We study the existence of weak solutions for the following class of problems { alpha triangle(2)u + beta triangle u = & micro;u + gamma h(x, u) in ohm, B(u) = 0 on partial derivative ohm, where ohm C R-N is a bounded smooth domain, N >= 1, alpha >= 0, -infinity < beta < alpha lambda(1), lambda(1) is the first eigenvalue of (-triangle, H-0(1)(ohm)), & micro; is an element of(0, & micro;), & micro; < & micro;(2), & micro;(2)is the second eigenvalue of the problem (alpha triangle(2)u+beta triangle u, H-0(1) (ohm)boolean AND H-2(ohm)), gamma=/ 0 is a real parameter and h : ohm & times;R -> R is a Carath & eacute;o dory function verifying some conditions, the boundary condition B(u) = 0 on partial derivative ohm means that u = triangle u = 0 on partial derivative ohm when alpha > 0 and u = 0 on partial derivative ohm when alpha = 0. In this article, we revisit the arguments of Landesman-Lazer, both in the global and local aspects.
D. Yost [Intersecting balls and proximinal subspaces in Banach spaces, Ph. D. Thesis, University of Edinburgh (1979)] introduced a notion called equability for subspaces of a Banach space, as a sufficient condition for the proximinality and lower semi-continuity of the metric projection. This notion was later modified by S. Lalithambigai [Ball proximinality of equable spaces, Collect. Math. 60/1 (2009) 79-88] to study ball proximinality. In this article, we study both the equability notions. We establish some geometric characterizations and strengthen the existing best approximation theoretic consequences of equability notions. Specifically, we prove that these equability notions are related to quasi uniform rotundity and uniform rotundity. We observe that equability notions are transitive through M-summands and further explore their stability under Pp-direct sum for 1 <= p <= infinity. Additionally, we show that equability can be lifted to the space of all vector valued bounded (and continuous) functions defined on topological spaces.
We study the triangle functionals induced by convex functions. To this end the vectorial majorization intended for comparing two tuples of vectors is used. A Hardy-Littlewood-P & oacute;lya-Karamata (HLPK) type theorem is proved for the triangle functionals. As applications, some refinements of the celebrated Jensen's inequality are established. In doing so, the specification of HLPK Theorem for two triples of vectors is utilized. Special attention is paid to convex functions that are simultaneously concave (affine) on some subset of their domains. In particular, functions that are both convex and piecewise affine are considered. Some applications for convex sequences are also provided.
Prox-regularity is a fundamental property of functions in second-order variational analysis that has come to be understood as marking the boundary for convexity-like behavior. Broad classes of examples have long been familiar, but there has not been any pointwise test based on generalized second derivatives to check for its presence. Such tests are developed here in terms of newly defined strict second subderivatives and strict second-order subdifferentials. They also identify the exact associated level of variational s-convexity, whether positive or negative. A recent criterion of Gfrerer for the level of variational s-convexity of a function already identified as prox-regular is shown to follow also from the theory of generalized twice differentiability of convex functions and their conjugates. Tests of the strong variational sufficient condition for local optimality are obtained as an application.
This paper is devoted to the optimization of the discrete and hyperbolic differential inclusions for polyhedral control problems with initial-boundary conditions. We formulate necessary and sufficient optimality conditions in the form of an Euler-Lagrange inclusion and transversality condition for the stated discrete problem. Then, to obtain optimality conditions for a discrete approximation problem, we approximate the discrete problem using the difference approximations of partial derivatives and grid functions on a uniform grid. The idea of that discretization method is to combine the discrete problem with the differential problem. Thus, the derivation of sufficient optimality conditions for the continuous problem is implemented by passing formally to the limit as the discrete steps tend to zero in the discrete-approximation problem. Finally, we conduct a numerical example to illustrate the efficiency of our results.
We give an explicit example of a compact convex set without relative interior. Moreover, we show that a Banach space in which every nonempty compact convex subset has nonempty relative interior is finite dimensional. The relation between several close variants of the notion of relative interior is discussed. Finally, we generalize the Borwein-Lewis theorem on the existence of quasi relative interior.
This paper is devoted to the study of variational analysis of the orthogonally invariant norm cone of symmetric matrices. For a general orthogonally invariant norm cone of symmetric matrices, formulas for the tangent cone, normal cone and second-order tangent set are established. The differentiability properties of the projection operator onto the orthogonally invariant norm cone are developed, including formulas for the directional derivative and the B-subdifferential. In particular, the directional derivative is characterized by the second-order derivative of the corresponding symmetric norm, which is convenient for computation.
C-p (Y|X) denotes the real-valued continuous functions on Y subset of X having continuous extensions to a Tychonoff space X, with pointwise topology inherited from Cp(Y). We recently proved C-p(Y) is distinguished negated left right arrow it is a large subspace of R-Y. We prove C-p (Y|X) is always a large subspace of C-p(Y). Thus Cp (Y|X) is always quasibarrelled; always has a feral strong dual; has a quasibarrelled countable enlargement double left right arrow Y is infinite; is distinguished negated left right arrow C-p(Y) is distinguished; is a Montel space double left right arrow Y is discrete and C-embedded in X. 'Nice' countable covers for C-p (Y|X) yield potent summary theorems that solve open problems, characterize P-spaces anew, and complete the list of Velichko variations. For example, Summary III: Assume Y is dense in X. Y is a P-space, or X is pseudocompact, or both negated left right arrow C-p (Y|X) is countably covered by sets that are, respectively, relatively sequentially complete in C-p(Y), or bounded, or both. Putting Y = X, one quickly comprehends Velichko variations & agrave; la Arkhangel'skii.
