Volume 17, Issue 2

A class of third-order nonlinear neutral differential equation ( r ( t )( y ″( t )) α )′ + q ( t ) ƒ( x ( σ ( t ))) = 0 is investigated in this paper, where y ( t ) = x ( t ) + p ( t ) x ( τ ( t )), and α > 0 is any quotient of odd integers. Using a new method, we obtain some sufficient conditions for the oscillation of the above equation, and some known oscillation criteria be extended. An example is inserted to illustrate the result.

Ellipses will be considered as subsets of suitably defined Minkowski planes in such a way that, additionally to the well-known area content property A ( r ) = π ( a,b ) r 2 , the number π ( a,b ) = abπ reflects a generalized circumference property U ( a,b ) ( r ) = 2 π ( a,b ) r of the ellipses E ( a,b ) ( r ) with main axes of lengths 2 ra and 2 rb , respectively. In this sense, the number π ( a,b ) is an ellipse number w.r.t. the Minkowski functional r of the reference set E ( a,b ) (1). This approach is closely connected with a generalization of the method of indivisibles and avoids elliptical integrals. Further, several properties of both a generalized arc-length measure and the ellipses numbers will be discussed, e.g. disintegration of the Lebesgue measure and an elliptically contoured Gaussian measure indivisiblen representation, wherein the ellipses numbers occur in a natural way as norming constants.

In this article, we study the duals of homogeneous weighted sequence Besov spaces , where the weight w is non-negative and locally integrable. In particular, when 0 < p < 1, we find a type of new sequence spaces which characterize the duals of . Also, we find the necessary and sufficient conditions for the boundedness of diagonal matrices acting on homogeneous weighted sequence Besov spaces. Using these results, we give some applications to characterize the boundedness of Fourier–Haar multipliers and paraproduct operators. In this situation, we need to require that the weight w is an A p weight.

We study bounded positive definite double sequences which are stationary with respect to a polynomial hypergroup structure generated by . Connected with bounded positive definite and R n -stationary double sequences is an R n -stationary sequence of elements in a Hilbert space. We derive an ergodic theorem for such R n -stationary sequences and we give a complete characterization of the space of multipliers defined by such an R n -stationary sequence. Further we give examples of bounded positive definite double sequences.

The purpose of this work is to extend the relative entropy S ( A | B ) = A 1/2 log( A –1/2 BA –1/2 ) A 1/2 from positive operators to convex functionals. Our functional approach implies immediately, in a fast way, some simplifications and improvements for that of positive operators already discussed in the literature.

We discuss several classes of linear second order initial-boundary value problems in which damping terms appear in the main wave equation and/or in the dynamic boundary condition. We investigate their well-posedness and describe some qualitative properties of their solutions, like boundedness and stability. In particular, we provide sufficient conditions for analyticity, boundedness, asymptotic almost periodicity and exponential stability of certain C 0 -semigroups associated to such problems.

We consider the Nemytskij operator, defined by ( Nφ )( x ) ≔ G ( x, φ ( x )), where G is a given set-valued function. It is shown that if N maps AC( I, C ), the space of all absolutely continuous functions on the interval I ≔ [0, 1] with values in a cone C in a reflexive Banach space, into AC( I , 𝒦), the space of all absolutely continuous set-valued functions on I with values in the set 𝒦, consisting of all compact intervals (including degenerate ones) on the real line ℝ, and N is uniformly continuous, then the generator G is of the form G ( x, y ) = A ( x )( y ) + B ( x ), where the function A ( x ) is additive and uniformly continuous for every x ∈ I and, moreover, the functions x ↦ A ( x )( y ) and B are absolutely continuous. Moreover, a condition, under which the Nemytskij operator maps the space AC( I, C ) into AC( I , 𝒦) and is Lipschitzian, is given.