{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:09:12Z","timestamp":1760058552830,"version":"build-2065373602"},"reference-count":44,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2025,4,10]],"date-time":"2025-04-10T00:00:00Z","timestamp":1744243200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12371184","IRT_16R16"],"award-info":[{"award-number":["12371184","IRT_16R16"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Program for Changjiang Scholars and Innovative Research Team in University","award":["12371184","IRT_16R16"],"award-info":[{"award-number":["12371184","IRT_16R16"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>The aim of this paper is to investigate the existence of positive solutions for a discrete Robin problem of the Kirchhoff type involving the p-Laplacian by the means of critical point theory. Our results demonstrate that the problem admits at least three solutions, or at least two solutions under different conditions on the nonlinear term f. We establish a strong maximum principle for the problem and obtain the existence and multiplicity of positive solutions. Finally, we give three examples to verify our results.<\/jats:p>","DOI":"10.3390\/axioms14040285","type":"journal-article","created":{"date-parts":[[2025,4,10]],"date-time":"2025-04-10T11:26:41Z","timestamp":1744284401000},"page":"285","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Positive Solutions for Discrete Robin Problem of Kirchhoff Type Involving p-Laplacian"],"prefix":"10.3390","volume":"14","author":[{"given":"Zhi","family":"Chen","sequence":"first","affiliation":[{"name":"School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zhan","family":"Zhou","sequence":"additional","affiliation":[{"name":"School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China"},{"name":"Guangzhou Center for Applied Mathematics, Guangzhou University, Guangzhou 510006, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,4,10]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Agarwal, R.P. (2000). Difference Equations and Inequalities: Theory, Methods, and Applications, CRC Press.","DOI":"10.1201\/9781420027020"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"718","DOI":"10.1137\/20M1368367","article-title":"Modeling and analysis of the implementation of the Wolbachia incompatible and sterile insect technique for mosquito population suppression","volume":"81","author":"Zheng","year":"2021","journal-title":"SIAM J. Appl. Math."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Yu, J., and Li, J. (2022). Discrete-time models for interactive wild and sterile mosquitoes with general time steps. Math. Biosci., 346.","DOI":"10.1016\/j.mbs.2022.108797"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"2544","DOI":"10.1109\/TAC.2022.3186827","article-title":"Stabilization of stochastic highly nonlinear delay systems with neutral term","volume":"68","author":"Zhao","year":"2023","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1749","DOI":"10.1007\/s11425-021-1891-7","article-title":"One discrete dynamical model on the Wolbachia infection frequency in mosquito populations","volume":"65","author":"Zheng","year":"2021","journal-title":"Sci. China Math."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"212","DOI":"10.1515\/anona-2020-0194","article-title":"Existence and uniqueness of periodic orbits in a discrete model on Wolbachia infection frequency","volume":"11","author":"Zheng","year":"2021","journal-title":"Adv. Nonlinear Anal."},{"key":"ref_7","first-page":"235","article-title":"Global stability of an economic model","volume":"95","author":"Yalcinkaya","year":"2014","journal-title":"Util. Math."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1768","DOI":"10.1080\/10236198.2019.1694014","article-title":"Multiple solutions for nonlinear functional difference equations by the invariant sets of descending flow","volume":"25","author":"Long","year":"2019","journal-title":"J. Differ. Equ. Appl."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1099","DOI":"10.1080\/10236190802332290","article-title":"Boundary value problems for second-order nonlinear difference equations with discrete \u03d5-Laplacian and singular \u03d5","volume":"14","author":"Bereanu","year":"2008","journal-title":"J. Differ. Equ. Appl."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"193","DOI":"10.1016\/j.cam.2003.08.004","article-title":"On the existence of positive solutions of p-Laplacian difference equations","volume":"161","author":"He","year":"2003","journal-title":"J. Comput. Appl. Math."