Nontrivial solutions for a fourth-order Riemann-Stieltjes integral boundary value problem

: In this paper we study a fourth-order di ﬀ erential equation with Riemann-Stieltjes integral boundary conditions. We consider two cases, namely when the nonlinearity satisﬁes superlinear growth conditions (we use topological degree to obtain an existence theorem on nontrivial solutions), when the nonlinearity satisﬁes a one-sided Lipschitz condition (we use the method of upper-lower solutions to obtain extremal solutions)


Introduction
In this paper we study the existence of solutions for the following integral boundary value problem of the fourth-order differential equation where f is a continuous function on [0, 1] × R, 1 0 u(t)dα(t) denotes the Riemann-Stieltjes integral, α is a function of bounded variation and satisfies the condition (H1) α(t) ≥ 0, t ∈ [0, 1] with 1 0 tdα(t) ∈ [0, 1).
Boundary value problems can describe many phenomena in the applied sciences such as nonlinear diffusion, thermal ignition of gases and concentration in chemical or biological problems.There are many papers in the literature considering the existence of solutions using Leray-Schauder degree, the method of upper-lower solutions and the Guo-Krasnoselskii fixed point theorem in cones; we refer the reader to  and the references cited therein.In [4] the authors used the Guo-Krasnoselskii fixed point theorem to study the existence of positive solutions of the fourth-order integral boundary value problem u (4) (t) + Mu(t) = f t, u(t), u (t) , t ∈ (0, 1), and in [13] the authors investigated monotone positive solutions for the nonlinear fourth-order boundary value problem with integral and multi-point boundary conditions u (4) (t) + f t, u(t), u (t) = 0, t ∈ (0, 1), where f ∈ C([0, 1] × R + × R + , R + ) satisfies some superlinear and sublinear growth conditions.In [14] the authors studied the existence and uniqueness of positive solutions for the fourth-order m-point boundary value problem where f ∈ C ([0, 1] × R + , R + ) satisfies the following conditions: (H) Hao1 lim u→∞ inf min t∈[0,1] f (t,u) u > λ * , lim u→0 + sup max t∈[0,1] f (t,u) u < λ * , and (H) Hao2 lim u→0 + inf min t∈[0,1] f (t,u) u > λ * , lim u→∞ sup max t∈[0,1] f (t,u) u < λ * , where λ * is the first eigenvalue of the eigenvalue problem u (4) (t) + αu − βu = λu with the boundary conditions in (1.2).
Note all integral boundary conditions include the two-point, three-point and multi-point boundary conditions as special cases and naturally this kind of problem has interested researchers; see for example [1,2,4,8,9,11,13,19,22,[24][25][26][27][28]30,31] and the references cited therein.In [11] the author studied the following nonlocal fractional boundary value problem with a Riemann-Stieltjes integral boundary condition where D α is the standard Caputo derivative, f : and in [31] the authors studied the eigenvalue problem for a class of singular p-Laplacian fractional differential equations involving the Riemann-Stieltjes integral boundary condition where D β t and D α t are the standard Riemann-Liouville derivatives, and f (t, x) : (0, 1) × (0, +∞) → R + is continuous.
As is well known, due to the non-locality of fractional calculus, more and more problems in physics, electromagnetism, electrochemistry, diffusion and general transport theory can be described by the fractional calculus approach.As a new modeling tool, it has a wide range of applications in many fields.However, in the process of research, more and more scholars have found that a variety of important dynamical problems exhibit fractional-order behavior that may vary with time, space or other conditions.This phenomenon indicates that variable-order fractional calculus is a natural choice, which provides an effective mathematical framework for the description of complex mathematics.For more definitions of fractional derivatives and physical understandings, we refer the reader to [3,[33][34][35].
Motivated by the aforementioned works, in this paper we use topological degree and the method of upper-lower solutions to study the fourth-order Riemann-Stieltjes integral boundary value problem (1.1), and obtain existence theorems for nontrivial solutions and extremal solutions.Moreover, we note that the conditions in this paper are more general than (H) Hao1 and (H) Hao2 .Finally, some appropriate examples to illustrate our main results are given.
) is a real Banach space and P a cone on E. Define a linear operator: Then B : E → E is a completely continuous, positive, linear operator, and its spectral radius, denoted by r(B), is 1 π 4 .Let an operator L ξ (ξ > 0) be given by Now L ξ : P → P is a completely continuous, linear, positive operator.Note that the spectral radius r L ξ ≥ ξr(B) > 0. Then the Krein-Rutman theorem [17] implies that there exists ϕ ξ ∈ P\{0} such that and (H1) implies that Therefore, the operator A can also be expressed as Then L Θ (P) ⊂ P 01 , where This completes the proof.
Proof.From (2.2) there exists Note that r(L ξ 1 ) ≥ 1.We multiply both sides of the above equation by dα(t) and integrate over [0, 1] (note (H1)) so we obtain and Consequently, we have where Let Indeed, from Lemma 2.1(iii) we have and . Therefore, by Lemma 2.3 we obtain L Θ (P) ⊂ P 02 .
(3.12) Suppose the contrary.Then there exist u 2 ∈ ∂B r 1 and µ 2 ≥ 1 such that where B r 1 = {u ∈ E : u < r 1 }.Consequently, we have exists, and Consequently, note that I − L ξ 2 −1 : P → P, and we have Hence, v 2 = 0 ⇒ u 2 = 0, and this contradicts u 2 ∈ ∂B r 1 .Thus, (3.12) holds, and Lemma 2.5 implies that deg Combining this with (3.11) we have Therefore the operator A has at least one fixed point in B R 1 \B r 1 .Equivalently, (1.1) has at least one nontrivial solution.This completes the proof.

