Existence Results Of Solutions Research Articles

Nonlinear Neumann Problems for Fully Nonlinear Elliptic PDEs on a Quadrant

We consider the nonlinear Neumann problem for fully nonlinear elliptic PDEs on a quadrant. We establish a comparison theorem for viscosity sub- and supersolutions of the nonlinear Neumann problem. The crucial argument in the proof of the comparison theorem is to build a $C^{1,1}$ test function which takes care of the nonlinear Neumann boundary condition. A similar problem has been treated on a general $n$-dimensional orthant by Biswas et al. [SIAM J. Control Optim., 55 (2017), pp. 365--396], where the functions ($H_i$ in the main text) describing the boundary condition are required to be positively one-homogeneous, and the result in this paper removes the positive homogeneity in two dimensions. An existence result for solutions is also presented.

Positive solutions for the Schrödinger–Poisson system with steep potential well

In this paper, we consider the following Schrödinger–Poisson system [Formula: see text] where [Formula: see text] are real parameters and [Formula: see text]. Suppose that [Formula: see text] represents a potential well with the bottom [Formula: see text], the system has been widely studied in the case [Formula: see text]. In contrast, no existence result of solutions is available for the case [Formula: see text] due to the presence of the nonlocal term [Formula: see text]. With the aid of the truncation technique and the parameter-dependent compactness lemma, we first prove the existence of positive solutions for [Formula: see text] large and [Formula: see text] small in the case [Formula: see text]. Then we obtain the nonexistence of nontrivial solutions for [Formula: see text] large and [Formula: see text] large in the case [Formula: see text]. Finally, we explore the decay rate of the positive solutions as [Formula: see text] as well as their asymptotic behavior as [Formula: see text] and [Formula: see text].

The regularity of weak solutions for certain n-dimensional strongly coupled parabolic systems

Abstract This paper is concerned with the n-dimensional strongly coupled parabolic systems with triangular form in the cylinder Ω × ( 0 , T ] \Omega \times (0,T] . We investigate L 2 {L}^{2} and Hölder regularity of the derivatives of weak solutions ( u 1 , u 2 ) \left({u}_{1},{u}_{2}) for the systems in the following two cases: one is that the boundedness of u 1 {u}_{1} and u 2 {u}_{2} has not been shown in existence result of solutions; the other is that the boundedness of u 1 {u}_{1} or u 2 {u}_{2} has been shown in existence result of solutions. By using difference ratios and Steklov averages methods and various estimates, we prove that if ( u 1 , u 2 ) \left({u}_{1},{u}_{2}) is a weak solution of the system, then for any Ω ′ ⊂ ⊂ Ω \Omega ^{\prime} \subset \hspace{-0.3em}\subset \hspace{0.33em}\Omega and t ′ ∈ ( 0 , T ) t^{\prime} \in \left(0,T) , u 1 , u 2 {u}_{1},{u}_{2} belong to C α ′ , α ′ / 2 ( Ω ¯ ′ × [ t ′ , T ] ) {C}^{\alpha ^{\prime} ,\alpha ^{\prime} \text{/}2}\left(\bar{\Omega }^{\prime} \times \left[t^{\prime} ,T]) and W 2 2 , 1 ( Ω ′ × ( t ′ , T ] ) {W}_{2}^{2,1}\left(\Omega ^{\prime} \times (t^{\prime} ,T]) under certain conditions, and u 1 , u 2 {u}_{1},{u}_{2} belong to C 2 + α ′ , 1 + α ′ / 2 ( Ω ¯ ′ × [ t ′ , T ] ) {C}^{2+\alpha ^{\prime} ,1+\alpha ^{\prime} \text{/}2}\left(\bar{\Omega }^{\prime} \times \left[t^{\prime} ,T]) under stronger assumptions. Applications of these results are given to two ecological models with cross-diffusion.

