Hindawi Publishing Corporation Advances in Difference Equations Volume 2010, Article ID 862016, 11 pages doi:10.1155/2010/862016 Research Article Three Solutions for a Discrete Nonlinear Neumann Problem Involving the p-Laplacian Pasquale Candito 1 and Giuseppina D’Agu`ı 2 1 DIMET University of Reggio Calabria, Via Graziella (Feo Di Vito), 89100 Reggio Calabria, Italy 2 Department of Mathematics of Messina, DIMET University of Reggio Calabria, 89100 Reggio Calabria, Italy Correspondence should be addressed to Giuseppina D’Agu ` ı, dagui@unime.it Received 26 October 2010; Accepted 20 December 2010 Academic Editor: E. Thandapani Copyright q 2010 P. Candito and G. D’Agu ` ı. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We investigate the existence of at least three solutions for a discrete nonlinear Neumann boundary value problem involving the p-Laplacian. Our approach is based on three critical points theorems. 1. Introduction In these last years, the study of discrete problems subject to various boundary value con- ditions has been widely approached by using different abstract methods as fixed point theorems, lower and upper solutions, a nd Brower degree see, e.g., 1–3 and the reference given therein. Recently, also the critical point theory has aroused the attention of many authors in the study of these problems 4–12. The main aim of this paper is to investigate different sets of assumptions which guarantee the existence and multiplicity of solutions for the following nonlinear Neumann boundary value problem −Δ φ p Δu k−1 q k φ p u k λf k, u k ,k∈ 1,N , Δu 0 Δu N 0, P f λ where N is a fixed positive integer, 1,N is the discrete interval {1, ,N}, q k > 0forall k ∈ 1,N, λ is a positive real parameter, Δu k : u k1 − u k , k 0, 1, ,N 1, is the forward difference operator, φ p s : |s| p−2 s,1<p<∞,andf : 1,N × → is a continuous function. 2AdvancesinDifference Equations In particular, for every λ lying in a suitable interval of parameters, at least three solutions are obtained under mutually independent conditions. First, we require that the primitive F of f is p-sublinear at infinity and satisfies appropriate local growth condition Theorem 3.1. Next, we obtain at least three positive solutions uniformly bounded with respect to λ, under a suitable sign hypothesis on f, an appropriate growth conditions on F in a bounded interval, and without assuming asymptotic condition at infinity on f Theorem 3.4, Corollary 3.6. Moreover, the existence of at least two nontrivial solutions for problem P f λ is obtained assuming that F is p-sublinear at zero and p-superlinear at infinity Theorem 3.5. It is worth noticing that it is the first time that this type of results are obtained for discrete problem with Neumann boundary conditions; instead of Dirichlet problem, similar results have been already given in 6, 9, 13.Moreover,in14, the existence of multiple solutions to problem P f λ is obtained assuming different hypotheses with respect to our assumptions see Remark 3.7. Investigation on the relation between continuous and discrete problems are available in the papers 15, 16. General references on difference equations Probability Distribution Function (PDF) for a Discrete Random Variable Probability Distribution Function (PDF) for a Discrete Random Variable By: OpenStaxCollege A discrete probability distribution function has two characteristics: Each probability is between zero and one, inclusive The sum of the probabilities is one A child psychologist is interested in the number of times a newborn baby's crying wakes its mother after midnight For a random sample of 50 mothers, the following information was obtained Let X = the number of times per week a newborn baby's crying wakes its mother after midnight For this example, x = 0, 1, 2, 3, 4, P(x) = probability that X takes on a value x x P(x) P(x = 0) = 50 P(x = 1) = 11 50 P(x = 2) = 23 50 P(x = 3) = 50 P(x = 4) = 50 P(x = 5) = 50 X takes on the values 0, 1, 2, 3, 4, This is a discrete PDF because: Each P(x) is between zero and one, inclusive 1/7 Probability Distribution Function (PDF) for a Discrete Random Variable The sum of the probabilities is one, that is, 11 23 + + + + + =1 50 50 50 50 50 50 Try It A hospital researcher is interested in the number of times the average post-op patient will ring the nurse during a 12-hour shift For a random sample of 50 patients, the following information was obtained Let X = the number of times a patient rings the nurse during a 12-hour shift For this exercise, x = 0, 1, 2, 3, 4, P(x) = the probability that X takes on value x Why is this a discrete probability distribution function (two reasons)? X P(x) P(x = 0) = 50 P(x = 1) = 50 P(x = 2) = 16 50 P(x = 3) = 14 50 P(x = 4) = 50 P(x = 5) = 50 Each P(x) is between and 1, inclusive, and the sum of the probabilities is 1, that is: 16 14 50 + 50 + 50 + 50 + 50 + 50 = Suppose Nancy has classes three days a week She attends classes three days a week 80% of the time, two days 15% of the time, one day 4% of the time, and no days 1% of the time Suppose one week is randomly selected a Let X = the number of days Nancy a Let X = the number of days Nancy attends class per week b X takes on what values? b 0, 1, 2, and 2/7 Probability Distribution Function (PDF) for a Discrete Random Variable c Suppose one week is randomly chosen Construct a probability distribution table (called a PDF table) like the one in [link] The table should have two columns labeled x and P(x) What does the P(x) column sum to? c x P(x) 0.01 0.04 0.15 0.80 Try It Jeremiah has basketball practice two days a week Ninety percent of the time, he attends both practices Eight percent of the time, he attends one practice Two percent of the time, he does not attend either practice What is X and what values does it take on? X is the number of days Jeremiah attends basketball practice per week X takes on the values 0, 1, and Chapter Review The characteristics of a probability distribution function (PDF) for a discrete random variable are as follows: Each probability is between zero and one, inclusive (inclusive means to include zero and one) The sum of the probabilities is one Use the following information to answer the next five exercises: A company wants to evaluate its attrition rate, in other words, how long new hires stay with the company Over the years, they have established the following probability distribution Let X = the number of years a new hire will stay with the company Let P(x) = the probability that a new hire will stay with the company x years Complete [link] using the data provided 3/7 Probability Distribution Function (PDF) for a Discrete Random Variable x P(x) 0.12 0.18 0.30 0.15 0.10 0.05 x P(x) 0.12 0.18 0.30 0.15 0.10 0.10 0.05 P(x = 4) = _ P(x ≥ 5) = _ 0.10 + 0.05 = 0.15 On average, how long would you expect a new hire to stay with the company? What does the column “P(x)” sum to? Use the following information to answer the next six exercises: A baker is 4/7 Probability Distribution Function (PDF) for a Discrete Random Variable deciding how many batches of muffins to make to sell in his bakery He wants to make enough to sell every one and no fewer Through observation, the baker has established a probability distribution x P(x) 0.15 0.35 0.40 0.10 Define the random variable X What is the probability the baker will sell more than one batch? P(x > 1) = _ 0.35 + 0.40 + 0.10 = 0.85 What is the probability the baker will sell exactly one batch? P(x = 1) = _ On average, how many batches should the baker make? 1(0.15) + 2(0.35) + 3(0.40) + 4(0.10) = 0.15 + 0.70 + 1.20 + 0.40 = 2.45 Use the following information to answer the next four exercises: Ellen has music practice three days a week She practices for all of the three days 85% of the time, two days 8% of the time, one day 4% of the time, and no days 3% of the time One week is selected at random Define the random variable X Construct a probability distribution table for the data x P(x) 0.03 5/7 Probability Distribution Function (PDF) for a Discrete Random Variable x P(x) 0.04 0.08 0.85 We know that for a probability distribution function to be ...Hindawi Publishing Corporation EURASIP Journal on Advances in Signal Processing Volume 2010, Article ID 575817, 8 pages doi:10.1155/2010/575817 Research Article Point Spread Function Estimation for a Terahertz Imaging System Dan C. Popescu and Andrew D. Hellicar Wireless and Networking Technologies Laboratory, CSIRO ICT Centre, Marsfield NSW 2121, Australia Correspondence should be addressed to Dan C. Popescu, dan.popescu@csiro.au Received 18 June 2010; Accepted 26 August 2010 Academic Editor: Enrico Capobianco Copyright © 2010 D. C. Popescu and A. D. Hellicar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We present a method for estimating the point spread function of a terahertz imaging system designed to operate in reflection mode. The method is based on imaging phantoms with known geometry, which have patterns with sharp edges at all orientations. The point spread functions are obtained by a deconvolution technique in the Fourier domain. We validate our results by using the estimated point spread functions to deblur several images of natural scenes and by direct comparison with a point source response. The estimations turn out to be robust and produce consistent deblurring quality over the entire depth of the focal region of the imaging system. 