Using the concept of $\mathcal {I}$-convergence we provide a Korovkin type approximation theorem by means of positive linear operators defined on an appropriate weighted space given with any interval of the real line. We also study rates of convergence by means of the modulus of continuity and the elements of the Lipschitz class.
The paper deals with the multivalued boundary value problem x ′ ∈ A(t,x)x + F(t,x) for a.a. t ∈ [a, b], Mx(a) + Nx(b) = 0, in a separable, reflexive Banach space E. The nonlinearity F is weakly upper semicontinuous in x. We prove the existence of global solutions in the Sobolev space W1,p([a, b], E) with 1 < p < ∞ endowed with the weak topology. We consider the case of multiple solutions of the associated homogeneous linearized problem. An example completes the discussion.
We present an algorithm for computing the greatest integer that is not a solution of the modular Diophantine inequality ax mod b ≤ x, with complexity similar to the complexity of the Euclid algorithm for computing the greatest common divisor of two integers.