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

Abstract

This paper investigates the existence of solutions for a weighted $p(t)$-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.

1 Introduction

In this paper, we consider the existence of solutions and nonnegative solutions for the following weighted $p(t)$-Laplacian integro-differential system:

$$-{△}_{p(t)}u+f(t,u,{(w(t))}^{\frac{1}{p(t)-1}}{u}^{\mathrm{\prime }},S(u),T(u))=0,t\in (0,1),t\ne t_i,$$

(1)

where $u:[0,1]\to {\mathbb{R}}^N$, $f(\cdot ,\cdot ,\cdot ,\cdot ,\cdot ):[0,1]\times {\mathbb{R}}^N\times {\mathbb{R}}^N\times {\mathbb{R}}^N\times {\mathbb{R}}^N\to {\mathbb{R}}^N$, $t_i\in (0,1)$, $i=1,\dots ,k$, with the following impulsive boundary value conditions:

$$\underset{t\to t_i^{+}}{lim}u(t)-\underset{t\to t_i^{-}}{lim}u(t)=A_i(\underset{t\to t_i^{-}}{lim}u(t),\underset{t\to t_i^{-}}{lim}{(w(t))}^{\frac{1}{p(t)-1}}{u}^{\mathrm{\prime }}(t)),i=1,\dots ,k,$$

(2)

$$\begin{array}{c}\underset{t\to t_i^{+}}{lim}w(t)|u^{\mathrm{\prime }}{|}^{p(t)-2}{u}^{\mathrm{\prime }}(t)-\underset{t\to t_i^{-}}{lim}w(t)|u^{\mathrm{\prime }}{|}^{p(t)-2}{u}^{\mathrm{\prime }}(t)\hfill \\ =B_i(\underset{t\to t_i^{-}}{lim}u(t),\underset{t\to t_i^{-}}{lim}{(w(t))}^{\frac{1}{p(t)-1}}{u}^{\mathrm{\prime }}(t)),i=1,\dots ,k,\hfill \end{array}$$

(3)

$$u(0)={\int }_0^{1}g(t)u(t)dt,u(1)=\sum _{\ell =1}^{m-2}{\alpha }_{\ell }u({\xi }_{\ell })-{\int }_0^{1}h(t)u(t)dt,$$

(4)

where $p\in C([0,1],\mathbb{R})$ and $p(t)>1$, $-{△}_{p(t)}u:=-{(w(t)|u^{\mathrm{\prime }}{|}^{p(t)-2}{u}^{\mathrm{\prime }})}^{\mathrm{\prime }}$ is called the weighted $p(t)$-Laplacian; $0<t_1<t_2<\cdots <t_k<1$, $0<{\xi }_1<\cdots <{\xi }_{m-2}<1$; ${\alpha }_{\ell }\ge 0$ ($\ell =1,\dots ,m-2$); $g\in L^1[0,1]$ is nonnegative, ${\int }_0^{1}g(t)dt=\sigma \in [0,1]$; $h\in L^1[0,1]$, ${\int }_0^{1}h(t)dt=\delta$; $A_i,B_i\in C({\mathbb{R}}^N\times {\mathbb{R}}^N,{\mathbb{R}}^N)$; T and S are linear operators defined by $(Su)(t)={\int }_0^1{h}_{\ast }(t,s)u(s)ds$, $(Tu)(t)={\int }_0^t{k}_{\ast }(t,s)u(s)ds$, $t\in [0,1]$, where $k_{\ast },h_{\ast }\in C([0,1]\times [0,1],\mathbb{R})$.

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, $o(1)$ 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 $J=[0,1]$, $J^{\mathrm{\prime }}=(0,1)\mathrm{\setminus }\{t_1,\dots ,t_k\}$, $J_0=[t_0,t_1]$, $J_i=(t_i,t_{i+1}]$, $i=1,\dots ,k$, where $t_0=0$, $t_{k+1}=1$. Denote by $J_i^o$ the interior of $J_i$, $i=0,1,\dots ,k$. Let

$$PC(J,{\mathbb{R}}^N)=\{x:J\to {\mathbb{R}}^N|\begin{array}{l}x\in C(J_i,{\mathbb{R}}^N),i=0,1,\dots ,k\\ \text{and }{lim}_{t\to t_i^{+}}x(t)\text{ exists for }i=1,\dots ,k\end{array}\},$$

$w\in PC(J,\mathbb{R})$ satisfy $0<w(t)$, $\mathrm{\forall }t\in (0,1)\mathrm{\setminus }\{t_1,\dots ,t_k\}$, and ${(w(t))}^{\frac{-1}{p(t)-1}}\in L^1(0,1)$,

$$P{C}^1(J,{\mathbb{R}}^N)=\{x\in PC(J,{\mathbb{R}}^N)|\begin{array}{l}{x}^{\mathrm{\prime }}\in C(J_i^o,{\mathbb{R}}^N),{lim}_{t\to t_i^{+}}{(w(t))}^{\frac{1}{p(t)-1}}{x}^{\mathrm{\prime }}(t)\\ \text{and }{lim}_{t\to t_{i+1}^{-}}{(w(t))}^{\frac{1}{p(t)-1}}{x}^{\mathrm{\prime }}(t)\text{ exists for }i=0,1,\dots ,k\end{array}\}.$$

For any $x=(x^1,\dots ,x^N)\in PC(J,{\mathbb{R}}^N)$, denote ${|x^i|}_0=sup\{|x^i(t)|\mid t\in J^{\mathrm{\prime }}\}$.

Obviously, $PC(J,{\mathbb{R}}^N)$ is a Banach space with the norm ${\parallel x\parallel }_0={({\sum }_{i=1}^N{|x^i|}_0^2)}^{\frac{1}{2}}$, and $P{C}^1(J,{\mathbb{R}}^N)$ is a Banach space with the norm ${\parallel x\parallel }_1={\parallel x\parallel }_0+{\parallel {(w(t))}^{\frac{1}{p(t)-1}}{x}^{\mathrm{\prime }}\parallel }_0$. Denote $L^1=L^1(J,{\mathbb{R}}^N)$ with the norm

$${\parallel x\parallel }_{L^1}={\left(\sum _{i=1}^N{\left|x^i\right|}_{L^1}^2\right)}^{\frac{1}{2}},\mathrm{\forall }x\in L^1,\text{ where }{\left|x^i\right|}_{L^1}={\int }_0^1|x^i(t)|dt.$$

In the following, $PC(J,{\mathbb{R}}^N)$ and $P{C}^1(J,{\mathbb{R}}^N)$ will be simply denoted by PC and $P{C}^1$, respectively. We denote

$$\begin{array}{c}u\left(t_i^{+}\right)=\underset{t\to t_i^{+}}{lim}u(t),u\left(t_i^{-}\right)=\underset{t\to t_i^{-}}{lim}u(t),\hfill \\ w(0)|u^{\mathrm{\prime }}{|}^{p(0)-2}{u}^{\mathrm{\prime }}(0)=\underset{t\to 0^{+}}{lim}w(t)|u^{\mathrm{\prime }}{|}^{p(t)-2}{u}^{\mathrm{\prime }}(t),\hfill \\ w(1)|u^{\mathrm{\prime }}{|}^{p(1)-2}{u}^{\mathrm{\prime }}(1)=\underset{t\to 1^{-}}{lim}w(t)|u^{\mathrm{\prime }}{|}^{p(t)-2}{u}^{\mathrm{\prime }}(t),\hfill \\ A_i=A_i(\underset{t\to t_i^{-}}{lim}u(t),\underset{t\to t_i^{-}}{lim}{(w(t))}^{\frac{1}{p(t)-1}}{u}^{\mathrm{\prime }}(t)),i=1,\dots ,k,\hfill \\ B_i=B_i(\underset{t\to t_i^{-}}{lim}u(t),\underset{t\to t_i^{-}}{lim}{(w(t))}^{\frac{1}{p(t)-1}}{u}^{\mathrm{\prime }}(t)),i=1,\dots ,k.\hfill \end{array}$$

The study of differential equations and variational problems with nonstandard $p(t)$-growth conditions has attracted more and more interest in recent years (see [1–4]). 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, [5–15]. Recently, the applications of variable exponent analysis in image restoration have attracted more and more attention [16–19]. If $p(t)\equiv p$ (a constant), (1)-(4) becomes the well-known p-Laplacian problem. If $p(t)$ is a general function, one can see easily $-{△}_{p(t)}cu\ne c^{p(t)-1}(-{△}_{p(t)}u)$ in general, but $-{△}_{p}cu=c^{p-1}(-{△}_{p}u)$, so $-{△}_{p(t)}$ represents a non-homogeneity and possesses more nonlinearity, thus $-{△}_{p(t)}$ is more complicated than $-{△}_p$. For example:

• (a) If $\mathrm{\Omega }\subset {\mathbb{R}}^N$ is a bounded domain, the Rayleigh quotient

  $${\lambda }_{p(x)}=\underset{u\in W_0^{1,p(x)}(\mathrm{\Omega })\mathrm{\setminus }\{0\}}{inf}\frac{{\int }_{\mathrm{\Omega }}\frac{1}{p(x)}|\mathrm{\nabla }u{|}^{p(x)}dx}{{\int }_{\mathrm{\Omega }}\frac{1}{p(x)}|u{|}^{p(x)}dx}$$

is zero in general, and only under some special conditions ${\lambda }_{p(x)}>0$ (see [9]), when $\mathrm{\Omega }\subset \mathbb{R}$ ($N=1$) is an interval, the results show that ${\lambda }_{p(x)}>0$ if and only if $p(x)$ 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.

• (b) If $w(t)\equiv 1$ and $p(t)\equiv p$ (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 $-{△}_{p(t)}$. It is another difference between $-{△}_p$ and $-{△}_{p(t)}$.

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, [22–29]. 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 [30–33]). In [34], using the coincidence degree method, the present author investigates the existence of solutions for $p(r)$-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, [35–38].

In this paper, when $p(t)$ is a general function, we investigate the existence of solutions and nonnegative solutions for the weighted $p(t)$-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:

$$\begin{array}{r}\underset{t\to t_i^{+}}{lim}{(w(t))}^{\frac{1}{p(t)-1}}{u}^{\mathrm{\prime }}(t)-\underset{t\to t_i^{-}}{lim}{(w(t))}^{\frac{1}{p(t)-1}}{u}^{\mathrm{\prime }}(t)\\ =D_i(\underset{t\to t_i^{-}}{lim}u(t),\underset{t\to t_i^{-}}{lim}{(w(t))}^{\frac{1}{p(t)-1}}{u}^{\mathrm{\prime }}(t)),i=1,\dots ,k,\end{array}$$

(5)

where $D_i\in C({\mathbb{R}}^N\times {\mathbb{R}}^N,{\mathbb{R}}^N)$, 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 ${\lambda }_{p(x)}=0$ in general and the $p(t)$-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-$(p^{-}-1)$ growth condition, but for the p-Laplacian impulsive problems, the nonlinear term only needs to satisfy the sub-$(p-1)$ growth condition.

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

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

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

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

  $$|f(t,x,y,s,z)|\le {\alpha }_R(t).$$

We say a function $u:J\to {\mathbb{R}}^N$ is a solution of (1) if $u\in P{C}^1$ with $w(t)|u^{\mathrm{\prime }}{|}^{p(t)-2}{u}^{\mathrm{\prime }}$ 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

$$z^{-}=\underset{t\in J}{inf}z(t),z^{+}=\underset{t\in J}{sup}z(t)\text{for any }z\in PC(J,\mathbb{R}).$$

We say f satisfies the sub-$(p^{-}-1)$ growth condition if f satisfies

$$\underset{|u|+|v|+|s|+|z|\to +\mathrm{\infty }}{lim}\frac{f(t,u,v,s,z)}{{(|u|+|v|+|s|+|z|)}^{q(t)-1}}=0\text{for }t\in J\text{ uniformly,}$$

where $q(t)\in PC(J,\mathbb{R})$ 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 $(t,x)\in J\times {\mathbb{R}}^N$, denote $\phi (t,x)=|x{|}^{p(t)-2}x$. Obviously, φ has the following properties.

