Prove Existence Results Research Articles

Special representatives of complexified Kähler classes

Motivated by constructions appearing in mirror symmetry, we study special representatives of complexified Kähler classes, which extend the notions of constant scalar curvature and extremal representatives for usual Kähler classes. In particular, we provide a moment map interpretation, discuss a possible correspondence with compactified Landau–Ginzburg models, and prove existence results for such special complexified Kähler forms and their large volume limits in certain toric cases.

Banach Contraction Principle-Type Results for Some Enriched Mappings in Modular Function Spaces

The idea of enriched mappings in normed spaces is relatively a newer idea. In this paper, we initiate the study of enriched mappings in modular function spaces. We first introduce the concepts of enriched ρ-contractions and enriched ρ-Kannan mappings in modular function spaces. We then establish some Banach Contraction Principle type theorems for the existence of fixed points of such mappings in this setting. Our results for enriched ρ-contractions are generalizations of the corresponding results from Banach spaces to modular function spaces and those from contractions to enriched ρ-contractions. We make a first ever attempt to prove existence results for enriched ρ-Kannan mappings and deduce the result for ρ-Kannan mappings. Note that even ρ-Kannan mappings in modular function spaces have not been considered yet. We validate our main results by examples.

Existence Results for an m-Point Mixed Fractional-Order Problem at Resonance on the Half-Line

This work considers the existence of solutions for a mixed fractional-order boundary value problem at resonance on the half-line. The Mawhin’s coincidence degree theory will be used to prove existence results when the dimension of the kernel of the linear fractional differential operator is equal to two. An example is given to demonstrate the main result obtained.

Existence and supercontractive estimates for parabolic–elliptic systems

In this paper we study existence and summability of the solutions of the following parabolic–elliptic system of partial differential equations ut−div(A(x,t)∇u)=−div(uM(x)∇ψ)in Ω×(0,T), −div(M(x)∇ψ)=|u|θin Ω×(0,T), ψ(x,t)=0on ∂Ω×(0,T), u(x,t)=0on ∂Ω×(0,T), u(x,0)=u0(x)in Ω where θ∈(0,1), Ω is a bounded subset of RN, N>2, and T>0. We will prove existence results for initial data u0 in L1(Ω). Moreover, despite the datum u0 is assumed to be only a summable function and although the function uM(x)∇ψ in the divergence term of the first equation is not regular enough, there exist solutions that immediately improve their summability and belong to every Lebesgue space. Finally, we study the behavior in time of such solutions and we prove estimates that describe their blow-up for t near zero.

Existence results for a nonlinear coupled radiative–conductive heat transfer system

In this paper, we present existence results for a one-dimensional coupled system of nonlinear integro-differential equations on Lp-spaces with 1≤p<∞. This system describes the steady-state combined radiative–conductive heat transfer. We reformulate the full coupled system as a fixed point problem. This formulation is applied to prove existence results for the radiation intensity and the temperature fields. We treat separately the cases p=1 and p>1. The analysis for p>1 uses the Krasnosel’skii fixed point theorem, while the existence results in the case p=1 are obtained via a new variant of that theorem for the weak topology established in Latrach et al. (2006).

Strongly singular problems in exterior domains

We prove existence and uniqueness of a weak solution of the singular problem{−Δpu=K(x)uδ in Ωe,u>0 in Ωe,u=0on ∂Ω,u(x)→0as|x|→∞, where Ω⊂RN(N>2) is a simply connected, bounded domain containing the origin with smooth boundary, Ωe=RN∖Ω‾ is the exterior domain, 1<p<N, K(x) is an appropriately decaying weight function and δ≥1. Additionally, we prove existence results and discuss decay rates of the solutions near ∂Ω and at infinity when 1uδ is replaced with f(u)uδ or 1uδ+λf(u) for some positive function f and bifurcation parameter λ>0. We use approximation scheme as well as sub- and supersolution method to prove our results. Finally, we establish that solutions corresponding to the nonlinearity 1uδ+λf(u) approach the unique solution corresponding to the singular term 1uδ as λ→0+.

