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Objective: This study aimed to verify miRNAs and the molecular mechanisms of diagnostic and prognostic biomarkers for rectal adenocarcinoma. background: - Methods: Two miRNA datasets of rectal adenocarcinoma were obtained from GEO and TCGA. GEO2R tool, Venn diagram, Kaplan-Meier survival analysis, KEGG pathway analyses, DIANA TOOLS, and Wilcoxon rank-sum test were used for biological information analysis. The diagnostic utility of miRNAs and immune infiltration of tumors in Chinese patients were validated by RT-qPCR and immunofluorescence analysis. objective: This study was aimed at verifying miRNAs and the molecular mechanisms of diagnostic and prognostic biomarkers for rectal adenocarcinoma. Results: MiR-21-5P and miR-455-5p were both found to have a significant correlation with poor prognosis and higher expression in rectal adenocarcinoma. Besides, the ability to prognosis was independent of the clinicopathological stage. MiR-21-5P and miR-455-5p were enriched in the TGF-beta, Wnt, MAKP, and PI3K-AKT signaling pathways. Meanwhile, the high expression phenotype of miR-21-5P and miR-455-5p decreased CD4+ and CD8+ T cells. Conclusion: In summary, we found two significant diagnostic and prognostic miRNAs of rectal adenocarcinoma via integrated bioinformatics approach and clinical trials, which might decrease CD4+ and CD8+ T cells. result: MiR-21-5P and miR-455-5p were both found to have a significant correlation with poor prognosis and higher expression in rectal adenocarcinoma. Besides, the ability to judge prognosis was independent of the clinicopathological stage. MiR-21-5P and miR-455-5p were enriched in the TGF-beta, Wnt, MAKP, and PI3K-AKT signaling pathway. Meanwhile, the high expression phenotype of miR-21-5P and miR-455-5p decreased CD4+ and CD8+ T cells. conclusion: In summary, we found two significant diagnostic and prognostic miRNAs of rectal adenocarcinoma via an integrated bioinformatics approach and clinical trials, which might decrease CD4+ and CD8+ T cells.

Materials with violent hydration reaction such as cement are used to solidify sandy soil slopes, which will cause destructive damage to the ecology of the slopes. In this paper, polyvinyl alcohol (PVA) and activated magnesium oxide (MgO) are used to improve sandy soil, and the effects of the dosage and curing age of modifiers on the mechanical properties of solidified sandy soil are studied. The dry–wet durability of the composite improved sandy soil is analyzed using a dry–wet cycle test, and the improvement mechanism of PVA and activated magnesium oxide is revealed using an electron microscope. The results show that the curing effect of polyvinyl alcohol and activated magnesium oxide on sand particles is better than that of polyvinyl alcohol alone. The compressive strength of improved soil samples increases with the increase of curing time, and magnesium oxide as an improved material needs appropriate reaction conditions to give full play to its role. The compressive strength of composite improved samples increases first and then decreases during the dry–wet cycle. Through the observation of microstructure, it can be seen that the cementing material wraps and connects the sand particles, and the cementing material of the sample after the dry–wet cycle develops more completely; if the magnesium oxide content is high, cracks may appear inside the sample.

