scholarly journals Existence of Positive Solutions for Some Superlinear Fourth-Order Boundary Value Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Imed Bachar ◽  
Habib Mâagli
Keyword(s):  
Green Function ◽  
Fourth Order ◽  
Integrability Condition ◽  
Order Equation ◽  
Global Behavior ◽  
Fourth Order Equation ◽  
Existence Of Positive Solutions ◽  
Nonnegative Continuous Function ◽  
The Green Function ◽  
Perturbation Arguments

We are concerned with the following superlinear fourth-order equationu4t+utφt,−ut=0,   t∈0, 1;−u0=u1=0,  −u′0=a,  −u′1=-b, wherea,−bare nonnegative constants such thata+b>0andφt,−sis a nonnegative continuous function that is required to satisfy some appropriate conditions related to a classKsatisfying suitable integrability condition. Our purpose is to prove the existence, uniqueness, and global behavior of a classical positive solution to the above problem by using a method based on estimates on the Green function and perturbation arguments.

scholarly journals Existence of Positive Solutions for a Pertubed Fourth-Order Equation

2021 ◽  
Vol 45 (4) ◽  
pp. 623-633
Author(s):  
MOHAMMAD REZA HEIDARI TAVANI ◽  
◽  
ABDOLLAH NAZARI ◽  
Keyword(s):  
Differential Equation ◽  
Fourth Order ◽  
Weak Solutions ◽  
Nonlinear Term ◽  
Point Theory ◽  
Order Equation ◽  
Fourth Order Equation ◽  
Equation Problem ◽  
Existence Of Positive Solutions ◽  
Fourth Order Problems

In this paper, a special type of fourth-order differential equations with a perturbed nonlinear term and some boundary conditions is considered which is very important in mechanical engineering. Therefore, the existence of a non-trivial solution for such equations is very important. Our goal is to ensure at least three weak solutions for a class of perturbed fourth-order problems by applying certain conditions to the functions that are available in the differential equation (problem (??)). Our approach is based on variational methods and critical point theory. In fact, using a fundamental theorem that is attributed to Bonanno, we get some important results. Finally, for some results, an example is presented.

scholarly journals Positive Solutions for a Singular Superlinear Fourth-Order Equation with Nonlinear Boundary Conditions

2020 ◽  
Vol 2020 ◽  
pp. 1-6
Author(s):  
Dongliang Yan
Keyword(s):  
Boundary Conditions ◽  
Positive Solutions ◽  
Fourth Order ◽  
Nonlinear Boundary ◽  
Nonlinear Boundary Conditions ◽  
Order Equation ◽  
Positive Parameter ◽  
Fourth Order Equation ◽  
Existence Of Positive Solutions ◽  
Small Positive Parameter

We show the existence of positive solutions for a singular superlinear fourth-order equation with nonlinear boundary conditions. u⁗x=λhxfux, x∈0,1,u0=u′0=0,u″1=0, u⁗1+cu1u1=0, where λ > 0 is a small positive parameter, f:0,∞⟶ℝ is continuous, superlinear at ∞, and is allowed to be singular at 0, and h: [0, 1] ⟶ [0, ∞) is continuous. Our approach is based on the fixed-point result of Krasnoselskii type in a Banach space.

A Geometrical Solution for the Inverse Kinematics of Three Revolute Manipulators

1993 ◽  
Author(s):  
Eugene F. Fichter
Keyword(s):  
Fourth Order ◽  
Inverse Kinematics ◽  
Solution Procedure ◽  
Order Equation ◽  
Algebraic Solution ◽  
Computer Implementation ◽  
Third Order ◽  
Fourth Order Equation ◽  
Inverse Kinematics Problem

Abstract Points of intersection of a circle and a torus are used to find a solution to the inverse kinematics problem for a three revolute manipulator. Both geometrical and algebraic solution procedures are discussed. The algebraic procedure begins with a third order equation instead of the usual fourth order equation. Since the procedure is basically geometrical it lends itself to a computer implementation which graphically displays each steps in the solution procedure. The potential of this approach for both design and pedagogy is discussed.

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