Existence results for a clamped beam equation with integral boundary conditions

Electronic Journal of Qualitative Theory of Differential Equations 2020, No. 70, 1–17; https://doi.org/10.14232/ejqtde.2020.1.70 www.math.u-szeged.hu/ejqtde/ Existence results for a clamped beam equation with integral boundary conditions Dedicated to Professor Jeffrey R. L. Webb on the occasion of his 75th birthday Alberto CabadaB1and Rochdi Jebari2, 3 1Departamento de Estatística, Análise Matemática e Optimización, Instituto de Matemáticas Facultade de Matemáticas, Universidade de Santiago de Compostela 2Department of Mathematics, College of Sciences and Humanities Al-Quwayiyah, Shaqra University, P.O.Box 33, Shaqra 11961, Riyadh, Kingdom of Saudi Arabia 3Faculté des Sciences de Tunis, Université de Tunis El-Manar Campus Universitaire 2092 - El Manar, Tunisie Received 13 March 2020, appeared 21 December 2020 Communicated by Gennaro Infante Abstract. In this paper we investigate the existence of positive solutions of fourthorder non autonomous differential equations with integral boundary conditions, the nonlinearity is a continuous function that depends on the spatial variable and its the second-order derivative. The approach relies an extension of Krasnoselskii’s fixed point theorem in a cone. Some examples are given to illustrate our results. Keywords: Green’s functions, Fourth-order boundary value problem, integral boundary conditions, positive solutions, extension of Krasnoselskii’s fixed point theorem in a cone. 2020 Mathematics Subject Classification: Primary: 34B15, 34B27; Secondary: 34B09, 34B10. . 1 Introduction Fourth-order boundary value problems with integral boundary conditions arises in the mathematical modeling of viscoelastic and inelastic flows, thermos-elasticity, deformation of beams and plate deflection theory [12,14,22]. In [2], Cabada and Enguiça characterized the inverse positive character of operator u(4)+ M u coupled with the, so called, clamped beam boundary conditions u(4)(t) + Mu(t) = σ(t),t∈I:= [0, 1](1.1) u(0) = u(1) = u0(0) = u0(1) = 0. (1.2) BCorresponding author. Email: [email protected] 2A. Cabada and R. Jebari Using oscillation theory [23], on [2] are obtained the exact values on the real parameter M∈(−m4 1,m4 0), for which the related Green’s function gMis strictly positive in (0, 1)×(0, 1). To be concise, m1∼ =4.73004 is the first positive root of equation cos mcosh m=1, and −m4 1coincides with the first negative eigenvalue of operator u(4)coupled to boundary conditions (1.2). Moreover, m0≈5.553 is the smaller positive solution of equation tanh m √2=tan m √2, (1.3) and, as it is showed at [4], m4 0is the first positive eigenvalue of operator u(4)coupled to boundary conditions u(0) = u0(0) = u00(0) = u(1) = 0. These results have been extended in [7] (and further in [8]) for any n-th order linear differential operator. The existence of positive solutions for nonlinear problems are deduced by using the upper and lower solutions method and fixed point theorems in cones. In those cases, the nonlinearity depends only on the function u. For these problems the dependence on the second derivative of their nonlinearity has taken less attention. In this work we will study the existence of positive solution of a more general fourth