In this paper, we consider the Lidstone boundary value problem $y^{(2m)}(t) = \lambda a(t)f(y(t), \dots, y^{(2j)}(t), \dots y^{(2(m-1))}(t), 0  0$ and $a$ is nonnegative. Growth conditions are imposed on $f$ and inequalities involving an associated Green's function are employed which enable us to apply a well-known cone theoretic fixed point theorem. This in turn yields a $\lambda$ interval on which there exists a nontrivial solution in a cone for each $\lambda$ in that interval. The methods of the paper are known. The emphasis here is that $f$ depends upon higher order derivatives. Applications are made to problems that exhibit superlinear or sublinear type growth

Eloe, Paul

Electronic journal of qualitative theory of differential equations

Eloe Paul W.

University of Szeged

NONLINEAR EIGENVALUE PROBLEMS FOR HIGHERORDER LIDSTONE BOUNDARY VALUE PROBLEMSPAUL W. ELOEAbstract. In this paper, we consider the Lidstone boundary valueproblem y(2m)(t) = λa(t)f(y(t), . . . , y(2j)(t), . . . y(2(m−1))(t)), 0 < t < 1,y(2i)(0) = 0 = y(2i)(1), i = 0, . . . , m − 1, where (−1)mf > 0 and ais nonnegative. Growth conditions are imposed on f and inequalitiesinvolving an associated Green’s function are employed which enable usto apply a well-known cone theoretic fixed point theorem. This in turnyields a λ interval on which there exists a nontrivial solution in a conefor each λ in that interval. The methods of the paper are known. Theemphasis here is that f depends upon higher order derivatives. Appli-cations are made to problems that exhibit superlinear or sublinear typegrowth.1. IntroductionWe consider the nonlinear Lidstone boundary value problem (BVP),y(2m)(t) = λa(t)f(y(t), . . . , y(2j)(t), . . . y(2(m−1))(t)), 0 < t < 1, (1.1)y(2i)(0) = 0 = y(2i)(1), i = 0, . . . m − 1, (1.2)where (−1)mf > 0 is continuous and a is nonnegative. For more preciseconditions on f and a, let (−1)j [a, b] = [a, b] if j is even and (−1)j [a, b] =[−b,−a] if j is odd. Letm−1∏j=0[aj , bj ] = [a0, b0] × · · · × [am−1, bm−1].We shall require that(A): (−1)mf :∏m−1j=0 (−1)j [0,∞) → [0,∞) is continuous,(B): a : [0, 1] → [0,∞) is continuous and does not vanish identically onany subinterval.This work is primarily motivated by the original work of Erbe and Wang[15] for m = 1. To our knowledge, Erbe and Wang [15] are the first to applythe methods employed here to the cases that f is superlinear or sublinear. Aflurry of extensions to nth order problems have been obtained in recent years1991 Mathematics Subject Classification. Primary: 34B10; Secondary: 34B15.Key words and phrases. nonlinear eigenvalue problem, Lidstone boundary value prob-lem, positive solutions, fixed points, Green’s function.EJQTDE, 2000 No. 2, p. 1(for example, see [1], [2], [22], [23], [11], [12]). Henderson and Wang [17] werethe first to introduce the problem as a nonlinear eigenvalue problem.In all the above cited works, the nonlinear term, f , only depends onposition. The primary interest of this paper is that the nonlinear term, f ,depends on position, acceleration and other even order derivatives of theunknown function. Recent related works in which dependence on higherorder derivatives is allowed can be found in [4] or [9].The Lidstone boundary value problem (BVP) was first studied by Lid-stone [21]; Agarwal and Wong’s work [3] has generated renewed interest inthe problem. Recently, Davis, Henderson, and Lamar ([6], [7], [10], [19])have studied the problem intensely. A feature of the Lidstone BVP thatis exploited in this paper is that it can be analyzed as a nested family ofsecond order conjugate BVPs. This feature has been employed by Davis,Eloe, Henderson, Islam and Thompson ([14], [8], and [13]). The primarycontribution of this paper is that this nested feature is exploited so thatthe methods employed by Erbe and Wang [15] can be be applyed to theBVP, (1.1), (1.2). Moreover, we indicate that the contribution is of interestby exhibiting applications to problems that exhibit superlinear or sublineartype growth.We close the introduction with one open question. Can the methods em-ployed here apply to a Lidstone BVP with nonlinear dependence on oddorder derivatives of the unknown function? That question is completelyopen. The problem is that large in norm does not imply large component-wise; by exploiting the nested feature of Lidstone BVPs in this paper, largein norm will, in fact, imply large in the appropriate components.2. The Fixed Point OperatorThe method developed by Erbe and Wang [15] employs an application ofthe cone theoretic fixed point theorem that we credit to Krasnosel’skii [18].Also see [16]. For simplicity we state the theorem here.Theorem 2.1. Let B be a Banach space, and let P ⊂ B be a cone in B.Assume Ω1,Ω2 are open subsets of B with 0 ∈ Ω1 ⊂ Ω1 ⊂ Ω2, and letT : P ∪ (Ω2 \ Ω1) → Pbe a completely continuous operator such that, either(i): ||T u|| ≤ ||u||, u ∈ P ∪ ∂Ω1, and ||T u|| ≥ ||u||, u ∈ P ∩ ∂Ω2, or(ii): ||T u|| ≥ ||u||, u ∈ P ∩ ∂Ω1, and ||T u|| ≤ ||u||, u ∈ P ∩ ∂Ω2.Then T has a fixed point in P ∩ (Ω2 \ Ω1).We now construct the fixed point operator upon which we apply the abovefixed point theorem. To do so, we exploit that the Lidstone BVP, (1.1),(1.2), can be constructed as a nested sequence of second order conjugateEJQTDE, 2000 No. 2, p. 2type BVPs. In