Nonlinear Elastic Beam Equation Research Articles

Multiple solutions of a nonlinear elastic beam equation supported at two ends

In this paper, several new existence theorems on pairs of solutions are obtained for the fourth order boundary value problem ${u^{(4)}}(t) = h(t,u(t))$ subject to $u(0) = u(1) = u''(0) = u''(1) = 0$. Further, if $h(t,u) = \mu f(t,u)$, we prove that there exists $ \mu ^ * > 0$ such that there is no solution to the above boundary value problem for $\mu > {\mu ^ * }$, and that there are multiple solutions of the above boundary value problem for $0 < \mu < {\mu^ * }$.

Relation-theoretic almost $ \phi $-contractions with an application to elastic beam equations

&lt;abstract&gt;&lt;p&gt;In this article, we prove some results on existence and uniqueness of fixed points for an almost $ \phi $-contraction mapping defined on a metric space endowed with an amorphous relation. Our results generalize and improve several well known fixed point theorems of the existing literature. To substantiate the credibility of our results, we construct some examples. We also apply our results to determine a unique solution of a boundary value problem associated with nonlinear elastic beam equations.&lt;/p&gt;&lt;/abstract&gt;

New fixed point theorems for the sum of two mixed monotone operators of Meir–Keeler type and their applications to nonlinear elastic beam equations

In this paper, we present a new fixed point theorem for the sum of two mixed monotone operators of Meir–Keeler type on ordered Banach spaces through projective metric, which extends the existing corresponding results. As applications, we utilize the results obtained in this paper to study the existence and uniqueness of positive solutions for nonlinear singular fourth-order elastic beam equations with nonlinear boundary conditions.

Common Fixed Points of(α,η)−(θ,ϝ)Rational Contractions with Applications

We present the notion of set valued(α,η)-(θ,ϝ)rational contraction mappings and then some common fixed point results of such mappings in the setting of metric spaces are established. Some examples are presented to support the concepts introduced and the results proved in this paper. These results unify, extend, and refine various results in the literature. Some fixed point results for both single and multivalued(θ,ϝ)rational contractions are also obtained in the framework of a space endowed with partial order. As application, we establish the existence of solutions of nonlinear elastic beam equations and first-order periodic problem.

Eigenvalue Interval for Singular Fractional-Order Elastic Beam Equation

The purpose of this paper, is to give the eigenvalue interval for nonlinear elastic beam equation and according to the range of the eigenvalue λ, establish sufficient conditions for the existence of positive solutions.

Eigenvalue problem for nonlinear elastic beam equation of fractional order

In this study, under some suitable assumptions, we determine an explicit eigenvalue interval for the existence of positive solution of singular fractional-order nonlinear elastic beam equation with bending term. Our analysis rely on cone theoretic techniques. Moreover, we consider some special cases and an example to affirm the applicability of the main result.

Existence of positive solutions for the nonlinear elastic beam equation via a mixed monotone operator

In this paper, we use a mixed monotone operator method to investigate the existence and uniqueness of positive solution to a nonlinear fourth-order boundary value problem which describes the deflection of an elastic beam with the left extreme fixed and the right extreme is attached to a bearing device given by a known function. Moreover, we present a concrete example illustrating the result obtained.

Fixed point theorems on ordered metric spaces and applications to nonlinear elastic beam equations

In this paper, we establish certain fixed point theorems in metric spaces with a partial ordering. Presented theorems extend and generalize several existing results in the literature. As application, we use the fixed point theorems obtained in this paper to study existence and uniqueness of solutions for fourth-order two-point boundary value problems for elastic beam equations.

On the Dirichlet problem for a nonlinear elastic beam equation

Using a direct variational approach with no global growth conditions on the nonlinear term, we consider the existence of solutions and their dependence on a functional parameter for the fourth order Dirichlet problem connected with the elastic beam equation. We investigate also the existence of an optimal process for such an optimal control problem in which the dynamics is described by the beam equation.

A sum operator equation and applications to nonlinear elastic beam equations and Lane–Emden–Fowler equations

This paper is concerned with an operator equation A x + B x + C x = x on ordered Banach spaces, where A is an increasing α-concave operator, B is an increasing sub-homogeneous operator and C is a homogeneous operator. The existence and uniqueness of its positive solutions is obtained by using the properties of cones and a fixed point theorem for increasing general β-concave operators. As applications, we utilize the fixed point theorems obtained in this paper to study the existence and uniqueness of positive solutions for two classes nonlinear problems which include fourth-order two-point boundary value problems for elastic beam equations and elliptic value problems for Lane–Emden–Fowler equations.

On the optimal control problem governed by the nonlinear elastic beam equation

We begin with using a dual variational method in proving the existence of solutions to a beam equation with Dirichlet boundary conditions and with a non-monotone nonlinear term. Later we investigate the dependence on a control parameter for the state equation. We find the optimal process for the optimal control problem in which the dynamics is governed by the nonlinear beam equation.

On the nonlinear elastic beam equation

We consider the existence of solutions and the continuous dependence on a parameter results for fourth-order Dirichlet problem connected with the elastic beam equation. We introduce a dual variational method.

Positive solutions of a nonlinear elastic beam equation rigidly fastened on the left and simply supported on the right

The purpose of this paper is to establish several local existence theorems concerned with n positive solutions for a fourth-order two-point boundary value problem, where n is an arbitrary positive integer and the nonlinear term may be singular. In mechanics, the problem describes deflection of an elastic beam rigidly fastened on the left and simply supported on the right.