Existence for Two-Point nth Order Boundary Value Problems under Barrier Strips

Abstract

Using barrier strip conditions, we study the solvability of two-point boundary value problems for the equation $x^{\left(n\right)}=f(t,x,x^{\prime },\dots ,x^{(n-1)}).$ In the case $n=4,$ we apply the used approach to obtain results guaranteeing positive or non-negative, monotone, convex solutions to boundary value problems with various boundary conditions.

1. Introduction

We consider boundary value problems (BVPs) for the equation

$$x^{\left(n\right)}=f(t,x,x^{\prime },\dots ,x^{(n-1)}),t\in [0,1],$$

(1)

$n\ge 4,$ with boundary conditions (BCs) either

$$\left\{\begin{array}{c}{x}^{\left(i\right)}\left(0\right)=A_i,i=\overline{0,k_1-1},x^{\left(i\right)}\left(1\right)=A_i,i=\overline{k_1,k_2-1},\dots ,\\ x^{\left(i\right)}\left(0\right)=A_i,i=\overline{k_l,n-3},x^{(n-2)}\left(0\right)=B,x^{(n-2)}\left(1\right)=C,\end{array}\right.$$

(2)

where $l\in \mathbf{N},1\le l\le n-3,$ and $k_j\in \mathbf{N},j=\overline{1,l},$ are fixed with $k_{j-1}+1\le k_j,j=\overline{2,l},$ $k_l\le n-3,$ or

$$\left\{\begin{array}{c}{x}^{\left(i\right)}\left(1\right)=A_i,i=\overline{0,m_1-1},x^{\left(i\right)}\left(0\right)=A_i,i=\overline{m_1,m_2-1},\dots ,\\ x^{\left(i\right)}\left(1\right)=A_i,i=\overline{m_s,n-3},x^{(n-2)}\left(0\right)=B,x^{(n-2)}\left(1\right)=C,\end{array}\right.$$

(3)

where $s\in \mathbf{N},1\le s\le n-3,$ and $m_j\in \mathbf{N},j=\overline{1,s},$ are fixed with $m_{j-1}+1\le m_j,j=\overline{2,s},$ $m_s\le n-3.$

In these problems, the function $f(t,p_0,\dots ,p_{n-1})$ is defined for $(t,p_0,\dots ,p_{n-1})\in [0,1]\times D_0\times D_1\times \cdots \times D_{n-1},D_i\subseteq \mathbf{R},i=\overline{0,n-1},$ and $A_i,i=\overline{0,n-3},B$ and C are real constants.

The solvability of two-point nth order BVPs has been studied, for example, by N. AL-Zaid et al. [1], P. Eloe and J. Henderson [2], M. El-Shahed [3], A. Granas et al. [4], I. Karaca and F. Fen [5], A. Lepin et al. [6], H. Lian et al. [7], M. Pei and S. Chang [8], K. Prasad et al. [9] and N. Vasil’ev et al. [10]. We will focus our attention in more detail on the most recent of these articles, in which additional literature can be found.

In [1], the authors employ the double decomposition method to obtain an approximate solution of BVPs for equations of the form

$$x^{\left(n\right)}+Rx\left(t\right)+Nx\left(t\right)=g\left(t\right),t\in (a,b),$$

where R is a linear differential operator, N is a nonlinear operator and the function g is continuous.

The BVP for (1), $t\in (a,b),$ with BCs

$$\begin{array}{c}\hfill x^{(i-1)}\left(T_1\right)=A_i,i=1,\dots ,n-1,x\left(T_2\right)=A_n,\end{array}$$

has been studied recently in [2]. Here, $a<T_1<T_2<b,$ and $f:(a,b)\times {\mathbf{R}}^n\to \mathbf{R}.$ The authors apply a sequential compactness argument to obtain the existence of solutions under the assumption that the solutions of initial value problems for (1) are unique and extend to $(a,b)$ and the solution of the BVPs for (1) with BCs

$$\begin{array}{c}\hfill x^{(i-1)}\left(T_1\right)=A_i,i=1,\dots ,n-1,x^{(j-1)}\left(T_2\right)=A_n,\end{array}$$

where $j\in \{1,2\}$ are unique if they exist.

A fixed point theorem is used in [5] to obtain the existence of at least two positive solutions for a BVP for

$$x^{\left(n\right)}=f(t,x),t\in (0,1),$$

with integral BCs; here, $f:[0,1]\times [0,\infty )\to [0,\infty )$ is continuous.

A result from [7] guarantees at least three solutions in $C^{n-1}[0,\infty )\cap C^n[0,\infty )$ to the problem

$$x^{\left(n\right)}=q\left(t\right)f(t,x,\dots ,x^{(n-1)}),0<t<+\infty ,$$

$$x^{\left(i\right)}\left(0\right)=A_i,i=0,1,\dots ,n-3,x^{(n-2)}\left(0\right)-a{x}^{(n-1)}\left(0\right)=B,\underset{t\to +\infty }{lim}{x}^{(n-1)}\left(t\right)=C,$$

where $q:(0,+\infty )\to (0,+\infty )$ and $f:[0,+\infty )\times {\mathbf{R}}^n\to \mathbf{R}$ are continuous, $a>0,A_i,B,C\in \mathbf{R},i=0,1,\dots ,n-3.$ Here, the authors use upper and lower solutions and a Nagumo-type condition to apply the Schauder fixed point theorem.

The shooting method together with maximum principle and Kneser–Hukahara continuum theorem have been used in [8] to establish an existence and uniqueness result for (1), $t\in (a,b),$ with BCs

$$g_i(x^{\left(i\right)}\left(a\right),x^{(i+1)}\left(a\right),\dots ,x^{(n-1)}\left(a\right))=0,i=0,1,\dots ,n-2,$$

$$h(x\left(a\right),x^{\prime }\left(a\right),\dots ,x^{(n-1)}\left(a\right);x\left(b\right),x^{\prime }\left(b\right),\dots ,x^{(n-1)}\left(b\right))=0,$$

where $f(t,p_0,p_1,\dots ,p_{n-1})$ is continuous on $[a,b]\times {\mathbf{R}}^n,$ $g_i(p_i,p_{i+1},\dots ,p_{n-1}),i=0,1,\dots ,n-2,$ are continuous on ${\mathbf{R}}^{n-i}$ and $h(p_0,p_1,\dots ,p_{n-1};q_0,q_1,\dots ,q_{n-1})$ is continuous on ${\mathbf{R}}^{2n}.$

We will prove the existence of solutions to (1), (2) and (1), (3) under the following assumptions:

Hypothesis 1 (H1).

For some constants $F_i,L_i,i=1,2,$ such that

$$F_2<F_1\le C-B\le L_1<L_2,[F_2,L_2]\subseteq D_{n-1},$$

it holds

$$\left\{\begin{array}{c}f(t,p_0,\dots ,p_{n-1})\ge 0\\ for(t,p_0,\dots ,p_{n-1})\in [0,1]\times D_0\times D_1\times \cdots \times D_{n-2}\times [L_1,L_2],\end{array}\right.$$

(4)

$$\left\{\begin{array}{c}f(t,p_0,\dots ,p_{n-1})\le 0\\ for(t,p_0,\dots ,p_{n-1})\in [0,1]\times D_0\times D_1\times \cdots \times D_{n-2}\times [F_2,F_1].\end{array}\right.$$

(5)

Hypothesis 2 (H2).

For some constants $F_i^{\prime },L_i^{\prime },i=1,2,$ such that

$$F_2^{\prime }<F_1^{\prime }\le C-B\le L_1^{\prime }<L_2^{\prime },[F_2^{\prime },L_2^{\prime }]\subseteq D_{n-1},$$

it holds

$$\left\{\begin{array}{c}f(t,p_0,\dots ,p_{n-1})\le 0\\ for(t,p_0,\dots ,p_{n-1})\in [0,1]\times D_0\times D_1\times \cdots \times D_{n-2}\times [L_1^{\prime },L_2^{\prime }],\end{array}\right.$$

(6)

$$\left\{\begin{array}{c}f(t,p_0,\dots ,p_{n-1})\ge 0\\ for(t,p_0,\dots ,p_{n-1})\in [0,1]\times D_0\times D_1\times \cdots \times D_{n-2}\times [F_2^{\prime },F_1^{\prime }].\end{array}\right.$$

(7)

Hypothesis 3 (H3).

For a sufficiently small $\delta >0$ and suitable constants $m_i\le M_i,i=\overline{0,n-1}$, it holds

$$[m_i-\delta ,M_i+\delta ]\subseteq D_i,i=\overline{0,n-1},$$

and $f(t,p_0,\dots ,p_{n-1})$ is continuous on $[0,1]\times J,$ where

$J=[m_0-\delta ,M_0+\delta ]\times [m_1-\delta ,M_1+\delta ]\times \cdots \times [m_{n-1}-\delta ,M_{n-1}+\delta ].$

The strips $[0,1]\times [L_1,L_2]$ and $[0,1]\times [F_2,F_1]$ from (H1) as well as $[0,1]\times [L_1^{\prime },L_2^{\prime }]$ and $[0,1]\times [F_2^{\prime },F_1^{\prime }]$ from (H2) are a kind of barriers for the $(n-1)$th derivative of each $C^n[0,1]$-solution to the families of BVPs for

$$x^{\left(n\right)}=\lambda f(t,x,x^{\prime },\dots ,x^{(n-1)}),\lambda \in [0,1],t\in (0,1),{\left(1\right)}_{\lambda }$$

with BCs (2) or (3). This property provides a priori bounds for the $(n-1)$th derivatives of the solutions to (1)${}_{\lambda },$ (2) and (1)${}_{\lambda },$ (3), see Lemma 2. Various other applications of barrier strips can be found for example in R. Agarwal et al. [11,12,13].

