Positive solutions for a system of fourth-order differential equations with integral boundary conditions and two parameters ∗

Abstract. In this work, we investigate a class of nonlinear fourth-order systems with coupled integral boundary conditions and two parameters. We give the Green’s functions for the system with boundary conditions, and then obtain some useful properties of the Green’s functions. By using the Guo–Krasnosel’skii fixed point theorem and the Green’s functions, some sufficient conditions for the existence of positive solutions are presented. As applications, two examples are presented to illustrate the application of our main results.

As we know, fourth-order ordinary differential equations are models for bending or deformation of elastic beams, therefore have important applications in engineering and physical sciences.Recently, fourth-order ordinary differential equations with different types of boundary conditions have been studied by many authors via many methods such as nonlinear alternatives of Leray-Schauder, the fixed point theory, the method of upper and lower solutions, Krasnoselskii fixed point theorem, bifurcation theory, the critical point theory, the shooting method, and fixed point theorems on cones.They can be seen in [2-5, 7, 8, 11, 16-18, 21, 24, 26-30] and the references therein.In [21], the authors considered a fully nonlinear fourth-order equation with integral boundary conditions of type x (4) (t) = f t, x(t), x (t), x (t), x (t) , t ∈ [0, 1], where , by using a fixed point theorem of cone expansion and compression of norm type, the existence and nonexistence of concave and monotone positive solutions for problem (3) was obtained.
In [11], the authors considered the nonlocal fourth-order boundary value problem with a parameter u (4) (t) + B(t)u (t) = λf t, u(t), u (t) , t ∈ (0, 1), where B ∈ C[0, 1], λ > 0 is a parameter.By using the Krasnoselskii's fixed point theorem and operator spectral theorem, the existence of positive solutions for problem (4) was given.Recently, there are some papers considered differential systems with coupled boundary conditions, see [6, 13-15, 19, 23, 25] for example.However, boundary value problems composed by systems of fourth-order differential equations are still scarce (see [1,9,12,20,22,31] for instance).In [31], the authors considered the existence of positive solutions for fourth-order nonlinear singular semipositone system The existing results were obtained by approximating the fourth-order system to a second-order singular one and using a fixed point index theorem on cones.In [9], the authors studied the existence of positive solutions for systems of the fourth-order singular semipositone Sturm-Liouville boundary value problems and by applying the fixed point index theorem, some sufficient conditions for positive solutions were established.
Motivated by the works mentioned above, we will study the existence of positive solutions for (1)-( 2).But we know, the main difficulty of studying fourth-order differential equations is the calculation of the Green's function for the problem, and it is more complicated than in the second-order and third-order cases.Therefore, we give the Green's functions for the fourth-order linear differential equation in Section 2 and then obtain some useful properties for the Green's functions.In Section 3, we define a proper cone and discuss several properties of the equivalent operator on the cone.By employing Green's functions and the Guo-Krasnosel'skii fixed point theorem, we establish some sufficient conditions on f , g, λ, µ for the existence of at least one positive solutions of (1)-(2) for appropriately chosen parameters.In Section 4, we present two examples to illustrate the application of our main results.

Auxiliary results
We consider the fourth-order coupled system with the coupled integral boundary conditions Nonlinear Anal.Model.Control, 23(3):401-422 , then the solution of problem (5)-( 6) is given by where and and https://www.mii.vu.lt/NA then we have Combining with u (0) = 1 0 h 1 (s)v (s) ds, one obtains that By the same method, we get then, by (10), the conclusion is established.
Nonlinear Anal.Model.Control, 23(3):401-422 Lemma 2. The functions g 1 and g 2 given by (9) have the properties: Proof.(ii) It is easy to get the conclusion, then we omit it.
and t ∈ [0, 1], from Lemma 4 we deduce Then we obtain the conclusion of this lemma.
Assume that Ω 1 and Ω 2 are bounded open subsets of X with 0 ∈ Ω 1 ⊂ Ω ⊂ Ω 2 , and let A : C ∩ ( Ω2 \ Ω 1 ) → C be a completely continuous operator such that either Then A has a fixed point in C ∩ ( Ω2 \ Ω 1 ).

Main results
In this section, we will give sufficient conditions on λ, µ, f , and g such that positive solutions with respect to a cone for our problem (1)-( 2) exist.We first present the assumptions, which we will use in the sequel: By using the functions G i (i = 1, 2, 3, 4) from Lemma 4, our problem ( 1)-( 2) can be written equivalently as the following nonlinear system of integral equations: We consider the Banach space X = C[0, 1] with supremum norm • and the Banach space Y = X × X with the norm (u, v) Y = u + v .We define the cone P ⊂ Y by . By Lemma 1 the positive solutions of our problem (1)-( 2) are fixed points of the operator Q. Lemma 6. Assume that (H1), (H2) hold, and σ ∈ (0, 1/2), then Q : P → P is a completely continuous operator.
For σ ∈ (0, 1/2), we denote by where G 1 (s), G 2 (s), G 3 (s), and G 4 (s) are defined in Lemma 4. We also introduce the extreme limits below In the following, we give our main results.
In order to get the other results, we introduce the extreme limits below g(t, u, v) u + v .