Solvability of the $\Phi$-Laplacian with nonlocal boundary conditions

Abstract

Rather mild sufficient conditions are provided for the existence of positive solutions of a boundary value problem of the form

$$\begin{array}{cc}\hfill & [\Phi (x^{\prime }(t)){]}^{\prime }+c(t)(\mathit{Fx})(t)=0\text{,}\text{a.a.}t\in (0\text{,}1)\text{,}\hfill \\ \hfill & x(0)-L_0(x)=x(1)-L_1(x)=0\text{,}\hfill \end{array}$$

which unify several cases discussed in the literature. In order to formulate these conditions one needs to know only properties of the homeomorphism $\Phi :\mathbb{R}\to \mathbb{R}$ and have information about the level of growth of the response operator F. No metric information concerning the linear operators $L_0\text{,}{L}_1$ in the boundary conditions is used, except that they are positive and continuous and such that $L_j(1)<1$ $j\in \{0\text{,}1\}$.

Introduction

In this paper we study the existence of positive solutions of the nonlocal boundary value problem of the form

$$[\Phi (x^{\prime }(t)){]}^{\prime }+c(t)(\mathit{Fx})(t)=0\text{,}\text{a.a.}t\in (0\text{,}1)\text{,}$$

$$x(0)-L_0(x)=x(1)-L_1(x)=0\text{.}$$

Here $L_0\text{,}{L}_1$ are linear operators, c is a measurable function, $\Phi$ is a specific (increasing) homeomorphism of the real line $\mathbf{R}$ onto itself (called sup-multiplicative-like function as it was defined in [23], [24]) and F is an operator having the property defined below.

First let us make a convention: C is the Banach space of all continuous functions $x:[0\text{,}1]\to \mathbb{R}$ endowed with the sup-norm $\Vert ·\Vert$; $C^{+}$ is the set of all nonnegative functions in C; for any $r>0$ the symbol $B_r$ represents the set of all points $x\in C^{+}$ belonging to the ball $B(0\text{,}r)$.

Property 1.1

There are (nonempty) sets $I\text{,}J\subseteq (0\text{,}1)$ with $\mathit{dist}(I\text{,}\{0\text{,}1\})=:\tau >0$ and J Borel with nonempty interior, satisfying the following condition:

$$(\forall M>0)(\exists m>0)(\forall x\in B_{\frac{m}{\tau }})\Rightarrow \underset{t\in I}{\inf }x(t)⩾m\Rightarrow \underset{t\in J}{\inf }(\mathit{Fx})(t)⩾M\text{.}$$

The infimum $m(M\text{;}I\text{,}J)$ of all such numbers m corresponding to M is a characteristic quantity which expresses the level of growth of the operator F with respect to the sets I and J. It is clear that the correspondence $M\to m(M\text{;}I\text{,}J)$ is a lower semicontinuous function and, as we shall see later, it plays an important role in our approach.

An operator F satisfying Property 1.1 is allowed to be singular¹ and its prototype is a function of the form given in the following example:

Example 1.2

Consider an operator of the form

$$(\mathit{Fu})(t):=\xi (t)+\frac{{\xi }_1(t)u^{{\rho }_1}(g_1(t))}{{\xi }_2(t)+u^{{\rho }_2}(g_2(t))}\text{,}u\in C^{+}\text{,}$$

where $\xi \text{,}{\xi }_1\text{,}{\xi }_2$ are nonnegative real valued functions defined on $[0\text{,}1]$, ${\xi }_2$ is bounded and $g_1\text{,}{g}_2$ are mappings of $[0\text{,}1]$ into itself. Assume that ${\xi }_1$ and $g_1$ are continuous functions and for some $t_0\in (0\text{,}1)$ it holds ${\xi }_1(t_0)>0$ and $0<g_1(t_0)<1$. If the exponents ${\rho }_1\text{,}{\rho }_2$ are such that ${\rho }_1>{\rho }_2>0$, then the function F satisfies Property 1.1.

