General Existence Theorem Research Articles

Null curves in $${\mathbb{C}^3}$$ and Calabi–Yau conjectures

For any open orientable surface M and convex domain $${\Omega\subset \mathbb{C}^3,}$$ there exist a Riemann surface N homeomorphic to M and a complete proper null curve F : N → Ω. This result follows from a general existence theorem with many applications. Among them, the followings: Any of the above surfaces can be chosen with hyperbolic conformal structure.

Smoothing positive currents and the existence of Kähler-Einstein metrics

In this paper, we prove a general existence theorem of Kahler-Einstein metrics on complete Kahler manifolds. We use the heat equation method smoothing certain positive (1, 1) current in the canonical class.

OPTIMAL DESIGN OF LOW-CONTRAST TWO-PHASE STRUCTURES FOR THE WAVE EQUATION

This paper is concerned with the following optimal design problem: find the distribution of two phases in a given domain that minimizes an objective function evaluated through the solution of a wave equation. This type of optimization problem is known to be ill-posed in the sense that it generically does not admit a minimizer among classical admissible designs. Its relaxation could be found, in principle, through homogenization theory but, unfortunately, it is not always explicit, in particular for objective functions depending on the solution gradient. To circumvent this difficulty, we make the simplifying assumption that the two phases have a low constrast. Then, a second-order asymptotic expansion with respect to the small amplitude of the phase coefficients yields a simplified optimal design problem which is amenable to relaxation by means of H-measures. We prove a general existence theorem in a larger class of composite materials and propose a numerical algorithm to compute minimizers in this context. As in the case of an elliptic state equation, the optimal composites are shown to be rank-one laminates. However, the proof that relaxation and small-amplitude limit commute is more delicate than in the elliptic case.

Small solutions of nonlinear differential equations near branching points

We construct parametric families of small branching solutions to nonlinear differential equations of the nth order near branching points. We use methods of the analytical theory of branching solutions of nonlinear equations and the theory of differential equations with a regular singular point. We illustrate the general existence theorems with an example of a nonlinear differential equation in a certain magnetic insulation problem.

A Model Problem for Conformal Parameterizations of the Einstein Constraint Equations

We investigate the possibility that the conformal and conformal thin sandwich (CTS) methods can be used to parameterize the set of solutions of the vacuum Einstein constraint equations. To this end we develop a model problem obtained by taking the quotient of certain symmetric data on conformally flat tori. Specializing the model problem to a three-parameter family of conformal data we observe a number of new phenomena for the conformal and CTS methods. Within this family, we obtain a general existence theorem so long as the mean curvature does not change sign. When the mean curvature changes sign, we find that for certain data solutions exist if and only if the transverse-traceless tensor is sufficiently small. When such solutions exist, there are generically more than one. Moreover, the theory for mean curvatures changing sign is shown to be extremely sensitive with respect to the value of a coupling constant in the Einstein constraint equations.

Supercritical Conformal Metrics on Surfaces with Conical Singularities

We study the problem of prescribing the Gaussian curvature on surfaces with conical singularities in supercritical regimes. Using a Morse-theoretical approach, we prove a general existence theorem on surfaces with positive genus, with a generic multiplicity result.

Edgeworth equilibrium in a model of interregional economic relations

We introduce an analog of an Edgeworth equilibrium for a class of multiregional economic systems. We analyze the game-theoretic aspects of the coalition stability of regional development plans and establish a quite general existence theorem for an Edgeworth equilibrium. We discuss the questions of coincidence of the set of these equilibria with the fuzzy core and the set of theWalrasian equilibria of the multiregional systemin question.Our methods rest on a systematic accounting for the polyhedrality of the sets of balanced coalition plans.

On nondegenerate coupling forms

The aim of the present paper is to investigate new classes of symplectically fat fibre bundles. We prove a general existence theorem for fat vectors with respect to the canonical invariant connections. Based on this result we give new proofs of some constructions of symplectic structures. This includes twistor bundles and locally homogeneous complex manifolds. The proofs are conceptually simpler and allow for obtaining more general results.

A Brézis–Browder principle on partially ordered spaces and related ordering theorems

Through a simple extension of Brézis–Browder principle to partially ordered spaces, a very general strong minimal point existence theorem on quasi ordered spaces, is proved. This theorem together with a generic quasi order and a new notion of strong approximate solution allow us to obtain two strong solution existence theorems, and three general Ekeland variational principles in optimization problems where the objective space is quasi ordered. Then, they are applied to prove strong minimal point existence results, generalizations of Bishop–Phelps lemma in linear spaces, and Ekeland variational principles in set-valued optimization problems through a set solution criterion.