Lemma 2.1 (see [34])

φ is a continuous function and satisfies:

• (i) For any $t\in [0,1]$, $\phi (t,\cdot )$ is strictly monotone, i.e.,

  $$〈\phi (t,x_1)-\phi (t,x_2),x_1-x_2〉>0\mathit{\text{for any}}{x}_1,x_2\in {\mathbb{R}}^N,x_1\ne x_2.$$

• (ii) There exists a function $\alpha :[0,+\mathrm{\infty })\to [0,+\mathrm{\infty })$, $\alpha (s)\to +\mathrm{\infty }$ as $s\to +\mathrm{\infty }$ such that

  $$〈\phi (t,x),x〉\ge \alpha (|x|)|x|\mathit{\text{for all}}x\in {\mathbb{R}}^N.$$

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

$${\phi }^{-1}(t,x)=|x{|}^{\frac{2-p(t)}{p(t)-1}}x\text{for }x\in {\mathbb{R}}^N\mathrm{\setminus }\{0\},{\phi }^{-1}(t,0)=0,\mathrm{\forall }t\in J.$$

It is clear that ${\phi }^{-1}(t,\cdot )$ 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):

$$\begin{array}{l}{(w(t)\phi (t,u^{\mathrm{\prime }}(t)))}^{\mathrm{\prime }}=f(t),t\in (0,1),t\ne t_i,\\ {lim}_{t\to t_i^{+}}u(t)-{lim}_{t\to t_i^{-}}u(t)=a_i,i=1,\dots ,k,\\ {lim}_{t\to t_i^{+}}w(t)|u^{\mathrm{\prime }}{|}^{p(t)-2}{u}^{\mathrm{\prime }}(t)-{lim}_{t\to t_i^{-}}w(t)|u^{\mathrm{\prime }}{|}^{p(t)-2}{u}^{\mathrm{\prime }}(t)=b_i,i=1,\dots ,k,\end{array}\}$$

(6)

where $a_i,b_i\in {\mathbb{R}}^N$; $f\in L^1$.

Denote $a=(a_1,\dots ,a_k)$, $b=(b_1,\dots ,b_k)$. 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

$$w(t)\phi (t,u^{\mathrm{\prime }}(t))=w(0)\phi (0,u^{\mathrm{\prime }}(0))+\sum _{t_i<t}{b}_i+{\int }_0^{t}f(s)ds,\mathrm{\forall }t\in J^{\mathrm{\prime }}.$$

(7)

Denote ${\rho }_1=w(0)\phi (0,u^{\mathrm{\prime }}(0))$. It is easy to see that ${\rho }_1$ is dependent on a, b and $f(\cdot )$. Define the operator $F:L^1\to PC$ as

$$F(f)(t)={\int }_0^{t}f(s)ds,\mathrm{\forall }t\in J,\mathrm{\forall }f\in L^1.$$

By solving for $u^{\mathrm{\prime }}$ in (7) and integrating, we find

$$u(t)=u(0)+\sum _{t_i<t}{a}_i+F\left\{{\phi }^{-1}[t,{(w(t))}^{-1}({\rho }_1+\sum _{t_i<t}{b}_i+F(f)(t))]\right\}(t),\mathrm{\forall }t\in J,$$

which together with boundary value condition (4) implies

$$u(0)=\frac{1}{(1-\sigma )}{\int }_0^{1}g(t)(F\left\{{\phi }^{-1}[t,{(w(t))}^{-1}({\rho }_1+\sum _{t_i<t}{b}_i+F(f)(t))]\right\}(t)+\sum _{t_i<t}{a}_i)dt,$$

and

$$\begin{array}{c}\sum _{\ell =1}^{m-2}{\alpha }_{\ell }\{\sum _{t_i<{\xi }_{\ell }}{a}_i+{\int }_0^{{\xi }_{\ell }}{\phi }^{-1}[t,{(w(t))}^{-1}({\rho }_1+\sum _{t_i<t}{b}_i+F(f)(t))]dt\}\hfill \\ -\sum _{i=1}^k{a}_i-{\int }_0^1{\phi }^{-1}[t,{(w(t))}^{-1}({\rho }_1+\sum _{t_i<t}{b}_i+F(f)(t))]dt\hfill \\ -{\int }_0^{1}h(t)(F\left\{{\phi }^{-1}[t,{(w(t))}^{-1}({\rho }_1+\sum _{t_i<t}{b}_i+F(f)(t))]\right\}(t)+\sum _{t_i<t}{a}_i)dt=0.\hfill \end{array}$$

Denote $W={\mathbb{R}}^{2kN}\times L^1$ with the norm

$$\parallel \omega \parallel =\sum _{i=1}^k|a_i|+\sum _{i=1}^k|b_i|+{\parallel \vartheta \parallel }_{L^1},\mathrm{\forall }\omega =(a,b,\vartheta )\in W,$$

then W is a Banach space.

For any $\omega \in W$, we denote

$$\begin{array}{rcl}{\mathrm{\Lambda }}_{\omega }({\rho }_1)& =& \sum _{\ell =1}^{m-2}{\alpha }_{\ell }\{\sum _{t_i<{\xi }_{\ell }}{a}_i+{\int }_0^{{\xi }_{\ell }}{\phi }^{-1}[t,{(w(t))}^{-1}({\rho }_1+\sum _{t_i<t}{b}_i+F(\vartheta )(t))]dt\}\\ -\sum _{i=1}^k{a}_i-{\int }_0^1{\phi }^{-1}[t,{(w(t))}^{-1}({\rho }_1+\sum _{t_i<t}{b}_i+F(\vartheta )(t))]dt\\ -{\int }_0^{1}h(t)(F\left\{{\phi }^{-1}[t,{(w(t))}^{-1}({\rho }_1+\sum _{t_i<t}{b}_i+F(\vartheta )(t))]\right\}(t)+\sum _{t_i<t}{a}_i)dt.\end{array}$$

Denote ${\xi }_{m-1}=1$. Then

$$\begin{array}{rcl}{\mathrm{\Lambda }}_{\omega }({\rho }_1)& =& -\sum _{\ell =1}^{m-2}{\alpha }_{\ell }\{\sum _{{\xi }_{\ell }\le t_i}{a}_i+{\int }_{{\xi }_{\ell }}^1{\phi }^{-1}[t,{(w(t))}^{-1}({\rho }_1+\sum _{t_i<t}{b}_i+F(\vartheta )(t))]dt\}\\ +{\int }_0^{1}h(t)({\int }_t^1{\phi }^{-1}[t,{(w(t))}^{-1}({\rho }_1+\sum _{t_i<t}{b}_i+F(\vartheta )(t))]dt+\sum _{t_i\ge t}{a}_i)dt\\ =& -\sum _{\ell =1}^{m-2}({\alpha }_{\ell }-{\int }_{{\xi }_{\ell }}^{{\xi }_{\ell +1}}h(t)dt){\int }_{{\xi }_{\ell }}^1{\phi }^{-1}[t,{(w(t))}^{-1}({\rho }_1+\sum _{t_i<t}{b}_i+F(\vartheta )(t))]dt\\ -\sum _{\ell =1}^{m-2}{\int }_{{\xi }_{\ell }}^{{\xi }_{\ell +1}}h(t){\int }_{{\xi }_{\ell }}^t{\phi }^{-1}[s,{(w(s))}^{-1}({\rho }_1+\sum _{s_i<s}{b}_i+F(\vartheta )(s))]dsdt\\ +{\int }_0^{{\xi }_1}h(t){\int }_t^1{\phi }^{-1}[s,{(w(s))}^{-1}({\rho }_1+\sum _{s_i<s}{b}_i+F(\vartheta )(s))]dsdt\\ -\sum _{\ell =1}^{m-2}{\alpha }_{\ell }\sum _{{\xi }_{\ell }\le t_i}{a}_i+{\int }_0^{1}h(t)\sum _{t_i\ge t}{a}_{i}dt.\end{array}$$

Throughout the paper, we denote

Lemma 2.2 Suppose that $h(t)\ge 0$ on $[{\xi }_1,1]$, ${\alpha }_{\ell }\ge {\int }_{{\xi }_{\ell }}^{{\xi }_{\ell +1}}h(t)dt$ ($\ell =1,\dots ,m-2$) and $h(t)\le 0$ on $[0,{\xi }_1]$. Then the function ${\mathrm{\Lambda }}_{\omega }(\cdot )$ has the following properties:

• (i) For any fixed $\omega \in W$, the equation

  $${\mathrm{\Lambda }}_{\omega }({\rho }_1)=0$$

  (8)

has a unique solution $\widetilde{{\rho }_1}(\omega )\in {\mathbb{R}}^N$.

• (ii) The function $\widetilde{{\rho }_1}:W\to {\mathbb{R}}^N$, defined in (i), is continuous and sends bounded sets to bounded sets. Moreover, for any $\omega =(a,b,\vartheta )\in W$, we have

  $$|\widetilde{{\rho }_1}(\omega )|\le 3N[{(2N)}^{p^{+}}{({\delta }^{\ast }\frac{E+1}{E}\sum _{i=1}^k|a_i|)}^{p^{\mathrm{\#}}-1}+\sum _{i=1}^k|b_i|+{\parallel \vartheta \parallel }_{L^1}],$$

where the notation $M^{p^{\mathrm{\#}}-1}$ means

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

Proof (i) From Lemma 2.1, it is immediate that

$$〈{\mathrm{\Lambda }}_{\omega }(x_1)-{\mathrm{\Lambda }}_{\omega }(x_2),x_1-x_2〉<0\text{for }{x}_1\ne x_2,\mathrm{\forall }{x}_1,x_2\in {\mathbb{R}}^N,$$

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

Set $R_0=3N[{(2N)}^{p^{+}}{({\delta }^{\ast }\frac{E+1}{E}{\sum }_{i=1}^k|a_i|)}^{p^{\mathrm{\#}}-1}+{\sum }_{i=1}^k|b_i|+{\parallel \vartheta \parallel }_{L^1}]$.

Suppose that $|{\rho }_1|>R_0$, it is easy to see that there exists some $j_0\in \{1,\dots ,N\}$ such that the absolute value of the $j_0$th component of ${\rho }_1$ satisfies

Thus the $j_0$th component of ${\rho }_1+{\sum }_{t_i<t}{b}_i+F(\vartheta )(t)$ 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 ${\mathrm{\Lambda }}_{\omega }({\rho }_1)$ keeps the same sign of . Thus,

$${\mathrm{\Lambda }}_{\omega }({\rho }_1)\ne 0.$$

Let us consider the equation

$$\lambda {\mathrm{\Lambda }}_{\omega }({\rho }_1)+(1-\lambda ){\rho }_1=0,\lambda \in [0,1].$$

(9)

According to the preceding discussion, all the solutions of (9) belong to $b(R_0+1)=\{x\in {\mathbb{R}}^N\mid |x|<R_0+1\}$. Therefore

$$d_B[{\mathrm{\Lambda }}_{\omega }({\rho }_1),b(R_0+1),0]=d_B[I,b(R_0+1),0]\ne 0,$$

it means the existence of solutions of ${\mathrm{\Lambda }}_{\omega }({\rho }_1)=0$.

In this way, we define a function $\widetilde{{\rho }_1}(\omega ):W\to {\mathbb{R}}^N$, which satisfies ${\mathrm{\Lambda }}_{\omega }(\widetilde{{\rho }_1}(\omega ))=0$.

