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We give a simple necessary and sufficient condition for the existence of star supremum for two arbitrary operators on a Hilbert space. It is shown that the results of Hartwig (1979) [15] , and Janowitz (1983) [18] , when applied to the ring of bounded operators on a Hilbert space, require much simpler conditions. We also give a complete answer to a question stated by Hartwig and Drazin (1982) in [16] , regarding the extremal values of range and null-space of the star infimum.

This paper deals with the following variational problem with variable exponents { − div ( ( 1 + | ∇ u | p ( x ) 1 + | ∇ u | 2 p ( x ) ) | ∇ u | p ( x ) − 2 ∇ u ) = f ( x , u ) , in Ω , u = 0 , on ∂ Ω , where Ω ⊆ R N ( N ≥ 2 ) be a bounded domain with the smooth boundary ∂Ω, p : Ω ‾ → R is a Lipschitz continuous function with 1 p − : = ess inf x ∈ Ω ⁡ p ( x ) ≤ p ( x ) ≤ p + : = ess sup x ∈ Ω ⁡ p ( x ) N and f ∈ C ( Ω × R , R ) is superlinear but does not satisfy the usual Ambrosetti-Rabinowitz type condition. Under three different superlinear conditions on f at infinity, we prove that the above equation has at least a nontrivial solution. Moreover, the existence of infinite many solutions is proved for odd nonlinearity. Especially, some new tricks are introduced to show the boundedness of Cerami sequences.

We consider a problem of minimization of convex function f ( x ) over the convex region R where the objective function and the feasible region have a common direction of recession. In cases when one of these directions is not in the constancy space of the objective function, then the minimal solution is not achieved even if the function f ( x ) is bounded below over the region R . Many algorithms, if applied to this class of programmes, do not guarantee convergence to the global infimum. Our approach to this problem leads to derivation of the equation of the feasible parametrized curve C ( t ), such that the infimum of the logarithmic penalty function along this curve is equal to the global infimum of the objective function over the region R . We show that if all functions defining the program are analytic, then C ( t ) is also an analytic function. The equation of the curve can be successfully used to determine the global infimum (in particular, unboundedness) of the convex constrained programmes in cases when the application of classical methods, such as the steepest descent method, fails to converge to the global infimum.

We give some sufficient conditions of deterioration of smoothness under the operation of convolution. We show that the convolution of two probability densities which are restrictions to R of entire functions can possess infinite essential supremum on each interval. © 1998 Academic Press.

In this paper, we prove that the bounded set of fuzzy numbers must exist supremum and infimum and give the concrete representation of supremum and infimum. As an application, we obtain that the continuous fuzzy-valued function on a closed interval exists supremum and infimum and give the precise representation. We also show that the bounded fuzzy-valued function on a closed interval can define the lower and upper sums and the lower and upper integrals of Riemann and Riemann–Stieltjes by the usual way.

Let (S, d) be a Polish space, M (S) the set of all probability measures on the σ-field of Borel subsets of S, and ⋁p = ⋁p(S) the set of all Pψ⋁(S) with ∝ d(o, ·)p dP D p (P,Q); = inf ∫d p dγ q γ ∈⋁(S 2 ) has marginal P,Q , if p ∞ inf{λ− ess sup d∥λψ⋁(S 2 has marginals P,Q}, ∞ where q:= min {1, 1 p )} . It is known, at least for 0

The maximum amplitude cost of a control function u(t) taken to be ess sup g(t, u(t)), where g(t, u) is a given function. (A particular example is g(t, u) = the norm of u.) We consider linear systems with this cost functional. The existence of optimal control is proved, and it is shown that the ess sup is uniformly essential with respect to the optimal controls. Properties of the extended attainable set are discussed and compared with the case of an integral cost. Finally, we show in what sense a cost functional of the form (∝ g(t, u(t)) q ) 1 q approximates the ess sup cost functional.

Let Z and R be the set of all integers and real numbers, respectively, and let Q = [0,27c] x [0, 2711. Let L’(Q) be the space of measurable real-valued functions U: Sz + R which are Lebesgue integrable over 52 with usual norm II.jI L1. Let L2(Q) be the space of measurable real-valued functions U: Sz + R which are Lebesgue square integrable over Q with usual inner product ( , ) and usual norm I/ . ljL2 and let L”(Q) be the space of measurable real-valued functions U: Q + R which are essentially bounded with the norm 1) u 11 Lm = ess sup 1 u( t, x) I . (I. x) E R Let C“(Q) be the space of all continuous functions u.: 52 --f R such that the partial derivatives up to order k with respect to both variables are continuous on 52, while C(Q) is used for Co(Q) with the usual norm 11. Iloo.

This paper is devoted to the study of four integral operators that are basic generalizations and modifications of fractional integrals of Hadamard in the space Xcp of functions f on R + =(0,∞) such that ∫ 0 ∞ u c f(u) p du u ess sup u>0 u c |f(u)| for c∈ R =(−∞,∞) , in particular in the space Lp(0,∞) (1⩽p⩽∞). The semigroup property and its generalizations are established for the generalized Hadamard-type fractional integration operators under consideration. Conditions are also given for the boundedness in Xcp of these operators; they involve Kummer confluent hypergeometric functions as kernels.