1. Introduction Imaging systems operating in the terahertz (THz) region of the spectrum have the potential to enable new applications due to the unique combination of properties that occur in this region, such as penetration through clothes, packaging and plastics, and also the fact that THz waves are nonionising and hence do not pose a health hazard for humans. Application domains such as security [1], medical imaging [2] and nondestructive testing [3]arelikelytobenefit from developments in this area. Despite these advantages, commercial systems in this spectral region have been slow to emerge, due to a lack of mature THz components and technology. Imaging at THz frequencies poses a challenge to the res- olution of the images that can be achieved [4], both because of the technology’s immaturity and the long wavelengths employed (relative to wavelengths at optical frequencies), which are typically around or over the millimetre range. Due to the expensive nature of terahertz imaging systems and the likelihood of long acquisition times, there is ample scope for employing image processing techniques, without increasing the system cost nor image acquisition time. Knowledge of the point spread function (PSF) of the imaging system is very important for improving image quality. The point spread function is the imaging system’s response to an ideal, point-like source. In practical situations it may not be easy to find such ideal sources, and methods relying on direct measurement of the point spread function from the response of a point-source approximation will face the challenge of balancing resolution ag ainst sensitivity. Examples of approximations for point sources include standard stars or quasars when calibrating astronomical instruments [ 5, 6], recording beads in microscopy [7, 8], and pinholes into opaque materials for various optical systems [9]. However, most practical computational methods used for the estimation of the point spread function are indirect and rely on some measured output of the system and possibly some additional knowledge of the scene being imaged and imaging system Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008, Article ID 385362, 11 pages doi:10.1155/2008/385362 Research Article Exponential Inequalities for Positively Associated Random Variables and Applications Guodong Xing, 1 Shanchao Yang, 2 and Ailin Liu 3 1 Department of Mathematics, Hunan University of Science and Engineering, Yongzhou, 425100 Hunan, China 2 Department of Mathematics, Guangxi Normal University, Guilin, 541004 Guangxi, China 3 Department of Physics, Hunan University of Science and Engineering, Yongzhou, 425100 Hunan, China Correspondence should be addressed to Guodong Xing, xingguod@163.com Received 1 January 2008; Accepted 6 March 2008 Recommended by Jewgeni Dshalalow We establish some exponential inequalities for positively associated random variables without the boundedness assumption. These inequalities improve the corresponding results obtained by Oliveira 2005. By one of the inequalities, we obtain the convergence rate n −1/2 log log n 1/2 log n 2 for the case of geometrically decreasing covariances, which closes to the optimal achievable conver- gence rate for independent random variables under the Hartman-Wintner law of the iterated log- arithm and improves the convergence rate n −1/3 log n 5/3 derived by Oliveira 2005 for the above case. Copyright q 2008 Guodong Xing et al. This is an open access article distributed under the Creative Commons Attribution License, which p ermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction A finite family of random variables {X i , 1 ≤ i ≤ n} is said to be positively associated PA if for every pair of disjoint subsets A 1 and A 2 of {1, 2, ,n}, Cov f 1 X i ,i∈ A 1 ,f 2 X j ,j ∈ A 2 ≥ 0 1.1 whenever f 1 and f 2 are coordinatewise increasing and the covariance exists. An infinite family is positively associated if every finite subfamily is positively associated. The exponential inequalities and moment inequalities for partial sum n i1 X i − EX i play a very important role in various proofs of limit theorems. For positively associated random variables, Birkel 1 seems the first to g et some moment inequalities. Shao and Yu 2 generalized later the previous results. Recently, Ioannides and Roussas 3 established aBernstein-Hoeffding-type inequality for stationary and positively associated random vari- ables being bounded; and Oliveira 4 gave a similar inequality dropping the boundedness 2 Journal of Inequalities and Applications assumption by the existence of Laplace transforms. By the inequality, he