particular, we shall construct a second order BVP that isequivalent to (1.1), (1.2). In Section 3 we shall apply the above fixed pointtheorem to the equivalent second order BVP.The Green’s function for y′′(t) = 0, 0 < t < 1, y(0) = y(1) = 0, isG(t, s) ={t(s − 1), 0 ≤ t ≤ s ≤ 1,s(t − 1), 0 ≤ s ≤ t ≤ 1.Set G1(t, s) := G(t, s), and for j = 2, . . . m, define recursivelyGj(t, s) =∫ 10G(t, r)Gj−1(r, s) dr.As a result, Gj(t, s) is the Green’s function for the BVP,y(2j)(t) = 0, 0 < t < 1,y(2i)(0) = 0 = y(2i)(1), i = 0, . . . , j − 1,for each j = 1 . . . m. One can verify this directly ([5], page 192) or see [13]or [19, 20].For each j = 1, . . . ,m − 1, define Aj : C[0, 1] → C[0, 1] byAjv(t) =∫ 10Gj(t, s)v(s)ds.By the construction of Aj it follows that(Ajv)(2j)(t) = v(t), 0 < t < 1,(Ajv)(2i)(0) = (Ajv)(2i)(1), i = 0, . . . , j − 1.Thus, it follows that the BVP, (1.1), (1.2), has a solution if, and only if, theBVP,v′′(t) = f(Am−1v(t), . . . , A1v(t), v(t)), 0 < t < 1,v(0) = v(1) = 0,has a solution. If y is a solution of the BVP, (1.1), (1.2), then v = y(2(m−1))is a solution of the second order BVP; conversely, if v is a solution of thesecond order BVP, then y = Am−1v is a solution of the BVP, (1.1), (1.2).Define T : C[0, 1] → C[0, 1] byT v(t) = λ∫ 10a(s)G(t, s)f(Am−1v(s), . . . , A1v(s), v(s))ds.The properties of each Gj readily imply that T : C[0, 1] → C[0, 1] is com-pletely continuous. It now follows that there exists a solution of the BVP,(1.1), (1.2), if, and only if, there exists a continuous fixed point of T . More-over, the relation between solutions, y, of the BVP, (1.1), (1.2), and fixedpoints, v, of T , is given by y(t) = Am−1v(t), or y(2(m−1))(t) = v(t).Note that G1 < 0 on (0, 1) × (0, 1), and (−1)jGj > 0 on (0, 1) × (0, 1).Thus, y is a positive solution of the BVP, (1.1), (1.2), if, and only if,EJQTDE, 2000 No. 2, p. 3(−1)m−1y(2(m−1)) = (−1)m−1v is positive, where v is the correspondingcontinuous fixed point of T . In the next section then, we restrict our anal-ysis to finding λ intervals upon which T generates a nontrivial fixed point,v, such that (−1)m−1v(t) > 0, 0 < t < 1.3. Existence of Positive SolutionsWe remind the reader of two fundamental bounds involving the Green’sfunction, G1.0 < −(G1(t, s)) ≤ s(1 − s), 0 < t, s < 1,−(G1(t, s)) ≥ (1/4)s(1 − s), (1/4) ≤ t ≤ (3/4), 0 ≤ s ≤ 1.From here, four estimates which we shall employ are readily obtained.max0≤t≤1∫ 10|G1(t, s)|ds ≤ (1/8), (3.1)min1/4≤t≤3/4∫ 3/41/4|G1(t, s)|ds ≥ (1/16). (3.2)If v ∈ C[0, 1], then||Ajv|| ≤ ||v||/8j , j = 1, . . . ,m − 1. (3.3)(|| · || denotes the usual supremum norm on [0, 1].) If (−1)(m−1)v(t) > 0, 0 <t < 1, and if (−1)(m−1)v(t) > ||v||/4 for 1/4 ≤ t ≤ 3/4, thenmin1/4≤t≤3/4|Ajv(t)| ≥ ||v||/(4(16)j ), j = 1, . . . ,m − 1. (3.4)(3.3) and (3.4) are readily obtained from (3.1) and (3.2). (3.3) and (3.4)motivate conditions (C1) and (C2) given below.(C1): There exist k0j ≥ (1/8)j−1, j = m, . . . , 2, such thatlim(−1)m−1v→0+(−1)mf((−1)m−1k0mv, . . . ,−k02v, v)/(−1)m−1v = f0,and there exist 0 < k∞j ≤ (1/16)j−1, j = m, . . . , 2, such thatlim(−1)m−1v→∞(−1)mf((−1)m−1k∞mv, . . . ,−k∞2v, v)/(−1)m−1v = f∞.(C2): There exist 0 < k0j ≤ (1/16)j−1, j = m, . . . , 2, such thatlim(−1)m−1v→0+(−1)mf((−1)m−1k0mv, . . . ,−k02v, v)/(−1)m−1v = f0,and there exist k∞j ≥ (1/8)j−1, j = m, . . . , 2, such thatlim(−1)m−1v→∞(−1)mf((−1)m−1k∞mv, . . . ,−k∞2v, v)/(−1)m−1v = f∞.(D): (−1)mf(u0, u1, . . . , um−1) is increasing in each u2j and decreasingin each u2j+1 for (u0, u1, . . . , um−1) ∈∏m−1j=0 (−1)j [0,∞).EJQTDE, 2000 No. 2, p. 4We shall now state and prove a typical result as an application of Theorem2.1. To apply Theorem 2.1 let B denote the Banach space C[0, 1] with thesupremum norm||v|| = max0≤t≤1|v(t)|and define the cone P ⊂ B by P :={v ∈ B : (−1)m−1v(t) ≥ 0, 0 ≤ t ≤ 1, min1/4≤t≤3/4(−1)m−1v(t) ≥ (1/4)||v||}.Theorem 3.1. Assume that conditions (A), (B), (C1), and (D) are satis-fied. Then, for each λ satisfying4/(∫ 3/41/4−G(1/2, s)a(s)dsf∞) < λ < 1/(∫ 10s(1 − s)a(s)dsf0), (3.5)there is at least one nontrivial solution, y, of the BVP, (1.1), (1.2), suchthat v = y(2(m−1)) belongs to P.Proof. Let λ satisfy (3.5) and let ε > 0 be such that4/(∫ 3/41/4−G(1/2, s)a(s)ds(f∞ − ε)) ≤ λ ≤ 1/(∫ 10s(1 − s)a(s)ds(f0 + ε)).Define T : P → B byT v(t) = λ∫ 10a(s)G(t, s)f(Am−1v(s), . . . , A1v(s), v(s))ds.To see that T : P → P apply Conditions (A) and (B) and recall that Gis negative on (0, 1) × (0, 1). Moreover, (−1)m(T v)′′(t) = a(t)(−1)mf ≥ 0,0 < t < 1, by Conditions (A) and (B), and T v(0) = T v(1) = 0; in particular,due to concavity,min1/4≤t≤3/4(−1)m−1T v(t) ≥ ||T v||/4.We now construct the domains, Ω1 and Ω2 in order to apply Theorem 2.1(i). Apply Condition (C1) and find H1 > 0 such that(−1)mf((−1)m−1k0mv, . . . ,−k02v, v) ≤ (f0 + ε)(−1)m−1v,for all 0 < (−1)m−1v ≤ H1. Let v ∈ P with ||v|| = H1. Apply (3.3) andCondition (D) to see that(−1)mf(Am−1v, . . . , v) ≤ (−1)mf(||v||/8m−1, . . . , (−1)m−1||v||)≤ (−1)mf(k0,m||v||, . . . , (−1)m−1||v||).Thus,|T v(t)| ≤ λ∫ 10s(1 − s)a(s)(−1)mf(Am−1v(s), . . . , v(s))ds≤ λ∫ 10s(1 − s)a(s)ds(f0 + ε)||v|| ≤ ||v||.EJQTDE, 2000 No. 2, p. 5DefineΩ1 = {x ∈ B : ||x|| < H1},and we have shown that||T v|| ≤ ||v||, v ∈ P ∩ ∂Ω1.To find Ω2, find H > 0 such that(−1)mf((−1)m−1k∞mv, . . . ,−k∞2v, v) ≥ (f∞ − ε)(−1)m−1v,for all 0 < (−1)m−1v ≥ H. Let H2 = max{2H1, 