Positive solutions of systems of perturbed Hammerstein integral equations with arbitrary order dependence

Abstract

Motivated by the study of systems of higher-order boundary value problems with functional boundary conditions, we discuss, by topological methods, the solvability of a fairly general class of systems of perturbed Hammerstein integral equations, where the nonlinearities and the functionals involved depend on some derivatives. We improve and complement earlier results in the literature. We also provide some examples in order to illustrate the applicability of the theoretical results.

This article is part of the theme issue ‘Topological degree and fixed point theories in differential and difference equations’.

1. Introduction

In this paper, we discuss the solvability of systems of perturbed Hammerstein integral equations of the form

$$\begin{array}{rl}{u}_i(t)& ={\lambda }_i{\int }_0^1{k}_i(t,s)f_i(s,u_1(s),\\ & \dots ,u_1^{(m_1)}(s),\dots ,u_n(s),\dots ,u_n^{(m_n)}(s))\hspace{0.17em}\text{d}s\\ & +\sum _{j=1}^{p_i}{\eta }_{ij}{\gamma }_{ij}(t)h_{ij}[u],\hspace{0.17em}t\in [0,1],\hspace{0.17em}i=1,2,\dots ,n,\end{array}$$ 1.1

where u = (u₁, …, u_n), the kernels k_i are sufficiently regular, f_i are continuous, γ_ij are sufficiently smooth, h_ij are compact functionals that are allowed to take into account higher-order derivatives and λ_i, η_ij are parameters.

One motivation for studying the kind of equations that occur in (1.1) is that these often occur in applications; we refer the reader to the Introduction of [1] and references therein. The case n = 1 has been studied recently by Goodrich [2,3], who complemented earlier works [1,4].

In particular, Goodrich studied the equation

$$u(t)=\lambda {\int }_0^{1}k(t,s)f(s,u(s))\hspace{0.17em}\text{d}s+\sum _{j=1}^2{\gamma }_j(t)h_j[u],$$

where the functionals h_j have the specific form

$$h_j[u]={\hat{h}}_j({\alpha }_j[u]).$$ 1.2

In (1.2), the functions ${\hat{h}}_j$ are continuous and α_j are linear functionals on the space C[0, 1] which can be represented as Stieltjes integrals, namely

$${\alpha }_j[u]:={\int }_0^{1}u(s)\hspace{0.17em}\text{d}{A}_j(s).$$ 1.3

The functional formulation (1.3) is well suited for handling, in a unified way, multi-point and integral boundary conditions (BCs). For an introduction to non-local BCs, we refer the reader to the reviews [5–11] and the articles [12–14].

The case n = 2 has been investigated in [1], where the authors studied the system

$$u_i(t)={\int }_0^1{k}_i(t,s)f_i(s,u_1(s),u_2(s))\hspace{0.17em}\text{d}s+\sum _{j=1}^2{\gamma }_{ij}(t)h_{ij}[(u_1,u_2)],\hspace{0.17em}i=1,2,$$

where the functionals h_ij act on the space C[0, 1] × C[0, 1].

We stress that functionals involving higher-order derivatives play an important role in applications. In order to illustrate this fact in a simple situation, consider the boundary value problem (BVP)

$$u^{(4)}(t)=f(t,u(t)),u(0)=h[u],\hspace{0.17em}{u}^{\mathrm{\prime }\mathrm{\prime }}(0)=u(1)=u^{\mathrm{\prime }\mathrm{\prime }}(1)=0.$$ 1.4

When h[u] ≡ 0 the BVP (1.4) can be used to describe the steady-state case of a simply supported beam of length 1. When the functional h is non-trivial the BVP (1.4) can be used to model a beam with a feedback control; for example, the case

$$h[u]=\hat{h}(u^{‴}(\xi ))$$ 1.5

models a beam with the right end simply supported and where the displacement in the left end is controlled (possibly in a nonlinear manner) by a sensor that measures the shear force in a point ξ placed along of the beam. The perturbed integral equation associated with (1.4) and (1.5) is

$$u(t)={\int }_0^{1}k(t,s)f(s,u_1(s))\hspace{0.17em}\text{d}s+(1-t)\hat{h}(u^{‴}(\xi )),$$

a case that cannot be handled with the theory developed in [1–4] due to the third-order term occurring in (1.5).

