Abstract:
We give a characterization of the boundaries of smooth strictly convex sets in the Euclidean plane R^{2？}based on the existence and uniqueness of inscribed triangles.

Abstract:
In this paper, new sufficient optimality theorems for a solution of a differentiable bilevel multiobjective optimization problem (BMOP) are established. We start with a discussion on solution concepts in bilevel multiobjective programming; a theorem giving necessary and sufficient conditions for a decision vector to be called a solution of the BMOP and a proposition giving the relations between four types of solutions of a BMOP are presented and proved. Then, under the pseudoconvexity assumptions on the upper and lower level objective functions and the quasiconvexity assumptions on the constraints functions, we establish and prove two new sufficient optimality theorems for a solution of a general BMOP with coupled upper level constraints. Two corollary of these theorems, in the case where the upper and lower level objectives and constraints functions are convex are presented.

Abstract:
we propose a discrete approximation scheme to a class of linear quadratic continuous time problems. it is shown, under positiveness of the matrix in the integral cost, that optimal solutions of the discrete problems provide a sequence of bounded variation functions which converges almost everywhere to the unique optimal solution. furthermore, the method of discretization allows us to derive a number of interesting results based on finite dimensional optimization theory, namely, karush-kuhn-tucker conditions of optimality and weak and strong duality. a number of examples are provided to illustrate the theory.

Abstract:
We show that an absolute normalized norm on is strictly convex if and only if the corresponding convex function on is strictly convex. In this context the monotonicity property of these norms is discussed. We also introduce the notion of the direct sum of Banach spaces and equipped with the associated norm with and characterize the strict convexity of .

Abstract:
We propose a discrete approximation scheme to a class of Linear Quadratic Continuous Time Problems. It is shown, under positiveness of the matrix in the integral cost, that optimal solutions of the discrete problems provide a sequence of bounded variation functions which converges almost everywhere to the unique optimal solution. Furthermore, the method of discretization allows us to derive a number of interesting results based on finite dimensional optimization theory, namely, Karush-Kuhn-Tucker conditions of optimality and weak and strong duality. A number of examples are provided to illustrate the theory.

Abstract:
In this paper a new duality mapping is defined, and it is our object to show that there is a similarity among these three types of characterizations of a strictly convex 2-normed space. This enables us to obtain more new results along each of two types of characterizations. We shall also investigate a strictly 2-convex 2-normed space in terms of the above two different types.

Abstract:
In this paper, we propose a refinement in the analytical definition of the s_{2}-convex classes of functions aiming to progress further in the direction of including s_{2}-convexity properly in the body of Real Analysis.

In this note, we analyze a few major claims about . As a consequence, we
rewrite a major theorem, nullify its proof and one remark of importance, and
offer a valid proof for it. The most important gift of this paper is probably
the reasoning involved in all: We observe that a constant, namely t, has been changed into a variable, and
we then tell why such a move could not have been made, we observe the
discrepancy between the claimed domain and the actual domain of a supposed
function that is created and we then explain why such a function
could not, or should not, have been created, along with others.

Abstract:
We studied the monotonicity and Convexity properties of the new functions involving the gamma function, and get the general conclusion that Minc-Sathre and C. P. Chen-G. Wang’s inequality are extended and refined.