Existence Results Research Articles

Existence and Multiplicity Results for an Elliptic Problem Involving Mixed Local and Nonlocal Operator

In this work, we consider the following mixed local-nonlocal quasilinear ellipic problem$$(P_\lambda)\left\{\begin{array}{rcll} -\Delta_p u+(-\Delta)_p^s u = \lambda f(u) \text{in}\Omega,\\ u 0 \text{in }\Omega,\\ u = 0 \text{in }\mathbb{R}^N \setminus\Omega,\end{array}\right.$$where Ω⊂RN is a bounded regular domain in RN with 0s1pN and f: R→R is a continuous function, that have a finite number of zeroes, changing sign between them. The main goal of this paper is to prove the existence and multiplicity of positive solutions for such problems by using variational methods.

On theoretical and numerical analysis of fractal--fractional non-linear hybrid differential equations

Abstract Recently, fractals and fractional calculus have received much attention from researchers of various fields of science and engineering. Because the said area has been found applicable in modeling various real-world processes and phenomena. Hybrid differential equations (HDEs) play significant roles in mathematical modeling of various processes because the aforesaid equations incorporate different dynamical systems as specific cases. For instance, it is possible to model and describe non-homogeneous physical phenomena on using the said equations. Therefore, this research work is concerned with studying a class of nonlinear hybrid fractal–fractional differential equations. We develop the existence result for the qualitative study using a hybrid fixed point theorem. For the mentioned goal, a fixed point theory for the product of two operators is applied to deduce appropriate conditions for the existence of exactly one solution. Additionally, the stability result based on Ulam–Hyers is also deduced. The said stability results play an important role in numerical investigations. In addition, a numerical method based on Euler procedure is utilized to approximate the solution of the proposed problems. Various computational test problems are given to demonstrate the results. Also, using various fractal–fractional order values, several graphical presentations are given for the examples. The concerned analysis will help in investigating many real-world problems modeled using HDEs with fractal–fractional orders in the near future.

Unified results for existence and compactness in the prescribed fractional Q-curvature problem

Unified results for existence and compactness in the prescribed fractional Q-curvature problem

Existence and Stability Results for Time-Dependent Impulsive Neutral Stochastic Partial Integrodifferential Equations with Rosenblatt Process and Poisson Jumps

Abstract In this paper, we discuss the existence, uniqueness and stability of mild solutions of time-dependent impulsive neutral stochastic partial integrodifferential equations with the Rosenblatt process and Poisson jumps. The existence of mild solutions for the equations is discussed by means of the semigroup theory and theory of the resolvent operator. Next, under some sufficient conditions, the results are obtained by using the method of successive approximation and Bihari’s inequality. Finally, an example is provided to illustrate our results.

A class of Hessian quotient equations in de Sitter space

Abstract In this paper, we consider the closed spacelike solution to a class of Hessian quotient equations in de Sitter space. Under mild assumptions, we obtain an existence result using standard degree theory based on a priori estimates.

Existence and Uniqueness of Some Unconventional Fractional Sturm–Liouville Equations

Existence and Uniqueness of Some Unconventional Fractional Sturm–Liouville Equations

On a class of Kirchhoff type logarithmic Schrödinger equations involving the critical or supercritical Sobolev exponent

In this paper, we study a class of Kirchhoff type logarithmic Schrödinger equations involving the critical or supercritical Sobolev exponent. Such problems cannot be studied by applying variational methods in a standard way, because the nonlinearities do not satisfy the Ambrosetti-Rabinowitz condition and change sign. Moreover, the appearance of the logarithmic term makes the associated energy functional lose differentiable in the sense of Gateaux. By analyzing the structure of the Nehari manifold and developing some analysis techniques, the above obstacles are overcome in subtle ways and several existence result are obtained. Furthermore, we investigate the regularity, the monotonicity, and the symmetric properties of the solutions via the iterative technique and the moving plane method.

Quasi-Periodic Traveling Waves on an Infinitely Deep Perfect Fluid Under Gravity

We consider the gravity water waves system with a periodic one-dimensional interface in infinite depth and we establish the existence and the linear stability of small amplitude, quasi-periodic in time, traveling waves. This provides the first existence result of quasi-periodic water waves solutions bifurcating from a completely resonant elliptic fixed point. The proof is based on a Nash–Moser scheme, Birkhoff normal form methods and pseudo differential calculus techniques. We deal with the combined problems of small divisors and the fully-nonlinear nature of the equations. The lack of parameters, like the capillarity or the depth of the ocean, demands a refined nonlinear bifurcation analysis involving several nontrivial resonant wave interactions, as the well-known “Benjamin-Feir resonances”. We develop a novel normal form approach to deal with that. Moreover, by making full use of the Hamiltonian structure, we are able to provide the existence of a wide class of solutions which are free from restrictions of parity in the time and space variables.

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Numerical solution of fuzzy fractional volterra integro differential equations with boundary conditions

In this study, the boundary value problem of fuzzy fractional nonlinear Volterra integro differential equations of order 1 < ϱ ≤ 2 is addressed. Fuzzy fractional derivatives are defined in the Caputo sense. To show the existence result, the Krasnoselkii theorem from the theory of fixed points is used, where as the well-known contraction mapping concept is utilized in order to show the solution is unique to the proposed problem. Moreover, a novel Adomian decomposition method is utilized to get numerical solution; the approach behind deriving the solution is from Adomian polynomials, and it is organized according to the recursive relation that is obtained. The proposed method significantly decreases the numerical computations by obtaining solutions without the need of discretization or constrictive assumptions. According to the results, there is substantial agreement between the series solutions produced by the fuzzy Adomian decomposition method. Finally, using MATLAB, the symmetry between the lower and upper-cut representations of the fuzzy solutions is demonstrated in the numerical result.