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Rao, R., Lin, Z., Ai, X., and Wu, J. (2022). Synchronization of epidemic systems with Neumann boundary value under delayed impulse. Mathematics, 10.","DOI":"10.3390\/math10122064"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1239","DOI":"10.1016\/S0898-1221(02)00095-0","article-title":"Existence of multiple solutions for second-order discrete boundary value problems","volume":"43","author":"Henderson","year":"2002","journal-title":"Comput. Math. Appl."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"35","DOI":"10.1515\/JAA.2005.35","article-title":"A generalized upper and lower solution method for singular discrete boundary value problems for the one-dimensional p-Laplacian","volume":"11","author":"Jiang","year":"2005","journal-title":"J. Appl. Anal."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"7020","DOI":"10.3934\/mbe.2023303","article-title":"Stability analysis of multi-point boundary conditions for fractional differential equation with non-instantaneous integral impulse","volume":"20","author":"Li","year":"2023","journal-title":"Math. Biosci. Eng."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"506","DOI":"10.1007\/BF02884022","article-title":"Existence of periodic and subharmonic solutions for second-order superlinear difference equations","volume":"46","author":"Guo","year":"2003","journal-title":"Sci. China Ser. A Math."},{"key":"ref_16","first-page":"3183","article-title":"Positive solutions of the discrete Robin problem with \u03d5-Laplacian","volume":"14","author":"Ling","year":"2021","journal-title":"Discrete Contin. Dyn. Syst. Ser. S"},{"key":"ref_17","first-page":"605","article-title":"Infinitely many solutions for a class of discrete nonlinear boundary value problems","volume":"88","author":"Bonanno","year":"2009","journal-title":"Int. J. Control"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"28","DOI":"10.1016\/j.aml.2018.11.016","article-title":"Infinitely many positive solutions for a discrete two point nonlinear boundary value problem with \u03d5c-Laplacian","volume":"91","author":"Zhou","year":"2019","journal-title":"Appl. Math. Lett."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"383","DOI":"10.1016\/j.jmaa.2016.10.023","article-title":"Positive solutions for a discrete two point nonlinear boundary value problem with p-Laplacian","volume":"447","author":"Mawhin","year":"2017","journal-title":"J. Math. Anal. Appl."},{"key":"ref_20","first-page":"144","article-title":"Second-order differential operators with non-local Ventcel\u2019s boundary conditions","volume":"2","year":"2019","journal-title":"Constr. Math. Anal."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"47","DOI":"10.3934\/cpaa.2021166","article-title":"Periodic solutions with prescribed minimal period for second order even Hamiltonian systems","volume":"21","author":"Kuang","year":"2022","journal-title":"Commun. Pure Appl. Anal."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"109123","DOI":"10.1016\/j.aml.2024.109123","article-title":"Periodic solutions with prescribed minimal period for second-order Hamiltonian systems with non-symmetric potentials","volume":"155","author":"Kuang","year":"2024","journal-title":"Appl. Math. Lett."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"117","DOI":"10.1007\/s12220-022-01166-w","article-title":"Homoclinic solutions for partial difference equations with mixed nonlinearities","volume":"33","author":"Mei","year":"2023","journal-title":"J. Geom. Anal."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"108006","DOI":"10.1016\/j.aml.2022.108006","article-title":"Homoclinic solutions of discrete prescribed mean curvature equations with mixed nonlinearities","volume":"130","author":"Mei","year":"2022","journal-title":"Appl. Math. Lett."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"108827","DOI":"10.1016\/j.aml.2023.108827","article-title":"Heteroclinic solutions for a difference equation involving the mean curvature operator","volume":"147","author":"Wang","year":"2024","journal-title":"Appl. Math. Lett."},{"key":"ref_26","unstructured":"Kirchhoff, G.M. (1883). Mechanik, Teubner."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"377","DOI":"10.1016\/j.aml.2009.11.001","article-title":"Nontrivial solutions of a class of nonlocal problems via local linking theory","volume":"23","author":"Yang","year":"2010","journal-title":"Appl. Math. Lett."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"2384","DOI":"10.1016\/j.jde.2016.04.032","article-title":"Ground state sign-changing solutions for Kirchhoff type problems in bounded domains","volume":"261","author":"Tang","year":"2016","journal-title":"J. Differ. Equ."