For the nth order nonlinear differential equation y (n)(t)=f(y(t)), t [0,1], satisfying the multipoint conjugate boundary conditions, y (j)(ai) = 0,1 i k, 0 j n i - 1, 0 =a 1 a 2 a k = 1, and i=1 k n i =n, where f: [0, ) is continuous, growth condtions are imposed on f which yield the existence of at least three solutions that belong to a cone

Davis, John M.

Eloe, Paul W.

Henderson, Johnny

English

University of Dayton

University of DaytoneCommonsMathematics Faculty Publications Department of Mathematics1999Triple positive solutions for multipoint conjugateboundary value problemsJohn M. DavisAuburn UniversityPaul W. EloeUniversity of Dayton, peloe1@udayton.eduJohnny HendersonBaylor UniversityFollow this and additional works at: https://ecommons.udayton.edu/mth_fac_pubPart of the Mathematics CommonsThis Article is brought to you for free and open access by the Department of Mathematics at eCommons. It has been accepted for inclusion inMathematics Faculty Publications by an authorized administrator of eCommons. For more information, please contact frice1@udayton.edu,mschlangen1@udayton.edu.eCommons CitationDavis, John M.; Eloe, Paul W.; and Henderson, Johnny, "Triple positive solutions for multipoint conjugate boundary value problems"(1999). Mathematics Faculty Publications. 109.https://ecommons.udayton.edu/mth_fac_pub/109GEORGIAN MATHEMATICAL JOURNAL: Vol. 6, No. 5, 1999, 415-420TRIPLE POSITIVE SOLUTIONS FOR MULTIPOINTCONJUGATE BOUNDARY VALUE PROBLEMSJOHN M. DAVIS, PAUL W. ELOE, AND JOHNNY HENDERSONAbstract. For the nth order nonlinear differential equation y(n)(t) =f(y(t)), t ∈ [0, 1], satisfying the multipoint conjugate boundary con-ditions, y(j)(ai) = 0, 1 ≤ i ≤ k, 0 ≤ j ≤ ni − 1, 0 = a1 < a2 <· · · < ak = 1, and∑ki=1 ni = n, where f : R→ [0,∞) is continuous,growth condtions are imposed on f which yield the existence of atleast three solutions that belong to a cone.1. IntroductionLet n ≥ 2 be an integer and k ∈ {2, 3, . . . , n}. Let 0 = a1 < a2 <· · · < ak = 1 be k points, ni ∈ {1, 2, . . . , n − 1}, 1 ≤ i ≤ k, be suchthat∑ki=1 ni = n and define αi =∑k−1j=i+1 nj where 1 ≤ i ≤ k. Weare concerned with the existence of multiple solutions for the nth ordermultipoint conjugate boundary value problemy(n)(t) = f(y(t)), 0 ≤ t ≤ 1, (1.1)y(j)(ai) = 0, 1 ≤ i ≤ k, 0 ≤ j ≤ ni − 1, (1.2)where f : R → [0,∞) is continuous. We will impose growth conditions onf which ensure the existence of at least three solutions of (1.1), (1.2) thatbelong to a cone.Recent attention has been directed toward obtaining multiple solutionsfor boundary value problems (BVPs) for ordinary differential equations(ODEs). While his methods are not the same as those applied here, Brykalov[1–3] has established the existence of multiple solutions for certain nonlin-ear BVPs for ODEs. Closer to this work, we refer the reader to the papersof Avery [4], Chyan and Davis [5], Chyan, Davis, and Yin [6], Davis and1991 Mathematics Subject Classification. Primary: 34B10; Secondary: 34B15.Key words and phrases. Multiple positive solutions, multipoint conjugate boundaryvalue problem, fixed points.4151072-947X/99/0900-0415$16.00/0 c© 1999 Plenum Publishing Corporation416 JOHN