Positive Solutions of a Fourth-Order Differential Equation with Multipoint Boundary Conditions

2014 ◽  
Vol 43 (1) ◽  
pp. 93-104
Author(s):  
Phan Dinh Phung
Keyword(s):  
Differential Equation ◽  
Boundary Conditions ◽  
Positive Solutions ◽  
Fourth Order ◽  
Order Differential Equation ◽  
Multipoint Boundary ◽  
Multipoint Boundary Conditions ◽  
Fourth Order Differential Equation

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 The Solution of Nonlinear Fourth-Order Differential Equation with Integral Boundary Conditions

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Yanli Fu ◽  
Huanmin Yao
Keyword(s):  
Differential Equation ◽  
Boundary Conditions ◽  
Fourth Order ◽  
Reproducing Kernel ◽  
Order Differential Equation ◽  
Integral Boundary Conditions ◽  
Reproducing Kernel Space ◽  
Integral Boundary ◽  
Derivatives Of ◽  
Fourth Order Differential Equation

An iterative algorithm is proposed for solving the solution of a nonlinear fourth-order differential equation with integral boundary conditions. Its approximate solutionun(x)is represented in the reproducing kernel space. It is proved thatun(x)converges uniformly to the exact solutionu(x). Moreover, the derivatives ofun(x)are also convergent to the derivatives ofu(x). Numerical results show that the method employed in the paper is valid.

scholarly journals Positive Solutions for the Eigenvalue Problem of Semipositone Fractional Order Differential Equation with Multipoint Boundary Conditions

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Ge Dong
Keyword(s):  
Differential Equation ◽  
Eigenvalue Problem ◽  
Boundary Conditions ◽  
Positive Solution ◽  
Fractional Order ◽  
Order Differential Equation ◽  
Sufficient Conditions ◽  
Fractional Order Differential Equation ◽  
Multipoint Boundary ◽  
Multipoint Boundary Conditions

We study the existence of positive solution for the eigenvalue problem of semipositone fractional order differential equation with multipoint boundary conditions by using known Krasnosel'skii's fixed point theorem. Some sufficient conditions that guarantee the existence of at least one positive solution for eigenvalues  λ>0sufficiently small andλ>0sufficiently large are established.

15.—A Class of Focal Points for Non-self-adjoint Fourth-order Differential Equations

1975 ◽  
Vol 72 (3) ◽  
pp. 207-214
Author(s):  
Kurt Kreith
Keyword(s):  
Differential Equation ◽  
Boundary Conditions ◽  
Fourth Order ◽  
Trivial Solution ◽  
General Equation ◽  
Order Differential Equation ◽  
Upper Bounds ◽  
Conjugate Points ◽  
Focal Points ◽  
Fourth Order Differential Equation

SynopsisConjugate points are defined in terms of solutions of a linear fourth-order differential equation satisfying two homogeneous boundary conditions at x = α and either u(β) = u′(β) = 0 or u′(γ) = u″(γ) = 0. The smallest β > α and γ > α such that these boundary conditions are satisfied by a non-trivial solution of the equation are denoted by η(α) and ῆ-(α), respectively. Upper bounds are established for min [η(α), ῆ(α)] relative to the conjugate points of a self-adjoint differential equation which is majorised by the more general equation under study.

The number of Dirichlet solutions of a fourth order differential equation

1982 ◽  
Vol 92 (3-4) ◽  
pp. 233-239 ◽  
Author(s):  
Thomas T. Read
Keyword(s):  
Differential Equation ◽  
Boundary Conditions ◽  
Differential Expression ◽  
Fourth Order ◽  
Dirichlet Form ◽  
Order Differential Equation ◽  
Dirichlet Integral ◽  
Linearly Independent ◽  
Nondecreasing Functions ◽  
Fourth Order Differential Equation

SynopsisIt is shown that the equation (p2y”)”–(p1y’)’+ p0y = 0 has exactly two linearly independent solutions on [0,∞) with finite Dirichlet integral when the coefficients are nonnegative and p2 satisfies a condition which includes all nondecreasing functions. An inequality for the Dirichlet form is derived and used to extend characterizations of the domains of certain self-adjoint operations associated with the differential expression to arbitrary symmetric boundary conditions at 0.

scholarly journals Asymptotic theory for a critical case for a general fourth-order differential equation

1998 ◽  
Vol 21 (3) ◽  
pp. 479-488
Author(s):  
A. S. A. Al-Hammadi
Keyword(s):  
Differential Equation ◽  
Fourth Order ◽  
Critical Case ◽  
Asymptotic Theory ◽  
Order Differential Equation ◽  
Fourth Order Equations ◽  
Fourth Order Differential Equation

In this paper we identify a relation between the coefficients that represents a critical case for general fourth-order equations. We obtained the forms of solutions under this critical case.

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