The paper deals with some existence results for an elliptic equation of Kirchhoff‐type with changing sign data and a logarithmic nonlinearity by direct variational method, Galerkin approach, and sub‐super solutions method. Our results are natural extension of Boulaaras' work in (Math Methods Appl Sci; 41(13):5203‐5210).

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This paper formulates a new optimization pickup and delivery problem with time windows which take into account CO2 emissions. This new NP-hard combinatorial optimization problem is called green pickup and delivery problem with time windows (GPDPTW), the recent development in the vehicle routing problem and its variants, which extends PDP and PDPTW with respect to several constraints. The objective is to find a set of routes for a fleet of vehicles in order to serve given transportation requests with a minimization of fuel consumption and CO2 emission to ensure the preservation of a clean and green environment. This paper presents a mathematical formulation and proposes a hybrid discrete artificial bee colony algorithm (HDABC) as a meta-heuristic algorithm which combines a discrete artificial bee colony with neighborhood operators to solve the GPDPTW model. To the best of our knowledge, this is the first time that an emission of CO2 for the PDPTW is proposed. We performed computational experiments to evaluate the effectiveness of the proposed method, which provides the best result and can effectively find an optimal tour. Our results show that, (1) the shortest route is not necessarily the route that consumes the least fuel; (2) the fuel consumption is affected by the load and the number of vehicles.Povzetek: Članek predstavi novo metodo za optimizacijo prevzema in dostave s časovnimi okni z minimalizacijo porabe goriva in emisij CO2.

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A Hermitian metric on a complex manifold is called SKT (strong Kähler with torsion) if the Bismut torsion 3-form H is closed. As the conformal generalization of the SKT condition, we introduce a new type of Hermitian structure, called locally conformal SKT (or shortly LCSKT). More precisely, a Hermitian structure (J, g) is said to be LCSKT if there exists a closed non-zero 1-form α such that dH = α ∧ H. In the paper we consider non-trivial LCSKT structures, i.e. we assume that dH = 0 and we study their existence on Lie groups and their compact quotients by lattices.In particular, we classify 6-dimensional nilpotent Lie algebras admitting a LCSKT structure and we show that, in contrast to the SKT case, there exists a 6-dimensional 3-step nilpotent Lie algebra admitting a non-trivial LCSKT structure. Moreover, we show a characterization of even dimensional almost abelian Lie algebras admitting a non-trivial LCSKT structure, which allows us to construct explicit examples of 6-dimensional unimodular almost abelian Lie algebras admitting a non-trivial LCSKT structure. The compatibility between the LCSKT and the balanced condition is also discussed, showing that a Hermitian structure on a 6-dimensional nilpotent or a 2n-dimensional almost abelian Lie algebra cannot be simultaneously LCSKT and balanced, unless it is Kähler.