order problem related to clamped beam: u(4)(t) + Mu(t) = f(t,u(t),u00(t)),t∈I, (1.4) subject to the perturbed functional boundary conditions: u(1) = u0(0) = u0(1) = 0, u(0) = λZ1 0u(s)ν(s)ds. (1.5) Where M∈(−m4 1, 4π4),ν∈L1(I)is a positive weight function a.e. on (0, 1)and λis a positive parameter bounded from above by a constant that will be introduced later. We suppose that the function fsatisfy the following regularity assumption (H0)f:I×[0, ∞)×R→[0, ∞)is a continuous function. Equation (1.4) models the stationary states of the deflection of an elastic beam. The boundary conditions (1.5) can be thought of as having the end at 1 clamped, and having some mechanism at end 0 that controls the displacement according to feedback from devices measuring the displacements along parts of the beam. This paper is a continuation of the work done in [5] for problem u(4)(t) + Mu(t) + f(t,u(t)) = 0, t∈I, subject to the perturbed functional boundary conditions: u(0) = u0(0) = u00(0) = 0, u(1) = λZ1 0u(s)ds. A standard approach to study positive solutions of a boundary value problem such as (1.4)–(1.5) consists of finding the corresponding Green’s function GMand seek solutions as Clamped beam equation with integral boundary conditions 3 fixed points of the Hammerstein integral equations with kernel GM. The majority of methods are based on classical fixed point index theory and Krasnoselskii’s fixed point theorem in a cone. The majority of authors work in a suitable cone Kin a Banach space which is made using the property of Green’s function. Sometimes the Green’s function associated to this integral equation can change its sign. In theses cases, the authors should work in a cone smaller than K(see [17–19,21]). The construction of a such cone requires more concise properties of the Green’s function (see [3,6,13]). We note that in our problem, the nonlinearity fdepends on the second order derivatives. Using the classical Krasnoselskii’s expansion/contraction theorem, we need to study the sign of the second order derivative of the Green’s function and look for a nonnegative function φ such that (C1)∂2GM ∂t2(t,s)≤φ(s),(t,s)∈I×I, and (C2)∂2GM ∂t2(t,s)≥cφ(s),(t,s)∈[a,b]×I, for some [a,b]⊂Iand c∈(0, 1). In our case, the explicit form of second derivative of Green’s function ∂2GM ∂t2is very complicated and the previous inequalities ((C1)and (C2)) become hard to be checked. So, we apply an extension of Krasnoselskii’s fixed point theorem that was used in [15,16,20,24]. With this result, we do not need to prove the inequalities (C1)and (C2). Here we need only the conditions (C1)and (C2)for the Green’s function GM. As far as we know, Problem (1.4)–(1.5) have not been previously studied. At the end of this paper, some examples are given to show that the theoretical results can be computed. This paper is organized as follows. In Section 2, we introduce some basic definitions and lemmas to prove our main results and through this section we prove that the Green’s function associated to (1.1), (1.5) satisfies some suitable properties. In Section 3, we show the existence of at least one positive solution. In section 4, some examples are presented to illustrate our main results. 