obtained that the rate of n i1 X i − EX i /n → 0 a.s. is n −1/3 log n 5/3 under the rate of covariances supposed to be geometrically decreasing, that is, ρ n for some 0 <ρ<1. The convergence rate is partially im- proved by Yang and Chen 5 only for positively associated random variables being bounded. Furthermore, the rate of convergence in 4 is even lower than that obtained by 3. These motivate us to establish some new exponential inequalities in order to improve the inequali- ties and the convergence rate which 4 obtained without the boundedness assumption. It is the main purpose of this paper. Our inequalities in Sections 3–5 improve the corresponding results in 4. Moreover, by Corollary 5.4 which can be seen in Section 5,wemaygetthe rate n −1/2 log log n 1/2 log n 2 if the rate of covariances is MULTIPLE PERIODIC SOLUTIONS FOR A DISCRETE TIME MODEL OF PLANKTON ALLELOPATHY JIANBAO ZHANG AND HUI FANG Received 19 May 2005; Revised 25 September 2005; Accepted 27 September 2005 We study a discrete time model of the growth of two species of plankton with compet- itive and allelopathic effects on each other N 1 (k +1)= N 1 (k)exp{r 1 (k) − a 11 (k)N 1 (k) − a 12 (k)N 2 (k) − b 1 (k)N 1 (k)N 2 (k)}, N 2 (k +1)= N 2 (k)exp{r 2 (k) − a 21 (k)N 1 (k) − a 22 (k) N 2 (k) − b 2 (k)N 1 (k)N 2 (k)}.Asetofsufficient conditions is obtained for the existence of multiple positive periodic solutions for this model. The approach is based on Mawhin’s continuation theorem of coincidence degree theory as well as some a prior i estimates. Some new results are obtained. Copyright © 2006 J. Zhang and H. Fang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, prov i ded the original work is properly cited. 1. Introduction Many researchers have noted that the increased population of one species of phytoplank- ton might affect the growth of one or several other species by the production of allelo- pathic toxins or stimulators, influencing bloom, pulses, and seasonal succession. The study of allelopathic interactions in the phytoplanktonic world has become an impor- tant subject in aquatic ecology. For detailed studies, we refer to [1, 2, 7, 9–11, 13]and references cited therein. Maynard-Smith [9] and Chattopadhyay [2] proposed the following two species Lotka- Volterra competition system, which descr ibes the changes of size and density of phyto- plankton: dN 1 dt = N 1 r 1 − a 11 N 1 (t) − a 12 N 2 (t) − b 1 N 1 (t)N 2 (t) , dN 2 dt = N 1 r 2 − a 21 N 1 (t) − a 22 N 2 (t) − b 2 N 1 (t)N 2 (t) , (1.1) where b 1 and b 2 are the rates of toxic inhibition of the first species by the second and vice versa, respectively . Hindawi Publishing Corporation Advances in Difference Equations Volume 2006, Article ID 90479, Pages 1–14 DOI 10.1155/ADE/2006/90479 2 Periodic solutions for a discrete plankton model Naturally, more realistic models require the inclusion of the periodic changing of envi- ronment (e.g., seasonal effects of weather, food supplies, etc). For such systems, as pointed out by Freedman and Wu [5] and Kuang [8], it would be of interest to study the existence of periodic solutions. This motivates us to modify system (1.1)totheform dN 1 dt = N 1 (t) r 1 (t) − a 11 (t)N 1 (t) − a 12 (t)N 2 (t) − b 1 (t)N 1 (t)N 2 (t) , dN 2 dt = N 1 (t) r 2 (t) − a 21 (t)N 1(t) − a 22 (t)N 2 (t) − b 2 (t)N 1 (t)N 2 (t) , (1.2) where r i (t), a ij (t) > 0, b i (t) > 0(i, j = 1,2) are continuous ω-periodic functions. The main purpose of this paper is to propose a discrete analogue of system (1.2)and to obtain sufficient conditions for the existence of multiple positive periodic solutions by employing the coincidence degree theory. To our knowledge, no work has been done for the existence of multiple positive periodic solutions for this model using this way. The paper is organized as follows. In Section 2, we propose a discrete analogue of sys- tem (1.2). In Section 3, motivated by the recent work of Fan and Wang [4]andChen[3], we study the existence of multiple positive periodic solutions of the difference equations derived in Section 2. 2. Discrete ... number of days Nancy attends class per week b X takes on what values? b 0, 1, 2, and 2/7 Probability Distribution Function (PDF) for a Discrete Random Variable c Suppose one week is randomly chosen... time, and no days 3% of the time One week is selected at random Define the random variable X Construct a probability distribution table for the data x P(x) 0.03 5/7 Probability Distribution Function. .. Javier volunteers for less than three events each month P(x < 3) = _ 6/7 Probability Distribution Function (PDF) for a Discrete Random Variable Find the probability that Javier volunteers for