Existence theorems for multidimensional Lagrange problems

This chapter focuses on the metric geometry of Teichmüller space. It first explains how one can think of Teich(Sɡ) as the space of complex structures on Sɡ. To this end, the chapter defines quasiconformal maps between surfaces and presents a solution to the resulting Teichmüller's extremal problem. It also considers the correspondence between complex structures and hyperbolic structures, along with the Teichmüller mapping, Teichmüller metric, and the proof of Teichmüller's uniqueness and existence theorems. The fundamental connection between Teichmüller's theorems, holomorphic quadratic differentials, and measured foliations is discussed as well. Finally, the chapter describes the Grötzsch's problem, whose solution is tied to the proof of Teichmüller's uniqueness theorem.

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Solution existence theorems for finite horizon optimal economic growth problems

Optimization ◽

10.1080/02331934.2021.1939340 ◽

2021 ◽

pp. 1-21

Author(s):

Vu Thi Huong

Keyword(s):

Economic Growth ◽

Existence Theorems ◽

Finite Horizon ◽

Solution Existence ◽

Optimal Economic Growth

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On Existence Theorems to Symmetric Functional Set-Valued Differential Equations

Symmetry ◽

10.3390/sym13071219 ◽

2021 ◽

Vol 13 (7) ◽

pp. 1219

Author(s):

Marek T. Malinowski

Keyword(s):

Differential Equations ◽

Existence And Uniqueness ◽

Existence Theorems ◽

Type Theorem ◽

Integral Representations ◽

Uniqueness Of The Solution ◽

Nonempty Compact ◽

Picard Type Theorem ◽

Convex Subsets

In this paper, we consider functional set-valued differential equations in their integral representations that possess integrals symmetrically on both sides of the equations. The solutions have values that are the nonempty compact and convex subsets. The main results contain a Peano type theorem on the existence of the solution and a Picard type theorem on the existence and uniqueness of the solution to such equations. The proofs are based on sequences of approximations that are constructed with appropriate Hukuhara differences of sets. An estimate of the magnitude of the solution’s values is provided as well. We show the closeness of the unique solutions when the equations differ slightly.

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Solutions for parametric double phase Robin problems

Asymptotic Analysis ◽

10.3233/asy-201598 ◽

2021 ◽

Vol 121 (2) ◽

pp. 159-170 ◽

Cited By ~ 2

Author(s):

Nikolaos S. Papageorgiou ◽

Calogero Vetro ◽

Francesca Vetro

Keyword(s):

Boundary Condition ◽

Existence Theorems ◽

Robin Boundary Condition ◽

Phase Problem ◽

Double Phase ◽

Robin Boundary ◽

Double Phase Problem

We consider a parametric double phase problem with Robin boundary condition. We prove two existence theorems. In the first the reaction is ( p − 1 )-superlinear and the solutions produced are asymptotically big as λ → 0 + . In the second the conditions on the reaction are essentially local at zero and the solutions produced are asymptotically small as λ → 0 + .

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Existence Theorems on Solvability of Constrained Inclusion Problems and Applications

Abstract and Applied Analysis ◽

10.1155/2018/6953649 ◽

2018 ◽

Vol 2018 ◽

pp. 1-10 ◽

Cited By ~ 1

Author(s):

Teffera M. Asfaw

Keyword(s):

Reflexive Banach Space ◽

Existence Theorems ◽

Maximal Monotone ◽

Existence Of Solution ◽

Existence Results ◽

Uniformly Convex ◽

Homotopy Invariance ◽

Inclusion Problems ◽

Compact Resolvents ◽

Nonlinear Variational Inequality

LetXbe a real locally uniformly convex reflexive Banach space with locally uniformly convex dual spaceX⁎. LetT:X⊇D(T)→2X⁎be a maximal monotone operator andC:X⊇D(C)→X⁎be bounded and continuous withD(T)⊆D(C). The paper provides new existence theorems concerning solvability of inclusion problems involving operators of the typeT+Cprovided thatCis compact orTis of compact resolvents under weak boundary condition. The Nagumo degree mapping and homotopy invariance results are employed. The paper presents existence results under the weakest coercivity condition onT+C. The operatorCis neither required to be defined everywhere nor required to be pseudomonotone type. The results are applied to prove existence of solution for nonlinear variational inequality problems.

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