Axioms

With a specific Darboux transformation, we construct solutions to the sine-Gordon equation. We use both the simple Darboux transformation as well as the multiple Darboux transformation, which enables the obtainment of compact solutions of this equation. We give a complete description of the method and the corresponding proofs. We explicitly construct some solutions for the first orders. Using particular generating functions, we give Wronskian representations of the solutions to the sine-Gordon equation. In this case, we give different solutions to this equation. We deduce generalized Wronskian representations of the solutions to the sine-Gordon equation. As an application, we give the general expression of the k-negaton-l-positon-n-soliton solutions of the sine-Gordon equation and we construct some explicit examples of these solutions as well as m complexitons.

Nonsmooth multiobjective mathematical programming problems with equilibrium constraints (NMMPEC) are studied in this article in the Hadamard manifold setting. In the context of $\left(\mathrm{NMMPEC}\right)$, the generalized Guignard constraint qualification (GGCQ) is formulated within the framework of Hadamard manifolds. Moreover, Karush–Kuhn–Tucker (KKT)-type necessary optimality conditions are derived for $\left(\mathrm{NMMPEC}\right)$. Thereafter, we explore constraint qualifications (CQ) tailored to $\left(\mathrm{NMMPEC}\right)$ in the Hadamard manifold setting. Interrelations between these constraint qualifications are subsequently derived. It is further demonstrated that the proposed constraint qualifications, when satisfied, ensure that GGCQ holds. It is noteworthy that constraint qualifications and optimality conditions for $\left(\mathrm{NMMPEC}\right)$ have not been investigated in the Hadamard manifold setting.

In this research article, we formulate and prove the multidimensional Widder–Arendt theorem and the integrated form of the multidimensional Widder–Arendt theorem for functions with values in sequentially complete locally convex spaces. Established results seem to be new even for scalar-valued functions.