Two homoclinic solutions for a nonperiodic fourth-order differential equation without coercive condition

2016 ◽  
Vol 40 (8) ◽  
pp. 3163-3172 ◽  
Author(s):  
Shiping Lu ◽  
Tao Zhong
Keyword(s):  
Differential Equation ◽  
Fourth Order ◽  
Order Differential Equation ◽  
Homoclinic Solutions ◽  
Coercive Condition ◽  
Fourth Order Differential Equation

scholarly journals Infinitely Many Homoclinic Solutions for Fourth Order p-Laplacian Differential Equations

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 505
Author(s):  
Stepan Tersian
Keyword(s):  
Differential Equation ◽  
Fourth Order ◽  
Variational Approach ◽  
Order Differential Equation ◽  
Growth Conditions ◽  
Homoclinic Solutions ◽  
Physical Models ◽  
Critical Point Theorem ◽  
Positive Functions ◽  
Fourth Order Differential Equation

The existence of infinitely many homoclinic solutions for the fourth-order differential equation φ p u ″ t ″ + w φ p u ′ t ′ + V ( t ) φ p u t = a ( t ) f ( t , u ( t ) ) , t ∈ R is studied in the paper. Here φ p ( t ) = t p − 2 t , p ≥ 2 , w is a constant, V and a are positive functions, f satisfies some extended growth conditions. Homoclinic solutions u are such that u ( t ) → 0 , | t | → ∞ , u ≠ 0 , known in physical models as ground states or pulses. The variational approach is applied based on multiple critical point theorem due to Liu and Wang.

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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