Positive solutions for singular third-order nonhomogeneous boundary value problems with nonlocal boundary conditions

Positive solutions for singular third-order nonhomogeneous boundary value problems with nonlocal boundary conditions Department of Mathematics Tianjin Polytechnic University No. 6 Chenglin RoadHedong District 6 Tianjin People s Republic of China jnkp98@6.com Abstract: Under various weaker conditions we establish various results on the existence and nonexistence of positive solutions for singular third-order nonhomogeneous boundary value problems with nonlocal boundary conditions. The arguments are based upon the fixed point theorem of cone expansion and compression. Finally we give two examples to demonstrate our results. Key Words: Positive solutions Fixed points Boundary value problems Nonhomogeneous Ordinary differential equations. Introduction The world is nonlinear in essence. Because nonlinear phenomena is studied by nonlinear theories and methods every field becomes nonlinear and then nonlinear mechanics nonlinear optics and nonlinear mathematics appear. Since the development of physics and applied mathematics calls for the global and high level development of the mathematics ability of analyzing and controlling objective phenomena nonlinear functional analysis which is one of the most important research fields in modern mathematics is formed by the continuously accumulation of nonlinear results. Until 95 s nonlinear functional analysis has initially formed a theory system. In recent years because nonlinear functional analysis has been an important tool for studying the nonlinear problem in mathematics physics aerospace engineering biology engineering it is greatly significant in the theory and application to study nonlinear functional analysis and its application. Since the th century the development of nonlinear functional analysis has achieved the great breakthrough. L. E. J. Brouwer had established the conception of topological degree for finite dimensional space in 9. Then J. Leray and J. Schauder extended the conception to completely continuous field of Banach space in 94 afterward E. Rothe M. A. Krasnosel skii P. H. Rabinowitz H. Amann and K. Deimling carried on embedded research on topological degree and cone theory. Many well known mathematicians in China for example Guo Dajun Zhang Gongqing Chen Wenyuan Ding Guanggui Sun Jingxian etc. had proud works in various fields of nonlinear functional analysissee [-]. The method to research nonlinear problems mainly has topological degree method critical point theory partial order method lower and upper solution method fixed point theory coincidence degree theory monotone iterative technique topological transversal degree and so on. The main questions to research are the existence of solution for nonlinear operator e- quation uniqueness of solution multi-solution structure of solution approximate solution divergent theory of solution iteration arithmetic nonlinear operator theory as well as the application for partial d- ifferential equation differential equation integral e- quation and differential-integral equation. All these problems are among the most active domain in analyzing mathematics at present. Among them firstly singular boundary value problem of nonlinear differential equations. It has resulted from the applied disciplines of nuclear physics hydromechanics boundary layer theory nonlinear optics and so on. It is an important research field of differential equations fields. Because it plays a very extensive and important role in the fields of physics mathematics aerospace engineering biology engineering and so on it has received high attention of numerous mathematicians. By applying the theories and methods of nonlinear functional analysis the numerous famous mathematicians in the world have deeply studied the existence uniqueness and multiplicity of solutions of singular boundary value problems and obtained lots of new results. However because there are lots of difficulties ISSN: 9-769 5 Issue Volume December

