positive solutions of singular boundary value problems

ABSTRACT Positive Solutions of Singular Boundary Value Problems Curtis J. Kunkel Advisor: Johnny Henderson, Ph.D. In this dis sertation, we focus on singular boundary value problems with mixed boundary conditions. We study a variety of types, to all of which we seek a positive solution. We begin by considering the discrete (or difference equation) case, from which we proceed to look at the continuous (or ordinary differential equation) case. In all cases, we make use of a lower and upper solutions method and the Brouwer fixed point theorem in conjunction with perturbation methods to approximate regular problems. Positive Solutions of Singular Boundary Value Problems by Curtis J. Kunkel A Dissertation Approved by the Department of Mathematics Robert Piziak, Ph.D., Chairperson Submitted to the Graduate Faculty of Baylor University in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy Approved by the Dissertation Committee Johnny Henderson, Ph.D., Chairperson John M. Davis, Ph.D. Dennis A. Johnston, Ph.D. Klaus Kirsten, Ph.D. Qin Sheng, Ph.D. Accepted by the Graduate School May 2007 J. Larry Lyon, Ph.D., Dean Page bearing signatures is kept on file in the Graduate School. UMI Number: 3247565 3247565 2007 UMI Microform Copyright All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. ProQuest Information and Learning Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, MI 48106-1346 by ProQuest Information and Learning Company. Copyright c  2007 by Curtis J. Kunkel All rights reserved TABLE OF CONTENTS ACKNOWLEDGMENTS v 1 Introduction 1 2 Singular Boundary Value Problems for Difference Equations 4 2.1 Third Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.1.2 Lower and Upper Solutions Method for Regular Problems . . 6 2.1.3 Existence Result . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2 Higher Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2.2 Lower and Upper Solutions Method for Regular Problems . . 18 2.2.3 Existence Result . . . . . . . . . . . . . . . . . . . . . . . . . 28 3 Singular Boundary Value Problems for Ordinary Differential Equations 33 3.1 Second Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.1.2 Lower and Upper Solutions Method for Regular Problems . . 34 3.1.3 Existence Result . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.2 Third Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.2.2 Lower and Upper Solutions Method for Regular Problems . . 43 3.2.3 Existence Result . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.3 Higher Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.3.2 Lower and Upper Solutions Method for Regular Problems . . 54 iii 3.3.3 Existence Result . . . . . . . . . . . . . . . . . . . . . . . . . 60 BIBLIOGRAPHY 64 iv ACKNOWLEDGMENTS I would like to thank Baylor University for giving me the opportunity to work on my dissertation and earn my doctorate. The professors at Baylor University will ever have my thanks in preparing me to do the research required for this dissertation and for preparing me to enter the field of mathematics as their colleague. Special thanks go to Dr. Johnny Henderson for advising me throughout this process. I am eternally grateful for the time he spent reading and re-reading this document. I would like to thank my family and friends for keeping me motivated throughout this process and for making my transition to Texas all the easier. I would like to thank my parents for always believing in me and helping me become the person I am today. Finally, I would like to thank my wife Barbara for supporting me in all that I do, the least of which is this dissertation. Thanks! v CHAPTER ONE Introduction In this dissertation we will be concerned with the existence of positive solutions of singular boundary value problems. We will begin in Chapter 2 by looking for solutions u of singular discrete boundary value problems of the form, (−1) n ∆ n u(t−(n−1))+f(t, u(t), . . . , ∆ n−1 u(t−(n−1))) = 0, t ∈ [n−1, T +1], (1.1) satisfying boundary conditions, ∆ n−1 u(0) = ∆ n−2 u(T + 2) = · · · = u(T + n) = 0, (1.2) where [n − 1, T + 1] = {n − 1, n, . . . , T, T + 1}, T ∈ N, n ≥ 2. We prove existence of positive solutions of (1.1), (1.2) by means of a lower and upper solutions method, the Brouwer fixed point theorem, and by perturbation methods to approximate regular problems. We will first look at the case where n = 3 in order to get a better under- standing