• (ii) By the proof of (i), we also obtain $\widetilde{{\rho }_1}$ sends bounded sets to bounded sets, and

  $$|\widetilde{{\rho }_1}(\omega )|\le 3N[{(2N)}^{p^{+}}{({\delta }^{\ast }\frac{E+1}{E}\sum _{i=1}^k|a_i|)}^{p^{\mathrm{\#}}-1}+\sum _{i=1}^k|b_i|+{\parallel \vartheta \parallel }_{L^1}].$$

It only remains to prove the continuity of $\widetilde{{\rho }_1}$. Let $\{{\omega }_n\}$ be a convergent sequence in W and ${\omega }_n\to \omega$, as $n\to +\mathrm{\infty }$. Since $\{\widetilde{{\rho }_1}({\omega }_n)\}$ is a bounded sequence, it contains a convergent subsequence $\{\widetilde{{\rho }_1}({\omega }_{n_j})\}$. Suppose that $\widetilde{{\rho }_1}({\omega }_{n_j})\to {\rho }_0$ as $j\to +\mathrm{\infty }$. Since ${\mathrm{\Lambda }}_{{\omega }_{n_j}}(\widetilde{{\rho }_1}({\omega }_{n_j}))=0$, letting $j\to +\mathrm{\infty }$, we have ${\mathrm{\Lambda }}_{\omega }({\rho }_0)=0$, which together with (i) implies ${\rho }_0=\widetilde{{\rho }_1}(\omega )$, it means $\widetilde{{\rho }_1}$ is continuous. This completes the proof. □

Now we denote by $N_f(u):[0,1]\times P{C}^1\to L^1$ the Nemytskii operator associated to f defined by

$$N_f(u)(t)=f(t,u(t),{(w(t))}^{\frac{1}{p(t)-1}}{u}^{\mathrm{\prime }}(t),S(u),T(u))\text{on }J.$$

(10)

We define ${\rho }_1:P{C}^1\to {\mathbb{R}}^N$ as

$${\rho }_1(u)=\widetilde{{\rho }_1}(A,B,N_f)(u),$$

(11)

where $A=(A_1,\dots ,A_k)$, $B=(B_1,\dots ,B_k)$.

It is clear that ${\rho }_1(\cdot )$ 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

$$u(t)=u(0)+\sum _{t_i<t}{a}_i+F\left\{{\phi }^{-1}[t,{(w(t))}^{-1}(\widetilde{{\rho }_1}(\omega )+\sum _{t_i<t}{b}_i+F(f)(t))]\right\}(t),\mathrm{\forall }t\in [0,1].$$

For fixed $a,b\in {\mathbb{R}}^{kN}$, we denote $K_{(a,b)}:L^1\to P{C}^1$ as

$$K_{(a,b)}(\vartheta )(t)=F\left\{{\phi }^{-1}[t,{(w(t))}^{-1}(\widetilde{{\rho }_1}(a,b,\vartheta )+\sum _{t_i<t}{b}_i+F(\vartheta )(t))]\right\}(t),\mathrm{\forall }t\in J.$$

Define $K_1:P{C}^1\to P{C}^1$ as

$$K_1(u)(t)=F\left\{{\phi }^{-1}[t,{(w(t))}^{-1}({\rho }_1(u)+\sum _{t_i<t}{B}_i+F(N_f(u))(t))]\right\}(t),\mathrm{\forall }t\in J.$$

Lemma 2.3 (i) The operator $K_{(a,b)}$ is continuous and sends equi-integrable sets in $L^1$ to relatively compact sets in $P{C}^1$.

• (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 $K_{(a,b)}(\vartheta )(\cdot )\in P{C}^1$, $\mathrm{\forall }\vartheta \in L^1$, $\mathrm{\forall }a,b\in {\mathbb{R}}^{kN}$. Since and

$$K_{(a,b)}{(\vartheta )}^{\mathrm{\prime }}(t)={\phi }^{-1}[t,{(w(t))}^{-1}(\widetilde{{\rho }_1}(a,b,\vartheta )+\sum _{t_i<t}{b}_i+F(\vartheta ))],\mathrm{\forall }t\in [0,1],$$

it is easy to check that $K_{(a,b)}(\cdot )$ 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

$$|u(t)|\le \alpha (t)\text{a.e. in }J\text{ for any }u\in L^1.$$

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

Let $\{u_n\}$ be a sequence in $K_{(a,b)}(U)$, then there exists a sequence $\{{\vartheta }_n\}\in U$ such that $u_n=K_{(a,b)}({\vartheta }_n)$. For any $t_1,t_2\in J$, we have

$$|F({\vartheta }_n)(t_1)-F({\vartheta }_n)(t_2)|=|{\int }_0^{t_1}{\vartheta }_n(t)dt-{\int }_0^{t_2}{\vartheta }_n(t)dt|=|{\int }_{t_1}^{t_2}{\vartheta }_n(t)dt|\le |{\int }_{t_1}^{t_2}\alpha (t)dt|.$$

Hence the sequence $\{F({\vartheta }_n)\}$ is uniformly bounded and equi-continuous. By the Ascoli-Arzela theorem, there exists a subsequence of $\{F({\vartheta }_n)\}$ (which we rename the same) which is convergent in PC. According to the bounded continuity of the operator $\widetilde{{\rho }_1}$, we can choose a subsequence of $\{\widetilde{{\rho }_1}(a,b,{\vartheta }_n)+F({\vartheta }_n)\}$ (which we still denote $\{\widetilde{{\rho }_1}(a,b,{\vartheta }_n)+F({\vartheta }_n)\}$) which is convergent in PC, then is convergent in PC.

Since

$$K_{(a,b)}({\vartheta }_n)(t)=F\left\{{\phi }^{-1}[t,{(w(t))}^{-1}(\widetilde{{\rho }_1}(a,b,{\vartheta }_n)+\sum _{t_i<t}{b}_i+F({\vartheta }_n))]\right\}(t),\mathrm{\forall }t\in [0,1],$$

it follows from the continuity of ${\phi }^{-1}$ and the integrability of in $L^1$ that $K_{(a,b)}({\vartheta }_n)$ is convergent in PC. Thus $\{u_n\}$ is convergent in $P{C}^1$.

• (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

$$P_1(u)=\frac{{\int }_0^{1}g(t)[K_1(u)(t)+{\sum }_{t_i<t}{A}_i]dt}{1-\sigma }.$$

It is easy to see that $P_1$ is compact continuous.

Lemma 2.4 Suppose that $\sigma <1$, ${\sum }_{\ell =1}^{m-2}{\alpha }_{\ell }-\delta =1$; $h(t)\ge 0$ on $[{\xi }_1,1]$, ${\alpha }_{\ell }\ge {\int }_{{\xi }_{\ell }}^{{\xi }_{\ell +1}}h(t)dt$ ($\ell =1,\dots ,m-2$) and $h(t)\le 0$ on $[0,{\xi }_1]$. Then u is a solution of (1)-(4) if and only if u is a solution of the following abstract operator equation:

$$u=P_1(u)+\sum _{t_i<t}{A}_i+K_1(u).$$

(12)

Proof Suppose that u is a solution of (1)-(4). By integrating (1) from 0 to t, we find that

$$w(t)\phi (t,u^{\mathrm{\prime }}(t))={\rho }_1(u)+\sum _{t_i<t}{B}_i+F(N_f(u))(t),\mathrm{\forall }t\in (0,1),t\ne t_1,\dots ,t_k.$$

(13)

It follows from (13) and (4) that

$$\begin{array}{c}u(t)=u(0)+\sum _{t_i<t}{A}_i\hfill \\ \phantom{u(t)=}+F\left\{{\phi }^{-1}[t,{(w(t))}^{-1}({\rho }_1(u)+\sum _{t_i<t}{B}_i+F(N_f(u)))]\right\}(t),\mathrm{\forall }t\in [0,1],\hfill \\ u(0)=\frac{1}{(1-\sigma )}\hfill \\ \phantom{u(0)=}\times {\int }_0^{1}g(t)(F\left\{{\phi }^{-1}[t,{(w(t))}^{-1}({\rho }_1(u)+\sum _{t_i<t}{B}_i+F(N_f(u)))]\right\}(t)+\sum _{t_i<t}{A}_i)dt\hfill \\ \phantom{u(0)}=\frac{{\int }_0^{1}g(t)[K_1(u)(t)+{\sum }_{t_i<t}{A}_i]dt}{1-\sigma }=P_1(u).\hfill \end{array}$$

(14)

Combining the definition of ${\rho }_1$, we can see

$$u=P_1(u)+\sum _{t_i<t}{A}_i+K_1(u).$$

Conversely, if u is a solution of (12), then (2) is satisfied. It is easy to check that

$$\begin{array}{c}u(0)=P_1(u)=\frac{{\int }_0^{1}g(t)[K_1(u)(t)+{\sum }_{t_i<t}{A}_i]dt}{1-\sigma },\hfill \\ u(0)=\sigma u(0)+{\int }_0^{1}g(t)[K_1(u)(t)+\sum _{t_i<t}{A}_i]dt={\int }_0^{1}g(t)u(t)dt,\hfill \end{array}$$

(15)

and

$$u(1)=P_1(u)+\sum _{i=1}^k{A}_i+K_1(u)(1).$$

By the condition of the mapping ${\rho }_1$, we have

$$\begin{array}{c}\sum _{\ell =1}^{m-2}{\alpha }_{\ell }\{\sum _{t_i<{\xi }_{\ell }}{A}_i+{\int }_0^{{\xi }_{\ell }}{\phi }^{-1}[t,{(w(t))}^{-1}({\rho }_1+\sum _{t_i<t}{B}_i+F(N_f(u))(t))]dt\}\hfill \\ -\sum _{i=1}^k{A}_i-{\int }_0^1{\phi }^{-1}[t,{(w(t))}^{-1}({\rho }_1+\sum _{t_i<t}{B}_i+F(N_f(u))(t))]dt\hfill \\ -{\int }_0^{1}h(t)(F\left\{{\phi }^{-1}[t,{(w(t))}^{-1}({\rho }_1+\sum _{t_i<t}{B}_i+F(N_f(u))(t))]\right\}(t)+\sum _{t_i<t}{A}_i)dt=0.\hfill \end{array}$$

Thus

$$u(1)=\sum _{\ell =1}^{m-2}{\alpha }_{\ell }u({\xi }_{\ell })-{\int }_0^{1}h(t)u(t)dt.$$

(16)

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

From (12), we have

$$\begin{array}{c}w(t)\phi (t,u^{\mathrm{\prime }}(t))={\rho }_1(u)+\sum _{t_i<t}{B}_i+F(N_f(u))(t),t\in (0,1),t\ne t_i,\hfill \\ {(w(t)\phi (t,u^{\mathrm{\prime }}))}^{\mathrm{\prime }}=N_f(u)(t),t\in (0,1),t\ne t_i.\hfill \end{array}$$

(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

$$w(t)\phi (t,u^{\mathrm{\prime }}(t))=w(0)\phi (0,u^{\mathrm{\prime }}(0))+\sum _{t_i<t}{b}_i+{\int }_0^{t}f(s)ds,\mathrm{\forall }t\in J^{\mathrm{\prime }}.$$

Denote ${\rho }_2=w(0)\phi (0,u^{\mathrm{\prime }}(0))$. It is easy to see that ${\rho }_2$ is dependent on a, b and $f(\cdot )$. Boundary value condition (4) implies that

$$\begin{array}{c}{\int }_0^{1}g(t)(F\left\{{\phi }^{-1}[t,{(w(t))}^{-1}({\rho }_2+\sum _{t_i<t}{b}_i+F(f)(t))]\right\}(t)+\sum _{t_i<t}{a}_i)dt=0,\hfill \\ u(0)=\frac{{\sum }_{\ell =1}^{m-2}{\alpha }_{\ell }\{{\sum }_{t_i<{\xi }_{\ell }}{a}_i+{\int }_0^{{\xi }_{\ell }}{\phi }^{-1}[t,{(w(t))}^{-1}({\rho }_2+{\sum }_{t_i<t}{b}_i+F(f)(t))]dt\}}{1-{\sum }_{i=1}^{m-2}{\alpha }_{\ell }+\delta }\hfill \\ \phantom{u(0)=}-\frac{{\sum }_{i=1}^k{a}_i+{\int }_0^1{\phi }^{-1}[t,{(w(t))}^{-1}({\rho }_2+{\sum }_{t_i<t}{b}_i+F(f)(t))]dt}{1-{\sum }_{\ell =1}^{m-2}{\alpha }_{\ell }+\delta }\hfill \\ \phantom{u(0)=}-\frac{{\int }_0^{1}h(t)(F\{{\phi }^{-1}[t,{(w(t))}^{-1}({\rho }_2+{\sum }_{t_i<t}{b}_i+F(f)(t))]\}(t)+{\sum }_{t_i<t}{a}_i)dt}{1-{\sum }_{\ell =1}^{m-2}{\alpha }_{\ell }+\delta }.\hfill \end{array}$$

For any $\omega \in W$, we denote

$${\mathrm{\Gamma }}_{\omega }({\rho }_2)={\int }_0^{1}g(t)(F\left\{{\phi }^{-1}[t,{(w(t))}^{-1}({\rho }_2+\sum _{t_i<t}{b}_i+F(\vartheta )(t))]\right\}(t)+\sum _{t_i<t}{a}_i)dt.$$

Throughout the paper, we denote .

