Abstract
In this article, we study a discrete fourth order boundary value problem. By making use of variational methods and critical point theory, we obtain some criteria for the existence and multiple solutions. Moreover, two examples are included to illustrate the applicability of the main results.
Highlights
Define a functional J on U by2un–2 2 + rn–1( un–1)2 – F(n, un). (rn–1 un–1) – f (n, un), n ∈ Z[1, k]
1 Introduction and statement of the main results In this article, we are interested in the existence and multiple solutions to the discrete fourth order nonlinear equation
Differential equations similar to (1.3) and special cases of it have been studied using a number of different methods in the literature, we refer the reader to papers [1, 2, 11,12,13,14, 24, 25] and the references contained therein
Summary
2un–2 2 + rn–1( un–1)2 – F(n, un). (rn–1 un–1) – f (n, un), n ∈ Z[1, k]. Λk be the eigenvalues of M. and λmax = max{λj | λj = 0, j = 1, 2, . Lemma 3.1 (Linking theorem [21, 23]) Let U be a real Banach space, U = U1 ⊕ U2, where U1 is finite dimensional. (J1) There are positive constants c and ρ such that J|∂Bρ(0)∩U2 ≥ c. (J2) There are μ ∈ ∂B1(0) ∩ U2 and a positive constant c ≥ ρ such that J|∂Ω ≤ 0, where. Lemma 3.2 (Clark theorem [21]) Let U be a real Banach space, J ∈ C1(U, R), with J being even, bounded from below and satisfying the Palais–Smale condition. The functional J satisfies the Palais–Smale condition.
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