Extremal solutions for (1.1)
In this section we use the method of upper-lower solutions to study the existence of extremal solutions for (1.1).We first provide the definitions of upper and lower solutions.Definition 4.1.We say that u ∈ E is an upper solution of (1.1) if Then u(t) ≥ 0, t ∈ [0, 1]; here c(t) satisfies the condition (H6) −π 4 < c(t) < c 0 , t ∈ [0, 1], and c 0 := 4k 4 0 with k 0 being the smallest positive solution of the equation tan k = tanh k (i.e., k 0 ≈ 3.9266 and c 0 ≈ 950.8843).Proof.From [6,7,32] we introduce a result.Let L c : W → C[0, 1] be defined by L c u = u (4) + c(t)u.Then by (H6), L c has a positive inverse, where is equivalent to where G is defined in [32, Lemma 2.1].
From (H8) and (4.4) we have and note that From Definition 4.2, v 1 is a lower solution for (1.1).From (H8) and (4.5) we have ≥ −c(t)(w 0 (t) − w 1 (t)) + c(t)w 0 (t) − c(t)w 1 (t) + f (t, w 1 ) = f (t, w 1 ), and From Definition 4.1, w 1 is an upper solution for (1.1).Therefore, for v n−1 , v n , w n−1 , w n we can use the method in Steps 1 and 2 to obtain and w n , v n ∈ E are upper and lower solutions of problem (1.1), respectively.Using mathematical induction, it is easy to verify that It is easy to conclude that {v n } ∞ n=0 and {w n } ∞ n=0 are uniformly bounded in E, and from the monotone bounded theorem we have Step 3. We prove (1.1) has solutions.These two integral equations can be transformed into the following boundary value problems i.e., v * , w * are solutions for (1.1).

Conclusions
In this paper we use topological degree and the method of upper-lower solutions to study the existence of solutions for (1.1).When the nonlinearity satisfies some superlinear growth conditions involving the first eigenvalue corresponding to the relevant linear operator we obtain nontrivial solutions.Also, when the nonlinearity satisfies a one-sided Lipschitz condition, we use the method of upper-lower solutions to obtain extremal solutions.We also provide two iterative sequences for these solutions.