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

• (i) For any fixed $\omega \in W$, the equation ${\mathrm{\Gamma }}_{\omega }({\rho }_2)=0$ has a unique solution $\widetilde{{\rho }_2}(\omega )\in {\mathbb{R}}^N$.

• (ii) The function $\widetilde{{\rho }_2}:W\to {\mathbb{R}}^N$, defined in (i), is continuous and sends bounded sets to bounded sets. Moreover, for any $\omega =(a,b,\vartheta )\in W$, we have

  $$|\widetilde{{\rho }_2}(\omega )|\le 3N[{(2N)}^{p^{+}}{(\frac{E_1+1}{E_1}\sum _{i=1}^k|a_i|)}^{p^{\mathrm{\#}}-1}+\sum _{i=1}^k|b_i|+{\parallel \vartheta \parallel }_{L^1}],$$

where the notation $M^{p^{\mathrm{\#}}-1}$ means

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

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 ${\rho }_2(u)=\widetilde{{\rho }_2}(A,B,N_f)(u)$, where $A=(A_1,\dots ,A_k)$, $B=(B_1,\dots ,B_k)$.

It is clear that ${\rho }_2(\cdot )$ 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 $K_{(a,b)}^{\ast }:L^1\to P{C}^1$ as

$$K_{(a,b)}^{\ast }(\vartheta )(t)=F\left\{{\phi }^{-1}[t,{(w(t))}^{-1}(\widetilde{{\rho }_2}(a,b,\vartheta )+\sum _{t_i<t}{b}_i+F(\vartheta )(t))]\right\}(t),\mathrm{\forall }t\in J.$$

Define $K_2:P{C}^1\to P{C}^1$ as

$$K_2(u)(t)=F\left\{{\phi }^{-1}[t,{(w(t))}^{-1}({\rho }_2(u)+\sum _{t_i<t}{B}_i+F(N_f(u))(t))]\right\}(t),\mathrm{\forall }t\in J.$$

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

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

• (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

$$\begin{array}{rcl}{P}_2(u)& =& \frac{{\sum }_{\ell =1}^{m-2}{\alpha }_{\ell }[{\sum }_{t_i<{\xi }_{\ell }}{A}_i+K_2(u)({\xi }_{\ell })]-{\sum }_{i=1}^k{A}_i}{1-{\sum }_{\ell =1}^{m-2}{\alpha }_{\ell }+\delta }\\ -\frac{K_2(u)(1)+{\int }_0^{1}h(t)[K_2(u)(t)+{\sum }_{t_i<t}{A}_i]dt}{1-{\sum }_{\ell =1}^{m-2}{\alpha }_{\ell }+\delta }.\end{array}$$

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:

$$u=P_2(u)+\sum _{t_i<t}{A}_i+K_2(u).$$

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

$$w(t)\phi (t,u^{\mathrm{\prime }}(t))=w(0)\phi (0,u^{\mathrm{\prime }}(0))+\sum _{t_i<t}{b}_i+{\int }_0^{t}f(s)ds,\mathrm{\forall }t\in J^{\mathrm{\prime }}.$$

Denote ${\rho }_3=w(0)\phi (0,u^{\mathrm{\prime }}(0))$. It is easy to see that ${\rho }_3$ is dependent on a, b and $f(\cdot )$.

From $u(0)={\int }_0^{1}g(t)u(t)dt$, we have

$$\begin{array}{rl}u(0)=& \frac{1}{(1-\sigma )}\\ \times {\int }_0^{1}g(t)(F\left\{{\phi }^{-1}[t,{(w(t))}^{-1}({\rho }_3+\sum _{t_i<t}{b}_i+F(f)(t))]\right\}(t)+\sum _{t_i<t}{a}_i)dt.\end{array}$$

(18)

From $u(1)={\sum }_{\ell =1}^{m-2}{\alpha }_{\ell }u({\xi }_{\ell })-{\int }_0^{1}h(t)u(t)dt$, we obtain

$$\begin{array}{rcl}u(0)& =& \frac{{\sum }_{\ell =1}^{m-2}{\alpha }_{\ell }\{{\sum }_{t_i<{\xi }_{\ell }}{a}_i+{\int }_0^{{\xi }_{\ell }}{\phi }^{-1}[t,{(w(t))}^{-1}({\rho }_3+{\sum }_{t_i<t}{b}_i+F(f)(t))]dt\}}{1-{\sum }_{\ell =1}^{m-2}{\alpha }_{\ell }+\delta }\\ -\frac{{\sum }_{i=1}^k{a}_i+{\int }_0^1{\phi }^{-1}[t,{(w(t))}^{-1}({\rho }_3+{\sum }_{t_i<t}{b}_i+F(f)(t))]dt}{1-{\sum }_{\ell =1}^{m-2}{\alpha }_{\ell }+\delta }\\ -\frac{{\int }_0^{1}h(t)(F\{{\phi }^{-1}[t,{(w(t))}^{-1}({\rho }_3+{\sum }_{t_i<t}{b}_i+F(f)(t))]\}(t)+{\sum }_{t_i<t}{a}_i)dt}{1-{\sum }_{\ell =1}^{m-2}{\alpha }_{\ell }+\delta }.\end{array}$$

(19)

For fixed $\omega \in W$, we denote

$$\begin{array}{rcl}{\mathrm{\Upsilon }}_{\omega }({\rho }_3)& =& \frac{1}{(1-\sigma )}{\int }_0^{1}g(t)(F\left\{{\phi }^{-1}[t,{(w(t))}^{-1}({\rho }_3+\sum _{t_i<t}{b}_i+F(\vartheta )(t))]\right\}(t)+\sum _{t_i<t}{a}_i)dt\\ -\frac{{\sum }_{\ell =1}^{m-2}{\alpha }_{\ell }\{{\sum }_{t_i<{\xi }_{\ell }}{a}_i+{\int }_0^{{\xi }_{\ell }}{\phi }^{-1}[t,{(w(t))}^{-1}({\rho }_3+{\sum }_{t_i<t}{b}_i+F(\vartheta )(t))]dt\}}{1-{\sum }_{\ell =1}^{m-2}{\alpha }_{\ell }+\delta }\\ +\frac{{\sum }_{i=1}^k{a}_i+{\int }_0^1{\phi }^{-1}[t,{(w(t))}^{-1}({\rho }_3+{\sum }_{t_i<t}{b}_i+F(\vartheta )(t))]dt}{1-{\sum }_{\ell =1}^{m-2}{\alpha }_{\ell }+\delta }\\ +\frac{{\int }_0^{1}h(t)(F\{{\phi }^{-1}[t,{(w(t))}^{-1}({\rho }_3+{\sum }_{t_i<t}{b}_i+F(\vartheta )(t))]\}(t)+{\sum }_{t_i<t}{a}_i)dt}{1-{\sum }_{\ell =1}^{m-2}{\alpha }_{\ell }+\delta },\\ \mathrm{\forall }{\rho }_3\in {\mathbb{R}}^N.\end{array}$$

From (18) and (19), we have ${\mathrm{\Upsilon }}_{\omega }({\rho }_3)=0$.

Obviously, ${\mathrm{\Upsilon }}_{\omega }({\rho }_3)$ can be rewritten as

$$\begin{array}{rcl}{\mathrm{\Upsilon }}_{\omega }({\rho }_3)& =& \frac{1}{(1-\sigma )}{\int }_0^{1}g(t)(F\left\{{\phi }^{-1}[t,{(w(t))}^{-1}({\rho }_3+\sum _{t_i<t}{b}_i+F(\vartheta )(t))]\right\}(t)+\sum _{t_i<t}{a}_i)dt\\ +\frac{{\sum }_{\ell =1}^{m-2}{\alpha }_{\ell }\{{\sum }_{{\xi }_{\ell }\le t_i}{a}_i+{\int }_{{\xi }_{\ell }}^1{\phi }^{-1}[t,{(w(t))}^{-1}({\rho }_3+{\sum }_{t_i<t}{b}_i+F(\vartheta )(t))]dt\}}{1-{\sum }_{\ell =1}^{m-2}{\alpha }_{\ell }+\delta }\\ +\frac{(1-{\sum }_{\ell =1}^{m-2}{\alpha }_{\ell }){\int }_0^1{\phi }^{-1}[t,{(w(t))}^{-1}({\rho }_3+{\sum }_{t_i<t}{b}_i+F(\vartheta )(t))]dt}{1-{\sum }_{\ell =1}^{m-2}{\alpha }_{\ell }+\delta }\\ +\frac{{\sum }_{i=1}^k{a}_i(1-{\sum }_{\ell =1}^{m-2}{\alpha }_{\ell })}{1-{\sum }_{\ell =1}^{m-2}{\alpha }_{\ell }+\delta }\\ +\frac{{\int }_0^{1}h(t)(F\{{\phi }^{-1}[t,{(w(t))}^{-1}({\rho }_3+{\sum }_{t_i<t}{b}_i+F(\vartheta )(t))]\}(t)+{\sum }_{t_i<t}{a}_i)dt}{1-{\sum }_{\ell =1}^{m-2}{\alpha }_{\ell }+\delta }.\end{array}$$

Denote ${\xi }_{m-1}=1$. Moreover, we also have

$$\begin{array}{c}{\mathrm{\Upsilon }}_{\omega }({\rho }_3)\hfill \\ =\frac{1}{(1-\sigma )}{\int }_0^{1}g(t)(F\left\{{\phi }^{-1}[t,{(w(t))}^{-1}({\rho }_3+\sum _{t_i<t}{b}_i+F(\vartheta )(t))]\right\}(t)+\sum _{t_i<t}{a}_i)dt\hfill \\ +\frac{{\sum }_{\ell =1}^{m-2}{\alpha }_{\ell }{\sum }_{{\xi }_{\ell }\le t_i}{a}_i}{1-{\sum }_{\ell =1}^{m-2}{\alpha }_{\ell }+\delta }\hfill \\ +\frac{{\sum }_{\ell =1}^{m-2}({\alpha }_{\ell }-{\int }_{{\xi }_{\ell }}^{{\xi }_{\ell +1}}h(t)dt){\int }_{{\xi }_{\ell }}^1{\phi }^{-1}[t,{(w(t))}^{-1}({\rho }_3+{\sum }_{t_i<t}{b}_i+F(\vartheta )(t))]dt}{1-{\sum }_{\ell =1}^{m-2}{\alpha }_{\ell }+\delta }\hfill \\ +\frac{{\sum }_{\ell =1}^{m-2}{\int }_{{\xi }_{\ell }}^{{\xi }_{\ell +1}}h(t){\int }_{{\xi }_{\ell }}^t{\phi }^{-1}[s,{(w(s))}^{-1}({\rho }_3+{\sum }_{s_i<s}{b}_i+F(\vartheta )(s))]dsdt}{1-{\sum }_{\ell =1}^{m-2}{\alpha }_{\ell }+\delta }\hfill \\ -\frac{{\int }_0^{{\xi }_1}h(t){\int }_t^1{\phi }^{-1}[s,{(w(s))}^{-1}({\rho }_3+{\sum }_{s_i<s}{b}_i+F(\vartheta )(s))]dsdt+{\int }_0^{1}h(t){\sum }_{t_i\ge t}{a}_{i}dt}{1-{\sum }_{\ell =1}^{m-2}{\alpha }_{\ell }+\delta }\hfill \\ +{\int }_0^1{\phi }^{-1}[t,{(w(t))}^{-1}({\rho }_3+\sum _{t_i<t}{b}_i+F(\vartheta )(t))]dt+\sum _{i=1}^k{a}_i.\hfill \end{array}$$

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

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

(2⁰) $h(t)\ge 0$ on $[{\xi }_1,1]$, ${\alpha }_{\ell }\ge {\int }_{{\xi }_{\ell }}^{{\xi }_{\ell +1}}h(t)dt$ ($\ell =1,\dots ,m-2$) and $h(t)\le 0$ on $[0,{\xi }_1]$.

