Sun, Jian-Ping

Zhang, Hai-E

Zhang Hai-E

Sun Jian-Ping

University of Szeged

Electronic Journal of Qualitative Theory of Differential Equations2012, No. 18, 1-9; http://www.math.u-szeged.hu/ejqtde/Existence and iteration of monotone positivesolutions for third-order nonlocal BVPsinvolving integral conditions∗Hai-E Zhang1†, Jian-Ping Sun21. Department of Basic Science, Tangshan College, Tangshan,Hebei 063000, People’s Republic of China2. Department of Applied Mathematics, Lanzhou University of Technology,Lanzhou, Gansu 730050, People’s Republic of ChinaAbstractThis paper is concerned with the existence of monotone positive solution for the follow-ing third-order nonlocal boundary value problem u′′′ (t)+f (t, u (t) , u′ (t)) = 0, 0 < t < 1;u (0) = 0, au′ (0) − bu′′ (0) = α[u], cu′ (1) + du′′ (1) = β[u], where f ∈ C([0, 1] × R+ ×R+, R+), α[u] =∫ 10u(t)dA(t) and β[u] =∫ 10u(t)dB(t) are linear functionals on C[0, 1]given by Riemann-Stieltjes integrals. By applying monotone iterative techniques, we notonly obtain the existence of monotone positive solution but also establish an iterativescheme for approximating the solution. An example is also included to illustrate the mainresults.Keywords: Monotone iterative method; Positive solutions; Nonlocal; Integral conditions2000 AMS Subject Classification: 34B10, 34B151 IntroductionThird-order differential equation arises in a variety of different areas of applied mathematicsand physics, e.g., in the deflection of a curved beam having a constant or varying cross section,a three layer beam, electromagnetic waves or gravity driven flows and so on [1].BVPs with Stieltjes integral boundary condition (BC for short) have been considered re-cently as both multipoint and integral type BCs are treated in a single framework. For∗Supported by the National Natural Science Foundation of China (10801068).†Corresponding author. E-mail: haiezhang@126.comEJQTDE, 2012 No. 18, p. 1more comments on Stieltjes integral BC and its importance, we refer the reader to the pa-pers by Webb and Infante [2, 3, 4] and their other related works. In recent years, third-order nonlocal BVPs have received much attention from many authors, see, for example[5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20] and the references therein. For fourth-order or higher-order nonlocal BVPs, one can refer to [3, 4, 21, 22]. In particular, it should bepointed out that Webb and Infante in [2], gave a unified approach for studying the existence ofmultiple positive solutions of second-order BVPs subject to various nonlocal BCs. In [3], theyextended their method to cover equations of order N with any number up to N of nonlocalBCs in a single theory.Recently, iterative methods have been successfully employed to prove the existence of pos-itive solutions of nonlinear BVPs for ordinary differential equations, see [23, 24, 25, 26, 27]and the references therein. It is worth