## Boundary Value Problems

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# Solutions and nonnegative solutions for a weighted variable exponent impulsive integro-differential system with multi-point and integral mixed boundary value problems

Boundary Value Problems20132013:161

DOI: 10.1186/1687-2770-2013-161

Accepted: 18 June 2013

Published: 5 July 2013

## Abstract

This paper investigates the existence of solutions for a weighted $p\left(t\right)$-Laplacian impulsive integro-differential system with multi-point and integral mixed boundary value problems via Leray-Schauder’s degree; sufficient conditions for the existence of solutions are given. Moreover, we get the existence of nonnegative solutions.

MSC:34B37.

### Keywords

weighted $p\left(t\right)$-Laplacian impulsive integro-differential system Leray-Schauder’s degree

## 1 Introduction

In this paper, we consider the existence of solutions and nonnegative solutions for the following weighted $p\left(t\right)$-Laplacian integro-differential system:
$-{△}_{p\left(t\right)}u+f\left(t,u,{\left(w\left(t\right)\right)}^{\frac{1}{p\left(t\right)-1}}{u}^{\mathrm{\prime }},S\left(u\right),T\left(u\right)\right)=0,\phantom{\rule{1em}{0ex}}t\in \left(0,1\right),t\ne {t}_{i},$
(1)
where $u:\left[0,1\right]\to {\mathbb{R}}^{N}$, $f\left(\cdot ,\cdot ,\cdot ,\cdot ,\cdot \right):\left[0,1\right]×{\mathbb{R}}^{N}×{\mathbb{R}}^{N}×{\mathbb{R}}^{N}×{\mathbb{R}}^{N}\to {\mathbb{R}}^{N}$, ${t}_{i}\in \left(0,1\right)$, $i=1,\dots ,k$, with the following impulsive boundary value conditions:
$\underset{t\to {t}_{i}^{+}}{lim}u\left(t\right)-\underset{t\to {t}_{i}^{-}}{lim}u\left(t\right)={A}_{i}\left(\underset{t\to {t}_{i}^{-}}{lim}u\left(t\right),\underset{t\to {t}_{i}^{-}}{lim}{\left(w\left(t\right)\right)}^{\frac{1}{p\left(t\right)-1}}{u}^{\mathrm{\prime }}\left(t\right)\right),\phantom{\rule{1em}{0ex}}i=1,\dots ,k,$
(2)
$\begin{array}{c}\underset{t\to {t}_{i}^{+}}{lim}w\left(t\right)|{u}^{\mathrm{\prime }}{|}^{p\left(t\right)-2}{u}^{\mathrm{\prime }}\left(t\right)-\underset{t\to {t}_{i}^{-}}{lim}w\left(t\right)|{u}^{\mathrm{\prime }}{|}^{p\left(t\right)-2}{u}^{\mathrm{\prime }}\left(t\right)\hfill \\ \phantom{\rule{1em}{0ex}}={B}_{i}\left(\underset{t\to {t}_{i}^{-}}{lim}u\left(t\right),\underset{t\to {t}_{i}^{-}}{lim}{\left(w\left(t\right)\right)}^{\frac{1}{p\left(t\right)-1}}{u}^{\mathrm{\prime }}\left(t\right)\right),\phantom{\rule{1em}{0ex}}i=1,\dots ,k,\hfill \end{array}$
(3)
$u\left(0\right)={\int }_{0}^{1}g\left(t\right)u\left(t\right)\phantom{\rule{0.2em}{0ex}}dt,\phantom{\rule{2em}{0ex}}u\left(1\right)=\sum _{\ell =1}^{m-2}{\alpha }_{\ell }u\left({\xi }_{\ell }\right)-{\int }_{0}^{1}h\left(t\right)u\left(t\right)\phantom{\rule{0.2em}{0ex}}dt,$
(4)

where $p\in C\left(\left[0,1\right],\mathbb{R}\right)$ and $p\left(t\right)>1$, $-{△}_{p\left(t\right)}u:=-{\left(w\left(t\right)|{u}^{\mathrm{\prime }}{|}^{p\left(t\right)-2}{u}^{\mathrm{\prime }}\right)}^{\mathrm{\prime }}$ is called the weighted $p\left(t\right)$-Laplacian; $0<{t}_{1}<{t}_{2}<\cdots <{t}_{k}<1$, $0<{\xi }_{1}<\cdots <{\xi }_{m-2}<1$; ${\alpha }_{\ell }\ge 0$ ($\ell =1,\dots ,m-2$); $g\in {L}^{1}\left[0,1\right]$ is nonnegative, ${\int }_{0}^{1}g\left(t\right)\phantom{\rule{0.2em}{0ex}}dt=\sigma \in \left[0,1\right]$; $h\in {L}^{1}\left[0,1\right]$, ${\int }_{0}^{1}h\left(t\right)\phantom{\rule{0.2em}{0ex}}dt=\delta$; ${A}_{i},{B}_{i}\in C\left({\mathbb{R}}^{N}×{\mathbb{R}}^{N},{\mathbb{R}}^{N}\right)$; T and S are linear operators defined by $\left(Su\right)\left(t\right)={\int }_{0}^{1}{h}_{\ast }\left(t,s\right)u\left(s\right)\phantom{\rule{0.2em}{0ex}}ds$, $\left(Tu\right)\left(t\right)={\int }_{0}^{t}{k}_{\ast }\left(t,s\right)u\left(s\right)\phantom{\rule{0.2em}{0ex}}ds$, $t\in \left[0,1\right]$, where ${k}_{\ast },{h}_{\ast }\in C\left(\left[0,1\right]×\left[0,1\right],\mathbb{R}\right)$.

If $\sigma <1$ and ${\sum }_{\ell =1}^{m-2}{\alpha }_{\ell }-\delta \ne 1$, we say the problem is nonresonant, but if $\sigma =1$ or ${\sum }_{\ell =1}^{m-2}{\alpha }_{\ell }-\delta =1$, we say the problem is resonant.

Throughout the paper, $o\left(1\right)$ means functions which are uniformly convergent to 0 (as $n\to +\mathrm{\infty }$); for any $v\in {\mathbb{R}}^{N}$, ${v}^{j}$ will denote the j th component of v; the inner product in ${\mathbb{R}}^{N}$ will be denoted by $〈\cdot ,\cdot 〉$, $|\cdot |$ will denote the absolute value and the Euclidean norm on ${\mathbb{R}}^{N}$. Denote $J=\left[0,1\right]$, ${J}^{\mathrm{\prime }}=\left(0,1\right)\mathrm{\setminus }\left\{{t}_{1},\dots ,{t}_{k}\right\}$, ${J}_{0}=\left[{t}_{0},{t}_{1}\right]$, ${J}_{i}=\left({t}_{i},{t}_{i+1}\right]$, $i=1,\dots ,k$, where ${t}_{0}=0$, ${t}_{k+1}=1$. Denote by ${J}_{i}^{o}$ the interior of ${J}_{i}$, $i=0,1,\dots ,k$. Let
$w\in PC\left(J,\mathbb{R}\right)$ satisfy $0, $\mathrm{\forall }t\in \left(0,1\right)\mathrm{\setminus }\left\{{t}_{1},\dots ,{t}_{k}\right\}$, and ${\left(w\left(t\right)\right)}^{\frac{-1}{p\left(t\right)-1}}\in {L}^{1}\left(0,1\right)$,

For any $x=\left({x}^{1},\dots ,{x}^{N}\right)\in PC\left(J,{\mathbb{R}}^{N}\right)$, denote ${|{x}^{i}|}_{0}=sup\left\{|{x}^{i}\left(t\right)|\mid t\in {J}^{\mathrm{\prime }}\right\}$.

Obviously, $PC\left(J,{\mathbb{R}}^{N}\right)$ is a Banach space with the norm ${\parallel x\parallel }_{0}={\left({\sum }_{i=1}^{N}{|{x}^{i}|}_{0}^{2}\right)}^{\frac{1}{2}}$, and $P{C}^{1}\left(J,{\mathbb{R}}^{N}\right)$ is a Banach space with the norm ${\parallel x\parallel }_{1}={\parallel x\parallel }_{0}+{\parallel {\left(w\left(t\right)\right)}^{\frac{1}{p\left(t\right)-1}}{x}^{\mathrm{\prime }}\parallel }_{0}$. Denote ${L}^{1}={L}^{1}\left(J,{\mathbb{R}}^{N}\right)$ with the norm
In the following, $PC\left(J,{\mathbb{R}}^{N}\right)$ and $P{C}^{1}\left(J,{\mathbb{R}}^{N}\right)$ will be simply denoted by PC and $P{C}^{1}$, respectively. We denote
$\begin{array}{c}u\left({t}_{i}^{+}\right)=\underset{t\to {t}_{i}^{+}}{lim}u\left(t\right),\phantom{\rule{2em}{0ex}}u\left({t}_{i}^{-}\right)=\underset{t\to {t}_{i}^{-}}{lim}u\left(t\right),\hfill \\ w\left(0\right)|{u}^{\mathrm{\prime }}{|}^{p\left(0\right)-2}{u}^{\mathrm{\prime }}\left(0\right)=\underset{t\to {0}^{+}}{lim}w\left(t\right)|{u}^{\mathrm{\prime }}{|}^{p\left(t\right)-2}{u}^{\mathrm{\prime }}\left(t\right),\hfill \\ w\left(1\right)|{u}^{\mathrm{\prime }}{|}^{p\left(1\right)-2}{u}^{\mathrm{\prime }}\left(1\right)=\underset{t\to {1}^{-}}{lim}w\left(t\right)|{u}^{\mathrm{\prime }}{|}^{p\left(t\right)-2}{u}^{\mathrm{\prime }}\left(t\right),\hfill \\ {A}_{i}={A}_{i}\left(\underset{t\to {t}_{i}^{-}}{lim}u\left(t\right),\underset{t\to {t}_{i}^{-}}{lim}{\left(w\left(t\right)\right)}^{\frac{1}{p\left(t\right)-1}}{u}^{\mathrm{\prime }}\left(t\right)\right),\phantom{\rule{1em}{0ex}}i=1,\dots ,k,\hfill \\ {B}_{i}={B}_{i}\left(\underset{t\to {t}_{i}^{-}}{lim}u\left(t\right),\underset{t\to {t}_{i}^{-}}{lim}{\left(w\left(t\right)\right)}^{\frac{1}{p\left(t\right)-1}}{u}^{\mathrm{\prime }}\left(t\right)\right),\phantom{\rule{1em}{0ex}}i=1,\dots ,k.\hfill \end{array}$
The study of differential equations and variational problems with nonstandard $p\left(t\right)$-growth conditions has attracted more and more interest in recent years (see [14]). The applied background of these kinds of problems includes nonlinear elasticity theory [4], electro-rheological fluids [1, 3], and image processing [2]. Many results have been obtained on these kinds of problems; see, for example, [515]. Recently, the applications of variable exponent analysis in image restoration have attracted more and more attention [1619]. If $p\left(t\right)\equiv p$ (a constant), (1)-(4) becomes the well-known p-Laplacian problem. If $p\left(t\right)$ is a general function, one can see easily $-{△}_{p\left(t\right)}cu\ne {c}^{p\left(t\right)-1}\left(-{△}_{p\left(t\right)}u\right)$ in general, but $-{△}_{p}cu={c}^{p-1}\left(-{△}_{p}u\right)$, so $-{△}_{p\left(t\right)}$ represents a non-homogeneity and possesses more nonlinearity, thus $-{△}_{p\left(t\right)}$ is more complicated than $-{△}_{p}$. For example:
1. (a)
If $\mathrm{\Omega }\subset {\mathbb{R}}^{N}$ is a bounded domain, the Rayleigh quotient
${\lambda }_{p\left(x\right)}=\underset{u\in {W}_{0}^{1,p\left(x\right)}\left(\mathrm{\Omega }\right)\mathrm{\setminus }\left\{0\right\}}{inf}\frac{{\int }_{\mathrm{\Omega }}\frac{1}{p\left(x\right)}|\mathrm{\nabla }u{|}^{p\left(x\right)}\phantom{\rule{0.2em}{0ex}}dx}{{\int }_{\mathrm{\Omega }}\frac{1}{p\left(x\right)}|u{|}^{p\left(x\right)}\phantom{\rule{0.2em}{0ex}}dx}$