The case of higher-order dependence within the equation has been in investigated recently, by means of the classical Krasnosel’skiĭ’s theorem of cone compression-expansion, by de Sousa & Minhós [15]. In particular, de Sousa & Minhós [15] consider the existence of non-trivial solutions for the system of Hammerstein equations

$$u_i(t)={\int }_0^1{k}_i(t,s)f_i(s,u_1(s),\dots ,u_1^{(m_1)}(s),\dots ,u_n(s),\dots ,u_n^{(m_n)}(s))\hspace{0.17em}\text{d}s,\hspace{0.17em}i=1,2,\dots ,n.$$

As an interesting application of their theory, de Sousa and Minhós apply their result to a system of BVPs of the form

$$\begin{array}{rl}& u_1^{\mathrm{\prime }\mathrm{\prime }}(t)+f_1(t,u_1(t),u_1^{\prime }(t),u_2(t),u_2^{\prime }(t),u_2^{″}(t),u_2^{‴}(t))=0,t\in (0,1),\\ & u_2^{(4)}(t)=f_2(t,u_1(t),u_1^{\prime }(t),u_2(t),u_2^{\prime }(t),u_2^{″}(t),u_2^{‴}(t)),t\in (0,1)\\ \text{and}& u_1(0)=u_1(1)=u_2(0)=u_2(1)=u_2^{\mathrm{\prime }\mathrm{\prime }}(0)=u_2^{\mathrm{\prime }\mathrm{\prime }}(1)=0.\end{array}\}$$ 1.6

The system (1.7) can be used as a model of the displacement of simply supported suspension bridge. In this model, the fourth-order equation describes the road bed and the second-order equation models the suspending cables, we refer to [15] for more details.

On the other hand, the case of equations of the form

$$u(t)=\lambda {\int }_0^{1}k(t,s)f(s,u(s),u^{\prime }(s))\hspace{0.17em}\text{d}s+\sum _{j=1}^2{\eta }_j{\gamma }_j(t)h_j[u],$$

where the functionals h_j act on the space C¹[0, 1], has been studied recently by Infante [16], by means of the classical fixed-point index. Here we develop further this approach and we extend the results of [16] to the case of systems and higher-order dependence in the nonlinearities and the functionals. We also improve the case n = 1 and m₁ = 1, by allowing more freedom in the growth of the nonlinearities near the origin; this is achieved by means of an eigenvalue comparison.

In order to illustrate the applicability of our theory, we discuss, merely as an example, the solvability of the system of the following model problem:

$$\begin{array}{rl}& u_1^{\mathrm{\prime }\mathrm{\prime }}(t)+{\lambda }_1{f}_1(t,u_1(t),u_1^{\prime }(t),u_2(t),u_2^{\prime }(t),u_2^{″}(t),u_2^{‴}(t))=0,t\in (0,1),\\ & u_2^{(4)}(t)={\lambda }_2{f}_2(t,u_1(t),u_1^{\prime }(t),u_2(t),u_2^{\prime }(t),u_2^{″}(t),u_2^{‴}(t)),t\in (0,1),\\ & u_1(0)=0,\hspace{0.17em}{u}_1(1)={\eta }_{11}{h}_{11}[(u_1,u_2)]\\ \text{and}& u_2(0)={\eta }_{21}{h}_{21}[(u_1,u_2)],\hspace{0.17em}{u}_2^{\mathrm{\prime }\mathrm{\prime }}(0)=u_2(1)=u_2^{\mathrm{\prime }\mathrm{\prime }}(1)=0,\end{array}\}$$ 1.7

where h₁₁, h₂₁ are non-negative, compact functionals defined on the space C¹[0, 1] × C³[0, 1]. The interest in (1.7) arises in the fact that it presents a coupling in the nonlinearities f₁ and f₂ and in the BCs and allows the presence derivatives of different order in the various components. The system (1.7) can be seen as a perturbation of the system (1.6) and is a generalization of some earlier ones studied in [17,18]. Here we discuss in detail the existence and non-existence of positive solutions of the system (1.7), illustrating how the constants that occur in our theory can be computed or estimated. Our results are new and complement the ones in [1,4,15,16,19–21].

2. Main results

In this section, we study the existence and non-existence of solutions of the system of perturbed Hammerstein equation of the type

$$\begin{array}{rl}{u}_i(t)& ={\lambda }_i{\int }_0^1{k}_i(t,s)f_i(s,u_1(s),\dots ,u_1^{(m_1)}(s),\dots ,u_n(s),\dots ,u_n^{(m_n)}(s))\hspace{0.17em}\text{d}s\\ & +\sum _{j=1}^{p_i}{\eta }_{ij}{\gamma }_{ij}(t)h_{ij}[u]:=T_{i}u(t),\hspace{0.17em}t\in [0,1],\hspace{0.17em}i=1,2,\dots ,n,\end{array}$$ 2.1

where u = (u₁, …, u_n). Throughout the paper, we make the following assumptions on the terms that occur in (2.1).

