Showing papers in "Nonlinear Analysis-theory Methods & Applications in 2003"
TL;DR: In this paper, several sufficient conditions for the existence of solutions for the Dirichlet problem of p(x)-Laplacian Laplacians were presented, and an existence criterion for infinite many pairs of solutions was obtained.
Abstract: This paper presents several sufficient conditions for the existence of solutions for the Dirichlet problem of p(x)-Laplacian {-div(|∇u|p(x)-2∇u) = f(x,u), x ∈ Ω, u = 0, x ∈ ∂Ω. Especially, an existence criterion for infinite many pairs of solutions for the problem is obtained. The discussion is based on the theory of the spaces Lp(x)(Ω) and W01, p(x)(Ω).
741 citations
TL;DR: In this paper, the notion of variational inequalities is extended to Hadamard manifolds and related to geodesic convex optimization problems, and a convexity property of the solution set of a variational inequality on a hadamard manifold is presented.
Abstract: The notion of variational inequalities is extended to Hadamard manifolds and related to geodesic convex optimization problems. Existence and uniqueness theorems for variational inequalities on Hadamard manifolds are proved. A convexity property of the solution set of a variational inequality on a Hadamard manifold is presented.
149 citations
TL;DR: In this paper, some recent applications of Ricceri's theorem to nonlinear boundary value problems are revisited by obtaining more precise conclusions, and some remarks on a strict minimax inequality are presented.
Abstract: Some remarks on a strict minimax inequality, which plays a fundamental role in Ricceri's three critical points theorem, are presented. As a consequence, some recent applications of Ricceri's theorem to nonlinear boundary value problems are revisited by obtaining more precise conclusions.
144 citations
TL;DR: In this article, the authors derived criteria on uniform ultimate boundedness for impulsive delay differential equations using Lyapunov functions and Razumikhin techniques and used them to obtain permanence results for population growth models.
Abstract: In this paper, criteria on uniform ultimate boundedness are derived for impulsive delay differential equations using Lyapunov functions and Razumikhin techniques It should be noted that the boundedness criteria establish global existence of solutions as well as boundedness without assuming, a priori, that solutions can necessarily be continued to infinity Those criteria are then used to obtain permanence results for population growth models Some examples are discussed to illustrate the theorems
132 citations
TL;DR: In this article, the authors consider equilibrium problems in the framework of the formulation proposed by Blum and Oettli, which includes variational inequalities, Nash equilibria in non-cooperative games, and vector optimization problems, for instance, as particular cases.
Abstract: We consider equilibrium problems in the framework of the formulation proposed by Blum and Oettli, which includes variational inequalities, Nash equilibria in noncooperative games, and vector optimization problems, for instance, as particular cases We establish new sufficient and/or necessary conditions for existence of solutions of such problems Our results are based upon the relation between equilibrium problems and certain auxiliary convex feasibility problems, together with extensions to equilibrium problems of gap functions for variational inequalities Then we apply our results to some particular instances of equilibrium problems, obtaining results which include, among others, a new lemma of the alternative for convex optimization problems
132 citations
TL;DR: In this paper, the authors investigated the global asymptotic behavior of solutions of the system of difference equations and showed that the stable manifold of this system separates the positive quadrant into basins of attraction of two types of asymPTotic behavior.
Abstract: We investigate the global asymptotic behavior of solutions of the system of difference equations xn+1 = xn/ a + cyn, yn+1 = yn/ b + dxn, n =0,1,..., where the parameters a and b are in (0, 1), c and d are arbitrary positive numbers and the initial conditions x0 and y0 are arbitrary nonnegative numbers. We show that the stable manifold of this system separates the positive quadrant into basins of attraction of two types of asymptotic behavior. In the case where a = b we find an explicit equation for the stable manifold.
118 citations
TL;DR: In this paper, the existence of mild and strong solutions of semilinear neutral functional differential evolution equations with nonlocal conditions is studied. And the results are a generalization and continuation of the recent results on this issue.
Abstract: In this paper, by using fractional power of operators and Sadovskii's fixed point theorem, we study the existence of mild and strong solutions of semilinear neutral functional differential evolution equations with nonlocal conditions. The results we obtained are a generalization and continuation of the recent results on this issue. In the end, an example is given to show the application of our results.
114 citations
TL;DR: In this article, it was shown that left-continuity of the system trajectories in time for each fixed state point and continuity of system trajectory in the state for every time in some dense subset of the semi-infinite interval are sufficient for establishing an invariance principle for hybrid and impulsive dynamical systems.