4H} and setΩ2 = {x ∈ B : ||x|| < H2}.Let v ∈ P, ||v|| = H2. Thenmin1/4≤t≤3/4(−1)m−1v(t) ≥ ||v||/4 ≥ H.Apply (3.4) and Condition (D), for s ∈ [1/4, 3/4], to obtain(−1)mf(Am−1v(s), . . . , v(s)) ≥ (−1)mf(||v||/4(16)m−1 , . . . , (−1)m−1||v||/4)≥ (−1)mf(k∞,m||v||/4, . . . , (−1)m−1||v||/4).Thus,|T v(1/2)| ≥ λ∫ 3/41/4−G(1/2, s)a(s)(−1)mf(Am−1v(s), . . . , v(s))ds≥ λ∫ 3/41/4−G(1/2, s)a(s)ds(f∞ − ε)(||v||/4) ≥ ||v||.In particular, defineΩ2 = {x ∈ B : ||x|| < H2},and||T v|| ≥ ||v||, v ∈ P ∩ ∂Ω2.This completes the proof of Theorem 3.1.We remark that if f is superlinear (i.e., f0 = 0, f∞ = ∞) then the proofof Theorem 3.1 is readily adapted to show that the BVP, (1.1), (1.2), hasa nontrivial solution, y, such that v = y(2(m−1)) belongs to P, for each0 < λ < ∞. To illustrate that this observation is of interest, set m = 2 andconsider the fourth order Lidstone BVP that relates to the cantilever beamproblem. Note that each off1(u, v) = u2 + v2, f2(u, v) = −uvsatisfy conditions (A), (C1) and (D).We will state without proof a second application of Theorem 2.1. Theproof, when f depends only on position is standard (see [15]) and the exten-sion to the problem addressed here is completely analogous to the extensionillustrated in the proof of Theorem 3.1.EJQTDE, 2000 No. 2, p. 6Theorem 3.2. Assume that conditions (A), (B), (C2), and (D) are satis-fied. Then, for each λ satisfying4/(∫ 3/41/4−G(1/2, s)a(s)dsf0) < λ < 1/(∫ 10s(1 − s)a(s)dsf∞),there is at least one nontrivial solution, y, of the BVP, (1.1), (1.2), suchthat v = y(2(m−1)) belongs to P.In the case that f is sublinear (i.e., f0 = ∞, f∞ = 0) the proof of Theorem3.2 is readily adapted to show that the BVP, (1.1), (1.2), has a nontrivialsolution, y, such that v = y(2(m−1)) belongs to P, for each 0 < λ < ∞. Toillustrate that this observation is of interest, again set m = 2 and note thateach off3(u, v) = u2/3 + v2/3, f4(u, v) = −(uv)1/3satisfy conditions (A), (C2) and (D).References[1] R.P. Agarwal and J. Henderson, Superlinear and sublinear focal boundary value prob-lems, Applic. Anal. 60 (1996), 189-200.[2] R.P. Agarwal, J. Henderson and P.J.Y. Wong, On superlinear and sublinear (n, p)boundary value problems for higher order difference equations, Nonlinear World 4(1997), 101-116.[3] R.P. Agarwal and P.J.Y. Wong, Lidstone polynomials and boundary value problems,Comput. Math. Appl. 17 (1989), 1397–1421.[4] R. I. Avery, J. Davis and J. Henderson, Three symmetric positive solutions for Lid-stone problems, preprint.[5] E. Coddington and N. Levinson, “Theory of Ordinary Differential Equations,”McGraw-Hill, New York, 1955.[6] J.M. Davis, Differentiability of solutions of differential equations with respect to non-linear Lidstone boundary conditions, Nonlinear Differential Equations, in press.[7] J.M. Davis, Differentiation of solutions of Lidstone boundary value problems withrespect to the boundary data, Math. Comput. Modelling, in press.[8] J.M. Davis, P.W. Eloe, and J. Henderson, Comparison of eigenvalues for discreteLidstone boundary value problems, Dynam. Systems Appl., 8 (1999), no. 3-4, 381-388.[9] J.M. Davis, P.W. Eloe, and J. Henderson, Triple positive solutions and dependenceon higher order derivatives, JMAA 237 (1999), 710-720.[10] J.M. Davis and J. Henderson, Uniqueness implies existence for fourth order Lidstoneboundary value problems, Panamer. Math. J. 8 (1998), 23-35.[11] P.W. Eloe and J. Henderson, Positive solutions for (n − 1, 1) conjugate boundaryvalue problems, Nonlinear Anal. 28 (1997), 1669-1680.[12] P.W. Eloe and J. Henderson, Positive solutions and nonlinear (k, n − k) conjugateeigenvalue problems, Differential Equations and Dynamical Systems 6 (1998), 309-317.[13] P.W. Eloe, J. Henderson and B. Thompson, Extremal points for impulsive Lidstoneboundary value problems, Mathematical and Computational Modeling, in press.[14] P.W. Eloe and M.N. Islam, Lidstone boundary value problems for ordinary differentialequations with impulse effects, Comm. Appl. Anal., in press.EJQTDE, 2000 No. 2, p. 7[15] L.H. Erbe and H. Wang, On the existence of positive solutions of ordinary differentialequations, Proc. Amer. Math. Soc. 120 (1994), 743-748.[16] D. Guo and V. Lakshmikantham, “Nonlinear Problems in Abstract Cones,” AcademicPress, San Diego, 1988.[17] J. Henderson and H. Wang, Positive solutions for a nonlinear eigenvalue problem, J.Math. Anal. Appl. 288 (1997), 252-259.[18] M.A. Krasnosel’skii, ”Positive Solutions of Operator Equations,” Noordhoff, Gronin-gen, 1964.[19] T.M. Lamar, Analysis of a 2nth Order Differential Equation with Lidstone BoundaryConditions, Ph.D. Dissertation, Auburn University, 1997.[20] T.M. Lamar, Comparison theory for eigenvalue problems with Lidstone boundaryconditions, preprint.[21] G.J. Lidstone, Notes on the extension of Aitken’s theorem (for polynomial interpola-tion) to the Everett types, Proc. Edinburgh Math. Soc. 2 (1929), 16-19.[22] P.J.Y. Wong and R.P. Agarwal, Multiple solutions of difference equations with Lid-stone conditions, preprint.[23] P.J.Y. Wong and R.P. Agarwal, Results and estimates on multiple solutions of Lid-stone boundary value problems, Acta Math. Hungar. 86 (2000), no.1-2, 137-168.Department of Mathematics, University of Dayton, Dayton, OH 45469-2316USAE-mail address: eloe@saber.udayton.eduEJQTDE, 2000 No. 2, p. 8

