Fourth-order Boundary Value Research Articles

Global structure of positive solutions for semipositone nonlinear Euler-Bernoulli beam equation with Neumann boundary conditions

In this paper, we focus on the existence of positive solutions for nonlinear fourth-order Neumann boundary value problem where k 1 and k 2 are constants, λ > 0 is the bifurcation parameter, f ∈ C([0, 1] × ℝ+, ℝ), ℝ+ := [0, ∞). We first discuss the sign properties of Green’s function for the elastic beam boundary value problem, and then we show that there exists a global branch of solutions emanating from infinity under some different growth conditions. In addition, we prove that for λ near the bifurcation points, solutions of large norm are indeed positive. The technique for dealing with this paper relies on the global bifurcation theory.

Existence of solutions for fourth-order nonlinear boundary value problems

In this paper, we discuss the existence and approximation of solutions for a fourth-order nonlinear boundary value problem by using a quasilinearization technique. In the presence of a lower solution α and an upper solution β in the reverse order alpha geq beta , we show the existence of (extreme) solution.

A Note on the Eigenvalue Criteria for Positive Solutions of a Cantilever Beam Equation with Free End

In this paper, we study the existence of at least one positive solution to the fourth-order two-point boundary value problem (BVP) which models a cantilever beam equation, where one end is kept free. Here $$f \in {\mathcal {C}}\left( [0,1] \times {\mathbb {R}}_{+}, {\mathbb {R}}_{+}\right) $$ , $$g \in {\mathcal {C}}\left( [0,1] , {\mathbb {R}}_{+}\right) $$ and $$\lambda $$ is a positive parameter. The sufficient conditions are interesting, new and easy to verify. We have used some inequalities on the nonlinear function f and eigenvalues of a linear integral operator as bounds for the parameter $$\lambda $$ to obtain our results. Our approach is based on a revised version of a fixed point theorem due to Gustafson and Schmitt.

New Fixed Point Theorem for Discontinuous Operators in Cones and Applications

We provide new fixed point theorems for a class of discontinuous operators by combining a new fixed point theorem of compression-expansion type for these discontinuous operators with monotone iterative methods. As an application we study the existence of positive solutions for a nonlinear fourth-order discontinuous boundary value problem.

Existence results and iterative method for a fully fourth-order nonlinear integral boundary value problem

We consider the boundary value problem $$ \begin{array}{@{}rcl@{}} u^{\prime\prime\prime\prime}(t)&=&f(t,u(t),u^{\prime}(t),u^{\prime\prime}(t),u^{\prime\prime\prime}(t)), \quad 0<t<1,\\ u^{\prime}(0)&=&u^{\prime\prime}(0)= u^{\prime}(1)=0, u(0)={{\int}_{0}^{1}} g(s)u(s) ds, \end{array} $$ where $f: [0, 1] \times \mathbb {R}^{4} \rightarrow \mathbb {R}^{+},\ a: [0, 1] \rightarrow \mathbb {R}^{+}$ are continuous functions. For f = f(u(t)), very recently in Benaicha and Haddouchi (An. Univ. Vest Timis. Ser. Mat.-Inform. 1(54): 73–86, 2016) the existence of positive solutions was studied by employing the fixed point theory on cones. In this paper, by the method of reducing the boundary value problem to an operator equation for the right-hand sides we establish the existence, uniqueness, and positivity of solution and propose an iterative method on both continuous and discrete levels for finding the solution. We also give error analysis of the discrete approximate solution. Some examples demonstrate the validity of the obtained theoretical results and the efficiency of the iterative method.

Existence and uniqueness of weak and classical solutions for a fourth-order semilinear boundary value problem

This paper is concerned with the problem of existence and uniqueness of weak and classical solutions for a fourth-order semilinear boundary value problem. The existence and uniqueness for weak solutions follows from standard variational methods, while similar uniqueness results for classical solutions are derived using maximum principles.&#x0D; &#x0D; &#x0D; &#x0D; &#x0D; &#x0D; &#x0D; doi:10.1017/S1446181119000129

Boundary value technique based on a fifth derivative method of order 10 for fourth order initial and boundary value problems

In this paper, the fourth order ordinary differential equation (ODE) of the form $$y^{(iv)}=f(x,y,y',y'',y''')$$ subject to initial or boundary conditions is solved directly without reducing the higher ODE into an equivalent first order system. This is accomplished by constructing a continuous fifth derivative method (CFIDM) which is used to produce the main and additional discrete fifth derivative methods (FIDMs). The FIDMs are then assembled and applied as a single block matrix equation to simultaneously provide all approximations on the entire interval of interest for the initial or boundary value problem. The convergence analysis of the method is discussed. The accuracy of the method is illustrated via numerical experiments.

Existence of solution for nonlinear fourth-order three-point boundary value problem

In this paper, we study the existence of nontrivial solution for the fourth-order three- point boundary value problem having the following formu(4) (t) + f (t, u(t)) = 0, 0 &lt; t &lt; 1,u(0) = α(η), u'(0) = u''(0) = 0, u(1) = βu(η),where η ∈ (0, 1), α, β ∈ R, f ∈ C ([0, 1] × R, R). We give sufficient conditions that allow us to obtain the existence of a nontrivial solution. And by using the Leray-Schauder nonlinear alternative we prove the existence of at least one solution of the posed problem. As an application, we also given some examples to illustrate the results obtained.

