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S -shaped component of nodal solutions for problem involving one-dimension mean curvature operator

Ruyun Ma, Zhiqian He, Xiaoxiao Su (2023)

Czechoslovak Mathematical Journal

Let E = { u ∈ C 1 [ 0 , 1 ] : u ( 0 ) = u ( 1 ) = 0 } . Let S k ν with ν = { + , - } denote the set of functions u ∈ E which have exactly k - 1 interior nodal zeros in (0, 1) and ν u be positive near 0 . We show the existence of S -shaped connected component of S k ν -solutions of the problem u ' 1 - u ' 2 ' + λ a ( x ) f ( u ) = 0 , x ∈ ( 0 , 1 ) , u ( 0 ) = u ( 1 ) = 0 , where λ > 0 is a parameter, a ∈ C ( [ 0 , 1 ] , ( 0 , ∞ ) ) . We determine the intervals of parameter λ in which the above problem has one, two or three S k ν -solutions. The proofs of the main results are based upon the bifurcation technique.

Scattered homoclinics to a class of time-recurrent Hamiltonian systems

Gregory S. Spradlin (2007)

ESAIM: Control, Optimisation and Calculus of Variations

A second-order Hamiltonian system with time recurrence is studied. The recurrence condition is weaker than almost periodicity. The existence is proven of an infinite family of solutions homoclinic to zero whose support is spread out over the real line.

Second order nonlinear differential equations with linear impulse and periodic boundary conditions

Aydin Huseynov (2011)

Applications of Mathematics

In this study, we establish existence and uniqueness theorems for solutions of second order nonlinear differential equations on a finite interval subject to linear impulse conditions and periodic boundary conditions. The results obtained yield periodic solutions of the corresponding periodic impulsive nonlinear differential equation on the whole real axis.

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