• (C₁)
  For every i = 1, …, n, k_i:[0, 1] × [0, 1] → [0, + ∞) is measurable in s for every t and continuous in t for almost every (a.e.) s, that is, for every τ ∈ [0, 1] we have

  $$\underset{t\to \tau }{lim}|k_i(t,s)-k_i(\tau ,s)|=0\hspace{0.17em}\text{for a.e.}\hspace{0.17em}s\in [0,1];$$

  furthermore, there exists a function Φ_i0 ∈ L¹(0, 1) such that 0 ≤ k_i(t, s) ≤ Φ_i0(s) for t ∈ [0, 1] and a.e. s ∈ [0, 1].
• (C₂)For every i = 1, …, n and for every $l_i\in \mathbb{N}$, with l_i < m_i, the partial derivative ${\mathrm{\partial }}^{l_i}{k}_i/\mathrm{\partial }{t}^{l_i}$ is measurable in s for every t, continuous in t for a.e. s, and there exists ${\mathit{\Phi }}_{i{l}_i}(s)\in L^1(0,1)$ such that $|({\mathrm{\partial }}^{l_i}{k}_i/\mathrm{\partial }{t}^{l_i})(t,s)|\le {\mathit{\Phi }}_{i{l}_i}(s)$ for t ∈ [0, 1] and a.e. s ∈ [0, 1].
• (C₃)For every i = 1, …, n, ${\mathrm{\partial }}^{m_i}{k}_i/\mathrm{\partial }{t}^{m_i}$ is measurable in s for every t, continuous in t except possibly at the point t = s where there can be a jump discontinuity, that is right and left limits both exist, and there exists ${\mathit{\Phi }}_{i{m}_i}(s)\in L^1(0,1)$ such that $|({\mathrm{\partial }}^{m_i}{k}_i/\mathrm{\partial }{t}^{m_i})(t,s)|\le {\mathit{\Phi }}_{i{m}_i}(s)$ for t ∈ [0, 1] and a.e. s ∈ [0, 1].
• (C₄)For every i = 1, …, n, $f_i:[0,1]\times \prod _{i=1}^n([0,+\mathrm{\infty })\times {\mathbb{R}}^{m_i})\to [0,+\mathrm{\infty })$ is continuous.
• (C₅)For every i = 1, …, n and j = 1, …, p_i, we have ${\gamma }_{ij}\in C^{m_i}[0,1]$ and ${\gamma }_{ij}(t)\ge 0\hspace{0.17em}\text{for every}\hspace{0.17em}t\in [0,1]$.
• (C₆)For every i = 1, …, n and j = 1, …, p_i, we have λ_i, η_ij, ∈ [0, + ∞).

Owing to the assumptions above, for every i = 1, …, n, the linear Hammerstein integral operator

$$L_{i}w(t):={\int }_0^1{k}_i(t,s)w(s)\hspace{0.17em}\text{d}s$$

is well defined and compact in the space C[0, 1], where we adopt the standard norm $||w|{|}_{\mathrm{\infty }}:=\underset{t\in [0,1]}{max}|w(t)|$. We recall that a cone K in a real Banach space X is a closed convex set such that λx ∈ K for every x ∈ K and for all λ ≥ 0 and satisfying $K\cap (-K)=\{0\}$. It is clear that the operator L_i leaves invariant the cone

$$\hat{P}:=\{w\in C[0,1]:\hspace{0.17em}w\ge 0\hspace{0.17em}\text{for every}\hspace{0.17em}t\in [0,1]\}.$$

We denote by r(L_i) the spectral radius of L_i and assume

• (C₇)For every i = 1, …, n, we have r(L_i) > 0.

Note that, since $\hat{P}$ is a reproducing cone in C[0, 1], the assumption (C₇) allows us to apply the well-known Krein–Rutman theorem and therefore r(L_i) is an eigenvalue of L_i with a corresponding eigenfunction ${\phi }_i\in \hat{P}\setminus \{0\}$, that is

$$L_i{\phi }_i(t)={\int }_0^1{k}_i(t,s){\phi }_i(s)\hspace{0.17em}\text{d}s=r(L_i){\phi }_i(t).$$ 2.2

In what follows we shall make use of the eigenfunction φ_i and the corresponding characteristic value

$${\mu }_i:=\frac{1}{r}(L_i).$$

Note that the non-negative eigenfunction φ_i inherits, from the kernel k_i, further regularity properties: indeed, since we have

$${\phi }_i(t)={\mu }_i{\int }_0^1{k}_i(t,s){\phi }_i(s)\hspace{0.17em}\text{d}s,$$ 2.3

and, due to the assumptions (C₁)–(C₃), the r.h.s. of (2.3) is, as a function of the variable t, in $C^{m_i}[0,1]$ we obtain

$${\phi }_i\in (\hat{P}\setminus \{0\})\cap C^{m_i}[0,1].$$

Remark 2.1. —

The assumption (C₇) is frequently satisfied in applications. A sufficient condition, for details see [22], is given by

• (C′₈)
  There exist a subinterval [a_i, b_i]⊆ [0, 1] and a constant c_i = c(a_i, b_i) ∈ (0, 1] such that

  $$k_i(t,s)\ge c_i{\mathit{\Phi }}_{i0}(s)\text{ for }t\in [a_i,b_i]\text{ and a. e. }\hspace{0.17em}s\in [0,1].$$