is zero in general, and only under some special conditions ${\lambda }_{p\left(x\right)}>0$ (see [9]), when $\mathrm{\Omega }\subset \mathbb{R}$ ($N=1$) is an interval, the results show that ${\lambda }_{p\left(x\right)}>0$ if and only if $p\left(x\right)$ is monotone. But the property of ${\lambda }_{p}>0$ is very important in the study of p-Laplacian problems, for example, in [20], the authors use this property to deal with the existence of solutions.
1. (b)

If $w\left(t\right)\equiv 1$ and $p\left(t\right)\equiv p$ (a constant) and $-{△}_{p}u>0$, then u is concave, this property is used extensively in the study of one-dimensional p-Laplacian problems (see [21]), but it is invalid for $-{△}_{p\left(t\right)}$. It is another difference between $-{△}_{p}$ and $-{△}_{p\left(t\right)}$.

In recent years, many results have been devoted to the existence of solutions for the Laplacian impulsive differential equation boundary value problems; see, for example, [2229]. There are some methods to deal with these problems, for example, sub-super-solution method, fixed point theorem, monotone iterative method, coincidence degree. Because of the nonlinear property of $-{△}_{p}$, results on the existence of solutions for p-Laplacian impulsive differential equation boundary value problems are rare (see [3033]). In [34], using the coincidence degree method, the present author investigates the existence of solutions for $p\left(r\right)$-Laplacian impulsive differential equation with multi-point boundary value conditions, when the problem is nonresonant. Integral boundary conditions for evolution problems have various applications in chemical engineering, thermo-elasticity, underground water flow and population dynamics. There are many papers on the differential equations with integral boundary value problems; see, for example, [3538].

In this paper, when $p\left(t\right)$ is a general function, we investigate the existence of solutions and nonnegative solutions for the weighted $p\left(t\right)$-Laplacian impulsive integro-differential system with integral and multi-point boundary value conditions. Results on these kinds of problems are rare. Our results contain both of the cases of resonance and nonresonance. Our method is based upon Leray-Schauder’s degree. The homotopy transformation used in [34] is unsuitable for this paper. Moreover, this paper will consider the existence of (1) with (2), (4) and the following impulsive condition:
$\begin{array}{r}\underset{t\to {t}_{i}^{+}}{lim}{\left(w\left(t\right)\right)}^{\frac{1}{p\left(t\right)-1}}{u}^{\mathrm{\prime }}\left(t\right)-\underset{t\to {t}_{i}^{-}}{lim}{\left(w\left(t\right)\right)}^{\frac{1}{p\left(t\right)-1}}{u}^{\mathrm{\prime }}\left(t\right)\\ \phantom{\rule{1em}{0ex}}={D}_{i}\left(\underset{t\to {t}_{i}^{-}}{lim}u\left(t\right),\underset{t\to {t}_{i}^{-}}{lim}{\left(w\left(t\right)\right)}^{\frac{1}{p\left(t\right)-1}}{u}^{\mathrm{\prime }}\left(t\right)\right),\phantom{\rule{1em}{0ex}}i=1,\dots ,k,\end{array}$
(5)

where ${D}_{i}\in C\left({\mathbb{R}}^{N}×{\mathbb{R}}^{N},{\mathbb{R}}^{N}\right)$, the impulsive condition (5) is called a linear impulsive condition (LI for short), and (3) is called a nonlinear impulsive condition (NLI for short). In general, p-Laplacian impulsive problems have two kinds of impulsive conditions, including LI and NLI; but Laplacian impulsive problems only have LI in general. It is another difference between p-Laplacian impulsive problems and Laplacian impulsive problems. Moreover, since the Rayleigh quotient ${\lambda }_{p\left(x\right)}=0$ in general and the $p\left(t\right)$-Laplacian is non-homogeneity, when we deal with the existence of solutions of variable exponent impulsive problems like (1)-(4), we usually need the nonlinear term that satisfies the sub-$\left({p}^{-}-1\right)$ growth condition, but for the p-Laplacian impulsive problems, the nonlinear term only needs to satisfy the sub-$\left(p-1\right)$ growth condition.

Let $N\ge 1$, the function $f:J×{\mathbb{R}}^{N}×{\mathbb{R}}^{N}×{\mathbb{R}}^{N}×{\mathbb{R}}^{N}\to {\mathbb{R}}^{N}$ is assumed to be Caratheodory, by which we mean:
1. (i)

For almost every $t\in J$, the function $f\left(t,\cdot ,\cdot ,\cdot ,\cdot \right)$ is continuous;

2. (ii)

For each $\left(x,y,s,z\right)\in {\mathbb{R}}^{N}×{\mathbb{R}}^{N}×{\mathbb{R}}^{N}×{\mathbb{R}}^{N}$, the function $f\left(\cdot ,x,y,s,z\right)$ is measurable on J;

3. (iii)
For each $R>0$, there is a ${\alpha }_{R}\in {L}^{1}\left(J,\mathbb{R}\right)$ such that, for almost every $t\in J$ and every $\left(x,y,s,z\right)\in {\mathbb{R}}^{N}×{\mathbb{R}}^{N}×{\mathbb{R}}^{N}×{\mathbb{R}}^{N}$ with $|x|\le R$, $|y|\le R$, $|s|\le R$, $|z|\le R$, one has
$|f\left(t,x,y,s,z\right)|\le {\alpha }_{R}\left(t\right).$

We say a function $u:J\to {\mathbb{R}}^{N}$ is a solution of (1) if $u\in P{C}^{1}$ with $w\left(t\right)|{u}^{\mathrm{\prime }}{|}^{p\left(t\right)-2}{u}^{\mathrm{\prime }}$ absolutely continuous on ${J}_{i}^{o}$, $i=0,1,\dots ,k$, which satisfies (1) a.e. on J.

In this paper, we always use ${C}_{i}$ to denote positive constants, if it cannot lead to confusion. Denote
We say f satisfies the sub-$\left({p}^{-}-1\right)$ growth condition if f satisfies

where $q\left(t\right)\in PC\left(J,\mathbb{R}\right)$ and $1<{q}^{-}\le {q}^{+}<{p}^{-}$.

We will discuss the existence of solutions for system (1)-(4) or (1) with (2), (4) and (5) in the following three cases:

Case (i): $\sigma <1$, ${\sum }_{\ell =1}^{m-2}{\alpha }_{\ell }-\delta =1$;

Case (ii): $\sigma =1$, ${\sum }_{\ell =1}^{m-2}{\alpha }_{\ell }-\delta \ne 1$;

Case (iii): $\sigma <1$, ${\sum }_{\ell =1}^{m-2}{\alpha }_{\ell }-\delta <1$.

This paper is organized as five sections. In Section 2, we present some preliminaries and give the operator equation which has the same solutions of (1)-(4) in the three cases, respectively. In Section 3, we give the existence of solutions for system (1)-(4) or (1) with (2), (4) and (5) when $\sigma <1$, ${\sum }_{\ell =1}^{m-2}{\alpha }_{\ell }-\delta =1$. In Section 4, we give the existence of solutions for system (1)-(4) or (1) with (2), (4) and (5) when $\sigma =1$, ${\sum }_{\ell =1}^{m-2}{\alpha }_{\ell }-\delta \ne 1$. Finally, in Section 5, we give the existence of solutions and nonnegative solutions for system (1)-(4) or (1) with (2), (4) and (5) when $\sigma <1$, ${\sum }_{\ell =1}^{m-2}{\alpha }_{\ell }-\delta <1$.

## 2 Preliminary

For any $\left(t,x\right)\in J×{\mathbb{R}}^{N}$, denote $\phi \left(t,x\right)=|x{|}^{p\left(t\right)-2}x$. Obviously, φ has the following properties.

Lemma 2.1 (see [34])

φ is a continuous function and satisfies:
1. (i)
For any $t\in \left[0,1\right]$, $\phi \left(t,\cdot \right)$ is strictly monotone, i.e.,
$〈\phi \left(t,{x}_{1}\right)-\phi \left(t,{x}_{2}\right),{x}_{1}-{x}_{2}〉>0\phantom{\rule{1em}{0ex}}\mathit{\text{for any}}\phantom{\rule{0.1em}{0ex}}{x}_{1},{x}_{2}\in {\mathbb{R}}^{N},{x}_{1}\ne {x}_{2}.$

2. (ii)
There exists a function $\alpha :\left[0,+\mathrm{\infty }\right)\to \left[0,+\mathrm{\infty }\right)$, $\alpha \left(s\right)\to +\mathrm{\infty }$ as $s\to +\mathrm{\infty }$ such that
$〈\phi \left(t,x\right),x〉\ge \alpha \left(|x|\right)|x|\phantom{\rule{1em}{0ex}}\mathit{\text{for all}}\phantom{\rule{0.1em}{0ex}}x\in {\mathbb{R}}^{N}.$

It is well known that $\phi \left(t,\cdot \right)$ is a homeomorphism from ${\mathbb{R}}^{N}$ to ${\mathbb{R}}^{N}$ for any fixed $t\in J$. Denote

It is clear that ${\phi }^{-1}\left(t,\cdot \right)$ is continuous and sends bounded sets to bounded sets.