Let S be a finite set in a real linear space and let JS be a family consisting of Sintervals in R. In this paper we deal with a convex operator co(S, JS) called the convex interval hull. This operator generalizes the familiar concepts of the convex hull, conv(S), and the affine hull, aff(S), of S. The set co(S, JS) is a convex subset of the linear space and can be either bounded or unbounded, depending on the families JS. In this paper we apply co(S, JS) to obtain unbounded images of a finite set S. As special images of co(S, JS) for finite S we obtain such unbounded objects as: hyperplanes, cylinders, cones, penumbras and wedges. We also apply co(S, JS) to study some properties of extreme points. In relation to co(S, JS) we introduce the so-called extreme interval operator Eco(S) and prove some analogues of the celebrated Minkowski-Krein-Milman's theorem.
The paper is a survey of available results concerning the generalized Bolza problem under fairly general assumptions on its component functions that can be extended-real-valued. Specifically, we consider the problem of existence of solutions, necessary conditions and characterization of the value function.
The Scaled Relative Graph (SRG) is a geometric tool that maps the action of a multi-valued nonlinear operator onto the 2D plane, used to analyze the convergence of a wide range of iterative methods. As the SRG includes the spectrum for linear operators, we can view the SRG as a generalization of the spectrum to multi-valued nonlinear operators. In this work, we further study the SRG of linear operators and characterize the SRG of block-diagonal and normal matrices.
We investigate how the subgradients of the value function of a discrete-time convex Bolza problem evolve over time. In particular, we develop a discrete-time version of the method of characteristics introduced by Rockafellar and Wolenski in the 2000s, by showing that the time-evolution of the subgradients of the value functions can be associated with trajectories of a discrete-time Hamiltonian system. To do so, we first prove that the value function has a dual counterpart, which corresponds to the conjugate of the value function of a suitable dual problem. We finally discuss about the qualification conditions required for our results, showing in particular that classical problems, such as the Linear Quadratic Regulator, satisfy these hypotheses.
We show that the Carath & eacute;o dory number of the joint numerical range of d many bounded selfadjoint operators is at most d-1, and even at most d-2 if the underlying Hilbert space has dimension at least 3. This extension of the classical convexity results for numerical ranges shows that also joint numerical ranges are significantly less non-convex than general sets.
We observe that the theorem of Hesselager may be used to obtain new proofs of Hermite-Hadamard inequalities for functions that are convex with respect to a Chebyshev system. Such inequalities were proved by Bessenyei and Pales for integrals with density function, in our approach, the first function from the system involved must be constant but the main theorem is valid for the integral with respect to any measure. Additionally, we present examples of inequalities that follow from Hesselager's theorem but are not of the Hermite-Hadamard type.
The aim of this paper is to investigate the dual problem of a generalized equilibrium problem in the framework of a reflexive Banach space. By means of the Fenchel duality, we introduce the dual problem defined via the Fitzpatrick transform of the bifunction involved, that turns out to be an equilibrium problem itself in the dual space. We present conditions which entail the solvability of both primal and dual problems.
Using advanced concepts and techniques from convex and variational analysis (such as the notion of twice epi-differentiability), we establish, under a set of appropriate conditions (including a polyhedricity assumption), that the solution to a parameterized nonlinear variational inequality associated with a positively homogeneous function is differentiable, and furthermore that its derivative is the solution to a corresponding complementarity problem. We illustrate our main result through a series of examples and counterexamples. In particular we introduce an example of a projection operator onto a nonempty closed convex cone that is not directionally differentiable, which is, to our best knowledge, new in the literature.
We characterize the closed convex sets K subset of R(n )which satisfy the following approximation condition: for any given point v is an element of rintK and a scalar epsilon > 0, there is a polyhedron P subset of R-n such that v is an element of rintP and P subset of K subset of v + (1 + epsilon)(P-v).
Let (Omega, Sigma, & micro;) be a measure space with at least two disjoint sets of finite and positive measure, and S+ = S+(Omega,Sigma,& micro;) denote the set of all & micro;-integrable simple functions x : Omega -> R+ having support Omega(x) of positive measure. Then, for an arbitrary bijection phi : (0, infinity) -> (0,infinity), the functional P-phi : S+ -> R+ given by P-phi (x) := phi(-1)(integral(ohm(x)) phi (degrees )xd & micro;) is well defined. The results presented support the conjecture that subadditivity of P-phi implies the convexity of phi. The case of superadditivity of P-phi is also discussed.
We establish the quantitative Brunn-Minkowski inequality and the quantitative Urysohn inequality for p-torsional rigidity. Here the p-torsional rigidity can be formulated by the weak solution of the boundary value problem of p-Laplace equation. In order to prove the main results, we establish the relation between the Borell-Brascamp-Lieb deficit and the Brunn-Minkowski deficit, and then use the relation to prove two different stability of the Borell-Brascamp-Lieb inequality for p-concave functions with p > 0. The constants in our stability results are independent on the parameter p and the integrals of given functions, and thus also improve the results given by Ghilli and Salani about the case p = 2.
We develop the P. Lions concentration-compactness principle for a sequence of Radon measures on R-n, the P. Lions principle is extended to variable exponent Lebesgue spaces L-p(.) (Omega), Omega subset of R-n, n >= 3. Employing this Lp((.))-extension of the concentration-compactness principle, we establish almost exact conditions under which the Dirichlet problem u|(partial derivative ohm) = 0 for variable exponent Laplace equation -div (|Vu|(p(x)-2) del u) + lambda|u|(p(x)-2) u = a (x) |u|(s(x)-2) u + f (x, u) has a weak solution in variable exponent Sobolev space W-1(p(.))(Omega), with critically grown coefficients.