Nonlinear eigenvalue problems for higher order Lidstone boundary value

problems

Paul Eloe

Crossref

Nonlinear eigenvalue problems for higher order Lidstone boundary value problems

Repository of the Academy's Library

NONLINEAR EIGENVALUE PROBLEMS FOR HIGHERORDER LIDSTONE BOUNDARY VALUE PROBLEMSPAUL W. ELOEAbstract. In this paper, we consider the Lidstone boundary valueproblem y(2m)(t) = λa(t)f(y(t), . . . , y(2j)(t), . . . y(2(m−1))(t)), 0 < t < 1,y(2i)(0) = 0 = y(2i)(1), i = 0, . . . , m − 1, where (−1)mf > 0 and ais nonnegative. Growth conditions are imposed on f and inequalitiesinvolving an associated Green’s function are employed which enable usto apply a well-known cone theoretic fixed point theorem. This in turnyields a λ interval on which there exists a nontrivial solution in a conefor each λ in that interval. The methods of the paper are known. Theemphasis here is that f depends upon higher order derivatives. Appli-cations are made to problems that exhibit superlinear or sublinear typegrowth.1. IntroductionWe consider the nonlinear Lidstone boundary value problem (BVP),y(2m)(t) = λa(t)f(y(t), . . . , y(2j)(t), . . . y(2(m−1))(t)), 0 < t < 1, (1.1)y(2i)(0) = 0 = y(2i)(1), i = 0, . . . m− 1, (1.2)where (−1)mf > 0 is continuous and a is nonnegative. For more preciseconditions on f and a, let (−1)j [a, b] = [a, b] if j is even and (−1)j [a, b] =[−b,−a] if j is odd. Letm−1∏j=0[aj , bj ] = [a0, b0]× · · · × [am−1, bm−1].We shall require that(A): (−1)mf :∏m−1j=0 (−1)j [0,∞) → [0,∞) is continuous,(B): a : [0, 1] → [0,∞) is continuous and does not vanish identically onany subinterval.This work is primarily motivated by the original work of Erbe and Wang[15] for m = 1. To our knowledge, Erbe and Wang [15] are the first to applythe methods employed here to the cases that f is superlinear or sublinear. Aflurry of extensions to nth order problems have been obtained in recent years1991 Mathematics Subject Classification. Primary: 34B10; Secondary: 34B15.Key words and phrases. nonlinear eigenvalue problem, Lidstone boundary value prob-lem, positive solutions, fixed points, Green’s function.EJQTDE, 2000 No. 2, p. 1(for example, see [1], [2], [22], [23], [11], [12]). Henderson and Wang [17] werethe first to introduce the problem as a nonlinear eigenvalue problem.In all the above cited works, the nonlinear term, f , only depends onposition. The primary interest of this paper is that the nonlinear term, f ,depends on position, acceleration and other even order derivatives of theunknown function. Recent related works in which dependence on higherorder derivatives is allowed can be found in [4] or [9].The Lidstone boundary value problem (BVP) was first studied by Lid-stone [21]; Agarwal and Wong’s work [3] has generated renewed interest inthe problem. Recently, Davis, Henderson, and Lamar ([6], [7], [10], [19])have studied the problem intensely. A feature of the Lidstone BVP thatis exploited in this paper is that it can be analyzed as a nested family ofsecond order conjugate BVPs. This feature has been employed by Davis,Eloe, Henderson, Islam and Thompson ([14], [8], and [13]). The primarycontribution of this paper is that this nested feature is exploited so thatthe methods employed by Erbe and Wang [15] can be be applyed to theBVP, (1.1), (1.2). Moreover, we indicate that the contribution is of interestby exhibiting applications to problems that exhibit superlinear or sublineartype growth.We close the introduction with one open question. Can the methods em-ployed here apply to a Lidstone BVP with nonlinear dependence on oddorder derivatives of the unknown function? That question is completelyopen. The problem is that large in norm does not imply large component-wise; by exploiting the nested feature of Lidstone BVPs in this paper, largein norm will, in fact, imply large in the appropriate components.2. The Fixed Point OperatorThe method developed by Erbe and Wang [15] employs an application ofthe cone theoretic fixed point theorem that we credit to Krasnosel’skii [18].Also see [16]. For simplicity we state the theorem here.Theorem 2.1. Let B be a Banach space, and let P ⊂ B be a cone in B.Assume Ω1,Ω2 are open subsets of B with 0 ∈ Ω1 ⊂ Ω1 ⊂ Ω2, and letT : P ∪ (Ω2 \ Ω1) → Pbe a completely continuous operator such that, either(i): ||T u|| ≤ ||u||, u ∈ P ∪ ∂Ω1, and ||T u|| ≥ ||u||, u ∈ P ∩ ∂Ω2, or(ii): ||T u|| ≥ ||u||, u ∈ P ∩ ∂Ω1, and ||T u|| ≤ ||u||, u ∈ P ∩ ∂Ω2.Then T has a fixed point in P ∩ (Ω2 \ Ω1).We now construct the fixed point operator upon which we apply the abovefixed point theorem. To do so, we exploit that the Lidstone BVP, (1.1),(1.2), can be constructed as a nested sequence of second order conjugateEJQTDE, 2000 No. 2, p. 2type BVPs. In particular, we shall construct a second order BVP that isequivalent to (1.1), (1.2). In Section 3 we shall apply the above fixed pointtheorem to the equivalent second order BVP.The Green’s function for y′′(t) = 0, 0 < t < 1, y(0) = y(1) = 0, isG(t, s) ={t(s− 1), 0 ≤ t ≤ s ≤ 1,s(t− 1), 0 ≤ s ≤ t ≤ 1.Set G1(t, s) := G(t, s), and for j = 2, . . . m, define recursivelyGj(t, s) =∫ 10G(t, r)Gj−1(r, s) dr.As a result, Gj(t, s) is the Green’s function for the BVP,y(2j)(t) = 0, 0 < t < 1,y(2i)(0) = 0 = y(2i)(1), i = 0, . . . , j − 1,for each j = 1 . . . m. One can verify this directly ([5], page 192) or see [13]or [19, 20].For each j = 1, . . . ,m− 1, define Aj : C[0, 1] → C[0, 1] byAjv(t) =∫ 10Gj(t, s)v(s)ds.By the construction of Aj it follows that(Ajv)(2j)(t) = v(t), 0 < t < 1,(Ajv)(2i)(0) = (Ajv)(2i)(1), i = 0, . . . , j − 1.Thus, it follows that the BVP, (1.1), (1.2), has a solution if, and only if, theBVP,v′′(t) = f(Am−1v(t), . . . , A1v(t), v(t)), 0 < t < 1,v(0) = v(1) = 0,has a solution. If y is a solution of the BVP, (1.1), (1.2), then v = y(2(m−1))is a solution of the second order BVP; conversely, if v is a solution of thesecond order BVP, then y = Am−1v is a solution of the BVP, (1.1), (1.2).Define T : C[0, 1] → C[0, 1] byT v(t) = λ∫ 10a(s)G(t, s)f(Am−1v(s), . . . , A1v(s), v(s))ds.The properties of each Gj readily imply that T : C[0, 1] → C[0, 1] is com-pletely continuous. It now follows that there exists a solution of the BVP,(1.1), (1.2), if, and only if, there exists a continuous fixed point of T . More-over, the relation between solutions, y, of the BVP, (1.1), (1.2), and fixedpoints, v, of T , is given by y(t) = Am−1v(t), or y(2(m−1))(t) = v(t).Note that G1 < 0 on (0, 1) × (0, 1), and (−1)jGj > 0 on (0, 1) × (0, 1).Thus, y is a positive solution of the BVP, (1.1), (1.2), if, and only if,EJQTDE, 2000 No. 2, p. 3(−1)m−1y(2(m−1)) = (−1)m−1v is positive, where v is the correspondingcontinuous fixed point of T . In the next section then, we restrict our anal-ysis to finding λ intervals upon which T generates a nontrivial fixed point,v, such that (−1)m−1v(t) > 0, 0 < t < 1.3. Existence of Positive SolutionsWe remind the reader of two fundamental bounds involving the Green’sfunction, G1.0 < −(G1(t, s)) ≤ s(1− s), 0 < t, s < 1,−(G1(t, s)) ≥ (1/4)s(1 − s), (1/4) ≤ t ≤ (3/4), 0 ≤ s ≤ 1.From here, four estimates which we shall employ are readily obtained.max0≤t≤1∫ 10|G1(t, s)|ds ≤ (1/8), (3.1)min1/4≤t≤3/4∫ 3/41/4|G1(t, s)|ds ≥ (1/16). (3.2)If v ∈ C[0, 1], then||Ajv|| ≤ ||v||/8j , j = 1, . . . ,m− 1. (3.3)(|| · || denotes the usual supremum norm on [0, 1].) If (−1)(m−1)v(t) > 0, 0 <t < 1, and if (−1)(m−1)v(t) > ||v||/4 for 1/4 ≤ t ≤ 3/4, thenmin1/4≤t≤3/4|Ajv(t)| ≥ ||v||/(4(16)j ), j = 1, . . . ,m− 1. (3.4)(3.3) and (3.4) are readily obtained from (3.1) and (3.2). (3.3) and (3.4)motivate conditions (C1) and (C2) given below.