IntroductionSubjective tinnitus is a common and intractable ear disease. The effectiveness of acupuncture in the treatment of subjective tinnitus has been confirmed, but its mechanism of action is not clear. The structures of the amygdala (AMYG) are mainly closely related to emotion in the human brain. This study aimed to investigate the changes in functional connectivity (FC) of AMYG in subjective tinnitus to elucidate the neural mechanism of acupuncture.MethodsCorrelation scale scores of 26 patients with subjective tinnitus were collected, including Tinnitus Evaluation Questionnaire (TEQ), Tinnitus Handicap Inventory (THI) and Visual Analog Scale (VAS). Meanwhile, rs-fMRI data were collected before and after acupuncture treatment in the patients, and in healthy controls (HC) matching the patient's gender and age. Then, AMYG was selected as region of interest to perform FC analysis. Finally, FC patterns of AMYG were first compared between patients with subjective tinnitus and HC, and then within subjects pre-acupuncture and post-acupuncture. Simple linear regression models between correlation scale scores and FC-values were established as well.ResultsAcupuncture treatment relieved the severity of tinnitus. With the acupuncture treatment, the total THI score, TEQ score, and VSA score of patients were significantly lower than before (p < 0.05). Compared with HC, FC of tinnitus patients between AMYG and right inferior temporal gyrus and right precuneus significantly decreased before acupuncture (voxel p < 0.001, cluster p < 0.05, corrected with GRF), while FC of tinnitus patients between AMYG and left superior frontal gyrus and right superior temporal gyrus significantly decreased after acupuncture treatment (voxel p < 0.001, cluster p < 0.05, corrected with GRF). FC of tinnitus patients between the AMYG and right superior frontal gyrus and left paracingulate gyrus showed significant decrease after acupuncture treatment (voxel p < 0.001, cluster p < 0.05, corrected with GRF). Besides, the linear regression models of the effect of THI on FC and VAS on FC performed were statistically significant (p < 0.05).DiscussionThe findings demonstrate that acupuncture can decrease FC of AMYG, which could be positively correlated with the relief of tinnitus symptoms. This result suggests that acupuncture stimulation can effectively relieve the severity of tinnitus by decreasing FC of AMYG in subjective tinnitus patients.

To investigate an environmentally benign stabilizer for coarse-grained soil in southeast Tibet, poly (vinyl alcohol) (PVA) and silica fume were used to improve the geotechnical properties of coarse-grained soil. Unconfined compressive strength (UCS) and wet–dry cycle tests were conducted on prepared samples to evaluate the effect of the additive content and curing age on the strength and durability of coarse-grained soil. The results reveal that the UCS of the samples increased with the additive content of PVA solution and the curing age. The optimal value for the additive content of PVA solution and the curing age is 12% and 7 days, respectively. With the optimal PVA solution content, the PVA solution combined with silica fume stabilizer exhibited better reinforcement compared with pristine PVA. The UCS of the samples stabilized by PVA solution and silica fume increased depending on the curing age, and plateaued after 14 days. Samples with 12% PVA solution and 6% silica fume achieved a satisfactory UCS of 1543.17 kPa after curing for 28 days. As the number of wet–dry cycles increased, the UCS of the samples stabilized by the PVA solution and silica fume exhibited an upward trend during the first three wet–dry cycles, owing to the filling of pores by the gel produced by the silica fume, but began to decline as the number of wet–dry cycles increased. All samples retained a high UCS value after 10 wet-dry cycles compared with the samples that were not subjected to wet–dry cycles.

Background Bone marrow stromal antigen 2 (BST2) is considered as a transmembrane glycoprotein and plays essential roles in innate immunity. It has been recently reported that up-regulation of BST2 was associated with the development of breast carcinoma. However, the clinical significance of BST2 in glioma has not been identified. The purpose of the present study is to explore the expression pattern and the role of BST2 in the progression of high-grade glioma. Methods Expression levels of BST2 were tested in glioma tissues by analyzing the GEO database and immunohistochemistry staining. The prognostic role of BST2 in glioma was evaluated through univariate and multivariate analyses. In vitro and in vivo assays were conducted to confirm the role of BST2 on promoting glioma proliferation. Results The mRNA level of BST2 was higher in glioma tissues than that in nontumorous brain tissues. High protein level of BST2 was correlated with larger tumor size and advanced WHO grade. Glioma patients with a high BST2 level had worse overall survival. In addition, BST2 was defined as an independent risk factor for glioma prognosis. Cellular and xenograft studies revealed that BST2 can significantly promote glioma proliferation. Conclusion Our study revealed that a high BST2 expression level was closely related to the unfavorable clinical features and poor prognosis of high-grade glioma patients. BST2 may serve as an invaluable prognostic indicator and novel therapeutic target for glioma treatment considering its membrane localization.