Abstract: In this paper we develop an invariance principle for dynamical systems possessing left-continuous flows. Specifically, we show that left-continuity of the system trajectories in time for each fixed state point and continuity of the system trajectory in the state for every time in some dense subset of the semi-infinite interval are sufficient for establishing an invariance principle for hybrid and impulsive dynamical systems. As a special case of this result we state and prove new invariant set stability theorems for a class of nonlinear impulsive dynamical systems; namely, state-dependent impulsive dynamical systems. These results provide less conservative stability conditions for impulsive systems as compared to classical results in the literature and allow us to address the stability of limit cycles and periodic orbits of impulsive systems.
113 citations
TL;DR: In this article, the data dependence of the fixed point set for a special class of weakly Picard operators is studied and an application to a Fredholm integral inclusion is given. And the existence and data dependence for some Reich-type multivalued operators are also proved.
Abstract: In this paper we study data dependence of the fixed point set for a special class of multivalued weakly Picard operators. Existence and data dependence of the common fixed points for some Reich-type multivalued operators are also proved. Finally, an application to a Fredholm integral inclusion is given.
109 citations
TL;DR: In this paper, an initial boundary value problem for systems of semilinear wave equations in a bounded domain is considered, and the global existence, uniqueness and blow-up of solutions by energy methods are proved.
Abstract: An initial boundary value problem for systems of semilinear wave equations in a bounded domain is considered. We prove the global existence, uniqueness and blow-up of solutions by energy methods and give some estimates for the lifespan of solutions.
92 citations
TL;DR: New conditions ensuring global asymptotic stability and global exponential stability for cellular neural networks with constant delay and variable delay are given by using the essence of piecewise linearity of the output function of cellular Neural networks and constructing Lyapunov functions and functionals.
Abstract: This paper gives new conditions ensuring global asymptotic stability and global exponential stability for cellular neural networks with constant delay and variable delay, respectively. These conditions are derived by using the essence of piecewise linearity of the output function of cellular neural networks and by constructing Lyapunov functions and functionals. Furthermore, these conditions are significantly weaker than those given in existing literature.
TL;DR: In this paper, the existence of solutions to equations of p-Laplacian type was studied and it was shown that at least one solution can be found and, under further assumptions, that infinitely many solutions can exist.
Abstract: This work is devoted to study the existence of solutions to equations of p-Laplacian type. We prove the existence of at least one solution, and under further assumptions, the existence of infinitely many solutions. In order to apply mountain pass results, we introduce a notion of uniformly convex functional that generalizes the notion of uniformly convex norm.
TL;DR: In this paper, it is shown that the iterates converge to Py, where P is the projection onto the intersection of the fixed point sets of the Tis, where p is some projection.
Abstract: The iteration scheme xn+1=λn+1y+(1−λn+1)Tn+1xn is first considered for infinitely many nonexpansive maps T1,T2,T3,… in a Hilbert space. A result of Shimizu and Takahashi (J. Math. Anal. Appl. 211 (1997) 71) is generalized, and it is shown that the sequence of iterates converges to Py, where P is some projection. For this same iteration scheme, with finitely many maps T1,T2,…,TN, a complementary result to a result of Bauschke (J. Math. Anal. Appl. 202 (1996) 150) is proved by introducing a new condition on the sequence of parameters (λn). This condition improves Lions’ condition (C.R. Acad. Sci. Paris Ser A-B 284 (1977) 1357). The iterates converge to Py, where P is the projection onto the intersection of the fixed point sets of the Tis.
TL;DR: In this paper, a Wintner-type result for fuzzy IVPs and superlinear-type results for fuzzy BVPs were proved. But they rely on a generalized Schauder theorem in metric spaces.
Abstract: We prove a Wintner-type result for fuzzy IVPs and a superlinear-type result for fuzzy BVPs. Both results rely on a generalized Schauder theorem in metric spaces.
TL;DR: An existence result is proved for set differential equation when the function involved is upper semicontinuous and the result is very general.
Abstract: An existence result is proved for set differential equation when the function involved is upper semicontinuous and the result is very general Then, the connection between the solutions of fuzzy differential equation and the set differential equation that is generated from it, is studied
TL;DR: In this paper, the authors studied the convergence to equilibrium of bounded solutions of first-order and second-order problems for diffusion, wave, Cahn-Hilliard and Kirchhoff-Carrier equations.
Abstract: We study the convergence to equilibrium of bounded solutions of the nonautonomous first-order problem u + M u=g(t), t∈ R + and of the second-order problem u + u + M u=g(t), t∈ R + . Applications to diffusion, wave, Cahn–Hilliard and Kirchhoff–Carrier equations are described.