Then the function ${\mathrm{\Upsilon }}_{\omega }(\cdot )$ has the following properties:

• (i) For any fixed $\omega \in W$, the equation ${\mathrm{\Upsilon }}_{\omega }({\rho }_3)=0$ has a unique solution $\widetilde{{\rho }_3}(\omega )\in {\mathbb{R}}^N$.

• (ii) The function $\widetilde{{\rho }_3}:W\to {\mathbb{R}}^N$, defined in (i), is continuous and sends bounded sets to bounded sets. Moreover, for any $\omega =(a,b,\vartheta )\in W$, we have

  $$\begin{array}{rcl}|\widetilde{{\rho }_3}(\omega )|& \le & 3N\{{(2N)}^{p^{+}}{[(\frac{E_1+1}{(1-\sigma )E_1}+({\delta }^{\ast }+1)\frac{E+1}{(1-{\sum }_{\ell =1}^{m-2}{\alpha }_{\ell }+\delta )E})\sum _{i=1}^k|a_i|]}^{p^{\mathrm{\#}}-1}\\ +\sum _{i=1}^k|b_i|+{\parallel \vartheta \parallel }_{L^1}\},\end{array}$$

where the notation $M^{p^{\mathrm{\#}}-1}$ means

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

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 ${\rho }_3(u)=\widetilde{{\rho }_3}(A,B,N_f)(u)$, where $A=(A_1,\dots ,A_k)$, $B=(B_1,\dots ,B_k)$.

It is clear that ${\rho }_3(\cdot )$ 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 $K_{(a,b)}^{\ast \ast }:L^1\to P{C}^1$ as

$$K_{(a,b)}^{\ast \ast }(\vartheta )(t)=F\left\{{\phi }^{-1}[t,{(w(t))}^{-1}(\widetilde{{\rho }_3}(a,b,\vartheta )+\sum _{t_i<t}{b}_i+F(\vartheta )(t))]\right\}(t),\mathrm{\forall }t\in J.$$

Define $K_3:P{C}^1\to P{C}^1$ as

$$K_3(u)(t)=F\left\{{\phi }^{-1}[t,{(w(t))}^{-1}({\rho }_3(u)+\sum _{t_i<t}{B}_i+F(N_f(u))(t))]\right\}(t),\mathrm{\forall }t\in J.$$

Similar to the proof of Lemma 2.3, we have

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

• (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

$$P_3(u)=\frac{{\int }_0^{1}g(t)[K_3(u)(t)+{\sum }_{t_i<t}{A}_i]dt}{1-\sigma }.$$

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

(2⁰) $h(t)\ge 0$ on $[{\xi }_1,1]$, ${\alpha }_{\ell }\ge {\int }_{{\xi }_{\ell }}^{{\xi }_{\ell +1}}h(t)dt$ ($\ell =1,\dots ,m-2$) and $h(t)\le 0$ on $[0,{\xi }_1]$.

Then u is a solution of (1)-(4) if and only if u is a solution of the following abstract operator equation:

$$u=P_3(u)+\sum _{t_i<t}{A}_i+K_3(u).$$

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-$(p^{-}-1)$ growth condition, we have the following theorem.

Theorem 3.1 Suppose that $\sigma <1$, ${\sum }_{\ell =1}^{m-2}{\alpha }_{\ell }-\delta =1$; $h(t)\ge 0$ on $[{\xi }_1,1]$, ${\alpha }_{\ell }\ge {\int }_{{\xi }_{\ell }}^{{\xi }_{\ell +1}}h(t)dt$ ($\ell =1,\dots ,m-2$) and $h(t)\le 0$ on $[0,{\xi }_1]$; f satisfies the sub-$(p^{-}-1)$ growth condition; and operators A and B satisfy the following conditions:

$$\begin{array}{r}\sum _{i=1}^k|A_i(u,v)|\le C_1{(1+|u|+|v|)}^{\frac{q^{+}-1}{p^{+}-1}},\\ \sum _{i=1}^k|B_i(u,v)|\le C_2{(1+|u|+|v|)}^{q^{+}-1},\mathrm{\forall }(u,v)\in {\mathbb{R}}^N\times {\mathbb{R}}^N,\end{array}$$

(20)

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

Proof First we consider the following problem:

$$(S_1)\{\begin{array}{l}-{\mathrm{△}}_{p(t)}u=\lambda N_f(u)(t),t\in (0,1),t\ne t_i,\\ {lim}_{t\to t_i^{+}}u(t)-{lim}_{t\to t_i^{-}}u(t)\\ =\lambda A_i({lim}_{t\to t_i^{-}}u(t),{lim}_{t\to t_i^{-}}{(w(t))}^{\frac{1}{p(t)-1}}{u}^{\mathrm{\prime }}(t)),i=1,\dots ,k,\\ {lim}_{t\to t_i^{+}}w(t)|u^{\mathrm{\prime }}{|}^{p(t)-2}{u}^{\mathrm{\prime }}(t)-{lim}_{t\to t_i^{-}}w(t)|u^{\mathrm{\prime }}{|}^{p(t)-2}{u}^{\mathrm{\prime }}(t)\\ =\lambda B_i({lim}_{t\to t_i^{-}}u(t),{lim}_{t\to t_i^{-}}{(w(t))}^{\frac{1}{p(t)-1}}{u}^{\mathrm{\prime }}(t)),i=1,\dots ,k,\\ u(0)={\int }_0^{1}g(t)u(t)dt,u(1)={\sum }_{\ell =1}^{m-2}{\alpha }_{\ell }u({\xi }_{\ell })-{\int }_0^{1}h(t)u(t)dt.\end{array}$$

Denote

where $N_f(u)$ is defined in (10).

Obviously, ($S_1$) has the same solution as the following operator equation when $\lambda =1$:

$$u={\mathrm{\Psi }}_f(u,\lambda ).$$

(21)

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

We claim that all the solutions of (21) are uniformly bounded for $\lambda \in [0,1]$. In fact, if it is false, we can find a sequence of solutions $\{(u_n,{\lambda }_n)\}$ 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

$${\parallel {(w(t))}^{\frac{1}{p(t)-1}}{u}_n^{\mathrm{\prime }}(t)\parallel }_0\le C_8{\parallel u_n\parallel }_1^{\alpha },n=1,2,\dots .$$

(23)

It follows from (14), (20) and (22) that

$$|u_n(0)|\le C_9{\parallel u_n\parallel }_1^{\alpha },\text{where }\alpha =\frac{q^{+}-1}{p^{-}-1}.$$

For any $j=1,\dots ,N$, we have

$$\begin{array}{rcl}|u_n^j(t)|& =& |u_n^j(0)+\sum _{t_i<t}{A}_i+{\int }_0^t{\left(u_n^j\right)}^{\mathrm{\prime }}(s)ds|\\ \le & |u_n^j(0)|+\left|\sum _{t_i<t}{A}_i\right|+\left|{\int }_0^t{(w(s))}^{\frac{-1}{p(s)-1}}\underset{t\in (0,1)}{sup}|{(w(t))}^{\frac{1}{p(t)-1}}{\left(u_n^j\right)}^{\mathrm{\prime }}(t)|ds\right|\\ \le & {\parallel u_n\parallel }_1^{\alpha }[C_{10}+C_{8}E]+\left|\sum _{t_i<t}{A}_i\right|\le C_{11}{\parallel u_n\parallel }_1^{\alpha },\mathrm{\forall }t\in J,n=1,2,\dots ,\end{array}$$

which implies that

$$|u_n^j{|}_0\le C_{12}{\parallel u_n\parallel }_1^{\alpha },j=1,\dots ,N;n=1,2,\dots .$$

Thus

$${\parallel u_n\parallel }_0\le N{C}_{12}{\parallel u_n\parallel }_1^{\alpha },n=1,2,\dots .$$

(24)

It follows from (23) and (24) that $\{{\parallel u_n\parallel }_1\}$ is uniformly bounded.

Thus, we can choose a large enough $R_0>0$ such that all the solutions of (21) belong to $B(R_0)=\{u\in P{C}^1\mid {\parallel u\parallel }_1<R_0\}$. Therefore the Leray-Schauder degree $d_{LS}[I-{\mathrm{\Psi }}_f(\cdot ,\lambda ),B(R_0),0]$ is well defined for $\lambda \in [0,1]$, and

$$d_{LS}[I-{\mathrm{\Psi }}_f(\cdot ,1),B(R_0),0]=d_{LS}[I-{\mathrm{\Psi }}_f(\cdot ,0),B(R_0),0].$$

It is easy to see that u is a solution of $u={\mathrm{\Psi }}_f(u,0)$ if and only if u is a solution of the following usual differential equation:

$$(S_2)\{\begin{array}{l}-{\mathrm{△}}_{p(t)}u=0,t\in (0,1),\\ u(0)={\int }_0^{1}g(t)u(t)dt,u(1)={\sum }_{\ell =1}^{m-2}{\alpha }_{\ell }u({\xi }_{\ell })-{\int }_0^{1}h(t)u(t)dt.\end{array}$$

Obviously, system ($S_2$) possesses a unique solution $u_0$. Since $u_0\in B(R_0)$, we have

$$d_{LS}[I-{\mathrm{\Psi }}_f(\cdot ,1),B(R_0),0]=d_{LS}[I-{\mathrm{\Psi }}_f(\cdot ,0),B(R_0),0]\ne 0,$$

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

Theorem 3.2 Suppose that $\sigma <1$, ${\sum }_{\ell =1}^{m-2}{\alpha }_{\ell }-\delta =1$; $h(t)\ge 0$ on $[{\xi }_1,1]$, ${\alpha }_{\ell }\ge {\int }_{{\xi }_{\ell }}^{{\xi }_{\ell +1}}h(t)dt$ ($\ell =1,\dots ,m-2$) and $h(t)\le 0$ on $[0,{\xi }_1]$; f satisfies the sub-$(p^{-}-1)$ growth condition; and operators A and $D=(D_1,\dots ,D_k)$ satisfy the following conditions:

$$\begin{array}{c}\sum _{i=1}^k|A_i(u,v)|\le C_1{(1+|u|+|v|)}^{\frac{q^{+}-1}{p^{+}-1}},\hfill \\ \sum _{i=1}^k|D_i(u,v)|\le C_2{(1+|u|+|v|)}^{{\alpha }_i^{+}},\mathrm{\forall }(u,v)\in {\mathbb{R}}^N\times {\mathbb{R}}^N,\hfill \end{array}$$

where ${\alpha }_i\le \frac{q^{+}-1}{p(t_i)-1}$, and $p(t_i)-1\le q^{+}-{\alpha }_i$, $i=1,\dots ,k$.

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

Proof Obviously, $B_i(u,v)=\phi (t_i,v+D_i(u,v))-\phi (t_i,v)$.

