Volume 37, Issue 4

In the present work, we give the definition of an s -convex functions for a convex real-valued function f defined on the set of integers ℤ{\mathbb{Z}}. We state and prove the discrete Hermite–Hadamard inequality for s -convex functions by using the basics of discrete calculus (i.e. the calculus on ℤ{\mathbb{Z}}). Finally, we state and prove the discrete fractional Hermite–Hadamard inequality for s -convex functions by using the basics of discrete fractional calculus.

In this paper, we compute the general solution and determine the Ulam–Hyers stability for a new form of hexadecic functional equations in multi-Banach spaces.

We prove the existence of variational solutions to equations of the form ∂t⁡u-div⁡(Dξ⁢f⁢(x,t,D⁢u))=0,\partial_{t}u-\operatorname{div}(D_{\xi}f(x,t,Du))=0, where the function f merely satisfies a p -growth condition and is convex with respect to the gradient variable. In particular, we do not require any regularity assumption with respect to time. We obtain an existence result for integrands that are Lipschitz continuous in time via the method of minimizing movements. For the general existence result, we show stability of solutions with respect to approximation of the integrands. In this context, we prove a result related to Γ-convergence that is also valid for functionals with (p,q){(p,q)}-growth.

We discuss certain quantitative geometric properties of energy stationary currents describing minimal surfaces under gravitational forces. Enclosure theorems give statements about the confinement of the support of currents to certain enclosing sets on the basis that one knows something about the position of their boundaries. These results are closely related to non-existence theorems for currents with connected support. Finally, we define a weak formulation in the theory of varifolds for the curvature flow associated to this energy functional. We extend the enclosure results to the flow and discuss several comparison principles.

In our paper in this journal, entitled “Remarks on L2L^{2} boundedness of Littlewood–Paley operators”, there are two incomplete statements and incompleteness in the proof of the main theorem. In this short note we will correct them.