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In this paper, we study the initial and boundary value problem of the Navier–Stokes equations in the half-space. We prove the existence of weak solution u ∈ L α q ( 0 , ∞ ; L p ( R + n ) ) , α = 1 2 ( 1 − n p − 2 q ) ≥ 0 , n < p < ∞ with ∇ u ∈ L loc q 2 ( 0 , ∞ ; L loc p 2 ( R + n ) ) for the solenoidal initial data h ∈ B ˙ p q − 1 + n p ( R + n ) and the boundary data g ∈ L α q ( 0 , ∞ ; B ˙ p p − 1 p ( R n − 1 ) ) when ‖ h ‖ B ˙ p q − 1 + n p ( R + n ) + ‖ g ‖ L α q ( 0 , ∞ ; B ˙ p p − 1 p ( R n − 1 ) ) is small enough. Moreover, the solution is unique in the class L α q ( 0 , T ; L p ( R + n ) ) for any T ≤ ∞ if α > 0 and for some T < ∞ if α = 0 . [ABSTRACT FROM AUTHOR]

In this paper, we establish existence results for positive solutions for the (n−1,1) three-point boundary value problems consisting of the equation u(n)+λa(t)fu(t)=0, t∈(0,1), with one of the following boundary value conditions: u(0)=αu(η), u(1)=βu(η), u(i)(0)=0 for i=1,2,…,n−2, and u(n−2)(0)=αu(n−2)(η), u(n−2)(1)=βu(n−2)(η), u(i)(0)=0 for i=0,1,…,n−3, where η∈(0,1), α&ges;0, β&ges;0, and a :(0,1)→R may change sign and R=(−∞,+∞). f(0)>0, λ>0 is a parameter. Our approach is based on the Leray–Schauder degree theory. This paper is motivated by Eloe and Henderson (Nonlinear Anal. 28 (1997) 1669–1680). [Copyright &y& Elsevier]

Abstract: Consider the higher order nonlinear partial difference equation of neutral type where , , and . In this paper, we first establish the discrete Arzela–Ascoli''s theorem. Next, we obtain some sufficient conditions for the existence of bounded and unbounded nonoscillatory solution of Eq. (∗). [Copyright &y& Elsevier]

Abstract: The g-Navier–Stokes equations in spatial dimension 2 were introduced by Roh as with the continuity equation where g is a suitable smooth real valued function. Roh proved the existence of global solutions and the global attractor, for the spatial periodic and Dirichlet boundary conditions. Roh also proved that the global attractor of the g-Navier–Stokes equations converges (in the sense of upper continuity) to the global attractor of the Navier–Stokes equations as in the proper sense. In this paper, we will estimate the dimension of the global attractor , for the spatial periodic and Dirichlet boundary conditions. Then, we will see that the upper bounds for the dimension of the global attractors converge to the corresponding upper bounds for the global attractor as in the proper sense. [Copyright &y& Elsevier]

Abstract: In this paper we study a class of inequality problems for the stationary Navier–Stokes type operators related to the model of motion of a viscous incompressible fluid in a bounded domain. The equations are nonlinear Navier–Stokes ones for the velocity and pressure with nonstandard boundary conditions. We assume the nonslip boundary condition together with a Clarke subdifferential relation between the pressure and the normal components of the velocity. The existence and uniqueness of weak solutions to the model are proved by using a surjectivity result for pseudomonotone maps. We also establish a result on the dependence of the solution set with respect to a locally Lipschitz superpotential appearing in the boundary condition. [Copyright &y& Elsevier]

Abstract: We discussed oscillating equations with Neumann boundary value in [Nonlinear Anal. 54 (2003) 431–443] and [J. Math. Anal. Appl. 298 (2004) 14–32] and prove the existence of infinitely many nonconstant solutions. However, it seems difficult to find infinitely many disjoint order intervals for oscillating equations with Dirichlet boundary value. To get rid of this difficulty, in this paper, we build up a mountain pass theorem in half-order intervals and use it to study oscillating problems with Dirichlet boundary value in which we only have the existence of super-solutions (or sub-solutions) and obtain new results on the exactly infinitely many solutions. [Copyright &y& Elsevier]

This paper is concerned with a class of quasilinear parabolic equations with singularity and arbitrary degeneracy. The existence and uniqueness of generalized solutions to a kind of boundary value problem is established. [Copyright &y& Elsevier]

In this paper, by means of a new twin fixed-point theorem in a cone, the existence of at least two positive solutions of m-point boundary value problem for second order dynamic equations on time scales is considered. [Copyright &y& Elsevier]

In this paper, Schaefer''s fixed point theorem and nonlinear alternative of Leray–Schauder type are used to investigate the existence of solutions for second order boundary value problem for impulsive dynamic equations on time scales. [Copyright &y& Elsevier]

In this paper, we study existence of solutions of first order dynamic inclusions on time scales with general boundary conditions. Both the ∇-derivative and Δ-derivative cases are considered. Examples are presented to illustrate that the Δ-derivative case needs more restrictive assumptions. [Copyright &y& Elsevier]

In this paper, we show that if L is a completely distributive commutative subspace lattice or a J-subspace lattice, then the space of all bounded derivations of algL is reflexive. We also study when local automorphisms on some algebras are automorphisms. [Copyright &y& Elsevier]

In this paper, we consider boundary value problems for nonlinear differential equations on the semi-axis (0,∞) and also on the whole axis (−∞,∞), under the assumption that the left-hand side being a second order linear differential expression belongs to the Weyl limit-circle case. The boundary value problems are considered in the Hilbert spaces L2(0,∞) and L2(−∞,∞), and include boundary conditions at infinity. The existence and uniqueness results for solutions of the considered boundary value problems are established. [Copyright &y& Elsevier]

In this paper, we consider the existence of at least three positive solutions for the 2nth order m-point boundary value problem y(2n)(t)=ft,y(t),y″(t),…,y(2(n−1))(t), 0&les;t&les;1,y(2i)(0)=0, y(2i)(1)=∑lower limit j=1, upper limit m−2 kijy(2i)(ξj), 0&les;i&les;n−1, where kij>0 (i=0,1,…,n−1; j=1,2,…,m−2), 0<ξ1<ξ2<⋯<ξm−2<1. The associated Green''s function for the higher order m-point boundary value problem is first given, and growth conditions are imposed on f which yield the existence of multiple positive solutions by using the Leggett–Williams fixed point theorem. [Copyright &y& Elsevier]

This paper deals with a nonlinear string-beam system describing the torsional-vertical oscillations of a suspension bridge. We consider the initial-boundary value problem and study the existence and uniqueness question. We assume time independent right hand sides, but allow quite general nonlinear terms. Using the Faedo–Galerkin method we prove the existence of a unique solution on an arbitrary large time interval. [Copyright &y& Elsevier]

This paper is devoted to the study of boundary value problems on infinite time intervals for nonlinear, discrete-time systems. For problems of the form x(k+1)=fk,x(k)+h(k), k=0,1,2,…, subject to constraints or nonlocal boundary conditions of the form ∑lower limit k=0, upper limit ∞ gk,x(k)=y we analyze how perturbations in h and y affect the existence of l∞-solutions of this problem. In this setting, h is an element of l∞ and y belongs to Rp. We also study the existence and behavior of bounded solutions to problems of the form x(k+1)=fλ,k,x(k), k=0,1,2,…, subject to ∑lower limit k=0, upper limit ∞ gλ,k,x(k)=0. We place particular importance on the behavior of the solutions as a function of the parameter λ. [Copyright &y& Elsevier]