For each $C^n[0,1]$-solution $x\left(t\right)$ to families (1)${}_{\lambda },$ (2) and (1)${}_{\lambda },$ (3) there is a $\mu \in (0,1)$ such that $x^{\prime \prime \prime }\left(\mu \right)=C-B.$ The intervals $[F_2,F_1],[F_2^{\prime },F_1^{\prime }],[L_1,L_2]$ and $[L_1^{\prime },L_2^{\prime }]$ are from the coordinate axis $O{p}_{n-1}.$ The first two are to the left of $p_{n-1}=x^{\prime \prime \prime }\left(\mu \right),$ and the others are to the right of $p_{n-1}=x^{\prime \prime \prime }\left(\mu \right).$ This creates the feeling of symmetry with respect to $p_{n-1}=x^{\prime \prime \prime }\left(\mu \right).$ As we will see in Lemma 3, the conditions (H1) and (H2) guarantee that, in the general case, the values of $x\left(t\right),x^{\prime }\left(t\right),\dots ,x^{n-2}$ are in intervals symmetrical to the coordinate origin of the axes $O{p}_0,O{p}_1,\dots ,O{p}_{n-2},$ respectively.

This paper is organized as follows. In Section 2, we obtain a basic existence theorem, our main tool. In Section 3, using (H1) and (H2), we first establish a priori bounds for the eventual $C^n[0,1]$-solutions of families (1)${}_{\lambda },$ (2) and (1)${}_{\lambda },$ (3). Next, we use the obtained bounds to apply the basic existence theorem to obtain a result guaranteeing the solvability of (1), (2) and (1), (3). In Section 4, on fourth-order BVPs with various two-point BCs, we show the possibility of using the applied approach to investigate the existence of solutions with important properties. Some examples to demonstrate our results are given in Section 5.

2. A general Existence Theorem

We will recall notations needed for our discussion.

Let E be a Banach space, Y be its convex subset and $U\subset Y$ be open in Y. The map $F:\overline{U}\to Y$ is called admissible if it is compact and does not have fixed points on $\partial U.$ The set of all such maps denote by $L_{\partial U}(\overline{U},Y).$

A map $F\in L_{\partial U}(\overline{U},Y)$ is called inessential if there exists a compact map $G:\overline{U}\to Y,$ which is fixed point free and coincides with F on $\partial U.$ A map $F\in L_{\partial U}(\overline{U},Y)$ is called essential if it is not inessential.

The maps $F,G\in L_{\partial U}(\overline{U},Y)$ are called homotopic, $F\sim G,$ if there exists a compact homotopy $H_{\lambda }:\overline{U}\to Y,\lambda \in [0,1],$ which is admissible and such that $F=H_0$ and $G=H_1.$

Theorem 1

([4], Chapter I, Theorem 2.2). Let $p\in U$ be fixed and $F\in L_{\partial U}(\overline{U},Y)$ be the constant map $F\left(x\right)=p$ for $x\in \overline{U}.$ Thus, F is essential.

The following lemma gives an important characteristic of the inessential maps.

Lemma 1

([4], Chapter I, Lemma 2.4). A map $F\in L_{\partial U}(\overline{U},Y)$ is inessential if and only if it is homotopic to a fixed point free map.

As a consequence of Lemma 1 we obtain the topological transversality theorem.

Theorem 2

([4], Chapter I, Theorem 2.5). Let F and G in $L_{\partial U}(\overline{U},Y)$ be homotopic maps. Then, one of these maps is essential if and only if the other is.

In fact, the following slightly different but equivalent form of this theorem is used.

Theorem 3

([4], Chapter I, Theorem 2.6). Suppose:

(i) 

$F,G:\overline{U}\to Y$ are compact maps.

(ii) 

$G\in L_{\partial U}(\overline{U},Y)$ is essential.

(iii) 

$H_{\lambda },\lambda \in [0,1],$ is a compact homotopy joining G and $F,$ i.e.,

$$H_0\left(x\right)=G\left(x\right)and{H}_1\left(x\right)=F\left(x\right).$$

(iv) 

$H_{\lambda },\lambda \in [0,1],$ is fixed point free on $\partial U.$

Then, $H_{\lambda },\lambda \in [0,1],$ has at least one fixed point in U and in particular there is a $x_0\in U$ such that $x_0=F\left(x_0\right).$

Now, consider the BVP

$$\left\{\begin{array}{c}{x}^{\left(n\right)}+\sum _{k=0}^{n-1}{s}_k\left(t\right)x^{\left(k\right)}=f(t,x,x^{\prime },\dots ,x^{(n-1)}),t\in [0,1],\\ V_i\left(x\right)=A_i,i=\overline{1,n},\end{array}\right.$$

(8)

where $s_k\left(t\right),k=\overline{0,n-1},$ are continuous on $[0,1],$ $f:[0,1]\times D_0\times D_1\times \cdots \times D_{n-1}\to \mathbf{R},$

$$V_i\left(x\right)\equiv \sum _{j=0}^{n-1}[a_{ij}{x}^{\left(j\right)}\left(0\right)+b_{ij}{x}^{\left(j\right)}\left(1\right)],i=\overline{1,n},$$

with constants $a_{ij}$ and $b_{ij}$ for which ${\sum }_{j=0}^{n-1}(a_{ij}^2+b_{ij}^2)>0,i=\overline{1,n},$ and $A_i\in \mathbf{R}.$

For $\lambda \in [0,1]$, consider also the family of BVPs

$$\left\{\begin{array}{c}{x}^{\left(n\right)}+\sum _{k=0}^{n-1}{s}_k\left(t\right)x^{\left(k\right)}=g(t,x,x^{\prime },\dots ,x^{(n-1)},\lambda ),t\in [0,1],\\ V_i\left(x\right)=A_i,i=\overline{1,n},\end{array}\right.$$

(9)

where $g:[0,1]\times D_0\times D_1\times \cdots \times D_{n-1}\times [0,1]\to \mathbf{R},$ and $s_k\left(t\right),k=\overline{0,n-1},V_i,A_i,i=\overline{1,n},$ are as above.

Let, as usual, $C[0,1]$ be the Banach space of continuous functions on $[0,1]$ with the sup norm ${\left|\right|x\left|\right|}_0=su{p}_{t\in [0,1]}\left|x\left(t\right)\right|,$ and $C^n[0,1]$ be the Banach space of n-times continuously differentiable functions with ${\left|\right|x\left|\right|}_n={max\left\{\right|\left|x\right||}_0,\dots ,\left|\right|x^{\left(n\right)}{\left|\right|}_0\}.$

Let $\mathcal{B}$ denote the set of functions that satisfy the BCs $V_i\left(x\right)=A_i,$ $i=\overline{1,n},$ and ${\mathcal{B}}_0$ be the set of functions satisfying $V_i\left(x\right)=0,i=\overline{1,n}.$ Finally, let $C_{\mathcal{B}}^n[0,1]=C^n[0,1]\cap \mathcal{B}$ and $C_{{\mathcal{B}}_0}^n[0,1]=C^n[0,1]\cap {\mathcal{B}}_0.$

We are now ready to formulate and prove our main theorem. It is a variant of [4], Chapter V, Theorem 1.1.

Theorem 4.

Assume that:

(i) 

For $\lambda =0$ problem (9) has a unique solution $x_0\in C^n[0,1].$

(ii) 

Problems (8) and (9) are equivalent when $\lambda =1.$

(iii) 

The map ${\mathsf{\Lambda }}_h:C_{{\mathcal{B}}_0}^n[0,1]\to C[0,1],$ defined by

$${\mathsf{\Lambda }}_{h}x=x^{\left(n\right)}+\sum _{k=0}^{n-1}{s}_k\left(t\right)x^{\left(k\right)},$$

is one-to-one.

(iv) 

Each solution $x\in C^n[0,1]$ to family (9) satisfies the bounds

$$m_i\le x^{\left(i\right)}\left(t\right)\le M_{i}fort\in [0,1],i=\overline{0,n},$$

where the constants $-\infty <m_i,M_i<\infty ,i=\overline{0,n},$ are independent of λ and x.

(v) 

There is a sufficiently small $\delta >0$ such that

$$[m_i-\delta ,M_i+\delta ]\subseteq D_i,i=\overline{0,n-1},$$

and the function $g(t,p_0,\dots ,p_{n-1},\lambda )$ is continuous on $[0,1]\times J\times [0,1],$ where$J=[m_0-\delta ,M_0+\delta ]\times [m_1-\delta ,M_1+\delta ]\times \cdots \times [m_{n-1}-\delta ,M_{n-1}+\delta ];$$m_i,M_i,i=\overline{0,n-1},$ are as in (iv).

Then, BVP (8) has at least one solution in $C^n[0,1].$

Proof. 

