General Existence Theorem Research Articles

A general existence theorem for a multi-valued control problem

A general existence theorem for a multi-valued control problem

On the bifurcation of solutions to Marguerre–von Kármán equations

The objective of this work is to study the bifurcation of solutions of the Marguerre–von Kármán equations, which constitute a mathematical model for the buckling of Marguerre–von Kármán shallow shells. More precisely, we reduce the Marguerre–von Kármán equations to a single equation with a cubic operator; its second member depends on the function that defines the middle surface of the shallow shell and the applied forces. Next, we prove a general existence theorem for the reduced equation, by using the main theorem on pseudomonotone operators. Then we study the bifurcation of solutions in the reduced equation, with a second member, is small, at neighborhood of the simple characteristic value of the linearized problem.

Detachment waves in frictional contact: I — A two-mass system

This work studies a mathematical model for the motion of a two mass–spring–damper system with friction, and has a two-fold purpose. The first is to start a mathematical study of the way detachment or slip waves, when the system transits from stick to slip motion, propagate in discrete and continuous systems. The introduction of friction changes the problems into systems of differential set-valued inclusions, which are mathematically interesting, but rather complex. The second is to use such a ‘simple’ system as an example of a very general existence and uniqueness theorem for systems described by set-valued pseudomonotone operators, Kuttler and Shillor (1999) and Andrews et al. (2020).

Combinatorial and accessible weak model categories

In a previous work, we have introduced a weakening of Quillen model categories called weak model categories. They still allow all the usual constructions of model category theory, but are easier to construct and are in some sense better behaved. In this paper we continue to develop their general theory by introducing combinatorial and accessible weak model categories. We give simple necessary and sufficient conditions under which such a weak model category can be extended into a left and/or right semi-model category. As an application, we recover Cisinski-Olschok theory and generalize it to weak and semi-model categories. We also provide general existence theorems for both left and right Bousfield localization of combinatorial and accessible weak model structures, which combined with the results above gives existence results for left and right Bousfield localization of combinatorial and accessible left and right semi-model categories, generalizing previous results of Barwick. Surprisingly, we show that any left or right Bousfield localization of an accessible or combinatorial Quillen model category always exists, without properness assumptions, and is simultaneously both a left and a right semi-model category, without necessarily being a Quillen model category itself.

A General Existence Theorem and Asymptotics for Non-self-adjoint Sturm-Liouville Problems

A General Existence Theorem and Asymptotics for Non-self-adjoint Sturm-Liouville Problems

Collapsed ancient solutions of the Ricci flow on compact homogeneous spaces

We prove a general existence theorem for collapsed ancient solutions to the Ricci flow on compact homogeneous spaces and we show that they converge in the Gromov–Hausdorff topology, under a suitable rescaling, to an Einstein metric on the base of a torus fibration. This construction generalizes all previous known examples in the literature.

Einstein-Type Elliptic Systems

In this paper, we analyse semi-linear systems of partial differential equations which are motivated by the conformal formulation of the Einstein constraint equations coupled with realistic physical fields on asymptotically Euclidean (AE) manifolds. In particular, electromagnetic fields give rise to this kind of system. In this context, under suitable conditions, we prove a general existence theorem for such systems, and, in particular, under smallness assumptions on the free parameters of the problem, we prove existence of far from CMC (near CMC) Yamabe positive (Yamabe non-positive) solutions for charged dust coupled to the Einstein equations, satisfying a trapped surface condition on the boundary. As a bypass, we prove a Helmholtz decomposition on AE manifolds with boundary, which extends and clarifies previously known results.

Combinatorial vs. classical dynamics: Recurrence

Establishing the existence of periodic orbits is one of the crucial and most intricate topics in the study of dynamical systems, and over the years, many methods have been developed to this end. On the other hand, finding closed orbits in discrete contexts, such as graph theory or in the recently developed field of combinatorial dynamics, is straightforward and computationally feasible. In this paper, we present an approach to study classical dynamical systems as given by semiflows or flows using techniques from combinatorial topological dynamics. More precisely, we present a general existence theorem for periodic orbits of semiflows which is based on suitable phase space decompositions, and indicate how combinatorial techniques can be used to satisfy the necessary assumptions. In this way, one can obtain computer-assisted proofs for the existence of periodic orbits and even certain chaotic behavior.