Owing to the hypotheses above, we work in the product space $\prod _{i=1}^n{C}^{m_i}[0,1]$ endowed with the norm

$$||u||:=\underset{i=1,\dots ,n}{max}\{||u_i|{|}_{C^{m_i}}\},$$

where $||u_i|{|}_{C^{m_i}}:=\underset{j=0,\dots ,m_i}{max}\{||u_i^{(j)}|{|}_{\mathrm{\infty }}\}$. We use the cone

$$P:=\{u\in \prod _{i=1}^n{C}^{m_i}[0,1]:\hspace{0.17em}{u}_i\ge 0\hspace{0.17em}\text{for every}\hspace{0.17em}t\in [0,1],\hspace{0.17em}i=1,\dots ,n\},$$

and we require the nonlinear functionals h_ij to act positively on the cone P and to be compact, that is:

• (C₈)For every i = 1, …, n and j = 1, …, p_i, h_ij: P → [0, + ∞) is continuous and maps bounded sets into bounded sets.

We define the operator T: P → P as

$$Tu:=(T_{i}u{)}_{i=1,\dots ,n}.$$ 2.4

We make use of the following basic properties of the fixed-point index (we refer the reader to [23,24] for more details).

Proposition 2.2 ([23,24]). —

Let K be a cone in a real Banach space X and let D be an open bounded set of X with 0 ∈ D_K and ${\overline{D}}_K\ne K$, where $D_K=D\cap K$. Assume that $\tilde{T}:{\overline{D}}_K\to K$ is a compact map such that $x\ne \tilde{T}x$ for x ∈ ∂D_K. Then the fixed-point index $i_K(\tilde{T},D_K)$ has the following properties:

• (1)If there exists $e\in K\setminus \{0\}$ such that $x\ne \tilde{T}x+\lambda e$ for all x ∈ ∂D_K and all λ > 0, then $i_K(\tilde{T},D_K)=0$.
• (2)If $\tilde{T}x\ne \lambda x$ for all x ∈ ∂D_K and all λ > 1, then $i_K(\tilde{T},D_K)=1$.
• (3)Let D¹ be open in X such that ${\overline{D^1}}_K\subset D_K$. If $i_K(\tilde{T},D_K)=1$ and $i_K(\tilde{T},D_K^1)=0$, then $\tilde{T}$ has a fixed point in $D_K\setminus \overline{D_K^1}$. The same holds if $i_K(\tilde{T},D_K)=0$ and $i_K(\tilde{T},D_K^1)=1$.

For ρ ∈ (0, ∞), we define the sets

$$P_{\rho }:=\{u\in P:||u||<\rho \},I_{\rho }:=[0,1]\times \prod _{i=1}^n([0,\rho ]\times {[-\rho ,\rho ]}^{m_i})$$

and the quantities

$${\overline{f}}_{i\rho }:=\underset{I_{\rho }}{max}{f}_i(t,x_{10},\dots ,x_{1m_1},\dots ,x_{n0},\dots ,x_{n{m}_n}),H_{ij\rho }:=\underset{u\in \mathrm{\partial }{P}_{\rho }}{sup}{h}_{ij}[u],$$

$$K_{il}:=\{\begin{array}{l}\underset{t\in [0,1]}{sup}{\int }_0^1{k}_i(t,s)\hspace{0.17em}ds,\hspace{0.17em}l=0,\\ \underset{t\in [0,1]}{sup}{\int }_0^1|\frac{{\mathrm{\partial }}^l{k}_i}{\mathrm{\partial }{t}^l}(t,s)|\hspace{0.17em}\text{d}s,\hspace{0.17em}l=1,\dots ,m_i.\end{array}$$

With these ingredients, we can state the following existence and localization result.

Theorem 2.3. —

Assume there exist r, R, δ ∈ (0, + ∞), with r < R, and i₀ ∈ {1, 2, …, n} such that the following three inequalities are satisfied:

$$\begin{array}{rl}& \underset{\begin{array}{c}i=1,\dots ,n\\ l=0,\dots ,m_i\end{array}}{max}\{{\lambda }_i{\overline{f}}_{iR}{K}_{il}+\sum _{j=1}^{p_i}{\eta }_{ij}||{\gamma }_{ij}^{(l)}|{|}_{\mathrm{\infty }}{H}_{ijR}\}\le R,\end{array}$$ 2.5

$$\begin{array}{rl}& {\lambda }_{i_0}\ge \frac{{\mu }_{i_0}}{\delta },f_{i_0}(t,x_{10},\dots ,x_{1m_1},\dots ,x_{n0},\dots ,x_{n{m}_n})\ge \delta x_{i_{0}0},\hspace{0.17em}\text{on}\hspace{0.17em}{I}_r.\end{array}$$ 2.6

Then the system (2.1) has a solution u ∈ P such that

$$r\le ||u||\le R.$$

Proof. —

With a careful use of the Ascoli–Arzelà theorem, it is can be proved that, under the assumptions (C₁)–(C₈), the operator T maps P into P and is compact.