(C1): There exist k0j ≥ (1/8)j−1, j = m, . . . , 2, such thatlim(−1)m−1v→0+(−1)mf((−1)m−1k0mv, . . . ,−k02v, v)/(−1)m−1v = f0,and there exist 0 < k∞j ≤ (1/16)j−1, j = m, . . . , 2, such thatlim(−1)m−1v→∞(−1)mf((−1)m−1k∞mv, . . . ,−k∞2v, v)/(−1)m−1v = f∞.(C2): There exist 0 < k0j ≤ (1/16)j−1, j = m, . . . , 2, such thatlim(−1)m−1v→0+(−1)mf((−1)m−1k0mv, . . . ,−k02v, v)/(−1)m−1v = f0,and there exist k∞j ≥ (1/8)j−1, j = m, . . . , 2, such thatlim(−1)m−1v→∞(−1)mf((−1)m−1k∞mv, . . . ,−k∞2v, v)/(−1)m−1v = f∞.(D): (−1)mf(u0, u1, . . . , um−1) is increasing in each u2j and decreasingin each u2j+1 for (u0, u1, . . . , um−1) ∈∏m−1j=0 (−1)j [0,∞).EJQTDE, 2000 No. 2, p. 4We shall now state and prove a typical result as an application of Theorem2.1. To apply Theorem 2.1 let B denote the Banach space C[0, 1] with thesupremum norm||v|| = max0≤t≤1|v(t)|and define the cone P ⊂ B by P :={v ∈ B : (−1)m−1v(t) ≥ 0, 0 ≤ t ≤ 1, min1/4≤t≤3/4(−1)m−1v(t) ≥ (1/4)||v||}.Theorem 3.1. Assume that conditions (A), (B), (C1), and (D) are satis-fied. Then, for each λ satisfying4/(∫ 3/41/4−G(1/2, s)a(s)dsf∞) < λ < 1/(∫ 10s(1− s)a(s)dsf0), (3.5)there is at least one nontrivial solution, y, of the BVP, (1.1), (1.2), suchthat v = y(2(m−1)) belongs to P.Proof. Let λ satisfy (3.5) and let ε > 0 be such that4/(∫ 3/41/4−G(1/2, s)a(s)ds(f∞ − ε)) ≤ λ ≤ 1/(∫ 10s(1− s)a(s)ds(f0 + ε)).Define T : P → B byT v(t) = λ∫ 10a(s)G(t, s)f(Am−1v(s), . . . , A1v(s), v(s))ds.To see that T : P → P apply Conditions (A) and (B) and recall that Gis negative on (0, 1) × (0, 1). Moreover, (−1)m(T v)′′(t) = a(t)(−1)mf ≥ 0,0 < t < 1, by Conditions (A) and (B), and T v(0) = T v(1) = 0; in particular,due to concavity,min1/4≤t≤3/4(−1)m−1T v(t) ≥ ||T v||/4.We now construct the domains, Ω1 and Ω2 in order to apply Theorem 2.1(i). Apply Condition (C1) and find H1 > 0 such that(−1)mf((−1)m−1k0mv, . . . ,−k02v, v) ≤ (f0 + ε)(−1)m−1v,for all 0 < (−1)m−1v ≤ H1. Let v ∈ P with ||v|| = H1. Apply (3.3) andCondition (D) to see that(−1)mf(Am−1v, . . . , v) ≤ (−1)mf(||v||/8m−1, . . . , (−1)m−1||v||)≤ (−1)mf(k0,m||v||, . . . , (−1)m−1||v||).Thus,|T v(t)| ≤ λ∫ 10s(1− s)a(s)(−1)mf(Am−1v(s), . . . , v(s))ds≤ λ∫ 10s(1− s)a(s)ds(f0 + ε)||v|| ≤ ||v||.EJQTDE, 2000 No. 2, p. 5DefineΩ1 = {x ∈ B : ||x|| < H1},and we have shown that||T v|| ≤ ||v||, v ∈ P ∩ ∂Ω1.To find Ω2, find H > 0 such that(−1)mf((−1)m−1k∞mv, . . . ,−k∞2v, v) ≥ (f∞ − ε)(−1)m−1v,for all 0 < (−1)m−1v ≥ H. Let H2 = max{2H1, 4H} and setΩ2 = {x ∈ B : ||x|| < H2}.Let v ∈ P, ||v|| = H2. Thenmin1/4≤t≤3/4(−1)m−1v(t) ≥ ||v||/4 ≥ H.Apply (3.4) and Condition (D), for s ∈ [1/4, 3/4], to obtain(−1)mf(Am−1v(s), . . . , v(s)) ≥ (−1)mf(||v||/4(16)m−1 , . . . , (−1)m−1||v||/4)≥ (−1)mf(k∞,m||v||/4, . . . , (−1)m−1||v||/4).Thus,|T v(1/2)| ≥ λ∫ 3/41/4−G(1/2, s)a(s)(−1)mf(Am−1v(s), . . . , v(s))ds≥ λ∫ 3/41/4−G(1/2, s)a(s)ds(f∞ − ε)(||v||/4) ≥ ||v||.In particular, defineΩ2 = {x ∈ B : ||x|| < H2},and||T v|| ≥ ||v||, v ∈ P ∩ ∂Ω2.This completes the proof of Theorem 3.1.We remark that if f is superlinear (i.e., f0 = 0, f∞ = ∞) then the proofof Theorem 3.1 is readily adapted to show that the BVP, (1.1), (1.2), hasa nontrivial solution, y, such that v = y(2(m−1)) belongs to P, for each0 < λ < ∞. To illustrate that this observation is of interest, set m = 2 andconsider the fourth order Lidstone BVP that relates to the cantilever beamproblem. Note that each off1(u, v) = u2 + v2, f2(u, v) = −uvsatisfy conditions (A), (C1) and (D).We will state without proof a second application of Theorem 2.1. Theproof, when f depends only on position is standard (see [15]) and the exten-sion to the problem addressed here is completely analogous to the extensionillustrated in the proof of Theorem 3.1.EJQTDE, 2000 No. 2, p. 6Theorem 3.2. Assume that conditions (A), (B), (C2), and (D) are satis-fied. Then, for each λ satisfying4/(∫ 3/41/4−G(1/2, s)a(s)dsf0) < λ < 1/(∫ 10s(1− s)a(s)dsf∞),there is at least one nontrivial solution, y, of the BVP, (1.1), (1.2), suchthat v = y(2(m−1)) belongs to P.In the case that f is sublinear (i.e., f0 = ∞, f∞ = 0) the proof of Theorem3.2 is readily adapted to show that the BVP, (1.1), (1.2), has a nontrivialsolution, y, such that v = y(2(m−1)) belongs to P, for each 0 < λ < ∞. Toillustrate that this observation is of interest, again set m = 2 and note thateach off3(u, v) = u2/3 + v2/3, f4(u, v) = −(uv)1/3satisfy conditions (A), (C2) and (D).References[1] R.P. Agarwal and J. Henderson, Superlinear and sublinear focal boundary value prob-lems, Applic. Anal. 60 (1996), 189-200.[2] R.P. Agarwal, J. Henderson and P.J.Y. Wong, On superlinear and sublinear (n, p)boundary value problems for higher order difference equations, Nonlinear World 4(1997), 101-116.[3] R.P. Agarwal and P.J.Y. Wong, Lidstone polynomials and boundary value problems,Comput. Math. Appl. 17 (1989), 1397–1421.[4] R. I. Avery, J. Davis and J. Henderson, Three symmetric positive solutions for Lid-stone problems, preprint.[5] E. Coddington and N. Levinson, “Theory of Ordinary Differential Equations,”McGraw-Hill, New York, 1955.[6] J.M. Davis, Differentiability of solutions of differential equations with respect to non-linear Lidstone boundary conditions, Nonlinear Differential Equations, in press.[7] J.M. Davis, Differentiation of solutions of Lidstone boundary value problems withrespect to the boundary data, Math. Comput. Modelling, in press.[8] J.M. Davis, P.W. Eloe, and J. Henderson, Comparison of eigenvalues for discreteLidstone boundary value problems, Dynam. Systems Appl., 8 (1999), no. 3-4, 381-388.[9] J.M. Davis, P.W. Eloe, and J. Henderson, Triple positive solutions and dependenceon higher order derivatives, JMAA 237 (1999), 710-720.[10] J.M. Davis and J. Henderson, Uniqueness implies existence for fourth order Lidstoneboundary value problems, Panamer. Math. J. 8 (1998), 23-35.[11] P.W. Eloe and J. Henderson, Positive solutions for (n − 1, 1) conjugate boundaryvalue problems, Nonlinear Anal. 28 (1997), 1669-1680.[12] P.W. Eloe and J. Henderson, Positive solutions and nonlinear (k, n − k) conjugateeigenvalue problems, Differential Equations and Dynamical Systems 6 (1998), 309-317.[13] P.W. Eloe, J. Henderson and B. Thompson, Extremal points for impulsive Lidstoneboundary value problems, Mathematical and Computational Modeling, in press.[14] P.W. Eloe and M.N. Islam, Lidstone boundary value problems for ordinary differentialequations with impulse effects, Comm. Appl. Anal., in press.EJQTDE, 2000 No. 2, p. 7[15] L.H. Erbe and H. Wang, On the existence of positive solutions of ordinary differentialequations, Proc. Amer. Math. Soc. 120 (1994), 743-748.[16] D. Guo and V. Lakshmikantham, “Nonlinear Problems in Abstract Cones,” AcademicPress, San Diego, 1988.[17] J. Henderson and H. Wang, Positive solutions for a nonlinear eigenvalue problem, J.Math. Anal. Appl. 288 (1997), 252-259.[18] M.A. Krasnosel’skii, ”Positive Solutions of Operator Equations,” Noordhoff, Gronin-gen, 1964.[19] T.M. Lamar, Analysis of a 2nth Order Differential Equation with Lidstone BoundaryConditions, Ph.D. Dissertation, Auburn University, 1997.[20] T.M. Lamar, Comparison theory for eigenvalue problems with Lidstone boundaryconditions, preprint.[21] G.J. Lidstone, Notes on the extension of Aitken’s theorem (for polynomial interpola-tion) to the Everett types, Proc. Edinburgh Math. Soc. 2 (1929), 16-19.[22] P.J.Y. Wong and R.P. Agarwal, Multiple solutions of difference equations with Lid-stone conditions, preprint.[23] P.J.Y. Wong and R.P. Agarwal, Results and estimates on multiple solutions of Lid-stone boundary value problems, Acta Math. Hungar. 86 (2000), no.1-2, 137-168.Department of Mathematics, University of Dayton, Dayton, OH 45469-2316USAE-mail address: eloe@saber.udayton.eduEJQTDE, 2000 No. 2, p. 8