M. DAVIS, PAUL W. ELOE, AND JOHNNY HENDERSONHenderson [7,8], Henderson and Thompson [9], and Guo and Lakshmikan-tham [10]. This paper can be considered a complete generalization of [7]which deals with triple positive solutions for two point conjugate BVPs (i.e.,when k = 2). Multipoint problems for higher order ODEs (specifically, con-jugate problems) and the existence of multiple positive solutions have beenstudied by Eloe and Henderson in [11].For the most part, each of the papers on the existence of triple positivesolutions makes an application of the fixed point theorem by Leggett andWilliams [12] which was developed using the fixed point index in orderedBanach spaces. Leggett and Williams [12] applied their fixed point theoremto prove the existence of three positive solutions for Hammerstein integralequations of the form y(x) =∫ΩG(x, s)g(s, y(s)) ds, Ω ⊂ RN , by makinguse of suitable inequalities imposed on G and g.In Section 2, we provide some definitions and background results, andwe state the Leggett–Williams Fixed Point Theorem. Then in Section 3,we impose growth conditions on f which allow us to apply the Leggett–Williams Fixed Point Theorem in obtaining three solutions of (1.1), (1.2)that lie in a cone.2. Background and DefinitionsOur main results will hinge on an application of the Legget–WilliamsFixed Point Theorem which deals with fixed points of a cone preservingoperator. For the convenience of the reader, we include here the necessarydefinitions from cone theory in Banach spaces.Definition 2.1. Let B be a Banach space over R. A nonempty, closedset P ⊂ B is said to be acone provided(a) αu+ βv ∈ P for all u,v ∈ P and all α, β ≥ 0, and(b) u,−u ∈ P implies u = 0.Definition 2.2. A Banach space B is called apartially ordered Banachspace if there exists a partial ordering  on B satisfying(a) u  v, for boldu,v ∈ B implies tu  tv, for all t ≥ 0, and(b) u1  v1 and u2  v2, for u1,u2,v1,v2 ∈ B imply u1+u2  v1+v2.Let P ⊂ B be a cone and define u  v if and only if v − u ∈ P . Then is a partial ordering on B and we will say that  is the partial orderinginduced by P. Moreover, B is a partially ordered Banach space with respectto .We also state the following definitions for future reference.Definition 2.3. The map α is anonnegative continuous concave func-tional on P provided α : P → [0,∞) is continuous andα(tx+ (1− t)y) ≥ tα(x) + (1− t)α(y)TRIPLE POSITIVE SOLUTIONS FOR MULTIPOINT CONJUGATE BVPS 417for all x, y ∈ P and 0 ≤ t ≤ 1.Definition 2.4. Let 0 < a < b be given and α be a nonnegative con-tinuous concave functional on the cone P. Define the convex sets Pr andP(α, a, b) byPr = {y ∈ P | ‖y‖ < r} and P(α, a, b) = {y ∈ P | a ≤ α(y), ‖y‖ ≤ b}.Next we state the Leggett–Williams Fixed Point Theorem. The proofcan be found in Deimling’s text [13] and utilizes the fixed point index inordered Banach spaces.Theorem 2.1 (Leggett–Williams Fixed Point Theorem). Let A :Pc → Pc be a completely continuous operator and let α be a nonnegativecontinuous concave functional on P such that α(y) ≤ ‖y‖ for all y ∈ Pc.Suppose there exist 0 < a < b < d ≤ c such that(C1) {y ∈ P(α, b, d) | α(y) > b} 6= ∅ and α(Ay) > b for y ∈ P(α, b, d),(C2) ‖Ay‖ < a for ‖y‖ ≤ a, and(C3) α(Ay) > b for y ∈ P(α, b, c) with ‖Ay‖ > d.Then A has at least three fixed points y1, y2, and y3 such that ‖y1‖ < a,b < α(y2), and ‖y3‖ > a with α(y3) < b.3. Triple Positive SolutionsIn this section, we will impose growth conditions on f which allow us toapply Theorem 2.1 in regard to obtaining three solutions of (1.1), (1.2). Wewill apply Theorem 2.1 to a completely continuous operator whose kernelis the Green’s function, G(t, s), fory(n)(t) = 0, (3.1)satisfying the boundary conditions (1.2). It is fairly well known [14] that(−1)αiG(t, s) > 0, for (t, s) ∈ (ai, ai+1)× (0, 1), 1 ≤ i ≤ k − 1. (3.2)For s ∈ (0, 1), define‖G(·, s)‖ = maxt∈[0,1] |G(t, s)|. (3.3)Eloe and Henderson [15] proved the following theorem which is a key esti-mate for our main result.Theorem 3.1. Suppose y ∈ C(n)[0, 1] is such that y(n)(t) ≥ 0, t ∈ [0, 1],and that y satisfies the multipoint conjugate boundary conditions (1.2).Then, for each 1 ≤ i ≤ k − 1,(−1)αiG(t, s) ≥ (a/4)m‖G(·, s)‖, t ∈ Si, s ∈ (0, 1), (3.4)418 JOHN M. DAVIS, PAUL W. ELOE, AND JOHNNY HENDERSONwhere Si = [3ai+ai+14 ,ai+3ai+14 ], a = min1≤i≤k−1{ai+1 − ai}, and m = max{n−n1, n− nk}.Next, we defineK =(maxt∈[0,1]∫ 10G(t, s) ds)−1, (3.5)L =(min1≤i≤k−1mint∈Si∫SiG(t, s) ds)−1. (3.6)Let B denote the Banach space C[0, 1] with the maximum norm ‖y‖ =maxt∈[0,1] |y(t)| and define the cone P ⊂ B byP := {y ∈ B | (−1)αiy(t) ≥ 0, t ∈ [ai, ai+1] for 1 ≤ i ≤ k − 1} .Let α : P → [0,∞) be the nonnegative continuous concave functionalα(y) = min1≤i≤k−1mint∈Si|y(t)|, for y ∈ P,and let A : B → B be the operatorAy(t) =∫ 10G(t, s)f(y(s)) ds.We now present the main result of the paper.Theorem 3.2. Let 0 < a < b <( 4a)mb ≤ c be such that f satisfies(i) f(w) < Ka, for 0 ≤ |w| ≤ a,(ii) f(w) ≥ Lb, for b ≤ |w| ≤ ( 4a)m b, and(iii) f(w) ≤ Kc, for 0 ≤ |w| ≤ c.Then, the boundary value problem (1.1), (1.2) has three positive solutionsy1, y2, and y3 satisfying‖y1‖ < a, b < min1≤i≤k−1mint∈Si|y2(t)|, ‖y3‖ > a with min1≤i≤k−1mint∈Si|y3(t)| < b.Proof. We seek fixed points of A which satisfy the conclusion of the theo-rem. We observe first from the positivity of f and (3.2) that, for y ∈ P,(−1)αiAy(t) ≥ 0 for t ∈ [ai, ai+1]. Thus, A : P → P .We now show that the conditions of Theorem 2.1 are satisfied. Choosey ∈ Pc. Then, ‖y‖ ≤ c and by assumption (iii), f(y(s)) ≤ Kc, s ∈ [0, 1].Thus, from (3.5)‖Ay‖ = maxt∈[0,1]∫ 10G(t, s)f(y(s)) ds ≤ maxt∈[0,1]∫ 10|G(t, s)|f(y(s)) ds≤ maxt∈[0,1]∫ 10|G(t, s)|Kcds = c.TRIPLE POSITIVE SOLUTIONS FOR MULTIPOINT CONJUGATE BVPS 419Hence, A : Pc → Pc. In a similar way, if y ∈ Pa, then assumption (i)yields f(y(s)) < Ka, s ∈ [0, 1], and it follows as above that A : Pa → Pa.Consequently, condition (C2) of Theorem 2.1 is fulfilled.To verify property (C1) of Theorem 2.1, we note that if we letx(t) ={(−1)αi(4/a)mb, t ∈ Si, 1 ≤ i ≤ k − 1,x(ai+1)−x(ai)ai+1−ai (t− ai) + x(ai), t ∈ [0, 1] \ Si, 1 ≤ i ≤ k − 1,then x(t) ∈ P(α, b, ( 4a)m b). Moreover, α(x) = ( 4a)m b > b. Hence{y ∈ P(α, b, (4/a)mb) |α(y)>b} 6=∅.Furthermore, if we choose y∈P(α, b, ( 4a)m b), thenα(y) = min1≤i≤k−1mint∈Si|y(t)| ≥ b,and so b ≤ |y(s)| ≤ ( 4a)m b, s ∈ Si, 1 ≤ i ≤ k − 1. Thus, for any y ∈P(α, b, ( 4a)m b), assumption (ii) yields f(y(s)) ≥ Lb, s ∈ Si, 1 ≤ i ≤ k − 1,and by (3.6) we haveα(Ay) = min1≤i≤k−1mint∈Si|Ay| == min1≤i≤k−1mint∈Si∫ 10(−1)αiG(t, s)f(y(s)) ds >> min1≤i≤k−1mint∈Si∫Si(−1)αiG(t, s)f(y(s)) ds ≥≥ min1≤i≤k−1mint∈Si∫Si(−1)αiG(t, s)Lb ds = b.Hence, condition (C1) of Theorem 2.1 is satisfied.We finally exhibit that (C3) of Theorem 2.1 is satisfied. (In particular,we show, if y ∈ P(α, b, c) and ‖Ay‖ > ( 4a)m b, then α(Ay) > b.) Thus wechoose y ∈ P(α, b, c) such that ‖Ay‖ > ( 4a)m b. Then, from (3.4),α(Ay) = min1≤i≤k−1mint∈Si∫ 10(−1)αiG(t, s)f(y(s)) ds ≥≥(a4)m ∫ 10‖G(·, s)‖f(y(s)) ds ≥≥(a4)mmaxt∈[0,1]∫ 10|G(t, s)|f(y(s)) ds ==(a4)m ‖Ay‖ > b,and (C3) of Theorem 2.1 is satisfied. Hence an application of Theorem 2.1completes the proof.420 JOHN M. DAVIS, PAUL W. ELOE, AND JOHNNY HENDERSONReferences1. S. A. Brykalov, Solvability of a nonlinear boundary value problem ina fixed set of functions. (Russian) Differentsial’nye Uravneniya 27(1991),2027–2933.2. S. A. Brykalov, A second-order nonlinear problem with two-point andintegral boundary conditions. Georgian Math. J. 1(1994), No. 3, 243–249.3. S. A. Brykalov, Solutions with a given maximum and minimum. (Rus-sian) Differentsial’nye Uravneniya 29(1993), 938–942.4. R. Avery, Existence of multiple positive solutions to a conjugateboundary value problem. MSR Hotline, 2(1998), No. 1.5. C. J. Chyan and J. M. Davis, Existence of triple positive solutions for(n, p) and (p, n) boundary value problems. Preprint.6. C. J. Chyan, J. M. Davis, andW. K. C. Yin, Existence of triple positivesolutions for (k, n− k) right focal boundary value problems. Preprint.7. J. M. Davis and J. Henderson, Triple positive solutions for (k, n− k)conjugate boundary value problems. Preprint.8. J. M. Davis and J. Henderson, Triple positive symmetric solutions fora Lidstone boundary value problem. Preprint.9. J. Henderson and H.B. Thompson, Existence of multiple solutions forsome nth order boundary value problems. Preprint.10. D. Guo and V. Lakshmikantham, Nonlinear problems in abstractcones. Academic Press, San Diego, 1988.11. P. W. Eloe and J. Henderson, Positive solutions and nonlinear mul-tipoint conjugate eigenvalue problems. Electron. J. Differential Equations,(1997), No. 3, 1–11.12. R. Leggett and L. Williams, Multiple positive fixed points of nonlin-ear operators on ordered Banach spaces. Indiana Univ. Math. J. 28(1979),673–688.13. K. Deimling, Nonlinear functional analysis. Springer-Verlag, NewYork, 1985.14. W. Coppel, Disconjugacy. Lecture Notes in Mathematics 220, Sprin-ger-Verlag, Berlin and New York, 1971.15. P. W. Eloe and J. Henderson, Inequalities for solutions of multipointboundary value problems. Rocky Mountain J. Math. (in press).(Received 14.12.1998)Authors’ addresses:John M. Davis, Johnny HendersonDepartment of MathematicsAuburn UniversityAuburn, AL 36849 USAPaul W. EloeDepartment of MathematicsUniversity of DaytonDayton, OH 45469 USAView publication stats

Triple positive solutions for multipoint conjugate boundary value problems

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Triple positive solutions for multipoint conjugate boundary value problems

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