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With the dual variational principle and the saddle point reduction we use the abstract bifurcation theory recently developed by author in previous work to prove many new bifurcation results for solutions of four types of Hamiltonian…
In this article, discrete variants of several results from vector calculus are studied for classical finite difference summation by parts operators in two and three space dimensions. It is shown that existence theorems for scalar/vector…
We investigate the existence of periodic solutions for a class of nonlocal continuity equations, which include mean-field equations derived from systems of coupled oscillators. While periodic solutions at the particle level have been…
We give a geometric description of variational principles in mechanics, with special attention to constrained systems. For the general case of nonholonomic constraints, a unified variational approach is given, and the equations of motion of…
In this paper we are concerned with existence results for a coupled system of quadratic functional differential equations. This system is reduced to a fixed point problem for a block operator matrix with nonlinear inputs. To prove the…
In this paper, we consider a class of nonlinear fractional differential equations involving Hilfer derivative with boundary conditions. First, we obtain an equivalent integral for the given boundary value problem in weighted space of…
A variational principle is introduced to provide a new formulation and resolution for several boundary value problems with a variational structure. This principle allows one to deal with problems well beyond the weakly compact structure. As…
We review some techniques from non-linear analysis in order to investigate critical paths for the action functional in the calculus of variations applied to physics. Previous attempts to analyse when these are minima ex- ist, but mainly…
At variance with fully inclusive quantities, which have been computed already at the two- or three-loop level, most exclusive observables are still known only at one-loop, as further progress was hampered so far by the greater computational…
For integrable systems in the sense of multidimensional consistency (MDC) we can consider the Lagrangian as a form, which is closed on solutions of the equations of motion. For 2-dimensional systems, described by partial difference…
The relationship between geometric and variational principles remains central to Nonlinear Analysis. This paper introduces the \textbf{Orbit-Summability Fixed Point Criterion}, a novel, purely dynamical condition, and establishes its…
The existence of stationary distributions to distribution dependent stochastic differential equations are investigated by using the ergodicity of the associated decoupled equation and the Schauder fixed point theorem. By using Zvonkin's…
We consider in this paper an area functional defined on submanifolds of fixed degree immersed into a graded manifold equipped with a Riemannian metric. Since the expression of this area depends on the degree, not all variations are…
In this paper, we focus on the existence and uniqueness of solutions of boundary value problems for a coupled system of fractional differential equations with four-point boundary conditions involving $\psi$-Caputo fractional derivatives.…
A modern notion of integrability is that of multidimensional consistency (MDC), which classically implies the coexistence of (commuting) dynamical flows in several independent variables for one and the same dependent variable. This property…
We introduce a fractional variant of the Cahn-Hilliard equation settled in a bounded domain and with a possibly singular potential. We first focus on the case of homogeneous Dirichlet boundary conditions, and show how to prove the existence…
We have established a coherent framework for applying variational methods to partial differential equations on hypergraphs, which includes the propositions of calculus and function spaces on hypergraphs. Several results related to the…
This paper presents some sufficient conditions for the existence of solutions of fractional differential equation with nonlocal multi-point boundary conditions involving Caputo fractional derivative and integral boundary conditions. Our…
We examine the total mixed scalar curvature of a fixed distribution as a functional of a pseudo-Riemannian metric. We develop variational formulas for quantities of extrinsic geometry of the distribution to find the critical points of this…
The difference variational bicomplex, which is the natural setting for systems of difference equations, is constructed and used to examine the geometric and algebraic properties of various systems. Exactness of the bicomplex gives a…