2 Preliminaries and Green’s function properties In this section we introduce some preliminary results which will be used along the paper. First, we provide some background definitions cited from cone theory in Banach spaces. After that, we introduce some definitions and properties of the Green’s function GMrelated to problem (1.1), (1.5). Definition 2.1. Let Ebe a real Banach space. A nonempty convex closed set P⊂Eis said to be a cone provided that (i)αu∈Pfor all u∈Pand all α≥0; (ii)u,−u∈Pimplies u=0. In the sequel, we enunciate the a fixed point theorem due to Guo and Ge [16]. Lemma 2.2 ([16, Theorem 2.1]).Let E be a Banach space and P ⊂E a cone. Suppose α,β:E→ [0, ∞)are two continuous convex functionals satisfying α(µu) = |µ|α(u),β(µu) = |µ|β(u),u∈E,µ∈R 4A. Cabada and R. Jebari and kuk ≤ Nmax{α(u),β(u)}, for u ∈E and α(u1)≤α(u2)for u1,u2∈P, u1≤u2, where N >0 is a constant. Let r2>r1>0, L >0be constants and Ωi={u∈E:α(u)<ri,β(u)<L}, i =1, 2. be two bounded open sets in E. Set Di={u∈E:α(u) = ri}. Assume that T :P→P is a completely continuous operator satisfying (C1)α(Tu)<r1, u ∈D1∩P; α(Tu)>r2, u ∈D2∩P, (C2)β(Tu)<L, u ∈P, (C3) there is a p ∈(Ω2∩P)\{0}such that α(p)6=0and α(u+µp)≥α(u)for all u ∈P and µ≥0. Then T has at least one fixed point in (Ω2\Ω1)∩P. Moreover, we enunciate the following result concerning the expression of the Green’s function gM, related to the linear Problem (1.1), (1.5). The proof can be found in [1,2]. To this end, we introduce the following condition: M<0 and cos 4 √−Mcosh 4 √−M=1. (2.1) Lemma 2.3. Let σ∈C(I)and M ∈R. Then problem (1.1)–(1.2)has a unique solution if and only if (2.1)does not hold. In such a case, it is given by the following expression: u(t) = Z1 0gM(t,s)σ(s)ds. Here, for M =−m4<0, we have gM(t,s) = (g1(t,s,m)if 0≤s≤t≤1 g1(s,t,m)if 0≤t≤s≤1, with g1(t,s,m) = 1 8m3((1+e2m)cos(m)−2em) ×e−m(4s+t)−2emt cos(mt) + e2mt +1e5ms −em(3s+2)cos(m) +e3sm+m−e5sm+m+e4ms −1+e2mcos(m−ms) + e5ms sin(m) +em(3s+2)sin(m)−2e4sm+msin(ms)−e4ms sin(m−ms)−e4sm+2msin(m−ms) −2em(s−t)+2em(t−s)1+e2mcos(m)−2em +e−m(4s+t)e5ms +em(3s+2)cos(m)−e3sm+m−e5sm+m−2e4sm+mcos(ms) +e4ms cos(m−ms) + e4sm+2mcos(m−ms)−e5ms sin(m) + em(3s+2)sin(m) +e4ms sin(m−ms)−e4sm+2msin(m−ms)2emt sin(mt)−e2mt +1 −4 sin(m(t−s)1+e2mcos(m)−2em. Clamped beam equation with integral boundary conditions 5 If M =0, it is given by g0(t,s) = −1 6(s2(t−1)2(2ts +s−3t)if 0≤s≤t≤1, t2(s−1)2(2ts +t−3s)if 0<t≤s≤1. Moreover, when M =m4>0it follows the expression gM(t,s) = (g2(t,s,m)if 0≤s≤t≤1 g2(s,t,m)if 0≤t≤s≤1, g2(t,s,m) = e−m(−6+3s+t) √2 2√2m31+e2√2m+2e√2m−2+cos √2m(−2−1+e√2mt ×e√2m(−2+s)−e2√2m(−1+s)cos m(−2+s) √2+−e√2m(−2+s)+e2√2m(−1+s) ×cos ms √2+e√2m(−2+s)−e√2m(−1+s)+e2√2m(−1+s)−e√2m(−3+2s) ×sin ms √2sin mt √2+e√2m(−2+s)+e2√2m(−1+s)cos m(−2+s) √2 +−2e√2m(−2+s)+e√2m(−1+s)−2e2√2m(−1+s)+e√2m(−3+2s)cos ms √2 +−e√2m(−2+s)+e2√2m(−1+s)sin m(−2+s) √2+e√2m(−1+s)−e√2m(−3+2s) ×sin ms √2−1+e√2mtcos mt √2−1+e√2mtsin mt √2). Using the expressions given in Lemma 2.3, coupled to the definition of a Green’s function [8] and, as a particular case of [8, Theorem 2.14 and Theorem 5.1], we deduce the following properties for function gM: Corollary 2.4. Assuming that condition (2.1)does not hold. Then, function gM, defined in Lemma 2.3, satisfies the following properties: 1. gMis symmetric, that is gM(t,s) = gM(s,t),for all t,s∈I. 2. gM(0, s) = ∂gM ∂t(0, s) = gM(1, s) = ∂gM ∂t(1, s) = 0, for all s ∈I. 3. gM(t, 1) = ∂gM ∂s(t, 1) = gM(t, 0) = ∂gM ∂s(t, 0) = 0, for all t ∈I. Moreover, if M ∈(−m4 1,m4 0)the following inequalities are fulfilled: 4. gM(t,s)>0for all t,s∈(0, 1). 