Open Access

Impulsive integral BVP for nonlinear integro-differential equation with monotone homomorphism in Banach spaces

Boundary Value Problems20162016:204

https://doi.org/10.1186/s13661-016-0710-8

Received: 2 August 2016

Accepted: 8 November 2016

Published: 18 November 2016

Abstract

In this paper, we consider a class of integral boundary value problems for nonlinear third-order impulsive integro-differential equation with a monotone homomorphism in real Banach space. By employing fixed point index theory, some sufficient criteria are obtained to ensure the existence of positive solutions. An example is given to demonstrate the application of our main results.

Keywords

integral boundary value problems impulsive integro-differential equation monotone homomorphism fixed point index theory Banach spaces

1 Introduction

This paper deals with the existence of positive solutions for the following nonlinear third-order impulsive integro-differential equation with monotone homomorphism and integral boundary value conditions in real Banach space (abbreviated by BVP (1.1) throughout this paper):
$$\begin{aligned} \left \{ \textstyle\begin{array}{l} (\varphi(-x''(t)))'=f(t,x(t),x'(t),(Tx)(t),(Sx)(t)), \quad t\in J, t\neq t_{k},\\ \Delta x|_{t=t_{k}}=-I_{k}(x(t_{k})), \quad k=1,\ldots,m,\\ \Delta x'|_{t=t_{k}}=-\overline{I}_{k}(x(t_{k}),x'(t_{k})), \quad k=1,\ldots,m,\\ x'(0)=x''(0)=\theta, \quad\quad x(1)=\int_{0}^{1}g(t)x(t)\,dt, \end{array}\displaystyle \right . \end{aligned}$$
(1.1)
where \(I=[0,1]\), \(J=(0,1)\), \(0< t_{1}< t_{2}<\cdots<t_{m}<1\), \(J'=J\backslash\{t_{1},t_{2},\ldots,t_{m}\}\), \(J_{0}=(0,t_{1}]\), \(J_{k}=(t_{k},t_{k+1}]\) (\(k=1,\ldots,m-1\)), \(J_{m}=(t_{m},1]\). \(\mathbb{E}\) is a real Banach space with the norm \(\Vert x\Vert \). \(\mathbb{P}\) is a positive cone in \(\mathbb{E}\). θ is a zero element of \(\mathbb{E}\). \(f\in C[J\times\mathbb{P}^{4}, \mathbb{P}]\), \(I_{k}\in C[\mathbb{P}, \mathbb{P}]\), \(\overline{I}_{k}\in C[\mathbb{P}^{2}, \mathbb{P}]\). \(g\in L^{1}[0,1]\) is nonnegative. \(\varphi:\mathbb{P}\rightarrow\mathbb{E}\) is an increasing and positive homomorphism (see Definition 2.3) and \(\varphi(\theta)=\theta\). \(\Delta x|_{t=t_{k}}=x(t_{k}^{+})-x(t_{k}^{-})\), \(\Delta x'|_{t=t_{k}}=x'(t_{k}^{+})-x'(t_{k}^{-})\). \(x(t_{k}^{+})\), \(x'(t_{k}^{+})\) and \(x(t_{k}^{-})\), \(x'(t_{k}^{-})\) represent the right-hand limits and left-hand limits of \(x(t)\) and \(x'(t)\) at \(t=t_{k}\), respectively. \((Tx)(t)\) and \((Sx)(t)\) are defined as
$$\begin{aligned} (Tx) (t)= \int_{0}^{t}K(t,s)x(s)\,ds, \quad\quad (Sx) (t)= \int_{0}^{1}F(t,s)x(s)\,ds, \end{aligned}$$
(1.2)
here \(K\in C[D,J]\), \(D=\{(t,s)\in J\times J:t\geq s\}\), \(H\in C[J\times J,J]\).
It is worth noting that BVP (1.1) is the extension and generalization of many boundary value problems. For example, if \(\varphi(x)=x\), then BVP (1.1) changes into a boundary value problem of the form
$$\begin{aligned} \left \{ \textstyle\begin{array}{l} -x'''(t)=f(t,x(t),x'(t),(Tx)(t),(Sx)(t)), \quad t\in J, t\neq t_{k},\\ \Delta x|_{t=t_{k}}=-I_{k}(x(t_{k})), \quad k=1,\ldots,m,\\ \Delta x'|_{t=t_{k}}=-\overline{I}_{k}(x(t_{k}),x'(t_{k})), \quad k=1,\ldots,m,\\ x'(0)=x''(0)=\theta, \quad\quad x(1)=\int_{0}^{1}g(t)x(t)\,dt. \end{array}\displaystyle \right . \end{aligned}$$
(1.3)
If \(\varphi(x)=\Phi_{p}(x):=\vert x\vert ^{p-2}x\) for some \(p>1\), then BVP (1.1) changes into the boundary value problem with p-Laplacian as follows:
$$\begin{aligned} \left \{ \textstyle\begin{array}{l} (\Phi_{p}(-x''(t)))'=f(t,x(t),x'(t),(Tx)(t),(Sx)(t)), \quad t\in J, t\neq t_{k},\\ \Delta x|_{t=t_{k}}=-I_{k}(x(t_{k})), \quad k=1,\ldots,m,\\ \Delta x'|_{t=t_{k}}=-\overline{I}_{k}(x(t_{k}),x'(t_{k})),\quad k=1,\ldots,m,\\ x'(0)=x''(0)=\theta, \quad\quad x(1)=\int_{0}^{1}g(t)x(t)\,dt. \end{array}\displaystyle \right . \end{aligned}$$
(1.4)

