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.
