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In this paper, we consider the existence of multiple periodic solutions for a class of second-order difference equations with quadratic–supquadratic growth condition at infinity. Moreover, we give three examples to illustrate our main result. [ABSTRACT FROM AUTHOR]

In the present study, we prepared a newly adsorbent, Chitosan (CTS) and sodium lignosulphonate (SLS) were added as monomers for the copolymerization of acrylic acid (AA), then ammonium persulfate (APS) bisacrylamide (NMBA) as a cross-linking agent, under the premise of ultrasound mixed uniformly prepared lignin-based hydrogel. The hydrogel samples were characterized by SEM, FT-IR and TG analysis. Finally, the samples were applied to the adsorption of heavy metal ions Cu2+ and Co2+. The influences of external environmental factors such as pH value, adsorption time, adsorbent dosage and initial concentration of heavy metal ions on the adsorption process were discussed. It was found that the optimum pH value for the adsorption of two heavy metal ions was close to 6, and the adsorption kinetics was found to fit the pseudo-second-order model and the isotherm model was fitted to the Freundlich model. Among them, the adsorption capacity of Co2+ is 385 mg g−1 and Cu2+ is 290 mg g−1. [ABSTRACT FROM AUTHOR]

Sorbitol, a major end-product of photosynthesis in many species of the Rosaceae family, accumulates in response to abiotic stressors. However, the relationship that arises between the expression of sorbitol transporters and sorbitol accumulation under abiotic stress remains unclear. In this study, micropropagated 'Fuji' apple plants ( Malus domestica Borkh. 'Fuji') were exposed to two varying degrees of osmotic stress and compared relative to an unstressed control. The osmotic stress was generated by adding PEG 6000 into full-strength Hoagland solution and adjusted the osmotic potential to either −0.75 MPa (mild drought stress [MIS]) or −1.5 MPa (severe drought stress [SES]). Analysis of sorbitol levels via high performance liquid chromatography (HPLC) showed that the sorbitol concentration was elevated in roots, phloem tissues and leaves in both the MIS and SES treatments compared to controls for the entire duration of the experiment. Three cDNA sequences, encoding sorbitol transporters ( MdSOT3, MdSOT4 and MdSOT5), were isolated from leaves. Real-time quantitative PCR (RT-qPCR) data suggests that the expression levels of MdSOT3 and MdSOT5 were higher under MIS and SES in roots, phloem tissues and leaves compared to unstressed controls. The average mRNA levels of MdSOT4 in phloem tissues declined under both drought treatments (with the exception being at 2 h of SES). In roots and leaves under SES, mRNA production was increased. These results indicate that the up-regulation of MdSOT3 and MdSOT5 expression is consistent with the accumulation of sorbitol under conditions of osmotic stress in apple plants. They enhanced drought tolerance in vegetative tissues. Increased MdSOT4 mRNA enhanced drought tolerance under SES. [ABSTRACT FROM AUTHOR]

Consider the following stochastic differential equation (SDE): X t = x + ∫ 0 t b (s , X s) d s + ∫ 0 t σ (s , X s) d B s , 0 ≤ t ≤ T , x ∈ R , where { B s } 0 ≤ s ≤ T is a 1-dimensional standard Brownian motion on [ 0 , T ] . Suppose that q ∈ (1 , ∞ ] , p ∈ (1 , ∞) , b = b 1 + b 2 , b 1 ∈ L q (0 , T ; L p (R)) such that 1 / p + 2 / q < 1 and b 2 is bounded measurable, with σ ∈ L ∞ (0 , T ; C u (R)) there being a real number δ > 0 such that σ 2 ≥ δ . Then there exists a weak solution to the above equation. Moreover, (i) if σ ∈ C ([ 0 , T ] ; C u (R)) , all weak solutions have the same probability law on 1-dimensional classical Wiener space on [ 0 , T ] and there is a density associated with the above SDE; (ii) if b 2 = 0 , p ∈ [ 2 , ∞) and σ ∈ L 2 (0 , T ; C b 1 / 2 (R)) , the pathwise uniqueness holds. [ABSTRACT FROM AUTHOR]

We consider nonlocal PDEs driven by additive white noises on R d . For L q integrable coefficients, we derive the existence and uniqueness, as well as Hölder continuity, of mild solutions. Precisely speaking, the unique mild solution is almost surely Hölder continuous with Hölder index 0 < θ < (1 / 2 − d / (q α)) (1 ∧ α) . Moreover, we show that any order γ (< q) moment of Hölder normal for u on every bounded domain of R + × R d is finite. [ABSTRACT FROM AUTHOR]