scholarly journals DIRICHLET BVP FOR THE SECOND ORDER NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS AT RESONANCE

2019 ◽  
Vol 24 (4) ◽  
pp. 585-597
Author(s):  
Sulkhan Mukhigulashvili ◽  
Mariam Manjikashvili
Keyword(s):  
Dirichlet Problem ◽  
Differential Equations ◽  
Nonlinear Equation ◽  
Linear Equation ◽  
Linear Problem ◽  
Sufficient Conditions ◽  
Nonlinear Ordinary Differential Equations ◽  
Nontrivial Solutions ◽  
At Resonance ◽  
Carathéodory Function

Landesman-Lazer’s type efficient sufficient conditions are established forthe solvability of the Dirichlet problem u′′(t) = p(t)u(t) + f(t, u(t)) + h(t),for a ≤ t ≤ b, u(a) = 0, u(b) = 0, where h, p ϵ L([a, b];R) and f is the L([a, b];R) Caratheodory function, in the casewhere the linear problem u′′(t) = p(t)u(t), u(a) = 0,u(b) = 0 has nontrivial solutions. The results obtained in the paper are optimal in the sense that if f ≡ 0,i.e., when nonlinear equation turns to the linear equation, from our results follows the first partof Fredholm’s theorem.

The problem of dirichlet for quasilinear elliptic differential equations with many independent variables

1969 ◽  
Vol 264 (1153) ◽  
pp. 413-496 ◽  
Keyword(s):  
Dirichlet Problem ◽  
Differential Equations ◽  
Elliptic Equations ◽  
Sufficient Conditions ◽  
Central Position ◽  
Second Primary ◽  
Elliptic Differential Equations ◽  
High Level ◽  
Quasilinear Elliptic

This paper is concerned with the existence of solutions of the Dirichlet problem for quasilinear elliptic partial differential equations of second order, the conclusions being in the form of necessary conditions and sufficient conditions for this problem to be solvable in a given domain with arbitrarily assigned smooth boundary data. A central position in the discussion is played by the concept of global barrier functions and by certain fundamental invariants of the equation. With the help of these invariants we are able to distinguish an important class of ‘ regularly elliptic5 equations which, as far as the Dirichlet problem is concerned, behave comparably to uniformly elliptic equations. For equations which are not regularly elliptic it is necessary to impose significant restrictions on the curvatures of the boundaries of the underlying domains in order for the Dirichlet problem to be generally solvable; the determination of the precise form of these restrictions constitutes a second primary aim of the paper. By maintaining a high level of generality throughout, we are able to treat as special examples the minimal surface equation, the equation for surfaces having prescribed mean curvature, and a number of other non-uniformly elliptic equations of classical interest.

scholarly journals Stability of some differential equations of the fourth-order and fifth-order

2019 ◽  
pp. 18-27
Author(s):  
Boris S. Kalitine
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Fourth Order ◽  
Classical Method ◽  
Nonlinear Differential Equations ◽  
Sufficient Conditions ◽  
Nonlinear Ordinary Differential Equations ◽  
Necessary And Sufficient ◽  
Positive Functions ◽  
Fifth Order

The article is devoted to the study of the problem of stability of nonlinear ordinary differential equations by the method of semi-definite Lyapunov’s functions. The types of fourth-order and fifth-order scalar nonlinear differential equations of general form are singled out, for which the sign-constant auxiliary functions are defined. Sufficient conditions for stability in the large are obtained for such equations. The results coincide with the necessary and sufficient conditions in the corresponding linear case. Studies emphasize the advantages in using the semi-positive functions in comparison with the classical method of applying Lyapunov’s definite positive functions.

scholarly journals On the Modified Boundary Value Problem of De La Vallée-Poussin for Nonlinear Ordinary Differential Equations

1994 ◽  
Vol 1 (4) ◽  
pp. 429-458
Author(s):  
G. Tskhovrebadze
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Ordinary Differential Equations ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Nonlinear Ordinary Differential Equation ◽  
Nonlinear Ordinary Differential Equations ◽  
De La Vallée Poussin

Abstract The sufficient conditions of the existence, uniqueness, and correctness of the solution of the modified boundary value problem of de la Vallée-Poussin have been found for a nonlinear ordinary differential equation u (n) = f(t, u, u′, … , u (n–1)), where the function f has nonitegrable singularities with respect to the first argument.

The Nonlinear Dynamics of Long, Slender Cylinders

1984 ◽  
Vol 106 (2) ◽  
pp. 250-256 ◽  
Author(s):  
Y. C. Kim ◽  
M. S. Triantafyllou
Keyword(s):  
Nonlinear Dynamics ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Geometric Nonlinearity ◽  
Linear Problem ◽  
Large Deformations ◽  
Nonlinear Ordinary Differential Equations ◽  
Wkb Method ◽  
Fluid Drag ◽  
Nonlinear Fluid

The nonlinear dynamics of long, slender cylinders for moderately large deformations are studied by projecting the solution along the set of eigenmodes of the linear problem. The resulting set of nonlinear ordinary differential equations is truncated on the basis of bandlimited response. The efficiency of the method is due to the derivation of asymptotic solutions for the linear problem in its general form, by using the WKB method. Applications for the dynamics of risers, including the effects of nonlinear fluid drag and geometric nonlinearity demonstrate the features of the method.

scholarly journals Remarks on superlinear boundary value problems

1977 ◽  
Vol 16 (2) ◽  
pp. 181-188 ◽  
Author(s):  
Svatopluk Fučík
Keyword(s):  
Dirichlet Problem ◽  
Differential Equations ◽  
Boundary Value Problems ◽  
Differential Operators ◽  
Null Space ◽  
Nonlinear Differential Equations ◽  
Sufficient Conditions ◽  
Positive Function ◽  
Necessary And Sufficient ◽  
Weak Solvability

Necessary and sufficient conditions for the weak solvability of the Dirichlet problem for nonlinear differential equations of the second order are proved. The differential operators considered are in the form of a sum of a linear noninvertible operator, with the null-space generated by a positive function, and a monotone nonlinear perturbation, the growth of which is more than linear.

scholarly journals Linearization of Fifth-Order Ordinary Differential Equations by Generalized Sundman Transformations

2018 ◽  
Vol 2018 ◽  
pp. 1-17
Author(s):  
Supaporn Suksern ◽  
Kwanpaka Naboonmee
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Differential Equations ◽  
Linear Equation ◽  
Ordinary Differential Equations ◽  
Sufficient Conditions ◽  
Order Ordinary Differential Equation ◽  
Linearization Problem ◽  
Necessary And Sufficient ◽  
Fifth Order

In this article, the linearization problem of fifth-order ordinary differential equation is presented by using the generalized Sundman transformation. The necessary and sufficient conditions which allow the nonlinear fifth-order ordinary differential equation to be transformed to the simplest linear equation are found. There is only one case in the part of sufficient conditions which is surprisingly less than the number of cases in the same part for order 2, 3, and 4. Moreover, the derivations of the explicit forms for the linearizing transformation are exhibited. Examples for the main results are included.

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