In this section, we will do some preparation and give the operator equation which has the same solutions of (1)-(4) in three cases, respectively. At first, let us now consider the following simple impulsive problem with boundary value condition (4):
$\begin{array}{l}{\left(w\left(t\right)\phi \left(t,{u}^{\mathrm{\prime }}\left(t\right)\right)\right)}^{\mathrm{\prime }}=f\left(t\right),\phantom{\rule{1em}{0ex}}t\in \left(0,1\right),t\ne {t}_{i},\\ {lim}_{t\to {t}_{i}^{+}}u\left(t\right)-{lim}_{t\to {t}_{i}^{-}}u\left(t\right)={a}_{i},\phantom{\rule{1em}{0ex}}i=1,\dots ,k,\\ {lim}_{t\to {t}_{i}^{+}}w\left(t\right)|{u}^{\mathrm{\prime }}{|}^{p\left(t\right)-2}{u}^{\mathrm{\prime }}\left(t\right)-{lim}_{t\to {t}_{i}^{-}}w\left(t\right)|{u}^{\mathrm{\prime }}{|}^{p\left(t\right)-2}{u}^{\mathrm{\prime }}\left(t\right)={b}_{i},\phantom{\rule{1em}{0ex}}i=1,\dots ,k,\end{array}\right\}$
(6)

where ${a}_{i},{b}_{i}\in {\mathbb{R}}^{N}$; $f\in {L}^{1}$.

Denote $a=\left({a}_{1},\dots ,{a}_{k}\right)$, $b=\left({b}_{1},\dots ,{b}_{k}\right)$. Obviously, $a,b\in {\mathbb{R}}^{kN}$.

We will discuss it in three cases, respectively.

### 2.1 Case (i)

Suppose that $\sigma <1$ and ${\sum }_{\ell =1}^{m-2}{\alpha }_{\ell }-\delta =1$. If u is a solution of (6) with (4), we have
$w\left(t\right)\phi \left(t,{u}^{\mathrm{\prime }}\left(t\right)\right)=w\left(0\right)\phi \left(0,{u}^{\mathrm{\prime }}\left(0\right)\right)+\sum _{{t}_{i}
(7)
Denote ${\rho }_{1}=w\left(0\right)\phi \left(0,{u}^{\mathrm{\prime }}\left(0\right)\right)$. It is easy to see that ${\rho }_{1}$ is dependent on a, b and $f\left(\cdot \right)$. Define the operator $F:{L}^{1}\to PC$ as
$F\left(f\right)\left(t\right)={\int }_{0}^{t}f\left(s\right)\phantom{\rule{0.2em}{0ex}}ds,\phantom{\rule{1em}{0ex}}\mathrm{\forall }t\in J,\mathrm{\forall }f\in {L}^{1}.$
By solving for ${u}^{\mathrm{\prime }}$ in (7) and integrating, we find
$u\left(t\right)=u\left(0\right)+\sum _{{t}_{i}
which together with boundary value condition (4) implies
$u\left(0\right)=\frac{1}{\left(1-\sigma \right)}{\int }_{0}^{1}g\left(t\right)\left(F\left\{{\phi }^{-1}\left[t,{\left(w\left(t\right)\right)}^{-1}\left({\rho }_{1}+\sum _{{t}_{i}
and
$\begin{array}{c}\sum _{\ell =1}^{m-2}{\alpha }_{\ell }\left\{\sum _{{t}_{i}<{\xi }_{\ell }}{a}_{i}+{\int }_{0}^{{\xi }_{\ell }}{\phi }^{-1}\left[t,{\left(w\left(t\right)\right)}^{-1}\left({\rho }_{1}+\sum _{{t}_{i}
Denote $W={\mathbb{R}}^{2kN}×{L}^{1}$ with the norm
$\parallel \omega \parallel =\sum _{i=1}^{k}|{a}_{i}|+\sum _{i=1}^{k}|{b}_{i}|+{\parallel \vartheta \parallel }_{{L}^{1}},\phantom{\rule{1em}{0ex}}\mathrm{\forall }\omega =\left(a,b,\vartheta \right)\in W,$

then W is a Banach space.

For any $\omega \in W$, we denote
$\begin{array}{rcl}{\mathrm{\Lambda }}_{\omega }\left({\rho }_{1}\right)& =& \sum _{\ell =1}^{m-2}{\alpha }_{\ell }\left\{\sum _{{t}_{i}<{\xi }_{\ell }}{a}_{i}+{\int }_{0}^{{\xi }_{\ell }}{\phi }^{-1}\left[t,{\left(w\left(t\right)\right)}^{-1}\left({\rho }_{1}+\sum _{{t}_{i}
Denote ${\xi }_{m-1}=1$. Then
$\begin{array}{rcl}{\mathrm{\Lambda }}_{\omega }\left({\rho }_{1}\right)& =& -\sum _{\ell =1}^{m-2}{\alpha }_{\ell }\left\{\sum _{{\xi }_{\ell }\le {t}_{i}}{a}_{i}+{\int }_{{\xi }_{\ell }}^{1}{\phi }^{-1}\left[t,{\left(w\left(t\right)\right)}^{-1}\left({\rho }_{1}+\sum _{{t}_{i}
Throughout the paper, we denote
Lemma 2.2 Suppose that $h\left(t\right)\ge 0$ on $\left[{\xi }_{1},1\right]$, ${\alpha }_{\ell }\ge {\int }_{{\xi }_{\ell }}^{{\xi }_{\ell +1}}h\left(t\right)\phantom{\rule{0.2em}{0ex}}dt$ ($\ell =1,\dots ,m-2$) and $h\left(t\right)\le 0$ on $\left[0,{\xi }_{1}\right]$. Then the function ${\mathrm{\Lambda }}_{\omega }\left(\cdot \right)$ has the following properties:
1. (i)
For any fixed $\omega \in W$, the equation
${\mathrm{\Lambda }}_{\omega }\left({\rho }_{1}\right)=0$
(8)

has a unique solution $\stackrel{˜}{{\rho }_{1}}\left(\omega \right)\in {\mathbb{R}}^{N}$.
1. (ii)
The function $\stackrel{˜}{{\rho }_{1}}:W\to {\mathbb{R}}^{N}$, defined in (i), is continuous and sends bounded sets to bounded sets. Moreover, for any $\omega =\left(a,b,\vartheta \right)\in W$, we have
$|\stackrel{˜}{{\rho }_{1}}\left(\omega \right)|\le 3N\left[{\left(2N\right)}^{{p}^{+}}{\left({\delta }^{\ast }\frac{E+1}{E}\sum _{i=1}^{k}|{a}_{i}|\right)}^{{p}^{\mathrm{#}}-1}+\sum _{i=1}^{k}|{b}_{i}|+{\parallel \vartheta \parallel }_{{L}^{1}}\right],$

where the notation ${M}^{{p}^{\mathrm{#}}-1}$ means
${M}^{{p}^{\mathrm{#}}-1}=\left\{\begin{array}{ll}{M}^{{p}^{+}-1},& M>1,\\ {M}^{{p}^{-}-1},& M\le 1.\end{array}$
Proof (i) From Lemma 2.1, it is immediate that

and hence, if (8) has a solution, then it is unique.

Set ${R}_{0}=3N\left[{\left(2N\right)}^{{p}^{+}}{\left({\delta }^{\ast }\frac{E+1}{E}{\sum }_{i=1}^{k}|{a}_{i}|\right)}^{{p}^{\mathrm{#}}-1}+{\sum }_{i=1}^{k}|{b}_{i}|+{\parallel \vartheta \parallel }_{{L}^{1}}\right]$.

Suppose that $|{\rho }_{1}|>{R}_{0}$, it is easy to see that there exists some ${j}_{0}\in \left\{1,\dots ,N\right\}$ such that the absolute value of the ${j}_{0}$th component of ${\rho }_{1}$ satisfies
Thus the ${j}_{0}$th component of ${\rho }_{1}+{\sum }_{{t}_{i} keeps sign on J, namely, for any $t\in J$, we have
Obviously, we have
then it is easy to see that the ${j}_{0}$th component of ${\mathrm{\Lambda }}_{\omega }\left({\rho }_{1}\right)$ keeps the same sign of . Thus,
${\mathrm{\Lambda }}_{\omega }\left({\rho }_{1}\right)\ne 0.$
Let us consider the equation
$\lambda {\mathrm{\Lambda }}_{\omega }\left({\rho }_{1}\right)+\left(1-\lambda \right){\rho }_{1}=0,\phantom{\rule{1em}{0ex}}\lambda \in \left[0,1\right].$
(9)
According to the preceding discussion, all the solutions of (9) belong to $b\left({R}_{0}+1\right)=\left\{x\in {\mathbb{R}}^{N}\mid |x|<{R}_{0}+1\right\}$. Therefore
${d}_{B}\left[{\mathrm{\Lambda }}_{\omega }\left({\rho }_{1}\right),b\left({R}_{0}+1\right),0\right]={d}_{B}\left[I,b\left({R}_{0}+1\right),0\right]\ne 0,$

it means the existence of solutions of ${\mathrm{\Lambda }}_{\omega }\left({\rho }_{1}\right)=0$.

In this way, we define a function $\stackrel{˜}{{\rho }_{1}}\left(\omega \right):W\to {\mathbb{R}}^{N}$, which satisfies ${\mathrm{\Lambda }}_{\omega }\left(\stackrel{˜}{{\rho }_{1}}\left(\omega \right)\right)=0$.
1. (ii)
By the proof of (i), we also obtain $\stackrel{˜}{{\rho }_{1}}$ sends bounded sets to bounded sets, and
$|\stackrel{˜}{{\rho }_{1}}\left(\omega \right)|\le 3N\left[{\left(2N\right)}^{{p}^{+}}{\left({\delta }^{\ast }\frac{E+1}{E}\sum _{i=1}^{k}|{a}_{i}|\right)}^{{p}^{\mathrm{#}}-1}+\sum _{i=1}^{k}|{b}_{i}|+{\parallel \vartheta \parallel }_{{L}^{1}}\right].$

It only remains to prove the continuity of $\stackrel{˜}{{\rho }_{1}}$. Let $\left\{{\omega }_{n}\right\}$ be a convergent sequence in W and ${\omega }_{n}\to \omega$, as $n\to +\mathrm{\infty }$. Since $\left\{\stackrel{˜}{{\rho }_{1}}\left({\omega }_{n}\right)\right\}$ is a bounded sequence, it contains a convergent subsequence $\left\{\stackrel{˜}{{\rho }_{1}}\left({\omega }_{{n}_{j}}\right)\right\}$. Suppose that $\stackrel{˜}{{\rho }_{1}}\left({\omega }_{{n}_{j}}\right)\to {\rho }_{0}$ as $j\to +\mathrm{\infty }$. Since ${\mathrm{\Lambda }}_{{\omega }_{{n}_{j}}}\left(\stackrel{˜}{{\rho }_{1}}\left({\omega }_{{n}_{j}}\right)\right)=0$, letting $j\to +\mathrm{\infty }$, we have ${\mathrm{\Lambda }}_{\omega }\left({\rho }_{0}\right)=0$, which together with (i) implies ${\rho }_{0}=\stackrel{˜}{{\rho }_{1}}\left(\omega \right)$, it means $\stackrel{˜}{{\rho }_{1}}$ is continuous. This completes the proof. □

Now we denote by ${N}_{f}\left(u\right):\left[0,1\right]×P{C}^{1}\to {L}^{1}$ the Nemytskii operator associated to f defined by
(10)
We define ${\rho }_{1}:P{C}^{1}\to {\mathbb{R}}^{N}$ as
${\rho }_{1}\left(u\right)=\stackrel{˜}{{\rho }_{1}}\left(A,B,{N}_{f}\right)\left(u\right),$
(11)

where $A=\left({A}_{1},\dots ,{A}_{k}\right)$, $B=\left({B}_{1},\dots ,{B}_{k}\right)$.