IntroductionKnee osteoarthritis is a common disease in the elderly. Patients suffer from long-term chronic pain and reduced life quality. Acupuncture has been proven to be an effective treatment for KOA. However, the neural mechanism of acupuncture is unclear, so far. Periaqueductal gray (PAG) and raphe nuclei (RPN) are essential structures associated with chronic pain in human brains. This study aims to investigate functional connectivity (FC) changes of PAG and RPN in KOA to interpret the neural mechanism of acupuncture.MethodsIn 15 patients with KOA and 15 healthy controls (HC), we acquired Visual Analog Scale (VAS) scores and resting-state fMRI images of each participant before and after acupuncture stimulation on EX-LE5 acupoint. Then, PAG and RPN were selected as seeds to perform FC analysis based on resting-state fMRI images. Finally, we compared FC patterns of PAG and RPN between patients with KOA and HC, then between pre-acupuncture and post-acupuncture. Correlations between FC values and VAS scores were calculated as well.ResultsFor PAG, FC of patients with KOA was lower in the right lingual gyrus at post-acupuncture compared with HC (p p p DiscussionOur study demonstrated that acupuncture enhanced FC between dorsal RPN and the right putamen in patients with KOA, which was associated with chronic pain intensity. This result suggests that acupuncture stimulation can enhance FC between dorsal raphe and striatum, illustrating a neural mechanism that acupuncture can drive the patients' brain, with KOA, to perceive pain.

In this paper, we concern with a 2nth-order discrete system. Using the critical point theory, we establish various sets of sufficient conditions for the existence of periodic solutions with prescribed minimal period. To the best of our knowledge, this is the first time to discuss the periodic solutions with prescribed minimal period for a 2nth-order discrete system.

To improve the stability of sand slopes in Southeast Tibet, the wet–dry cycle test, immersion test, and permeability test were carried out using polyvinyl alcohol (PVA) as the improving material. The improvement effect was evaluated by considering the unconfined compressive strength. The results have revealed that the unconfined compressive strength of the improved soil significantly increased with the PVA content, while the degradation of the improved soil by wet–dry cycling and water immersion weakened. However, the permeability of the improved soil slightly decreased compared with that of unmodified soil. Through scanning electron microscopy (SEM), it was found that PVA only forms an elastic network structure in the sand to wind and connect the soil particles, but pores remain in the sand. This indicates that the addition of PVA does not affect the permeability to a great extent but greatly increases the strength. The findings of this study provide a useful reference for the practical application of PVA.

The occurrence of bakanae disease of rice caused by the fungus Fusarium fujikuroi in Zhejiang Province has become increasingly aggravated in recent years, concomitant with the development of resistance to the widely applied fungicides, prochloraz and phenamacril. In this study, the activity of a novel succinate dehydrogenase inhibitor (SDHI) fungicide, penflufen, against different fungi was evaluated in addition to the potential of penflufen in controlling F. fujikuroi infections. Penflufen exhibited good bioactivity against F. fujikuroi, but weak activity against Fusarium spp. and other investigated plant-pathogenic fungi including Colletotrichum spp. In addition to inhibiting mycelial growth, penflufen effectively inhibited F. fujikuroi conidium production. For germination, penflufen could effectively inhibit that of small conidia, but only delay the germination of large conidia. In addition, the sensitivity to penflufen among 100 F. fujikuroi isolates that were collected in areas that were never exposed to SDHIs was determined based on mycelium growth. Sensitivities surprisingly exhibited bimodal distributions, indicating the presence of natural resistance. Cross-resistance was not observed between penflufen in F. fujikuroi and two fungicides that have been extensively applied in field including prochloraz (a DMI) and phenamacril (a 2-cyanoacrylate fungicide), nor with the three SDHIs, fluopyram, benzovindiflupyr, and pydiflumetofen. Additional analysis identified five different point mutations in SDH-A (i.e., at residues 46, 225, 283, 430, and 586) of naturally resistant isolates. These results inform the potential application of the new SDHI fungicide penflufen for managing crop diseases and understanding possible resistance mechanisms among pathogens.

