Duffing equation with potential Landesman-Lazer condition

Size: px

Start display at page:

Download "Duffing equation with potential Landesman-Lazer condition"

Sharon Small
7 years ago
Views:

1 Duffing equation with potential Landesman-Lazer condition Petr Tomiczek Faculty of Applied Science West Bohemia University Hejnice 27

2 Linear equation Nonlinear equation with c = Nonlinear equation with c R Introduction Let us consider the nonlinear problem u (x) + c u + (m 2 c2 ) u + g(u) = f (x), x, π, 4 u() = u(π) =, (1) where c R, m N, a nonlinearity g : R R is a continuous function and f L 1 (, π).

3 Linear equation Nonlinear equation with c = Nonlinear equation with c R Content 1 Linear equation Eigenvalue problem Fredholm alternative 2 Nonlinear equation with c = Landesman-Lazer condition Generalized Landesman-Lazer condition Varitional method Potential Landesman-Lazer condition 3 Nonlinear equation with c R Potential form

4 Linear equation Nonlinear equation with c = Nonlinear equation with c R content 1 Linear equation Eigenvalue problem Fredholm alternative 2 Nonlinear equation with c = Landesman-Lazer condition Generalized Landesman-Lazer condition Varitional method Potential Landesman-Lazer condition 3 Nonlinear equation with c R Potential form

5 Linear equation Nonlinear equation with c = Nonlinear equation with c R Eigenvalue problem First we solve the homogenous equation u (x) + λ u(x) =, x (, π), u() = u(π) =. (2) We obtain a nontrivial solution to (2) if λ = m 2, m N. The number λ = m 2 is called an eigenvalue of (2) and corresponding solution sin mx is called eigenfunction of (2). We will investigate the problem (1) on the space H spanned by functions sin x, sin 2x, sin 3x,....

6 Linear equation Nonlinear equation with c = Nonlinear equation with c R Fredholm alternative Now we investigate the linear problem u (x) + λ u(x) = f (x), x (, π), u() = u(π) =. (3) We obtain the solution to (3) for all f L 1 (, π) if λ m 2, m N. If λ = m 2 then f (x) sin mx dx =. (4)

7 Linear equation Nonlinear equation with c = Nonlinear equation with c R Fredholm alternative Now we investigate the linear problem u (x) + λ u(x) = f (x), x (, π), u() = u(π) =. (3) We obtain the solution to (3) for all f L 1 (, π) if λ m 2, m N. If λ = m 2 then f (x) sin mx dx =. (4)

8 Linear equation Nonlinear equation with c = Nonlinear equation with c R content 1 Linear equation Eigenvalue problem Fredholm alternative 2 Nonlinear equation with c = Landesman-Lazer condition Generalized Landesman-Lazer condition Varitional method Potential Landesman-Lazer condition 3 Nonlinear equation with c R Potential form

9 Linear equation Nonlinear equation with c = Nonlinear equation with c R Landesman-Lazer condition We define problem lim g(s) = g + and lim g(s) = g. Then the s s u (x) + m 2 u + g(u) = f (x), x, π, u() = u(π) =, (5) has at least one solution provided that g (sin mx) + g + (sin mx) ] dx (6) < f (x) sin mx dx < g+ (sin mx) + g (sin mx) ] dx, (sin mx) + = max{sin mx, }, (sin mx) = max{ sin mx, }.