in studying singular ordinary differential equations at present it is still the advance orientation in the s- tudy of nonlinear analysis. Secondly nonlocal boundary value problems for ordinary differential equations. The meaning of the nonlocal problems is that the definite condition of definite problem of ordinary differential equations not only depends on the value of solution in the end of interval but also depends on the value of solution in some points of the interior of interval. Although lots of problems in theory and application can be reduced to nonlocal boundary value problems for ordinary differential equations people started to fairly late study the nonlocal problems for the difficulties of nonlocal problems itself. Kiguradze Lomtatidze984 Il in and Moiseev987 began to discuss the existence of solutions of nonlinear multi-point boundary value problems for ordinary d- ifferential equations. Within the following ten years the study on nonlocal boundary value problems for ordinary differential equations has been made great progress. However it is not good enough and it is also a research topic to have a strong interest and maybe obtain some new significant achievements. Thirdly system of nonlinear ordinary differential equations. S- ince lots of higher order differential-integral equations and implicit form equations can be reduced to the system of differential-integral equations by the appropriate variable substitution the research of the system of equations plays a very important role in studying those equations. The purpose of this paper is to establish the existence and nonexistence of positive solutions for the following singular third-order nonhomogeneous boundary value problems BVPs for short with nonlocal boundary conditions : u t + atft ut u t = u = g usdαs u = g u sdβs u = λ where t λ is a parameter a C [ + and may be singular at t = or t = ; f : [ ] [ + [ + [ + g g : [ + [ + are continuous; usdαs and u sdβs denote the Riemann-Stieltjes integrals α β are increasing nonconstant functions defined on [] with α = β =. Here we call a function u a positive solution of BVP if u satisfies BVP and u t > for any t. In the last years third-order ordinary differential equations with a two-point or multi-point boundary value problem have been studied widely in the literaturesee [-7] and [-]. For example Guo et al. in [] discussed the following nonlinear third-order three-point boundary value problem: { u t + atft ut u t = u = u = u = αu η where t < η < and < α < η. The authors established the existence of at least a positive solution for the above problem when f is superlinear or sublinear. Zhang et al. in [6] studied the following third-order eigenvalue problems: { u t = λft ut u t u = u η = u = where t λ > is a parameter and η < is a constant f : [ ] R R R is continuous R = +. By using Leray-Schauder nonlinear alternative the authors obtain the existence and uniqueness of nontrivial solution of. when λ in some interval. However to our knowledge the corresponding results for third-order nonhomogeneous boundary value problems especially in the case that the BVPs with nonlocal boundary conditions are rarely seen see for example [7-8] and references therein. Sun et al. in [7] studied the existence and nonexistence of positive solutions of nonhomogeneous BVPs of thirdorder ordinary differential equations. Du et al. in [8] consider the following third-order nonlocal BVPs: u t = ft ut u t u t u = u = u = u sdgs 4 where t f : [ ] R R R is a continuous function g : [ ] [ is a nondecreasing function with g =. Under the resonance condition g = an existence results is given by using the coincidence degree theory. Obviously what we consider is more different from those in [-8]. Firstly we will consider the boundary conditions which is nonlocal. Secondly f and g i i = satisfy the limit conditions which are more extensive than the superlinear and sublinear conditions and the nonexistence of positive solutions of BVP is also studied. The paper is organized as follows. In Section we present some preliminaries and lemmas that will be used to prove our main results. In Section various conditions on the existence and nonexistence of positive solutions for the BVP are discussed. In Section 4 we give two examples to demonstrate our results. ISSN: 9-769 5 Issue Volume December

Preliminaries and Lemmas In this Section we present some lemmas that will be used in the proof of our main results. Throughout this paper we assume that: H a C [ + may be singular at t = or t = and at does not vanish identically on any subinterval of with s sasds < + ; H f : [ ] [ + [ + [ + and g i i = : [ + [ + are continuous. Lemma Let h C [ + with s shsds < + and u C [ ] be the function from the set {u : ut u t t } with ut = g usdαs + Gt shsds g u sdβs + t t λ + t 5 then u is the unique solution of the following boundary value problem u t + ht = < t < u = g usdαs u = g u sdβs u = λ where Gt s = st s t s t t s t s. 6 7 Proof: First suppose that ut is a solution of problem 6. Then we may suppose that ut = t t s hsds + At + Bt + C. 8 By the boundary condition 6 we get A = λ g u sdβs s hsds B = g u sdβs C = g usdαs. 9 Substituting 9 into 8 we obtain ut = g u sdβs t + g usdαs t t s hsds + t λ t g u sdβs t = g usdαs t g t u sdβs s hsds s hsds + tg u sdβs + t λ t t s hsds = g usdαs + t t g u sdβs + t λ + Gt shsds. Conversely directing differentiation of 5 we can obtain 6. This completes our proof. From 7 we can prove that Gt s have the following properties. Lemma For all t s [ ] [ ] we have where G t t s = G t t s ss. { s t s t t s t s. Lemma For all t s [/ /] [ ] we have 8 s s Gt s ss Proof: For t So G tts G sss G t ts G s ss G t t s ss. = s t s s = t s t s = t s s s = t s t s G t t s G ss s = s s for t s. Lemma 4 In Lemma the solution ut of boundary value problem 6 and u t are nonnegative and satisfy min ut t 9 u min u t t u. ISSN: 9-769 5 Issue Volume December

