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Nonlinear Functional Analysis And Its Applications

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Difference Equations An Introduction With Applications

Partial Difference Equations.

Roughly speaking, the basic idea behind the so-called minimax method is the following: Find a critical value of a functional ϕ ∈ C1 (X, ℝ) as a minimax (or maximin) value c ∈ ℝ of ϕ over a suitable class A of subsets of X: 

$$ c = \mathop {\inf }\limits_{A \in \mathcal{A}} \mathop {\sup }\limits_{u \in A} \phi \left( u \right). $$

The Mountain-Pass Theorem

Variational And Non Variational Methods Variational Bayesian methods are a family of techniques for approximating intractable integrals arising in Bayesian inference and machine learning.They are typically used in complex statistical models consisting of observed variables (usually termed "data") as well as unknown parameters and latent variables, with various sorts of relationships among the three types of random variables, as ...

Variational and Non-Variational Methods in Nonlinear Analysis and Boundary Value Problems

In this paper we are interested in the existence of infinitely many solutions for a partial discrete Dirichlet problem depending on a real parameter. More precisely, we determine unbounded intervals of parameters such that the treated problems admit either an unbounded sequence of solutions, provided that the nonlinearity has a suitable behaviour at infinity, or a pairwise distinct sequence of solutions that strongly converges to zero if a similar behaviour occurs at zero. Finally, the attained solutions are positive when the nonlinearity is supposed to be nonnegative thanks to a discrete maximum principle.

Discrete Elliptic Dirichlet Problems and Nonlinear Algebraic Systems

In this paper we establish the existence of multiple solutions for a partial discrete Dirichlet problem depending on a real parameter. More precisely, under appropriate assumptions on the nonlinearities, we determine exact collections of parameters such that the treated problems admit at least three solutions.

Multiple solutions for partial discrete Dirichlet problems depending on a real parameter

In this work we deal with the symmetric algebra of monomial ideals that arise from graphs, the edge ideals. The notion of s-sequence is explored for such ideals in order to compute standard algebraic invariants of their symmetric algebra in terms of the corresponding invariants of special quotients of the polynomial ring related to the graphs.

https://journals.tubitak.gov.tr/math/issues/mat-12-36-3/mat-36-3-1-1010-68.pdf

Invariants of symmetric algebras associated to graphs

We investigate, using the notion of linear quotients, significative classes of connected graphs whose monomial edge ideals, not necessarily squarefree, have linear resolution, in order to compute standard algebraic invariants of the polynomial ring related to these graphs modulo such ideals. Moreover we are able to determine the structure of the ideals of vertex covers for the edge ideals associated to the previous classes of graphs which can have loops on any vertex. Lastly, it is showed that these ideals are of linear type.

Monomial ideals of graphs with loops

In this paper, we study the existence of at least three distinct solutions for a perturbed anisotropic discrete Dirichlet problem. Our approach is based on recent variational methods for smooth functionals defined on reflexive Banach spaces. Some examples are presented to demonstrate the application of our main results.

Maurizio Imbesi

Papers

Discrete Elliptic Dirichlet Problems and Nonlinear Algebraic Systems

Multiple solutions for partial discrete Dirichlet problems depending on a real parameter

Invariants of symmetric algebras associated to graphs

Monomial ideals of graphs with loops

Existence of three weak solutions for a perturbed anisotropic discrete Dirichlet problem