Showing papers in "Nonlinear Analysis-theory Methods & Applications in 2005"
TL;DR: In this article, an iterative scheme for finding a common element of the set of fixed points of a nonexpansive mapping and the solutions of the variational inequality for an inverse-strongly monotone mapping in a Hilbert space is introduced.
Abstract: In this paper, we introduce an iterative scheme for finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality for an inverse-strongly monotone mapping in a Hilbert space. Then we show that the sequence converges strongly to a common element of two sets. Using this result, we consider the problem of finding a common fixed point of a nonexpansive mapping and a strictly pseudocontractive mapping and the problem of finding a common element of the set of fixed points of a nonexpansive mapping and the set of zeros of an inverse-strongly monotone mapping.
293 citations
TL;DR: In this article, the authors studied the localization properties of weak solutions to the Dirichlet problem for the degenerate parabolic equation u t - div ( | u | γ ( x, t ) ∇ u ) = f, with variable exponent of nonlinearity γ.
Abstract: We study the localization properties of weak solutions to the Dirichlet problem for the degenerate parabolic equation u t - div ( | u | γ ( x , t ) ∇ u ) = f , with variable exponent of nonlinearity γ . We prove the existence and uniqueness of weak solutions and establish conditions on the problem data and the exponent γ ( x , t ) sufficient for the existence of such properties as finite speed of propagation of disturbances, the waiting time effect, finite time vanishing of the solution. It is shown that the solution may instinct in a finite time even if γ ≡ γ ( x ) ⩽ 0 in the problem domain but max γ = 0 .
249 citations
TL;DR: In this paper, two modifications of the Mann iterations in a uniformly smooth Banach space were proposed, one for nonexpansive mappings and the other for the resolvent of accretive operators.
Abstract: The Mann iterations for nonexpansive mappings have only weak convergence even in a Hilbert space. We propose two modifications of the Mann iterations in a uniformly smooth Banach space, one for nonexpansive mappings and the other for the resolvent of accretive operators. The two modified Mann iterations are proved to have strong convergence.
238 citations
TL;DR: A short survey on nonlocal elliptic boundary value problems is given in this paper, where a smooth bounded domain of R N is considered, and f is a positive function with subcritical growth.
Abstract: We present a short survey on the nonlocal elliptic boundary value problem - M ∫ Ω | ∇ u | 2 d x Δ u = f ( x , u ) in Ω , u = 0 on ∂ Ω , where Ω is a smooth bounded domain of R N , M is a positive function, and f has subcritical growth.
179 citations
TL;DR: In this article, it was shown that every bounded solution to ( * ) is a pseudo-almost periodic function, and the theory of the invariant subspaces for unbounded linear operators was used to show that any bounded solution is also a pseudo almost periodic function.
Abstract: This paper is concerned with the pseudo almost periodic solutions to the abstract differential equations of the form: d d s w ( s ) = Aw ( s ) + Bw ( s ) + f ( s ) ( * ) , where A , B are densely defined closed linear operators acting in a Hilbert space H , and f : R ↦ H is a H -valued pseudo almost periodic function. Using the theory of the invariant subspaces for unbounded linear operators, we show that every bounded solution to ( * ) is a pseudo almost periodic function.
168 citations
TL;DR: In this paper, the authors proposed a model for stochastic hybrid systems (SHSs) where transitions between discrete modes are triggered by discrete events much like transitions between states of a continuous-time Markov chains, but the rate at which transitions occur depends both on the continuous and the discrete states of the SHS.