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"488","DOI":"10.1016\/j.jmaa.2012.04.025","article-title":"New existence and multiplicity of nontrivial solutions for nonlocal elliptic Kirchhoff type problems","volume":"394","author":"Cheng","year":"2012","journal-title":"J. Math. Anal. Appl."},{"key":"ref_30","first-page":"1","article-title":"Existence of three solutions for a Kirchhoff-type boundary-value problem","volume":"2011","author":"Heidarkhani","year":"2011","journal-title":"Electron. J. Differ. Equ."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"220","DOI":"10.1007\/s000130050496","article-title":"On a three critical points theorem","volume":"75","author":"Ricceri","year":"2000","journal-title":"Arch. Math."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"3084","DOI":"10.1016\/j.na.2008.04.010","article-title":"A three critical points theorem revisited","volume":"70","author":"Ricceri","year":"2009","journal-title":"Nonlinear Anal. Theory Methods Appl."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"4151","DOI":"10.1016\/j.na.2009.02.074","article-title":"A further three critical points theorem","volume":"71","author":"Ricceri","year":"2009","journal-title":"Nonlinear Anal. Theory Methods Appl."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1515\/acv-2015-0032","article-title":"Existence and multiplicity results for the fractional Laplacian in bounded domains","volume":"10","author":"Mugnai","year":"2017","journal-title":"Adv. Calc. Var."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"107817","DOI":"10.1016\/j.aml.2021.107817","article-title":"Existence and multiplicity solutions for discrete Kirchhoff type problems","volume":"126","author":"Long","year":"2022","journal-title":"Appl. Math. Lett."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"959","DOI":"10.1016\/j.camwa.2008.01.025","article-title":"Multiple solutions for a discrete boundary value problem involving p-Laplacian","volume":"56","author":"Bonanno","year":"2008","journal-title":"Comput. Math. Appl."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"2281","DOI":"10.1016\/j.na.2006.03.019","article-title":"Three solutions for a Dirichlet boundary value problem involving the p-Laplacian","volume":"66","author":"Afrouzi","year":"2007","journal-title":"Nonlinear Anal. Theory Methods Appl."},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Xiong, F. (2023). Infinitely many solutions for partial discrete Kirchhoff type problems involving p-Laplacian. Mathematics, 11.","DOI":"10.3390\/math11153288"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"917","DOI":"10.1080\/10236198.2017.1306061","article-title":"Variational approaches to p-Laplacian discrete problems of Kirchhoff-type","volume":"23","author":"Heidarkhani","year":"2017","journal-title":"J. Differ. Equ. Appl."},{"key":"ref_40","first-page":"58","article-title":"Critical point approaches for discrete anisotropic Kirchhoff type problems","volume":"13","author":"Heidarkhani","year":"2022","journal-title":"Adv. Appl. Math."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"3031","DOI":"10.1016\/j.jde.2008.02.025","article-title":"Non-differentiable functionals and applications to elliptic problems with discontinuous nonlinearities","volume":"244","author":"Bonanno","year":"2008","journal-title":"J. Differ. Equ."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"915","DOI":"10.1515\/ans-2014-0406","article-title":"Variational methods on finite dimensional Banach spaces and discrete problems","volume":"14","author":"Bonanno","year":"2014","journal-title":"Adv. Nonlinear Stud."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"449","DOI":"10.4171\/zaa\/1573","article-title":"Two non-zero solutions for elliptic Dirichlet problems","volume":"35","author":"Bonanno","year":"2016","journal-title":"Z. Anal. Anwend."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"297","DOI":"10.3906\/mat-1303-6","article-title":"Existence and multiplicity of positive solutions for discrete anisotropic equations","volume":"38","author":"Galewski","year":"2014","journal-title":"Turk. J. Math."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/4\/285\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T17:12:28Z","timestamp":1760029948000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/4\/285"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,4,10]]},"references-count":44,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2025,4]]}},"alternative-id":["axioms14040285"],"URL":"https:\/\/doi.org\/10.3390\/axioms14040285","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2025,4,10]]}}}