary mean-field games, that is mean-field games which depend on the velocity field of the players. We prove various a-priori estimates which generalize the results for quasi-variational mean-field games in Gomes et al. (2012). In addition we use adjoint method techniques to obtain higher regularity bounds. Then we establish the existence of smooth solutions under fairly general conditions by applying the continuity method. When applied to standard stationary mean-field games as in Lasry and Lions (2006), Gomes and Sanchez-Morgado (2011) or Gomes et al. (2012) this paper yields various new estimates and regularity properties not available previously. We discuss additionally several examples where the existence of classical solutions can be proved.

In two and three space dimensions, and under suitable assumptions on the initial data, we show global existence for a damped wave equation which approaches, in some sense, the Navier-Stokes problem. The proofs are based on a refined energy method. In this paper, we improve the results in two papers by Y. Brenier, R. Natalini and M. Puel and by M. Paicu and G. Raugel. We relax the regularity of the initial data of the former, even though we still use energy methods as a principal tool. Regarding the second paper, the improvement consists in the simplicity of the proofs since we do not use any Strichartz estimate and in requiring less regularity for the convergence to the Navier-Stokes problem. Indeed, the convergence result we obtain is near-optimal regularity., Comment: 30 pages, submitted (May 2012)

The aim of this paper is to improve the results on stability in distribution of nonlinear stochastic differential equations in Yuan and Mao (2003). Both conditions for the stability in distribution given in that paper are weakened.

In this paper, consideration is given to the 3-coupled nonlinear Schrodinger system i ∂ ∂ t u j + ∂ 2 ∂ x 2 u j + ∑ i = 1 3 b i j | u i | 2 u j = 0 , where u j are complex-valued functions of ( x , t ) ∈ R 2 , j = 1 , 2 , 3 , and b i j are positive constants satisfying b i j = b j i . It will be shown first that if the symmetric matrix B = ( b i j ) satisfies certain conditions, then ground-state solutions of the 3-coupled nonlinear Schrodinger system exist, and moreover, they are orbitally stable. The theory is then extended to include solitary waves as well. In particular, it will be shown that when a solitary wave is perturbed, the perturbed solution must stay close to a solitary-wave profile in which the translation and phase parameters are prescribed functions of time. Properties of these functions are then studied. This is a continuous work of our previous paper where the 2-coupled nonlinear Schrodinger system was considered.

If S denotes the class of functions h ( z ) = z + ∑ n = 2 ∞ a n z n which are analytic and univalent in the unit disk | z | 1 , then the classical result of Szego shows that every section s n ( h ) ( z ) = ∑ k = 1 n a k z k of h is univalent in | z | 1 / 4 . Exact (largest) radius of univalence r n of s n ( h ) remains an open problem, although the corresponding results for sections of various geometric subclasses of S have been obtained which include those h ∈ S for which Re h ′ ( z ) > 0 holds in the unit disk. However, no attempt has been made to derive harmonic analog of these results. The central object in this case is the class S H 0 of sense-preserving harmonic univalent mappings f = h + g ¯ defined on the unit disk, normalized so that h ( 0 ) = g ( 0 ) = h ′ ( 0 ) − 1 = g ′ ( 0 ) = 0 . Our primary objective in this paper is to solve the univalency of every section of a harmonic function in the class P H 0 ( α ) , where P H 0 ( α ) denotes the class of normalized univalent harmonic mapping f = h + g ¯ in the unit disk | z | 1 satisfying the condition Re ( h ′ ( z ) − α ) > | g ′ ( z ) | for | z | 1 , where g ′ ( 0 ) = 0 and 0 ≤ α 1 . In this paper, we first present sharp bounds for the moduli of the coefficients a n , b n of f ∈ P H 0 ( α ) and then determine the value of r so that the partial sums of f ∈ P H 0 are close-to-convex in | z | r .

A Riemannian metric with a local contraction property can be used to prove existence and uniqueness of a periodic orbit and determine a subset of its basin of attraction. While the existence of such a contraction metric is equivalent to the existence of an exponentially stable periodic orbit, the explicit construction of the metric is a difficult problem. In this paper, the construction of such a contraction metric is achieved by formulating it as an equivalent problem, namely a feasibility problem in semidefinite optimization. The contraction metric, a matrix-valued function, is constructed as a continuous piecewise affine (CPA) function, which is affine on each simplex of a triangulation of the phase space. The contraction conditions are formulated as conditions on the values at the vertices. The paper states a semidefinite optimization problem. We prove on the one hand that a feasible solution of the optimization problem determines a CPA contraction metric and on the other hand that the optimization problem is always feasible if the system has an exponentially stable periodic orbit and the triangulation is fine enough. An objective function can be used to obtain a bound on the largest Floquet exponent of the periodic orbit.