From Theorem 3.1, it suffices to show that

$$\sum _{i=1}^k|B_i(u,v)|\le C_2{(1+|u|+|v|)}^{q^{+}-1},\mathrm{\forall }(u,v)\in {\mathbb{R}}^N\times {\mathbb{R}}^N.$$

(25)

• (a) Suppose that $|v|\le M^{\ast }|D_i(u,v)|$, where $M^{\ast }$ is a large enough positive constant. From the definition of D, we have

  $$|B_i(u,v)|\le C_1|D_i(u,v){|}^{p(t_i)-1}\le C_2{(1+|u|+|v|)}^{{\alpha }_i(p(t_i)-1)}.$$

Since ${\alpha }_i<\frac{q^{+}-1}{p(t_i)-1}$, we have ${\alpha }_i(p(t_i)-1)\le q^{+}-1$. Thus (25) is valid.

• (b) Suppose that $|v|>M^{\ast }|D_i(u,v)|$, we can see that

  $$|B_i(u,v)|\le C_3|v{|}^{p(t_i)-1}\frac{|D_i(u,v)|}{|v|}=C_4|v{|}^{p(t_i)-2}|D_i(u,v)|.$$

There are two cases: Case (i): $p(t_i)-1\ge 1$; Case (ii): $p(t_i)-1<1$.

Case (i): Since $p(t_i)-1\le q^{+}-{\alpha }_i$, we have $p(t_i)-2+{\alpha }_i\le q^{+}-1$, and

$$|B_i(u,v)|\le C_5|v{|}^{p(t_i)-2}|D_i(u,v)|\le C_6{(1+|u|+|v|)}^{p(t_i)-2+{\alpha }_i}\le C_6{(1+|u|+|v|)}^{q^{+}-1}.$$

Thus (25) is valid.

Case (ii): Since ${\alpha }_i<\frac{q^{+}-1}{p(t_i)-1}$, we have ${\alpha }_i(p(t_i)-1)\le q^{+}-1$, and

$$|B_i(u,v)|\le C_7|v{|}^{p(t_i)-2}|D_i(u,v)|\le C_8|D_i(u,v){|}^{p(t_i)-1}\le C_9{(1+|u|+|v|)}^{{\alpha }_i(p(t_i)-1)}.$$

Thus (25) is valid.

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

Let us consider

$$-{(w(t)|u^{\mathrm{\prime }}{|}^{p(t)-2}{u}^{\mathrm{\prime }})}^{\mathrm{\prime }}+\varphi (t,u,{(w(t))}^{\frac{1}{p(t)-1}}{u}^{\mathrm{\prime }},S(u),T(u),\epsilon )=0,t\in (0,1),t\ne t_i,$$

(26)

where ε is a parameter, and

$$\begin{array}{c}\varphi (t,u,{(w(t))}^{\frac{1}{p(t)-1}}{u}^{\mathrm{\prime }},S(u),T(u),\epsilon )\hfill \\ =f(t,u,{(w(t))}^{\frac{1}{p(t)-1}}{u}^{\mathrm{\prime }},S(u),T(u))+\epsilon h(t,u,{(w(t))}^{\frac{1}{p(t)-1}}{u}^{\mathrm{\prime }},S(u),T(u)),\hfill \end{array}$$

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

Theorem 3.3 Suppose that $\sigma <1$, ${\sum }_{\ell =1}^{m-2}{\alpha }_{\ell }-\delta =1$; $h(t)\ge 0$ on $[{\xi }_1,1]$, ${\alpha }_{\ell }\ge {\int }_{{\xi }_{\ell }}^{{\xi }_{\ell +1}}h(t)dt$ ($\ell =1,\dots ,m-2$) and $h(t)\le 0$ on $[0,{\xi }_1]$; f satisfies the sub-$(p^{-}-1)$ growth condition; and we assume that

$$\begin{array}{c}\sum _{i=1}^k|A_i(u,v)|\le C_1{(1+|u|+|v|)}^{\frac{q^{+}-1}{p^{+}-1}},\hfill \\ \sum _{i=1}^k|B_i(u,v)|\le C_2{(1+|u|+|v|)}^{q^{+}-1},\mathrm{\forall }(u,v)\in {\mathbb{R}}^N\times {\mathbb{R}}^N,\hfill \end{array}$$

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

Proof Denote

$$\begin{array}{c}{\varphi }_{\lambda }(t,u,{(w(t))}^{\frac{1}{p(t)-1}}{u}^{\mathrm{\prime }},S(u),T(u),\epsilon )\hfill \\ =f(t,u,{(w(t))}^{\frac{1}{p(t)-1}}{u}^{\mathrm{\prime }},S(u),T(u))+\lambda \epsilon h(t,u,{(w(t))}^{\frac{1}{p(t)-1}}{u}^{\mathrm{\prime }},S(u),T(u)).\hfill \end{array}$$

We consider the existence of solutions of the following equation with (2)-(4)

$$-{(w(t)|u^{\mathrm{\prime }}{|}^{p(t)-2}{u}^{\mathrm{\prime }})}^{\mathrm{\prime }}+{\varphi }_{\lambda }(t,u,{(w(t))}^{\frac{1}{p(t)-1}}{u}^{\mathrm{\prime }},S(u),T(u),\epsilon )=0,t\in (0,1),t\ne t_i.$$

(27)

Denote

$$\begin{array}{c}{\rho }_{1,\lambda }^{\mathrm{\#}}(u,\epsilon )=\widetilde{{\rho }_1}(A,B,N_{{\varphi }_{\lambda }})(u),\hfill \\ K_{1,\lambda }^{\mathrm{\#}}(u,\epsilon )=F\left\{{\phi }^{-1}[t,{(w(t))}^{-1}({\rho }_{1,\lambda }^{\mathrm{\#}}(u,\epsilon )+\sum _{t_i<t}{B}_i+F(N_{{\varphi }_{\lambda }}(u))(t))]\right\},\hfill \\ P_{1,\lambda }^{\mathrm{\#}}(u,\epsilon )=\frac{{\int }_0^{1}g(t)[K_{1,\lambda }^{\mathrm{\#}}(u,\epsilon )(t)+{\sum }_{t_i<t}{A}_i]dt}{(1-\sigma )},\hfill \\ {\mathrm{\Phi }}_{\epsilon }(u,\lambda )=P_{1,\lambda }^{\mathrm{\#}}(u,\epsilon )+\sum _{t_i<t}{A}_i+K_{1,\lambda }^{\mathrm{\#}}(u,\epsilon ),\hfill \end{array}$$

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

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

Obviously, ${\varphi }_0=f$. So ${\mathrm{\Phi }}_{\epsilon }(u,0)=$ ${\mathrm{\Psi }}_f(u,1)$. As in the proof of Theorem 3.1, we know that all the solutions of $u={\mathrm{\Phi }}_{\epsilon }(u,0)$ are uniformly bounded, then there exists a large enough $R_0>0$ such that all the solutions of $u={\mathrm{\Phi }}_{\epsilon }(u,0)$ belong to $B(R_0)=\{u\in P{C}^1\mid {\parallel u\parallel }_1<R_0\}$. Since ${\mathrm{\Phi }}_{\epsilon }(\cdot ,0)$ is compact continuous from $P{C}^1$ to $P{C}^1$, we have

$$\underset{u\in \partial B(R_0)}{inf}{\parallel u-{\mathrm{\Phi }}_{\epsilon }(u,0)\parallel }_1>0.$$

(28)

Since f and h are Caratheodory, we have

$$\begin{array}{c}{\parallel F(N_{{\varphi }_{\lambda }}(u))-F(N_{{\varphi }_0}(u))\parallel }_0\to 0\text{for }(u,\lambda )\in \overline{B(R_0)}\times [0,1]\text{ uniformly, as }\epsilon \to 0,\hfill \\ |{\rho }_{1,\lambda }^{\mathrm{\#}}(u,\epsilon )-{\rho }_{1,0}^{\mathrm{\#}}(u,\epsilon )|\to 0\text{for }(u,\lambda )\in \overline{B(R_0)}\times [0,1]\text{ uniformly, as }\epsilon \to 0,\hfill \\ {\parallel K_{1,\lambda }^{\mathrm{\#}}(u,\epsilon )-K_{1,0}^{\mathrm{\#}}(u,\epsilon )\parallel }_1\to 0\text{for }(u,\lambda )\in \overline{B(R_0)}\times [0,1]\text{ uniformly, as }\epsilon \to 0,\hfill \\ |P_{1,\lambda }^{\mathrm{\#}}(u,\epsilon )-P_{1,0}^{\mathrm{\#}}(u,\epsilon )|\to 0\text{for }(u,\lambda )\in \overline{B(R_0)}\times [0,1]\text{ uniformly, as }\epsilon \to 0.\hfill \end{array}$$

Thus

$${\parallel {\mathrm{\Phi }}_{\epsilon }(u,\lambda )-{\mathrm{\Phi }}_0(u,\lambda )\parallel }_1\to 0\text{for }(u,\lambda )\in \overline{B(R_0)}\times [0,1]\text{ uniformly, as }\epsilon \to 0.$$

Obviously, ${\mathrm{\Phi }}_0(u,\lambda )={\mathrm{\Phi }}_{\epsilon }(u,0)={\mathrm{\Phi }}_0(u,0)$. We obtain

$${\parallel {\mathrm{\Phi }}_{\epsilon }(u,\lambda )-{\mathrm{\Phi }}_{\epsilon }(u,0)\parallel }_1\to 0\text{for }(u,\lambda )\in \overline{B(R_0)}\times [0,1]\text{ uniformly, as }\epsilon \to 0.$$

Thus, when ε is small enough, from (28), we can conclude that

$$\begin{array}{c}\underset{(u,\lambda )\in \partial B(R_0)\times [0,1]}{inf}{\parallel u-{\mathrm{\Phi }}_{\epsilon }(u,\lambda )\parallel }_1\hfill \\ \ge \underset{u\in \partial B(R_0)}{inf}{\parallel u-{\mathrm{\Phi }}_{\epsilon }(u,0)\parallel }_1-\underset{(u,\lambda )\in \overline{B(R_0)}\times [0,1]}{sup}{\parallel {\mathrm{\Phi }}_{\epsilon }(u,0)-{\mathrm{\Phi }}_{\epsilon }(u,\lambda )\parallel }_1>0.\hfill \end{array}$$

Thus $u={\mathrm{\Phi }}_{\epsilon }(u,\lambda )$ has no solution on $\partial B(R_0)$ for any $\lambda \in [0,1]$, when ε is small enough. It means that the Leray-Schauder degree $d_{LS}[I-{\mathrm{\Phi }}_{\epsilon }(\cdot ,\lambda ),B(R_0),0]$ is well defined for any $\lambda \in [0,1]$, and

$$d_{LS}[I-{\mathrm{\Phi }}_{\epsilon }(u,\lambda ),B(R_0),0]=d_{LS}[I-{\mathrm{\Phi }}_{\epsilon }(u,0),B(R_0),0].$$

Since ${\mathrm{\Phi }}_{\epsilon }(u,0)=$ ${\mathrm{\Psi }}_f(u,1)$, 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 $\sigma <1$, ${\sum }_{\ell =1}^{m-2}{\alpha }_{\ell }-\delta =1$; $h(t)\ge 0$ on $[{\xi }_1,1]$, ${\alpha }_{\ell }\ge {\int }_{{\xi }_{\ell }}^{{\xi }_{\ell +1}}h(t)dt$ ($\ell =1,\dots ,m-2$) and $h(t)\le 0$ on $[0,{\xi }_1]$; f satisfies the sub-$(p^{-}-1)$ growth condition; and we assume that

$$\begin{array}{c}\sum _{i=1}^k|A_i(u,v)|\le C_1{(1+|u|+|v|)}^{\frac{q^{+}-1}{p^{+}-1}},\hfill \\ \sum _{i=1}^k|D_i(u,v)|\le C_2{(1+|u|+|v|)}^{{\alpha }_i^{+}},\mathrm{\forall }(u,v)\in {\mathbb{R}}^N\times {\mathbb{R}}^N,\hfill \end{array}$$

where ${\alpha }_i\le \frac{q^{+}-1}{p(t_i)-1}$, and $p(t_i)-1\le q^{+}-{\alpha }_i$, $i=1,\dots ,k$, 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-$(p^{-}-1)$ 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-$(p^{-}-1)$ growth condition; and operators A and B satisfy the following conditions:

$$\begin{array}{c}\sum _{i=1}^k|A_i(u,v)|\le C_1{(1+|u|+|v|)}^{\frac{q^{+}-1}{p^{+}-1}},\hfill \\ \sum _{i=1}^k|B_i(u,v)|\le C_2{(1+|u|+|v|)}^{q^{+}-1},\mathrm{\forall }(u,v)\in {\mathbb{R}}^N\times {\mathbb{R}}^N,\hfill \end{array}$$