Proof: It is obvious that ut and u t are nonnegative. Step. We will prove that min ut t 9 u. For any t [ ] by 5 and Lemma it follows that ut = g usdαs + t t g u sdβs + t λ + Gt shsds g usdαs + g + λ + s shsds u sdβs thus u g + usdαs g u sdβs + λ + s shsds. On the other hand 5 and Lemma imply that for any t [ ] ut = g usdαs + Gt shsds t t + g u sdβs + t λ g usdαs + 5 8 g 9 + + 9 9 u. Therefore 8 λ + 8 s shsds [ g + usdαs g [ λ + ] s shsds Step. We will prove that min ut t 9 u. min u t t u. u sdβs u sdβs For any t [ ] by 5 and Lemma it follows that u t = t g u sdβs +tλ + G tt shsds g u sdβs +λ + s shsds ] thus u g u sdβs +λ + s shsds. On the other hand 5 and Lemma imply that for any t [ ] u t = t g u sdβs Therefore +tλ + G tt shsds g u sdβs + λ + s shsds u. min u t t u. This completes our proof. We shall discuss the existence of a positive solution of the BVP by using the following fixed-point theorem of cone expansion and compression. Lemma 5 [9] Let K be a cone of the real Banach space E Ω Ω E be bounded open sets of E. θ Ω Ω Ω A : K Ω \Ω K is a completely continuous mapping such that one of the following two conditions is satisfied: Au u u K Ω ; Au u u K Ω. Au u u K Ω ; Au u u K Ω. Then A has a fixed point in K Ω \Ω. We shall consider the Banach space C [ ] e- quipped with the standard norm u = u + u where u = max ut. t Define the cone P by P = u C [ ] Define an operator T by ut u t ut 9 u min t min t u t u. T ut = g usdαs + t λ + t t g u sdβs + Gt sasfs us u sds. ISSN: 9-769 5 Issue Volume December

By Lemma it is clear that the existence of a positive solution of BVP is equivalent to the existence of a nontrivial fixed point of T. Lemma 6 Assume that H and H hold. Then T : P P is completely continuous. Proof: Since the proof of the completed continuity is standard we need only to prove T P P. In fact for x y P t J by H H and it is obvious that T ut and T u t are nonnegative. Step. We will prove that min T ut t 9 u. For any t [ ] by and Lemma it follows that T ut = g usdαs thus + t t + t λ + g u sdβs Gt shsds g + usdαs g + λ + s shsds u g usdαs + g + λ + s shsds. u sdβs u sdβs On the other hand and Lemma imply that for any t [ ] T ut = g usdαs + t t g u sdβs + t λ + Gt shsds g usdαs + 5 8 g + 8 λ + 8 s shsds [ 9 g usdαs + g [ + λ 9 + ] s shsds 9 T u. Therefore min T ut T u. t 9 u sdβs ] u sdβs Step. We will prove that min T u t t T u. For any t [ ] by and Lemma it follows that T u t = t g u sdβs thus +tλ + G tt shsds g u sdβs u g u sdβs + λ + s shsds +λ+ s shsds. On the other hand and Lemma imply that for any t [ ] T u t = t g u sdβs +tλ + G tt shsds [ g u sdβs T u. Therefore min T u t t T u. + λ + s shsds ] From the above discussion we assert that T x y P. Therefore T : P P is completely continuous. This completes our proof. Main results In the following for convenience we denote J = [ ] J = [ ] and set f α ft x y = lim sup max x+y α t J x + y f α = lim inf x+y α min t J ft x y x + y g α i = lim sup x α g i x x where α is + or + i =. Throughout this section we assume that p i i = 4 are four positive numbers satisfying p + ISSN: 9-769 54 Issue Volume December