The proof utilizes Theorem 3. Let

$$\overline{U}=\{x\in C_{\mathcal{B}}^n[0,1]:m_i-\delta \le x^{\left(i\right)}\left(t\right)\le M_i+\delta \mathrm{on}[0,1],i=\overline{0,n}\}$$

and define the maps

$$j:C_{\mathcal{B}}^n[0,1]\to C^{n-1}[0,1]\mathrm{by}jx=x,$$

$$\mathsf{\Lambda }:C_{\mathcal{B}}^n[0,1]\to C[0,1]\mathrm{by}\mathsf{\Lambda }x=x^{\left(n\right)}+\sum _{k=0}^{n-1}{s}_k\left(t\right)x^{\left(k\right)},$$

and for $\lambda \in [0,1]$

$${\mathsf{\Phi }}_{\lambda }:C^{n-1}[0,1]\to C[0,1]\mathrm{by}\left({\mathsf{\Phi }}_{\lambda }x\right)\left(t\right)=g(t,x\left(t\right),x^{\prime }\left(t\right),...,x^{(n-1)}\left(t\right),\lambda ),x\in j\left(\overline{U}\right).$$

We need to establish that ${\mathsf{\Lambda }}^{-1}:C[0,1]\to C_{\mathcal{B}}^n[0,1]$ exists and is continuous. First, we will use (iii) which actually means that for each $y\in C[0,1]$ the BVP

$$x^{\left(n\right)}+\sum _{k=0}^{n-1}{s}_k\left(t\right)x^{\left(k\right)}=y\left(t\right),$$

(10)

$$V_i\left(x\right)=0,i=\overline{1,n},$$

has a unique $C^n[0,1]$-solution $x\left(t\right).$ It has the form

$$x\left(t\right)={\tilde{C}}_1{x}_1\left(t\right)+{\tilde{C}}_2{x}_2\left(t\right)+\cdots +{\tilde{C}}_n{x}_n\left(t\right)+\overline{x}\left(t\right),$$

where $x_i\left(t\right),i=\overline{1,n},$ are linearly independend solutions to the homogeneous equation

$$x^{\left(n\right)}+\sum _{k=0}^{n-1}{s}_k\left(t\right)x^{\left(k\right)}=0,$$

(11)

$\overline{x}\left(t\right)$ is a solution to (10), and $({\tilde{C}}_1,{\tilde{C}}_2,\dots ,{\tilde{C}}_n)$ is the unique solution to the system

$$C_1{V}_i\left(x_1\right)+C_2{V}_i\left(x_2\right)+\cdots +C_n{V}_i\left(x_n\right)=-V_i\left(\overline{x}\right),i=\overline{1,n}.$$

However, the last means that the determinant of this system is not zero. Thus, the inhomogeneous system

$$C_1{V}_i\left(x_1\right)+C_2{V}_i\left(x_2\right)+\cdots +C_n{V}_i\left(x_n\right)=A_i,i=\overline{1,n},$$

also has a unique solution, to say $({\breve{C}}_1,{\breve{C}}_2,\dots ,{\breve{C}}_n),$ and so

$$l\left(t\right)={\breve{C}}_1{x}_1\left(t\right)+{\breve{C}}_2{x}_2\left(t\right)+\cdots +{\breve{C}}_n{x}_n\left(t\right)$$

is the unique $C^n[0,1]$-solution to (11) which satisfies the inhomogeneous BCs

$$V_i\left(x\right)=A_i,i=\overline{1,n}.$$

Then, it is not difficult to verify that the inverse map of $\mathsf{\Lambda }$ exists and can be defined by ${\mathsf{\Lambda }}^{-1}y={\mathsf{\Lambda }}_h^{-1}y+l.$ On the other hand, for $x\in C_{{\mathcal{B}}_0}^n[0,1]$, we have

$$\left|\right|{\mathsf{\Lambda }}_h{x\left|\right|}_0\le \left|\right|x^{\left(n\right)}{\left|\right|}_0+\sum _{k=0}^{n-1}\left|\right|s_k{\left|\right|}_0\left|\right|x^{\left(k\right)}{\left|\right|}_0\le {\left|\right|x\left|\right|}_n+\sum _{k=0}^{n-1}\left|\right|s_k{\left|\right|}_0{\left|\right|x\left|\right|}_n=\left(1+\sum _{k=0}^{n-1}\left|\right|s_k{\left|\right|}_0\right){\left|\right|x\left|\right|}_n.$$

Setting $S=1+{\sum }_{k=0}^{n-1}\left|\right|s_k{\left|\right|}_0,$ we obtain

$$\left|\right|{\mathsf{\Lambda }}_h{x\left|\right|}_0\le {S\left|\right|x\left|\right|}_n,$$

which means that the map ${\mathsf{\Lambda }}_h$ is bounded, and so it is continuous because is linear. Thus, ${\mathsf{\Lambda }}_h^{-1}$ is continuous, which implies that ${\mathsf{\Lambda }}^{-1}$ is also continuous.

Now, by

$$H_{\lambda }={\mathsf{\Lambda }}^{-1}{\mathsf{\Phi }}_{\lambda }j$$

define a homotopy $H_{\lambda }:\overline{U}\to C_{\mathcal{B}}^n[0,1].$

It is easily seen that the solutions to (9) are precisely the fixed points of $H_{\lambda }.$ Consequently, by (iv), the homotopy is fixed-point-free on $\partial U.$ Since U is bounded, it follows, from the complete continuity of the embedding $j,$ that $j\left(\overline{U}\right)$ is a compact set. However, in view of (v), the map ${\mathsf{\Phi }}_{\lambda }$ is continuous on $j\left(\overline{U}\right)$ for each $\lambda \in [0,1].$ Additionally, we have already established that ${\mathsf{\Lambda }}^{-1}$ is continuous. Thus, the homotopy $H_{\lambda }$ is compact. Hence $H_0\sim H_1.$ Finally, in view of (i), we have $H_0=x_0.$ Since $x_0\in U,$ $H_0$ is essential by Theorem 1. Then, because $H_0\sim H_1,$ $H_1$ is also essential by Theorem 2. In particular, $H_1$ has a fixed point, (9) has a solution in $C^n[0,1],$ and, by (ii), problem (8) has a solution in $C^n[0,1].$ □

3. Existence Results

With a few auxiliary results, we will prepare the application of Theorem 4.

Lemma 2.

Let (H1) and (H2) hold. Then every solution $x\in C^n[0,1]$ to (1)${}_{\lambda },$ (2) or (1)${}_{\lambda },$ (3) satisfies the bounds

$$min\{F_1,F_1^{\prime }\}\le x^{(n-1)}\left(t\right)\le max\{L_1,L_1^{\prime }\},t\in [0,1].$$

Proof. 

By the Lagrange mean value theorem, there is a $\tau \in (0,1)$ such that

$$x^{(n-1)}\left(\tau \right)=x^{(n-2)}\left(1\right)-x^{(n-2)}\left(0\right)=C-B.$$

Reasoning by contradiction, assume that $x^{(n-1)}\left(t\right)>L_1$ for some $t\in [0,\tau ],$ that is, assume that

$$S_{+}=\{t\in [0,\tau ]:L_1<x^{(n-1)}\left(t\right)\le L_2\}$$

is a non-empty set. However, by the assumption, $x^{(n-1)}\left(\tau \right)\le L_1$ and $x^{(n-1)}\left(t\right)$ is continuous on $[0,\tau ],$ from where it follows that $x^{(n-1)}\left(t\right)$ decreases on any subset of $S_{+}.$ This means that there exists an $\alpha \in S_{+}$ such that

$$x^{\left(n\right)}\left(\alpha \right)<0.$$

On the other hand, in particular for $t=\alpha$, we have

$$x^{\left(n\right)}\left(\alpha \right)=\lambda f(\alpha ,x\left(\alpha \right),x^{\prime }\left(\alpha \right),\dots ,x^{(n-1)}\left(\alpha \right)),$$

because $x\left(t\right)$ is a $C^n[0,1]$-solution to (1)${}_{\lambda }.$ Additionally,

$$(\alpha ,x\left(\alpha \right),x^{\prime }\left(\alpha \right),\dots ,x^{(n-1)}\left(\alpha \right))\in S_{+}\times D_0\times \cdots \times D_{n-2}\times (L_1,L_2].$$

Therefore, we can use (4) to establish

$$\lambda f(\alpha ,x\left(\alpha \right),x^{\prime }\left(\alpha \right),\dots ,x^{(n-1)}\left(\alpha \right))\ge 0,\lambda \in [0,1],$$

which means

$$x^{\left(n\right)}\left(\alpha \right)\ge 0,$$

which is a contradiction. So, the set $S_{+}$ is empty and

$$x^{(n-1)}\left(t\right)\le L_1\mathrm{on}[0,\tau ].$$

Let, again on the contrary, the set

$$S_{-}=\{t\in [0,\tau ]:F_2\le x^{(n-1)}\left(t\right)<F_1\}$$

not be empty. Then, using (5) and proceeding analogously to above, we arrive at a contradiction, which implies that

$$F_1\le x^{(n-1)}\left(t\right)\mathrm{on}[0,\tau ].$$

Now, assume that the set

$$S_{+}^{\prime }=\{t\in [\tau ,1]:F_2^{\prime }\le x^{(n-1)}\left(t\right)<F_1^{\prime }\}$$

is not empty. However, $x^{(n-1)}\left(\tau \right)=C-B\ge F_1^{\prime }$ and $x^{(n-1)}\left(t\right)$ is continuous on $[\tau ,1].$ Hence, there is a $\beta \in S_{+}^{\prime }$ such that

$$x^{\left(n\right)}\left(\beta \right)<0.$$

We have also

$$x^{\left(n\right)}\left(\beta \right)=\lambda f(\beta ,x\left(\beta \right),x^{\prime }\left(\beta \right),\dots ,x^{(n-1)}\left(\beta \right))$$

and $(\beta ,x\left(\beta \right),x^{\prime }\left(\beta \right),\dots ,x^{(n-1)}\left(\beta \right))\in S_{+}^{\prime }\times D_0\times \cdots \times D_{n-2}\times [F_2^{\prime },F_1^{\prime }),$ from where, in view of (7), it follows that

$$x^{\left(n\right)}\left(\beta \right)\ge 0,$$

which is a contradiction. Consequently, the set $S_{+}^{\prime }$ is empty and

$$F_1^{\prime }\le x^{(n-1)}\left(t\right)\mathrm{on}[\tau ,1].$$

Finally, using (6), we establish similarly that the set

$$S_{-}^{\prime }=\{t\in [\tau ,1]:L_1^{\prime }<x^{(n-1)}\left(t\right)\le L_2^{\prime }\}$$

is empty and so

$$x^{(n-1)}\left(t\right)\le L_1^{\prime }\mathrm{on}[\tau ,1].$$

The bounds obtained at the intervals $[0,\tau ]$ and $[\tau ,1]$ yield the lemma. □

Lemma 3.