of the methods involved in the higher order case. Following the n = 3 case, we will give the generalization. Discrete boundary value problems (sometimes referred to as boundary value problems for difference equations) have bee n studied extensively in the recent past. Most notable is the work done by Agarwal and his collaborators. For a few of their works, see [1], [6] and [8]. Other mathematicians have also studied discrete boundary value problems. The interested reader should refer to [3], [16], [17], [21], [22], [24], [27], [28], [29], [34], [35], [36] and [37]. The class of singular discrete boundary value problems has been studied in [27], [28] and [32]. As is stated above, we employ the use of a lower and upper solutions method to prove the existence of positive solutions. The lower and upper solutions methods have been employed by many others, and a brief list of works include [2], [9], [12], [13], 1 2 [38] and [40]. If one were to look to study positive solutions, one may look towards [7], [10], [19], [20] and [41] as a starting point. It was also stated that we will use a perturbation method to approximate our regular problem. Again, others have used this technique, and a brief summary of their works can be found in [1], [31] and [33]. Primary motivation for the results of this dissertation is the recent paper on the existence of singular boundary value problems by Rach˙unkov´a and Rach˙unek [39]. They studied a second order boundary value problem for the discrete p-Laplacian, ϕ p (x) = |x| p−2 x, p > 1. In particular, Rach˙unkov´a and Rach˙unek dealt with the discrete boundary value problem, ∆ (ϕ p (∆u(t − 1))) + f (t, u(t), ∆u(t − 1)) = 0, t ∈ [1, T + 1], ∆u(0) = u(T + 2) = 0. (1.3) We begin our study by focussing on p = 2 for (1.3), or in particular, for (1.1), (1.2), when n = 2. We do not include this case in the dissertation because it would effectively amount to the removal of the p-Laplacian in their paper. The discrete p-Laplacian in reference to boundary value problems is an active field, and some other work in the area can be found in [11], [26] and of course in [39]. Moving along to Chapter 3, we make a transition and consider singular bound- ary value problems for ordinary differential equations of the form, (−1) n u (n) (t) + f(t, u(t), . . . , u (n−2) (t)) = 0, t ∈ (0, 1), (1.4) satisfying boundary conditions, u (n−1) (0) = u (n−2) (1) = · · · = u(1) = 0, (1.5) where n ≥ 2. We use similar methods to acheive existence of positive solutions of problem (1.4), (1.5). That is, we again use the lower and upper solutions method, the Brouwer fixed point theorem, and perturbation methods to approximate regular problems. We begin the chapter by giving the second order case. Certain technical 3 difficulties arose for higher order problems which were not present in the second order case. In particular, we were forced to use a slightly different approach to the lower and upper s olution method for the higher order results. In view of this, we include the third order results as a means to more fully understand the methods of pro of involved in the generalization. We then conclude the chapter with the higher order generalization that will in fact work for all values of n ≥ 2, however the initial method for n = 2 will be a slightly stronger version than this generalization. Boundary value problems for ordinary differential equations have been studied extensively in the recent past. To see just a few of these recent works, the interested reader should refer to [1], [5], [6], [14], [15] and [25]. The class of singular boundary value problems also has been studied a great deal; see [5], [4], [23], [25] and [32]. As was stated, we again employ the use of a lower and upper solutions method to prove the existence of positive solutions. This method has been employed by many others, and a brief list of works includes [14], [15] and [18]. For works somewhat related to those of this dissertation in the study of positive solutions, one may look towards [4], [23] and [30]. Again, we will use a perturbation method to approximate our regular problem. We mention that others have also used this technique in the differential equations setting, and a few such works can be found in [1], [4] and [30]. [...]