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-$(p^{-}-1)$ growth condition; and operators A and $D=(D_1,\dots ,D_k)$ satisfy the following conditions:

$$\begin{array}{c}\sum _{i=1}^k|A_i(u,v)|\le C_1{(1+|u|+|v|)}^{\frac{q^{+}-1}{p^{+}-1}},\hfill \\ \sum _{i=1}^k|D_i(u,v)|\le C_2{(1+|u|+|v|)}^{{\alpha }_i^{+}},\mathrm{\forall }(u,v)\in {\mathbb{R}}^N\times {\mathbb{R}}^N,\hfill \end{array}$$

where

$${\alpha }_i\le \frac{q^{+}-1}{p(t_i)-1}\mathit{\text{and}}p(t_i)-1\le q^{+}-{\alpha }_i,i=1,\dots ,k,$$

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-$(p^{-}-1)$ growth condition; and we assume that

$$\begin{array}{c}\sum _{i=1}^k|A_i(u,v)|\le C_1{(1+|u|+|v|)}^{\frac{q^{+}-1}{p^{+}-1}},\hfill \\ \sum _{i=1}^k|B_i(u,v)|\le C_2{(1+|u|+|v|)}^{q^{+}-1},\mathrm{\forall }(u,v)\in {\mathbb{R}}^N\times {\mathbb{R}}^N,\hfill \end{array}$$

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-$(p^{-}-1)$ growth condition; and we assume that

$$\begin{array}{c}\sum _{i=1}^k|A_i(u,v)|\le C_1{(1+|u|+|v|)}^{\frac{q^{+}-1}{p^{+}-1}},\hfill \\ \sum _{i=1}^k|D_i(u,v)|\le C_2{(1+|u|+|v|)}^{{\alpha }_i^{+}},\mathrm{\forall }(u,v)\in {\mathbb{R}}^N\times {\mathbb{R}}^N,\hfill \end{array}$$

where ${\alpha }_i\le \frac{q^{+}-1}{p(t_i)-1}$, and $p(t_i)-1\le q^{+}-{\alpha }_i$, $i=1,\dots ,k$, 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-$(p^{-}-1)$ 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) ${\sum }_{\ell =1}^{m-2}{\alpha }_{\ell }\le 1$, $g(t)(1-{\sum }_{\ell =1}^{m-2}{\alpha }_{\ell }+\delta )+h(t)(1-\sigma )\ge 0$;

(2⁰) $h(t)\ge 0$ on $[{\xi }_1,1]$, ${\alpha }_{\ell }\ge {\int }_{{\xi }_{\ell }}^{{\xi }_{\ell +1}}h(t)dt$ ($\ell =1,\dots ,m-2$) and $h(t)\le 0$ on $[0,{\xi }_1]$;

when f satisfies the sub-$(p^{-}-1)$ growth condition; and operators A and B satisfy the following conditions:

$$\begin{array}{c}\sum _{i=1}^k|A_i(u,v)|\le C_1{(1+|u|+|v|)}^{\frac{q^{+}-1}{p^{+}-1}},\hfill \\ \sum _{i=1}^k|B_i(u,v)|\le C_2{(1+|u|+|v|)}^{q^{+}-1},\mathrm{\forall }(u,v)\in {\mathbb{R}}^N\times {\mathbb{R}}^N,\hfill \end{array}$$

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

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

Theorem 5.2 Suppose that $\sigma <1$, ${\sum }_{\ell =1}^{m-2}{\alpha }_{\ell }-\delta <1$ and ${\alpha }_{\ell }$, g, h satisfy one of the following:

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

(2⁰) $h(t)\ge 0$ on $[{\xi }_1,1]$, ${\alpha }_{\ell }\ge {\int }_{{\xi }_{\ell }}^{{\xi }_{\ell +1}}h(t)dt$ ($\ell =1,\dots ,m-2$) and $h(t)\le 0$ on $[0,{\xi }_1]$;

when f satisfies the sub-$(p^{-}-1)$ growth condition; and operators A and $D=(D_1,\dots ,D_k)$ satisfy the following conditions:

$$\begin{array}{c}\sum _{i=1}^k|A_i(u,v)|\le C_1{(1+|u|+|v|)}^{\frac{q^{+}-1}{p^{+}-1}},\hfill \\ \sum _{i=1}^k|D_i(u,v)|\le C_2{(1+|u|+|v|)}^{{\alpha }_i^{+}},\mathrm{\forall }(u,v)\in {\mathbb{R}}^N\times {\mathbb{R}}^N,\hfill \end{array}$$

where

$${\alpha }_i\le \frac{q^{+}-1}{p(t_i)-1}\mathit{\text{and}}p(t_i)-1\le q^{+}-{\alpha }_i,i=1,\dots ,k,$$

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

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

(2⁰) $h(t)\ge 0$ on $[{\xi }_1,1]$, ${\alpha }_{\ell }\ge {\int }_{{\xi }_{\ell }}^{{\xi }_{\ell +1}}h(t)dt$ ($\ell =1,\dots ,m-2$) and $h(t)\le 0$ on $[0,{\xi }_1]$;

when f satisfies the sub-$(p^{-}-1)$ growth condition; and we assume that

$$\begin{array}{c}\sum _{i=1}^k|A_i(u,v)|\le C_1{(1+|u|+|v|)}^{\frac{q^{+}-1}{p^{+}-1}},\hfill \\ \sum _{i=1}^k|B_i(u,v)|\le C_2{(1+|u|+|v|)}^{q^{+}-1},\mathrm{\forall }(u,v)\in {\mathbb{R}}^N\times {\mathbb{R}}^N,\hfill \end{array}$$

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

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

(2⁰) $h(t)\ge 0$ on $[{\xi }_1,1]$, ${\alpha }_{\ell }\ge {\int }_{{\xi }_{\ell }}^{{\xi }_{\ell +1}}h(t)dt$ ($\ell =1,\dots ,m-2$) and $h(t)\le 0$ on $[0,{\xi }_1]$;

when f satisfies the sub-$(p^{-}-1)$ growth condition; and we assume that

$$\begin{array}{c}\sum _{i=1}^k|A_i(u,v)|\le C_1{(1+|u|+|v|)}^{\frac{q^{+}-1}{p^{+}-1}},\hfill \\ \sum _{i=1}^k|D_i(u,v)|\le C_2{(1+|u|+|v|)}^{{\alpha }_i^{+}},\mathrm{\forall }(u,v)\in {\mathbb{R}}^N\times {\mathbb{R}}^N,\hfill \end{array}$$

where ${\alpha }_i\le \frac{q^{+}-1}{p(t_i)-1}$, and $p(t_i)-1\le q^{+}-{\alpha }_i$, $i=1,\dots ,k$, 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. □

In the following, we will consider the existence of nonnegative solutions. For any $x=(x^1,\dots ,x^N)\in {\mathbb{R}}^N$, the notation $x\ge 0$ means $x^j\ge 0$ for any $j=1,\dots ,N$.

Theorem 5.5 Suppose that $\sigma <1$, ${\sum }_{\ell =1}^{m-2}{\alpha }_{\ell }-\delta <1$, ${\sum }_{\ell =1}^{m-2}{\alpha }_{\ell }\le 1$, $g(t)(1-{\sum }_{\ell =1}^{m-2}{\alpha }_{\ell }+\delta )+h(t)(1-\sigma )\ge 0$. We also assume:

(1⁰) $f(t,x,y,s,z)\le 0$, $\mathrm{\forall }(t,x,y,s,z)\in J\times {\mathbb{R}}^N\times {\mathbb{R}}^N\times {\mathbb{R}}^N\times {\mathbb{R}}^N$;

(2⁰) For any $i=1,\dots ,k$, $B_i(u,v)\le 0$, $\mathrm{\forall }(u,v)\in {\mathbb{R}}^N\times {\mathbb{R}}^N$;

(3⁰) For any $i=1,\dots ,k$, $j=1,\dots ,N$, $A_i^j(u,v)v^j\ge 0$, $\mathrm{\forall }(u,v)\in {\mathbb{R}}^N\times {\mathbb{R}}^N$;

(4⁰) $h(t)\le 0$.

Then every solution of (1)-(4) is nonnegative.

Proof Let u be a solution of (1)-(4). From Lemma 2.10, we have

$$u(t)=u(0)+\sum _{t_i<t}{A}_i+F\left\{{\phi }^{-1}[t,{(w(t))}^{-1}({\rho }_3(u)+\sum _{t_i<t}{B}_i+F(N_f(u)))]\right\}(t),\mathrm{\forall }t\in J.$$

We claim that ${\rho }_3(u)\ge 0$. If it is false, then there exists some $j\in \{1,\dots ,N\}$ such that ${\rho }_3^j(u)<0$.

It follows from (1⁰) and (2⁰) that

$${[{\rho }_3(u)+\sum _{t_i<t}{B}_i+F(N_f(u))(t)]}^j<0,\mathrm{\forall }t\in J.$$

(29)

Thus (29) and condition (3⁰) hold

$$A_i^j\le 0,i=1,\dots ,k.$$

(30)

Similar to the proof before Lemma 2.8, from the boundary value conditions, we have

$$\begin{array}{rcl}0& =& \frac{1}{(1-\sigma )}{\int }_0^{1}g(t)(F\left\{{\phi }^{-1}[t,{(w(t))}^{-1}({\rho }_3+\sum _{t_i<t}{B}_i+F(N_f(u)))]\right\}(t)+\sum _{t_i<t}{A}_i)dt\\ +\frac{{\sum }_{\ell =1}^{m-2}{\alpha }_{\ell }\{{\sum }_{{\xi }_{\ell }\le t_i}{A}_i+{\int }_{{\xi }_{\ell }}^1{\phi }^{-1}[t,{(w(t))}^{-1}({\rho }_3+{\sum }_{t_i<t}{B}_i+F(N_f(u)))]dt\}}{1-{\sum }_{i=1}^{m-2}{\alpha }_{\ell }+\delta }\\ +\frac{{\sum }_{i=1}^k{A}_i(1-{\sum }_{\ell =1}^{m-2}{\alpha }_{\ell })}{1-{\sum }_{i=1}^{m-2}{\alpha }_{\ell }+\delta }\\ +\frac{(1-{\sum }_{\ell =1}^{m-2}{\alpha }_{\ell }){\int }_0^1{\phi }^{-1}[t,{(w(t))}^{-1}({\rho }_3+{\sum }_{t_i<t}{B}_i+F(N_f(u)))]dt}{1-{\sum }_{\ell =1}^{m-2}{\alpha }_{\ell }+\delta }\\ +\frac{{\int }_0^{1}h(t)(F\{{\phi }^{-1}[t,{(w(t))}^{-1}({\rho }_3+{\sum }_{t_i<t}{B}_i+F(N_f(u)))]\}(t)+{\sum }_{t_i<t}{A}_i)dt}{1-{\sum }_{\ell =1}^{m-2}{\alpha }_{\ell }+\delta }.\end{array}$$

(31)

From (29) and (30), we get a contradiction to (31). Thus ${\rho }_3(u)\ge 0$.