Abstract: We propose a model for stochastic hybrid systems (SHSs) where transitions between discrete modes are triggered by stochastic events much like transitions between states of a continuous-time Markov chains. However, the rate at which transitions occur is allowed to depend both on the continuous and the discrete states of the SHS. Based on results available for piecewise-deterministic Markov process (PDPs), we provide a formula for the extended generator of the SHS, which can be used to compute expectations and the overall distribution of the state. As an application, we construct a stochastic model for on-off TCP flows that considers both the congestion-avoidance and slow-start modes and takes directly into account the distribution of the number of bytes transmitted. Using the tools derived for SHSs, we model the dynamics of the moments of the sending rate by an infinite system of ODEs, which can be truncated to obtain an approximate finite-dimensional model. This model shows that, for transfer-size distributions reported in the literature, the standard deviation of the sending rate is much larger than its average. Moreover, the later seems to vary little with the probability of packet drop. This has significant implications for the design of congestion control mechanisms.
155 citations
TL;DR: In this article, a generalization of the mean value theorem is considered in the case of functions defined on an invex set with respect to η (which is not necessarily connected).
Abstract: In this paper, a generalization of the mean value theorem is considered in the case of functions defined on an invex set with respect to η (which is not necessarily connected).
149 citations
TL;DR: In this article, a weak and strong convergence of an iterative scheme in a uniformly convex Banach space under a condition weaker than compactness was studied. But the convergence of the scheme was not considered.
Abstract: In this paper, we are concerned with the study of an iterative scheme with errors involving two nonexpansive mappings. We approximate the common fixed points of these two mappings by weak and strong convergence of the scheme in a uniformly convex Banach space under a condition weaker than compactness.
137 citations
TL;DR: In this article, the authors considered the N-dimensional Cauchy problem in R N for a semilinear damped wave equation with a power-type nonlinearity, and derived a global in time existence result in the case when the power of the nonlinear term satisfies 1 + 2 / N p ⩽ N / [ N - 2 ] +.
Abstract: We consider the N-dimensional Cauchy problem in R N for a semilinear damped wave equation with a power-type nonlinearity | u | p For a noncompactly supported initial data, which has a small energy, we shall derive a global in time existence result in the case when the power of the nonlinear term satisfies 1 + 2 / N p ⩽ N / [ N - 2 ] + This generalizes a previous result due to Todorova–Yordanov (J Differential Equations 174 (2001) 464–489), which dealt with a solution in the framework of compactly supported initial data
136 citations
TL;DR: In this article, the authors provide variational formulas characterizing the speed of travelling front solutions of the following nonlocal diffusion equation: ∂ u ∂ t = J * u - u + f ( u ), where J is a dispersion kernel and f is any of the nonlinearities commonly used in various models ranging from combustion theory of ecology.
Abstract: The object of this paper is to provide variational formulas characterizing the speed of travelling front solutions of the following nonlocal diffusion equation: ∂ u ∂ t = J * u - u + f ( u ) , Where J is a dispersion kernel and f is any of the nonlinearities commonly used in various models ranging from combustion theory of ecology. In several situations, such as population dynamics, it is indeed natural to model the dispersion of a population using such operators. Furthermore, since travelling front solutions are expected to give the asymptotic behaviour in large time for solutions of the above equation, it is of the interest to characterize their speed. Our results, based on elementary techniques, generalize known results obtained for models involving local diffusion operators.
134 citations
Abstract: The dynamics of a Nicholson's blowflies equation with a finite delay are investigated. We prove that a sequence of Hopf bifurcations occur at the positive equilibrium as the delay increases. Explicit algorithm for determining the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions are derived, using the theory of normal form and center manifold. Global existence of periodic solutions are established using a global Hopf bifurcation result of Wu (Trans. Amer. Math. Soc. 350 (1998) 4799), and a Bendixson criterion for higher dimensional ordinary differential equations due to Li and Muldowney (J. Differential Equations 106 (1994) 27).
TL;DR: By means of Riccati transformation technique, the authors established new oscillation criteria for a second-order delay differential equation on time scales in terms of the coefficients and established some new oscillations criteria for the second order delay differential equations.
Abstract: By means of Riccati transformation technique, we establish some new oscillation criteria for a second-order delay differential equation on time scales in terms of the coefficients
TL;DR: The asymptotic behavior of the composition of two resolvents in a Hilbert space is investigated in this paper, where connections are made between the solutions of associated monotone inclusion problems and their dual versions.