This paper is concerned with the semilinear Schrodinger equation ( S ) − Δ u + V ( x ) u = f ( x , u ) , u ∈ H 1 ( R N ) , where V and f are periodic in the x -variables, f is a superlinear and subcritical nonlinearity, and 0 lies in a spectral gap of − Δ u + V . It is shown that, if f is odd in u then (S) has infinitely many large energy solutions. The proof relies on a generalized variant fountain theorem for strongly indefinite functionals, established in this paper.

The aim of this paper is to verify the Strauss Conjecture for semilinear wave equations with variable coefficients and for the subcritical and critical cases on the exterior domain for when the number of space dimensions n = 2 . The main ingredients include generalized Sideris ODE blow-up results, and the existence and a priori estimates of the test functions (see Section 2 ). For this purpose, we study the following initial–boundary value problem for two-dimensional semilinear wave equations with variable coefficients on the exterior domain with subcritical and critical exponents: (0.1) u t t − ∑ i , j = 1 2 ∂ i ( a i j ( x ) ∂ j u ) = | u | p , ( x , t ) ∈ Ω c × ( 0 , + ∞ ) , n = 2 , where u = u ( x , t ) is a real-valued scalar unknown function in Ω c × [ 0 , + ∞ ) , where Ω is a smooth compact obstacle in R 2 and Ω c is its complement. Two-dimensional initial–boundary value problems for semilinear wave equations with variable coefficients on the exterior domain are difficult to deal with because the fundamental solution of the harmonic equation is different from that for the higher dimensional case, and thus there will be a growth factor in the logarithmic term in the key test function ϕ 0 (see Section 2 ). One of the key points of our paper is the improvement of a well-known ODE blow-up result: Lemma 2.1 of [12, p. 386] , for when the differential inequality involves a logarithmic term; and we obtain an estimate of the upper bound for the blow-up time. On the basis of the key results Lemma 2.3 , Lemma 2.4 obtained, we will establish a blow-up result for the above initial–boundary value problem; it is proved that there can be no global solutions, no matter how small the initial data are, and a lifespan estimate for the solutions for this problem will also be given.

The paper considers the following nonhomogeneous Schrodinger–Maxwell system: (SM) { − Δ u + u + λ ϕ ( x ) u = | u | p − 1 u + g ( x ) , x ∈ R 3 , − Δ ϕ = u 2 , x ∈ R 3 , where λ > 0 , p ∈ ( 1 , 5 ) , and 0 ≤ g ( x ) = g ( | x | ) ∈ L 2 ( R 3 ) . There seem to be no results on the existence of multiple solutions to problem (SM) for p ∈ ( 1 , 3 ) . In this paper, we find that there is a constant C p > 0 such that problem (SM) has at least two solutions for all p ∈ ( 1 , 5 ) provided that ‖ g ‖ L 2 ≤ C p , however, for p ∈ ( 1 , 2 ] we need λ > 0 is small. Moreover, C p = ( p − 1 ) 2 p [ ( p + 1 ) S p + 1 2 p ] 1 / ( p − 1 ) , where S is the Sobolev constant.

This paper continues the work contained in two previous papers of the authors, devoted to the qualitative study of the dynamical system generated by a delay differential equation that models leukemia. The problem depends on five parameters and has two equilibria. As already known, at the non-zero equilibrium solution, for certain values of the parameters, Hopf bifurcation occurs. Our aim here is to investigate the Bautin bifurcation for the considered model. By using a second order approximation of the center manifold, we compute the first Lyapunov coefficient for Hopf bifurcation points. We explore the space of parameters in a zone of biological interest and find, by direct computation, points where the first Lyapunov coefficient equals zero. These are candidates for Bautin type bifurcation. For these points we compute the second Lyapunov coefficient, that determines the type of Bautin bifurcation. The computation of the second Lyapunov coefficient requires a fourth order approximation of the center manifold, that we determine. The points with null first Lyapunov coefficient obtained are given in tables and are also plotted on surfaces of Hopf bifurcation obtained by fixing two parameters. Next we vary two parameters around a point in the space of parameters with l 1 = 0 and numerically explore the behavior of the solution. The results confirm the Bautin type-bifurcation.

In this paper we study the existence of mild solutions of the semilinear nonlocal initial value problem on a bounded and on an unbounded interval. In our considerations we apply a new calculation method based on the technique of modulus of continuity in conjunction with idea of the family of measures of noncompactness in Frechet space. The results obtained in this paper generalize several ones obtained earlier by many authors.

The aim of this paper is to discuss the existence and uniqueness of a local solution for a class of isentropic compressible full non-Newtonian models in one-dimensional bounded intervals. The first important point in this paper is that we allow the initial vacuum; another one is that the viscosity term and the Newtonian potential term are fully nonlinear.