If T has a fixed point either on ∂P_r or ∂P_R we are done. Assume now that T is fixed point free on $\mathrm{\partial }{P}_r\cup \mathrm{\partial }{P}_R$. We are going to prove that T has a fixed point in $P_R\setminus \overline{P_r}$.

We firstly prove that $\sigma u\ne Tu\hspace{0.17em}\text{for every}\hspace{0.17em}u\in \mathrm{\partial }{P}_R\hspace{0.17em}\text{and every}\hspace{0.17em}\sigma >1.$ If this does not hold, then there exist u ∈ ∂P_R and σ > 1 such that σu = Tu. Note that if $||u||=R$ then there exist i₀, l₀ such that $||u_{i_0}^{(l_0)}|{|}_{\mathrm{\infty }}=R$. We show the case l₀ ≠ 0 (the case l₀ = 0 is simpler, hence omitted). Thus we have, for t ∈ [0, 1],

$$\begin{array}{r}\sigma u_{i_0}^{(l_0)}(t)={\lambda }_i{\int }_0^1\frac{{\mathrm{\partial }}^{l_0}{k}_{i_0}}{\mathrm{\partial }{t}^{l_0}}(t,s)f_{i_0}(s,u_1(s),\dots ,u_1^{(m_1)}(s),\dots ,u_n(s),\dots ,u_n^{(m_n)}(s))\hspace{0.17em}\text{d}s+\sum _{j=1}^{p_i}{\eta }_{i_{0}j}{\gamma }_{i_{0}j}^{(l_0)}(t)h_{i_{0}j}[u].\end{array}$$ 2.7

From (2.7) we obtain, for t ∈ [0, 1],

$$\begin{array}{rl}\sigma |u_{i_0}^{(l_0)}(t)|& \le {\lambda }_i{\int }_0^1|\frac{{\mathrm{\partial }}^{l_0}{k}_{i_0}}{\mathrm{\partial }{t}^{l_0}}(t,s)|f_{i_0}(s,u_1(s),\dots ,u_1^{(m_1)}(s),\dots ,u_n(s),\dots ,u_n^{(m_n)}(s))\hspace{0.17em}\text{d}s\\ & +\sum _{j=1}^{p_i}{\eta }_{i_{0}j}|{\gamma }_{i_{0}j}^{(l_0)}(t)|h_{i_{0}j}[u]\le {\lambda }_{i_0}{\overline{f}}_{i_{0}R}{K}_{i_0{l}_0}+\sum _{j=1}^{p_i}{\eta }_{i_{0}j}||{\gamma }_{i_{0}j}^{(l_0)}|{|}_{\mathrm{\infty }}{H}_{i_{0}jR}\le R.\end{array}$$ 2.8

Taking in (2.8) the supremum for t ∈ [0, 1] yields σ ≤ 1, a contradiction.

Therefore, we have i_P(T, P_R) = 1.

We now consider the function φ(t): = (φ₁(t), …, φ_n(t)), where t ∈ [0, 1] and φ_i is given by (2.2). Note that $\phi \in P\setminus \{0\}$. We show that

$$u\ne Tu+\sigma \phi \hspace{0.17em}\text{for every}\hspace{0.17em}u\in \mathrm{\partial }{P}_r\hspace{0.17em}\text{and every}\hspace{0.17em}\sigma >0.$$

If not, there exist u ∈ ∂P_r and σ > 0 such that u = Tu + σφ. In particular, we have $u_{i_0}(t)=T_{i_0}u(t)+\sigma {\phi }_{i_0}(t)$ for every t ∈ [0, 1] and therefore $u_{i_0}(t)\ge \sigma {\phi }_{i_0}(t)$ in [0, 1]. Observe that we have $r\ge ||u_{i_0}|{|}_{\mathrm{\infty }}\ge \sigma ||{\phi }_{i_0}|{|}_{\mathrm{\infty }}>0$. For every t ∈ [0, 1], we have

$$\begin{array}{rl}{u}_{i_0}(t)& ={\lambda }_i{\int }_0^1{k}_{i_0}(t,s)f_{i_0}(s,u_1(s),\dots ,u_1^{(m_1)}(s),\dots ,u_n(s),\dots ,u_n^{(m_n)}(s))\hspace{0.17em}\text{d}s\\ & +\sum _{j=1}^{p_i}{\eta }_{i_{0}j}{\gamma }_{i_{0}j}(t)h_{i_{0}j}[u]+\sigma {\phi }_{i_0}(t)\ge {\lambda }_i{\int }_0^1{k}_{i_0}(t,s)\delta u_{i_0}(s)\hspace{0.17em}\text{d}s+\sigma {\phi }_{i_0}(t)\\ & \ge {\lambda }_i{\int }_0^1{k}_{i_0}(t,s)\delta \sigma {\phi }_{i_0}(s)\hspace{0.17em}\text{d}s+\sigma {\phi }_{i_0}(t)\\ & =\frac{\sigma {\lambda }_{i_0}\delta }{{\mu }_{i_0}}{\phi }_{i_0}(t)+\sigma {\phi }_{i_0}(t)\ge 2\sigma {\phi }_{i_0}(t).\end{array}$$

By iteration we obtain, for t ∈ [0, 1],

$$u_{i_0}(t)\ge n\sigma {\phi }_{i_0}(t)\hspace{0.17em}\text{for every}\hspace{0.17em}n\in \mathbb{N},$$

which contradicts the fact that $||u_{i_0}|{|}_{\mathrm{\infty }}\le r$.

