we prove the generalized hyers--ulam stability  of n--th order linear differential equation of the form $y^{(n)}+p_{1}(x)y^{(n-1)}+ cdots+p_{n-1}(x)y^{prime}+p_{n}(x)y=f(x)$, with condition that there exists a non--zero solution of corresponding homogeneous equation. our main results extend and improve the corresponding results obtained by many authors.

– in this paper we consider some (α ,β ) -metrics such as generalized kropina, matsumoto and f (α β )2α = + metrics, and obtain necessary and sufficient conditions for them to be einstein metrics when βis a constant killing form. then we prove with this assumption that the mentioned einstein metrics must beriemannian or ricci flat.

The generalized Fibonacci cube Qd(f) is the graph obtained from the d-cube Qd by removing all vertices that contain a given binary word f as a factor; the generalized Lucas cube Qd( ↽Ð f ) is obtained from Qd by removing all the vertices that have a circulation containing f as a factor. In this paper the Wiener index of Qd(1) and the Wiener index of Qd( ↽Ð 1) are expressed as functions of the o...

Let P ,X and Y be Banach spaces. Suppose that f : P ×X → Y is continuously Fréchet differentiable function depend on the point (p, x) and F : X ⇒ 2 is a set-valued mapping with closed graph. Consider the following parametric generalized equation of the form: 0 ∈ f(p, x) + F (x). (1) In the present paper, we study an extended Newton-type method for solving parametric generalized equation (1). In...

We find necessary and sufficient conditions for a Banach space operator T to satisfy the generalized Browder’s theorem. We also prove that the spectral mapping theorem holds for the Drazin spectrum and for analytic functions on an open neighborhood of σ(T ). As applications, we show that if T is algebraically M -hyponormal, or if T is algebraically paranormal, then the generalized Weyl’s theore...

Let f be a function meromorphic on the open unit disk D, with angular boundary limits bounded by one in modulus almost everywhere on the unit circle. We give sufficient conditions in terms of boundary asymptotics at finitely many points on the unit circle T for f to be a ratio of two finite Blaschke products. A necessary condition is that f has finitely many poles in D, i.e., that f is a genera...