In this paper, we consider the Lidstone boundary value problem y(t) = λa(t)f(y(t), . . . , y(t), . . . y(t)), 0 \u3c t \u3c 1, y(0) = 0 = y(1), i = 0, . . . , m − 1, where (−1)f \u3e 0 and a is nonnegative. Growth conditions are imposed on f and inequalities involving an associated Green’s function are employed which enable us to apply a well-known cone theoretic fixed point theorem. This in turn yields a λ interval on which there exists a nontrivial solution in a cone for each λ in that interval. The methods of the paper are known. The emphasis here is that f depends upon higher order derivatives. Applications are made to problems that exhibit superlinear or sublinear type growth

Eloe, Paul W.

University of Dayton

University of DaytoneCommonsMathematics Faculty Publications Department of Mathematics2000Nonlinear eigenvalue problems for higher orderLidstone boundary value problemsPaul W. EloeUniversity of Dayton, peloe1@udayton.eduFollow 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 CitationEloe, Paul W., "Nonlinear eigenvalue problems for higher order Lidstone boundary value problems" (2000). Mathematics FacultyPublications. 103.https://ecommons.udayton.edu/mth_fac_pub/103NONLINEAR EIGENVALUE PROBLEMS FOR HIGHERORDER LIDSTONE BOUNDARY VALUE PROBLEMSPAUL W. ELOEAbstract. In this paper, we consider the Lidstone boundary valueproblem y(2m)(t) = λa(t)f(y(t), . . . , y(2j)(t), . . . y(2(m−1))(t)), 0 < t < 1,y(2i)(0) = 0 = y(2i)(1), i = 0, . . . , m − 1, where (−1)mf > 0 and ais nonnegative. Growth conditions are imposed on f and inequalitiesinvolving an associated Green’s function are employed which enable usto apply a well-known cone theoretic fixed point theorem. This in turnyields a λ interval on which there exists a nontrivial solution in a conefor each λ in that interval. The methods of the paper are known. Theemphasis here is that f depends upon higher order derivatives. Appli-cations are made to problems that exhibit superlinear or sublinear typegrowth.1. IntroductionWe consider the nonlinear Lidstone boundary value problem (BVP),y(2m)(t) = λa(t)f(y(t), . . . , y(2j)(t), . . . y(2(m−1))(t)), 0 < t < 1, (1.1)y(2i)(0) = 0 = y(2i)(1), i = 0, . . . m− 1, (1.2)where (−1)mf > 0 is continuous and a is nonnegative. For more preciseconditions on f and a, let (−1)j [a, b] = [a, b] if j is even and (−1)j [a, b] =[−b,−a] if j is odd. Letm−1∏j=0[aj , bj ] = [a0, b0]× · · · × [am−1, bm−1].We shall require that(A): (−1)mf :∏m−1j=0 (−1)j [0,∞) → [0,∞) is continuous,(B): a : [0, 1] → [0,∞) is continuous and does not vanish identically onany subinterval.This work is primarily motivated by the original work of Erbe and Wang[15] for m = 1. To our knowledge, Erbe and Wang [15] are the first to applythe methods employed here to the cases that f is superlinear or sublinear. Aflurry of extensions to nth order problems have been obtained in recent years1991 Mathematics Subject Classification. Primary: 34B10; Secondary: 34B15.Key words and phrases. nonlinear eigenvalue problem, Lidstone boundary value prob-lem, positive solutions, fixed points, Green’s function.EJQTDE, 2000 No. 2, p. 1(for example, see [1], [2], [22], [23], [11], [12]). Henderson and Wang [17] werethe first to introduce the problem as a nonlinear eigenvalue problem.In all the above cited works, the nonlinear term, f , only depends onposition. The primary interest of this paper is that the nonlinear term, f ,depends on position, acceleration and other even order derivatives of theunknown function. Recent related works in which dependence on higherorder derivatives is allowed can be found in [4] or [9].The Lidstone boundary value problem (BVP) was first studied by Lid-stone [21]; Agarwal and Wong’s work [3] has generated renewed interest inthe problem. Recently, Davis, Henderson, and Lamar ([6], [7], [10], [19])have studied the problem intensely. A feature of the Lidstone BVP thatis exploited in this paper is that it can be analyzed as a nested family ofsecond order conjugate BVPs. This feature has been employed by Davis,Eloe, Henderson, Islam and Thompson ([14], [8], and [13]). The primarycontribution of this paper is that this nested feature is exploited so thatthe methods employed by Erbe and Wang [15] can be be applyed to theBVP, (1.1), (1.2). Moreover, we indicate that the contribution is of interestby exhibiting applications to problems that exhibit superlinear or sublineartype growth.We close the introduction with one open question. Can the methods em-ployed here apply to a Lidstone BVP with nonlinear dependence on oddorder derivatives of the unknown function? That question is completelyopen. The problem is that large in norm does not imply large component-wise; by exploiting the nested feature of Lidstone BVPs in this paper, largein norm will, in fact, imply large in the appropriate components.2. The Fixed Point OperatorThe method developed by Erbe and Wang [15] employs an application ofthe cone theoretic fixed point theorem that we credit to Krasnosel’skii [18].Also see [16]. For simplicity we state the theorem here.Theorem 2.1. Let B be a Banach space, and let P ⊂ B be a cone in B.Assume Ω1,Ω2 are open subsets of B with 0 ∈ Ω1 ⊂ Ω1 ⊂ Ω2, and letT : P ∪ (Ω2 \ Ω1) → Pbe a completely continuous operator such that, either(i): ||T u|| ≤ ||u||, u ∈ P ∪ ∂Ω1, and ||T u|| ≥ ||u||, u ∈ P ∩ ∂Ω2, or(ii): ||T u|| ≥ ||u||, u ∈ P ∩ ∂Ω1, and ||T u|| ≤ ||u||, u ∈ P ∩ ∂Ω2.Then T has a fixed point in P ∩ (Ω2 \ Ω1).We now construct the fixed point operator