It is clear that ${\rho }_{1}\left(\cdot \right)$ is continuous and sends bounded sets of $P{C}^{1}$ to bounded sets of ${\mathbb{R}}^{N}$, and hence it is compact continuous.

If u is a solution of (6) with (4), we have
$u\left(t\right)=u\left(0\right)+\sum _{{t}_{i}
For fixed $a,b\in {\mathbb{R}}^{kN}$, we denote ${K}_{\left(a,b\right)}:{L}^{1}\to P{C}^{1}$ as
${K}_{\left(a,b\right)}\left(\vartheta \right)\left(t\right)=F\left\{{\phi }^{-1}\left[t,{\left(w\left(t\right)\right)}^{-1}\left(\stackrel{˜}{{\rho }_{1}}\left(a,b,\vartheta \right)+\sum _{{t}_{i}
Define ${K}_{1}:P{C}^{1}\to P{C}^{1}$ as
${K}_{1}\left(u\right)\left(t\right)=F\left\{{\phi }^{-1}\left[t,{\left(w\left(t\right)\right)}^{-1}\left({\rho }_{1}\left(u\right)+\sum _{{t}_{i}
Lemma 2.3 (i) The operator ${K}_{\left(a,b\right)}$ is continuous and sends equi-integrable sets in ${L}^{1}$ to relatively compact sets in $P{C}^{1}$.
1. (ii)

The operator ${K}_{1}$ is continuous and sends bounded sets in $P{C}^{1}$ to relatively compact sets in $P{C}^{1}$.

Proof (i) It is easy to check that ${K}_{\left(a,b\right)}\left(\vartheta \right)\left(\cdot \right)\in P{C}^{1}$, $\mathrm{\forall }\vartheta \in {L}^{1}$, $\mathrm{\forall }a,b\in {\mathbb{R}}^{kN}$. Since and
${K}_{\left(a,b\right)}{\left(\vartheta \right)}^{\mathrm{\prime }}\left(t\right)={\phi }^{-1}\left[t,{\left(w\left(t\right)\right)}^{-1}\left(\stackrel{˜}{{\rho }_{1}}\left(a,b,\vartheta \right)+\sum _{{t}_{i}

it is easy to check that ${K}_{\left(a,b\right)}\left(\cdot \right)$ is a continuous operator from ${L}^{1}$ to $P{C}^{1}$.

Let now U be an equi-integrable set in ${L}^{1}$, then there exists $\alpha \in {L}^{1}$ such that

We want to show that $\overline{{K}_{\left(a,b\right)}\left(U\right)}\subset P{C}^{1}$ is a compact set.

Let $\left\{{u}_{n}\right\}$ be a sequence in ${K}_{\left(a,b\right)}\left(U\right)$, then there exists a sequence $\left\{{\vartheta }_{n}\right\}\in U$ such that ${u}_{n}={K}_{\left(a,b\right)}\left({\vartheta }_{n}\right)$. For any ${t}_{1},{t}_{2}\in J$, we have
$|F\left({\vartheta }_{n}\right)\left({t}_{1}\right)-F\left({\vartheta }_{n}\right)\left({t}_{2}\right)|=|{\int }_{0}^{{t}_{1}}{\vartheta }_{n}\left(t\right)\phantom{\rule{0.2em}{0ex}}dt-{\int }_{0}^{{t}_{2}}{\vartheta }_{n}\left(t\right)\phantom{\rule{0.2em}{0ex}}dt|=|{\int }_{{t}_{1}}^{{t}_{2}}{\vartheta }_{n}\left(t\right)\phantom{\rule{0.2em}{0ex}}dt|\le |{\int }_{{t}_{1}}^{{t}_{2}}\alpha \left(t\right)\phantom{\rule{0.2em}{0ex}}dt|.$

Hence the sequence $\left\{F\left({\vartheta }_{n}\right)\right\}$ is uniformly bounded and equi-continuous. By the Ascoli-Arzela theorem, there exists a subsequence of $\left\{F\left({\vartheta }_{n}\right)\right\}$ (which we rename the same) which is convergent in PC. According to the bounded continuity of the operator $\stackrel{˜}{{\rho }_{1}}$, we can choose a subsequence of $\left\{\stackrel{˜}{{\rho }_{1}}\left(a,b,{\vartheta }_{n}\right)+F\left({\vartheta }_{n}\right)\right\}$ (which we still denote $\left\{\stackrel{˜}{{\rho }_{1}}\left(a,b,{\vartheta }_{n}\right)+F\left({\vartheta }_{n}\right)\right\}$) which is convergent in PC, then is convergent in PC.

Since
${K}_{\left(a,b\right)}\left({\vartheta }_{n}\right)\left(t\right)=F\left\{{\phi }^{-1}\left[t,{\left(w\left(t\right)\right)}^{-1}\left(\stackrel{˜}{{\rho }_{1}}\left(a,b,{\vartheta }_{n}\right)+\sum _{{t}_{i}
it follows from the continuity of ${\phi }^{-1}$ and the integrability of in ${L}^{1}$ that ${K}_{\left(a,b\right)}\left({\vartheta }_{n}\right)$ is convergent in PC. Thus $\left\{{u}_{n}\right\}$ is convergent in $P{C}^{1}$.
1. (ii)

It is easy to see from (i) and Lemma 2.2.

This completes the proof. □

Let us define ${P}_{1}:P{C}^{1}\to P{C}^{1}$ as
${P}_{1}\left(u\right)=\frac{{\int }_{0}^{1}g\left(t\right)\left[{K}_{1}\left(u\right)\left(t\right)+{\sum }_{{t}_{i}

It is easy to see that ${P}_{1}$ is compact continuous.

Lemma 2.4 Suppose that $\sigma <1$, ${\sum }_{\ell =1}^{m-2}{\alpha }_{\ell }-\delta =1$; $h\left(t\right)\ge 0$ on $\left[{\xi }_{1},1\right]$, ${\alpha }_{\ell }\ge {\int }_{{\xi }_{\ell }}^{{\xi }_{\ell +1}}h\left(t\right)\phantom{\rule{0.2em}{0ex}}dt$ ($\ell =1,\dots ,m-2$) and $h\left(t\right)\le 0$ on $\left[0,{\xi }_{1}\right]$. Then u is a solution of (1)-(4) if and only if u is a solution of the following abstract operator equation:
$u={P}_{1}\left(u\right)+\sum _{{t}_{i}
(12)
Proof Suppose that u is a solution of (1)-(4). By integrating (1) from 0 to t, we find that
$w\left(t\right)\phi \left(t,{u}^{\mathrm{\prime }}\left(t\right)\right)={\rho }_{1}\left(u\right)+\sum _{{t}_{i}
(13)
It follows from (13) and (4) that
$\begin{array}{c}u\left(t\right)=u\left(0\right)+\sum _{{t}_{i}
(14)
Combining the definition of ${\rho }_{1}$, we can see
$u={P}_{1}\left(u\right)+\sum _{{t}_{i}
Conversely, if u is a solution of (12), then (2) is satisfied. It is easy to check that
$\begin{array}{c}u\left(0\right)={P}_{1}\left(u\right)=\frac{{\int }_{0}^{1}g\left(t\right)\left[{K}_{1}\left(u\right)\left(t\right)+{\sum }_{{t}_{i}
(15)
and
$u\left(1\right)={P}_{1}\left(u\right)+\sum _{i=1}^{k}{A}_{i}+{K}_{1}\left(u\right)\left(1\right).$
By the condition of the mapping ${\rho }_{1}$, we have
$\begin{array}{c}\sum _{\ell =1}^{m-2}{\alpha }_{\ell }\left\{\sum _{{t}_{i}<{\xi }_{\ell }}{A}_{i}+{\int }_{0}^{{\xi }_{\ell }}{\phi }^{-1}\left[t,{\left(w\left(t\right)\right)}^{-1}\left({\rho }_{1}+\sum _{{t}_{i}
Thus
$u\left(1\right)=\sum _{\ell =1}^{m-2}{\alpha }_{\ell }u\left({\xi }_{\ell }\right)-{\int }_{0}^{1}h\left(t\right)u\left(t\right)\phantom{\rule{0.2em}{0ex}}dt.$
(16)

It follows from (15) and (16) that (4) is satisfied.

From (12), we have
$\begin{array}{c}w\left(t\right)\phi \left(t,{u}^{\mathrm{\prime }}\left(t\right)\right)={\rho }_{1}\left(u\right)+\sum _{{t}_{i}
(17)

It follows from (17) that (3) is satisfied.

Hence u is a solution of (1)-(4). This completes the proof. □

### 2.2 Case (ii)

Suppose that $\sigma =1$ and ${\sum }_{\ell =1}^{m-2}{\alpha }_{\ell }-\delta \ne 1$. If u is a solution of (6) with (4), we have
$w\left(t\right)\phi \left(t,{u}^{\mathrm{\prime }}\left(t\right)\right)=w\left(0\right)\phi \left(0,{u}^{\mathrm{\prime }}\left(0\right)\right)+\sum _{{t}_{i}
Denote ${\rho }_{2}=w\left(0\right)\phi \left(0,{u}^{\mathrm{\prime }}\left(0\right)\right)$. It is easy to see that ${\rho }_{2}$ is dependent on a, b and $f\left(\cdot \right)$. Boundary value condition (4) implies that
$\begin{array}{c}{\int }_{0}^{1}g\left(t\right)\left(F\left\{{\phi }^{-1}\left[t,{\left(w\left(t\right)\right)}^{-1}\left({\rho }_{2}+\sum _{{t}_{i}
For any $\omega \in W$, we denote
${\mathrm{\Gamma }}_{\omega }\left({\rho }_{2}\right)={\int }_{0}^{1}g\left(t\right)\left(F\left\{{\phi }^{-1}\left[t,{\left(w\left(t\right)\right)}^{-1}\left({\rho }_{2}+\sum _{{t}_{i}

Throughout the paper, we denote .

Lemma 2.5 The function ${\mathrm{\Gamma }}_{\omega }\left(\cdot \right)$ has the following properties:
1. (i)

For any fixed $\omega \in W$, the equation ${\mathrm{\Gamma }}_{\omega }\left({\rho }_{2}\right)=0$ has a unique solution $\stackrel{˜}{{\rho }_{2}}\left(\omega \right)\in {\mathbb{R}}^{N}$.