Thus we obtain i_P(T, P_r) = 0.

Therefore, we have

$$i_P(T,P_R\setminus \overline{P_r})=i_P(T,P_R)-i_P(T,P_r)=1,$$

which proves the result. ▪

We now illustrate the applicability of theorem 2.3.

Example 2.4. —

We focus on the system

$$\begin{array}{rl}& u_1^{\mathrm{\prime }\mathrm{\prime }}(t)+{\lambda }_1{f}_1(t,u_1(t),u_1^{\prime }(t),u_2(t),u_2^{\prime }(t),u_2^{″}(t),u_2^{‴}(t))=0,t\in (0,1),\\ & u_2^{(4)}(t)={\lambda }_2{f}_2(t,u_1(t),u_1^{\prime }(t),u_2(t),u_2^{\prime }(t),u_2^{″}(t),u_2^{‴}(t)),t\in (0,1),\\ & u_1(0)=0,\hspace{0.17em}{u}_1(1)={\eta }_{11}{h}_{11}[(u_1,u_2)]\\ \text{and}& u_2(0)={\eta }_{21}{h}_{21}[(u_1,u_2)],\hspace{0.17em}{u}_2^{\mathrm{\prime }\mathrm{\prime }}(0)=u_2(1)=u_2^{\mathrm{\prime }\mathrm{\prime }}(1)=0,\end{array}\}$$ 2.9

where h₁₁, h₂₁ are non-negative, compact functionals acting on the cone

$$P=\{(u_1,u_2)\in C^1[0,1]\times C^3[0,1]:\hspace{0.17em}{u}_1,u_2\ge 0\hspace{0.17em}\text{for every}\hspace{0.17em}t\in [0,1]\}.$$

With our methodology, we could study a more complicated version of this BVP, by adding more functional terms in the BCs, but we refrain from doing so for the sake of clarity.

It is routine to show that the solutions of (2.9) can be written in the form

$$\begin{array}{rl}{u}_1(t)& ={\eta }_{11}t{h}_{11}[(u_1,u_2)]+{\lambda }_1{\int }_0^1{k}_1(t,s)f_1(s,u_1(s),u_1^{\prime }(s),u_2(s),u_2^{\prime }(s),u_2^{″}(s),u_2^{‴}(s))\hspace{0.17em}\text{d}s\\ \text{and}{u}_2(t)& ={\eta }_{21}(1-t)h_{21}[(u_1,u_2)]+{\lambda }_2{\int }_0^1{k}_2(t,s)f_2(s,u_1(s),u_1^{\prime }(s),u_2(s),u_2^{\prime }(s),u_2^{″}(s),u_2^{‴}(s))\hspace{0.17em}\text{d}s,\end{array}\}$$ 2.10

where

$$k_1(t,s)=\{\begin{array}{ll}s(1-t),& s\le t,\\ t(1-s),& s>t,\end{array}\text{and}{k}_2(t,s)=\{\begin{array}{ll}\frac{1}{6}s(1-t)(2t-s^2-t^2),& s\le t,\\ \frac{1}{6}t(1-s)(2s-t^2-s^2),& s>t.\end{array}$$ 2.11

It is known that the kernels k₁ and k₂ that occur in (2.11) are continuous, non-negative, satisfy condition (C₇) and (e.g. [22,25,26])

$$K_{10}=\frac{1}{8},\hspace{0.17em}{\mu }_1={\pi }^2,\hspace{0.17em}{K}_{20}=\frac{5}{384},\hspace{0.17em}{\mu }_2={\pi }^4.$$

By direct calculation, we obtain

$$\begin{array}{rl}\frac{\mathrm{\partial }{k}_1}{\mathrm{\partial }t}(t,s)& =\{\begin{array}{ll}-s,& s<t,\\ (1-s),& s>t,\end{array}\frac{\mathrm{\partial }{k}_2}{\mathrm{\partial }t}(t,s)=\frac{1}{6}\{\begin{array}{ll}s(-6t+s^2+3t^2+2),& s\le t,\\ (1-s)(-s^2+2s-3t^2),& s>t,\end{array}\\ \frac{{\mathrm{\partial }}^2{k}_2}{\mathrm{\partial }{t}^2}(t,s)& =\{\begin{array}{ll}s(t-1),& s\le t,\\ t(s-1),& s>t.\end{array}\text{and}\frac{{\mathrm{\partial }}^3{k}_2}{\mathrm{\partial }{t}^3}(t,s)=\{\begin{array}{ll}s,& s<t,\\ (s-1),& s>t.\end{array}\end{array}$$