A new construction of compact torsion-free $G_2$-manifolds by gluing families of Eguchi–Hanson spaces

We give a new construction of compact Riemannian 7-manifolds with holonomy $G_2$. Let $M$ be a torsion-free $G_2$-manifold (which can have holonomy a proper subgroup of $G_2$) such that $M$ admits an involution $\iota$ preserving the $G_2$-structure. Then $M/{\langle \iota \rangle}$ is a $G_2$-orbifold, with singular set $L$ an associative submanifold of $M$, where the singularities are locally of the form $\mathbb R^3 \times (\mathbb R^4 / \{\pm 1\})$. We resolve this orbifold by gluing in a family of Eguchi-Hanson spaces, parametrized by a nonvanishing closed and coclosed $1$-form $\lambda$ on $L$. Much of the analytic difficulty lies in constructing appropriate closed $G_2$-structures with sufficiently small torsion to be able to apply the general existence theorem of the first author. In particular, the construction involves solving a family of elliptic equations on the noncompact Eguchi-Hanson space, parametrized by the singular set $L$. We also present two generalizations of the main theorem, and we discuss several methods of producing examples from this construction.

A unified study of existence theorems in topologically based settings and applications in optimization

ABSTRACT A unified study of necessary and sufficient conditions for the existence of intersection points and other important points in mathematical analysis is presented. The results are established in topologically based settings and shown to have equivalent formulations for various kinds of important points. For set-valued maps from a set to a topological space, instead of a convexity structure on this set, we use two general structures. The first structure is based on continuous single-valued maps from simpleces to the aforementioned set in order to extend the ideas of Knaster–Kuratowski–Mazurkiewicz and Fan for convex hulls. The second one is based on single-valued maps from the unit interval of the real line to the considered set and extends the classical connectedness properties. The general existence theorems of the above types are crucial tools in studies of the solution existence in optimization and other areas of applied mathematics. Here we choose minimax problems for applications of our general theorems. Our results are new, or improve or include as special cases many known results.

Renormalising SPDEs in regularity structures

The formalism recently introduced in [BHZ19] allows one to assign a regularity structure, as well as a corresponding “renormalisation group”, to any subcritical system of semilinear stochastic PDEs. Under very mild additional assumptions, it was shown in [CH16] that large classes of driving noises exhibiting the relevant small-scale behaviour can be lifted to such a regularity structure in a robust way, following a renormalisation procedure reminiscent of the BPHZ procedure arising in perturbative QFT. The present work completes this programme by constructing an action of the renormalisation group on a suitable class of stochastic PDEs which is intertwined with its action on the corresponding space of models. This shows in particular that solutions constructed from the BPHZ lift of a smooth driving noise coincide with the classical solutions of a modified PDE. This yields a very general black box type local existence and stability theorem for a wide class of singular non-linear SPDEs.

Generalized Compactness for Finite Perimeter Sets and Applications to the Isoperimetric Problem

For a complete noncompact Riemannian manifold with bounded geometry, we prove a “generalized” compactness result for sequences of finite perimeter sets with uniformly bounded volume and perimeter in a larger space obtained by adding limit manifolds at infinity. We extend previous results contained in Nardulli (Asian J Math 18(1):1–28, 2014), in such a way that the main theorem is a generalization of the generalized existence theorem, i.e., Theorem 1 of Nardulli (Asian J Math 18(1):1–28, 2014). We replace C2,α locally asymptotic bounded geometry with C0 locally asymptotic bounded geometry.

Differential variational\u2013hemivariational inequalities: existence, uniqueness, stability, and convergence

The goal of this paper is to study a comprehensive system called differential variational–hemivariational inequality which is composed of a nonlinear evolution equation and a time-dependent variational–hemivariational inequality in Banach spaces. Under the general functional framework, a generalized existence theorem for differential variational–hemivariational inequality is established by employing KKM principle, Minty’s technique, theory of multivalued analysis, the properties of Clarke’s subgradient. Furthermore, we explore a well-posedness result for the system, including the existence, uniqueness, and stability of the solution in mild sense. Finally, using penalty methods to the inequality, we consider a penalized problem-associated differential variational–hemivariational inequality, and examine the convergence result that the solution to the original problem can be approached, as a parameter converges to zero, by the solution of the penalized problem.

Convergence of a generalized penalty method for variational–hemivariational inequalities

The aim the paper is to study a large class of variational-hemivariational inequalities involving constraints in a Banach space. First, we establish a general existence theorem for this class. Second, we introduce a sequence of penalized problems without constraints. Under the suitable assumptions, we prove that the Kuratowski upper limit with respect to the weak topology of the sets of solutions to penalized problems, w--lim supn→∞Sn, is nonempty and is contained in the set of solutions to original inequality problem. Also, we prove the identity, w--lim supn→∞Sn=s--lim supn→∞Sn, when operator A satisfies (S)+-property. Finally, we illustrate the applicability of the theoretical results and we explore two complicated partial differential systems of elliptic type, which are an elliptic mixed boundary value problem involving a nonlinear nonhomogeneous differential operator with an obstacle effect, and a nonlinear elastic contact problem in mechanics with unilateral constraints, respectively.