Let (H1) and (H2) hold. Then, every solution $x\in C^n[0,1]$ to (1)${}_{\lambda },$ (2) or (1)${}_{\lambda },$ (3) satisfies the bounds

$$|x^{(n-2)}\left(t\right)|\le |B|+max\{|F_1|,|F_1^{\prime }|,|L_1|,|L_1^{\prime }\left|\right\},t\in [0,1],$$

$$|x^{\left(i\right)}\left(t\right)|\le |B|+max\{|F_1|,|F_1^{\prime }|,|L_1|,|L_1^{\prime }\left|\right\}+\sum _{j=i}^{n-3}\left|A_j\right|,t\in [0,1],i=\overline{0,n-3}.$$

Proof. 

Let $x\left(t\right)$ be a solution to (1)${}_{\lambda },$ (2). We will establish the truth of the statement by repeatedly applying the mean value theorem. At first, for each $t\in (0,1]$, there is a $\xi \in (0,t)$ such that

$$x^{(n-2)}\left(t\right)-x^{(n-2)}\left(0\right)=x^{(n-1)}\left(\xi \right)t,t\in (0,1],$$

and therefore,

$$|x^{(n-2)}\left(t\right)|\le |x^{(n-2)}\left(0\right)|+|x^{(n-1)}\left(\xi \right)|,t\in [0,1].$$

(12)

However, Lemma 2 implies

$$|x^{(n-1)}\left(t\right)|\le max\{|F_1|,|F_1^{\prime }|,|L_1|,|L_1^{\prime }\left|\right\}\mathrm{for}t\in [0,1]$$

and in particular

$$|x^{(n-1)}\left(\xi \right)|\le max\{|F_1|,|F_1^{\prime }|,|L_1|,|L_1^{\prime }\left|\right\}.$$

Substituting into (12) gives the bound for $|x^{(n-2)}\left(t\right)|.$

Again, from the mean value theorem, for each $t\in (0,1]$, there is an $\eta \in (0,t)$ with the property

$$x^{(n-3)}\left(t\right)-x^{(n-3)}\left(0\right)=x^{(n-2)}\left(\eta \right)t,t\in [0,1),$$

from where, using the obtained bound for $|x^{(n-2)}\left(t\right)|$ and $x^{(n-3)}\left(0\right)=A_{n-3},$ we can establish

$$|x^{(n-3)}\left(t\right)|\le |B|+max\{|F_1|,|F_1^{\prime }|,|L_1|,|L_1^{\prime }\left|\right\}+|A_{n-3}|,t\in [0,1].$$

Likewise, we obtain

$$|x^{\left(i\right)}\left(t\right)|\le |B|+max\{|F_1|,|F_1^{\prime }|,|L_1|,|L_1^{\prime }\left|\right\}+\sum _{j=i}^{n-3}\left|A_j\right|,t\in [0,1],i=\overline{k_l,n-3}.$$

Next, keeping in mind $x^{\left(i\right)}\left(1\right)=A_i,i=\overline{k_{l-1},k_l-1},$ we apply the mean value theorem on intervals of the form $(t,1)$; that is, for each $t\in [0,1)$, there is an $\zeta \in (t,1)$ such that

$$x^{(k_l-1)}\left(1\right)-x^{(k_l-1)}\left(t\right)=x^{\left(k_l\right)}\left(\zeta \right)(1-t),t\in [0,1),$$

which gives successively first

$$|x^{(k_l-1)}\left(t\right)|\le |B|+max\{|F_1|,|F_1^{\prime }|,|L_1|,|L_1^{\prime }\left|\right\}+\sum _{j=k_l}^{n-3}|A_j|+|A_{k_l-1}|,t\in [0,1],$$

and then

$$|x^{\left(i\right)}\left(t\right)|\le |B|+max\{|F_1|,|F_1^{\prime }|,|L_1|,|L_1^{\prime }\left|\right\}+\sum _{j=i}^{n-3}\left|A_j\right|,t\in [0,1],i=\overline{k_{l-1},n-3}.$$

Further, because $x^{\left(i\right)}\left(0\right)=A_i,i=\overline{k_{l-2},k_{l-1}-1},$ we apply $k_{l-1}-k_{l-2}$ times the mean value theorem on intervals of the form $(0,t)$ for each $t\in (0,1]$ to establish

$$|x^{\left(i\right)}\left(t\right)|\le |B|+max\{|F_1|,|F_1^{\prime }|,|L_1|,|L_1^{\prime }\left|\right\}+\sum _{j=i}^{n-3}\left|A_j\right|,t\in [0,1],i=\overline{k_{l-2},n-3}.$$

Continuing this process leads to the establishment of the truth of the lemma. In a similar way, we can prove it if $x\left(t\right)$ is a solution to (1)${}_{\lambda },$ (3). □

Lemma 4.

Each of the BVPs for the equation

$$x^{\left(n\right)}=0$$

(13)

with BCs (2) or (3) has a unique solution.

Proof. 

It is well known that the solutions to (13) are polynomials of the form

$$P_{n-1}\left(t\right)=a_{n-1}{t}^{n-1}+a_{n-2}{t}^{n-2}+\cdots +a_{m_s}{t}^{m_s}+\cdots +a_{m_1}{t}^{m_1}+\cdots +a_3{t}^3+a_2{t}^2+a_{1}t+a_0.$$

(14)

The BCs (3), for example, are satisfied by those polynomials whose coefficients $a_i,i=\overline{0,n-1}$ are a solution of the system:

$$\left\{\begin{array}{ccc}{a}_{n-1}+a_{n-2}+\dots \dots +a_3+a_2+a_1+a_0\hfill & =& A_0\hfill \\ (n-1)a_{n-1}+(n-2)a_{n-2}+\dots ..+3a_3+2a_2+a_1\hfill & =& A_1\hfill \\ (n-1)(n-2)a_{n-1}+(n-2)(n-3)a_{n-2}+\dots .+3.2a_3+2a_2\hfill & =& A_2\hfill \\ ⋮\hfill & & ⋮\hfill \\ m_1!a_{m_1}\hfill & =& A_{m_1}\hfill \\ (m_1+1)!a_{m_1+1}\hfill & =& A_{m_1+1}\hfill \\ ⋮\hfill & & ⋮\hfill \\ (n-1)\dots (n-m_s)a_{n-1}+(n-2)\dots (n-m_s-1)a_{n-2}+\dots +m_s!a_{m_s}\hfill & =& A_{m_s}\hfill \\ ⋮\hfill & & ⋮\hfill \\ (n-1)\dots 3a_{n-1}+(n-2)\dots 2a_{n-2}+(n-3)!a_{n-3}\hfill & =& A_{n-3}\hfill \\ (n-2)!a_{n-2}\hfill & =& B\hfill \\ (n-1)\dots 2a_{n-1}+(n-2)!a_{n-2}\hfill & =& C.\hfill \end{array}\right.$$

From the last two equations, we find successively

$$a_{n-2}=\frac{B}{(n-2)!}\mathrm{and}{a}_{n-1}=\frac{C-B}{(n-1)!}.$$

By substituting $a_{n-1}$ and $a_{n-2}$ in the remaining equations, we obtain a system with unknowns $a_i,i=\overline{0,n-3}.$ The determinant of the new system is:

$$\left|\begin{array}{c}1\dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots 1111\hfill \\ (n-3)\dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots 3210\hfill \\ (n-3)(n-4)\dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots ..3.2200\hfill \\ ⋮⋮\hfill \\ m_1!0\dots \dots \dots .00\hfill \\ (m_1+1)!00\dots \dots \dots .00\hfill \\ ⋮⋮\hfill \\ ⋮⋮\hfill \\ (n-3)\dots (n-m_s-2)\dots \dots m_s!0\dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots .00\hfill \\ ⋮⋮\hfill \\ (n-3)!0\dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots 00\hfill \end{array}\right|=$$

$$=2!3!\dots m_1!(m_1+1)!\dots m_s!\dots (n-3)!,$$

obviously non-zero. Hence, the reduced system has a unique solution. Then, the original system also has a unique solution, say $(a_{n-1}^{*},a_{n-2}^{*},\dots ,a_1^{*},a_0^{*}),$ and so the polynomial

$$P_{n-1}^{*}\left(t\right)=a_{n-1}^{*}{t}^{n-1}+a_{n-2}^{*}{t}^{n-2}+\cdots +a_3^{*}{t}^3+a_2^{*}{t}^2+a_1^{*}t+a_0^{*}$$

is the unique solution of (13), (3).

The validity of the lemma for problem (13), (2) is established by similar reasoning. □

By the end of this part, ${\mathcal{B}}_0$ will be the set of functions satisfying the homogeneous boundary conditions (2) or (3); obviously they are of the form $V_i\left(x\right)=0,i=\overline{1,n}.$

Lemma 5.

The map ${\mathsf{\Lambda }}_h:C_{{\mathcal{B}}_0}^n[0,1]\to C[0,1],$ defined by ${\mathsf{\Lambda }}_{h}x=x^{\left(n\right)},$ is one-to-one.

Proof. 

Let ${\mathcal{B}}_0$ be the set of functions satisfying the homogeneous boundary conditions (3); similar considerations can be applied to the homogeneous conditions (2) as well.