In recent years, the boundary value problems has aroused great attention due to its important applications in the fields of science and engineering such as chemical engineering, heat conduction, thermo-elasticity, plasma physics, underground water flow, and so on. Many scholars began with a study of the dynamics of boundary value problems by some nonlinear analysis methods such as fixed point theorems, shooting method, iterative method with upper and lower solutions, and so forth. The fixed point principle in cone is one of the important methods by applying to investigate the existence and multiplicity of positive solutions for boundary value problems.

In order to describe the dynamics of populations subject to abrupt changes and other phenomena such as harvesting, diseases, and so on, some authors have used an impulsive differential system to describe these kinds of phenomena since the last century. Some scholars have begun to study the boundary value problems of impulsive differential equations and obtained many good results (see [122]). The main tools of our study of the existence and multiplicity of positive solutions for this problem are the Schauder fixed point theorem, fixed point index theory, upper and lower solutions together with the monotone iterative technique, etc. In recent years, there have appeared some papers dealing with impulsive integro-differential equations in Banach spaces (see [418]).

In addition, the inspiration of this paper comes from the following two systems (see [4, 23]). In [23], Fu and Ding considered the existence of positive solutions of the boundary value problems with integral boundary conditions in Banach spaces of the form
$$\begin{aligned} \left \{ \textstyle\begin{array}{l} (\varphi(-x''(t)))'=f(t,x(t)), \quad t\in J,\\ x(0)=x''(0)=\theta, \quad\quad x(1)=\int_{0}^{1}g(t)x(t)\,dt, \end{array}\displaystyle \right . \end{aligned}$$
where θ is the zero element of \(\mathbb{E}\). \(g\in L^{1}[0,1]\) is nonnegative. \(\varphi: \mathbb{R}\rightarrow\mathbb{R}\) is an increasing and positive homomorphism and \(\varphi(0)=\theta_{1}\), \(\mathbb{R}:=(-\infty,+\infty)\).

Remark 1.1

In [23], the author defined the mapping \(\varphi:\mathbb{R}\rightarrow\mathbb{R}\) being an increasing and positive homomorphism and \(\varphi(0)=\theta_{1}\). It seems that this definition is incorrect since \(x(t)\) is an element of Banach space \(\mathbb{E}\) but not \(x(t)\in\mathbb{R}\). So we modify it as \(\varphi:\mathbb{P}\rightarrow\mathbb{E}\) is an increasing and positive homomorphism (see Definition 2.3) and \(\varphi(\theta)=\theta\).

In [4], Sun considered the positive solutions of the Sturm-Liouville boundary value problems for singular nonlinear second-order impulsive integro-differential equation in Banach spaces of the form
$$\begin{aligned} \left \{ \textstyle\begin{array}{l} y''=f(t,y(t),y'(t),(Ty)(t),(Sy)(t)), \quad t\in J, t\neq t_{k},\\ \Delta y|_{t=t_{k}}=I_{k}(y(t_{k})), \quad k=1,\ldots,m,\\ -\Delta y'|_{t=t_{k}}=\overline{I}_{k}(y(t_{k}),y'(t_{k})), \quad k=1,\ldots,m,\\ \alpha y(0)-\beta y'(0)=0, \quad\quad \gamma y(1)-\delta y'(1)=0, \end{array}\displaystyle \right . \end{aligned}$$
where \(\alpha,\beta,\gamma,\delta\geq0\), \(\rho=\beta\gamma+\alpha\gamma+\alpha\delta>0\), \(I=[0,1]\), \(J=(0,1)\), \(0< t_{1}< t_{2}<\cdots<t_{m}<1\), \(J'=J\backslash\{t_{1},t_{2},\ldots,t_{m}\}\), \(\overline{J}=[0,1]\), \(J_{0}=(0,t_{1}]\), \(J_{k}=(t_{k},t_{k}+1]\) (\(k=1,\ldots,m-1\)), \(J_{m}=(t_{m},1]\), \(f\in C[J\times\mathbb{P}^{4}, \mathbb{P}]\). \(\mathbb{P}\) is a positive cone in \(\mathbb{E}\). θ is a zero element of E. \(I_{k}\in C[\mathbb{P}, \mathbb{P}]\), \(\overline{I}_{k}\in C[\mathbb{P}^{2}, \mathbb{P}]\).