TL;DR: In this paper, the existence theorem for monotone positive solutions of nonlinear second-order ODEs was obtained by using the Schauder-Tikhonov fixed point theorem.
Abstract: We obtain an existence theorem for monotone positive solutions of nonlinear second-order ordinary differential equations by using the Schauder-Tikhonov fixed point theorem. The result can also be applied to prove the existence of positive solutions of certain semilinear elliptic equations in R-n (n greater than or equal to 3).
TL;DR: In this article, the existence of at least three weak Dirichlet problems involving the p-Laplacian by a variational approach is established. But the existence is not proven.
Abstract: In this paper, we establish some results on the existence of at least three weak solutions for Dirichlet problems involving the p -Laplacian by a variational approach.
TL;DR: In this paper, the authors investigated a nonlinear partial differential equation arising from a model of cellular proliferation, which describes the production of blood cells in the bone marrow and proved that the behaviour of primitive cells influences the global behaviour of the population.
Abstract: In this paper, we investigate a nonlinear partial differential equation, arising from a model of cellular proliferation. This model describes the production of blood cells in the bone marrow. It is represented by a partial differential equation with a retardation of the maturation variable and a distributed temporal delay. Our aim is to prove that the behaviour of primitive cells influences the global behaviour of the population.
TL;DR: Using the results of set differential equations obtained in V. Laksmikantham, Setvalued hybrid differential equations and stability in terms of two measures, a new approach is suggested in order to capture the vagueness and the rich properties of solutions without fuzziness.
Abstract: The original formulation of fuzzy differential equations suffers from the disadvantage since the solutions increases as time increases. In this paper, employing the results of set differential equations obtained in V. Laksmikantham, S. Leela, A.S. Vatsala, Setvalued hybrid differential equations and stability in terms of two measures, J. Hybrid Systems 2 (2) (2002) 169–188, a new approach is suggested in order to capture the vagueness and the rich properties of solutions without fuzziness.
TL;DR: In this article, the existence of multiple periodic solutions and subharmonic solutions to discrete Hamiltonian systems has been studied using critical point theory, where x 1, x 2 ∈ R d, H ∈ C 1 (R × R d × R D, R ).
Abstract: In this paper, some new results are obtained for the existence of multiple periodic solutions and subharmonic solutions to discrete Hamiltonian systems Δ x 1 (n)=−H x 2 (n,x 1 (n+1),x 2 (n)), Δ x 2 (n)=H x 1 (n,x 1 (n+1),x 2 (n)) by using critical point theory, where x1, x 2 ∈ R d , H∈C 1 ( R × R d × R d , R ) .
TL;DR: In this paper, the authors consider two different classes of nonlinear impulsive systems, one driven purely by Dirac measures at a fixed set of points and the second driven by signed measures.
Abstract: In this paper we consider two different classes of nonlinear impulsive systems one driven purely by Dirac measures at a fixed set of points and the second driven by signed measures. The later class is easily extended to systems driven by general vector measures. The principal nonlinear operator is monotone hemicontinuous and coercive with respect to certain triple of Banach spaces called Gelfand triple. The other nonlinear operators are more regular, non-monotone continuous operators with respect to suitable Banach spaces. We present here a new result on compact embedding of the space of vector-valued functions of bounded variation and then use this result to prove two new results on existence and regularity properties of solutions for impulsive systems described above. The new embedding result covers the well-known embedding result due to Aubin.
TL;DR: In this paper, it was shown that the fixed point set of a piecewise affine map f : Rn → Rn is isomorphic to a convex inf-subsemilattice of Rn, whose dimension is at most equal to the number of strongly connected components of a critical graph defined from the tangent affine maps of f.
Abstract: We consider convex maps f : Rn → Rn that are monotone (i.e., that preserve the product ordering of Rn), and nonexpansive for the sup-norm. This includes convex monotone maps that are additively homogeneous (i.e., that commute with the addition of constants). We show that the fixed point set of f, when it is nonempty, is isomorphic to a convex inf-subsemilattice of Rn, whose dimension is at most equal to the number of strongly connected components of a critical graph defined from the tangent affine maps of f. This yields in particular an uniqueness result for the bias vector of ergodic control problems. This generalizes results obtained previously by Lanery, Romanovsky, and Schweitzer and Federgruen, for ergodic control problems with finite state and action spaces, which correspond to the special case of piecewise affine maps f. We also show that the length of periodic orbits of f is bounded by the cyclicity of its critical graph, which implies that the possible orbit lengths of f are exactly the orders of elements of the symmetric group on n letters.