Reproducing kernel method in Hilbert spaces for solving the linear and nonlinear four-point boundary value problems

ABSTRACTAn analysis was performed in this paper to study the problem of magnetohydrodynamic squeezing flow of nanofluid between parallel disks. This paper also proposed a reproducing kernel method for solving fourth-order four-point boundary value problems. In order to eliminate the singularity of the equation, a transform was used. Convergence analysis and error estimation for the present method in -space were also discussed. The exact solution was represented in the form of series in the reproducing kernel Hilbert space which proved higher accuracy in numerical computation. The algorithm derived from this approach can be easily implemented. The ideas and techniques presented in this paper will be useful for solving many other problems.

Solving Nonlinear Fourth-Order Boundary Value Problems Using a Numerical Approach: (m+1)th-Step Block Method

Nonlinear boundary value problems (BVPs) are more tedious to solve than their linear counterparts. This is observed in the extra computation required when determining the missing conditions in transforming BVPs to initial value problems. Although a number of numerical approaches are already existent in literature to solve nonlinear BVPs, this article presents a new block method with improved accuracy to solve nonlinear BVPs. A m+1th-step block method is developed using a modified Taylor series approach to directly solve fourth-order nonlinear boundary value problems (BVPs) where m is the order of the differential equation under consideration. The schemes obtained were combined to simultaneously produce solution to the fourth-order nonlinear BVPs at m+1 points iteratively. The derived block method showed improved accuracy in comparison to previously existing authors when solving the same problems. In addition, the suitability of the m+1th-step block method was displayed in the solution for magnetohydrodynamic squeezing flow in porous medium.

Computational Approach for Fourth-Order Self-Adjoint Singularly Perturbed Boundary Value Problems via Non-polynomial Quintic Spline

This paper proposes a non-polynomial quintic spline method for approximate solutions of fourth-order self-adjoint singular perturbation boundary value problems, which comprises the polynomial part of degree three and two non-polynomial terms, i.e., sine and cosine. It is explicitly shown that the free parameter \(\tau\) of the non-polynomial part can be used to raise the order of accuracy of the scheme. The method has been proved for the second and fourth-order convergence. The proposed scheme is tested on two examples. The experimental results are compared with the existing method.

Existence and multiplicity of positive solutions to a fourth-order impulsive integral boundary value problem with deviating argument

In this paper, we study the existence of multiple positive solutions for fourth-order impulsive differential equation with integral boundary conditions and deviating argument. The main tool is based on the Avery and Peterson fixed point theorem. Meanwhile, an example to demonstrate the main results is given.

New results for positive solutions of singular fourth-order four-point p-Laplacian problem

The existence and uniqueness of positive solutions are obtained for singular fourth-order four-point boundary value problem with p-Laplace operator $[\varphi_{p}(u''(t))]''=f(t,u(t))$ , $0< t<1$ , $u(0)=0$ , $u(1)=au(\xi)$ , $u''(0)=0$ , $u''(1)=bu''(\eta)$ , where $f(t,u)$ is singular at $t=0,1$ and $u=0$ . A fixed point theorem for mappings that are decreasing with respect to a cone in a Banach space plays a key role in the proof.

Existence results for a kind of fourth-order impulsive integral boundary value problems

In this paper we investigate the existence of solutions to a kind of fourth-order impulsive differential equations with integral boundary value conditions. By employing the Schauder fixed point theorem, we obtain sufficient conditions which ensure the system has at lease one solution. Also by using the contraction mapping theorem we get the uniqueness result. Finally an example is given to illustrate the effectiveness of our results.

On the nonexistence and existence of solutions for a fourth-order discrete Dirichlet boundary value problem

This paper is concerned with a class of fourth-order nonlinear difference equations. By using the critical point theory, we establish various sets of sufficient conditions of the nonexistence and existence of solutions for a Dirichlet boundary value problem and give some new results. Results obtained generalize and complement the existing ones.

A nonlinear boundary value problem for fourth-order elastic beam equations

By using an infinitely many critical points theorem, we study the existence of infinitely many solutions for a fourth-order nonlinear boundary value problem, depending on two real parameters. No symmetric condition on the nonlinear term is assumed. Some recent results are improved and extended.

Existence and nonexistence results for a fourth-order discrete neumann boundary value problem

In this paper, a fourth-order nonlinear difference equation is considered. By making use of the critical point theory, we establish various sets of sufficient conditions for the existence and nonexistence of solutions for Neumann boundary value problem and give some new results. Results obtained generalize and complement the existing ones.

Nonexistence and existence of solutions for a fourth-order discrete mixed boundary value problem

In this paper, a fourth-order nonlinear difference equation is considered. By using the critical point theory, we establish various sets of sufficient conditions of the nonexistence and existence of solutions for mixed boundary value problem and give some new results. Results obtained generalize and complement the existing ones.

The Solvability and Numerical Simulation for the Elastic Beam Problems with Nonlinear Boundary Conditions

We study the existence of multiple solutions for a fourth-order nonlinear boundary value problem. We give some new criteria for guaranteeing that the fourth-order elastic beam equation with a perturbed term has at least three solutions. The proof is based on some three critical points theorems of B. Ricceri. Furthermore, numerical simulations are also presented.

Nontrivial Solutions of a Fully Fourth-Order Periodic Boundary Value Problem

We investigate the solvability of a fully fourth-order periodic boundary value problem of the formx(4)=f(t,x,x′,x′′,x′′′), x(i)(0)=x(i)(T), i=0,1,2,3,wheref:[0,T]×R4→Rsatisfies Carathéodory conditions. By using the coincidence degree theory, the existence of nontrivial solutions is obtained. Meanwhile, as applications, some examples are given to illustrate our results.