In this article we study discrete nonlinear Schrodinger equations without periodicity assumptions. We show the existence of multiple solutions of the form $u_n e^{-i\omega t}$ (called breathers) by using Clark's Theorem in critical point theory.

The authors explore the boundary value problems of a discrete generalized beam equation. Using the critical point theory, some sufficient conditions for the existence of the solutions are obtained. Several consequences of the main results are also presented. Examples are given to illustrate the theorems.

Laboratory freezing experiments were conducted to evaluate the effect of polyacrylamide (PAM) and lignocellulose on the mechanical properties and microstructural characteristics of Tibetan clay. Direct shear and unconfined compressive tests and field emission scanning electron microscopy analyses were performed on clay samples with different contents of stabilizers. The test results show that the addition of PAM can improve the unconfined compressive strength and cohesion of Tibetan clay, but an excessive amount of PAM reduces the internal friction angle. After several freeze-thaw cycles, the unconfined compressive strength and cohesion of samples stabilized by PAM decrease significantly, while the internal friction angle increases. Samples stabilized by PAM and lignocellulose have higher internal friction angles, cohesion, and unconfined compressive strength and can retain about 80% of the original strength after 10 freeze-thaw cycles. PAM fills the pores between soil particles and provides adhesion. The addition of lignocellulose can form a network, restrict the expansion of pores caused by freeze-thaw cycles, and improve the integrity of PAM colloids. It is postulated that the addition of a composite stabilizer with a PAM content of 0.4% and a lignocellulose content of 2% may be a technically feasible method to increase the strength of Tibetan clay.

In this paper, by using the critical point theory, some new results of the existence of at least two nontrivial periodic solutions with prescribed minimal period to a class of 2 n th-order nonlinear discrete system are obtained. The main approach used in our paper is variational technique and the linking theorem. The problem is to solve the existence of periodic solutions with prescribed minimal period of 2 n th-order discrete systems.

Background Bone marrow stromal antigen 2 (BST2) is considered as a transmembrane glycoprotein and plays essential roles in innate immunity. It has been recently reported that up-regulation of BST2 was associated with the development of breast carcinoma. However, the clinical significance of BST2 in glioma has not been identified. The purpose of the present study is to explore the expression pattern and the role of BST2 in the progression of high-grade glioma. Methods Expression levels of BST2 were tested in glioma tissues by analyzing the GEO database and immunohistochemistry staining. The prognostic role of BST2 in glioma was evaluated through univariate and multivariate analyses. In vitro and in vivo assays were conducted to confirm the role of BST2 on promoting glioma proliferation. Results The mRNA level of BST2 was higher in glioma tissues than that in nontumorous brain tissues. High protein level of BST2 was correlated with larger tumor size and advanced WHO grade. Glioma patients with a high BST2 level had worse overall survival. In addition, BST2 was defined as an independent risk factor for glioma prognosis. Cellular and xenograft studies revealed that BST2 can significantly promote glioma proliferation. Conclusion Our study revealed that a high BST2 expression level was closely related to the unfavorable clinical features and poor prognosis of high-grade glioma patients. BST2 may serve as an invaluable prognostic indicator and novel therapeutic target for glioma treatment considering its membrane localization.

In this article, some new existence criteria of at least one solution to boundary value problems for a fourth-order difference equation are obtained by using the critical point theory. In a special case, a necessary and sufficient condition for the existence and uniqueness of solution is also established. An example of the main result is given.

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.

This paper is devoted to investigate a question of the existence of solutions to boundary value problems for a class of nonlinear difference systems. The proof is based on the notable mountain pass lemma in combination with variational technique. By using the critical point theory, some new existence criteria are obtained.