p + p + p 4. And p A = B = 6 s sasds s sasds. In this section we are concerned with the existence and nonexistence of positive solutions of BVP. We obtain the following theorems. Theorem 7 Assume that H and H hold. In addition assume that f g g satisfy: f < A f > B g < p α g < p β. Then BVP has at least one positive solution for λ small enough and has no positive solution for λ large enough. Proof: Take a sufficiently small positive number ϵ : < ϵ < min{ p α p β } such that g < p α ϵ g < p β ϵ. Then there exists an < η < such that g x p α ϵ x g x p β ϵ x < x η. By f < A we have that there exists η > such that ft x y Ax + y t J x + y η. { } η Let r = min η α η β and Ω = {u P : u < r }. Then for any u Ω P g usdαs p α ϵ usdαs p α αr = r g u sdβs p p β ϵ u sdβs p β βr = r p. Therefore for any u Ω P let < λ r p 4 we have T u + T u g + usdαs λ+ g u sdβs + s sasfs us u sds r p + r p + λ + s sasadsr p + p + p + p 4 r r Consequently T u u u Ω P. On the other hand by f > B there exists r > r such that ft x y Bx + y t J x + y 9 r. 4 Let Ω = {u P : u < r }. Then for any u Ω P we have T u = g usdαs + 5 8 g u sdβs + 8 λ + G sasfs us u sds 8 s sasfs us u sds 8 8 6 Consequently s sasbus + u sds s sasb 9 r ds s sasbr ds = r T u u u Ω P. 5 Thus 5 and Lemma 5 imply T has a fixed point u P such that < r < u < r and hence u is a positive solution of the BVP. Next we prove that problem has no positive solution for λ large enough. Otherwise there exist < λ < λ < < λ n < with lim λ n = + such that for any positive integer n n the BVP < t < u t + atft ut u t = u = g usdαs u = g u sdβs u = λ n 6 ISSN: 9-769 55 Issue Volume December

has a positive solution u n t. By we have Thus u n = g u nsdαs + λ n + g u nsdβs + G sasfs u ns u nsds λ n n. u n n. By f > B there exists R > such that 7 ft x y Bx + y t J x + y R. 8 9 Let n large enough such that u n R. Then u n > G sasfs u ns u nsds B 8 s sasu n s + u nsds B 8 B 6 = u n s sas 9 u n ds s sas u n ds which is a contradiction. The proof of Theorem 7 is completed. Similar to the proof of Theorem 7 we can also prove the following results. Corollary 8 Assume that H and H hold. In addition assume that f g g satisfy: f = f = g = g =. Then BVP has at least one positive solution for λ small enough and has no positive solution for λ large enough. Theorem 9 Assume that H and H hold. In addition assume that f g g satisfy: f > B f < A g < p α g < p β. Then BVP has at least one positive solution for any λ. Proof: By f > B there exists R > such that ft x y Bx + y t J x + y R. 9 Let Ω = {u P : u < R }. Then for any u Ω P we have T u = g usdαs + 8 λ + 5 8 g u sdβs + G sasfs us u sds 8 s sasfs us u sds B 8 B 8 B 6 R Consequently s sasus + u sds s sas 9 R ds s sasr ds T u u u Ω P. Take a sufficiently small positive numberϵ : such that g < < ϵ < min{ p α p β } p α ϵ g < p β ϵ. This together with f < A implies that there exists an C > such that for x y [ + Let ft x y Ax + y + C t J g x p α ϵ x + C g x p β ϵ x + C. R > max { R 5C + λ + C } s sasds p 4 and Ω 4 = {u P : u < R }. Then for any u Ω 4 P g usdαs p α ϵ usdαs + C p α αr + C = R p + C ISSN: 9-769 56 Issue Volume December

