Boundary Value Problems

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List of Papers (Total 1,633)

Positive solutions of conformable fractional differential equations with integral boundary conditions

In this paper, we discuss the existence of positive solutions of the conformable fractional differential equation T α x ( t ) + f ( t , x ( t ) ) = 0 $T_{\alpha }x(t)+f(t,x(t))=0$ , t ∈ [ 0 , 1 ] $t\in [0,1]$ , subject to the boundary conditions x ( 0 ) = 0 $x(0)=0$ and x ( 1 ) = λ ∫ 0 1 x ( t ) d t $x(1)= \lambda \int_{0}^{1}x(t)\,\mathrm{d}t$ , where the order α belongs to ( 1...

Existence of a regular solution for 1D Green–Naghdi equations with surface tension at a large time instant

In this paper the model 1D-GNσ is considered, which concerns the 1D Green–Naghdi equations with non-flat bottom and under the influence of surface tension, to be widely used in coastal oceanography to describe the propagation of large-wave amplitudes. The purpose of this paper is to show that the solution of 1D-GNσ can be made by the Picard iterative scheme, which proves that...

New homoclinic solutions for a class of second-order Hamiltonian systems with a mixed condition

In this paper, we introduce a new mixed condition to obtain a new compact embedding theorem. Under this theorem, we study the existence and multiplicity of nontrivial homoclinic solutions for a class of second-order Hamiltonian systems with variable separated type nonlinear terms.

Hölder continuity of weak solution to a nonlinear problem with non-standard growth conditions

We study the Hölder continuity of weak solution u to an equation arising in the stationary motion of electrorheological fluids. To this end, we first obtain higher integrability of Du in our case by establishing a reverse Hölder inequality. Next, based on the result of locally higher integrability of Du and difference quotient argument, we derive a Nikolskii type inequality; then...

Adaptive stabilized finite volume method and convergence analysis for the Oseen equations

In this paper, based on the pressure project method, we consider an adaptive stabilized finite volume method for the Oseen equations with the lowest equal order finite element pair. Firstly, we develop the discrete forms in both finite element and finite volume methods, and establish the existence and uniqueness of numerical solutions by establishing the equivalence of linear...

On a class of stationary loops on SO ( n ) $\mathbf{SO}(n)$ and the existence of multiple twisting solutions to a nonlinear elliptic system subject to a hard incompressibility constraint

In this paper we consider the second order nonlinear elliptic system in divergence and variational form { div [ F ξ ( | x | , | ∇ u | 2 ) ∇ u ] = [ cof ∇ u ] ∇ P in  U , det ∇ u = 1 in  U , u = φ on  ∂ U , $$\begin{aligned} \textstyle\begin{cases} \operatorname{div}[ F_{\xi }(\vert x\vert ,\vert \nabla u\vert ^{2})\nabla u ] = [ \operatorname{cof}\nabla u] \nabla \mathscr{P...

Nontrivial solutions of second-order singular Dirichlet systems

We study the existence of nontrivial solutions for second-order singular Dirichlet systems. The proof is based on a well-known fixed point theorem in cones and the Leray-Schauder nonlinear alternative principle. We consider a very general singularity and generalize some recent results.

Existence of positive solutions for a fractional elliptic problems with the Hardy-Sobolev-Maz’ya potential and critical nonlinearities

In this paper, we consider the study of a fractional elliptic problem with the Hardy-Sobolev-Maz’ya potential and critical nonlinearities. By means of variational methods and suitable technique, a positive solution to this problem is obtained.

Boundary value problems for strongly nonlinear equations under a Wintner-Nagumo growth condition

We study the following strongly nonlinear differential equation: ( a ( t , x ( t ) ) Φ ( x ′ ( t ) ) ) ′ = f ( t , x ( t ) , x ′ ( t ) ) , a.e. in  [ 0 , T ] $$\bigl(a \bigl(t,x(t) \bigr)\Phi\bigl(x'(t) \bigr) \bigr)

New approximation methods for solving elliptic boundary value problems via Picard-Mann iterative processes with mixed errors

In this paper, we introduce and study a class of new Picard-Mann iterative methods with mixed errors for common fixed points of two different nonexpansive and contraction operators. We also give convergence and stability analysis of the new Picard-Mann iterative approximation and propose numerical examples to show that the new Picard-Mann iteration converges more effectively than...