In this paper some sufficient conditions are obtained to guarantee the existence of nontrivial 4T+ 2 periodic solutions of asymptotically linear delay difference equations. The approach used is based on Morse theory.

This article concerns a second-order nonlinear difference equation. By using critical point theory, the existence of two heteroclinic orbits is obtained. The main method used is variational.

Glioma is the most common type of malignancy in the central nervous system, which has a poor prognosis due to its rapid progression and diffuse invasion. Identification of novel biomarkers for glioma would be invaluable for studying disease mechanism and improving prognosis. Orphan G protein-coupled receptor 110 (GPR110) belongs to the subfamily VI of adhesion GPCR. The knowledge of the ligand, signaling pathway or physiology function of GPR110 is poorly elucidated. The potential role of GPR110 as an oncogene in mouse has been recently reported by mutagenesis screen. However, its expression and role in human glioma hasn't been identified. Here in the current study, we initially explored the RNA and protein expression of GPR110 in patients with glioma. Statistical analysis proved that GPR110 was highly expressed in some patients, which was correlated with advanced disease stages. Furthermore, univariate and multivariate analyses revealed its role as an independent prognostic biomarker for the overall survival of glioma patients. Interestingly, cellular studies showed that overexpression or knockdown of GPR110 in U87 cells didn't affect cell proliferation and migration. However, the invasion of U87 cells was significantly enhanced by GPR110-overepxression, while inhibited by GPR110-knockdown. The detailed mechanisms remain further investigation although our results suggested the possible participation of STAT3 instead of ERK in the GPR110 signaling pathways.

By making use of the critical point theory, we establish some new existence criteria to guarantee that a 2nth-order nonlinear difference equation containing both advance and retardation with p-Laplacian has a nontrivial homoclinic orbit. Our conditions on the potential are rather relaxed, and some existing results in the literature are improved.

Piezoresistance, induced by an applied mechanical stress, has been widely used in various sensors. Herein, individual PbS micro/nanowire-based devices were constructed on flexible insulating plastic substrates. They not only show dynamic strain-evoked giant piezoresistance (GPR) effects, but also show excellent functions for application in non-volatile piezoresistance random access memory (PRRAM). The trapped electron hopping mechanism, tuned by exerting external strain and an electric field, is proposed. Under dynamic compression/tension with a strain of ±0.26%, the gauge factor can approach four orders of magnitude, dominantly originating from the strain-induced variation of the trap barrier height within the PbS micro/nanowires. After loading both compressive and tensile strains at a low operation bias voltage, additionally, the emptying of trap states results in a down-shift of the PbS Fermi level, and correspondingly the device resistivity increases, indicating that the stress-related data can be written/set by loading different strains. Subsequently, the emptied trap states can be filled up under a relatively high external electric field so that in turn the Fermi level of PbS up-shifts, and accordingly the device resistivity restores to the initial low resistance state, that is, the stored stress-related information can be effectively erased/reset by applying a relatively high external bias voltage. The repeatable writing/erasing characteristics of nanostructure-based devices offer an avenue to develop low power and reliable non-volatile PRRAM for applications.

It is a challenge to realize nonvolatile stress-writing memory. Herein, we propose a strategy to construct rewritable stress information storage devices, consisting of deliberately designing individual sandwiched ZnS/Si/SiO2 radial heterostructure nanotubes synthesized by one-step thermal evaporation method. A bulk trap-related Poole–Frenkel hopping mechanism is proposed. Carriers are localized in a narrow bandgap Si intermediate layer; moreover, incorporated impurities and heterointerface defects can serve as charge trap centers or storage mediators. Compressive strain can induce trap barrier height to decrease at relatively low operation bias voltage, whereas tensile strain can induce it to increase, resulting in a giant piezoresistance effect. After both loading compressive and tensile strains at low bias voltage, additionally, the emptying of trap states results in a high resistance state. However, the emptied trap states can be filled by applying a relatively high bias voltage without strains and, co...

