First boundary-value problem for a certain fourth-order differential equation

1983 ◽  
Vol 24 (1) ◽  
pp. 102-105 ◽  
Author(s):  
V. A. Malovichko
Keyword(s):  
Differential Equation ◽  
Boundary Value Problem ◽  
Fourth Order ◽  
Order Differential Equation ◽  
Boundary Value ◽  
Fourth Order Differential Equation

scholarly journals NONCLASSICAL FOURTH ORDER DISCONTINUOUS BOUNDARY VALUE PROBLEM WITH ABSTRACT LINEAR FUNCTIONALS

2016 ◽  
Vol 12 (11) ◽  
pp. 6812-6820
Author(s):  
Mustafa KANDEMÄ°R
Keyword(s):  
Differential Equation ◽  
Boundary Value Problem ◽  
Fourth Order ◽  
Linear Functional ◽  
Order Differential Equation ◽  
Boundary Value ◽  
Point Interaction ◽  
Linear Functionals ◽  
Discontinuous Boundary ◽  
Fourth Order Differential Equation

In this study, we have considered a nonstandard boundary-value problem for fourth order differential equation on two disjoint intervals The boundary conditions contains not only endpoints  and but also a point interaction and abstract linear functional So our problem is nota pure differential one. We investigate such some important properties as isomorphism, Fredholmness and coerciveness with respect to the spectral parameter.

scholarly journals Solvability of a Fourth-Order Boundary Value Problem with Integral Boundary Conditions

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Hui Li ◽  
Libo Wang ◽  
Minghe Pei
Keyword(s):  
Differential Equation ◽  
Boundary Conditions ◽  
Positive Solutions ◽  
Fourth Order ◽  
Order Differential Equation ◽  
Existence Of Solutions ◽  
Boundary Value ◽  
Integral Boundary Conditions ◽  
Integral Boundary ◽  
Fourth Order Differential Equation

We investigate the existence of solutions and positive solutions for a nonlinear fourth-order differential equation with integral boundary conditions of the formx(4)(t)=f(t,x(t),x′(t),x′′(t),x′′′(t)),t∈[0,1],x(0)=x′(1)=0,x′′(0)=∫01h(s,x(s),x′(s),x′′(s))ds,x′′′(1)=0, wheref∈C([0,1]×ℝ4),h∈C([0,1]×ℝ3). By using a fixed point theorem due to D. O'Regan, the existence of solutions and positive solutions for the previous boundary value problems is obtained. Meanwhile, as applications, some examples are given to illustrate our results.

scholarly journals Positive Solutions for Two-Point Boundary Value Problems for Fourth-Order Differential Equations with Fully Nonlinear Terms

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Yixin Zhang ◽  
Yujun Cui
Keyword(s):  
Differential Equation ◽  
Boundary Value Problem ◽  
Positive Solutions ◽  
Fourth Order ◽  
Order Differential Equation ◽  
Boundary Value ◽  
Linear Operators ◽  
Continuous Space ◽  
Fixed Point Index Theory ◽  
Existence Of Positive Solutions

In this paper, we consider the existence of positive solutions for the fully fourth-order boundary value problem u 4 t = f t , u t , u ′ t , u ″ t , u ‴ t ,   0 ≤ t ≤ 1 , u 0 = u 1 = u ″ 0 = u ″ 1 = 0 , where f : 0,1 × 0 , + ∞ × − ∞ , + ∞ × − ∞ , 0 × − ∞ , + ∞ ⟶ 0 , + ∞ is continuous. This equation can simulate the deformation of an elastic beam simply supported at both ends in a balanced state. By using the fixed-point index theory and the cone theory, we discuss the existence of positive solutions of the fully fourth-order boundary value problem. We transform the fourth-order differential equation into a second-order differential equation by order reduction method. And then, we examine the spectral radius of linear operators and the equivalent norm on continuous space. After that, we obtain the existence of positive solutions of such BVP.

An extremum variational principle and error estimate procedure for qIV = f(q, x)

1982 ◽  
Vol 92 (2) ◽  
pp. 307-316 ◽  
Author(s):  
Dj. S. Djukic ◽  
T. M. Atanackovic
Keyword(s):  
Differential Equation ◽  
Variational Principle ◽  
Error Estimate ◽  
Boundary Value Problems ◽  
Fourth Order ◽  
Order Differential Equation ◽  
Boundary Value ◽  
Approximate Solutions ◽  
Estimate Procedure ◽  
Fourth Order Differential Equation

AbstractAn extremum variational principle for boundary value problems described by a non-linear fourth order differential equation qIV = f(q, x) is constructed. For approximate solutions an error estimate procedure, based on the value of the functional, is developed and applied to two concrete problems.

On a Singular Two-Point Boundary Value Problem for the Nonlinear mth-Order Differential Equation with Deviating Arguments

1997 ◽  
Vol 4 (6) ◽  
pp. 557-566
Author(s):  
B. Půža
Keyword(s):  
Differential Equation ◽  
Boundary Value Problem ◽  
Vector Function ◽  
Unique Solvability ◽  
Order Differential Equation ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Point Boundary ◽  
Deviating Arguments ◽  
Measurable Functions

Abstract Sufficient conditions of solvability and unique solvability of the boundary value problem u (m)(t) = f(t, u(τ 11(t)), . . . , u(τ 1k (t)), . . . , u (m–1)(τ m1(t)), . . . . . . , u (m–1)(τ mk (t))), u(t) = 0, for t ∉ [a, b], u (i–1)(a) = 0 (i = 1, . . . , m – 1), u (m–1)(b) = 0, are established, where τ ij : [a, b] → R (i = 1, . . . , m; j = 1, . . . , k) are measurable functions and the vector function f : ]a, b[×Rkmn → Rn is measurable in the first and continuous in the last kmn arguments; moreover, this function may have nonintegrable singularities with respect to the first argument.

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