We claim that

$${\rho }_3(u)+\sum _{i=1}^k{B}_i+F(N_f)(1)\le 0.$$

(32)

If it is false, then there exists some $j\in \{1,\dots ,N\}$ such that

$${[{\rho }_3(u)+\sum _{i=1}^k{B}_i+F(N_f)(1)]}^j>0.$$

It follows from (1⁰) and (2⁰) that

$${[{\rho }_3(u)+\sum _{t_i<t}{B}_i+F(N_f(u))(t)]}^j>0,\mathrm{\forall }t\in J.$$

(33)

Thus (33) and condition (3⁰) hold

$$A_i^j\ge 0,i=1,\dots ,k.$$

(34)

From (33), (34), we get a contradiction to (31). Thus (32) is valid.

Denote $\mathrm{\Theta }(t)={\rho }_3(u)+{\sum }_{t_i<t}{B}_i+F(N_f(u))(t)$, $\mathrm{\forall }t\in J^{\mathrm{\prime }}$.

Obviously, $\mathrm{\Theta }(0)={\rho }_3\ge 0$, $\mathrm{\Theta }(1)\le 0$, and $\mathrm{\Theta }(t)$ is decreasing, i.e., $\mathrm{\Theta }(t^{\mathrm{\prime }})\le \mathrm{\Theta }(t^{\mathrm{\prime }\mathrm{\prime }})$ for any $t^{\mathrm{\prime }},t^{\mathrm{\prime }\mathrm{\prime }}\in J$ with $t^{\mathrm{\prime }}\ge t^{\mathrm{\prime }\mathrm{\prime }}$. For any $j=1,\dots ,N$, there exist ${\zeta }_j\in J$ such that

$${\mathrm{\Theta }}^j(t)\ge 0,\mathrm{\forall }t\in (0,{\zeta }_j),\text{and}{\mathrm{\Theta }}^j(t)\le 0,\mathrm{\forall }t\in ({\zeta }_j,T).$$

It follows from condition (3⁰) that $u^j(t)$ is increasing on $[0,{\zeta }_j]$ and $u^j(t)$ is decreasing on $({\zeta }_j,T]$. Thus $min\{u^j(0),u^j(1)\}={inf}_{t\in J}{u}^j(t)$, $j=1,\dots ,N$.

For any fixed $j\in \{1,\dots ,N\}$, if

$$u^j(0)=\underset{t\in J}{inf}{u}^j(t),$$

(35)

from (4) and (35), we have $(1-\sigma )u^j(0)\ge 0$. Then $u^j(0)\ge 0$.

If

$$u^j(1)=\underset{t\in J}{inf}{u}^j(t),$$

(36)

from (4), (36) and condition (4⁰), we have $(1-{\sum }_{i=1}^{m-2}{\alpha }_{\ell }+\delta )u^j(1)\ge 0$. Then $u^j(1)\ge 0$.

Thus $u(t)\ge 0$, $\mathrm{\forall }t\in [0,T]$. The proof is completed. □

Corollary 5.6 Under the conditions of Theorem  5.1, we also assume:

(1⁰) $f(t,x,y,s,z)\le 0$, $\mathrm{\forall }(t,x,y,s,z)\in J\times {\mathbb{R}}^N\times {\mathbb{R}}^N\times {\mathbb{R}}^N\times {\mathbb{R}}^N$ with $x,s,z\ge 0$;

(2⁰) For any $i=1,\dots ,k$, $B_i(u,v)\le 0$, $\mathrm{\forall }(u,v)\in {\mathbb{R}}^N\times {\mathbb{R}}^N$ with $u\ge 0$;

(3⁰) For any $i=1,\dots ,k$, $j=1,\dots ,N$, $A_i^j(u,v)v^j\ge 0$, $\mathrm{\forall }(u,v)\in {\mathbb{R}}^N\times {\mathbb{R}}^N$ with $u\ge 0$;

(4⁰) $h(t)\le 0$;

(5⁰) For any $t\in [0,1]$ and $s\in [0,1]$, $k_{\ast }(t,s)\ge 0$, $h_{\ast }(t,s)\ge 0$.

Then (1)-(4) has a nonnegative solution.

Proof Define $M(u)=(M_{\mathrm{\#}}(u^1),\dots ,M_{\mathrm{\#}}(u^N))$, where

$$M_{\mathrm{\#}}(u)=\{\begin{array}{ll}u,& u\ge 0,\\ 0,& u<0.\end{array}$$

Denote

$$\tilde{f}(t,u,v,S(u),T(u))=f(t,M(u),v,S(M(u)),T(M(u))),\mathrm{\forall }(t,u,v)\in J\times {\mathbb{R}}^N\times {\mathbb{R}}^N,$$

then $\tilde{f}(t,u,v,S(u),T(u))$ satisfies the Caratheodory condition, and $\tilde{f}(t,u,v,S(u),T(u))\le 0$ for any $(t,u,v)\in J\times {\mathbb{R}}^N\times {\mathbb{R}}^N$.

For any $i=1,\dots ,k$, we denote

$${\tilde{A}}_i(u,v)=A_i(M(u),v),{\tilde{B}}_i(u,v)=B_i(M(u),v),\mathrm{\forall }(u,v)\in {\mathbb{R}}^N\times {\mathbb{R}}^N,$$

then ${\tilde{A}}_i$ and ${\tilde{B}}_i$ are continuous and satisfy

$$\begin{array}{c}{\tilde{B}}_i(u,v)\le 0,\mathrm{\forall }(u,v)\in {\mathbb{R}}^N\times {\mathbb{R}}^N\text{ for any }i=1,\dots ,k,\hfill \\ {\tilde{A}}_i^j(u,v)v^j\ge 0,\mathrm{\forall }(u,v)\in {\mathbb{R}}^N\times {\mathbb{R}}^N\text{ for any }i=1,\dots ,k,j=1,\dots ,N.\hfill \end{array}$$

It is not hard to check that

(2⁰)′ ${lim}_{|u|+|v|\to +\mathrm{\infty }}(\tilde{f}(t,u,v,S(u),T(u))/{(|u|+|v|)}^{q(t)-1})=0$ for $t\in J$ uniformly, where $q(t)\in C(J,\mathbb{R})$, and $1<q^{-}\le q^{+}<p^{-}$;

(3⁰)′ ${\sum }_{i=1}^k|{\tilde{A}}_i(u,v)|\le C_1{(1+|u|+|v|)}^{\frac{q^{+}-1}{p^{+}-1}}$, $\mathrm{\forall }(u,v)\in {\mathbb{R}}^N\times {\mathbb{R}}^N$;

(4⁰)′ ${\sum }_{i=1}^k|{\tilde{B}}_i(u,v)|\le C_2{(1+|u|+|v|)}^{q^{+}-1}$, $\mathrm{\forall }(u,v)\in {\mathbb{R}}^N\times {\mathbb{R}}^N$.

Let us consider

$$\begin{array}{l}{(w(t){\phi }_{p(t)}(u^{\mathrm{\prime }}(t)))}^{\mathrm{\prime }}=\tilde{f}(t,u,{(w(t))}^{\frac{1}{p(t)-1}}{u}^{\mathrm{\prime }},S(u),T(u)),t\in J^{\mathrm{\prime }},\\ {lim}_{t\to t_i^{+}}u(t)-{lim}_{t\to t_i^{-}}u(t_i)\\ ={\tilde{A}}_i({lim}_{t\to t_i^{-}}u(t),{lim}_{t\to t_i^{-}}{(w(t))}^{\frac{1}{p(t)-1}}{u}^{\mathrm{\prime }}(t)),i=1,\dots ,k,\\ {lim}_{t\to t_i^{+}}w(t){\phi }_{p(t)}(u^{\mathrm{\prime }}(t))-{lim}_{t\to t_i^{-}}w(t){\phi }_{p(t)}(u^{\mathrm{\prime }}(t))\\ ={\tilde{B}}_i({lim}_{t\to t_i^{-}}u(t),{lim}_{t\to t_i^{-}}{(w(t))}^{\frac{1}{p(t)-1}}{u}^{\mathrm{\prime }}(t)),i=1,\dots ,k,\\ u(0)={\int }_0^{1}g(t)u(t)dt,u(1)={\sum }_{\ell =1}^{m-2}{\alpha }_{\ell }u({\xi }_{\ell })-{\int }_0^{1}h(t)u(t)dt.\end{array}\}$$

(37)

It follows from Theorem 5.1 and Theorem 5.5 that (37) has a nonnegative solution u. Since $u\ge 0$, we have $M(u)=u$, and then

$$\begin{array}{c}\tilde{f}(t,u,{(w(t))}^{\frac{1}{p(t)-1}}{u}^{\mathrm{\prime }},S(u),T(u))=f(t,u,{(w(t))}^{\frac{1}{p(t)-1}}{u}^{\mathrm{\prime }},S(u),T(u)),\hfill \\ {\tilde{A}}_i(\underset{t\to t_i^{-}}{lim}u(t),\underset{t\to t_i^{-}}{lim}{(w(t))}^{\frac{1}{p(t)-1}}{u}^{\mathrm{\prime }}(t))=A_i(\underset{t\to t_i^{-}}{lim}u(t),\underset{t\to t_i^{-}}{lim}{(w(t))}^{\frac{1}{p(t)-1}}{u}^{\mathrm{\prime }}(t)),\hfill \\ {\tilde{B}}_i(\underset{t\to t_i^{-}}{lim}u(t),\underset{t\to t_i^{-}}{lim}{(w(t))}^{\frac{1}{p(t)-1}}{u}^{\mathrm{\prime }}(t))=B_i(\underset{t\to t_i^{-}}{lim}u(t),\underset{t\to t_i^{-}}{lim}{(w(t))}^{\frac{1}{p(t)-1}}{u}^{\mathrm{\prime }}(t)).\hfill \end{array}$$

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

Note (i) Similarly, we can get the existence of nonnegative solutions of (26) with (2)-(4).

• (ii) Similarly, under the conditions of Case (ii), we can discuss the existence of nonnegative solutions.

6 Examples

Example 6.1 Consider the existence of solutions of (1)-(4) under the following assumptions:

$$\begin{array}{c}f(t,u,{(w(t))}^{\frac{1}{p(t)-1}}{u}^{\mathrm{\prime }},S(u),T(u))\hfill \\ =|u{|}^{q(t)-2}u+{(w(t))}^{\frac{q(t)-1}{p(t)-1}}|u^{\mathrm{\prime }}{|}^{q(t)-2}{u}^{\mathrm{\prime }}\hfill \\ +{(S(u))}^{q(t)-1}+{(T(u))}^{q(t)-1},t\in (0,1),t\ne t_i=\frac{i}{k+\pi },\hfill \\ A_i(u,v)=|u{|}^{-1/2}u+|v{|}^{-1/2}v,i=1,\dots ,k,\hfill \\ B_i(u,v)=|u{|}^{2}u+|v{|}^{2}v,i=1,\dots ,k,\hfill \\ g(t)=\frac{1}{1+t^2},{\alpha }_{\ell }=\frac{\ell +1}{\ell },{\xi }_{\ell }=\frac{\ell }{m},h(t)=\{\begin{array}{ll}0,& 0\le t\le \frac{1}{m},\\ \frac{1}{1+t},& \frac{1}{m}\le t\le 1,\end{array}\hfill \end{array}$$

where $(Su)(t)={\int }_0^1{e}^{t+s}u(s)ds$, $(T(u))(t)={\int }_0^t(t^2+s^2)u(s)ds$, $p(t)=6+3^{-t}cos3t$, $q(t)=3+2^{-t}cost$.

Obviously, $q(t)\le 4<5\le p(t)$; $h(t)=0$ when $0\le t\le \frac{1}{m}={\xi }_1$; ${\alpha }_{\ell }\ge {\int }_{{\xi }_{\ell }}^{{\xi }_{\ell +1}}h(t)dt$ ($\ell =1,\dots ,m-2$); then the conditions of Theorem 3.1 are satisfied, then (1)-(4) has a solution.