Abstract: The asymptotic behavior of the composition of two resolvents in a Hilbert space is investigated. Connections are made between the solutions of associated monotone inclusion problems and their dual versions. The applications provided include a study of an alternating minimization procedure and a new proof of von Neumann's classical result on the method of alternating projections.
TL;DR: In this paper, the existence of positive solutions to a system of second-order nonlocal boundary value problems by using fixed point index theory in a cone is studied. But their results cannot be reliably deduced from the ones for single nonlocal problems in the literature.
Abstract: In this paper we study the existence of positive solutions to a system of second-order nonlocal boundary value problems by using fixed point index theory in a cone. Our hypotheses imposed on nonlinearities are those which characterize systems of nonlocal boundary value problems, and our boundary value conditions are expressed in terms of possibly nonlinear functions of Riemann–Stieltjes integrals, thus generalizing and unifying the boundary value conditions in the literature. Therefore our results cannot be routinely deduced from the ones for single nonlocal problem in the literature.
TL;DR: In this paper, the erosion curves for the Helmholtz, the Duffing and rigid block oscillators are constructed to investigate the loss of integrity in different mechanical systems, and various definitions of safe basin and integrity measures are reviewed, compared with each other and applied to the considered cases.
Abstract: The erosion curves for the Helmholtz, the Duffing and rigid block oscillators are constructed to investigate the loss of integrity in different mechanical systems. Various definitions of safe basin and integrity measures are reviewed, compared with each other and applied to the considered cases. A control method aimed at shifting the erosion profiles towards larger excitation amplitudes is considered, too, and its main features are discussed in depth.
TL;DR: In this article, the authors prove the strong convergence of an implicit iteration process to a common fixed point for a finite family of nonexpansive mappings, and give an affirmative response to a question raised by Xu and Ori.
Abstract: The purpose of this paper is to prove the strong convergence of an implicit iteration process to a common fixed point for a finite family of nonexpansive mappings. Our theorems give an affirmative response to a question raised by [Xu and Ori, Numer. Funct. Anal. Optim. 22 (2001) 767–773].
TL;DR: Complexiton solutions (or complexitons for short) are exact solutions newly introduced to integrable equations are newly introduced as discussed by the authors, starting with a solution classification for a linear differential equation, the Korteweg-de Vries equation and the Toda lattice equation are considered as examples to exhibit complexiton structures of nonlinear integrably equations, the crucial step in the solution process is to apply the Wronskian and Casoratian techniques for Hirota's bilinear equations.
Abstract: Complexiton solutions (or complexitons for short) are exact solutions newly introduced to integrable equations are newly introduced. Starting with the solution classification for a linear differential equation, the Korteweg–de Vries equation and the Toda lattice equation are considered as examples to exhibit complexiton structures of nonlinear integrable equations. The crucial step in the solution process is to apply the Wronskian and Casoratian techniques for Hirota's bilinear equations. Correspondence between complexitons of the Korteweg–de Vries equation and complexitons of the Toda lattice equation is provided.
TL;DR: In this paper, the authors considered a linear scalar differential equation with variable delays and gave conditions to ensure that the zero solution is asymptotically stable by means of fixed point theory.
Abstract: In this paper we consider a linear scalar differential equation with variable delays and give conditions to ensure that the zero solution is asymptotically stable by means of fixed point theory. These conditions do not require the boundedness of delays, nor do they ask for a fixed sign on the coefficient functions. An asymptotic stability theorem with a necessary and sufficient condition is proved.
TL;DR: In this article, the existence and non-existence of non-trivial solutions to quasilinear Brezis-Nirenberg-type problems with singular weights were investigated.
