Abstract
In this paper we deal with the existence and multiplicity of nontrivial solutions to a class of operator equation. By using infinite dimensional Morse theory, we establish some conditions which guarantee that the equation has many nontrivial solutions.
Highlights
Let E = C[, ] be the usual real Banach space with the norm u = maxt∈[, ] |u(t)| for all u ∈ C[, ], and H = L [, ] be the usual real Hilbert space with the inner product (·, ·) and the norm ·
This paper is concerned with the existence of nontrivial solutions for the following operator equation of the form u = K fu, ( . )
There have been many papers to study the existence of nontrivial solutions on higher order boundary value problems, see [ – ]
Summary
This paper is concerned with the existence of nontrivial solutions for the following operator equation of the form u = K fu,. In a later paper [ ], by applying the strongly monotone operator principle and the critical point theory, some new existence theorems on unique, at least one nontrivial and infinitely many solutions were established. The mountain pass lemma is applied to obtain a critical point v+ ∈ H of J+, with critical value c+ > , which satisfies. The mountain pass lemma is applied to obtain a critical point u = θ of J which solves the equation u(t) = K f u(t) , t ∈ [ , ]. We shall prove that I – K f (u (t)) has a bounded inverse operator on H, i.e., u is a nondegenerate critical point of J.
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