On irreversible investment

This paper presents a new and general approach to the theory of irreversible investment. We show that the optimal policy is a base capacity policy and derive general monotone comparative statics results. When the operating profit function is supermodular, the base capacity increases monotonically with the exogenous shock; and firm size is decreasing in the user cost of capital. Last but not least, the paper provides a general existence theorem for optimal policies.

A generalized existence theorem of backward doubly stochastic differential equations

In this paper, we deal with a class of one-dimensional backward doubly stochastic differential equations (BDSDEs). We obtain a generalized comparison theorem and a generalized existence theorem of BDSDEs.

Solution of Volterra operator-integral equations in the nonregular case by the successive approximation method

The branches of solutions of a nonlinear integral equation of Volterra type in a Banach space are constructed by the successive approximation method. We consider the case in which a solution may have an algebraic branching point. We reduce the equation to a system regular in a neighborhood of the branching point. Continuous and generalized solutions are considered. General existence theorems are used to study an initial-boundary value problem with degeneration in the leading part.

Existence of a positive solution to a class of fractional differential equations

In this paper, we consider a (continuous) fractional boundary value problem of the form − D 0 + ν y ( t ) = f ( t , y ( t ) ) , y ( i ) ( 0 ) = 0 , [ D 0 + α y ( t ) ] t = 1 = 0 , where 0 ≤ i ≤ n − 2 , 1 ≤ α ≤ n − 2 , ν > 3 satisfying n − 1 < ν ≤ n , n ∈ N , is given, and D 0 + ν is the standard Riemann–Liouville fractional derivative of order ν . We derive the Green’s function for this problem and show that it satisfies certain properties. We then use cone theoretic techniques to deduce a general existence theorem for this problem. Certain of our results improve on recent work in the literature, and we remark on the consequences of this improvement.

Flat covers in abelian and in non-abelian categories

A general existence theorem for flat covers in (e.g., quasi-abelian) locally finitely presented categories is obtained from an additive Ramsey type theorem. In the abelian case, it is shown that flat covers always exist. Applications to categories of separated presheaves or sheaves, localizations of Bousfield type, torsion-free classes of finite type, and categories of filtered objects or complexes, are given.

A general existence theorem for symmetric floating drops

An existence theorem for floating drops due to Elcrat, Neel, and Siegel is generalized. The theorem applies to all radially symmetric domains, and to both light and heavy floating drops, and utilizes new results in annular capillary theory.

Abstract Differential and Difference Equations

This special issue of Advances in Difference Equations is devoted to highlight some recent developments in abstract differential equations, fractional differential equations, and difference equations and their applications to mathematical physics, engineering, and biology. It consists of 20 papers carefully selected through a rigorous peer review. The first category of papers deals with the asymptotic and oscillatory behavior of solutions to various abstract differential equations and fractional differential equations. Periodic problems involving the scalar p-Laplacian equation on time scales, or n-species nonautonomous food chains with harvesting terms are studied using the Mawhin’s continuation theorem. Some new oscillation criteria for the second-order quasilinear neutral delay dynamic equations and nonlinear delay dynamic equations on a time scale T, are established, improving some known results for oscillation of second-order nonlinear delay dynamic equations on time scales. The study of almost automorphic functions in Bochner’s sense have attracted several mathematicians since the publication of N’Guerekata’s book in 2001. Recently, their applications to fractional differential equations have become an emerging field. A new and general existence and uniqueness theorem of almost automorphic solutions for the semilinear fractional differential equation

Almost Automorphic Solutions to Abstract Fractional Differential Equations

A new and general existence and uniqueness theorem of almost automorphic solutions is obtained for the semilinear fractional differential equation , in complex Banach spaces, with Stepanov-like almost automorphic coefficients. Moreover, an application to a fractional relaxation-oscillation equation is given.

Mild Solutions for Fractional Differential Equations with Nonlocal Conditions

This paper is concerned with the existence and uniqueness of mild solution of the fractional differential equations with nonlocal conditions , in a Banach space , where . General existence and uniqueness theorem, which extends many previous results, are given.

General existence theorems, alternative theorems and applications to minimax problems

We establish general theorems on maximal elements, coincidence points and nonempty intersections for set-valued mappings on GFC-spaces and show their equivalence. Applying them we derive equivalent forms of alternative theorems. As applications, we develop in detail general types of minimax theorems. The results obtained improve or include as special cases several recent ones in the literature.

Self-similar solutions to a generalized Davey–Stewartson system

In this paper we study the global solvability and existence of self-similar solutions to a generalized Davey–Stewartson system. We first use the Banach fixed point theorem to establish a general global existence theorem. By using this general result to a class of special initial data we obtain the existence of global self-similar solutions.