We present some results and open problems in the Calculus of Variations.

Let E be a separable real inner product space of dimension at least 2 and V be a metrizable and separable linear topological space. We show that the set of all orthogonally additive functions mapping E into V and having big graphs is dense in the space of all orthogonally additive functions from E into V with the Tychonoff topology.

On a commutative ring R we study outer measures induced by measures on Spec(R). The focus is on examples of such outer measures and on subsets of R that satisfy the Carathéodory condition.

Let E be a complete uniform topological algebra with Arens-Michael normed factors (E_α)_α∈Λ. Then M(E)≅lim_←M(E_α) within an algebra isomorphism ϕ. If each factor E_α is complete, then every multiplier of E is continuous and ϕ is a topological algebra isomorphism where M(E) is endowed with its seminorm topology.

In this article, quadratic B-spline Galerkin method has been employed to solve the time fractional order Schrödinger equation. Numerical solutions and error norms L₂ and L_∞ are presented in tables.

Based on a characterization of logarithmic functions due to Heuvers we develop analogous results for multiplicative, exponential and additive functions, respectively.

Functional inequalities generalizing m-convexity are considered. A result of a sandwich type is proved. Some applications are indicated.

In this paper we study some generalization of invariant means on Banach spaces. We give some sufficient condition for the existence of the invariant mean and some examples where we have not it.

In this paper we provide some conditions under which a Lie derivation on a trivial extension algebra is proper, that is, it can be expressed as a sum of a derivation and a center valued map vanishing at commutators. We then apply our results for triangular algebras. Some illuminating examples are also included.

We give some stability results for the functional equation a(xy) = a(x) + a(y), where a: G→E, G being a groupoid and E a Banach space.

In this expository paper, we present a new and easier proof of the Polar Decomposition Theorem. Unlike in classical proofs, we do not use the square root of a positive matrix. The presented proof is accessible to a broad audience.

In our previous work titled “Fixed Point Results Satisfying Rational type Contractive Conditions in Complex Valued Metric Spaces”[Ann. Math. Sil. 30 (2016), 89–110], some errors has been made in the main results (Theorem 3.1, Theorem 3.7 and Theorem 3.22), that may misguide the readers. This note provides corrections of these errors.

Report of Meeting. The Seventeenth Katowice–Debrecen Winter Seminar Zakopane (Poland) , February 1–4, 2017.