We may use

$$\begin{array}{rl}{\mathit{\Phi }}_{20}(s)& =\{\begin{array}{ll}\frac{\sqrt{3}}{27}s{(1-s^2)}^{\frac{3}{2}},& \text{for}\hspace{0.17em}0\le s\le \frac{1}{2},\\ \frac{\sqrt{3}}{27}(1-s)s^{\frac{3}{2}}{(2-s)}^{\frac{3}{2}},& \text{for}\hspace{0.17em}\frac{1}{2}<s\le 1,\end{array}[26],\\ {\mathit{\Phi }}_{21}(s)& =\frac{1}{6}s(2+s^2),[15],\end{array}$$

and, by direct calculation, we take

$${\mathit{\Phi }}_{10}(s)={\mathit{\Phi }}_{22}(s)=s(1-s),{\mathit{\Phi }}_{11}(s)={\mathit{\Phi }}_{23}(s)=|s-\frac{1}{2}|+\frac{1}{2}.$$

Therefore the assumptions (C₁)–(C₃) are satisfied. By direct computation, we obtain

$$K_{22}=\frac{1}{8},\hspace{0.17em}{K}_{11}=K_{23}=\frac{1}{2},\hspace{0.17em}{K}_{21}\le \frac{5}{24}.$$

Note that we have

$${\gamma }_{11}(t)=t,\hspace{0.17em}{{\gamma }_{11}}^{\prime }(t)=1,\hspace{0.17em}{\gamma }_{21}(t)=(1-t),\hspace{0.17em}{{\gamma }_{21}}^{\prime }(t)=-1,$$

and therefore we get

$$||{\gamma }_{11}|{|}_{\mathrm{\infty }}=||{\gamma }_{11}^{\prime }|{|}_{\mathrm{\infty }}=||{\gamma }_{21}|{|}_{\mathrm{\infty }}=||{{\gamma }_{21}}^{\prime }|{|}_{\mathrm{\infty }}=1,\hspace{0.17em}||{{\gamma }_{21}}^{″}|{|}_{\mathrm{\infty }}=||{{\gamma }_{21}}^{‴}|{|}_{\mathrm{\infty }}=0,$$

Thus the condition (2.5) is satisfied if

$$max\{\frac{1}{2}{\lambda }_1{\overline{f}}_{1R}+{\eta }_{11}{H}_{11R},\hspace{0.17em}\frac{5}{24}{\lambda }_2{\overline{f}}_{2R}+{\eta }_{21}{H}_{21R},\frac{1}{2}{\lambda }_2{\overline{f}}_{2R}\}\le R.$$ 2.12

Let us now fix the nonlinearities f_i and the functionals h_i1, say

$$\begin{array}{r}{f}_1(t,u_1(t),u_1^{\prime }(t),u_2(t),u_2^{\prime }(t),u_2^{″}(t),u_2^{‴}(t))=u_1^2(t)(2-t\sin (u_1^{\prime }(t)+u_2^{″}(t)),\\ f_2(t,u_1(t),u_1^{\prime }(t),u_2(t),u_2^{\prime }(t),u_2^{″}(t),u_2^{‴}(t))=\sqrt{u_2(t)}{e}^{t(u_1(t)+u_2^{‴}(t))},\\ h_{11}[(u_1,u_2)]={\int }_0^1{(u_1^{\prime }(t)+u_2^{‴}(t))}^2\hspace{0.17em}\text{d}t\\ \text{and}{h}_{21}[(u_1,u_2)]={(u_1^{\prime }(1/4))}^2+{(u_2^{″}(3/4))}^4,\end{array}$$

and prove the existence of solutions in u ∈ P with $||u||\le 1$. Thus we fix R = 1. Since ${\overline{f}}_{11}\le 3$, ${\overline{f}}_{21}\le e^2$, H₁₁₁ ≤ 4, H₂₁₁ ≤ 2, the condition (2.12) is satisfied if the inequality

$$max\{\frac{3}{2}{\lambda }_1+4{\eta }_{11},\hspace{0.17em}\frac{5e^2}{24}{\lambda }_2+2{\eta }_{21},\frac{e^2}{2}{\lambda }_2\}\le 1$$ 2.13

holds. Note that f₂ satisfies condition (2.6) for every fixed λ₂ > 0, by choosing r sufficiently small. Therefore, for the range of parameters that satisfy the inequality (2.13) with λ₂ > 0, theorem 2.3 provides the existence of a solution of the system (2.10) in P, with $0<||u||\le 1$; this occurs, for example, for λ₁ = 1/10, λ₂ = 1/5, η₁₁ = 1/5, η₂₁ = 1/3.

We now use an elementary argument to prove a non-existence result.