TL;DR: In this paper, the authors prove the existence of weak solutions for a reaction-diusion system with general anisotropic di"usivities and transport eects, supplemented with either mixed boundary conditions or no-5ux boundary conditions.
Abstract: In this paper, we prove the existence of weak solutions for a reaction-di"usion system with general anisotropic di"usivities and transport e"ects, supplemented with either mixed boundary conditions or no-5ux boundary conditions. Initial conditions and external forcing terms are in L 1 ; this does not allow us to use classical variational formulations. Our motivation is a mathematical model describing the spatial propagation in heterogeneous environments of Feline Immunode7ciency Virus, a feline retrovirus. ? 2003 Elsevier Science Ltd. All rights reserved. MSC: 35K57; 35K55; 92D30
TL;DR: In this paper, the existence results of positive solutions are obtained for fourth-order periodic boundary value problem u (4) −βu″+αu =f(t, u), 0⩽t⩾1, u (i) (0)=u (i), (1), i=0,1,2,3, where f : [0, 1]× R + → R + is continuous, α, β∈ R and satisfy 0 −2π2,α/π4+β/π2+1>0.
Abstract: In this paper the existence results of positive solutions are obtained for fourth-order periodic boundary value problem u (4) −βu″+αu=f(t, u), 0⩽t⩽1, u (i) (0)=u (i) (1), i=0,1,2,3, where f : [0, 1]× R + → R + is continuous, α, β∈ R and satisfy 0 −2π2,α/π4+β/π2+1>0. The discussion is based on a new maximum principle for operator L 4 u=u (4) −βu″+αu in periodic boundary condition and fixed point index theory in cones.
TL;DR: In this article, the authors prove nonexistence results for the Cauchy problem for the abstract hyperbolic equation in a Banach space X, where utt = f'(u), t > 0; u(0) = u0, ut = u1, where f : X → R is a C1 function.
Abstract: We prove nonexistence results for the Cauchy problem for the abstract hyperbolic equation in a Banach space X, utt = f'(u), t > 0; u(0) = u0, ut(0) = u1, where f : X → R is a C1-function Several applications to the second- and higher-order hyperbolic equations with local and nonlocal nonlinearities are presented We also describe an approach to Kato's and John's critical exponents for the semilinear equations ut = Δu + b(x,t)|u|p, p > 1, which are responsible for phenomena of stability, unstability, blow-up and asymptotic behaviour
TL;DR: In this paper, the stability and boundedness problems of some Volterra nonlinear difference equations are investigated in terms of the characteristics of the equations and the stability conditions are formulated.
Abstract: Volterra difference equations arise in the modeling of many real phenomena. In this survey stability and boundedness problems of some Volterra nonlinear difference equations are investigated. Stability conditions and boundedness are formulated in terms of the characteristics of equations.
TL;DR: Local Lipschitz continuity of local minimizers of vectorial integrals is proved when f satisfies p−q growth condition and ξ↦f(x,ξ) is convex.
Abstract: Local Lipschitz continuity of local minimizers of vectorial integrals
∫Ωf(x,Du(x))dx
is proved when f satisfies p−q growth condition and ξ↦f(x,ξ) is convex. The uniform convexity and the radial structure condition with respect to the last variable are assumed only at infinity.
TL;DR: In this paper, the authors use an algebraic topological argument due to Bahri and Coron to show how the topology of the domain influences the existence of positive solutions of the following problem involving the bilaplacian operator with the critical Sobolev exponent.
Abstract: In this paper we use an algebraic topological argument due to Bahri and Coron to show how the topology of the domain influences the existence of positive solutions of the following problem involving the bilaplacian operator with the critical Sobolev exponent Δ2u = un+4/(n-4) in Ω, u > 0 in Ω, u = Δu = 0 on ∂Ω, where (Ω is a bounded domain of Rn (n≥ 5) with a smooth boundary ∂Ω.
TL;DR: In this article, nonimprovable effective sufficient conditions for solvability and uniqueness of the boundary value problem were established for continuous operators satisfying the Caratheodory condition, where F : C([a,b];R) → L([a and b];R] is a continuous operator satisfying the CARA condition.
Abstract: Nonimprovable effective sufficient conditions for solvability and unique solvability of the boundary value problem u'(t) = F(u)(t), u(a) = h(u), where F : C([a,b];R) → L([a,b];R) is a continuous operator satisfying the Caratheodory condition and h : C([a,b];R) → R is a continuous functional, are established.