Existence and Uniqueness Results for Fractional (p, q)-Difference Equations with Separated Boundary Conditions

In this paper, we study the existence of solutions to a fractional (p, q)-difference equation equipped with separate local boundary value conditions. The uniqueness of solutions is established by means of Banach’s contraction mapping principle, while the existence results of solutions are obtained by applying Krasnoselskii’s fixed-point theorem and the Leary–Schauder alternative. Some examples illustrating the main results are also presented.

On the novel existence results of solutions for a class of fractional boundary value problems on the cyclohexane graph

A branch of mathematical science known as chemical graph theory investigates the implications of connectedness in chemical networks. A few researchers have looked at the solutions of fractional differential equations using the concept of star graphs. Their decision to use star graphs was based on the assumption that their method requires a common point linked to other nodes but not to each other. Our goal is to broaden the scope of the method by defining the idea of a cyclohexane graph, which is a cycloalkane with the molecular formula C_{6}H_{12} and CAS number 110-82-7. It consists of a ring of six carbon atoms, each bonded with two hydrogen atoms above and below the plane with multiple junction nodes. This article examines the existence of fractional boundary value problem’ solutions on such graphs in the sense of the Caputo fractional derivative by using the well-known fixed point theorems. In addition, an example is given to support our key findings.

Existence and regularization of solutions for nonlinear fractional Rayleigh–Stokes problem with final condition

In this paper, we study a fractional nonlinear Rayleigh–Stokes problem with final value condition. By means of the finite dimensional approximation, we first obtain the compactness of solution operators. Moreover, we handle the problem in weighted continuous function spaces, and then the existence result of solutions is established. Finally, because of the ill posedness of backward problem in the sense of Hadamard, the quasi‐boundary value method is utilized to get the regularized solutions, and the corresponding convergence rate is obtained.

Lower and upper solutions method to the fully elastic cantilever beam equation with support

The aim of this paper is to consider a fully cantilever beam equation with one end fixed and the other connected to a resilient supporting device, that is, \t\t\t{u(4)(t)=f(t,u(t),u′(t),u″(t),u‴(t)),t∈[0,1],u(0)=u′(0)=0,u″(1)=0,u‴(1)=g(u(1)),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\textstyle\\begin{cases} u^{(4)}(t)=f(t,u(t),u'(t),u''(t),u'''(t)), \\quad t\\in [0,1], \\\\ u(0)=u'(0)=0, \\\\ u''(1)=0,\\qquad u'''(1)=g(u(1)), \\end{cases} $$\\end{document} where f:[0,1]times mathbb{R}^{4}rightarrow mathbb{R}, g: mathbb{R}rightarrow mathbb{R} are continuous functions. Under the assumption of monotonicity, two existence results for solutions are acquired with the monotone iterative technique and the auxiliary truncated function method.

Nonuniqueness of Solutions to the Lp Dual Minkowski Problem

Abstract The $L_p$ dual Minkowski problem with $p<0<q$ is investigated in this paper. By proving a new existence result of solutions and constructing an example, we obtain the nonuniqueness of solutions to this problem.