Abstract: In this paper, we consider the existence and non-existence of non-trivial solutions to quasilinear Brezis–Nirenberg-type problems with singular weights. First, we shall obtain a compact imbedding theorem which is an extension of the classical Rellich–Kondrachov compact imbedding theorem, and consider the corresponding eigenvalue problem. Secondly, we deduce a Pohozaev-type identity and obtain a non-existence result. Thirdly, thanks to the generalized concentration compactness principle, we will give some abstract conditions when the functional satisfies the (PS) c condition. Finally, basing on the explicit form of the extremal function, we will obtain some existence results.
TL;DR: In this article, the properties of set-optimization are investigated and conditions for existence of solutions are established for set-valued mappings, and necessary and sufficient conditions in the existence of solution are showed with directional derivatives.
Abstract: Problems in set-valued optimization can be solved via set-optimization. In this paper properties of set-optimization are investigated. Conditions for existence of solutions are established. Directional derivatives are studied for set-valued mappings. Necessary and sufficient conditions in the existence of solutions are showed with directional derivatives.
TL;DR: In this article, a non-empty closed convex nonexpansive retract of a real uniformly convex Banach space E with P as a nonexpan-ansive retraction is considered.
Abstract: Suppose K is a nonempty closed convex nonexpansive retract of a real uniformly convex Banach space E with P as a nonexpansive retraction. Let T : K → E be a nonexpansive non-self map with F ( T ) := { x ∈ K : Tx = x } ≠ ∅ . Suppose { x n } is generated iteratively by x 1 ∈ K , x n + 1 = P ( ( 1 - α n ) x n + α n TP [ ( 1 - β n ) x n + β n Tx n ] ) , n ⩾ 1 , where { α n } and { β n } are real sequences in [ e , 1 - e ] for some e ∈ ( 0 , 1 ) . (1) If the dual E * of E has the Kadec–Klee property, then weak convergence of { x n } to some x * ∈ F ( T ) is proved; (2) If T satisfies condition ( A ) , then strong convergence of { x n } to some x * ∈ F ( T ) is obtained.
TL;DR: In this paper, the existence of multiple nontrivial solutions for some fourth order semilinear elliptic boundary value problems is considered and weak solutions are sought by means of Morse theory and local linking.
Abstract: In this paper, we consider the existence of multiple nontrivial solutions for some fourth order semilinear elliptic boundary value problems. The weak solutions are sought by means of Morse theory and local linking.
TL;DR: In this article, existence and multiplicity results to the following nonlinear elliptic equation were established, where Δ p u denotes the p-Laplacian operator, 1 q p N, p * = Np / ( N - p ) and λ is a positive real parameter.
Abstract: In this paper, existence and multiplicity results to the following nonlinear elliptic equation: - Δ p u = λ | u | q - 2 u + | u | p * - 2 u , u > 0 in Ω ⊂ R N , together with mixed Dirichlet–Neumann or Neumann boundary conditions, are established. Here, Δ p u denotes the p-Laplacian operator, 1 q p N , p * = Np / ( N - p ) and λ is a positive real parameter. The study is based on the extraction of Palais–Smale sequences in the Nehari manifold.
TL;DR: In this article, the existence and multiplicity results of the solutions are obtained for the fourth-order boundary value problem u ( 4 ) (t ) = f ( t, u ( t ) ) for all t ∈ [ 0, 1 ] subject to u ( 0 ) = u ( 1 ) was u ″ ( 0) = u � ( 1) = 0, where f is continuous.
Abstract: In this paper, the existence and multiplicity results of the solutions are obtained for the fourth-order boundary value problem u ( 4 ) ( t ) = f ( t , u ( t ) ) for all t ∈ [ 0 , 1 ] subject to u ( 0 ) = u ( 1 ) = u ″ ( 0 ) = u ″ ( 1 ) = 0 , where f is continuous. The monotone operator theory and critical point theory are employed to discuss this problem, respectively.
TL;DR: In this article, the authors investigate the existence and method of construction of solutions for a general class of strongly coupled elliptic systems by the method of upper and lower solutions and its associated monotone iterations.