In fact, it is enough to show that, for an arbitrary $y\left(t\right)\in C[0,1]$, the BVP

$$x^{\left(n\right)}=y\left(t\right)$$

(15)

$$\left\{\begin{array}{c}{x}^{\left(i\right)}\left(1\right)=0,i=\overline{0,m_1-1},x^{\left(i\right)}\left(0\right)=0,i=\overline{m_1,m_2-1},\dots ,x^{\left(i\right)}\left(1\right)=0,i=\overline{m_s,n-3},\\ x^{(n-2)}\left(0\right)=0,x^{(n-2)}\left(1\right)=0,\end{array}\right.$$

(16)

has a unique solution in $C^n[0,1].$ We know that the solutions of (15) have the form

$$x\left(t\right)=P_{n-1}\left(t\right)+\eta \left(t\right),$$

where $P_{n-1}\left(t\right)$ is the polynomial (14), and $\eta \left(t\right)$ is a solution to (15). To isolate a solution to (15), (16), we should determine the coefficients $a_i,i=\overline{0,n-1}$ of the polynomial $P_{n-1}\left(t\right)$ from a system similar to that of the proof of Lemma 4, but with

$$A_i=-{\eta }^{\left(i\right)}\left(1\right),i=\overline{0,m_1-1},A_i=-{\eta }^{\left(i\right)}\left(0\right),\overline{m_1,m_2-1},\dots ,A_i=-{\eta }^{\left(i\right)}\left(1\right),i=\overline{m_s,n-3},$$

$$B=-{\eta }^{(n-2)}\left(0\right),C=-{\eta }^{(n-2)}\left(1\right).$$

However, from the proof of Lemma 4, we known that the determinant of this system is nonzero. Therefore, problem (15), (16) has a unique solution which means that ${\mathsf{\Lambda }}_h:C_{{\mathcal{B}}_0}^n[0,1]\to C[0,1]$ is one-to-one. □

Now, we can present the main result of this part.

Theorem 5.

Let (H1) and (H2) hold. Let in addition (H3) hold for

$$M_i=\left|B\right|+max\left\{\right|F_1|,|F_1^{\prime }|,|L_1|,|L_1^{\prime }\left|\right\}+\sum _{j=i}^{n-3}\left|A_j\right|,m_i=-M_i,i=\overline{0,n-3},$$

$$M_{n-2}=\left|B\right|+max\left\{\right|F_1|,|F_1^{\prime }|,|L_1|,|L_1^{\prime }\left|\right\},m_{n-2}=-M_{n-2},$$

$$m_{n-1}=min\{F_1,F_1^{\prime }\},M_{n-1}=max\{L_1,L_1^{\prime }\}.$$

Then, each of the BVPs (1), (2) and (1), (3) has at least one solution in $C^n[0,1].$

Proof. 

We will prove the assertion for (1), (2); essentially the same considerations apply to (1), (3) as well. Moreover, we will check that all hypotheses of Theorem 4 are fullfiled for BVP (1), (2) and family (1)${}_{\lambda },$ (2). According to Lemma 4, (i) is satisfied. Additionally, (ii) is obvious. It follows from Lemma 5 that (iii) holds. Furthermore, for each solution $x\left(t\right)\in C^n[0,1]$ to (1)${}_{\lambda },$ (2), we have

$$m_i\le x^{\left(i\right)}\left(t\right)\le M_i,t\in [0,1],i=\overline{0,n-2},\mathrm{by}\mathrm{Lemma}3,$$

$$m_{n-1}\le x^{(n-1)}\left(t\right)\le M_{n-1},t\in [0,1],\mathrm{by}\mathrm{Lemma}2,$$

Because of the continuity of f on $[0,1]\times J,$ there are constants $m_n$ and $M_n$ such that

$$m_n\le \lambda f(t,p_0,\dots ,p_{n-1})\le M_n\mathrm{for}\lambda \in [0,1]\mathrm{and}(t,p_0,\dots ,p_{n-1})\in [0,1]\times J.$$

However, from above, we have $(x\left(t\right),x^{\prime }\left(t\right),...x^{(n-1)}\left(t\right))\in J$ for $t\in [0,1].$ Thus, Equation (1)${}_{\lambda }$ implies

$$m_n\le x^{\left(n\right)}\left(t\right)\le M_n\mathrm{for}t\in [0,1].$$

Hence, (iv) also holds. Finally, (v) follows again from the continuity of f on the set $J.$ Therefore, the assertion is true by Theorem 4. □

4. Fourth-Order BVPs

The idea from the previous part is also applicable to BVPs with boundary conditions that do not include set values for $x^{(n-2)}\left(0\right)$ and/or $x^{(n-2)}\left(1\right).$ We will demonstrate this on fourth-order BVPs.

So, we consider the equation

$$x^{\left(4\right)}=f(t,x,x^{\prime },x^{″},x^{\prime \prime \prime }),t\in [0,1],$$

(17)

with BCs either

$$\begin{array}{c}\hfill x^{″}\left(0\right)=A,x^{\prime }\left(0\right)=B,x\left(0\right)=C,x^{\prime }\left(1\right)=D,\end{array}$$

(18)

$$\begin{array}{c}\hfill x^{\prime }\left(0\right)=A,x\left(0\right)=B,x\left(1\right)=C,x^{″}\left(1\right)=D,\end{array}$$

(19)

or

$$\begin{array}{c}\hfill x^{\prime }\left(0\right)=A,x\left(0\right)=B,x\left(1\right)=C,x^{\prime }\left(1\right)=D.\end{array}$$

(20)

For convenience, we will reformulate the hypotheses for the case under consideration. In them, the constant K will be further specified for each of the considered BVPs.

Hypothesis 4 (H4).

For some constants $F_i,L_i,i=1,2,$ such that $[F_2,L_2]\subseteq D_3,$ $F_2<F_1\le K\le L_1<L_2,$ it holds

$$\begin{array}{c}\hfill f(t,p_0,p_1,p_2,p_3)\ge 0for(t,p_0,p_1,p_2,p_3)\in [0,1]\times D_0\times D_1\times D_2\times [L_1,L_2],\\ \hfill f(t,p_0,p_1,p_2,p_3)\le 0for(t,p_0,p_1,p_2,p_3)\in [0,1]\times D_0\times D_1\times D_2\times [F_2,F_1].\end{array}$$

Hypothesis 5 (H5).

For some constants $F_i^{\prime },L_i^{\prime },i=1,2,$ such that $[F_2^{\prime },L_2^{\prime }]\subseteq D_3,$ $F_2^{\prime }<F_1^{\prime }\le K\le L_1^{\prime }<L_2^{\prime },$ it holds

$$\begin{array}{c}\hfill f(t,p_0,p_1,p_2,p_3)\le 0for(t,p_0,p_1,p_2,p_3)\in [0,1]\times D_0\times D_1\times D_2\times [L_1^{\prime },L_2^{\prime }],\\ \hfill f(t,p_0,p_1,p_2,p_3)\ge 0for(t,p_0,p_1,p_2,p_3)\in [0,1]\times D_0\times D_1\times D_2\times [F_2^{\prime },F_1^{\prime }].\end{array}$$

Hypothesis 6 (H6).

For a sufficiently small $\delta >0$ and suitable constants $m_i\le M_i,i=\overline{0,3}$, it holds

$$[m_i-\delta ,M_i+\delta ]\subseteq D_i,i=\overline{0,3},$$

and $f(t,p_0,p_1,p_2,p_3)$ is continuous on $[0,1]\times J,$ where

$J=[m_0-\delta ,M_0+\delta ]\times [m_1-\delta ,M_1+\delta ]\times [m_2-\delta ,M_2+\delta ]\times [m_3-\delta ,M_3+\delta ].$

Consider also the families of BVPs for

$$x^{\left(4\right)}=\lambda f(t,x,x^{\prime },x^{″},x^{\prime \prime \prime }),\lambda \in [0,1],t\in (0,1),{\left(17\right)}_{\lambda }$$

with BCs (18)–(20).

Lemma 6.

Let (H4) and (H5) hold for $K=D-B-A.$ Then every solution $x\in C^4[0,1]$ to (17)${}_{\lambda },$ (18) satisfies the bounds

$$min\{F_1,F_1^{\prime }\}\le x^{\prime \prime \prime }\left(t\right)\le max\{L_1,L_1^{\prime }\},t\in [0,1].$$

Proof. 

According to the mean value theorem, there is a $\nu \in (0,1)$ such that $x^{″}\left(\nu \right)=D-B$ and there is a $\tau \in (0,\nu )$ such that $x^{\prime \prime \prime }\left(\tau \right)=x^{″}\left(\nu \right)-x^{″}\left(0\right)=D-B-A.$ Further, the proof follows the proof of Lemma 2, of course for $n=4.$ □

Lemma 7.

Let (H4) and (H5) hold for $K=D-B-A.$ Then, every solution $x\in C^4[0,1]$ to (17)${}_{\lambda },$ (18) satisfies the bounds

$$\left|x\left(t\right)\right|\le \left|A\right|+\left|B\right|+\left|C\right|+max\left\{\right|F_1|,|F_1^{\prime }|,|L_1|,|L_1^{\prime }\left|\right\},t\in [0,1],$$

$$|x^{\prime }\left(t\right)|\le |A|+|B|+max\{|F_1|,|F_1^{\prime }|,|L_1|,|L_1^{\prime }\left|\right\},t\in [0,1],$$

$$|x^{″}\left(t\right)|\le |A|+max\{|F_1|,|F_1^{\prime }|,|L_1|,|L_1^{\prime }\left|\right\},t\in [0,1].$$

Proof. 