2. (ii)
The function $\stackrel{˜}{{\rho }_{2}}:W\to {\mathbb{R}}^{N}$, defined in (i), is continuous and sends bounded sets to bounded sets. Moreover, for any $\omega =\left(a,b,\vartheta \right)\in W$, we have
$|\stackrel{˜}{{\rho }_{2}}\left(\omega \right)|\le 3N\left[{\left(2N\right)}^{{p}^{+}}{\left(\frac{{E}_{1}+1}{{E}_{1}}\sum _{i=1}^{k}|{a}_{i}|\right)}^{{p}^{\mathrm{#}}-1}+\sum _{i=1}^{k}|{b}_{i}|+{\parallel \vartheta \parallel }_{{L}^{1}}\right],$

where the notation ${M}^{{p}^{\mathrm{#}}-1}$ means
${M}^{{p}^{\mathrm{#}}-1}=\left\{\begin{array}{ll}{M}^{{p}^{+}-1},& M>1,\\ {M}^{{p}^{-}-1},& M\le 1.\end{array}$

Proof Similar to the proof of Lemma 2.2, we omit it here. □

We define ${\rho }_{2}:P{C}^{1}\to {\mathbb{R}}^{N}$ as ${\rho }_{2}\left(u\right)=\stackrel{˜}{{\rho }_{2}}\left(A,B,{N}_{f}\right)\left(u\right)$, where $A=\left({A}_{1},\dots ,{A}_{k}\right)$, $B=\left({B}_{1},\dots ,{B}_{k}\right)$.

It is clear that ${\rho }_{2}\left(\cdot \right)$ is continuous and sends bounded sets of $P{C}^{1}$ to bounded sets of ${\mathbb{R}}^{N}$, and hence it is compact continuous.

For fixed $a,b\in {\mathbb{R}}^{kN}$, we denote ${K}_{\left(a,b\right)}^{\ast }:{L}^{1}\to P{C}^{1}$ as
${K}_{\left(a,b\right)}^{\ast }\left(\vartheta \right)\left(t\right)=F\left\{{\phi }^{-1}\left[t,{\left(w\left(t\right)\right)}^{-1}\left(\stackrel{˜}{{\rho }_{2}}\left(a,b,\vartheta \right)+\sum _{{t}_{i}
Define ${K}_{2}:P{C}^{1}\to P{C}^{1}$ as
${K}_{2}\left(u\right)\left(t\right)=F\left\{{\phi }^{-1}\left[t,{\left(w\left(t\right)\right)}^{-1}\left({\rho }_{2}\left(u\right)+\sum _{{t}_{i}

Similar to the proof of Lemma 2.3, we have the following.

Lemma 2.6 (i) The operator ${K}_{\left(a,b\right)}^{\ast }$ is continuous and sends equi-integrable sets in ${L}^{1}$ to relatively compact sets in $P{C}^{1}$.
1. (ii)

The operator ${K}_{2}$ is continuous and sends bounded sets in $P{C}^{1}$ to relatively compact sets in $P{C}^{1}$.

Let us define ${P}_{2}:P{C}^{1}\to P{C}^{1}$ as
$\begin{array}{rcl}{P}_{2}\left(u\right)& =& \frac{{\sum }_{\ell =1}^{m-2}{\alpha }_{\ell }\left[{\sum }_{{t}_{i}<{\xi }_{\ell }}{A}_{i}+{K}_{2}\left(u\right)\left({\xi }_{\ell }\right)\right]-{\sum }_{i=1}^{k}{A}_{i}}{1-{\sum }_{\ell =1}^{m-2}{\alpha }_{\ell }+\delta }\\ -\frac{{K}_{2}\left(u\right)\left(1\right)+{\int }_{0}^{1}h\left(t\right)\left[{K}_{2}\left(u\right)\left(t\right)+{\sum }_{{t}_{i}

It is easy to see that ${P}_{2}$ is compact continuous.

Lemma 2.7 Suppose that $\sigma =1$, ${\sum }_{\ell =1}^{m-2}{\alpha }_{\ell }-\delta \ne 1$, then u is a solution of (1)-(4) if and only if u is a solution of the following abstract operator equation:
$u={P}_{2}\left(u\right)+\sum _{{t}_{i}

Proof Similar to the proof of Lemma 2.4, we omit it here. □

### 2.3 Case (iii)

Suppose that $\sigma <1$ and ${\sum }_{\ell =1}^{m-2}{\alpha }_{\ell }-\delta <1$. If u is a solution of (6) with (4), we have
$w\left(t\right)\phi \left(t,{u}^{\mathrm{\prime }}\left(t\right)\right)=w\left(0\right)\phi \left(0,{u}^{\mathrm{\prime }}\left(0\right)\right)+\sum _{{t}_{i}

Denote ${\rho }_{3}=w\left(0\right)\phi \left(0,{u}^{\mathrm{\prime }}\left(0\right)\right)$. It is easy to see that ${\rho }_{3}$ is dependent on a, b and $f\left(\cdot \right)$.

From $u\left(0\right)={\int }_{0}^{1}g\left(t\right)u\left(t\right)\phantom{\rule{0.2em}{0ex}}dt$, we have
$\begin{array}{rl}u\left(0\right)=& \frac{1}{\left(1-\sigma \right)}\\ ×{\int }_{0}^{1}g\left(t\right)\left(F\left\{{\phi }^{-1}\left[t,{\left(w\left(t\right)\right)}^{-1}\left({\rho }_{3}+\sum _{{t}_{i}
(18)
From $u\left(1\right)={\sum }_{\ell =1}^{m-2}{\alpha }_{\ell }u\left({\xi }_{\ell }\right)-{\int }_{0}^{1}h\left(t\right)u\left(t\right)\phantom{\rule{0.2em}{0ex}}dt$, we obtain
$\begin{array}{rcl}u\left(0\right)& =& \frac{{\sum }_{\ell =1}^{m-2}{\alpha }_{\ell }\left\{{\sum }_{{t}_{i}<{\xi }_{\ell }}{a}_{i}+{\int }_{0}^{{\xi }_{\ell }}{\phi }^{-1}\left[t,{\left(w\left(t\right)\right)}^{-1}\left({\rho }_{3}+{\sum }_{{t}_{i}
(19)
For fixed $\omega \in W$, we denote
$\begin{array}{rcl}{\mathrm{\Upsilon }}_{\omega }\left({\rho }_{3}\right)& =& \frac{1}{\left(1-\sigma \right)}{\int }_{0}^{1}g\left(t\right)\left(F\left\{{\phi }^{-1}\left[t,{\left(w\left(t\right)\right)}^{-1}\left({\rho }_{3}+\sum _{{t}_{i}

From (18) and (19), we have ${\mathrm{\Upsilon }}_{\omega }\left({\rho }_{3}\right)=0$.

Obviously, ${\mathrm{\Upsilon }}_{\omega }\left({\rho }_{3}\right)$ can be rewritten as
$\begin{array}{rcl}{\mathrm{\Upsilon }}_{\omega }\left({\rho }_{3}\right)& =& \frac{1}{\left(1-\sigma \right)}{\int }_{0}^{1}g\left(t\right)\left(F\left\{{\phi }^{-1}\left[t,{\left(w\left(t\right)\right)}^{-1}\left({\rho }_{3}+\sum _{{t}_{i}
Denote ${\xi }_{m-1}=1$. Moreover, we also have
$\begin{array}{c}{\mathrm{\Upsilon }}_{\omega }\left({\rho }_{3}\right)\hfill \\ \phantom{\rule{1em}{0ex}}=\frac{1}{\left(1-\sigma \right)}{\int }_{0}^{1}g\left(t\right)\left(F\left\{{\phi }^{-1}\left[t,{\left(w\left(t\right)\right)}^{-1}\left({\rho }_{3}+\sum _{{t}_{i}

Lemma 2.8 Suppose that ${\alpha }_{\ell }$, g, h satisfy one of the following:

(10) ${\sum }_{\ell =1}^{m-2}{\alpha }_{\ell }\le 1$, $g\left(t\right)\left(1-{\sum }_{\ell =1}^{m-2}{\alpha }_{\ell }+\delta \right)+h\left(t\right)\left(1-\sigma \right)\ge 0$;

(20) $h\left(t\right)\ge 0$ on $\left[{\xi }_{1},1\right]$, ${\alpha }_{\ell }\ge {\int }_{{\xi }_{\ell }}^{{\xi }_{\ell +1}}h\left(t\right)\phantom{\rule{0.2em}{0ex}}dt$ ($\ell =1,\dots ,m-2$) and $h\left(t\right)\le 0$ on $\left[0,{\xi }_{1}\right]$.

Then the function ${\mathrm{\Upsilon }}_{\omega }\left(\cdot \right)$ has the following properties:
1. (i)

For any fixed $\omega \in W$, the equation ${\mathrm{\Upsilon }}_{\omega }\left({\rho }_{3}\right)=0$ has a unique solution $\stackrel{˜}{{\rho }_{3}}\left(\omega \right)\in {\mathbb{R}}^{N}$.

2. (ii)
The function $\stackrel{˜}{{\rho }_{3}}:W\to {\mathbb{R}}^{N}$, defined in (i), is continuous and sends bounded sets to bounded sets. Moreover, for any $\omega =\left(a,b,\vartheta \right)\in W$, we have
$\begin{array}{rcl}|\stackrel{˜}{{\rho }_{3}}\left(\omega \right)|& \le & 3N\left\{{\left(2N\right)}^{{p}^{+}}{\left[\left(\frac{{E}_{1}+1}{\left(1-\sigma \right){E}_{1}}+\left({\delta }^{\ast }+1\right)\frac{E+1}{\left(1-{\sum }_{\ell =1}^{m-2}{\alpha }_{\ell }+\delta \right)E}\right)\sum _{i=1}^{k}|{a}_{i}|\right]}^{{p}^{\mathrm{#}}-1}\\ +\sum _{i=1}^{k}|{b}_{i}|+{\parallel \vartheta \parallel }_{{L}^{1}}\right\},\end{array}$

where the notation ${M}^{{p}^{\mathrm{#}}-1}$ means
${M}^{{p}^{\mathrm{#}}-1}=\left\{\begin{array}{ll}{M}^{{p}^{+}-1},& M>1,\\ {M}^{{p}^{-}-1},& M\le 1.\end{array}$

Proof Similar to the proof of Lemma 2.2, we omit it here. □

We define ${\rho }_{3}:P{C}^{1}\to {\mathbb{R}}^{N}$ as ${\rho }_{3}\left(u\right)=\stackrel{˜}{{\rho }_{3}}\left(A,B,{N}_{f}\right)\left(u\right)$, where $A=\left({A}_{1},\dots ,{A}_{k}\right)$, $B=\left({B}_{1},\dots ,{B}_{k}\right)$.

It is clear that ${\rho }_{3}\left(\cdot \right)$ is continuous and sends bounded sets of $P{C}^{1}$ to bounded sets of ${\mathbb{R}}^{N}$, and hence it is compact continuous.