By using critical point theory, we prove the existence of a nontrivial homoclinic orbit for a class of nonlinear functional difference equations. Our conditions on the nonlinear term do not need to satisfy the well-known global Ambrosetti-Rabinowitz superquadratic condition.

Jia Duan,* Wei Huang,* Haiping Shi Department of Neurosurgery, Suining Central Hospital, Suining, Sichuan, People’s Republic of China *These authors contributed equally tothis work Abstract: Glioma patients have a poor overall survival; however, patients can show distinct clinical outcomes due to the high heterogeneity of the tumor, which may be indicated by certain clinicobiological parameters. Kinesin family member 20A (KIF20A), which participates incytokinesis and intracellular transportation, has been recently reported to be upregulated in pancreatic cancer, breast cancer, and bladder cancer. In the current study, we investigated the expression of KIF20A in gliomas and its significance in predicting the prognosis after surgery. We found that KIF20A positive expression in glioma tissues correlated significantly with Ki67 protein expression and advanced World Health Organization grade. Univariate and multivariate analysis revealed that KIF20A can act as an independent prognostic factor for predicting the overall survival of glioma patients. Moreover, we demonstrated that KIF20A can positively regulate the expression of Ki67 in glioma cell lines. Correspondingly, overexpression of KIF20A can promote cell proliferation and invasion, whereas knockdown of KIF20A can inhibit cell viability and invasion capacity. In vitro study also showed that under the treatment of plumbagin, an anticancer drug, KIF20A expression decreased in a dose-dependent manner. In addition, the overexpression of KIF20A can also increase the drug resistance toward plumbagin, which provided the possibility that KIF20A may contribute to the chemotherapy resistance of gliomas. Keywords: glioma, invasion, KIF20A, prognosis, proliferation

We investigate the discrete nonlinear Schrodinger equations with indefinite sign subquadratic potentials. Making use of the critical point theory, we obtain a new result concerning the existence of a standing wave solution.

By using the Mountain Pass Lemma, we establish sufficient conditions for the existence of solitons for the discrete nonlinear Schrodinger equations.

By making use of the critical point theory, the existence of periodic solutions for fourth-order nonlinear p-Laplacian difference equations is obtained. The main approach used in our paper is a variational technique and the Saddle Point Theorem. The problem is to solve the existence of periodic solutions of fourth-order nonlinear p-Laplacian difference equations. The results obtained successfully generalize and complement the existing one.

The discrete non-linear Schrödinger equation is one of the most important inherently discrete models, having a crucial role in the modelling of a great variety of phenomena, ranging from solid-state and condensed-matter physics to biology. In this paper, a class of discrete non-linear Schrödinger equations are considered. Using critical point theory in combination with periodic approximations, we establish some new sufficient conditions on the existence results for solitons of the equation. The classical Ambrosetti–Rabinowitz superlinear condition is improved.

For nanostructures with a large surface-to-volume ratio, quantities of dangling bonds can create affluent surface states due to the breaking of lattice periodicity on their surfaces. Herein, surface states are utilized for the development of high-performance photodetectors based on individual SnO2 nanowires. The photodetectors not only show ultrahigh selectivity and sensitivity to ultraviolet (UV) light of about 340 nm with and without bias, but also show weak response to sub-bandgap visible (VIS) and near infrared (NIR) light. Moreover, their detectivity is strongly dependent on externally applied bias voltage and illumination light intensity. The abundant surface states result in the formation of back-to-back diodes for two-terminal nanowire-based devices. The photoconductive mechanism is controlled by a photovoltaic effect of a surface state related reverse biased diode in contact with a negative electrode. Under light illumination, the efficient separation of photoexcited carriers at the Ag and SnO2 interface and the fast drift of electrons separated into the core of n-type SnO2 nanowires toward another end lead to ultrahigh gain, a large photo-to-dark current ratio, and high-speed response for the nanostructure photodetectors at a relatively high bias voltage. These results demonstrate that the surface states of nanostructures are promising for the development of high performance photodetectors, and meanwhile render an interpretation particularly valuable for the photoconductive mechanism of nanostructures.