Periodic solutions for one-dimensional nonlinear nonlocal problem with drift including singular nonlinearities

In this paper, we prove existence results of a one-dimensional periodic solution to equations with the fractional Laplacian of order $s\in (1/2,1)$, singular nonlinearity and gradient term under various situations, including nonlocal contra-part of classical Lienard vector equations, as well other nonlocal versions of classical results know only in the context of second-order ODE. Our proofs are based on degree theory and Perron's method, so before that we need to establish a variety of priori estimates under different assumptions on the nonlinearities appearing in the equations. Besides, we obtain also multiplicity results in a regime where a priori bounds are lost and bifurcation from infinity occurs.

Optimal control for obstacle problems involving time-dependent variational inequalities with Liouville\u2013Caputo fractional derivative

We consider an optimal control problem for a time-dependent obstacle variational inequality involving fractional Liouville–Caputo derivative. The obstacle is considered as the control, and the corresponding solution to the obstacle problem is regarded as the state. Our aim is to find the optimal control with the properties that the state is closed to a given target profile and the obstacle is not excessively large in terms of its norm. We prove existence results and establish necessary conditions of obstacle problems via the approximated time fractional-order partial differential equations and their adjoint problems. The result in this paper is a generalization of the obstacle problem for a parabolic variational inequalities as the Liouville–Caputo fractional derivatives were used instead of the classical derivatives.

Infinite combinatorics in mathematical biology

Is it possible to apply infinite combinatorics and (infinite) set theory in theoretical biology? We do not know the answer yet but in this article we try to present some techniques from infinite combinatorics and set theory that have been used over the last decades in order to prove existence results and independence theorems in algebra and that might have the flexibility and generality to be also used in theoretical biology. In particular, we will introduce the theory of forcing and an algebraic construction technique based on trees and forests using infinite binary sequences. We will also present an overview of the theory of circular codes. Such codes had been found in the genetic information and are assumed to play an important role in error detecting and error correcting mechanisms during the process of translation. Finally, examples and constructions of infinite mixed circular codes using binary sequences hopefully show some similarity between these theories - a starting point for future applications.

$rho$-Attractive Elements in Modular Function Spaces

In this paper, we introduce the notion of ρ-attractive elements in modular function spaces. A new class of mappings called ρ-k-nonspreading mappings is also introduced. Making a good use of the two notions, we first prove existence results and then some approximation results in the setup of modular function spaces. An example is presented to support the results proved herein.

Optimization of nonlinear eigenvalues under measure or perimeter constraint

In this paper we recall some recent results about variational eigenvalues of the p-Laplacian, we show new applications and point out some open problems. We focus on the continuity properties of the eigenvalues under the gamma_p-convergence of capacitary measures, which are needed to prove existence results for the minimization of nonlinear eigenvalues in the class of p-quasi open sets contained in a box under a measure constraint. Finally, the new contribution of this paper is to show that these continuity results can be employed to prove existence of minimizers for nonlinear eigenvalues among measurable sets contained in a box and under a perimeter constraint.

On sequential fractional Caputo $ (p, q) $-integrodifference equations via three-point fractional Riemann-Liouville $ (p, q) $-difference boundary condition

&lt;abstract&gt;&lt;p&gt;In this paper, we aim to study the problem of a sequential fractional Caputo $ (p, q) $-integrodifference equation with three-point fractional Riemann-Liouville $ (p, q) $-difference boundary condition. We use some properties of $ (p, q) $-integral in this study and employ Banach fixed point theorems and Schauder's fixed point theorems to prove existence results of this problem.&lt;/p&gt;&lt;/abstract&gt;

Control and optimal stopping Mean Field Games: a linear programming approach

We develop the linear programming approach to mean-field games in a general setting. This relaxed control approach allows to prove existence results under weak assumptions, and lends itself well to numerical implementation. We consider mean-field game problems where the representative agent chooses both the optimal control and the optimal time to exit the game, where the instantaneous reward function and the coefficients of the state process may depend on the distribution of the other agents. Furthermore, we establish the equivalence between mean-field games equilibria obtained by the linear programming approach and the ones obtained via the controlled/stopped martingale approach, another relaxation method used in a few previous papers in the case when there is only control.