Existence conditions for symmetric strong vector quasi-equilibrium problems

In the following paper, the symmetric strong vector quasi-equilibrium problems will be studied thoroughly. Afterward, the existence conditions of solution sets for these problems has been established. The results which are presented in this paper improve and extend the main results mentioned in the literature. Our results can be illustrated by some interesting examples. In 1994, Noor and Oettli introduced the following the symmetric scalar quasi-equilibrium problem. This problem is one of the generalization of the symmetric scalar quasi-equilibrium problem which is presented by Noor and Oettli. Since then, the symmetric vector quasi-equilibrium problem has been investigated by a huge number of authors in different ways. The research works mentioned above are one of our motivation to improve and extend the problem. So, in this paper, we will introduce the vector quasi-equilibrium problems. Afterward, some existence conditions of solution sets for these problems will be established. The symmetric vector quasi-equilibrium problems consist of many optimization - related models namely symmetric vector quasi-variational inequality problems, fixed point problems, coincidence-point problems and complementarity problems, etc. In recent years, a lot of results for existence of solutions for symmetric vector quasi-equilibrium problems, vector quasi-equilibrium problems, vector quasivariational inequality problems and optimization problems have been established by many authors in different ways. We will present our work in the following steps. In the first section of our paper, we will introduce the model of symmetric vector quasi-equilibrium problems. In the following section, we recall definitions, lemmas which can be used for the main results. In the last section, we will establish some conditions for existence and closedness of the solutions set by applying fixed-point theorem for symmetric vector quasi-equilibrium problems. The results presented in this paper improve and extend the main results in the literature. Some examples are given to illustrate our results. Hence our results, Theorem 3.1 and Theorem 3.6 have significant improvements.

Nonlinear boundary value problems for mixed-type fractional equations and Ulam-Hyers stability

Abstract In this article, we discuss the nonlinear boundary value problems involving both left Riemann-Liouville and right Caputo-type fractional derivatives. By using some new techniques and properties of the Mittag-Leffler functions, we introduce a formula of the solutions for the aforementioned problems, which can be regarded as a novelty item. Moreover, we obtain the existence result of solutions for the aforementioned problems and present the Ulam-Hyers stability of the fractional differential equation involving two different fractional derivatives. An example is given to illustrate our theoretical result.

Global exponential stability analysis of anti-periodic solutions of discontinuous bidirectional associative memory (BAM) neural networks with time-varying delays

Abstract This paper is concerned with a class of bidirectional associative memory (BAM) neural networks with discontinuous activations and time-varying delays. Under the basic framework of differential inclusions theory, the existence result of solutions in sense of Filippov solution is firstly established by using the fundamental solution matrix of coefficients and inequality analysis technique. Also, the boundness of the solutions can be estimated. Secondly, based on the non-smooth Lyapunov-like approach and by construsting suitable Lyapunov–Krasovskii functionals, some new sufficient criteria are given to ascertain the globally exponential stability of the anti-periodic solutions for the proposed neural network system. Furthermore, we have collated our effort with some previous existing ones in the literatures and showed that it can take more advantages. Finally, two examples with numerical simulations are exploited to illustrate the correctness.

Existence and asymptotic behavior of positive solutions for Kirchhoff type problems with steep potential well

In this paper, we consider the following Kirchhoff type problem{−(a+b∫R3|∇u|2dx)Δu+λV(x)u=|u|p−2uinR3,u∈H1(R3), where a>0 is a constant, b and λ are positive parameters, and 2<p<6. Suppose that the nonnegative continuous potential V represents a potential well with the bottom V−1(0), the equation has been extensively studied in the case 4≤p<6. In contrast, no existence result of solutions is available for the case 2<p<4 due to the presence of the term (∫R3|∇u|2dx)Δu. By combining the truncation technique and the parameter-dependent compactness lemma, we prove the existence of positive solutions for b small and λ large in the case 2<p<4. Moreover, we also explore the decay rate of the positive solutions as |x|→∞ as well as their asymptotic behavior as b→0 and λ→∞.

Nonlinear fractional differential equation involving two mixed fractional orders with nonlocal boundary conditions and Ulam\u2013Hyers stability

In this paper, we study a nonlinear fractional differential equation involving two mixed fractional orders with nonlocal boundary conditions. By using some new techniques, we introduce a formula of solutions for above problem, which can be regarded as a novelty item. Moreover, under the weak assumptions and using Leray–Schauder degree theory, we obtain the existence result of solutions for above problem. Furthermore, we discuss the Ulam–Hyers stability of the above fractional differential equation. Three examples illustrate our results.