For fixed $a,b\in {\mathbb{R}}^{kN}$, we denote ${K}_{\left(a,b\right)}^{\ast \ast }:{L}^{1}\to P{C}^{1}$ as
${K}_{\left(a,b\right)}^{\ast \ast }\left(\vartheta \right)\left(t\right)=F\left\{{\phi }^{-1}\left[t,{\left(w\left(t\right)\right)}^{-1}\left(\stackrel{˜}{{\rho }_{3}}\left(a,b,\vartheta \right)+\sum _{{t}_{i}
Define ${K}_{3}:P{C}^{1}\to P{C}^{1}$ as
${K}_{3}\left(u\right)\left(t\right)=F\left\{{\phi }^{-1}\left[t,{\left(w\left(t\right)\right)}^{-1}\left({\rho }_{3}\left(u\right)+\sum _{{t}_{i}

Similar to the proof of Lemma 2.3, we have

Lemma 2.9 (i) The operator ${K}_{\left(a,b\right)}^{\ast \ast }$ is continuous and sends equi-integrable sets in ${L}^{1}$ to relatively compact sets in $P{C}^{1}$.
1. (ii)

The operator ${K}_{3}$ is continuous and sends bounded sets in $P{C}^{1}$ to relatively compact sets in $P{C}^{1}$.

Let us define ${P}_{3}:P{C}^{1}\to P{C}^{1}$ as
${P}_{3}\left(u\right)=\frac{{\int }_{0}^{1}g\left(t\right)\left[{K}_{3}\left(u\right)\left(t\right)+{\sum }_{{t}_{i}

It is easy to see that ${P}_{3}$ is compact continuous.

Lemma 2.10 Suppose that $\sigma <1$, ${\sum }_{\ell =1}^{m-2}{\alpha }_{\ell }-\delta <1$ and ${\alpha }_{\ell }$, g, h satisfy one of the following:

(10) ${\sum }_{\ell =1}^{m-2}{\alpha }_{\ell }\le 1$, $g\left(t\right)\left(1-{\sum }_{\ell =1}^{m-2}{\alpha }_{\ell }+\delta \right)+h\left(t\right)\left(1-\sigma \right)\ge 0$;

(20) $h\left(t\right)\ge 0$ on $\left[{\xi }_{1},1\right]$, ${\alpha }_{\ell }\ge {\int }_{{\xi }_{\ell }}^{{\xi }_{\ell +1}}h\left(t\right)\phantom{\rule{0.2em}{0ex}}dt$ ($\ell =1,\dots ,m-2$) and $h\left(t\right)\le 0$ on $\left[0,{\xi }_{1}\right]$.

Then u is a solution of (1)-(4) if and only if u is a solution of the following abstract operator equation:
$u={P}_{3}\left(u\right)+\sum _{{t}_{i}

Proof Similar to the proof of Lemma 2.4, we omit it here. □

## 3 Existence of solutions in Case (i)

In this section, we apply Leray-Schauder’s degree to deal with the existence of solutions for system (1)-(4) or (1) with (2), (4) and (5) when $\sigma <1$, ${\sum }_{\ell =1}^{m-2}{\alpha }_{\ell }-\delta =1$.

When f satisfies the sub-$\left({p}^{-}-1\right)$ growth condition, we have the following theorem.

Theorem 3.1 Suppose that $\sigma <1$, ${\sum }_{\ell =1}^{m-2}{\alpha }_{\ell }-\delta =1$; $h\left(t\right)\ge 0$ on $\left[{\xi }_{1},1\right]$, ${\alpha }_{\ell }\ge {\int }_{{\xi }_{\ell }}^{{\xi }_{\ell +1}}h\left(t\right)\phantom{\rule{0.2em}{0ex}}dt$ ($\ell =1,\dots ,m-2$) and $h\left(t\right)\le 0$ on $\left[0,{\xi }_{1}\right]$; f satisfies the sub-$\left({p}^{-}-1\right)$ growth condition; and operators A and B satisfy the following conditions:
$\begin{array}{r}\sum _{i=1}^{k}|{A}_{i}\left(u,v\right)|\le {C}_{1}{\left(1+|u|+|v|\right)}^{\frac{{q}^{+}-1}{{p}^{+}-1}},\\ \sum _{i=1}^{k}|{B}_{i}\left(u,v\right)|\le {C}_{2}{\left(1+|u|+|v|\right)}^{{q}^{+}-1},\phantom{\rule{1em}{0ex}}\mathrm{\forall }\left(u,v\right)\in {\mathbb{R}}^{N}×{\mathbb{R}}^{N},\end{array}$
(20)

then problem (1)-(4) has at least a solution.

Proof First we consider the following problem:
$\left({S}_{1}\right)\phantom{\rule{1em}{0ex}}\left\{\begin{array}{l}-{\mathrm{△}}_{p\left(t\right)}u=\lambda {N}_{f}\left(u\right)\left(t\right),\phantom{\rule{1em}{0ex}}t\in \left(0,1\right),t\ne {t}_{i},\\ {lim}_{t\to {t}_{i}^{+}}u\left(t\right)-{lim}_{t\to {t}_{i}^{-}}u\left(t\right)\\ \phantom{\rule{1em}{0ex}}=\lambda {A}_{i}\left({lim}_{t\to {t}_{i}^{-}}u\left(t\right),{lim}_{t\to {t}_{i}^{-}}{\left(w\left(t\right)\right)}^{\frac{1}{p\left(t\right)-1}}{u}^{\mathrm{\prime }}\left(t\right)\right),\phantom{\rule{1em}{0ex}}i=1,\dots ,k,\\ {lim}_{t\to {t}_{i}^{+}}w\left(t\right)|{u}^{\mathrm{\prime }}{|}^{p\left(t\right)-2}{u}^{\mathrm{\prime }}\left(t\right)-{lim}_{t\to {t}_{i}^{-}}w\left(t\right)|{u}^{\mathrm{\prime }}{|}^{p\left(t\right)-2}{u}^{\mathrm{\prime }}\left(t\right)\\ \phantom{\rule{1em}{0ex}}=\lambda {B}_{i}\left({lim}_{t\to {t}_{i}^{-}}u\left(t\right),{lim}_{t\to {t}_{i}^{-}}{\left(w\left(t\right)\right)}^{\frac{1}{p\left(t\right)-1}}{u}^{\mathrm{\prime }}\left(t\right)\right),\phantom{\rule{1em}{0ex}}i=1,\dots ,k,\\ u\left(0\right)={\int }_{0}^{1}g\left(t\right)u\left(t\right)\phantom{\rule{0.2em}{0ex}}dt,\phantom{\rule{2em}{0ex}}u\left(1\right)={\sum }_{\ell =1}^{m-2}{\alpha }_{\ell }u\left({\xi }_{\ell }\right)-{\int }_{0}^{1}h\left(t\right)u\left(t\right)\phantom{\rule{0.2em}{0ex}}dt.\end{array}$
Denote

where ${N}_{f}\left(u\right)$ is defined in (10).

Obviously, (${S}_{1}$) has the same solution as the following operator equation when $\lambda =1$:
$u={\mathrm{\Psi }}_{f}\left(u,\lambda \right).$
(21)

It is easy to see that the operator is compact continuous for any $\lambda \in \left[0,1\right]$. It follows from Lemma 2.2 and Lemma 2.3 that ${\mathrm{\Psi }}_{f}\left(\cdot ,\lambda \right)$ is compact continuous from $P{C}^{1}$ to $P{C}^{1}$ for any $\lambda \in \left[0,1\right]$.

We claim that all the solutions of (21) are uniformly bounded for $\lambda \in \left[0,1\right]$. In fact, if it is false, we can find a sequence of solutions $\left\{\left({u}_{n},{\lambda }_{n}\right)\right\}$ for (21) such that ${\parallel {u}_{n}\parallel }_{1}\to +\mathrm{\infty }$ as $n\to +\mathrm{\infty }$ and ${\parallel {u}_{n}\parallel }_{1}>1$ for any $n=1,2,\dots$ .

From Lemma 2.2, we have
Thus
(22)
From (${S}_{1}$), we have
It follows from (11) and Lemma 2.2 that
Denote $\alpha =\frac{{q}^{+}-1}{{p}^{-}-1}$. If the above inequality holds then
${\parallel {\left(w\left(t\right)\right)}^{\frac{1}{p\left(t\right)-1}}{u}_{n}^{\mathrm{\prime }}\left(t\right)\parallel }_{0}\le {C}_{8}{\parallel {u}_{n}\parallel }_{1}^{\alpha },\phantom{\rule{1em}{0ex}}n=1,2,\dots .$
(23)
It follows from (14), (20) and (22) that
For any $j=1,\dots ,N$, we have
$\begin{array}{rcl}|{u}_{n}^{j}\left(t\right)|& =& |{u}_{n}^{j}\left(0\right)+\sum _{{t}_{i}
which implies that
$|{u}_{n}^{j}{|}_{0}\le {C}_{12}{\parallel {u}_{n}\parallel }_{1}^{\alpha },\phantom{\rule{1em}{0ex}}j=1,\dots ,N;n=1,2,\dots .$
Thus
${\parallel {u}_{n}\parallel }_{0}\le N{C}_{12}{\parallel {u}_{n}\parallel }_{1}^{\alpha },\phantom{\rule{1em}{0ex}}n=1,2,\dots .$
(24)

It follows from (23) and (24) that $\left\{{\parallel {u}_{n}\parallel }_{1}\right\}$ is uniformly bounded.

Thus, we can choose a large enough ${R}_{0}>0$ such that all the solutions of (21) belong to $B\left({R}_{0}\right)=\left\{u\in P{C}^{1}\mid {\parallel u\parallel }_{1}<{R}_{0}\right\}$. Therefore the Leray-Schauder degree ${d}_{LS}\left[I-{\mathrm{\Psi }}_{f}\left(\cdot ,\lambda \right),B\left({R}_{0}\right),0\right]$ is well defined for $\lambda \in \left[0,1\right]$, and
${d}_{LS}\left[I-{\mathrm{\Psi }}_{f}\left(\cdot ,1\right),B\left({R}_{0}\right),0\right]={d}_{LS}\left[I-{\mathrm{\Psi }}_{f}\left(\cdot ,0\right),B\left({R}_{0}\right),0\right].$
It is easy to see that u is a solution of $u={\mathrm{\Psi }}_{f}\left(u,0\right)$ if and only if u is a solution of the following usual differential equation:
$\left({S}_{2}\right)\phantom{\rule{1em}{0ex}}\left\{\begin{array}{l}-{\mathrm{△}}_{p\left(t\right)}u=0,\phantom{\rule{1em}{0ex}}t\in \left(0,1\right),\\ u\left(0\right)={\int }_{0}^{1}g\left(t\right)u\left(t\right)\phantom{\rule{0.2em}{0ex}}dt,\phantom{\rule{2em}{0ex}}u\left(1\right)={\sum }_{\ell =1}^{m-2}{\alpha }_{\ell }u\left({\xi }_{\ell }\right)-{\int }_{0}^{1}h\left(t\right)u\left(t\right)\phantom{\rule{0.2em}{0ex}}dt.\end{array}$
Obviously, system (${S}_{2}$) possesses a unique solution ${u}_{0}$. Since ${u}_{0}\in B\left({R}_{0}\right)$, we have
${d}_{LS}\left[I-{\mathrm{\Psi }}_{f}\left(\cdot ,1\right),B\left({R}_{0}\right),0\right]={d}_{LS}\left[I-{\mathrm{\Psi }}_{f}\left(\cdot ,0\right),B\left({R}_{0}\right),0\right]\ne 0,$

which implies that (1)-(4) has at least one solution. This completes the proof. □