To the best of our knowledge, there is less research dealing with integral BVPs for nonlinear third-order impulsive integro-differential equation with monotone homomorphism in Banach spaces. Therefore, we will investigate the existence of positive solutions for BVP (1.1) under some further conditions by making use of fixed point index theory. Our main results in essence improve and generalize the corresponding results of [4, 23].

The rest of this paper is organized as follows. In Section 2, we present some known results and introduce conditions to be used in the next section. We give some sufficient conditions for the existence of positive solutions for BVP (1.1) in Section 3. Finally, one example is also provided to illustrate the validity of our main results in Section 4.

2 Preliminaries and statements

In this section, we shall state some necessary definitions and preliminaries results.

Definition 2.1

Let \((\mathbb{E},\Vert \cdot \Vert )\) be a Banach space. A nonempty closed convex set \(\mathbb{P}\subset\mathbb{E}\) is said to be a cone provided that the following conditions are satisfied:
  1. (i)

    \(y\in\mathbb{P}\), \(\lambda\geq0\) implies \(\lambda y\in\mathbb{P}\);

     
  2. (ii)

    \(y\in\mathbb{P}\), \(-y\in\mathbb{P}\) implies \(y=\theta\).

     

Let be the partial order on \(\mathbb{E}\) induced by the cone \(\mathbb{P}\) in \(\mathbb{E}\). That is, \(x\preceq y\) if and only if \(y-x\in\mathbb{P}\). If \(x, y\in\mathbb{E}\), the notation \(x\prec y\) means that \(x\preceq y\) and \(x\neq y\). \(\mathbb{P}\) is said to be normal if there exists a positive constant N such that \(\theta\preceq x\preceq y\) implies \(\Vert x\Vert \leq N\Vert y\Vert \), where θ denotes the zero element of \(\mathbb{E}\), and N is called the normal constant of \(\mathbb{P}\) (it is clear that \(N\geq1\)).

Definition 2.2

Let \(\mathbb{E}\) be a Banach space and \(\mathbb{P}\subset\mathbb{E}\) be a cone in \(\mathbb{E}\). Let \(\mathbb{C}\) be a convex subset in \(\mathbb{E}\). An operator \(\Phi: \mathbb{C}\rightarrow\mathbb{E}\) is said to be an increasing operator if
$$\begin{aligned} x, y\in\mathbb{C},\quad x\preceq y\quad \Rightarrow\quad \Phi x\preceq\Phi y. \end{aligned}$$
Similarly, An operator \(\Phi: \mathbb{C}\rightarrow\mathbb{E}\) is called a decreasing operator if
$$\begin{aligned} x, y\in\mathbb{C},\quad x\preceq y\quad \Rightarrow\quad \Phi x\succeq\Phi y. \end{aligned}$$

Definition 2.3

see [24]

Let \(\mathbb{E}\) be a Banach space and \(\mathbb{P}\subset\mathbb{E}\) be a positive cone in \(\mathbb{E}\). A projection \(\varphi: \mathbb{P}\rightarrow\mathbb{E}\) is called an increasing and positive homomorphism if and only if
  1. (i)

    if \(x\preceq y\), then \(\varphi(x)\preceq \varphi(y)\) for all \(x, y\in\mathbb{P}\);

     
  2. (ii)

    φ is a continuous bijection and its inverse mapping \(\varphi^{-1}\) is also continuous;

     
  3. (iii)

    \(\varphi(xy)=\varphi(x)\varphi(y)\) for all \(x, y\in\mathbb{P}\).

     
It is easy to verify that \(\varphi^{-1}\) is also an increasing and positive homomorphism.