According to the mean value theorem, for each $t\in (0,1]$, there is a $\xi \in (0,t)$ such that

$$x^{″}\left(t\right)-x^{″}\left(0\right)=x^{\prime \prime \prime }\left(\xi \right)t,t\in (0,1],$$

and therefore,

$$|x^{″}\left(t\right)|\le |x^{″}\left(0\right)|+|x^{\prime \prime \prime }\left(\xi \right)|,t\in [0,1].$$

(21)

But, from Lemma 6, we have

$$|x^{\prime \prime \prime }\left(t\right)|\le max\{|F_1|,|F_1^{\prime }|,|L_1|,|L_1^{\prime }\left|\right\}fort\in [0,1]$$

and in particular

$$|x^{\prime \prime \prime }\left(\xi \right)|\le max\{|F_1|,|F_1^{\prime }|,|L_1|,|L_1^{\prime }\left|\right\}.$$

Substituting into (21) gives the bound for $|x^{″}\left(t\right)|.$

Again, from the mean value theorem, for each $t\in (0,1]$, there is an $\eta \in (0,t)$ with the property

$$x^{\prime }\left(t\right)-x^{\prime }\left(0\right)=x^{″}\left(\eta \right)t,t\in (0,1],$$

which yields the bound for $|x^{\prime }\left(t\right)|.$

The bound for $\left|x\right(t\left)\right|$ follows similarly by applying the mean value theorem on $x\left(t\right)$ in intervals of the form $(0,t)$ for each $t\in (0,1].$ □

Lemma 8.

Let $A,B\ge 0,D>A+B$ and (H4) and (H5) hold for $K=D-B-A$ with $F_1,F_1^{\prime }\ge 0.$ Then, each solution $x\in C^4[0,1]$ to (17)${}_{\lambda },$ (18) satisfies the bounds

$$C\le x\left(t\right)\le A+B+C+max\{L_1,L_1^{\prime }\},t\in [0,1],$$

$$B\le x^{\prime }\left(t\right)\le A+B+max\{L_1,L_1^{\prime }\},t\in [0,1],$$

(22)

$$A\le x^{″}\left(t\right)\le A+max\{L_1,L_1^{\prime }\},t\in [0,1].$$

(23)

Proof. 

By Lemma 6,

$$0\le min\{F_1,F_1^{\prime }\}\le x^{\prime \prime \prime }\left(t\right)\le max\{L_1,L_1^{\prime }\},t\in [0,1].$$

Then, for $t\in (0,1]$, we have

$$0\le {\int }_0^t{x}^{\prime \prime \prime }\left(s\right)ds\le {\int }_0^{t}max\{L_1,L_1^{\prime }\}ds,$$

which yields consequtively

$$0\le x^{″}\left(t\right)-A\le max\{L_1,L_1^{\prime }\}t,t\in [0,1],$$

$$0\le x^{″}\left(t\right)-A\le max\{L_1,L_1^{\prime }\},t\in [0,1],$$

and (23). Next, by integration of (23) from 0 to $t\in (0,1]$ we obtain (22), and a new integration from 0 to $t\in (0,1]$ gives the bound for $x\left(t\right).$ □

Lemma 9.

Let (H4) and (H5) hold for $K=A+B+D-C.$ Then, every solution $x\in C^4[0,1]$ to (17)${}_{\lambda },$ (19) satisfies the bounds

$$min\{F_1,F_1^{\prime }\}\le x^{\prime \prime \prime }\left(t\right)\le max\{L_1,L_1^{\prime }\},t\in [0,1].$$

Proof. 

By the mean value theorem, there is a $\mu \in (0,1)$ such that $x^{\prime }\left(\mu \right)=C-B$ and there is a $\nu \in (0,\mu )$ such that $x^{″}\left(\nu \right)=x^{\prime }\left(\mu \right)-x^{\prime }\left(0\right)=C-B-A.$ Finally, there is a $\tau \in (\nu ,1)$ for which $x^{\prime \prime \prime }\left(\tau \right)=x^{″}\left(1\right)-x^{″}\left(\nu \right)=A+B+D-C.$ Now the proof continues following the line of proof of Lemma 2. □

Lemma 10.

Let (H4) and (H5) hold for $K=A+B+D-C.$ Then every solution $x\in C^4[0,1]$ to (17)${}_{\lambda },$ (19) satisfies the bounds

$$\left|x\left(t\right)\right|\le \left|A\right|+\left|B\right|+\left|D\right|+max\left\{\right|F_1|,|F_1^{\prime }|,|L_1|,|L_1^{\prime }\left|\right\},t\in [0,1],$$

$$|x^{\prime }\left(t\right)|\le |A|+|D|+max\{|F_1|,|F_1^{\prime }|,|L_1|,|L_1^{\prime }\left|\right\},t\in [0,1],$$

(24)

$$|x^{″}\left(t\right)|\le |D|+max\{|F_1|,|F_1^{\prime }|,|L_1|,|L_1^{\prime }\left|\right\},t\in [0,1].$$

Proof. 

By the mean value theorem for each $t\in [0,1)$ there is a $\xi \in (t,1)$ with the property

$$x^{″}\left(1\right)-x^{″}\left(t\right)=x^{\prime \prime \prime }\left(\xi \right)(1-t),t\in [0,1),$$

from where, using Lemma 9, the bound for $|x^{″}\left(t\right)|$ follows. Further, for each $t\in (0,1]$, there is an $\eta \in (0,t)$ such that

$$x^{\prime }\left(t\right)-x^{\prime }\left(0\right)=x^{″}\left(\eta \right)t,t\in (0,1],$$

which gives (24). Finally, for each $t\in (0,1]$ there is a $\zeta \in (0,t)$ such that

$$x\left(t\right)-x\left(0\right)=x^{\prime }\left(\zeta \right)t,$$

which together with the obtained bound for $|x^{\prime }\left(t\right)|$ yields the bound for $\left|x\right(t\left)\right|.$ □

Lemma 11.

Let $A,D\ge 0,$ $A+B+D<C$ and (H4) and (H5) hold for $K=A+B+D-C$ with $L_1,L_1^{\prime }\le 0.$ Then, each solution $x\in C^4[0,1]$ to (17)${}_{\lambda },$ (19) satisfies the bounds

$$B\le x\left(t\right)\le A+B+D-min\{F_1,F_1^{\prime }\},t\in [0,1],$$

$$A\le x^{\prime }\left(t\right)\le A+D-min\{F_1,F_1^{\prime }\},t\in [0,1],$$

$$D\le x^{″}\left(t\right)\le D-min\{F_1,F_1^{\prime }\},t\in [0,1].$$

(25)

Proof. 

From Lemma 9, we have

$$min\{F_1,F_1^{\prime }\}\le x^{\prime \prime \prime }\left(t\right)\le max\{L_1,L_1^{\prime }\}\le 0,t\in [0,1],$$

which yields

$${\int }_t^{1}min\{F_1,F_1^{\prime }\}ds\le {\int }_t^1{x}^{\prime \prime \prime }\left(s\right)ds\le 0,t\in [0,1),$$

and

$$min\{F_1,F_1^{\prime }\}(1-t)\le D-x^{″}\left(t\right)\le 0,t\in [0,1],$$

from where (25) follows. Then, integrating (25) twice from 0 to $t\in (0,1]$ gives us successively the bounds for $x^{\prime }\left(t\right)$ and $x\left(t\right).$ □

Lemma 12.

Let (H4) and (H5) hold for $K=A+2B+D-2C.$ Then, every solution $x\in C^4[0,1]$ to (17)${}_{\lambda },$ (20) satisfies the bounds

$$min\{F_1,F_1^{\prime }\}\le x^{\prime \prime \prime }\left(t\right)\le max\{L_1,L_1^{\prime }\},t\in [0,1].$$

Proof. 

From the mean value theorem, it follows that there exists a $\mu \in (0,1)$ such that $x^{\prime }\left(\mu \right)=C-B.$ Additionally, there exists a ${\nu }_1\in (0,\mu )$ and a ${\nu }_2\in (\mu ,1)$ with the properties

$$x^{″}\left({\nu }_1\right)=x^{\prime }\left(\mu \right)-x^{\prime }\left(0\right)=C-B-A\text{and}{x}^{″}\left({\nu }_2\right)=x^{\prime }\left(1\right)-x^{\prime }\left(\mu \right)=D-C+B.$$

Finally, there is a $\tau \in ({\nu }_1,{\nu }_2)$ for which $x^{\prime \prime \prime }\left(\tau \right)=x^{″}\left({\nu }_2\right)-x^{″}\left({\nu }_1\right)=A+2B+D-2C.$ In the next part of the proof, we proceed analogously to the proof of Lemma 2. □

Lemma 13.

Let (H4) and (H5) hold for $K=A+2B+D-2C.$ Then, every solution $x\in C^4[0,1]$ to (17)${}_{\lambda },$ (20) satisfies the bounds

$$\left|x\left(t\right)\right|\le \left|A\right|+\left|B\right|+|D-A|+max\left\{\right|F_1|,|F_1^{\prime }|,|L_1|,|L_1^{\prime }\left|\right\},t\in [0,1],$$

$$|x^{\prime }\left(t\right)|\le |A|+|D-A|+max\{|F_1|,|F_1^{\prime }|,|L_1|,|L_1^{\prime }\left|\right\},t\in [0,1],$$

$$|x^{″}\left(t\right)|\le |D-A|+max\{|F_1|,|F_1^{\prime }|,|L_1|,|L_1^{\prime }\left|\right\},t\in [0,1].$$

Proof. 

It is clear, there is a $\mu \in (0,1)$ for which $x^{″}\left(\mu \right)=D-A.$ Then, for each $t\in [0,\mu )$ there is a $\xi \in (t,\mu )$ such that

$$x^{″}\left(\mu \right)-x^{″}\left(t\right)=x^{\prime \prime \prime }\left(\xi \right)(\mu -t),$$

from where, using Lemma 12, we obtain

$$|x^{″}\left(t\right)|\le |D-A|+max\{|F_1|,|F_1^{\prime }|,|L_1|,|L_1^{\prime }\left|\right\},t\in [0,\mu ].$$

We can proceed analogously to see that the same bound is valid for $t\in [\mu ,1].$ Next, the bounds for $|x^{\prime }\left(t\right)|$ and $\left|x\right(t\left)\right|$ follow as in the proof of Lemma 10. □

Lemma 14.