Set-valued Leader type contractions, periodic point and endpoint theorems, quasi-triangular spaces, Bellman and Volterra equations

Set-valued contractions of Leader type in quasi-triangular spaces are constructed, conditions guaranteeing the existence of nonempty sets of periodic points, fixed points and endpoints of such contractions are established, convergence of dynamic processes of these contractions are studied, uniqueness properties are derived, and single-valued cases are considered. Investigated dynamic systems are not necessarily continuous and spaces are not necessarily sequentially complete or Hausdorff. Obtained results suggest, in particular, strategies to new studies of functional Bellman equations and variable discounted Bellman equations in metric spaces and integral Volterra equations in locally convex spaces. Results in this direction are also presented in this paper. More precisely, without continuity of Bellman and Volterra appropriate operators, the sets of solutions of these equations (which are periodic points of these operators) are studied and new and general convergence, existence and uniqueness theorems concerning such equations are proved.

A class of generalized mixed variational–hemivariational inequalities I: Existence and uniqueness results

We investigate a generalized Lagrange multiplier system in a Banach space, called a mixed variational–hemivariational inequality (MVHVI, for short), which contains a hemivariational inequality and a variational inequality. First, we employ the Minty technique and a monotonicity argument to establish an equivalence theorem, which provides three different equivalent formulations of the inequality problem. Without compactness for one of operators in the problem, a general existence theorem for (MVHVI) is proved by using the Fan–Knaster–Kuratowski–Mazurkiewicz principle combined with methods of nonsmooth analysis. Furthermore, we demonstrate several crucial properties of the solution set to (MVHVI) which include boundedness, convexity, weak closedness, and continuity. Finally, a uniqueness result with respect to the first component of the solution for the inequality problem is proved by using the Ladyzhenskaya–Babuška–Brezzi (LBB) condition. All results are obtained in a general functional framework in reflexive Banach spaces.

Solutions to fractional Sobolev-type integro-differential equations in Banach spaces with operator pairs and impulsive conditions

We are concerned with fractional Sobolev-type integro-differential equations in Banach spaces with operator pairs and impulsive conditions, where the operator pairs generate propagation families. With the help of the theory of propagation family and Laplace transforms, along with an estimate for a special sequence improved in this article, we introduce a definition of mild solutions to the impulsive problem for these abstract fractional Sobolev-type integro-differential equations and we establish general existence theorems and a continuous dependence theorem, which essentially extend some previous conclusions. In our results, the operator B could be unbounded, and the existence of an operator B−1 is not necessarily needed. Moreover, we give some examples to illustrate our main results.

The smooth Riemannian extension problem

Given a metrically complete Riemannian manifold $(M,g)$ with smooth nonempty boundary and assuming that one of its curvatures is subject to a certain bound, we address the problem of whether it is possibile to realize $(M,g)$ as a domain inside a geodesically complete Riemannian manifold $(M',g')$ without boundary, by preserving the same curvature bounds. In this direction we provide three kind of results: (1) a general existence theorem showing that it is always possible to obtain a geodesically complete Riemannian extension without curvature constraints; (2) various topological obstructions to the existence of a complete Riemannian extension with prescribed sectional and Ricci curvature bounds; (3) some existence results of complete Riemannian extensions with sectional and Ricci curvature bounds, mostly in the presence of a convexity condition on the boundary.

Properly embedded minimal annuli in $$\mathbb {H}^2 \times \mathbb {R}$$

In this paper we study the moduli space of properly Alexandrov-embedded, minimal annuli in $$\mathbb {H}^2 \times \mathbb {R}$$ with horizontal ends. We say that the ends are horizontal when they are graphs of $${\mathcal {C}}^{2, \alpha }$$ functions over $$\partial _\infty \mathbb {H}^2$$ . Contrary to expectation, we show that one can not fully prescribe the two boundary curves at infinity, but rather, one can prescribe one of the boundary curves, but the other one only up to a translation and a tilt, along with the position of the neck and the vertical flux of the annulus. We also prove general existence theorems for minimal annuli with discrete groups of symmetries.

The biharmonic homotopy problem for unit vector fields on 2-tori

The bienergy of smooth maps between Riemannian manifolds, when restricted to unit vector fields, yields two different variational problems depending on whether one takes the full functional or just the vertical contribution. Their critical points, called biharmonic unit vector fields and biharmonic unit sections, form different sets. Working with surfaces, we first obtain general characterizations of biharmonic unit vector fields and biharmonic unit sections under conformal change of the metric. In the case of a 2-dimensional torus, this leads to a proof that biharmonic unit sections are always harmonic and a general existence theorem, in each homotopy class, for biharmonic unit vector fields.