mentioning that, Sun et al. [24] obtained the existenceof monotone positive solutions for third-order three-point BVPs, the main tools used weremonotone iterative techniques. Inspired by the above mentioned excellent works, the aim ofthis paper is to investigate the existence and iteration of monotone positive solution for thefollowing BVPu′′′ (t) + f (t, u (t) , u′ (t)) = 0, 0 < t < 1,u (0) = 0,au′ (0) − bu′′ (0) = α[u],cu′ (1) + du′′ (1) = β[u],(1.1)where f ∈ C([0, 1] × R+ × R+, R+), α[u] =∫ 10u(t)dA(t) and β[u] =∫ 10u(t)dB(t) are linearfunctionals on C[0, 1] given by Riemann-Stieltjes integrals and a, b, c, d are nonnegativeconstants with ρ := ac + ad + bc > 0. By a positive solution of BVP (1.1), we understand asolution u(t) which is positive on t ∈ (0, 1) and satisfies BVP (1.1). By applying monotoneiterative techniques, we construct a successive iterative scheme whose starting point is a zerofunction, which is very useful and feasible for computational purpose. An example is alsoincluded to illustrate the main results.2 Preliminary lemmasIn this section, the ideas and the method we will adopt, which have been widely used, aredue to Webb and Infante in [2, 3].In our case, the existence of positive solutions of nonlocal BVP (1.1) with two nonlocalboundary terms α[u], β[u], can be studied, via a perturbed Hammerstein integral equation ofthe typeu(t) = γ(t)α[u] + δ(t)β[u] +∫ 10G (t, s) f (s, u(s), u′(s)) ds =: Tu(t). (2.1)Here γ(t), δ(t) are linearly independent and given by−γ′′′ (t) = 0, γ (0) = 0, aγ′ (0) − bγ′′ (0) = 1, cγ′ (1) + dγ′′ (1) = 0,−δ′′′ (t) = 0, δ (0) = 0, aδ′ (0) − bδ′′ (0) = 0, cδ′ (1) + dδ′′ (1) = 1,EJQTDE, 2012 No. 18, p. 2which imply γ(t) = 2ct+2dt−ct22ρand δ(t) = at2+2bt2ρ, t ∈ [0, 1]. A direct calculation shows thatfor t ∈ [θ, 1], γ(t) ≥ c1‖γ‖∞ and δ(t) ≥ c2‖δ‖∞ (‖·‖∞ is the usual supremum norm in C[0, 1]),where c1 =2cθ+2dθ−cθ2c+2dand c2 =aθ2+2bθa+2b; G(t, s) is the Green’s function for the correspondingproblem with local terms when α[u] and β[u] are identically 0.We first make the following hypotheses on the Green’s function:(H1) The kernel G is measurable, non-negative, and for every τ ∈ [0, 1] satisfieslimt→τ|G(t, s) − G(τ, s)| = 0 for s ∈ [0, 1].(H2) There exist a subinterval [a, b] ⊆ [0, 1], a measurable function Φ, and a constantc3 ∈ (0, 1] such thatG(t, s) ≤ Φ(s) for t ∈ [0, 1], s ∈ [0, 1],G(t, s) ≥ c3Φ(s) for t ∈ [a, b], s ∈ [0, 1].(H3) A, B are functions of bounded variation, and KA(s),KB(s) ≥ 0 for s ∈ [0, 1], whereKA(s) :=∫ 10G (t, s) dA(t) and KB(s) :=∫ 10G (t, s) dB(t).In the remainder of this paper, we always assume that 0 ≤ α[γ], β[δ] < 1, α[δ], β[γ] ≥ 0and D := (1 − α[γ])(1 − β[δ]) − α[δ]β[γ] > 0.As shown in Theorem 2.3 in [3], if u is a fixed point of T in (2.1), then u is a fixed point ofS, which is now given bySu(t) :=γ(t)D((1 − β[δ])∫ 10KA(s)f (s, u (s) , u′ (s)) ds + α[δ]∫ 10KB(s)f (s, u (s) , u′ (s)) ds)+δ(t)D(β[γ]∫ 10KA(s)f (s, u (s) , u′ (s)) ds + (1 − α[γ])∫ 10KB(s)f (s, u (s) , u′ (s)) ds)+∫ 10G(t, s)f (s, u (s) , u′ (s)) ds =:∫ 10GS(t, s)f (s, u (s) , u′ (s)) dsin our case. The kernel GS is the Green’s function corresponding to the BVP (1.1).Lemma 2.1 Let ρ := ac + ad + bc > 0. Then the Green’s function G(t, s) satisfies (H1), (H2)with [a, b] = [θ, 1], c3 =ρRθ0Φ(τ)dτ(a+b)(c+d), 0 < θ < 1.Proof. A direct calculation shows that,G (t, s) ={(at2+2bt)(c(1−s)+d)2ρ− (t−s)22, 0 ≤ s ≤ t ≤ 1,(at2+2bt)(c(1−s)+d)2ρ, 0 ≤ t ≤ s ≤ 1.For any fixed s ∈ [0, 1], it is easy to see thatG1(t, s) :=∂G (t, s)∂t=1ρ{(b + as) (d + c(1 − t)) , 0 ≤ s ≤ t ≤ 1,(b + at) (d + c(1 − s)) , 0 ≤ t ≤ s ≤ 1,(2.2)EJQTDE, 2012 No. 18, p. 3which shows that0 ≤ G1(t, s) ≤1ρ(b + as) (d + c(1 − s)) =: Φ(s), for (t, s) ∈ [0, 1] × [0, 1] , (2.3)and so,G (t, s) =∫ t0G1 (τ, s) dτ ≤∫ t0Φ(s)dτ = Φ(s)t ≤ Φ(s), for (t, s) ∈ [0, 1] × [0, 1] . (2.4)On the other hand,G1(t, s)Φ(s)={(b+as)(d+c(1−t))(b+as)(d+c(1−s))= (b+at)(d+c(1−t))(b+at)(d+c(1−s))≥ ρΦ(t)(a+b)(c+d), 0 ≤ s ≤ t ≤ 1,(b+at)(d+c(1−s))(b+as)(d+c(1−s))= (b+at)(d+c(1−t))(b+as)(d+c(1−t))≥ ρΦ(t)(a+b)(c+d), 0 ≤ t ≤ s ≤ 1,(2.5)so,G1 (t, s) ≥ρΦ(t)(a + b)(c + d)Φ(s), for (t, s) ∈ [0, 1] × [0, 1] . (2.6)Thus,G (t, s) =∫ t0G1 (τ, s) dτ ≥∫ t0ρΦ(τ)(a + b)(c + d)Φ(s)dτ ≥ρ∫ θ0Φ(τ)dτ(a + b)(c + d)Φ(s), for (t, s) ∈ [θ, 1]×[0, 1] .(2.7)Lemma 2.2 GS(t, s) satisfies (H1), (H2) for a function Φ1, the same interval [θ, 1], and theconstant c0 = min {c1, c2, c3}.Proof. Let Φ1(s) :=‖γ‖∞D((1 − β[δ])KA(s) + α[δ]KB(s))+‖δ‖∞D(β[γ]KA(s) + (1 − α[γ])KB(s))+Φ(s), s ∈ [0, 1]. The proof is same to the Theorem 2.4 in [2], so omitted. Moreover, we easily know that0 ≤∂GS (t, s)∂t≤ Φ2(s), t, s ∈ [0, 1] × [0, 1] (2.8)for a function Φ2, i.e., for s ∈ [0, 1],Φ2(s) :=‖γ′‖∞D[(1 − β[δ])KA(s) + α[δ]KB(s)] +‖δ′‖∞D[β[γ]KA(s) + (1 − α[γ])KB(s)] + Φ(s).We will use the classical Banach space E = C1 [0, 1] equipped with the norm ‖u‖ =max {‖u‖∞ , ‖u′ ‖∞}, where ‖u‖∞ is the usual supremum norm in C[0, 1].LetP = {u ∈ E : u (t) ≥ 0}and let c0 be same as in Lemma 2.2, then defineK ={u ∈ P : mint∈[θ,1]u (t) ≥ c0 ‖u‖∞ and u′ (t) ≥ 0, t ∈ [0, 1]}.Then it is to verify that P and K are cones in E. Note that this induces an order relation in E by defining u  v if and only if v − u ∈ K.Similar to the proofs of lemma 2.6, 2.7 and 2.8 in [2], we can get the following lemmas.EJQTDE, 2012 No. 18, p. 4Lemma 2.3 The maps T, S : P → E are compact.Lemma 2.4 T : K → K and S : P → K.Lemma 2.5 T and S have the same fixed points (in K).3 Main resultsNow we apply monotone iterative techniques to seek solution of BVP (1.1) as fixed pointof the integral operator S.Theorem 3.1 Let σ = max{maxs∈[0,1]Φ1(s), maxs∈[0,1]Φ2(s)}. Assume that f(t, 0, 0) 6≡ 0 for t ∈ [0, 1]and there exists a constant r > 0 such thatf(t, u1, v1) ≤ f(t, u2, v2) ≤rσ, 0 ≤ t ≤ 1, 0 ≤ u1 ≤ u2 ≤ r, 0 ≤ v1 ≤ v2 ≤ r. (3.1)If we construct an iterative sequence vn+1 = Svn, n = 0, 1, 