Existence of Solutions for Fractional Boundary Value Problems with a Quadratic Growth of Fractional Derivative

In this paper, we deal with two fractional boundary value problems which have linear growth and quadratic growth about the fractional derivative in the nonlinearity term. By using variational methods coupled with the iterative methods, we obtain the existence results of solutions. To the best of the authors’ knowledge, there are no results on the solutions to the fractional boundary problem which have quadratic growth about the fractional derivative in the nonlinearity term.

A characterization related to Schrödinger equations on Riemannian manifolds

In this paper, we consider the following problem: [Formula: see text] where [Formula: see text] is a [Formula: see text]-dimensional ([Formula: see text], non-compact Riemannian manifold with asymptotically non-negative Ricci curvature, [Formula: see text] is a real parameter, [Formula: see text] is a positive coercive potential, [Formula: see text] is a bounded function and [Formula: see text] is a suitable nonlinearity. By using variational methods, we prove a characterization result for existence of solutions for [Formula: see text].

On evolution quasi-variational inequalities and implicit state-dependent sweeping processes

In this paper, we study a variant of the state-dependent sweeping process with velocity constraint. The constraint \begin{document}$ {C(\cdot, u)} $\end{document} depends upon the unknown state \begin{document}$ u $\end{document} , which causes one of the main difficulties in the mathematical treatment of quasi-variational inequalities. Our aim is to show how a fixed point approach can lead to an existence theorem for this implicit differential inclusion. By using Schauder's fixed point theorem combined with a recent existence and uniqueness theorem in the case where the moving set \begin{document}$ C $\end{document} does not depend explicitly on the state \begin{document}$ u $\end{document} (i.e. \begin{document}$ C: = C(t) $\end{document} ) given in [ 4 ], we prove a new existence result of solutions of the quasi-variational sweeping process in the infinite dimensional Hilbert spaces with a velocity constraint. Contrary to the classical state-dependent sweeping process, no conditions on the size of the Lipschitz constant of the moving set, with respect to the state, is required.

RETRACTED ARTICLE: Existence results for the general Schr\xf6dinger\t\t\tequations with a superlinear Neumann boundary value problem

The main goal of this paper is to study the general Schrödinger\t\t\t\tequations with a superlinear Neumann boundary value problem in domains with conical\t\t\t\tpoints on the boundary of the bases. First the formulation and the complex form of\t\t\t\tthe problem for the equations are given, and then the existence result of solutions\t\t\t\tfor the above problem is proved by the complex analytic method and the fixed point\t\t\t\tindex theory, where we absorb the advantages of the methods in recent works and give\t\t\t\tsome improvement and development. Finally, we are also interested in the asymptotic\t\t\t\tbehavior of solutions of the mentioned equation. These results generalize some\t\t\t\tprevious results concerning the asymptotic behavior of solutions of non-delay\t\t\t\tsystems of Schrödinger equations or of delay Schrödinger equations.

Periodic Perturbations of Linear Systems at Resonance

Second-order linear Hamiltonian systems at resonance with periodic nonlinearity is investigated. An existence result of solutions for such systems is obtained by means of variational methods, saddle point theorem, and an index theory for second-order linear Hamiltonian systems. Meanwhile, two examples and two extensions are presented.

Caputo–Hadamard fractional differential Cauchy problem in Fréchet spaces

This article deals with some existence results of solutions for a class of differential equations involving the Caputo–Hadamard fractional derivative in Frechet spaces. These results are based on a generalization of the classical Darbo fixed point theorem for Frechet spaces associated with the concept of measure of noncompactness. We illustrate our results by an example.

Existence of Self-similar Solutions to the Anisotropic Affine Curve-shortening Flow

Abstract In this paper the existence of positive $2\pi $-periodic solutions to the ordinary differential equation $$\begin{equation*} u^{\prime\prime}+u=\frac{f}{u^3} \ \textrm{ in } \mathbb{R} \end{equation*}$$is studied, where $f$ is a positive $2\pi $-periodic smooth function. By virtue of a new generalized Blaschke–Santaló inequality, we obtain a new existence result of solutions.