Theorem 3.2 Suppose that $\sigma <1$, ${\sum }_{\ell =1}^{m-2}{\alpha }_{\ell }-\delta =1$; $h\left(t\right)\ge 0$ on $\left[{\xi }_{1},1\right]$, ${\alpha }_{\ell }\ge {\int }_{{\xi }_{\ell }}^{{\xi }_{\ell +1}}h\left(t\right)\phantom{\rule{0.2em}{0ex}}dt$ ($\ell =1,\dots ,m-2$) and $h\left(t\right)\le 0$ on $\left[0,{\xi }_{1}\right]$; f satisfies the sub-$\left({p}^{-}-1\right)$ growth condition; and operators A and $D=\left({D}_{1},\dots ,{D}_{k}\right)$ satisfy the following conditions:
$\begin{array}{c}\sum _{i=1}^{k}|{A}_{i}\left(u,v\right)|\le {C}_{1}{\left(1+|u|+|v|\right)}^{\frac{{q}^{+}-1}{{p}^{+}-1}},\hfill \\ \sum _{i=1}^{k}|{D}_{i}\left(u,v\right)|\le {C}_{2}{\left(1+|u|+|v|\right)}^{{\alpha }_{i}^{+}},\phantom{\rule{1em}{0ex}}\mathrm{\forall }\left(u,v\right)\in {\mathbb{R}}^{N}×{\mathbb{R}}^{N},\hfill \end{array}$

where ${\alpha }_{i}\le \frac{{q}^{+}-1}{p\left({t}_{i}\right)-1}$, and $p\left({t}_{i}\right)-1\le {q}^{+}-{\alpha }_{i}$, $i=1,\dots ,k$.

Then problem (1) with (2), (4) and (5) has at least a solution.

Proof Obviously, ${B}_{i}\left(u,v\right)=\phi \left({t}_{i},v+{D}_{i}\left(u,v\right)\right)-\phi \left({t}_{i},v\right)$.

From Theorem 3.1, it suffices to show that
$\sum _{i=1}^{k}|{B}_{i}\left(u,v\right)|\le {C}_{2}{\left(1+|u|+|v|\right)}^{{q}^{+}-1},\phantom{\rule{1em}{0ex}}\mathrm{\forall }\left(u,v\right)\in {\mathbb{R}}^{N}×{\mathbb{R}}^{N}.$
(25)
1. (a)
Suppose that $|v|\le {M}^{\ast }|{D}_{i}\left(u,v\right)|$, where ${M}^{\ast }$ is a large enough positive constant. From the definition of D, we have
$|{B}_{i}\left(u,v\right)|\le {C}_{1}|{D}_{i}\left(u,v\right){|}^{p\left({t}_{i}\right)-1}\le {C}_{2}{\left(1+|u|+|v|\right)}^{{\alpha }_{i}\left(p\left({t}_{i}\right)-1\right)}.$

Since ${\alpha }_{i}<\frac{{q}^{+}-1}{p\left({t}_{i}\right)-1}$, we have ${\alpha }_{i}\left(p\left({t}_{i}\right)-1\right)\le {q}^{+}-1$. Thus (25) is valid.
1. (b)
Suppose that $|v|>{M}^{\ast }|{D}_{i}\left(u,v\right)|$, we can see that
$|{B}_{i}\left(u,v\right)|\le {C}_{3}|v{|}^{p\left({t}_{i}\right)-1}\frac{|{D}_{i}\left(u,v\right)|}{|v|}={C}_{4}|v{|}^{p\left({t}_{i}\right)-2}|{D}_{i}\left(u,v\right)|.$

There are two cases: Case (i): $p\left({t}_{i}\right)-1\ge 1$; Case (ii): $p\left({t}_{i}\right)-1<1$.

Case (i): Since $p\left({t}_{i}\right)-1\le {q}^{+}-{\alpha }_{i}$, we have $p\left({t}_{i}\right)-2+{\alpha }_{i}\le {q}^{+}-1$, and
$|{B}_{i}\left(u,v\right)|\le {C}_{5}|v{|}^{p\left({t}_{i}\right)-2}|{D}_{i}\left(u,v\right)|\le {C}_{6}{\left(1+|u|+|v|\right)}^{p\left({t}_{i}\right)-2+{\alpha }_{i}}\le {C}_{6}{\left(1+|u|+|v|\right)}^{{q}^{+}-1}.$

Thus (25) is valid.

Case (ii): Since ${\alpha }_{i}<\frac{{q}^{+}-1}{p\left({t}_{i}\right)-1}$, we have ${\alpha }_{i}\left(p\left({t}_{i}\right)-1\right)\le {q}^{+}-1$, and
$|{B}_{i}\left(u,v\right)|\le {C}_{7}|v{|}^{p\left({t}_{i}\right)-2}|{D}_{i}\left(u,v\right)|\le {C}_{8}|{D}_{i}\left(u,v\right){|}^{p\left({t}_{i}\right)-1}\le {C}_{9}{\left(1+|u|+|v|\right)}^{{\alpha }_{i}\left(p\left({t}_{i}\right)-1\right)}.$

Thus (25) is valid.

Thus problem (1) with (2), (4) and (5) has at least a solution. This completes the proof. □

Let us consider
$-{\left(w\left(t\right)|{u}^{\mathrm{\prime }}{|}^{p\left(t\right)-2}{u}^{\mathrm{\prime }}\right)}^{\mathrm{\prime }}+\varphi \left(t,u,{\left(w\left(t\right)\right)}^{\frac{1}{p\left(t\right)-1}}{u}^{\mathrm{\prime }},S\left(u\right),T\left(u\right),\epsilon \right)=0,\phantom{\rule{1em}{0ex}}t\in \left(0,1\right),t\ne {t}_{i},$
(26)
where ε is a parameter, and
$\begin{array}{c}\varphi \left(t,u,{\left(w\left(t\right)\right)}^{\frac{1}{p\left(t\right)-1}}{u}^{\mathrm{\prime }},S\left(u\right),T\left(u\right),\epsilon \right)\hfill \\ \phantom{\rule{1em}{0ex}}=f\left(t,u,{\left(w\left(t\right)\right)}^{\frac{1}{p\left(t\right)-1}}{u}^{\mathrm{\prime }},S\left(u\right),T\left(u\right)\right)+\epsilon h\left(t,u,{\left(w\left(t\right)\right)}^{\frac{1}{p\left(t\right)-1}}{u}^{\mathrm{\prime }},S\left(u\right),T\left(u\right)\right),\hfill \end{array}$

where $h,f:J×{\mathbb{R}}^{N}×{\mathbb{R}}^{N}×{\mathbb{R}}^{N}×{\mathbb{R}}^{N}\to {\mathbb{R}}^{N}$ are Caratheodory. We have the following theorem.

Theorem 3.3 Suppose that $\sigma <1$, ${\sum }_{\ell =1}^{m-2}{\alpha }_{\ell }-\delta =1$; $h\left(t\right)\ge 0$ on $\left[{\xi }_{1},1\right]$, ${\alpha }_{\ell }\ge {\int }_{{\xi }_{\ell }}^{{\xi }_{\ell +1}}h\left(t\right)\phantom{\rule{0.2em}{0ex}}dt$ ($\ell =1,\dots ,m-2$) and $h\left(t\right)\le 0$ on $\left[0,{\xi }_{1}\right]$; f satisfies the sub-$\left({p}^{-}-1\right)$ growth condition; and we assume that
$\begin{array}{c}\sum _{i=1}^{k}|{A}_{i}\left(u,v\right)|\le {C}_{1}{\left(1+|u|+|v|\right)}^{\frac{{q}^{+}-1}{{p}^{+}-1}},\hfill \\ \sum _{i=1}^{k}|{B}_{i}\left(u,v\right)|\le {C}_{2}{\left(1+|u|+|v|\right)}^{{q}^{+}-1},\phantom{\rule{1em}{0ex}}\mathrm{\forall }\left(u,v\right)\in {\mathbb{R}}^{N}×{\mathbb{R}}^{N},\hfill \end{array}$

then problem (26) with (2)-(4) has at least one solution when parameter ε is small enough.

Proof Denote
$\begin{array}{c}{\varphi }_{\lambda }\left(t,u,{\left(w\left(t\right)\right)}^{\frac{1}{p\left(t\right)-1}}{u}^{\mathrm{\prime }},S\left(u\right),T\left(u\right),\epsilon \right)\hfill \\ \phantom{\rule{1em}{0ex}}=f\left(t,u,{\left(w\left(t\right)\right)}^{\frac{1}{p\left(t\right)-1}}{u}^{\mathrm{\prime }},S\left(u\right),T\left(u\right)\right)+\lambda \epsilon h\left(t,u,{\left(w\left(t\right)\right)}^{\frac{1}{p\left(t\right)-1}}{u}^{\mathrm{\prime }},S\left(u\right),T\left(u\right)\right).\hfill \end{array}$
We consider the existence of solutions of the following equation with (2)-(4)
$-{\left(w\left(t\right)|{u}^{\mathrm{\prime }}{|}^{p\left(t\right)-2}{u}^{\mathrm{\prime }}\right)}^{\mathrm{\prime }}+{\varphi }_{\lambda }\left(t,u,{\left(w\left(t\right)\right)}^{\frac{1}{p\left(t\right)-1}}{u}^{\mathrm{\prime }},S\left(u\right),T\left(u\right),\epsilon \right)=0,\phantom{\rule{1em}{0ex}}t\in \left(0,1\right),t\ne {t}_{i}.$
(27)
Denote
$\begin{array}{c}{\rho }_{1,\lambda }^{\mathrm{#}}\left(u,\epsilon \right)=\stackrel{˜}{{\rho }_{1}}\left(A,B,{N}_{{\varphi }_{\lambda }}\right)\left(u\right),\hfill \\ {K}_{1,\lambda }^{\mathrm{#}}\left(u,\epsilon \right)=F\left\{{\phi }^{-1}\left[t,{\left(w\left(t\right)\right)}^{-1}\left({\rho }_{1,\lambda }^{\mathrm{#}}\left(u,\epsilon \right)+\sum _{{t}_{i}

where ${N}_{{\varphi }_{\lambda }}\left(u\right)$ is defined in (10).

We know that (27) with (2)-(4) has the same solution of $u={\mathrm{\Phi }}_{\epsilon }\left(u,\lambda \right)$.

