Nonlinear Fourth-order Two-point Boundary Value Research Articles

Existence-uniqueness and fixed-point iterative method for general nonlinear fourth-order boundary value problems

Existence and uniqueness results are established for the general nonlinear fourth-order two-point boundary value problem with general linear homogeneous boundary conditions. The proofs are based on the Banach fixed-point theorem under the conditions that the nonlinear term is bounded and Lipschitz in a specific region. A fixed-point iterative method for finding the solution of the problem is also proposed with error estimates. The resulting iterative solutions automatically satisfy the boundary conditions provided that the initial iterate does so. Two examples are given to illustrate the applicability of the proposed approach and the efficiency of the iterative method.

A Computationally Efficient Iterative Scheme for Solving Fourth-Order Boundary Value Problems

A new rapid-converging analytical scheme is introduced to obtain the approximate analytical solutions of nonlinear fourth-order two-point boundary value problems, which appear in various physical phenomena. The idea of the method to obtain the solution of such problems is essentially based on reducing the solution of the main problem to the solution of an integral problem. The introduced technique consists of two steps. First, construct an integral operator by introducing Green’s function, and then, the Normal-S iterative scheme is applied to this integral operator to construct the iterative approach for such problems, which yields a simple way to improve the convergence of the iterative solutions to the problem. We also discuss the convergence of the introduced iterative method. To exhibit the performance of the method, we consider some numerical test examples. The obtained results are compared with the existing analytical and numerical approaches to reveal the superiority and computational efficiency of the proposed approach. In fact, it is a direct recursive and computationally cost-effective method for dealing with strong nonlinearity. The numerical simulations signify the applicability and effectiveness of the present work.

Positive Solutions of a Nonlinear Fourth-order Integral Boundary Value Problem

Abstract In this paper, the existence of positive solutions for a nonlinear fourth-order two-point boundary value problem with integral condition is investigated. By using Krasnoselskii’s fixed point theorem on cones, sufficient conditions for the existence of at least one positive solutions are obtained.

Positive solutions and eigenvalue intervals of a nonlinear singular fourth-order boundary value problem

We consider the classical nonlinear fourth-order two-point boundary value problem Open image in new window . In this problem, the nonlinear term h(t)f(t, u(t), u′(t), u″(t)) contains the first and second derivatives of the unknown function, and the function h(t)f(t, x, y, z) may be singular at t = 0, t = 1 and at x = 0, y = 0, z = 0. By introducing suitable height functions and applying the fixed point theorem on the cone, we establish several local existence theorems on positive solutions and obtain the corresponding eigenvalue intervals.

Existence of nodal solutions of a nonlinear fourth-order two-point boundary value problem

In this article, we give conditions on parameters k, l that the generalized eigenvalue problem x″″ + kx″ + lx = λh(t)x, 0 < t < 1, x(0) = x(1) = x′(0) = x′(1) = 0 possesses an infinite number of simple positive eigenvalues { λ k } k = 1 ∞ and to each eigenvalue there corresponds an essential unique eigenfunction ψ k which has exactly k - 1 simple zeros in (0,1) and is positive near 0. It follows that we consider the fourth-order two-point boundary value problem x″″ + kx″ + lx = f(t,x), 0 < t < 1, x(0) = x(1) = x′(0) = x′(1) = 0, where f(t, x) ∈ C([0,1] × ℝ, ℝ) satisfies f(t, x)x > 0 for all x ≠ 0, t ∈ [0,1] and lim |x|→0 f(t,x)/x = a(t), lim |x|→+∞ f(t,x)/x = b(t) or lim x→-∞ f(t,x)/x = 0 and lim x→+∞ f(t,x)/x = c(t) for some a(t), b(t), c(t) ∈ C([0,1], (0,+∞)) and t ∈ [0,1]. Furthermore, we obtain the existence and multiplicity results of nodal solutions for the above problem. The proofs of our main results are based upon disconjugate operator theory and the global bifurcation techniques.MSC (2000): 34B15.

Positive solutions of nonlinear beam equations with time and space singularities

We consider the existence and multiplicity of positive solutions to a nonlinear fourth-order two-point boundary value problem. The nonlinear term may be singular with respect to both the time and space variables. In mechanics, the problem describes the deformation of an elastic beam fixed at the left and supported at the right by sliding clamps. By introducing height functions of the nonlinear term on some bounded sets and considering integrations of these height functions, several local existence theorems are obtained.

Existence of solutions for a fully nonlinear fourth-order two-point boundary value problem

In this paper, we investigate the existence of solutions of a fully nonlinear fourth-order differential equation $$x^{(4)}=f(t,x,x',x'',x'''),\quad t\in [0,1]$$ with one of the following sets of boundary value conditions; $$x'(0)=x(1)=a_{0}x''(0)-b_{0}x'''(0)=a_{1}x''(1)+b_{1}x'''(1)=0,$$ $$x(0)=x'(1)=a_{0}x''(0)-b_{0}x'''(0)=a_{1}x''(1)+b_{1}x'''(1)=0.$$ By using the Leray-Schauder degree theory, the existence of solutions for the above boundary value problems are obtained. Meanwhile, as an application of our results, an example is given.

Triple positive solutions of a class of fourth-order two-point boundary value problems

In this paper, by using the Leggett–Williams fixed point theorem, we establish an existence criterion for triple positive solutions of the nonlinear fourth-order two-point boundary value problem { u ( 4 ) ( t ) = g ( t ) f ( t , u ( t ) , u ′ ( t ) ) , t ∈ ( 0 , 1 ) u ( 0 ) = u ′ ( 1 ) = u ″ ( 0 ) = u ‴ ( 1 ) = 0 . An example is also included to demonstrate the result we obtained.

Monotonically iterative method of nonlinear cantilever beam equations

The positive solution is studied for a nonlinear fourth-order two-point boundary value problem. In mechanics, the problem is called cantilever beam equation. By making use of the monotonically iterative technique, a successively iterative sequence is constructed. By applying the successively approximate method, an existence theorem is proved.

Existence of [formula omitted] solutions and/or positive solutions to a semipositone elastic beam equation

The existence of n solutions and/or positive solutions is established for a nonlinear fourth-order two-point boundary value problem where n is an arbitrary natural number and the nonlinear term has a nonpositive lower bound. In mechanics, the problem describes deformation of the elastic beam whose two ends are simply supported. The main results show that the problem has at least n solutions and/or positive solutions provided that the “heights” of the nonlinear term are appropriate on some bounded sets.

Existence of solutions and positive solutions to a fourth-order two-point BVP with second derivative.

Several existence theorems were established for a nonlinear fourth-order two-point boundary value problem with second derivative by using Leray-Schauder fixed point theorem, equivalent norm and technique on system of integral equations. The main conditions of our results are local. In other words, the existence of the solution can be determined by considering the "height" of the nonlinear term on a bounded set. This class of problems usually describes the equilibrium state of an elastic beam which is simply supported at both ends.

Spline Solutions for Nonlinear Fourth-Order Two-Point Boundary Value Problems

Methods of order 2 and 4 are developed for the continuous approximation of thesolution of a nonlinear fourth-order two-point boundary value problem. Numerical results are briefly summarized to demonstrate the practical usefulness of the methods.