A nonintersection property for extremals of variational problems with vector-valued functions

In this work we study the structure of extremals of variational problems with vector-valued functions on [0,∞). We show that if an extremal is not periodic, then the corresponding curve in the phase space does not intersect itself.     

Turnpike property for extremals of variational problems with vector-valued functions

In this paper we study the structure of extremals of variational problems with large enough , fixed end points and an integrand from a complete metric space of functions. We will establish the turnpike property for a generic integrand . Namely, we will show that for a generic integrand , any small and an extremal of the variational problem with large enough , fixed end points and the integrand , for each the set is equal to a set up to in the Hausdorff metric. Here is a compact set depending only on the integrand and are constants which depend only on and , .

     

On variational problems with obstacles and integral constraints for vector-valued functions

The authors prove existence and regularity for vectorvalued solutions of n-dimensional variational problems with boundary conditions, integral constraints, and obstacles as side conditions. Main emphasis is given to the regularity proof in the case n=2 generalizing a well known technique due to C. B. Morrey. In addition, a regularity result is stated for the general n-dimensional case.Dedicated to Hans Lewy and Charles B. Morrey, Jr.     

A geometric maximum principle for variational problems in spaces of vector-valued functions of bounded variation

Variational integrals with density having linear growth on spaces of vector-valued BV -functions are discussed; it is proved that Im (u) ⊂ K for minimizers u, provided that the boundary data take their values in a closed convex set K; it is assumed in addition that the integrand satisfies natural structure conditions. Bibliography: 14 titles.     

A minimax theorem for vector-valued functions

In this work, as usual in vector-valued optimization, we consider the partial ordering induced in a topological vector space by a closed and convex cone. In this way, we define maximal and minimal sets of a vector-valued function and consider minimax problems in this setting. Under suitable hypotheses (continuity, compactness, and special types of convexity), we prove that, for every $$\alpha \varepsilon Max\bigcup\limits_{s\varepsilon X_o } {Min_w } f(s,Y_0 ),$$ there exists $$\beta \varepsilon Min\bigcup\limits_{r\varepsilon Y_o } {Max} f(X_0 ,t),$$ such that β ≤ α (the exact meanings of the symbols are given in Section 2).     

A characterization of quasiconvex vector-valued functions

The aim of this paper is to characterize in terms of scalar quasiconvexity the vector-valued functions which are -quasiconvex with respect to a closed convex cone in a Banach space. Our main result extends a well-known characterization of -quasiconvexity by means of extreme directions of the polar cone of , obtained by Dinh The Luc in the particular case when is a polyhedral cone generated by exactly linearly independent vectors in the Euclidean space .

     

A minimax theorem for vector-valued functions,part 2

We deal with the minimax problem relative to a vector-valued functionf: X ₀×Y ₀ »V, where a partial ordering in the topological vector spaceV is induced by a closed and convex coneC. In Ref. 1, under suitable hypotheses, we proved that $$Max\bigcup\limits_{s\varepsilon X_0 } {Min_w f(s,Y_0 )} \subset Min\bigcup\limits_{t\varepsilon Y_0 } {Maxf(X_0 ,t) + C;}$$ the exact meaning of the symbols is given in Section 2. In this work, we prove that, under a reasonable setting of hypotheses, the previous inclusion holds and also we have that $$Min_w \bigcup\limits_{t\varepsilon Y_0 } {Max} f(X_0 ,t) \subset Max\bigcup\limits_{s\varepsilon X_0 } {Min_w } f(s,Y_0 ) - C.$$      

A minimal set-valued strong derivative for vector-valued Lipschitz functions

A set-valued derivative for a function at a point is a set of linear transformations whichapproximates the function near the point. This is stated precisely, and it is shown that, in general, there is not a unique minimal set-valued derivative for functions in the family of closed convex sets of linear transformations. For Lipschitz functions, a construction is given for a specific set-valued derivative, which reduces to the usual derivative when the function is strongly differentiable, and which is shown to be the unique minimal set-valued derivative within a certain subfamily of the family of closed convex sets of linear transformations. It is shown that this constructed set may be larger than Clarke's and Pourciau's set-valued derivatives, but that no irregularity is introduced.The author would like to thank Professor H. Halkin for numerous discussions of the material contained here.     

A minimax theorem for vector-valued functions in lexicographic order

In this paper, a minimax theorem and a saddle point theorem are obtained for vector-valued functions in the sense of lexicographic order, respectively. An equivalent relationship between the minimax inequality and the saddle point is established. Some examples are given to illustrate our results.     

Interpolation and vector-valued functions

Let A be a subspace of C(X), and let K ? X be an interpolation set for A. Let F be a Banach space. We study the following question: When is K a set of interpolation for A ? F, a space of vector-valued functions naturally associated with A ?     

Haar's condition for systems of vector-valued functions

Translated from Matematicheskie Zametki, Vol. 45, No. 1, pp. 10–15, January–February, 1989.     

Decomposition of vector-valued additive functions

From the additive nonnegative fonctions, defined on a - algebra of subsets, the property of countable additivity separates a class of regular objects, namely measures. Among additive Banach-valued functions, countable additivity is already not necessary for non-pathology. In the paper one isolates a class of regular vector-valued additive functions (measures) and one proves a theorem on the decomposition of a vector-valued additive function into the sum of a regular component (measure) and a purely pathological additive function, similar to a purely finitely additive measure.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 177, pp. 148–156, 1989.     

Multiplication of continuous vector-valued functions

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Chebyshev subspaces of vector-valued functions

It is shown that if on a compact space Q any polynomial 0} $$ " align="middle" border="0"> , in a system of continuous vector functions with real coefficients such that N=n·s and s=2p +1 has at most n–1 zeros, then Q is homeomorphic to a circle or a part of one.Translated from Matematicheskie Zametki, Vol. 19, No. 3, pp. 347–352, March 1976.In conclusion, the author thanks S. B. Stechkin for stating the problem globally.     

Hardy spaces of vector-valued functions

The purpose of this paper is to consider the relations among the various definitions of the spaces H^p of analytic functions with values in a Banach space and to investigate the problem of the structure of the conjugates of these spaces. In particular, one constructs an example of a reflexive separable Banach space χ, for which the equality H^p(X)^*=H^p′(X^*) (1<p<∞,1/p+1/p′=1) ails. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 65, pp. 5–16, 1976.