Obviously, ${\varphi }_{0}=f$. So ${\mathrm{\Phi }}_{\epsilon }\left(u,0\right)=$ ${\mathrm{\Psi }}_{f}\left(u,1\right)$. As in the proof of Theorem 3.1, we know that all the solutions of $u={\mathrm{\Phi }}_{\epsilon }\left(u,0\right)$ are uniformly bounded, then there exists a large enough ${R}_{0}>0$ such that all the solutions of $u={\mathrm{\Phi }}_{\epsilon }\left(u,0\right)$ belong to $B\left({R}_{0}\right)=\left\{u\in P{C}^{1}\mid {\parallel u\parallel }_{1}<{R}_{0}\right\}$. Since ${\mathrm{\Phi }}_{\epsilon }\left(\cdot ,0\right)$ is compact continuous from $P{C}^{1}$ to $P{C}^{1}$, we have
$\underset{u\in \partial B\left({R}_{0}\right)}{inf}{\parallel u-{\mathrm{\Phi }}_{\epsilon }\left(u,0\right)\parallel }_{1}>0.$
(28)
Since f and h are Caratheodory, we have
Thus
Obviously, ${\mathrm{\Phi }}_{0}\left(u,\lambda \right)={\mathrm{\Phi }}_{\epsilon }\left(u,0\right)={\mathrm{\Phi }}_{0}\left(u,0\right)$. We obtain
Thus, when ε is small enough, from (28), we can conclude that
$\begin{array}{c}\underset{\left(u,\lambda \right)\in \partial B\left({R}_{0}\right)×\left[0,1\right]}{inf}{\parallel u-{\mathrm{\Phi }}_{\epsilon }\left(u,\lambda \right)\parallel }_{1}\hfill \\ \phantom{\rule{1em}{0ex}}\ge \underset{u\in \partial B\left({R}_{0}\right)}{inf}{\parallel u-{\mathrm{\Phi }}_{\epsilon }\left(u,0\right)\parallel }_{1}-\underset{\left(u,\lambda \right)\in \overline{B\left({R}_{0}\right)}×\left[0,1\right]}{sup}{\parallel {\mathrm{\Phi }}_{\epsilon }\left(u,0\right)-{\mathrm{\Phi }}_{\epsilon }\left(u,\lambda \right)\parallel }_{1}>0.\hfill \end{array}$
Thus $u={\mathrm{\Phi }}_{\epsilon }\left(u,\lambda \right)$ has no solution on $\partial B\left({R}_{0}\right)$ for any $\lambda \in \left[0,1\right]$, when ε is small enough. It means that the Leray-Schauder degree ${d}_{LS}\left[I-{\mathrm{\Phi }}_{\epsilon }\left(\cdot ,\lambda \right),B\left({R}_{0}\right),0\right]$ is well defined for any $\lambda \in \left[0,1\right]$, and
${d}_{LS}\left[I-{\mathrm{\Phi }}_{\epsilon }\left(u,\lambda \right),B\left({R}_{0}\right),0\right]={d}_{LS}\left[I-{\mathrm{\Phi }}_{\epsilon }\left(u,0\right),B\left({R}_{0}\right),0\right].$

Since ${\mathrm{\Phi }}_{\epsilon }\left(u,0\right)=$ ${\mathrm{\Psi }}_{f}\left(u,1\right)$, from the proof of Theorem 3.1, we can see that the right-hand side is nonzero. Thus (26) with (2)-(4) has at least one solution when ε is small enough. This completes the proof. □

Theorem 3.4 Suppose that $\sigma <1$, ${\sum }_{\ell =1}^{m-2}{\alpha }_{\ell }-\delta =1$; $h\left(t\right)\ge 0$ on $\left[{\xi }_{1},1\right]$, ${\alpha }_{\ell }\ge {\int }_{{\xi }_{\ell }}^{{\xi }_{\ell +1}}h\left(t\right)\phantom{\rule{0.2em}{0ex}}dt$ ($\ell =1,\dots ,m-2$) and $h\left(t\right)\le 0$ on $\left[0,{\xi }_{1}\right]$; f satisfies the sub-$\left({p}^{-}-1\right)$ growth condition; and we assume that
$\begin{array}{c}\sum _{i=1}^{k}|{A}_{i}\left(u,v\right)|\le {C}_{1}{\left(1+|u|+|v|\right)}^{\frac{{q}^{+}-1}{{p}^{+}-1}},\hfill \\ \sum _{i=1}^{k}|{D}_{i}\left(u,v\right)|\le {C}_{2}{\left(1+|u|+|v|\right)}^{{\alpha }_{i}^{+}},\phantom{\rule{1em}{0ex}}\mathrm{\forall }\left(u,v\right)\in {\mathbb{R}}^{N}×{\mathbb{R}}^{N},\hfill \end{array}$

where ${\alpha }_{i}\le \frac{{q}^{+}-1}{p\left({t}_{i}\right)-1}$, and $p\left({t}_{i}\right)-1\le {q}^{+}-{\alpha }_{i}$, $i=1,\dots ,k$, then problem (26) with (2), (4) and (5) has at least one solution when parameter ε is small enough.

Proof Similar to the proof of Theorem 3.2 and Theorem 3.3, we omit it here. □

## 4 Existence of solutions in Case (ii)

In this section, we apply Leray-Schauder’s degree to deal with the existence of solutions for system (1)-(4) or (1) with (2), (4) and (5) when $\sigma =1$, ${\sum }_{\ell =1}^{m-2}{\alpha }_{\ell }-\delta \ne 1$.

When f satisfies the sub-$\left({p}^{-}-1\right)$ growth condition, we have the following.

Theorem 4.1 Suppose that $\sigma =1$, ${\sum }_{\ell =1}^{m-2}{\alpha }_{\ell }-\delta \ne 1$; f satisfies the sub-$\left({p}^{-}-1\right)$ growth condition; and operators A and B satisfy the following conditions:
$\begin{array}{c}\sum _{i=1}^{k}|{A}_{i}\left(u,v\right)|\le {C}_{1}{\left(1+|u|+|v|\right)}^{\frac{{q}^{+}-1}{{p}^{+}-1}},\hfill \\ \sum _{i=1}^{k}|{B}_{i}\left(u,v\right)|\le {C}_{2}{\left(1+|u|+|v|\right)}^{{q}^{+}-1},\phantom{\rule{1em}{0ex}}\mathrm{\forall }\left(u,v\right)\in {\mathbb{R}}^{N}×{\mathbb{R}}^{N},\hfill \end{array}$

then problem (1)-(4) has at least a solution.

Proof Similar to the proof of Theorem 3.1, we omit it here. □

Theorem 4.2 Suppose that $\sigma =1$, ${\sum }_{\ell =1}^{m-2}{\alpha }_{\ell }-\delta \ne 1$; f satisfies the sub-$\left({p}^{-}-1\right)$ growth condition; and operators A and $D=\left({D}_{1},\dots ,{D}_{k}\right)$ satisfy the following conditions:
$\begin{array}{c}\sum _{i=1}^{k}|{A}_{i}\left(u,v\right)|\le {C}_{1}{\left(1+|u|+|v|\right)}^{\frac{{q}^{+}-1}{{p}^{+}-1}},\hfill \\ \sum _{i=1}^{k}|{D}_{i}\left(u,v\right)|\le {C}_{2}{\left(1+|u|+|v|\right)}^{{\alpha }_{i}^{+}},\phantom{\rule{1em}{0ex}}\mathrm{\forall }\left(u,v\right)\in {\mathbb{R}}^{N}×{\mathbb{R}}^{N},\hfill \end{array}$
where
${\alpha }_{i}\le \frac{{q}^{+}-1}{p\left({t}_{i}\right)-1}\phantom{\rule{1em}{0ex}}\mathit{\text{and}}\phantom{\rule{1em}{0ex}}p\left({t}_{i}\right)-1\le {q}^{+}-{\alpha }_{i},\phantom{\rule{1em}{0ex}}i=1,\dots ,k,$

then problem (1) with (2), (4) and (5) has at least a solution.

Proof Similar to the proof of Theorem 3.2, we omit it here. □

Theorem 4.3 Suppose that $\sigma =1$, ${\sum }_{\ell =1}^{m-2}{\alpha }_{\ell }-\delta \ne 1$; f satisfies the sub-$\left({p}^{-}-1\right)$ growth condition; and we assume that
$\begin{array}{c}\sum _{i=1}^{k}|{A}_{i}\left(u,v\right)|\le {C}_{1}{\left(1+|u|+|v|\right)}^{\frac{{q}^{+}-1}{{p}^{+}-1}},\hfill \\ \sum _{i=1}^{k}|{B}_{i}\left(u,v\right)|\le {C}_{2}{\left(1+|u|+|v|\right)}^{{q}^{+}-1},\phantom{\rule{1em}{0ex}}\mathrm{\forall }\left(u,v\right)\in {\mathbb{R}}^{N}×{\mathbb{R}}^{N},\hfill \end{array}$

then problem (26) with (2)-(4) has at least one solution when parameter ε is small enough.

Proof Similar to the proof of Theorem 3.3, we omit it here. □

Theorem 4.4 Suppose that $\sigma =1$, ${\sum }_{\ell =1}^{m-2}{\alpha }_{\ell }-\delta \ne 1$; f satisfies the sub-$\left({p}^{-}-1\right)$ growth condition; and we assume that
$\begin{array}{c}\sum _{i=1}^{k}|{A}_{i}\left(u,v\right)|\le {C}_{1}{\left(1+|u|+|v|\right)}^{\frac{{q}^{+}-1}{{p}^{+}-1}},\hfill \\ \sum _{i=1}^{k}|{D}_{i}\left(u,v\right)|\le {C}_{2}{\left(1+|u|+|v|\right)}^{{\alpha }_{i}^{+}},\phantom{\rule{1em}{0ex}}\mathrm{\forall }\left(u,v\right)\in {\mathbb{R}}^{N}×{\mathbb{R}}^{N},\hfill \end{array}$

where ${\alpha }_{i}\le \frac{{q}^{+}-1}{p\left({t}_{i}\right)-1}$, and $p\left({t}_{i}\right)-1\le {q}^{+}-{\alpha }_{i}$, $i=1,\dots ,k$, then problem (26) with (2), (4) and (5) has at least one solution when parameter ε is small enough.

Proof Similar to the proof of Theorem 3.2 and Theorem 3.3, we omit it here. □

## 5 Existence of solutions in Case (iii)

In this section, we apply Leray-Schauder’s degree to deal with the existence of solutions and nonnegative solutions for system (1)-(4) or (1) with (2), (4) and (5) when $\sigma <1$, ${\sum }_{\ell =1}^{m-2}{\alpha }_{\ell }-\delta <1$.

When f satisfies the sub-$\left({p}^{-}-1\right)$ growth condition, we have the following theorem.

Theorem 5.1 Suppose that $\sigma <1$, ${\sum }_{\ell =1}^{m-2}{\alpha }_{\ell }-\delta <1$ and ${\alpha }_{\ell }$, g, h satisfy one of the following:

(10) ${\sum }_{\ell =1}^{m-2}{\alpha }_{\ell }\le 1$, $g\left(t\right)\left(1-{\sum }_{\ell =1}^{m-2}{\alpha }_{\ell }+\delta \right)+h\left(t\right)\left(1-\sigma \right)\ge 0$;

(20) $h\left(t\right)\ge 0$ on $\left[{\xi }_{1},1\right]$, ${\alpha }_{\ell }\ge {\int }_{{\xi }_{\ell }}^{{\xi }_{\ell +1}}h\left(t\right)\phantom{\rule{0.2em}{0ex}}dt$ ($\ell =1,\dots ,m-2$