5. ∂2gM ∂t2(0, s)>0and ∂2gM ∂t2(1, s)>0, for all s ∈(0, 1). 6. ∂2gM ∂s2(t, 1)>0and ∂2gM ∂s2(t, 0)>0, for all t ∈(0, 1). To obtain the expression of the solution of Problem (1.1),(1.5), we must study the solution of a suitable non-homogeneous boundary value problem as follows 6A. Cabada and R. Jebari Lemma 2.5 ([2, Theorem 3.12]).The following problem:        u(4)(t) + Mu(t) = 0, t∈I, u(1) = u0(0) = u0(1) = 0, u(0) = 1, (2.2) has no solution if and only (2.1)holds. In any other case, it has a unique solution, denoted by wM, which is given by the following expression: wM(t) =                                                                            cos mt √2cosh m(t−2) √2−sin mt √2sinh m(t−2) √2 cos √2m+cosh √2m−2 +cos(m(t−2) √2)−2 cos(mt √2)cosh mt √2 cos √2m+cosh √2m−2 +sin m(t−2) √2sinh mt √2 cos √2m+cosh √2m−2 if m >0and M =m4, (t−1)2(1+2t)if M =0, −cos(m−mt) + cosh(m)(cos(mt)−cosh(mt)) 2 cos(m)cosh(m)−2 +cos(m)cosh(mt)−sin(mt)sinh(m) 2 cos(m)cosh(m)−2 +(sin(m) + sinh(m)) sinh(mt)) 2 cos(m)cosh(m)−2if m >0and M =−m4. (2.3) In [2, Theorem 3.12] it is proved that if M>0, then wM(t)>0 for all t∈[0, 1)if and only M∈(0, 4π4]. It is obvious that w0(t)>0 for all t∈[0, 1). To study the sign in the negative case, M=−m4, we must introduce the concept of disconjugate equation given in [10]. Definition 2.6. Let ak∈Cn−k(I)for k=1, . . . , n. The general n-th order linear differential equation u(n)(t) + a1(t)u(n−1)(t) + ···+an−1(t)u0(t) + an(t)u(t) = 0 defined on any arbitrary interval [a,b]is said to be disconjugate on an interval J⊂[a,b]if every non trivial solution has, at most, n−1 zeros on J, multiple zeros being counted according to their multiplicity. Moreover, we use the characterization for an equation to be disconjugate given in [9, Theorem 2.1] for a general nth order linear equation. Next, we enunciate the particular case for operator u(4)+M u. Lemma 2.7. The linear equation u(4)(t) + M u(t) = 0is disconjugate on the interval I if and only if M∈(−m4 1,m4 0). As a consequence, due to the continuity of the expression of wMwith respect to M, since w00 0(1) = 6, we have that if there is some ¯ M∈(−m4 1, 0)for which w¯ Mtakes some negative values on (0, 1), then it must exists M∗∈(¯ M, 0)such that one of the two following situations holds: Clamped beam equation with integral boundary conditions 7 There is t0∈(0, 1)such that wM∗(t0) = w0M∗(t0) = wM∗(1) = w0M∗(1) = 0, which contradicts Lemma 2.7, or wM∗(1) = w0M∗(1) = w00 M∗(1) = 0. But, in this last case, we have that w00 −m4(1) = m2(cos(m)−cosh(m)) cos(m)cosh(m)−1, which never takes the value zero for m>0. Therefore, if M∈(−m4 1, 0)then wM(t)>0 for all t∈[0, 1). From the expression of w00 M(1)we have that wM<0 in a neighborhood of t=1 for M smaller and close enough to −m4 1. Now, suppose that there is some M1<−m4 1for wich wM1>0 on [0, 1). Let −m4 1<M2<0, we have that for all t∈[0, 1), the following property is fulfilled: w(4) M2(t)−w(4) M1(t) = −M2(wM2−wM1)(t)−(M2−M1)wM1(t)<−M2(wM2−wM1)(t). Now, since wM2−wM1satisfies the boundary conditions (1.2), from Corollary 2.4, we deduce that 0 <wM2<wM1on (0, 1). But this contradicts the fact that lim M→−m4+ 1{wM(t)}= +∞, for all t∈(0, 1). So, we have proved the following result: Lemma 2.8. wM>0on [0, 1)if and only if M ∈(−m4 1, 4π4). Now, by denoting CM=Z1 0wM(τ)ν(τ)dτ, (2.4) we are in a position to obtain the explicit expression of the Green’s function related to the equation (1.1) coupled to boundary conditions (1.5). The result is the following. Lemma 2.9. Let σ∈L1(I),λ>0and M ∈Rbe such that (2.1)does not hold. Then problem          u(4)(t) + Mu(t) = σ(t),t∈I, u0(0) = u(1) = u0(1) = 0, u(0) = λZ1 0u(s)ν(s)ds (2.5) has a unique solution if and only if λCM6=1. In such a case, it is given by the following expression uM(t) = Z1 0GM(t,s)σ(s)ds where GM(t,s) = gM(t,s) + λwM(t) 1−λCMZ1 0gM(τ,s)ν(τ)dτ, (2.6) wMand CMare defined in (2.3)and (2.4)respectively and gMis showed in Lemma 2.3. 8A. Cabada and R. Jebari Proof. Since (2.1) does not hold, we have that Problems (1.1)–(1.2) and (2.2) are uniquely solvable. Let vMand wMbe the unique solutions of each problem respectively. Then, it is clear that uM(t) = vM(t) + λwM(t)Z1 0uM(s)ν(s)ds is the unique solution of problem (2.5). As a consequence, for all t∈I, the following equalities are fulfilled: uM(t) = Z1 0gM(t,s)σ(s)ds +λwM(t)Z1 0uM(s)ν(s)ds. (2.7) Let AM=R1 0uM(τ)ν(τ)dτ, then, from the previous equality, we deduce that AM=Z1 0Z1 0gM(τ,s)ν(τ)σ(s)ds dτ+λAMZ1 0wM(τ)ν(τ)dτ or, which is the same, AM=Z1 0σ(s)Z1 0gM(τ,s)ν(τ)dτds 1−λR1 0wM(τ)ν(τ)dτ. Replacing this value in (2.7), we arrive at the following expression for function uM: uM(t) = Z1 0gM(t,s)σ(s)ds +λwM(t)Z1 0σ(s)Z1 0gM(τ,s)ν(τ)dτds 1−λR1 0wM(τ)ν(τ)dτ, and the proof is concluded. Assuming that (2.1) does not hold, let zMbe the unique solution of the following boundary value problem: z(4)(t) + Mz(t) = ν(t)t∈I,z(0) = z(1) = z0(0) = z0(1) = 0, (2.8) which is given by the following expression zM(t) = Z1 0gM(t,s)ν(s)ds. Moreover, if M∈(−m4 1,m4 0), since ν(t)>0 a.e. t∈(0, 1), from Corollary 2.4, we have that zM(t)>0 for all t∈(0, 1),z00 M(0)>0 and z00 M(1)>0. We point out that, by direct computations, it is possible to obtain the explicit expression of function zMfor any particular choice of function ν. A careful analysis of the Green’s function GMallows us to deduce the following result: Theorem 2.10. Let GM(t,s)be the Green’s function related to problem (1.1),(1.5)given by expression (2.6). Then if M ∈(−m4 1, 4π4)and λ∈(0, 1/CM)we have that GM(t,s)>0for all (t,s)∈ (0, 1)×(0, 1). Moreover there exist R >0and h ∈C(I), such that h(1) = 0and h >0on [0, 1), for which the following inequalities are fulfilled: h(t)λ 1−λCM zM(s)≤GM(t,s)≤Rλ 1−λCM zM(s),for all (t,s)∈I×I. (2.9) Clamped beam equation with integral boundary conditions 9 Proof. First, notice that 4π4<m4 0. So, since M∈(−m4 1, 4π4)we have, from Corollary 2.4, that gM>0 on (0, 1)×(0, 1)and, as a direct consequence of λ∈(0, 1/CM)and the fact that wM>0 on [0, 1)for all M∈(−m4 0, 4π4)(Lemma 2.8), we conclude, from (2.6), that GM(t,s)>0 for all (t,s)∈(0, 1)×(0, 1). Now, we denote by ϕ(t,s) = GM(t,s) GM(0, s)=1−λCM λ gM(t,s) Z1 0gM(s,r)ν(r)dr +wM(t). (2.10) It is clear that function ϕis continuous on [0, 1]×(0, 1),ϕ(0, s) = 1 and ϕ(1, s) = 0 for all s∈I. Using the properties of gMshowed in Lemma 2.3 and those of zMpreviously explained, by