upon which we apply the abovefixed point theorem. To do so, we exploit that the Lidstone BVP, (1.1),(1.2), can be constructed as a nested sequence of second order conjugateEJQTDE, 2000 No. 2, p. 2type BVPs. In particular, we shall construct a second order BVP that isequivalent to (1.1), (1.2). In Section 3 we shall apply the above fixed pointtheorem to the equivalent second order BVP.The Green’s function for y′′(t) = 0, 0 < t < 1, y(0) = y(1) = 0, isG(t, s) ={t(s− 1), 0 ≤ t ≤ s ≤ 1,s(t− 1), 0 ≤ s ≤ t ≤ 1.Set G1(t, s) := G(t, s), and for j = 2, . . . m, define recursivelyGj(t, s) =∫ 10G(t, r)Gj−1(r, s) dr.As a result, Gj(t, s) is the Green’s function for the BVP,y(2j)(t) = 0, 0 < t < 1,y(2i)(0) = 0 = y(2i)(1), i = 0, . . . , j − 1,for each j = 1 . . . m. One can verify this directly ([5], page 192) or see [13]or [19, 20].For each j = 1, . . . ,m− 1, define Aj : C[0, 1] → C[0, 1] byAjv(t) =∫ 10Gj(t, s)v(s)ds.By the construction of Aj it follows that(Ajv)(2j)(t) = v(t), 0 < t < 1,(Ajv)(2i)(0) = (Ajv)(2i)(1), i = 0, . . . , j − 1.Thus, it follows that the BVP, (1.1), (1.2), has a solution if, and only if, theBVP,v′′(t) = f(Am−1v(t), . . . , A1v(t), v(t)), 0 < t < 1,v(0) = v(1) = 0,has a solution. If y is a solution of the BVP, (1.1), (1.2), then v = y(2(m−1))is a solution of the second order BVP; conversely, if v is a solution of thesecond order BVP, then y = Am−1v is a solution of the BVP, (1.1), (1.2).Define T : C[0, 1] → C[0, 1] byT v(t) = λ∫ 10a(s)G(t, s)f(Am−1v(s), . . . , A1v(s), v(s))ds.The properties of each Gj readily imply that T : C[0, 1] → C[0, 1] is com-pletely continuous. It now follows that there exists a solution of the BVP,(1.1), (1.2), if, and only if, there exists a continuous fixed point of T . More-over, the relation between solutions, y, of the BVP, (1.1), (1.2), and fixedpoints, v, of T , is given by y(t) = Am−1v(t), or y(2(m−1))(t) = v(t).Note that G1 < 0 on (0, 1) × (0, 1), and (−1)jGj > 0 on (0, 1) × (0, 1).Thus, y is a positive solution of the BVP, (1.1), (1.2), if, and only if,EJQTDE, 2000 No. 2, p. 3(−1)m−1y(2(m−1)) = (−1)m−1v is positive, where v is the correspondingcontinuous fixed point of T . In the next section then, we restrict our anal-ysis to finding λ intervals upon which T generates a nontrivial fixed point,v, such that (−1)m−1v(t) > 0, 0 < t < 1.3. Existence of Positive SolutionsWe remind the reader of two fundamental bounds involving the Green’sfunction, G1.0 < −(G1(t, s)) ≤ s(1− s), 0 < t, s < 1,−(G1(t, s)) ≥ (1/4)s(1 − s), (1/4) ≤ t ≤ (3/4), 0 ≤ s ≤ 1.From here, four estimates which we shall employ are readily obtained.max0≤t≤1∫ 10|G1(t, s)|ds ≤ (1/8), (3.1)min1/4≤t≤3/4∫ 3/41/4|G1(t, s)|ds ≥ (1/16). (3.2)If v ∈ C[0, 1], then||Ajv|| ≤ ||v||/8j , j = 1, . . . ,m− 1. (3.3)(|| · || denotes the usual supremum norm on [0, 1].) If (−1)(m−1)v(t) > 0, 0 <t < 1, and if (−1)(m−1)v(t) > ||v||/4 for 1/4 ≤ t ≤ 3/4, thenmin1/4≤t≤3/4|Ajv(t)| ≥ ||v||/(4(16)j ), j = 1, . . . ,m− 1. (3.4)(3.3) and (3.4) are readily obtained from (3.1) and (3.2). (3.3) and (3.4)motivate conditions (C1) and (C2) given below.(C1): There exist k0j ≥ (1/8)j−1, j = m, . . . , 2, such thatlim(−1)m−1v→0+(−1)mf((−1)m−1k0mv, . . . ,−k02v, v)/(−1)m−1v = f0,and there exist 0 < k∞j ≤ (1/16)j−1, j = m, . . . , 2, such thatlim(−1)m−1v→∞(−1)mf((−1)m−1k∞mv, . . . ,−k∞2v, v)/(−1)m−1v = f∞.(C2): There exist 0 < k0j ≤ (1/16)j−1, j = m, . . . , 2, such thatlim(−1)m−1v→0+(−1)mf((−1)m−1k0mv, . . . ,−k02v, v)/(−1)m−1v = f0,and there exist k∞j ≥ (1/8)j−1, j = m, . . . , 2, such thatlim(−1)m−1v→∞(−1)mf((−1)m−1k∞mv, . . . ,−k∞2v, v)/(−1)m−1v = f∞.(D): (−1)mf(u0, u1, . . . , um−1) is increasing in each u2j and decreasingin each u2j+1 for (u0, u1, . . . , um−1) ∈∏m−1j=0 (−1)j [0,∞).EJQTDE, 2000 No. 2, p. 4We shall now state and prove a typical result as an application of Theorem2.1. To apply Theorem 2.1 let B denote the Banach space C[0, 1] with thesupremum norm||v|| = max0≤t≤1|v(t)|and define the cone P ⊂ B by P :={v ∈ B : (−1)m−1v(t) ≥ 0, 0 ≤ t ≤ 1, min1/4≤t≤3/4(−1)m−1v(t) ≥ (1/4)||v||}.Theorem 3.1. Assume that conditions (A), (B), (C1), and (D) are satis-fied. Then, for each λ satisfying4/(∫ 3/41/4−G(1/2, s)a(s)dsf∞) < λ < 1/(∫ 10s(1− s)a(s)dsf0), (3.5)there is at least one nontrivial solution, y, of the BVP, (1.1), (1.2), suchthat v = y(2(m−1)) belongs to P.Proof. Let λ satisfy (3.5) and let ε > 0 be such that4/(∫ 3/41/4−G(1/2, s)a(s)ds(f∞ − ε)) ≤ λ ≤ 1/(∫ 10s(1− s)a(s)ds(f0 + ε)).Define T : P → B byT v(t) = λ∫ 10a(s)G(t, s)f(Am−1v(s), . . . , A1v(s), v(s))ds.To see that T : P → P apply Conditions (A) and (B) and recall that Gis negative on (0, 1) × (0, 1). Moreover, (−1)m(T v)′′(t) = a(t)(−1)mf ≥ 0,0 < t < 1, by Conditions (A) and (B), and T v(0) = T v(1) = 0; in particular,due to concavity,min1/4≤t≤3/4(−1)m−1T v(t) ≥ ||T v||/4.We now construct the domains, Ω1 and Ω2 in order to apply Theorem 2.1(i). Apply Condition (C1) and find H1 > 0 such that(−1)mf((−1)m−1k0mv, . . . ,−k02v, v) ≤ (f0 + ε)(−1)m−1v,for all 0 < (−1)m−1v ≤ H1. Let v ∈ P with ||v|| = H1. Apply (3.3) andCondition (D) to see that(−1)mf(Am−1v, . . . , v) ≤ (−1)mf(||v||/8m−1, . . . , (−1)m−1||v||)≤ (−1)mf(k0,m||v||, . . . , (−1)m−1||v||).Thus,|T v(t)| ≤ λ∫ 10s(1− s)a(s)(−1)mf(Am−1v(s), . . . , v(s))ds≤ λ∫ 10s(1− s)a(s)ds(f0 + ε)||v|| ≤ ||v||.EJQTDE, 2000 No. 2, p. 5DefineΩ1 = {x ∈ B : ||x|| < H1},and we have shown that||T v|| ≤ ||v||, v ∈ P ∩ ∂Ω1.To find Ω2, find H > 0 such that(−1)mf((−1)m−1k∞mv, . . . ,−k∞2v, v) ≥ (f∞ − ε)(−1)m−1v,for all 0 < (−1)m−1v ≥ H. Let