scholarly journals Lyapunov's Type Inequalities for Fourth-Order Differential Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-25 ◽  
Author(s):  
Samir H. Saker
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Nontrivial Solution ◽  
Lower Bounds ◽  
Fourth Order ◽  
Order Differential Equation ◽  
Higher Order ◽  
Hardy’S Inequality ◽  
Hardy's Inequality ◽  
Fourth Order Differential Equation

For a fourth-order differential equation, we will establish some new Lyapunov-type inequalities, which give lower bounds of the distance between zeros of a nontrivial solution and also lower bounds of the distance between zeros of a solution and/or its derivatives. The main results will be proved by making use of Hardy’s inequality and some generalizations of Opial-Wirtinger-type inequalities involving higher-order derivatives. Some examples are considered to illustrate the main results.

scholarly journals Oscillation Theorems for Nonlinear Differential Equations of Fourth-Order

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 520 ◽  
Author(s):  
Osama Moaaz ◽  
Ioannis Dassios ◽  
Omar Bazighifan ◽  
Ali Muhib
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Fourth Order ◽  
Order Differential Equation ◽  
Nonlinear Differential Equations ◽  
Sufficient Conditions ◽  
Oscillatory Behavior ◽  
Middle Term ◽  
Oscillation Theorems ◽  
Fourth Order Differential Equation

We study the oscillatory behavior of a class of fourth-order differential equations and establish sufficient conditions for oscillation of a fourth-order differential equation with middle term. Our theorems extend and complement a number of related results reported in the literature. One example is provided to illustrate the main results.

Asymptotic methods for fourth-order differential equations

1980 ◽  
Vol 87 (1-2) ◽  
pp. 53-63 ◽  
Author(s):  
C. G. M. Grudniewicz
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Fourth Order ◽  
Asymptotic Form ◽  
Order Differential Equation ◽  
Asymptotic Methods ◽  
Index Theory ◽  
New Method ◽  
Image Position ◽  
Fourth Order Differential Equation

SynopsisA new method is developed for obtaining the asymptotic form of solutions of the fourth-order differential equationwherem, nare integers and 1 ≦m,n≦ 2. The method gives new, shorter proofs of the well-known results of Walker in deficiency index theory and covers the cases not considered by Walker.

Instability results for fourth order differential equations

1980 ◽  
Vol 85 (3-4) ◽  
pp. 247-250 ◽  
Author(s):  
W. A. Skrapek
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Fourth Order ◽  
Order Differential Equation ◽  
Sufficient Conditions ◽  
Fourth Order Differential Equation

SYNOPSISIn this paper we give sufficient conditions (Theorems 1 and 2) for the instability of the fourth order differential equation

The limit-4 case of fourth-order self-adjoint differential equations

1977 ◽  
Vol 79 (1-2) ◽  
pp. 51-59 ◽  
Author(s):  
M. S. P. Eastham
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Fourth Order ◽  
Asymptotic Theory ◽  
Order Differential Equation ◽  
New Method ◽  
All Solutions ◽  
Image Position ◽  
Fourth Order Differential Equation

SynopsisA new method is developed for identifying real-valued coefficientsr(x),p(x), andq(x)for which all solutions of the fourth-order differential equationareL2(0, ∞). The results are compared with those derived from the asymptotic theory of Devinatz, Walker, Kogan and Rofe-Beketov.

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.

25.—On the Eigenfunction Expansion associated with a Singular Complex-valued Fourth-order Differential Equation

1976 ◽  
Vol 75 (4) ◽  
pp. 325-332
Author(s):  
Jyoti Chaudhuri ◽  
V. Krishna Kumar
Keyword(s):  
Differential Equation ◽  
Fourth Order ◽  
Representation Theorem ◽  
Eigenfunction Expansion ◽  
Order Differential Equation ◽  
Convergence Theory ◽  
Infinite Intervals ◽  
Complex Valued ◽  
Class Of Functions ◽  
Fourth Order Differential Equation

SynopsisThe direct convergence theory of eigenfunction expansions associated with boundry value problems, not necessarily self-adjoint, generated from complex-valued fourth-order symmetric ordinary differential expressions on semi-infinite intervals, is discussed. An admissible class of functions for the expansion is characterised. Also a generalisation of Stieltjes representation theorem for analytic functions discussed in [13, §§ 22.23 and 24] is obtained.

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