2, . . . , where v0(t) = 0 for t ∈ [0, 1],then {vn}∞n=0 converges to v∗ in C1[0, 1], which is a monotone positive solution of the BVP(1.1) and satisfies0 < v∗(t) ≤ r for t ∈ (0, 1], 0 ≤ (v∗)′(t) ≤ r for t ∈ [0, 1].Proof. Let Kr = {u ∈ K : ‖u‖ < r}. We assert that S : Kr → Kr. In fact, if u ∈ Kr, then0 ≤ u(s) ≤ ‖u‖∞ ≤ ‖u‖ ≤ r, 0 ≤ u′(s) ≤ ‖u′‖∞ ≤ ‖u‖ ≤ r, for s ∈ [0, 1],which together with the condition (3.1) and Lemma 2.2 and (2.8) implies that0 ≤ (Su) (t) =∫ 10GS (t, s) f (s, u (s) , u′(s)) ds ≤ r, t ∈ [0, 1] ,0 ≤ (Su)′ (t) =∫ 10∂GS (t, s)∂tf (s, u (s) , u′(s)) ds ≤ r, t ∈ [0, 1] .Hence, we have shown that S : Kr → Kr.Now, we assert that {vn}∞n=0 converges to v∗ in C1[0, 1], which is a monotone positive solutionof the BVP (1.1) and satisfies0 < v∗(t) ≤ r for t ∈ (0, 1], 0 ≤ (v∗)′(t) ≤ r for t ∈ [0, 1].In fact, in view of v0 ∈ Kr and S : Kr → Kr, we have that vn ∈ Kr, n = 1, 2, . . .. Sincethe set {vn}∞n=0 is bounded and T is completely continuous, we know that {vn}∞n=0 is relativelycompact.EJQTDE, 2012 No. 18, p. 5In what follows, we prove that {vn}∞n=0 is monotone by induction. Firstly, by v0 = 0 andS : P → K, we easily know v1 − v0 ∈ K, which shows that v0  v1. Next, we assume thatvk−1  vk. Then, in view of Lemma 2.2 and (3.1), we have0 ≤ vk+1(t) − vk(t) =∫ 10GS (t, s)[f (s, vk(s), v′k(s)) − f(s, vk−1(s), v′k−1(s))]ds≤∫ 10Φ1(s)[f (s, vk(s), v′k(s)) − f(s, vk−1(s), v′k−1(s))]ds, t ∈ [0, 1]andvk+1(t) − vk(t) =∫ 10GS (t, s)[f (s, vk(s), v′k(s)) − f(s, vk−1(s), v′k−1(s))]ds≥ c0∫ 10Φ1(s)[f (s, vk(s), v′k(s)) − f(s, vk−1(s), v′k−1(s))]ds, t ∈ [θ, 1],which imply thatvk+1(t) − vk(t) ≥ c0 ‖vk+1 − vk‖∞ , t ∈ [θ, 1]. (3.2)At the same time, by Lemma 2.2, (2.8) and (3.1), we also havev′k+1(t) − v′k(t) =∫ 10∂GS (t, s)∂t[f (s, vk(s), v′k(s)) − f(s, vk−1(s), v′k−1(s))]ds ≥ 0, t ∈ [0, 1].(3.3)It follows from (3.2) and (3.3) that vk+1(t)− vk(t) ∈ K, which shows that vk  vk+1. Thus, wehave shown that vn  vn+1, n = 0, 1, 2 . . . .Since {vn}∞n=0 is relatively compact and monotone, there exists a v∗ ∈ Kr such that‖vn − v∗‖ → 0 (n → ∞), which together with the continuity of S and the fact that vn+1 = Svnimplies that v∗ = Sv∗. Moreover, in view of f(t, 0, 0) 6≡ 0 for t ∈ (0, 1), we know that the zerofunction is not a solution of BVP (1.1). Thus, ‖v∗‖∞ > 0. So, it follows from v∗ ∈ Kr that0 < v∗(t) ≤ r for t ∈ (0, 1], 0 ≤ (v∗)′(t) ≤ r for t ∈ [0, 1].4 An exampleConsider the BVPu′′′ (t) + 12tu + 18u′2 + 1 = 0, 0 < t < 1,u (0) = 0,u′ (0) = α[u],u′ (1) = β[u],(4.1)where α[u] =∫ 10(1−s)u(s)ds and β[u] =∫ 10su(s)ds are nonlocal BCs of integral type. For thisBCs the corresponding γ(t) = 2t−t22and δ(t) = t22. By simple calculation shows thatα[γ] =18, α[δ] =124, β[γ] =524, β[δ] =18, D = (1 − α[γ])(1 − β[δ]) − α[δ]β[γ] =109144,EJQTDE, 2012 No. 18, p. 6KA(s) :=∫ 10G (t, s) (1 − t)dt =s8−s24+s36−s424, KB(s) :=∫ 10G (t, s) tdt =5s24−s24+s424,Φ1(s) =265s218−145s2109+13s3109−s4218, Φ2(s) =156s109−181s2109+26s3109−s4109,and