By using the critical point method, some new criteria are obtained for the existence and multiplicity of homoclinic solutions to a 2nth-order nonlinear difference equation. The proof is based on the Mountain Pass Lemma in combination with periodic approximations. Our results extend and improve some known ones.

In this paper, we obtain a new sufficient condition on the existence of breathers for the discrete nonlinear Schrodinger equations by using critical point theory in combination with periodic approximations. The classical Ambrosetti–Rabinowitz superlinear condition is improved.

For nanostructure SnO2, it is very difficult for its electric properties to accurately control due to the presence of abundant surface states. The introduction of Sm can improve the traps in surface space charge region of SnO2 nanowires, resulting in a controllable storage charge effect. For the single nanowire-based two-terminal device, two surface state-related back-to-back diodes are formed. At a relatively large voltage, electrons can be injected into the traps in surface space charge region from negative electrode, resulting in a decrease of surface barrier connected with negative electrode, and contrarily electrons can be extracted from the traps in surface space charge region into positive electrode, resulting in an increase of surface barrier connected with positive electrode. The reversible injection and extraction induce a nonvolatile resistive switching memory effect.

This paper is concerned with boundary value problems for a fourth-order nonlinear difference equation. Via variational methods and critical point theory, sufficient conditions are obtained for the existence of at least two nontrivial solutions, the existence ofndistinct pairs of nontrivial solutions, and nonexistence of solutions. Some examples are provided to show the effectiveness of the main results.

A fourth-order nonlinear difference equation is considered. By making use of critical point theory, some new criteria are obtained for the existence of periodic solutions with minimal period. The main methods used are a variational technique and the Linking Theorem.

In this paper, we investigate the solutions of boundary value problems for second-order p-Laplacian difference equations. By using the critical point theory, the existence and multiple results are obtained.

We establish existence criteria for homoclinic orbits of second-order nonlinear difference equations by using the critical point theory in combination with periodic approximations.

Using the critical point theory, we obtain the existence of a nontrivial homoclinic orbit for second order \(p\)-Laplacian difference equations containing both advance and retardation. The proof is based on the Mountain Pass Lemma in combination with periodic approximations. One of our results generalizes and improves the results in the literature.

By using the critical point theory, the existence of a nontrivial homoclinic orbit which decays exponentially at infinity for difference equations containing both advance and retardation is obtained. The proof is based on the Mountain Pass Lemma in combination with periodic approximations. Our results obtained successfully extend and complement the results in the literature.

In this paper, a class of fourth-order nonlinear difference equations are considered. By making use of the critical point method, we establish various sets of sufficient conditions for the nonexistence and existence of solutions for Neumann boundary value problems and give some new results. Our results successfully complement the existing ones.

Using the critical point theory, the existence of periodic solutions for a 2 $$n$$ th-order nonlinear difference equation containing both advance and retardation involving $$p$$ -Laplacian is obtained. The main approaches used in our paper are variational techniques and the Saddle Point Theorem. The problem is to solve the existence of periodic solutions for a 2 $$n$$ th-order $$p$$ -Laplacian difference equation. The obtained results successfully generalize and complement the existing ones.

By making use of the critical point theory, the existence of periodic solutions for a class of nonlinear difference equations is obtained. The proof is based on the saddle point theorem in combination with variational technique. The problem is to solve the existence of periodic solutions of a class of nonlinear difference equations. Results obtained complement the existing one.

In this paper, a 2\(n\)th-order nonlinear difference equation is considered. By using the critical point theory, we establish various sets of sufficient conditions of the nonexistence and existence of solutions for mixed boundary value problem and give some new results. Recent results in the literature are generalized and significantly complemented.