Indeed, to show this fact, take a $\tau \in (0\text{,}\frac{1}{2})$ with $\tau <g_1(t_0)<1-\tau$ and for some $r\in (0\text{,}{\xi }_1(t_0))$ define

$$J:=\{t\in (0\text{,}1):{\xi }_1(t)⩾r\text{,}\tau ⩽g_1(t)⩽1-\tau \}$$

and

$$I:=g_1(J)\text{.}$$

Now for each $M>0$ we let $m_0$ be a positive real number satisfying the inequality

$${\mathit{rm}}_0^{{\rho }_1}⩾M\left(\Vert {\xi }_2{\Vert }_{\infty }+{\left(\frac{m_0}{\tau }\right)}^{{\rho }_2}\right)\text{,}$$

where (here and in the sequel) $\Vert {\xi }_2{\Vert }_{\infty }$ is the sup-norm of the function ${\xi }_2$. Also let us take any $x\in C^{+}$ with ${\mathit{sup}}_{t\in [0\text{,}1]}x(t)⩽\frac{m_0}{\tau }$ and ${\inf }_{t\in I}x(t)⩾m_0$. Then we see that for all $t\in J$ it holds $g_1(t)\in I$ and

$$(\mathit{Fx})(t)⩾\frac{{\xi }_1(t)x^{{\rho }_1}(g_1(t))}{{\xi }_2(t)+x^{{\rho }_2}(g_2(t))}⩾\frac{{\mathit{rm}}_0^{{\rho }_1}}{\Vert {\xi }_2{\Vert }_{\infty }+{\left(\frac{m_0}{\tau }\right)}^{{\rho }_2}}⩾M\text{.}$$

This argument shows that Property 1.1 is satisfied. Obviously, in this example, the quantity $m(M\text{;}I\text{,}J)$ is a positive real number satisfying $m(M\text{;}I\text{,}J)⩽m_1$, where $m_1$ is the least (positive) root of the equation

$${\mathit{rm}}_1^{{\rho }_1}=M\left(\underset{t\in J}{\sup }{\xi }_2(t)+{\left(\frac{m_1}{\tau }\right)}^{{\rho }_2}\right)\text{.}$$

(Notice that ${\rho }_1>{\rho }_2$.) Another example of a general operator which satisfies Property 1.1 will be given later in Section 2.

During the last two decades, multi-point boundary value problems have been extensively studied and many results have been established. Boundary value problems having closed connection or are specific cases of the problem (1.1), (1.2) are investigated elsewhere in the literature, see, for instance, [1], [2], [3], [4], [5], [8], [9], [10], [11], [12], [13], [14], [15], [17], [16], [18], [19], [20], [21], [28], [30], [31], [33], [34], [35], [36], [37], [38] and the references therein. Among others a short, but informative, survey on this subject was presented by Liu [26], but our interest is focused mainly on the papers [4], [22], [38], which are more closely related to our subject under investigation.

In [4] Bai and Fang, deal with the problem

$$\begin{array}{cc}\hfill & ({\varphi }_p(u^{\prime }){)}^{\prime }+a(t)f(t\text{,}u)=0\text{,}t\in (0\text{,}1)\text{,}\hfill \\ \hfill & u(0)=0\text{,}u(1)=\sum _{i=1}^{m-2}{\alpha }_{i}u({\xi }_i)\text{,}\hfill \end{array}$$

where ${\varphi }_p(s)=|s{|}^{p-2}s$, $p>1$, $0<{\xi }_1<{\xi }_2<\cdots <{\xi }_{m-2}<1$, ${\alpha }_i⩾0$ for $i=1\text{,}\cdots m-3$ and ${\alpha }_{m-2}>0$. They, in order to apply a fixed point theorem in cones based on index theory to get the existence of multiple positive solutions, among others, assume that ${\sum }_1^{m-2}{\alpha }_i{\xi }_i<1$ and there exists $x_0\in ({\xi }_{m-2}\text{,}1)$ such that $a(x_0)>0$.