σ = max{maxs∈[0,1]Φ1(s), maxs∈[0,1]Φ2(s)}≈ 0.3303. Then all the hypotheses of Theorem 3.1 arefulfilled with r = 1. It follows from Theorem 3.1 that the BVP (4.1) has a monotone positivesolution v∗ satisfying0 < v∗(t) ≤ 1 for t ∈ (0, 1], 0 ≤ (v∗)′(t) ≤ 1 for t ∈ [0, 1].Moreover, the iterative scheme isv0(t) = 0, t ∈ [0, 1],vn+1(t) =∫ t0[2ts − t2s − s22+ g(t, s)](12svn(s) +18(v′n(s))2+ 1)ds+∫ 1t[t2(1 − s)2+ g(t, s)](12svn(s) +18(v′n(s))2+ 1)ds, t ∈ [0, 1], n = 1, 2, . . . ,v′n+1(t) =∫ t0[s(1 − t) + g′t(t, s)](12svn(s) +18(v′n(s))2+ 1)ds+∫ 1t[t(1 − s) + g′t(t, s)](12svn(s) +18(v′n(s))2+ 1)ds, t ∈ [0, 1], n = 1, 2, . . . .whereg(t, s) =(126t109−48t2109) (s8−s24+s36−s424)+(6t109+60t2109) (5s24−s24+s424),g′t(t, s) =(126109−96t109) (s8−s24+s36−s424)+(6109+120t109) (5s24−s24+s424),for t, s ∈ [0, 1] × [0, 1].The first, second and third terms of the scheme vn and v′n are as follows:v0(t) = 0,v1(t) =7436t +142545t2 −16t3,v2(t) =25540644751715664670784000t +57295606951217564872000t2 −5069393041536t3−4975702880t4 −379849570288000t5 −71130800t6 +14032t7,v′0(t) = 0,v′1(t) =7436+284545t −12t2,v′2(t) =25540644751715664670784000+57295606951108782436000t −15208173041536t2−4971425720t3 −379849114057600t4 −7121800t5 +1576t6.EJQTDE, 2012 No. 18, p. 7References[1] M. Gregus, Third Order Linear Differential Equations, Math. Appl., Reidel, Dordrecht,1987.[2] J. R. L. Webb and G. Infante, Positive solutions of nonlocal boundary value problems: aunified approach, J. London Math. Soc., (2) 74 (2006), 673-693.[3] J. R. L. Webb and G. Infante, Nonlocal boundary value problems of arbitrary order, J.London Math. Soc., (2) 79 (2009), 238-258.[4] J. R. L. Webb, Positive solutions of some higher order nonlocal boundary value problems,Electron. J. Qual. Theory Differ. Equ., 29 (2009), 1-15.[5] D. R. Anderson, Green’s function for a third-order generalized right focal problem, J.Math. Anal. Appl., 288 (2003), 1-14.[6] D. R. Anderson and J. M. Davis, Multiple solutions and eigenvalues for third-order rightfocal boundary value problems, J. Math. Anal. Appl., 267 (2002), 135-157.[7] D. R. Anderson and C. C. Tisdell, Third-order nonlocal problems with sign-changingnonlinearity on time scales, Electron. J. Differential Equations, 19 (2007), 1-12.[8] A. Boucherif, S. M. Bouguima, N. Al-Malki and Z. Benbouziane, Third order differentialequations with integral boundary conditions, Nonlinear Anal., 71 (2009) e1736-e1743[9] Z. Du, W. Ge and M. Zhou, Singular perturbations for third-order nonlinear multi-pointboundary value problem, J. Differential Equations, 218 (2005), 69-90.[10] M. El-Shahed, Positive solutions for nonlinear singular third order boundary value prob-lems, Commun. Nonlinear Sci. Numer. Simul., 14 (2009) 424-429.[11] J. R. Graef and J. R. L. Webb, Third order boundary value problems with nonlocal bound-ary conditions, Nonlinear Anal., 71 (2009), 1542-1551.