means of L’Hôpital’s rule, we deduce, for all t∈(0, 1): lim s→0+ gM(t,s) Z1 0gM(s,r)ν(r)dr =lim s→0+ gM(t,s) zM(s)=lim s→0+ ∂2gM ∂s2(t,s) z00 M(s)= ∂2gM ∂s2(t, 0) z00 M(0)>0. Thus, lim s→0+ϕ(t,s) = 1−λCM λ  ∂2gM ∂s2(t, 0) z00 M(0) +wM(t):=l1(t)>0 for all t∈[0, 1). Analogously, if t∈(0, 1), we have lim s→1− gM(t,s) Z1 0gM(s,r)ν(r)dr =lim s→1− gM(t,s) zM(s)=lim s→1− ∂2gM ∂s2(t,s) z00 M(s)= ∂2gM ∂s2(t, 1) z00 M(1)>0 and lim s→1−ϕ(t,s) = 1−λCM λ  ∂2gM ∂s2(t, 1) z00 M(1) +wM(t):=l2(t)>0 for all t∈[0, 1). The limits l1(t)and l2(t)exist and are finite, so ϕhas removable discontinuities at s=0, 1, and we can extend it to a function e ϕ∈C(I×I). Therefore h(t) = mins∈[0,1]e ϕ(t,s)is a continuous function such that h(1) = 0 and 0 <h(t)≤e ϕ(t,s)≤Rfor all (t,s)∈[0, 1)×[0, 1], where R=max(t,s)∈I×Ie ϕ(t,s). Corollary 2.11. Let GM(t,s)be Green’s function related to problem (1.1),(1.5)given by expression (2.6). Then if M ∈(−m4 1, 4π4)and λ∈(0, 1/CM)we have that for all positive constant δ∈(0, 1) there exists γ(δ)∈(0, 1)for which the following inequality is fulfilled: γ(δ)λ 1−λCM zM(s)≤GM(t,s),for all (t,s)∈[0, δ]×I. (2.11) Proof. The result follows from the fact that function his continuous on Iand strictly positive on [0, 1). 16 A. Cabada and R. Jebari References [1] A. Cabada, J. A. Cid, B. Máquez-Villamarín, Computation of Green’s functions for boundary value problems with Mathematica, Appl. Math. Comput.219(2012), 1919–1936. https://doi.org/10.1016/j.amc.2012.08.035;MR2983897 [2] A. Cabada, R. R. Enguiça, Positive solutions of fourth order problems with clamped beam boundary conditions, Nonlinear Anal.74(2011), 3112–3122. https://doi.org/10. 1016/j.na.2011.01.027;MR2793550 [3] A. Cabada, R. Enguiça, L. López-Somoza, Positive solutions for second-order boundaryvalue problems with sign changing Green’s functions, Electronic J. Differential Equations 2017, No. 245, 1–17. MR3711198 [4] A. Cabada, C. Fernández-Gómez, Constant sign solutions of two-point fourth order problems; Appl. Math. Comput. 263(2015), 122–133. https://doi.org/10.1016/j.amc. 2015.03.112;MR3348530 [5] A. Cabada, R. Jebari, Multiplicity results for fourth order problems related to the theory of deformations beams, Discrete Contin. Dyn. Syst. Ser. B 25(2020), No. 2, 489–505. https: //doi.org/10.3934/dcdsb.2019250;MR4043575 [6] A. Cabada, L. López-Somoza, F. Minhós, Existence, non-existence and multiplicity results for a third order eigenvalue three-point boundary value problem, J. Nonlinear Sci. Appl. 10(2017), 5445–5463. https://doi.org/10.22436/jnsa.010.10.28;MR3725419 [7] A. Cabada, L. Saavedra, The eigenvalue characterization for the constant sign Green’s functions of (k,n−k)problems, Bound. Value Probl.2016, No. 44, 35 pp. https://doi. org/10.1186/s13661-016-0547-1;MR3459347 [8] A. Cabada, L. Saavedra, Characterization of constant sign Green’s function for a twopoint boundary-value problem by means of spectral theory, Electron. J. Differential Equations 2017, No. 146, 1–96. MR3690173 [9] A. Cabada, L. Saavedra, Characterization of non-disconjugacy for a one parameter family of nth-order linear differential equations, Appl. Math. Lett.65(2017), 98–105. https://doi.org/10.1016/j.aml.2016.10.01;MR3575178 [10] W. A. Coppel,Disconjugacy, Lecture Notes in Mathematics, Vol. 220, Springer-Verlag, Berlin-New York, 1971. https://doi.org/10.1007/BFb0058618;MR0460785 [11] K. Deimling,Nonlinear functional analysis, Springer, Berlin, 1985. https://doi.org/10. 