Let $A,D\ge 0,$ $A+2B+D<2C$ and (H4) and (H5) hold for $K=A+2B+D-2C$ with $L_1,L_1^{\prime }\le 0$ and $D-A\ge -min\{F_1,F_1^{\prime }\}.$ Then each solution $x\in C^4[0,1]$ to (17)${}_{\lambda },$ (20) satisfies the bounds

$$B\le x\left(t\right)\le B+D-min\{F_1,F_1^{\prime }\},t\in [0,1],$$

$$A\le x^{\prime }\left(t\right)\le D-min\{F_1,F_1^{\prime }\},t\in [0,1],$$

$$D-A+min\{F_1,F_1^{\prime }\}\le x^{″}\left(t\right)\le D-A-min\{F_1,F_1^{\prime }\},t\in [0,1].$$

(26)

Proof. 

There is a $\mu \in (0,1)$ such that $x^{″}\left(\mu \right)=D-A.$ Now, integrating from $t\in [0,\mu )$ to $\mu$ the bounds

$$min\{F_1,F_1^{\prime }\}\le x^{\prime \prime \prime }\left(t\right)\le max\{L_1,L_1^{\prime }\}\le 0,t\in [0,1],$$

(27)

from Lemma 12, we establish

$$min\{F_1,F_1^{\prime }\}(\mu -t)\le x^{″}\left(\mu \right)-x^{″}\left(t\right)\le 0,t\in [0,\mu ),$$

$$-(D-A)+min\{F_1,F_1^{\prime }\}\le -x^{″}\left(t\right)\le -(D-A),t\in [0,\mu ),$$

$$D-A\le x^{″}\left(t\right)\le D-A-min\{F_1,F_1^{\prime }\},t\in [0,\mu ].$$

Then, integrating (27) from $\mu$ to $t\in (\mu ,1],$ we establish similarly

$$D-A+min\{F_1,F_1^{\prime }\}\le x^{″}\left(t\right)\le D-A,t\in (\mu ,1].$$

As a result, keeping in mind that $min\{F_1,F_1^{\prime }\}\le 0,$ we obtain (26). Integrating (26) from 0 to $t\in (0,1]$ and using $D-A+min\{F_1,F_1^{\prime }\}\ge 0,$ we obtain the bounds for $x^{\prime }\left(t\right).$ A new integration from 0 to $t\in (0,1]$ gives the bounds for $x\left(t\right).$ □

We are ready to obtain the results guaranteeing the solvability of the BVPs under consideration. We will use again Theorem 4, applying it to the families of BVPs for (17)${}_{\lambda }$ with BCs either (18), (19) or (20), which are of the form (9).

Theorem 6.

Let (H4) and (H5) hold for $K=D-B-A.$ Let, in addition, (H6) hold for

$$M_0=\left|A\right|+\left|B\right|+\left|C\right|+max\left\{\right|F_1|,|F_1^{\prime }|,|L_1|,|L_1^{\prime }\left|\right\},m_0=-M_0,$$

$$M_1=\left|A\right|+\left|B\right|+max\left\{\right|F_1|,|F_1^{\prime }|,|L_1|,|L_1^{\prime }\left|\right\},m_1=-M_1,$$

$$M_2=\left|A\right|+max\left\{\right|F_1|,|F_1^{\prime }|,|L_1|,|L_1^{\prime }\left|\right\},m_2=-M_2$$

and

$$m_3=min\{F_1,F_1^{\prime }\},M_3=max\{L_1,L_1^{\prime }\}.$$

Then, BVP (17), (18) has at least one solution in $C^4[0,1].$

Proof. 

It is not difficult to check that (17)${}_0,$ (18), that is, the BVP

$$x^{\left(4\right)}=0,t\in (0,1),$$

$$x^{″}\left(0\right)=A,x^{\prime }\left(0\right)=B,x\left(0\right)=C,x^{\prime }\left(1\right)=D,$$

has a unique solution in $C^4[0,1].$ Thus, we establish that (i) of Theorem 4 is fulfilled. Obviously, (17), (18) and (17)${}_1$ (18) are the same problem, and so (ii) is satisfied. To show that the map ${\mathsf{\Lambda }}_h:C_{{\mathcal{B}}_0}^4[0,1]\to C[0,1],$ defined by ${\mathsf{\Lambda }}_{h}x=x^{\left(4\right)},$ is one-to-one, we simply establish that for each $y\in C[0,1]$, the BVP

$$x^{\left(4\right)}=y\left(t\right),t\in (0,1),$$

$$x^{″}\left(0\right)=0,x^{\prime }\left(0\right)=0,x\left(0\right)=0,x^{\prime }\left(1\right)=0,$$

has a unique solution $x\left(t\right)$ in $C^4[0,1],$and (iii) holds. Next, by Lemma 7 and Lemma 6, for each solution $x\in C^4[0,1]$ to (17)${}_{\lambda },$ (18), we have

$$m_i\le x^{\left(i\right)}\left(t\right)\le M_i,t\in [0,1],i=0,1,2,3.$$

From the continuity of $f(t,p_0,p_1,p_2,p_3)$ on $[0,1]\times J$, it follows that there exist constants $m_4$ and $M_4$ such that

$$m_4\le \lambda f(t,p_0,p_1,p_2,p_3)\le M_4\mathrm{for}(t,p_0,p_1,p_2,p_3)\in [0,1]\times J\mathrm{and}\lambda \in [0,1],$$

which, together with the obtained above $(x\left(t\right),x^{\prime }\left(t\right),x^{″}\left(t\right),x^{\prime \prime \prime }\left(t\right))\in J$ for $t\in [0,1]$ and Equation (17)${}_{\lambda }$, implies

$$m_4\le x^{\left(4\right)}\left(t\right)\le M_4,t\in [0,1].$$

This implies that (iv) holds for (17)${}_{\lambda }$, (18). Again, from the continuity of f on the set J, it follows that (v) is satisfied, and so the assertion is true by Theorem 4. □

Theorem 7.

Let $A,B,C\ge 0(B,C>0),D>A+B,$ (H4) and (H5) hold for $K=D-B-A$ with $F_1,F_1^{\prime }\ge 0$ and (H6) hold for

$$m_0=C,M_0=A+B+C+max\{L_1,L_1^{\prime }\},$$

$$m_1=B,M_1=A+B+max\{L_1,L_1^{\prime }\},$$

$$m_2=A,M_2=A+max\{L_1,L_1^{\prime }\},m_3=min\{F_1,F_1^{\prime }\},M_3=max\{L_1,L_1^{\prime }\}.$$

Then, BVP (17), (18) has at least one non-negative (positive), non-decreasing (increasing), convex solution in $C^4[0,1].$

Proof. 

Essentially the same reasoning as in the proof of Theorem 6 establishes that (17), (18) has a solution $x\left(t\right)$ in $C^4[0,1]$. Moreover, Lemma 8 guarantees the bounds

$$m_0\le x^{\left(i\right)}\left(t\right)\le M_0,t\in [0,1],i=0,1,2,$$

and Lemma 6 yields

$$m_3\le x^{\prime \prime \prime }\left(t\right)\le M_3,t\in [0,1].$$

Since for $t\in [0,1]$ we have $x\left(t\right)\ge C\ge 0(x\left(t\right)\ge C>0),x^{\prime }\left(t\right)\ge B\ge 0(x^{\prime }\left(t\right)\ge B>0)$ and $x^{″}\left(t\right)\ge A\ge 0$, this solution has the desired properties. □

Theorem 8.

Let (H4) and (H5) hold for $K=A+B+D-C$ and (H6) hold for

$$M_0=\left|A\right|+\left|B\right|+\left|D\right|+max\left\{\right|F_1|,|F_1^{\prime }|,|L_1|,|L_1^{\prime }\left|\right\},m_0=-M_0,$$

$$M_1=\left|A\right|+\left|D\right|+max\left\{\right|F_1|,|F_1^{\prime }|,|L_1|,|L_1^{\prime }\left|\right\},m_1=-M_1,$$

$$M_2=\left|D\right|+max\left\{\right|F_1|,|F_1^{\prime }|,|L_1|,|L_1^{\prime }\left|\right\},m_2=-M_2,m_3=min\{F_1,F_1^{\prime }\},M_3=max\{L_1,L_1^{\prime }\}.$$

Then, BVP (17), (19) has at least one solution in $C^4[0,1].$

Proof. 

It is similar to the proof of Theorem 6. Now Lemma 10 guarantees

$$m_i\le x^{\left(i\right)}\left(t\right)\le M_i,t\in [0,1],i=0,1,2,$$

and Lemma 9 yields

$$m_3\le x^{\prime \prime \prime }\left(t\right)\le M_3,t\in [0,1].$$

□

Theorem 9.

Let $A,B,D\ge 0(A,B>0),A+B+D<C,$ (H4) and (H5) hold for $K=A+B+D-C$ with $L_1,L_1^{\prime }\le 0$ and (H6) hold for

$$m_0=B,M_0=A+B+D-min\{F_1,F_1^{\prime }\},$$

$$m_1=A,M_1=A+D-min\{F_1,F_1^{\prime }\},$$

$$m_2=D,M_2=D-min\{F_1,F_1^{\prime }\},m_3=min\{F_1,F_1^{\prime }\},M_3=max\{L_1,L_1^{\prime }\}.$$

Then, BVP (17), (19) has at least one non-negative (positive), non-decreasing (increasing), convex solution in $C^4[0,1].$

Proof. 

