Abstract
In this paper we investigate a class of impulsive differential equations with Dirichlet boundary conditions. Firstly, we define new inner product of and prove that the norm which is deduced by the inner product is equivalent to the usual norm. Secondly, we construct the lower and upper solutions of (1.1). Thirdly, we obtain the existence of a positive solution, a negative solution and a sign-changing solution by using critical point theory and variational methods. Finally, an example is presented to illustrate the application of our main result.
Highlights
This paper is mainly concerned with the following second order impulsive differential equations with Dirichlet boundary conditions −−∆( p((t ) x′(t ))′ + q p = x′(ti ))= (t ) x (t ) f (t, x α= i x, i 1,(t)), 2, t∈ k, J, t ≠ ti (1.1)x= (0) x= (1) 0, where J ⊂ [0,1], p ∈ C1 J,[β, +∞), β > 0, q ∈ C J,[0, +∞),0 = t0 < t1 < < tk < tk+1 = 1, α1,α2,αk are nonnegative constants with ( ) ( ) ( ) ( ) ( ) k ( ) ∑αi < 4β . −∆ p= x′(ti ) p ti+ x′ ti+ − p ti− x′ ti−
We obtain the existence of a positive solution, a negative solution and a sign-changing solution by using critical point theory and variational methods
Second order impulsive differential Equation (1.1) of this paper happens to be the mathematical model of this kind of problem
Summary
X= (0) x= (1) 0, where J ⊂ [0,1] , p ∈ C1 J ,[β , +∞) , β > 0 , q ∈ C J ,[0, +∞) ,. Second order impulsive differential Equation (1.1) of this paper happens to be the mathematical model of this kind of problem. Motivated by the papers mentioned above, we study the existence of sign-changing solution for second order impulsive differential equations with Dirichlet boundary conditions. To the best of our knowledge, there are few papers concerned with the existence of sign-changing solution for impulsive differential equations. We obtain the existence of a positive solution, a negative solution and a sign-changing solution of (1.1) by using critical point theory and variational methods. Let x (t, x0 ) and x (t, x0 ) be the unique solution of this initial value problem considered in H and E respectively, with 0,η ( x0 )) and 0,η ( x0 )) the right maximal interval of existence. Problem (1.1) has at least three solutions: one positive, one negative, and one sign-changing
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