H2 = max{2H1, 4H} and setΩ2 = {x ∈ B : ||x|| < H2}.Let v ∈ P, ||v|| = H2. Thenmin1/4≤t≤3/4(−1)m−1v(t) ≥ ||v||/4 ≥ H.Apply (3.4) and Condition (D), for s ∈ [1/4, 3/4], to obtain(−1)mf(Am−1v(s), . . . , v(s)) ≥ (−1)mf(||v||/4(16)m−1 , . . . , (−1)m−1||v||/4)≥ (−1)mf(k∞,m||v||/4, . . . , (−1)m−1||v||/4).Thus,|T v(1/2)| ≥ λ∫ 3/41/4−G(1/2, s)a(s)(−1)mf(Am−1v(s), . . . , v(s))ds≥ λ∫ 3/41/4−G(1/2, s)a(s)ds(f∞ − ε)(||v||/4) ≥ ||v||.In particular, defineΩ2 = {x ∈ B : ||x|| < H2},and||T v|| ≥ ||v||, v ∈ P ∩ ∂Ω2.This completes the proof of Theorem 3.1.We remark that if f is superlinear (i.e., f0 = 0, f∞ = ∞) then the proofof Theorem 3.1 is readily adapted to show that the BVP, (1.1), (1.2), hasa nontrivial solution, y, such that v = y(2(m−1)) belongs to P, for each0 < λ < ∞. To illustrate that this observation is of interest, set m = 2 andconsider the fourth order Lidstone BVP that relates to the cantilever beamproblem. Note that each off1(u, v) = u2 + v2, f2(u, v) = −uvsatisfy conditions (A), (C1) and (D).We will state without proof a second application of Theorem 2.1. Theproof, when f depends only on position is standard (see [15]) and the exten-sion to the problem addressed here is completely analogous to the extensionillustrated in the proof of Theorem 3.1.EJQTDE, 2000 No. 2, p. 6Theorem 3.2. Assume that conditions (A), (B), (C2), and (D) are satis-fied. Then, for each λ satisfying4/(∫ 3/41/4−G(1/2, s)a(s)dsf0) < λ < 1/(∫ 10s(1− s)a(s)dsf∞),there is at least one nontrivial solution, y, of the BVP, (1.1), (1.2), suchthat v = y(2(m−1)) belongs to P.In the case that f is sublinear (i.e., f0 = ∞, f∞ = 0) the proof of Theorem3.2 is readily adapted to show that the BVP, (1.1), (1.2), has a nontrivialsolution, y, such that v = y(2(m−1)) belongs to P, for each 0 < λ < ∞. Toillustrate that this observation is of interest, again set m = 2 and note thateach off3(u, v) = u2/3 + v2/3, f4(u, v) = −(uv)1/3satisfy conditions (A), (C2) and (D).References[1] R.P. Agarwal and J. Henderson, Superlinear and sublinear focal boundary value prob-lems, Applic. Anal. 60 (1996), 189-200.[2] R.P. Agarwal, J. Henderson and P.J.Y. Wong, On superlinear and sublinear (n, p)boundary value problems for higher order difference equations, Nonlinear World 4(1997), 101-116.[3] R.P. Agarwal and P.J.Y. Wong, Lidstone polynomials and boundary value problems,Comput. Math. Appl. 17 (1989), 1397–1421.[4] R. I. Avery, J. Davis and J. Henderson, Three symmetric positive solutions for Lid-stone problems, preprint.[5] E. Coddington and N. Levinson, “Theory of Ordinary Differential Equations,”McGraw-Hill, New York, 1955.[6] J.M. Davis, Differentiability of solutions of differential equations with respect to non-linear Lidstone boundary conditions, Nonlinear Differential Equations, in press.[7] J.M. Davis, Differentiation of solutions of Lidstone boundary value problems withrespect to the boundary data, Math. Comput. Modelling, in press.[8] J.M. Davis, P.W. Eloe, and J. Henderson, Comparison of eigenvalues for discreteLidstone boundary value problems, Dynam. Systems Appl., 8 (1999), no. 3-4, 381-388.[9] J.M. Davis, P.W. Eloe, and J. Henderson, Triple positive solutions and dependenceon higher order derivatives, JMAA 237 (1999), 710-720.[10] J.M. Davis and J. Henderson, Uniqueness implies existence for fourth order Lidstoneboundary value problems, Panamer. Math. J. 8 (1998), 23-35.[11] P.W. Eloe and J. Henderson, Positive solutions for (n − 1, 1) conjugate boundaryvalue problems, Nonlinear Anal. 28 (1997), 1669-1680.[12] P.W. Eloe and J. Henderson, Positive solutions and nonlinear (k, n − k) conjugateeigenvalue problems, Differential Equations and Dynamical Systems 6 (1998), 309-317.[13] P.W. Eloe, J. Henderson and B. Thompson, Extremal points for impulsive Lidstoneboundary value problems, Mathematical and Computational Modeling, in press.[14] P.W. Eloe and M.N. Islam, Lidstone boundary value problems for ordinary differentialequations with impulse effects, Comm. Appl. Anal., in press.EJQTDE, 2000 No. 2, p. 7[15] L.H. Erbe and H. Wang, On the existence of positive solutions of ordinary differentialequations, Proc. Amer. Math. Soc. 120 (1994), 743-748.[16] D. Guo and V. Lakshmikantham, “Nonlinear Problems in Abstract Cones,” AcademicPress, San Diego, 1988.[17] J. Henderson and H. Wang, Positive solutions for a nonlinear eigenvalue problem, J.Math. Anal. Appl. 288 (1997), 252-259.[18] M.A. Krasnosel’skii, ”Positive Solutions of Operator Equations,” Noordhoff, Gronin-gen, 1964.[19] T.M. Lamar, Analysis of a 2nth Order Differential Equation with Lidstone BoundaryConditions, Ph.D. Dissertation, Auburn University, 1997.[20] T.M. Lamar, Comparison theory for eigenvalue problems with Lidstone boundaryconditions, preprint.[21] G.J. Lidstone, Notes on the extension of Aitken’s theorem (for polynomial interpola-tion) to the Everett types, Proc. Edinburgh Math. Soc. 2 (1929), 16-19.[22] P.J.Y. Wong and R.P. Agarwal, Multiple solutions of difference equations with Lid-stone conditions, preprint.[23] P.J.Y. Wong and R.P. Agarwal, Results and estimates on multiple solutions of Lid-stone boundary value problems, Acta Math. Hungar. 86 (2000), no.1-2, 137-168.Department of Mathematics, University of Dayton, Dayton, OH 45469-2316USAE-mail address: eloe@saber.udayton.eduEJQTDE, 2000 No. 2, p. 8

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Nonlinear eigenvalue problems for higher order Lidstone boundary value problems

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