If \((\mathbb{E},\Vert \cdot \Vert )\) is a real Banach space and \(\mathbb{P}\subset\mathbb{E}\) is a cone in \(\mathbb{E}\), then the common relationship of less than or equal to ≤ decides a partial order on \(\mathbb{E}\) induced by the cone \(\mathbb{P}\) in \(\mathbb{E}\). It is worth noting that the partial order ≤ is used throughout the paper.

Definition 2.4

Let \(\mathbb{E}\) be a metric space and \(\mathbb{S}\) be a bounded subset of \(\mathbb{E}\). The (Kuratowski) measure of noncompactness \(\alpha(\mathbb{S})\) of \(\mathbb{S}\) is defined by
$$\alpha(\mathbb{S})=\inf\bigl\{ \delta>0: \mathbb{S} \text{ admits a finite cover by subsets of } \mathbb{S}_{i}\subset\mathbb{S} \text{ such that } \operatorname {diam}(\mathbb{S}_{i})\leq\delta\bigr\} , $$
where \(\operatorname {diam}(\mathbb{S}_{i})\) denotes the diameter of the set \(\mathbb{S}_{i}\).

Definition 2.5

An operator \(\Phi: \mathbb{D}\rightarrow\mathbb{E}\) is said to be completely continuous if it is continuous and compact.

Lemma 2.1

Fixed-point index theorem; see [25]

Let \(\mathbb{E}\) be a Banach space, \(\mathbb{P}\subset\mathbb{E}\) is a cone. For \(r>0\), define \(\Omega_{r}=\{u\in\mathbb{P}:\Vert u\Vert < r\}\). Assume that \(A:\overline{\Omega}_{r}{\rightarrow}\mathbb{P}\) is a completely continuous operator such that \(Au\neq u\) for \(u\in\partial\Omega_{r}=\{u\in\mathbb{P}:\Vert u\Vert =r\}\).
  1. (1)

    If \(\Vert Au\Vert \geq \Vert u\Vert \), for \(u\in\partial\Omega_{r}\), then \(i(A,\Omega_{r},\mathbb{P})=0\).

     
  2. (2)

    If \(\Vert Au\Vert \leq \Vert u\Vert \), for \(u\in\partial\Omega_{r}\), then \(i(A,\Omega_{r},\mathbb{P})=1\).

     

Let \((\mathbb{E},\Vert \cdot \Vert )\) be a real Banach space and denote \(\operatorname {PC}[J,\mathbb{E}]:= \{x| x:J \rightarrow\mathbb{E}\text{ is continuous at} t\neq t_{k}, \text{left continuous at } {t=t_{k}}\text{ and } x(t^{+}_{k})\text{ exists for }k=1,\ldots,m \}\) and \(\operatorname {PC}[J,\mathbb{P}]:= \{x\in \operatorname {PC}[J, \mathbb{E}]: x(t)\geq\theta \}\). It is easy to verify that \(\operatorname {PC}[J,\mathbb{E}]\) is a Banach space with norm \(\Vert x\Vert _{\operatorname {PC}}=\sup_{t\in J}\Vert x(t)\Vert \). Obviously, \(\operatorname {PC}[J,\mathbb{P}]\) is a cone in Banach space \(\operatorname {PC}[J,\mathbb{E}]\).

Denote \(\operatorname {PC}^{1}[J,\mathbb{E}]:= \{x\in \operatorname {PC}[J,\mathbb{E}]: x'(t) \text{ is continuous at }t\neq t_{k}, \text{left continuous at }t=t_{k}\text{ and } x'(t^{+}_{k})\text{ exists for }k=1,\ldots,m \}\) and \(\operatorname {PC}^{1}[J,\mathbb{P}]:= \{x\in \operatorname {PC}^{1}[J,\mathbb{E}]: x(t)\geq\theta, x'(t)\geq\theta\}\). Clearly, \(\operatorname {PC}[J,\mathbb{E}]\) is a Banach space equipped with the norm \(\Vert x\Vert _{\operatorname {PC}^{1}}=\max \{\Vert x\Vert _{\operatorname {PC}},\Vert x'\Vert _{\operatorname {PC}} \}\) and \(\Vert x\Vert _{\operatorname {PC}^{1}}\leq \Vert x\Vert _{\operatorname {PC}}+\Vert x'\Vert _{\operatorname {PC}}\). \(\operatorname {PC}^{1}[J,\mathbb{P}]\) is a cone in \(\operatorname {PC}^{1}[J,\mathbb{E}]\).