Existence of Walrasian equilibria with discontinuous, non-ordered, interdependent preferences, without free disposal, and with an infinite-dimensional commodity space

AbstractA new proof of the existence of a Walrasian equilibrium with an infinite dimensional commodity space is provided, which allows agents’ preferences to be discontinuous. The new theorems include as corollaries the existence results of Mas-Collel, Yannelis and Zame, Araujo and Monteiro, and Mas-Collel and Richard, among others.

Blow up at Infinite Time of Solutions for a Plate Equation with Delay Term

In this article, we investigate the viscoelastic plate equation with both delay and source terms. Initially, we give the local-global existence results. Later, we establish the blow-up results at infinite time by utilizing the energy method when E (0) &lt; 0 under suitable conditions. Delays effect generally seems in many practical problems for instance medicine, biological, chemical, physical, thermal, economic phenomena, electrical engineering systems and mechanical applications.

A VARIATIONAL FORMULATION GOVERNED BY TWO BIPOTENTIALS FOR A FRICTIONLESS CONTACT MODEL

We consider a frictionless contact model whose constitutive law and contact condition are described by means of subdifferential inclusions. For this model, we deliver a variational formulation based on two bipotentials. Our formulation envisages the computation of a three-field unknown consisting of the displacement vector, the stress tensor and the normal stress on the contact zone, the contact being described by a generalized Winkler condition. Subsequently, we obtain existence and uniqueness results. Some properties of the solution are also discussed, focusing on the data dependence.

Upper and lower solutions method for a class of second-order coupled systems

This paper provides a class of upper and lower solution definitions for second-order coupled systems by transforming the fourth-order differential equation into a second-order differential system. Then, by constructing a homotopy parameter and utilizing the maximum principle, we propose an upper and lower solutions method for studying a class of second-order coupled systems with Dirichlet boundary conditions and obtain an existence result.

Global solutions of two‐dimensional inviscid and resistive magnetohydrodynamics system near an equilibrium

We investigate the global existence of classical solutions for two‐dimensional incompressible, inviscid, and resistive magnetohydrodynamics (MHD) system. Assuming that the initial magnetic field is close to an equilibrium state of constant magnetic vorticity, the global existence result could be obtained using a time‐weighted energy estimate.

Existence of harmonic maps and eigenvalue optimization in higher dimensions

AbstractWe prove the existence of nonconstant harmonic maps of optimal regularity from an arbitrary closed manifold $(M^{n},g)$ ( M n , g ) of dimension $n&gt;2$ n &gt; 2 to any closed, non-aspherical manifold $N$ N containing no stable minimal two-spheres. In particular, this gives the first general existence result for harmonic maps from higher-dimensional manifolds to a large class of positively curved targets. In the special case of the round spheres $N=\mathbb{S}^{k}$ N = S k , $k\geqslant 3$ k ⩾ 3 , we obtain a distinguished family of nonconstant harmonic maps $M\to \mathbb{S}^{k}$ M → S k of index at most $k+1$ k + 1 , with singular set of codimension at least 7 for $k$ k sufficiently large. Furthermore, if $3\leqslant n\leqslant 5$ 3 ⩽ n ⩽ 5 , we show that these smooth harmonic maps stabilize as $k$ k becomes large, and correspond to the solutions of an eigenvalue optimization problem on $M$ M , generalizing the conformal maximization of the first Laplace eigenvalue on surfaces.

On uniform stability and numerical simulations of complex valued neural networks involving generalized Caputo fractional order

The dynamics and existence results of generalized Caputo fractional derivatives have been studied by several authors. Uniform stability and equilibrium in fractional-order neural networks with generalized Caputo derivatives in real-valued settings, however, have not been extensively studied. In contrast to earlier studies, we first investigate the uniform stability and equilibrium results for complex-valued neural networks within the framework of a generalized Caputo fractional derivative. We investigate the intermittent behavior of complex-valued neural networks in generalized Caputo fractional-order contexts. Numerical results are supplied to demonstrate the viability and accuracy of the presented results. At the end of the article, a few open questions are posed.

Locating network trees by a bilevel scheme

AbstractIn this paper we investigate how to choose an optimal position of a specific facility that is constrained to a network tree connecting some given demand points in a given area. A bilevel formulation is provided and existence results are given together with some properties when a density describes the construction cost of the networks in the area. This includes the presence of an obstacle or of free regions. To prove existence of a solution of the bilevel problem, that is framed in Euclidean spaces, a lower semicontinuity property is required. This is obtained proving an extension of Goła̧b’s theorem in the general setting of metric spaces, which allows for considering a density function.

Existence of entire solutions for k-th derivative of certain type of non-linear differential-difference equations

The paper discusses the existence of entire solutions for k th derivative of certain type of non-linear differential-difference equations. In addition to, the paper evaluates some previous results for existence of finite order transcendental entiresolutions and extends to the higher order. Particularly, we obtain the conditions for occurrence and attainable forms of entire solutions.

On inverse variational–hemivariational inequalities and its regularization

In this article, we propose the inverse variational-hemivariational inequalities problems (IVHIP) and its Tikhonov regularization theory. We establish the existence results for IVHIP and its regularized problems. Moreover, we also discuss the nonemptiness, boundedness and the asymptotic analysis of regularized solutions. Furthermore, we also investigate the existence results of the regularized solutions on the unbounded constrained through the coercive conditions. Several examples are given to support the new findings.