...CHAPTER TWO Singular Boundary Value Problems for Difference Equations This chapter is devoted to the study of solutions of singular boundary value problems for difference equations of the form, (−1)n ∆n u(t−(n−1))+f (t, u(t), , ∆n−1 u(t−(n−1))) = 0, t ∈ [n−1, T +1], (2.1) satisfying boundary conditions, ∆n−1 u(0) = ∆n−2 u(T + 2) = · · · = u(T + n)... the boundary value problem −∆3 u(t − 2) + 1 = 0, t ∈ [2, T + 1], u(t) satisfying mixed boundary conditions, ∆2 u(0) = ∆u(T + 2) = u(T + 3) = 0 yield a positive solution, u(t) 2.2 2.2.1 Higher Order Preliminaries We now consider a generalization of the results given in the previous section concerning the singular discrete boundary value problem (2.1), (2.2) when n = 3 Referring back to the definitions of. .. of boundary conditions (2.23), (2.28) and (2.26), along with summing both sides of ∆n−2 β(l) ≤ ∆n−2 u(l), we see that u(t) ≤ β(t) A similar argument shows that α(t) ≤ u(t) At this, we note from the definition of h that u(t) is a solution of (2.24), (2.23) Thus, the conclusion of the theorem holds and our proof is complete 2.2.3 Existence Result In this section, we make use of Theorem 2.3 to obtain positive. .. ∆(∆n−1 u(t)) In this chapter, we will derive the existence of a positive solution to the third order version of the singular problem (2.1), (2.2) in hopes of better understanding the method that will be used in deriving the higher order generalization 2.1 Third Order 2.1.1 Preliminaries To give us some insight into the methods of proof for the singular discrete (2.1), (2.2), we will first present results... definition of h that u(t) is a solution of (2.5), (2.4) Hence, T +3 T +3 ∆β(l) ≤ l=t ∆u(l) l=t This, in turn, yields β(T + 3) − β(t) ≤ u(T + 3) − u(t), β(T + 3) − β(t) ≤ −u(t), u(t) ≤ β(t) − β(T + 3), u(t) ≤ β(t) A similar argument shows that α(t) ≤ u(t) Thus, the conclusion of the theorem holds and our proof is complete 2.1.3 Existence Result We now make use of Theorem 2.1 to obtain positive solutions of. .. Existence Result In this section, we make use of Theorem 2.3 to obtain positive solutions of the singular problem (2.22), (2.23) In particular, in applying Theorem 2.3, we deal with a sequence of regular perturbations of (2.22), (2.23) Ultimately, we obtain a desired solution of (2.22), (2.23) by passing to the limit on a sequence of solutions for the perturbations Theorem 2.4 Assume conditions (A), (B), and... for the singular third order nonlinear difference equation, −∆3 u(t − 2) + f (t, u(t), ∆u(t − 1), ∆2 u(t − 2)) = 0, t ∈ [2, T + 1], (2.3) with mixed boundary conditions, ∆2 u(0) = ∆u(T + 2) = u(T + 3) = 0 (2.4) Our goal is to prove the existence of a positive solution of the problem (2.3), (2.4) To this end, we must define what we refer to as a positive solution Definition 2.3 By a solution u of problem... − 2)) Therefore, u satisfies (2.3) and by (2.21), our theorem holds 17 We have therefore proved the existence of a positive solution of the third order boundary value problem, and as such, have exhibited the methods to be used in the higher order case, which is to come in the next section of this chapter We first, however consider an example to illustrate the third order result Example: 1 Consider the... f has singularities on the boundary of D, ∂D, then problem (2.3), (2.4) is singular We will assume throughout this section that the following assumptions hold: A: D = (0, ∞) × R2 B: f is continuous on [2, T + 1] × D C: f (t, x0 , x1 , x2 ) has a singularity at x0 = 0, i.e lim sup |f (t, x0 , x1 , x2 )| = ∞ x0 →0+ 2 for each t ∈ [2, T + 1] and for some (x1 , x2 ) ∈ R 6 2.1.2 Lower and Upper Solutions. .. Method) Let α and β be lower and upper solutions of (2.24), (2.23), respectively, and α ≤ β on [n−1, T +1] Let h(t, x0 , , xn−1 ) be continuous on [n − 1, T + 1] × Rn and nonincreasing in its xn−1 variable Then (2.24), (2.23) has a solution u satisfying, α(t) ≤ u(t) ≤ β(t), t ∈ [0, T + n] (2.29) Proof We proceed through a sequence of steps involving modifications of the function h 20 Step 1 For t ∈ . ABSTRACT Positive Solutions of Singular Boundary Value Problems Curtis J. Kunkel Advisor: Johnny Henderson, Ph.D. In this dis sertation, we focus on singular boundary value problems with mixed boundary. [30]. CHAPTER TWO Singular Boundary Value Problems for Difference Equations This chapter is devoted to the study of solutions of singular boundary value problems for difference equations of the form, (−1) n ∆ n u(t−(n−1))+f(t,. the least of which is this dissertation. Thanks! v CHAPTER ONE Introduction In this dissertation we will be concerned with the existence of positive solutions of singular boundary value problems.