1007/978-3-662-00547-7;MR787404 [12] E. Feireisl, Nonzero time periodic solutions to an equation of Petrovsky type with nonlinear boundary conditions: slow oscillations of beams on elastic bearings, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 20(1993), 133–146. MR1216001 [13] C. Gao, F. Zhang, R. Ma, Existence of positive solutions of second-order periodic boundary value problems with sign-changing Green’s function, Acta Math. Appl. Sin. Engl. Ser. 33(2017), 263–268. https://doi.org/10.1007/s10255-017-0657-2;MR3646983 Clamped beam equation with integral boundary conditions 17 [14] M. R. Grossinho, T. F. Ma, Symmetric equilibria for a beam with a nonlinear elastic foundation, Portugal. Math. 51(1994), 375–393. MR1295208 [15] Y. Ji, Y. Guo, Chang-long Yu, Positive solutions to (n−1, 1)m-point boundary value problems with dependence on the first order derivative, Appl. Math. Mech. (English Ed.) 30(2009), No. 4, 527–536. https://doi.org/10.1007/s10483-009-0413-x;MR2513432 [16] Y. Guo, W. Ge, Positive solutions for three-point boundary value problems with dependence on the first order derivative, J. Math. Anal. Appl. 290(2004), 291–301. https: //doi.org/10.1016/j.jmaa.2003.09.061;MR2032241 [17] G. Infante, Positive solutions of nonlocal boundary value problems with singularities, Discrete Contin. Dyn. Syst., Dynamical systems, differential equations and applications. 7th AIMS Conference, suppl. (2009), 377–384. https://doi.org/10.3934/proc.2009. 2009.377;MR2641414 [18] G. Infante, F. Minhós, Nontrivial solutions of systems of Hammerstein integral equations with first derivative dependence, Mediterr. J. Math. 14(2017), Paper No. 242, 18 pp. https://doi.org/10.1007/s00009-017-1044-1;MR3735472 [19] G. Infante, J. R. L. Webb, Nonzero solutions of Hammerstein integral equations with discontinuous kernels, J. Math. Anal. Appl.272(2002), 30–42. https://doi.org/10.1016/ S0022-247X(02)00125-7;MR1930701 [20] Y. Liu, Z. Weiguo, L. Xiping, A sufficient condition for the existence of a positive solution for a nonlinear fractional differential equation with the Riemann–Liouville derivative, Appl. Math. Lett. 25(2012), 1986–1992. https://doi.org/10.1016/j.aml.2012.03.018; MR2957792 [21] F. Minhós, R. de Sousa, On the solvability of third-order three point systems of differential equations with dependence on the first derivative, Bull. Braz. Math. Soc. (N.S.) 48(2017), 485–503. https://doi.org/10.1007/s00574-016-0025-5;MR3712344 [22] G. Perla Menzala, E. Zuazua, Explicit exponential decay rates for solutions of von Kármán’s system of thermoelastic plates, C. R. Acad. Sci. Paris Sér. I Math. 324(1997), 49– 54. https://doi.org/10.1016/S0764-4442(97)80102-4;MR1435586 [23] J. Schröder, On linear differential inequalities, J. Math. Anal. Appl. 22(1968), 188–216. https://doi.org/10.1016/0022-247X(68)90169-8;MR223935 [24] X. Yang, Z. Wei, W. Dong, Existence of positive solutions for the boundary value problem of nonlinear fractional differential equations, Commun. Nonlinear Sci. Numer. Simul. 17(2012), 85–92. https://doi.org/10.1016/j.cnsns.2011.05.007;MR2825989 View publication stats