Also a paper, which was our main motivation to investigate equation (1.1), is the paper [38] due to Zhang and Wang, where the multi-point boundary value problem

$$\begin{array}{cc}\hfill & (p(t)x^{\prime }(t){)}^{\prime }+f(t\text{,}x(t))=0\text{,}t\in (0\text{,}1)\text{,}\hfill \\ \hfill & x(0)-\sum _{i=1}^m{\alpha }_{i}x({\xi }_i)=x(1)-\sum _{i=1}^n{\beta }_{i}x({\xi }_i)=0\hfill \end{array}$$

is investigated. Here $0<{\xi }_i<\cdots <{\xi }_m<1$, ${\alpha }_i\text{,}{\beta }_i\in [0\text{,}+\infty )$ and moreover with

$$0<\sum _{i=1}^m{\alpha }_i<1$$

and

$$0<\sum _{i=1}^m{\beta }_i<1\text{.}$$

Under certain conditions on f existence results for positive solutions of (1.5) are established. A little more general version including the p-Laplacian case of (1.5) was discussed in [22] as well as in its closely related [9].

It is clear that in case $\Phi$ is a function of the form $\Phi (u):=|u{|}^{m-2}u$, Eq. (1.1) may be produced by a nonautonomous m-Laplacian elliptic equation in the n-dimensional space which has radially symmetric solutions. Also (1.1) includes the form of the equation

$$x^{″}+p(x^{\prime })f(t\text{,}x)=0\text{,}$$

where $\inf \{p(u):|u|⩽r\}>0$, for all $r>0$, by setting

$$\Phi (u):={\int }_0^u\frac{d\xi }{p(\xi )}\text{.}$$

Wang [32], motivated by Erbe and Wang [7], has considered the function $\Phi (u)=|u{|}^{p-2}u$, $p>1$ and he studied existence of solutions of an equation of the form satisfying some specific boundary conditions.

In [6], where an equation of the form (1.1) is discussed (but without deviating arguments and with simple Dirichlet conditions), the leading factor depends on an odd homeomorphism $\Phi$, which, in order to guarantee the nonexistence of solutions, actually, satisfies the condition

$$\underset{u>0}{\sup }\frac{\Phi (\mathit{uv})}{\Phi (u)}<+\infty$$

for all $v>0$.

It is well known that in order to seek for positive solutions of operator equations Krasnoselskii presented in [25] a fixed point theorem, which is stated below and which has been proved as a powerful tool in investigating the existence of positive solutions of boundary value problems, see, e. g., most of the papers cited above.

Theorem 1.3 Krasnoselskii [25]

Let $\mathbf{B}$ be a Banach space and let $\mathbf{K}$ be a cone in $\mathbf{B}$. Assume that ${\Omega }_1$, ${\Omega }_2$ are open, bounded subsets of E, with $0\in {\Omega }_1\subset {\overline{\Omega }}_1\subset {\Omega }_2$, and let

$$A:\mathbf{K}\cap ({\overline{\Omega }}_2⧹{\Omega }_1)\to \mathbf{K}$$

be a completely continuous operator such that either

$$\Vert \mathit{Au}\Vert ⩽\Vert u\Vert \text{,}u\in \mathbf{K}\cap \partial {\Omega }_1\text{<mi mathvariant="italic" is="true">and</mi>}\Vert \mathit{Au}\Vert ⩾\Vert u\Vert \text{,}u\in \mathbf{K}\cap \partial {\Omega }_2\text{,}$$

or

$$\Vert \mathit{Au}\Vert ⩾\Vert u\Vert \text{,}u\in \mathbf{K}\cap \partial {\Omega }_1\text{<mi mathvariant="italic" is="true">and</mi>}\Vert \mathit{Au}\Vert ⩽\Vert u\Vert \text{,}u\in \mathbf{K}\cap \partial {\Omega }_2\text{.}$$

Then A has a fixed point in $\mathbf{K}\cap ({\overline{\Omega }}_2⧹{\Omega }_1)$.

The paper is organized as follows: Section 2 is devoted to the conditions of our situation as well as to the formulation of the main theorem concerning the problem. Also we present two examples to illustrate the results. Finally, in Section 3, we give the proof of the main results.