Existence results for closed Finsler geodesics via spherical complexities

We apply topological methods and a Lusternik-Schnirelmann-type approach to prove existence results for closed geodesics of Finsler metrics on spheres and projective spaces. The main tool in the proofs are spherical complexities, which have been introduced in earlier work of the author. Using them, we show how pinching conditions and inequalities between a Finsler metric and a globally symmetric metric yield the existence of multiple closed geodesics as well as upper bounds on their lengths.

On the existence of the solution of non-linear coupled ordinary differential equation arising in swirling flow in rotating frame of references

In this article, the steady, incompressible fluid above a radially stretching rotating disk is investigated regarding rotating frame of reference. The three dimensional Navier-Stokes equations are transformed into ordinary differential equation group by a suitable similarity transformation. In particular, an attempt has been made to prove existence results of the solution by using topological shooting argument. At the end, numerical results have been described with all possible physical interpretation to validate our existence results.

Existence of entropy solutions to nonlinear degenerate parabolic problems with variable exponent and L 1-data

Abstract In the present paper, we prove existence results of entropy solutions to a class of nonlinear degenerate parabolic p(·)-Laplacian problem with Dirichlet-type boundary conditions and L 1 data. The main tool used here is the Rothe method combined with the theory of variable exponent Sobolev spaces.

Existence results for fractional order boundary value problem with nonlocal non-separated type multi-point integral boundary conditions

In this article, we discuss the existence of solutions of a fractional boundary value problem of order m ∈ (1, 2], with nonlocal non-separated type integral multipoint boundary conditions. Shaefer type and Krasnoselskii's fixed point theorems are used to prove existence results for the given problem. To establish the uniqueness of solutions Banach contraction principle is used. The criteria for HyersUlam stability of the given boundary value problem is also discussed. Some examples are included for the illustration of our results.

Existence and Regularity Results for Some Elliptic Equations with Degenerate Coercivity and Singular Quadratic Lower-Order Terms

In this paper, we study the existence and regularity results for some elliptic equations with degenerate coercivity and singular quadratic lower-order terms with natural growth with respect to the gradient. The model problem is 0.1 $$\begin{aligned} \left\{ \begin{array}{ll} - \mathrm {div}\left( \frac{\nabla u}{(1+|u|)^{\gamma }}\right) +\frac{\vert \nabla u\vert ^{2}}{u^{\theta }}=f+u^{r} &{} \quad \text{ in }\ \Omega ,\\ u=0&{} \quad \text{ on }\ \partial \Omega , \end{array}\right. \end{aligned}$$ where $$\Omega $$ is a bounded open subset in $$\mathbb {R}^{N}$$ , $$0<\theta <1$$ , $$\gamma >0$$ and $$0<r<2-\theta $$ . We will prove existence results for solutions under various assumptions on the summability of the source f.

Gradient Ricci almost soliton warped product

We present the necessary and sufficient conditions for constructing gradient Ricci almost solitons that are realized as warped products. This will be done by means of Bishop–O’Neill’s formulas and a particular study of Riemannian manifolds satisfying a Ricci–Hessian type equation. We prove existence results and give an example of particular solutions of the PDEs that arise from our construction. We also prove a rigidity result for a gradient Ricci soliton Riemannian product in the class of gradient Ricci almost soliton warped products under some natural geometric assumptions on the warping function.

Hardy-Sobolev type inequality and supercritical extremal problem

This paper deals with Hardy-Sobolev type inequalities involving variable exponents. Our approach also enables us to prove existence results for a wide class of quasilinear elliptic equations with supercritical power-type nonlinearity with variable exponent.