This paper is devoted to the stability problem of a nonlinear system in the critical case of q pairs of purely imaginary eigenvalues. The center manifold theory and the normal form method are exploited in this study. The main result of the paper is a power estimate for the solutions in the case when stability is ensured by third order forms. A Lyapunov function is constructed explicitly as a part of the proof of this estimate. The result obtained is illustrated by an example of a double pendulum with partial dissipation.

In this paper, we consider the following Schrodinger–Poisson equation: { − Δ u + V u + μ ϕ u = f ( u ) , in R 3 , − Δ ϕ = μ u 2 , in R 3 , where f has critical growth. The purpose of this paper is to study the existence of solutions for the Schrodinger–Poisson equation with a general nonlinearity in the critical growth.

In this paper, some properties of the solutions are considered to the two-component b -family systems, which cover the two-component Camassa–Holm equation and the two-component Degasperis–Procesi equation. This paper is concerned with some aspects of the existence of global solutions, persistence properties and propagation speed. One first discusses the local well-posedness and blow-up scenario, then, establishes the sufficient conditions on the global existence of the solutions. Finally, the persistence properties and propagation speed of strong solutions will also be investigated.

a b s t r a c t Since the celebrated Mackey–Glass model of respiratory dynamics was introduced in 1977, many results on its qualitative behavior have been obtained, including oscillation, stability and chaos. The paper reviews some known properties and presents new results for more general models: equations with time-dependent parameters, several delays, a positive periodic equilibrium and distributed delays. The problems considered in the paper involve existence, positivity and permanence of solutions, oscillation and global asymptotic stability. In addition, some general approaches to the study of nonlinear nonautonomous scalar delay equations are outlined. The paper generalizes and unifies existing results and provides an outlook on further studies.

In the paper Nakauchi and Takenaka (2011) [3] the second author introduced a stationary map for the functional of the L 2 -norm of the pullbacks of metrics. We call it a symphonic map. In this paper, using Morrey’s argument, we give a regularity of (locally) minimizing symphonic maps into the Euclidean space.

In this paper we provide a complete characterization of fully nonlinear differential operators of any integer order on R n , which exhibit conformal invariance of exponential type. In this way we intend to complete the work that we undertook in Li et al. (2011) [2] , where we introduced the family of elementary conformal tensors { T m , α u } in order to describe all fully nonlinear differential operators of any integer order on R n which are conformally invariant of degree α ≠ 0 . Examples of the differential operators that we study in this paper are those related to the Q –curvature equation on R 4 and to the Gauss equation on R 2 .

In this paper, we investigate a class of nonlinear damped stochastic hyperbolic equations with jumps. The jump component considered here is described as a Poisson point process. This paper is divided into two parts. The first part deals with existence and uniqueness of global weak and strong solutions to this type of equations, based on the energy approach. The second part devotes to the existence and support of invariant measures corresponding to the weak solution semi-group, based on Markov property of the solution.

In this paper, existence results are derived for the unique, smallest, greatest, minimal, and/or maximal solutions of finite systems of nonlinear distributional Cauchy problems. The dependence of solutions on the data is also studied. Applications to systems of distributional differential equations with impulses and to higher-order distributional Cauchy problems are presented. The main tools are fixed point results in function spaces and a primitive integral of distributions defined in the paper, including recently introduced regulated and continuous primitive integrals on compact intervals.

There are two types of criteria of solutions for the set-valued optimization problem, the vectorial criterion and set optimization criterion. The first criterion consists of looking for efficient points of set valued map and is called set-valued vector optimization problem. On the other hand, Kuroiwa–Tanaka–Ha started developing a new approach to set-valued optimization which is based on comparison among values of the set-valued map. In this paper, we treat the second type criterion and call set optimization problem. The aim of this paper is to investigate four types of nonlinear scalarizing functions for set valued maps and their relationships. These scalarizing functions are generalization of Tammer–Weidner’s scalarizing functions for vectors. As applications of the scalarizing functions for sets, we present nonconvex separation type theorems, Gordan’s type alternative theorems for set-valued map, optimality conditions for set optimization problem and Takahashi’s minimization theorems for set-valued map.

In this paper, we will give conditions which will guarantee the existence of positive solutions for a variety of second order boundary value problems, using the well-known Krasnoselskii fixed point theorem. We will deal with a specific differential equation meeting a specific initial condition and use a general boundary condition, involving a not necessarily linear functional. Our purpose is to pose conditions on that functional, which will guarantee that the Krasnoselskii fixed point theorem can be applied. It is important to notice that only the values of this functional on two specific functions are involved in the conditions we pose. This paper unifies the way we deal with a wide variety of boundary value problems and provides results which, to the best of our knowledge, are new.

