Employing variational methods in conjunction with the technique of invariant sets of descending flow, we study the existence and multiplicity of solutions for a second-order difference equation subject to a three-point boundary condition. By imposing suitable growth and sign conditions on the nonlinearity, we establish sufficient criteria for the existence of at least three nontrivial solutions. These solutions are characterized by their nodal properties: one positive, one negative, and one sign-changing solution, in addition to the trivial solution. Our results generalize and extend previous work on discrete boundary value problems (BVPs), notably encompassing and broadening known results for Robin boundary conditions. Furthermore, two demonstrative examples are provided not only to validate the theoretical results but also to illustrate the applicability of the problem in modeling phenomenological processes.
Citation: Zhi Zhang. Existence and multiplicity of sign-changing solutions for a discrete three-point boundary value problem[J]. Electronic Research Archive, 2026, 34(1): 209-231. doi: 10.3934/era.2026011
Employing variational methods in conjunction with the technique of invariant sets of descending flow, we study the existence and multiplicity of solutions for a second-order difference equation subject to a three-point boundary condition. By imposing suitable growth and sign conditions on the nonlinearity, we establish sufficient criteria for the existence of at least three nontrivial solutions. These solutions are characterized by their nodal properties: one positive, one negative, and one sign-changing solution, in addition to the trivial solution. Our results generalize and extend previous work on discrete boundary value problems (BVPs), notably encompassing and broadening known results for Robin boundary conditions. Furthermore, two demonstrative examples are provided not only to validate the theoretical results but also to illustrate the applicability of the problem in modeling phenomenological processes.
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Zhi Zhang. Existence and multiplicity of sign-changing solutions for a discrete three-point boundary value problem[J]. Electronic Research Archive, 2026, 34(1): 209-231. doi: 10.3934/era.2026011
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