Nontrivial solutions of second-order singular Dirichlet systems

We study the existence of nontrivial solutions for second-order singular Dirichlet systems. The proof is based on a well-known fixed point theorem in cones and the Leray-Schauder nonlinear alternative principle. We consider a very general singularity and generalize some recent results.

Existence of positive solutions for a fractional elliptic problems with the Hardy-Sobolev-Maz’ya potential and critical nonlinearities

In this paper, we consider the study of a fractional elliptic problem with the Hardy-Sobolev-Maz’ya potential and critical nonlinearities. By means of variational methods and suitable technique, a positive solution to this problem is obtained.

A natural boundary element method for the Sobolev equation in the 2D unbounded domain

In this article, we devote ourselves to establishing a natural boundary element (NBE) method for the Sobolev equation in the 2D unbounded domain. To this end, we first constitute the time semi-discretized super-convergence format for the Sobolev equation by means of the Newmark method. Then, using the principle of natural boundary reduction, we establish a fully discretized NBE...

Nontrivial solutions of second-order singular Dirichlet systems

We study the existence of nontrivial solutions for second-order singular Dirichlet systems. The proof is based on a well-known fixed point theorem in cones and the Leray-Schauder nonlinear alternative principle. We consider a very general singularity and generalize some recent results.

Existence of positive solutions for a fractional elliptic problems with the Hardy-Sobolev-Maz’ya potential and critical nonlinearities

In this paper, we consider the study of a fractional elliptic problem with the Hardy-Sobolev-Maz’ya potential and critical nonlinearities. By means of variational methods and suitable technique, a positive solution to this problem is obtained.

On the Wiener criterion in higher dimensions

The Wiener criterion is a sufficient and necessary condition for the solvability of the Dirichlet problem. However, its geometric interpretation is not clear. In the case that the domain satisfies an exterior spine condition, the requirement for the spine is clear in dimension 3. In this note, we intend to obtain the condition that the exterior spine should satisfy in higher...

A natural boundary element method for the Sobolev equation in the 2D unbounded domain

In this article, we devote ourselves to establishing a natural boundary element (NBE) method for the Sobolev equation in the 2D unbounded domain. To this end, we first constitute the time semi-discretized super-convergence format for the Sobolev equation by means of the Newmark method. Then, using the principle of natural boundary reduction, we establish a fully discretized NBE...

Nontrivial solutions of second-order singular Dirichlet systems

We study the existence of nontrivial solutions for second-order singular Dirichlet systems. The proof is based on a well-known fixed point theorem in cones and the Leray-Schauder nonlinear alternative principle. We consider a very general singularity and generalize some recent results.

On the Wiener criterion in higher dimensions

The Wiener criterion is a sufficient and necessary condition for the solvability of the Dirichlet problem. However, its geometric interpretation is not clear. In the case that the domain satisfies an exterior spine condition, the requirement for the spine is clear in dimension 3. In this note, we intend to obtain the condition that the exterior spine should satisfy in higher...

A natural boundary element method for the Sobolev equation in the 2D unbounded domain

In this article, we devote ourselves to establishing a natural boundary element (NBE) method for the Sobolev equation in the 2D unbounded domain. To this end, we first constitute the time semi-discretized super-convergence format for the Sobolev equation by means of the Newmark method. Then, using the principle of natural boundary reduction, we establish a fully discretized NBE...

Nontrivial solutions of second-order singular Dirichlet systems

We study the existence of nontrivial solutions for second-order singular Dirichlet systems. The proof is based on a well-known fixed point theorem in cones and the Leray-Schauder nonlinear alternative principle. We consider a very general singularity and generalize some recent results.