Following the proof of Theorem 6 establishes that (17), (19) has a solution $x\left(t\right)$ in $C^4[0,1]$. Now, Lemma 11 provides the bounds

$$m_0\le x^{\left(i\right)}\left(t\right)\le M_0,t\in [0,1],i=0,1,2,$$

and again Lemma 9 yields

$$m_3\le x^{\prime \prime \prime }\left(t\right)\le M_3,t\in [0,1].$$

Since for $t\in [0,1]$ we have $x\left(t\right)\ge B\ge 0(x\left(t\right)\ge B>0),x^{\prime }\left(t\right)\ge A\ge 0(x^{\prime }\left(t\right)\ge A>0)$ and $x^{″}\left(t\right)\ge D\ge 0$, this solution possesses the specified properties. □

Theorem 10.

Let (H4) and (H5) hold for $K=A+2B+D-2C$ and (H6) hold for

$$M_0=\left|A\right|+\left|B\right|+|D-A|+max\left\{\right|F_1|,|F_1^{\prime }|,|L_1|,|L_1^{\prime }\left|\right\},m_0=-M_0,$$

$$M_1=\left|A\right|+|D-A|+max\left\{\right|F_1|,|F_1^{\prime }|,|L_1|,|L_1^{\prime }\left|\right\},m_1=-M_1,$$

$$M_2=|D-A|+max\left\{\right|F_1|,|F_1^{\prime }|,|L_1|,|L_1^{\prime }\left|\right\},m_2=-M_2,$$

$$m_3=min\{F_1,F_1^{\prime }\},M_3=max\{L_1,L_1^{\prime }\}.$$

Then BVP (17), (20) has at least one solution in $C^4[0,1].$

Proof. 

From Lemmas 13 and 12 we have respectively

$$m_i\le x^{\left(i\right)}\left(t\right)\le M_i,t\in [0,1],i=0,1,2,$$

and

$$m_3\le x^{\prime \prime \prime }\left(t\right)\le M_3,t\in [0,1].$$

The rest of the proof does not differ from that of Theorem 6. □

Theorem 11.

Let $A,B,D\ge 0(A,B>0),A+2B+D<2C,$ (H4) and (H5) hold for $K=A+2B+D-2C$ with $L_1,L_1^{\prime }\le 0$ and

$$\begin{array}{c}\hfill D-A\ge -min\{F_1,F_1^{\prime }\},\end{array}$$

(28)

and (H6) hold for

$$m_0=B,M_0=B+D-min\{F_1,F_1^{\prime }\},$$

$$m_1=A,M_1=D-min\{F_1,F_1^{\prime }\},$$

$$m_2=D-A+min\{F_1,F_1^{\prime }\},M_2=D-A-min\{F_1,F_1^{\prime }\},$$

$$m_3=min\{F_1,F_1^{\prime }\},M_3=max\{L_1,L_1^{\prime }\}.$$

Then, BVP (17), (20) has at least one non-negative (positive), non-decreasing (increasing), convex solution in $C^4[0,1].$

Proof. 

As undertaken in the proof of Theorem 6, we establish that (17), (20) has a solution $x\left(t\right)$ in $C^4[0,1]$. Now, the bounds

$$m_0\le x^{\left(i\right)}\left(t\right)\le M_0,t\in [0,1],i=0,1,2,$$

follow from Lemma 14, and Lemma 12 yields

$$m_3\le x^{\prime \prime \prime }\left(t\right)\le M_3,t\in [0,1].$$

Since for $t\in [0,1]$ we have $x\left(t\right)\ge B\ge 0(x\left(t\right)\ge B>0),x^{\prime }\left(t\right)\ge A\ge 0(x^{\prime }\left(t\right)\ge A>0)$ and $x^{″}\left(t\right)\ge D-A+min\{F_1,F_1^{\prime }\}\ge 0$, this solution possesses the specified properties. □

5. Examples

Example 1.

Consider the BVP

$$x^{\left(6\right)}=\phi (t,x,x^{\prime },x^{\prime \prime },x^{\prime \prime \prime },x^{\left(4\right)})P_n\left(x^{\left(5\right)}\right),t\in [0,1],$$

$$x^{\left(4\right)}\left(0\right)=B,x\left(0\right)=A_0,x^{\prime }\left(0\right)=A_1,x^{″}\left(1\right)=A_2,x^{\prime \prime \prime }\left(0\right)=A_3,x^{\left(4\right)}\left(1\right)=C,$$

where $\phi :[0,1]\times {\mathbb{R}}^5\to \mathbb{R}$ is continuous and does not change its sign, and the polynomial $P_n\left(p_5\right),n\ge 2,$ has simple zeros $z_1$ and $z_2$ such that $z_1<C-B<z_2.$

The BCs are of the form (2) with $k_1=2$ and $k_2=3.$

Clearly, there is a $\theta >0$ such that $z_1+\theta \le B-C\le z_2-\theta$ and

$$P_n\left(p_5\right)\ne 0\mathrm{for}{p}_5\in {\cup }_{i=1}^2\left((z_i-\theta ,z_i+\theta )\setminus \left\{z_i\right\}\right).$$

Now, let for concreteness $\phi (t,p_0,p_1,p_2,p_3,p_4)\ge 0$ on $[0,1]\times {\mathbb{R}}^5$ and

$$P_n\left(p_5\right)<0\mathrm{for}{p}_5\in (z_1-\theta ,z_1)\mathrm{and}{P}_n\left(p_5\right)>0\mathrm{for}{p}_5\in (z_2,z_2+\theta );$$

the other cases can be considered analogously. It is easy to verify that (H4) holds for $F_2=z_1-\theta ,F_1=z_1,L_1=z_2,L_2=z_2+\theta$ and (H5) holds for $F_2^{\prime }=z_1,F_1^{\prime }=z_1+\theta ,L_1^{\prime }=z_2-\theta ,L_2^{\prime }=z_2.$ Additionally, since the right-hand side $\phi (t,p_0,p_1,p_2,p_3,p_4)P_n\left(p_5\right)$ of the equation is continuous on $[0,1]\times {\mathbb{R}}^6$, (H6) also holds.

So, we can apply Theorem 5 to conclude that the considered problem has at least one solution in $C^6[0,1].$

Example 2.

Consider the BVP

$$\begin{array}{c}\hfill x^{\left(4\right)}=\frac{(x^{\prime \prime \prime }+9)(x^{\prime \prime \prime }+3)\sqrt{225-x^{\prime \prime 2}}}{\sqrt{900-x^{\prime 2}}},t\in [0,1],\end{array}$$

(29)

$$x^{″}\left(0\right)=-1,x^{\prime }\left(0\right)=4,x\left(0\right)=2,x^{\prime }\left(1\right)=-3.$$

Let us pay attention first that here, $D_0,D_3=(-\infty ,\infty ),$ but $D_1=(-30,30)$ and $D_2=[-15,15]$ are bounded. Since the BCs are of the form (18), $K=D-B-A=-6.$ Then, (H4) and (H5) are satisfied for $F_2=-8,F_1=-7,L_1=-2,L_2=-1,$ $F_2^{\prime }=-11,$ $F_1^{\prime }=-10,$ $L_1^{\prime }=-5,L_2^{\prime }=-4.$ Finally, considering that $m_0=-17,M_0=17,m_1=-15,M_1=15,m_2=-11,M_2=11,m_3=-10$ and $M_3=-2,$ we easily check that (H6) holds for a sufficiently small $\delta >0.$ So, this problem has a solution in $C^4[0,1]$ by Theorem 6.

Example 3.

Consider the BVP for (29) with BCs

$$x^{\prime }\left(0\right)=1,x\left(0\right)=1,x\left(1\right)=7,x^{″}\left(1\right)=0.$$

The BCs are of the form (19). Although now $K=A+B+D-C=-5,$ we can choose the same constants as in Example 2, namely $F_2=-8,F_1=-7,L_1=-2,L_2=-1,$ $F_2^{\prime }=-11,$ $F_1^{\prime }=-10,$ $L_1^{\prime }=-5$ and $L_2^{\prime }=-4,$ to see that (H4) and (H5) are satysfied with $L_1,L_1^{\prime }<0$. Additionally, (H6) holds for $m_0=1,M_0=12,m_1=1,M_1=11,m_2=0,M_2=10,m_3=-10,M_3=-2$ and a sufficiently small $\delta >0.$ So, by Theorem 9, this problem has a positive, increasing, convex solution in $C^4[0,1].$

Example 4.

Consider the BVP for (19) with BCs

$$x^{\prime }\left(0\right)=1,x\left(0\right)=1,x\left(1\right)=10,x^{\prime }\left(1\right)=12.$$

The BCs are of the form (20). Now $K=A+2B+D-2C=-5$ and so we can choose again $F_2=-8,F_1=-7,L_1=-2,L_2=-1,$ $F_2^{\prime }=-11,$ $F_1^{\prime }=-10,$ $L_1^{\prime }=-5$ and $L_2^{\prime }=-4,$ to see that (H4) and (H5) are satysfied with $L_1,L_1^{\prime }<0$ and (28). (H6) also holds for $m_0=1,M_0=23,m_1=1,M_1=22,m_2=1,M_2=21,m_3=-10,M_3=-2$ and a sufficiently small $\delta >0.$ So, by Theorem 11, this problem has a positive, increasing, convex solution in $C^4[0,1].$

6. Conclusions

This paper shows how the barrier strip technique (based here on conditions (H1) and (H2)) can be used to study the solvability of two-point BVPs for nth-order nonlinear differential equations. The idea is realized on two BVPs, but it can be applied to others problems as well. This possibility is demonstrated on fourth-order BVPs for which results guaranteeing positive or non-negative, monotone, convex solutions have been also obtained.