In this paper, we study the optimal decay rates of solutions for the generalized Benjamin–Bona–Mahony equation in multi-dimensional space ( n ≥ 3 ). By using Fourier transform and the energy method, we obtain the L q ( 2 ≤ q ≤ ∞ ) convergence rates of the solutions under the condition that the initial data is small. The optimal decay rates obtained in this paper are found to be the same as the decay rate for the Heat equation.

This paper, together with a recent paper by the second author on convex singular kernels, establishes a base for further investigation of mildly singular equations with Liapunov theory. We study the two nonlinear scalar integral equations x ( t ) = a ( t ) − ∫ 0 t D ( t , s ) [ x ( s ) + G ( s , x ( s ) ) ] d s and z ( t ) = a ( t ) − ∫ 0 t D ( t , s ) g ( s , z ( s ) ) d s where D has a singularity at t = s . The first equation is decomposed into three other simpler equations. We then construct a Liapunov functional for each of the equations which will yield L p properties of the solutions.

Fractional stochastic differential equations have gained considerable importance due to their application in various fields of science and engineering. This paper is concerned with the square-mean pseudo almost automorphic solutions for a class of fractional stochastic differential equations in a Hilbert space. The main objective of this paper is to establish the existence and uniqueness of square-mean pseudo almost automorphic mild solutions to a linear and semilinear case of these equations. A new set of sufficient conditions is obtained to achieve the required result by using the stochastic analysis theory and fixed point strategy. Finally, an example is provided to illustrate the obtained theory.

This paper is concerned with viscosity solutions for a class of degenerate quasilinear parabolic equations in a bounded domain with homogeneous Dirichlet boundary condition. The equation under consideration arises from a number of practical model problems including reaction–diffusion processes in a porous medium. The degeneracy of the problem appears on the boundary and possibly in the interior of the domain. The goal of this paper is to establish some comparison properties between viscosity upper and lower solutions and to show the existence of a continuous viscosity solution between them. An application of the above results is given to a porous-medium type of reaction–diffusion model which demonstrates some distinctive properties of the solution when compared with the corresponding semilinear problem.

It is known that the concept of dissipativeness is fundamental for understanding the asymptotic behavior of solutions to evolutionary problems. In this paper we investigate the dissipative mechanism for some semilinear fourth-order parabolic equations in the spaces of Bessel potentials and discuss some weak conditions that lead to the existence of a compact global attractor. While for second-order reaction-diffusion equations the dissipativeness mechanism has already been satisfactorily understood (see Arrieta et al. (2004), doi:10.1142/S0218202504003234), for higher order problems in unbounded domains it has not yet been fully developed. As shown throughout the paper, one of the main differences from the case of reaction-diffusion equations stems from the lack of a maximum principle. Thus we have to rely here on suitable energy estimates for the solutions. As in the case of second-order reaction-diffusion equations, we show here that both linear and nonlinear terms have to collaborate in order to produce dissipativeness. Thus, the dissipative mechanisms in second-order and fourth-order equations are similar, although the lack of a maximum principle makes the proofs more difficult and the results not as complete. Finally, we make essential use of the sharp results of Cholewa and Rodriguez-Bernal (2012), doi:10.1016/j.na.2011.08.022, on linear fourth-order equations with a very large class of linear potentials.

In a recent paper published in Nonlinear Analysis: Theory, Methods & Applications, Argaez and Melgaard have studied excited states for pseudo-relativistic multi-configuration methods. Their paper follows a previous work of mine in the non-relativistic case [M. Lewin, Solutions of the multiconfiguration equations in quantum chemistry, Arch. Ration. Mech. Anal. 171 (1) (2004) 83–114. doi:10.1007/s00205003-0281-6]. The main results of the paper of Argaez and Melgaard are correct, but the proofs are both wrong and incomplete.

The purpose of this article is to propose a new hybrid algorithm with variable coefficients and prove convergence theorems for asymptotically pseudocontractive mappings in the intermediate sense on unbounded domains. The results of the paper improve and extend the recent results of Qin et al. [X.L. Qin, S.Y. Cho, J.K. Kim, Convergence theorems on asymptotically pseudocontractive mappings in the intermediate sense, Fixed Point Theory Appl. 2010, doi:10.1155/2010/186874 , Article ID 186874, 14 pages] and several others. The algorithm with variable coefficients introduced in this paper is of independent interest.

The purpose of this paper is to present some new fixed point theorems for mixed monotone operators with perturbation by using the properties of cones and a fixed point theorem for mixed monotone operators. As applications, we utilize the results obtained in this paper to study the existence and uniqueness of positive solutions for nonlinear fractional differential equation boundary value problems.