Abstract: The aim of this paper is to investigate the existence and method of construction of solutions for a general class of strongly coupled elliptic systems by the method of upper and lower solutions and its associated monotone iterations. The existence problem is for nonquasimonotone functions arising in the system, while the monotone iterations require some mixed monotone property of these functions. Applications are given to three Lotka–Volterra model problems with cross-diffusion and self-diffusion which are some extensions of the classical competition, prey–predator, and cooperating ecological systems. The monotone iterative schemes lead to some true positive solutions of the competition system, and to quasisolutions of the prey–predator and cooperating systems. Also given are some sufficient conditions for the existence of a unique positive solution to each of the three model problems.
TL;DR: In this article, the multiplicity of positive solutions of Ω ∋ 0 was studied and the effect of the coefficient of the critical nonlinearity on the existence of multiple positive solutions was investigated by means of variational method.
Abstract: Let Ω ∋ 0 be an open bounded domain, Ω ⊂ R N ( N > p 2 ) . We are concerned with the multiplicity of positive solutions of - Δ p u - μ | u | p - 2 u | x | p = λ | u | p - 2 u + Q ( x ) | u | p * - 2 u , u ∈ W 0 1 , p ( Ω ) , where - Δ p u = - div ( | ∇ u | p - 2 ∇ u ) , 1 p N , p * = Np N - p , 0 μ N - p p p , λ > 0 and Q ( x ) is a nonnegative function on Ω ¯ . By investigating the effect of the coefficient of the critical nonlinearity, we, by means of variational method, prove the existence of multiple positive solutions.
TL;DR: It is proved the existence and uniqueness of solution for nonlinearities satisfying a Lipschitz condition and the obtained results are applied to the particular case of linear fuzzy problems.
Abstract: We consider nth-order fuzzy differential equations with initial value conditions. We prove the existence and uniqueness of solution for nonlinearities satisfying a Lipschitz condition. We apply the obtained results to the particular case of linear fuzzy problems.
TL;DR: In this paper, it was shown that the 2D quasi-geostrophic equation with α ⩽ 1 2 has a unique local in time solution corresponding to any initial datum in the space C r ∩ L q for r > 1 and q > 1.
Abstract: The 2D quasi-geostrophic equation ∂ t θ + u · ∇ θ + κ ( - Δ ) α θ = 0 , u = R ⊥ ( θ ) is a two-dimensional model of the 3D hydrodynamics equations. When α ⩽ 1 2 , the issue of existence and uniqueness concerning this equation becomes difficult. It is shown here that this equation with either κ = 0 or κ > 0 and 0 ⩽ α ⩽ 1 2 has a unique local in time solution corresponding to any initial datum in the space C r ∩ L q for r > 1 and q > 1 .
TL;DR: In this article, a finite element approach (FEM) is proposed to handle the strong electromechanical coupling in micro-electromechanical systems (MEMS), which can reveal complex dynamical behaviors of MEMS such as dynamic pull-in.
Abstract: In micro-electromechanical systems (MEMS), coupling of structures through electrostatic forces is a primordial phenomenon. Simulating the dynamics of MEMS and taking into account such strong coupling effects allows one to predict dynamical performance and stability, and is therefore an essential issue in the design of highly effective and reliable devices. Analysis techniques for such systems require special attention in order to provide to the designer accurate and fast tools. We propose a finite element approach (FEM) that properly handles the strong electromechanical coupling in MEMS. In the simulation example of a micro-bridge, we show that such simulation techniques can reveal complex dynamical behaviors of MEMS such as dynamic pull-in.
TL;DR: In this article, the existence of integral solutions for nonlinear differential equations with nonlocal initial conditions in Banach spaces was studied and conditions under which the integral solutions exist were derived.
Abstract: In this paper we study the existence of integral solutions for nonlinear differential equations with nonlocal initial conditions in Banach spaces. We derive conditions under which the integral solutions exist.