Theorem 2.5. —

Assume that there exist τ_i, ξ_ij ∈ (0, + ∞) such that

$$0\le f_i(t,x_{10},\dots ,x_{1m_1},\dots ,x_{n0},\dots ,x_{n{m}_n})\le {\tau }_i{x}_{i0},\hspace{0.17em}\text{on}\hspace{0.17em}[0,1]\times \prod _{i=1}^n([0,+\mathrm{\infty })\times {\mathbb{R}}^{m_i}),$$

$$h_{ij}[u]\le {\xi }_{ij}||u_i|{|}_{\mathrm{\infty }},\hspace{0.17em}\text{for every}\hspace{0.17em}u\in P,\hspace{0.17em}i=1\dots n,\hspace{0.17em}j=1\dots p_i,$$

$$\underset{i=1,\dots ,n}{max}\{{\lambda }_i{\tau }_i{K}_{i0}+\sum _{j=1}^{p_i}{\eta }_{ij}{\xi }_{ij}||{\gamma }_{ij}|{|}_{\mathrm{\infty }}\}<1.$$ 2.14

Then the system (2.1) has at most the zero solution in P.

Proof. —

Assume that there exist $u\in P\setminus \{0\}$ such that Tu = u. Then there exists i₀ ∈ {1, …, n} such that $||u_{i_0}|{|}_{\mathrm{\infty }}=\rho$, for some ρ > 0. Then, for every t ∈ [0, 1], we have

$$\begin{array}{rl}{u}_{i_0}(t)& ={\lambda }_{i_0}{\int }_0^1{k}_{i_0}(t,s)f_{i_0}(s,u_1(s),\dots ,u_1^{(m_1)}(s),\dots ,u_n(s),\dots ,u_n^{(m_n)}(s))\hspace{0.17em}\text{d}s+\sum _{j=1}^{p_i}{\eta }_{i_{0}j}{\gamma }_{i_{0}j}(t)h_{i_{0}j}[u]\\ & \le {\lambda }_{i_0}{\int }_0^1{k}_{i_0}(t,s){\tau }_{i_0}{u}_{i0}\hspace{0.17em}\text{d}s+\sum _{j=1}^{p_i}{\eta }_{i_{0}j}{\gamma }_{i_{0}j}(t)h_{i_{0}j}[u]\\ & \le {\lambda }_{i_0}{\int }_0^1{k}_{i_0}(t,s){\tau }_{i_0}\rho \hspace{0.17em}\text{d}s+\sum _{j=1}^{p_i}{\eta }_{i_{0}j}{\gamma }_{i_{0}j}(t){\xi }_{i_{0}j}\rho \\ & \le {\lambda }_{i_0}{\tau }_{i_0}{K}_{i_{0}0}\rho +\sum _{j=1}^{p_i}{\eta }_{i_{0}j}||{\gamma }_{i_{0}j}|{|}_{\mathrm{\infty }}{\xi }_{i_{0}j}\rho .\end{array}$$ 2.15

Taking the supremum for t ∈ [0, 1] in (2.15) gives ρ < ρ, a contradiction. ▪

We conclude by illustrating the applicability of theorem 2.5.

Example 2.6. —

Let us now consider the system

$$\begin{array}{rl}& u_1^{\mathrm{\prime }\mathrm{\prime }}(t)+{\lambda }_1{u}_1(t)(2-t\sin (u_2(t)u_1^{\prime }(t)))=0,t\in (0,1),\\ & u_2^{(4)}(t)={\lambda }_2{u}_2(t)(2-t\cos (u_1(t)u_2^{‴}(t))),t\in (0,1),\\ & u_1(0)=0,\hspace{0.17em}{u}_1(1)={\eta }_{11}{u}_1(1/4){\cos }^2(u_1^{\prime }(3/4)u_2^{″}(1/4))\\ \text{and}& u_2(0)={\eta }_{21}{u}_2(3/4){\sin }^2(u_1^{\prime }(1/4)u_2^{‴}(3/4)),\hspace{0.17em}{u}_2^{\mathrm{\prime }\mathrm{\prime }}(0)=u_2(1)=u_2^{\mathrm{\prime }\mathrm{\prime }}(1)=0.\end{array}\}$$ 2.16

In this case, we may take τ₁ = τ₂ = 3, ξ₁₁ = ξ₂₁ = 1. Then the condition (2.14) reads

$$max\{\frac{3}{8}{\lambda }_1+{\eta }_{11},\hspace{0.17em}\frac{5}{128}{\lambda }_2+{\eta }_{21}\}<1.$$ 2.17

Since (0, 0) is a solution of the system (2.16), for the range of parameters that satisfy the inequality (2.17), theorem 2.5 guarantees that the only possible solution in P of the BVP (2.16) is the trivial one; this occurs, for example, for λ₁ = 1, λ₂ = 5, η₁₁ = 1/2, η₂₁ = 1/3.

Data accessibility

This article does not contain any additional data.

Competing interests

I declare I have no competing interests.

Funding

The author was partially supported by G.N.A.M.P.A. - INdAM (Italy).