10 Linear equation Nonlinear equation with c = Nonlinear equation with c R Generalized Landesman-Lazer condition It is worth mentioning that if we suppose g < g(s) < g + for any s R, the conditon (6) is also necessary for the solvability of (5). We suppose g(s) lim =. s s We can generalized the condition (6) if we define lim inf g(s) = g + and lim sup g(s) = g. s s

11 Linear equation Nonlinear equation with c = Nonlinear equation with c R Generalized Landesman-Lazer condition It is worth mentioning that if we suppose g < g(s) < g + for any s R, the conditon (6) is also necessary for the solvability of (5). We suppose g(s) lim =. s s We can generalized the condition (6) if we define lim inf g(s) = g + and lim sup g(s) = g. s s

12 Linear equation Nonlinear equation with c = Nonlinear equation with c R Varitional method We study (5) by using of varitional method. More precisely, we look for critical points of the functional J, which is defined by J(u) = 1 2 where (u ) 2 m 2 u 2] ] dx G(u) f u dx, G(s) = s g(t) dt. Every critical point u H of the functional J satisfies J (u), v = u v m 2 uv ] ] dx g(u)v f v dx = for all v H. Then u is also a weak solution of (5) and vice versa.

13 Linear equation Nonlinear equation with c = Nonlinear equation with c R Potential Landesman-Lazer condition We define G(s) G(s) G + = lim inf, G = lim sup. s + s s s We assume that the following potential Landesman-Lazer type condition holds: < G (sin mx) + G + (sin mx) ] dx (7) f (x) sin mx dx < G+ (sin mx) + G (sin mx) ] dx. Then the problem (5) has at least one solution.

14 Linear equation Nonlinear equation with c = Nonlinear equation with c R content 1 Linear equation Eigenvalue problem Fredholm alternative 2 Nonlinear equation with c = Landesman-Lazer condition Generalized Landesman-Lazer condition Varitional method Potential Landesman-Lazer condition 3 Nonlinear equation with c R Potential form

15 Linear equation Nonlinear equation with c = Nonlinear equation with c R Potential form We multiply (1) u (x) + c u + (m 2 c2 4 ) u + g(u) = f (x) by the function e c 2 x. We put w(x) = e c 2 x u(x) and obtain an equivalent Dirichlet problem w (x) + m 2 w(x) + e c 2 x g( w e c 2 x ) = e c 2 x f (x), w() = w(π) =. (8) Now we investigate the functional J : H R, which is defined by J(w) = 1 2 (w ) 2 m 2 w 2] dx e cx G( w ] e ) e c c 2 x 2 x f w dx.

16 Linear equation Nonlinear equation with c = Nonlinear equation with c R Potential form We multiply (1) u (x) + c u + (m 2 c2 4 ) u + g(u) = f (x) by the function e c 2 x. We put w(x) = e c 2 x u(x) and obtain an equivalent Dirichlet problem w (x) + m 2 w(x) + e c 2 x g( w e c 2 x ) = e c 2 x f (x), w() = w(π) =. (8) Now we investigate the functional J : H R, which is defined by J(w) = 1 2 (w ) 2 m 2 w 2] dx e cx G( w ] e ) e c c 2 x 2 x f w dx.

17 Linear equation Nonlinear equation with c = Nonlinear equation with c R Potential form Generalized potential Landesman-Lazer condition We will suppose that g is bounded and for the potential G the following condition holds : < e c 2 x (G (x)(sin mx) + G + (x)(sin mx) ) ] dx < e c 2 x f (x) sin mx] dx e c 2 x (G + (x)(sin mx) + G (x)(sin mx) ) ] dx. (9) Then the problem (1) has at least one solution in H.

18 Linear equation Nonlinear equation with c = Nonlinear equation with c R Potential form Literature Saddle point theorem and Fredholm alternative, Proceedings of Equadiff 11, July 25-29, 25, Bratislava. Potential Landesman-Lazer type conditions and the Fucik spectrum, EJDE Vol. 25(25), No. 94, pp Remark on Duffing Equation with the Dirichlet Boundary Condition, EJDE Vol. 27(27), No. 81, pp. 1-3.

The idea to study a boundary value problem associated to the scalar equation

The idea to study a boundary value problem associated to the scalar equation Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 23, 2004, 73 88 DEGREE COMPUTATIONS FOR POSITIVELY HOMOGENEOUS DIFFERENTIAL EQUATIONS Christian Fabry Patrick Habets