On the Cauchy-Rassias stability of a generalized additive functional equation

Let X and Y be Banach spaces and f:X→Y an odd mapping. We solve the following generalized additive functional equation

     

Numerical radius and zero pattern of matrices

Let A be an n×n complex matrix and r be the maximum size of its principal submatrices with no off-diagonal zero entries. Suppose A has zero main diagonal and x is a unit n-vector. Then, letting ‖A‖ be the Frobenius norm of A, we show that

|〈Ax,x〉^|2?(1−1/2r−1/2n)‖A^‖2.     

Higher order duality in multiobjective fractional programming with support functions

In this paper a new class of higher order (F,ρ,σ)-type I functions for a multiobjective programming problem is introduced, which subsumes several known studied classes. Higher order Mond-Weir and Schaible type dual programs are formulated for a nondifferentiable multiobjective fractional programming problem where the objective functions and the constraints contain support functions of compact convex sets in Rn. Weak and strong duality results are studied in both the cases assuming the involved functions to be higher order (F,ρ,σ)-type I. A number of previously studied problems appear as special cases.     

Well-posedness of a class of perturbed optimization problems in Banach spaces

Let X be a Banach space and Z a nonempty subset of X. Let J:Z→R be a lower semicontinuous function bounded from below and p?1. This paper is concerned with the perturbed optimization problem of finding z₀∈Z such that ‖x−z₀p^‖+J(z₀)=infz∈Z{‖x−zp^‖+J(z)}, which is denoted by minJ(x,Z). The notions of the J-strictly convex with respect to Z and of the Kadec with respect to Z are introduced and used in the present paper. It is proved that if X is a Kadec Banach space with respect to Z and Z is a closed relatively boundedly weakly compact subset, then the set of all x∈X for which every minimizing sequence of the problem minJ(x,Z) has a converging subsequence is a dense Gδ-subset of X?Z₀, where Z₀ is the set of all points z∈Z such that z is a solution of the problem minJ(z,Z). If additionally p>1 and X is J-strictly convex with respect to Z, then the set of all x∈X for which the problem minJ(x,Z) is well-posed is a dense Gδ-subset of X?Z₀.     

The Independence Number for De Bruijn networks and Kautz networks

Let D be a directed graph; the (l,ω)-Independence Number of graph D, denoted by α_l,ω(D), is an important performance parameter for interconnection networks. De Bruijn networks and Kautz networks, denoted by B(d,n) and K(d,n) respectively, are versatile and efficient topological structures of interconnection networks. For l=1,2,…,n, this paper shows that α_l,d−1(B(d,n))=dⁿ,α_l,d−1(K(d,n))=α_l,d(K(d,n))=dⁿ+dⁿ⁻¹ if d≥3 and n≤d−2. In particular, the paper shows the exact value of the Independence Number for B(d,1) and B(d,2) for any d. For the generalized situation, the paper obtains a lower bound α_l,d−1(B(d,n))≥d² if n≥3 and d≥5.     

An algorithm based on resolvent operators for solving variational inequalities in Hilbert spaces

In this paper, a new monotonicity, M$M$-monotonicity, is introduced, and the resolvent operator of an M$M$-monotone operator is proved to be single valued and Lipschitz continuous. With the help of the resolvent operator, an equivalence between the variational inequality VI(C,F+G)$(C,F+G)$ and the fixed point problem of a nonexpansive mapping is established. A proximal point algorithm is constructed to solve the fixed point problem, which is proved to have a global convergence under the condition that F$F$ in the VI problem is strongly monotone and Lipschitz continuous. Furthermore, a convergent path Newton method, which is based on the assumption that the projection mapping _∏C(⋅)${\prod }_C(\cdot )$ is semismooth, is given for calculating ε$\epsilon$-solutions to the sequence of fixed point problems, enabling the proximal point algorithm to be implementable.     

Gradient estimates in Orlicz spaces for quasilinear elliptic equation

In this paper we generalize classical L^q$L^q$, q≥p$q\ge p$, estimates of the gradient to the Orlicz space for weak solutions of quasilinear elliptic equations of p-Laplacian type.