In this paper, we prove a minimization theorem for a proper lower semicontinuous convex function in a real Banach space, applying Takahashi’s nonconvex minimization theorem. Then we give another proof of Bishop-Phelps’ theorem.

We consider integral functionals of the type F (u) := R Ω f(x, u, Du) dx exhibiting a gap between the coercivity and the growth exponent: L−1|Du|p ≤ f(x, u, Du) ≤ L(1 + |Du|q) 1 < p < q 1 ≤ L < +∞ . We give lower semicontinuity results and conditions ensuring that the relaxed functional F is equal to R Ω Qf(x, u, Du) dx, where Qf denotes the usual quasi-convex envelope; our conditions are sharp...

We study the problem of minimizing the second Dirichlet eigenvalue for the Laplacian operator among sets of given perimeter. In two dimensions, we prove that the optimum exists, is convex, regular, and its boundary contains exactly two points where the curvature vanishes. In N dimensions, we prove a more general existence theorem for a class of functionals which is decreasing with respect to se...

The purpose of this article is to prove strong convergence theorems for total asymptotically strict quasi-φ-pseudo-contractions by using a hybrid projection algorithm in Banach spaces. As applications, we apply our main results to find minimizers of proper, lower semicontinuous, and convex functionals and solutions of equilibrium problems.

In the paper, the notion of a generalized convexity was defined and studied from the view-point of the selection and approximation theory of set-valued maps. We study the simultaneous existence of continuous relative selections and graph-approximations of lower semicontinuous and upper semicontinuous set-valued maps with α-convex values having nonempty intersection.

Our aim in the current article is to extend the developments in Kruger, Ngai & Théra, SIAM J. Optim. 20(6), 3280–3296 (2010) and, more precisely, to characterize, in the Banach space setting, the stability of the local and global error bound property of inequalities determined by proper lower semicontinuous under data perturbations. We propose new concepts of (arbitrary, convex and linear) pert...

We give a categorical treatment, in the spirit of Baez and Fritz, of relative entropy for probability distributions defined on standard Borel spaces. We define a category called SbStat suitable for reasoning about statistical inference on standard Borel spaces. We define relative entropy as a functor into Lawvere’s category [0,∞] and we show convexity, lower semicontinuity and uniqueness.

We show that in a Banach space X every closed convex subset is strongly proximinal if and only if the dual norm is strongly sub differentiable and for each norm one functional f in the dual space X∗, JX(f) the set of norm one elements in X where f attains its norm is compact. As a consequence, it is observed that if the dual norm is strongly sub differentiable then every closed convex subset of...

Research on underwater robots is attracting increased attention around the world. Various kinds of underwater robots have been developed, using an assortment of shapes, sizes, weights, and propulsion methods. In this paper, we propose a novel underwater robot, employing a spherical hull and equipped with multiple vectored water-jet-based thrusters. The overall design of the robot is first intro...

Radzik (1991) showed that two-player games on compact intervals of the real line have ε – equilibria for all ε > 0, provided that payoff functions are upper semicontinuous and strongly quasi-concave. In an attempt to generalize this theorem, Ziad (1997) stated that the same is true for n-player games on compact, convex subsets of Rm, m ≥ 1 provided that we strengthen the upper semicontinuity co...