Existence of at Least One Homoclinic Solution for a Nonlinear Second-Order Difference Equation

Abstract

This paper presents sufficient conditions for the existence of at least one homoclinic solution for a nonlinear second-order difference equation with p-Laplacian. Our technical approach is based on variational methods. An example is offered to demonstrate the applicability of our main results.

Multiplicity of heteroclinic solutions for the discrete p(k)-Laplace Kirchhoff type equations with a parameter

“…Our first tool and approach is based on the variational principle of Ricceri [41, Theorem 2.1], which is recalled below (see [3,Theorem 2.1(a)] or [4, Theorem 3.1 and Remark 3.1] for a different version of the result). We refer to the works [1,5,6,14,15,20,21,22,33] in which Theorem 2.9 below has been successfully applied to show the existence of at least one nontrivial solution for boundary value problems.…”

Existence of homoclinic solutions for difference equations on integers via variational method

“…There are various methods such fixed point, variational methods, critical point theory, Morse theory and the mountain-pass theorem. For background and recent results, we refer the reader to [1,2,3,4,5,6,7,8,12,13,19,20,22,23,24,26,27,28,29,30,32,33]. For example, Henderson and Thompson investigated existence multiple solutions for second order discrete boundary value problems in [19].…”

“…Stegliński in [28] determined a concrete interval of positive parameter λ, for prove the existence of homoclinic solutions for the problem (1.1), while in [30] dealt with the problem (P f,g,h λ,µ ), in the case µ = 0 and h ≡ 0, and using both the general variational principle of Ricceri and the direct method introduced by Faraci and Kristály proved the existence of infinitely many homoclinic solutions for a the problem where the nonlinear term f has an appropriate oscillatory behavior at zero. In [4], sufficient conditions for the existence of at least one homoclinic solution for a nonlinear second-order difference equation with p-Laplacian were presented.…”

Existence and multiplicity of homoclinic solutions for a difference equation

“…There is an increasing interest in the existence and multiplicity of homoclinic solutions to discrete nonlinear problems. The existence and multiplicity of homoclinic solutions have been investigated using various methods by many authors; see [3,10,13,14,15,19,20,21,22] and the references therein. For example, Iannizzotto and Tersian [13] used critical point theory, and proved the existence of at least two nontrivial homoclinic solutions for the nonlinear second-order difference equation…”

“…Stegliński [21] by using both the general variational principle of Ricceri and the direct method introduced by Faraci and Kristály [9] obtained infinitely many solutions for the problem (1.1), when h ≡ 0. In [3] using variational methods and critical point theory, sufficient conditions for the existence of at least one homoclinic solution for the problem (1.1), in the case h ≡ 0 have been presented.…”