[12] J. R. Graef and Bo Yang, Positive solutions of a third order nonlocal boundary valueproblem, Discrete Contin. Dyn. Syst. Ser. S, 1 (2008), 89-97.[13] J. Henderson and C. C. Tisdell, Five-point boundary value problems for third-order dif-ferential equations by solution matching, Math. Comput. Model., 42 (2005), 133-137.[14] B. Hopkins and N. Kosmatov, Third-order boundary value problems with sign-changingsolutions, Nonlinear Anal., 67 (2007), 126-137.[15] R. Ma, Multiplicity results for a third order boundary value problem at resonance, Non-linear Anal., 32 (1998), 493-499.EJQTDE, 2012 No. 18, p. 8[16] J. P. Sun and H. E Zhang, Existence of solutions to third-order m-point boundary valueproblems, Electron. J. Differential Equations, 125 (2008), 1-9.[17] J. P. Sun and H. B. Li, Monotone positive solution of nonlinear third-order BVP withintegral boundary conditions, Bound. Value Probl., Volume 2010, Article ID 874959, 12pages doi:10.1155/2010/874959.[18] Y. Sun, Positive solutions for third-order three-point nonhomogeneous boundary valueproblems, Appl. Math. Lett., 22 ( 2009), 45-51.[19] Q. Yao, Positive solutions of singular third-order three-point boundary value problems, J.Math. Anal. Appl., 354 (2009), 207-212.[20] Y. Wang and W. Ge, Existence of solutions for a third order differential equation withintegral boundary conditions, Comput. Math. Appl., 53 (2007) 144-154.[21] J. R. L. Webb, Nonlocal conjugate type boundary value problems of higher order, NonlinearAnal., 29 (2009), 1-15.[22] J. R. L. Webb, G. Infante and D. Franco, Positive solutions of nonlinear fourth-orderboundary value problems with local and non-local boundary conditions, Proc. Roy. Soc.Edinburgh Sect. A, 138 (2008), 427-446.[23] D. Ma, Z. Du and W. Ge, Existence and iteration of monotone positive solutions formultipoint boundary value problems with p-Laplacian operator, Comput. Math. Appl., 50(2005), 729-739.[24] J. P. Sun, K. Cao, Y. H. Zhao and X. Q. Wang, Existence and iteration of monotonepositive solutions for third-order three-point BVPs, J. Appl. Math. Informatics, 29 (2011),417-426.[25] Q. Yao, Monotone iterative technique and positive solutions of Lidstone boundary valueproblems, Appl. Math. Comput., 138 (2003), 1-9.[26] Q. Yao, Monotonically iterative method of nonlinear cantilever beam equations, Appl.Math. Comput., 205 (2008), 432-437.[27] X. Zhang, Existence and iteration of monotone positive solutions for an elastic beamequation with a corner, Nonlinear Anal., 10 (2009), 2097-2103.(Received December 3, 2011)EJQTDE, 2012 No. 18, p. 9

Existence and iteration of monotone positive solutions for third-order nonlocal BVPs involving integral conditions

http://acta.bibl.u-szeged.hu/42419/1/ejqtde_2012_018_001-009.pdf

Existence and iteration of monotone positive solutions for third-order nonlocal BVPs involving integral conditions

Abstract

Similar works

Full text

Available Versions

University of Szeged