g u sdβs p β ϵ u sdβs + C p β βr + C = R p + C. Therefore for any u Ω 4 P we have T u + T u g usdαs + g u sdβs + λ + s sasfs us u sds R p + R + p + λ + 5 C s sasar + Cds p + p + p + p 4 R R Consequently T u u u Ω 4 P. Thus and Lemma 5 imply T has a fixed point u P such that < R < u < R and hence u is a positive solution of BVP. The proof of Theorem 9 is completed. Similar to the proof of Theorem. we can also prove the following results. Corollary Assume that H and H hold. In addition assume that f g g satisfy: f = f = g = g =. Then BVP has at least one positive solution for any λ. 4 Example Example Consider the following singular thirdorder nonhomogeneous boundary value problems: u t + atft ut u t = u = g usdαs u = g u sdβs u = λ 4 where t at = αs = s βs = t s g t = t g t = t and ft x y = tx + y x y t. Then BVP 4 has at least one positive solution for λ small enough and has no positive solution for λ large enough. It is easy to check that H and H hold. In addition assume that f g g satisfy: f = f = and g = g = By Corollary 8 our conclusion follows. Example Consider the following singular thirdorder nonhomogeneous boundary value problems: u t + atft ut u t = u = g usdαs 5 u = g u sdβs u = λ where < t < at = t αs = s s βs = s g t = t g t = t and ft x y = x + y + t x + y x y t. Then BVP 5 has at least one positive solution for any λ. It is easy to check that H and H hold. In addition assume that f g g satisfy: f = f = and g = g =. By Corollary our conclusion follows. Acknowledgements: The authors are very grateful to the referee who read the manuscript carefully and provided a lot of valuable suggestions and useful comments. It should be said that we could not have polished the final version of this paper well without his or her outstanding efforts. The research was supported financially by the NNSF-China 779 and Tianjin City High School Science and Technology Fund Planning Project 98. References: [] L. J. Guo J. P. Sun and Y. H. Zhao Existence of positive solutions for nonlinear third-order three-point boundary value problems Nonlinear Anal. 68 8 pp. 5-58. [] Y. P. Sun Positive solutions for singular thirdorder three-point boundary value problem J. Math. Anal. Appl. 6 5 pp. 589-6. ISSN: 9-769 57 Issue Volume December

[] Q. L. Yao Successive iteration of positive solution of a discontinuous third-order boundary value problem Comp. Math. Appl. 5 7 p- p. 74-749. [4] A. Boucherif and N. Al-Malki Nonlinear threepoint third-order boundary value problems Appl. Math. Comput. 9 7 pp. 68-77. [5] J. R. Graef and J. R. L. Webb Third order boundary value problems with nonlocal boundary conditions Nonlinear Anal. 7 9 p- p. 54-55. [6] X. G. Zhang L. S. Liu and C. X. Wu Nontrivial solution of third-order nonlinear eigenvalue problems Appl. Math. Comput. 76 6 p- p. 74-7. [7] Y. P. Sun Positive solutions for third-order three-point nonhomogeneous boundary value problem Appl. Math. Lett. 9 pp. 45-5. [8] Z. J. Du X. J. Lin and W. G. Ge Solvability of a third order nonlocal boundary value problems at resonance Acta Math. Sin. 49 6 pp. 87-94 in Chinese. [9] D. Guo and V. Lakshmikantham Nonlinear Problems in Abstract Cones Academic Press Inc. New York 988. [] D. Anderson Multiple positive solutions for a three-point boundary value problem Math. Comput. Modelling 7 998 pp. 49-57. [] D. Anderson Green s functions for a third-order generalized right focal problem J. Math. Anal. Appl. 67 pp. 5-57. [] D. Anderson and J. M. Davis Multiple positive solutions and eigenvalues for third-order right focal boundary value problem J. Math. Anal. Appl. 9 6 pp. 47-444. [] H. Chen Positive solutions for the nonhomogeneous three-point boundary value problem of second order differential equations Math. Comput. Modelling 45 7 pp. 844-85. [4] J. R. Graef and B. Yang Multiple positive solutions to a three-point third order boundary value problems Discrete and Continuous ynmical Systems Supplement Volume 5 pp. -8. [5] M. R. Grossinho and F. M. Minhos Existence result for some third-order separated boundary value problems Nonlinear Anal. 47 p- p. 47-48. [6] L. Kong and Q. Kong Multi-point boundary value problems of second order differential equationsi Nonlinear Anal. 58 4 pp. 99-9. [7] L. Kong and Q. Kong Multi-point boundary value problems of second order differential equationsii Commun. Appl. Nonl Anal. 4 7 pp. 9-. [8] M. A. Krasnosel skii Positive solutions of operator equations Noordhoff Groningen Nethland 964. [9] R. Ma Positive solutions for a second order three-point boundary value problems Appl. Math. Lett. 4 pp. -5. [] Q. L. Yao The existence and multipilicity of positive solutions for a third-order three-point boundary value problem Acta Math. Appl. Sinica 9 pp. 7-. [] H. Yu H. Lv and Y. Liu Multiple positive solutions to third-order three-point singular semipositone boundary value problem Proc. Indian Acad. Sci.Math. Sci. 4 4 pp. 49-4. ISSN: 9-769 58 Issue Volume December