Existence To Solutions For 4th-order Differential Equation Dirichlet Boundary Value Problems

Differential equation Dirichlet boundary value problems are more general ones in bound-ary value problems. Many researchers studied these problems by employing the topological degree theory and fixed point index theory (see [4-8,30-32]) or by using Morse theory (see [16-23]) or the critical point theory (see [9], [13-15,24-29]), and obtained the existence and multiplicity of solutions.In paper [9], the author used the local linking theory [10] to discuss a class of su-perquadratic second order Hamiltonian systems and obtained at least one nontrival T-periodic solution. Motivated by the paper [9] and the result, in Chapter one, we apply the method to the following 4th-order differential equation Dirichlet boundary value problem: where A(t),B(t) is an N x N symmetric matrix, A(·) is continuous, B(·) is continuously differentiable and (B(t)x,x)≥｜x｜2 for (t,x)∈[0,1] xRn, F:[0,1]×RNâ†'R satisfies the following assumption:(A) F(t, x) is measurable in t for every x∈RN and continuously differentiable in x for a.e. t∈[0,1], and there exist a∈C(R+, R+), b∈L1([0,1], R+) such that for all x∈RN and a.e. t∈[0,1].Using the local linking theory [10], we obtained the following mainly result:Theorem Suppose that F satisfies (A) and the following condition: uniformly for a.e. t∈[0,1]; uniformly for a.e. t∈[0,1];(F3) Assume that there existλ> 2 and d1> 0, such that uniformly for a.e. t∈[0,1];(F4) Assume that there existβ>λ-1 and d2> 0 such that uniformly for a.e. tκ[0,1];(F5) F(t,x)≥0 for all (t,x)∈[0,1] x RN. Then the problem (1.1.1) has at least one nontrival solution. In paper [14], the author used the critical point theory and Morse theory to discuss Kirchhoff type problems and obtained the existence and multiplicity of solutions. In paper [13], the author used the critical point theory to discuss a class of Kirchhoff type equation with Dirichlet boundary condition and obtained the existence and multiplicity of weak solutions. Particularly, in paper [12], the author used Krasnoselskill genus to discuss the existence of solution for p-Kirchhoff type equation. Motivated by paper [12] and [13], in chapter two, we apply the method to the following 4th-order Kirchhoff equation with Dirichlet boundary value problem: where f:[0,1]×Râ†'R and M:R+â†'R+are continuous functions and satisfy the following conditions:(M) There are positive constants A, B and a such that Ata≤M(t)≤Bta for t∈R+;(f1)｜f(x,t)｜≤c(｜tlq-1+1) for all (x,t)∈[0,1]×R where c is a positive constant q∈(2,2*) and a>q+1/2;(f2)(?) f(x,t)/t=+∞uniformly for a.e. t∈[0,1];(f3) f(x,t)=-f(x,-t) for all (x,t)∈[0,1]×R:(f4) There existsμ> 2(a+1), R> 0 such that 0<μF(x,t)≤f(x,t)t for all|t|> R.Theorem Assume M and f satisfy these conditions (M), (f1), (f2), (fa), and (f4), Then the problem (2.1.1) has infinitely many solutions.