Let \(C^{3}[J',\mathbb{E}]:= \{x:J'\rightarrow\mathbb{E} |x'''(t)\text{ is continuous in }J' \}\). A functional \(x\in \operatorname {PC}^{1}[J,\mathbb{E}]\cap C^{3}[J',\mathbb{E}]\) is called a nonnegative solution of (1.1) if \(x(t)\geq\theta\), \(x'(t)\geq\theta\) for \(t\in J\) and \(x(t)\) satisfies BVP (1.1). A functional \(x\in \operatorname {PC}^{1}[J,\mathbb{E}]\cap C^{3}[J',\mathbb{E}]\) is called a positive solution of BVP (1.1) if x is a nonnegative solution of BVP (1.1) and \(x(t)\neq\theta\).

Next we state the integral equation associated with BVP (1.1).

Lemma 2.2

Denote \(\sigma:=\int_{0}^{1}g(s)\,ds\). If \(\sigma\neq1\), then \(x\in \operatorname {PC}^{1}[\overline{J},\mathbb{E}]\cap C^{3}[J',\mathbb{E}]\) is a solution of (1.1) if and only if \(x\in C^{3}[\overline{J},\mathbb{E}]\) is a solution of the impulsive integral equation of the form
$$\begin{aligned} x(t)&= \int_{0}^{1}H(t,s)\varphi^{-1} \biggl( \int_{0}^{s}f\bigl(\tau ,x(\tau),x'( \tau),(Tx) (\tau),(Sx) (\tau)\bigr)\,d\tau \biggr)\,ds+\phi (t)+\psi(t) \\ &\quad{} +\frac{1}{1-\sigma} \int_{0}^{1}g(s)\phi(s)\,ds+\frac{1}{1-\sigma } \int_{0}^{1}g(s)\psi(s)\,ds, \end{aligned}$$
(2.1)
where
$$\begin{aligned}& H(t,s)=G(t,s)+\frac{1}{1-\sigma} \int_{0}^{1}g(\tau)G(\tau,s)\,d\tau, \end{aligned}$$
(2.2)
$$\begin{aligned}& G(t,s)=\left \{ \textstyle\begin{array}{l@{\quad}l} 1-t,&0\leq s\leq t\leq1;\\ 1-s,&0\leq t\leq s\leq1, \end{array}\displaystyle \right . \end{aligned}$$
(2.3)
$$\begin{aligned}& \phi(t)=(1-t)\sum_{0< t_{k}< t} \overline{I}_{k}\bigl(x(t_{k}),x'(t_{k}) \bigr), \end{aligned}$$
(2.4)
$$\begin{aligned}& \psi(t)=\sum_{t\leq t_{k}< 1} \bigl[I_{k}\bigl(x(t_{k})\bigr)+(1-t_{k}) \overline{I}_{k}\bigl(x(t_{k}),x'(t_{k}) \bigr) \bigr]. \end{aligned}$$
(2.5)

Proof

Now we prove the necessity of Lemma 2.2. Suppose that \(x\in \operatorname {PC}^{1}[\overline{J},\mathbb{E}]\cap C^{3}[J',\mathbb{E}]\) is a solution of BVP (1.1). Integrating both sides of the first equation of (1.1) from 0 to t, we have
$$\begin{aligned} \varphi\bigl(-x^{\prime\prime}(t)\bigr)-\varphi\bigl(-x^{\prime\prime}(0)\bigr)= \int _{0}^{t}f\bigl(\tau,x(\tau),x'( \tau),(Tx) (\tau),(Sx) (\tau)\bigr)\,d\tau. \end{aligned}$$
By \(\varphi(\theta)=\theta\) and the boundary value condition \(x''(0)=\theta\), we get
$$\begin{aligned} x^{\prime\prime}(t)=-\varphi^{-1} \biggl( \int_{0}^{t}f\bigl(\tau,x(\tau ),x'( \tau),(Tx) (\tau),(Sx) (\tau)\bigr)\,d\tau \biggr). \end{aligned}$$
(2.6)
By taking the integral of (2.6) on \([0,t]\) and noting that \(x^{\prime}(0)=\theta\), we obtain
$$\begin{aligned} x^{\prime}(t)&=- \int_{0}^{t}\varphi^{-1} \biggl( \int_{0}^{s}f\bigl(\tau ,x(\tau),x'( \tau),(Tx) (\tau),(Sx) (\tau)\bigr)\,d\tau \biggr)\,ds +\sum _{0< t_{k}< t} \bigl[x'\bigl(t_{k}^{+} \bigr)-x'\bigl(t_{k}^{-}\bigr) \bigr] \\ &=- \int_{0}^{t}\varphi^{-1} \biggl( \int_{0}^{s}f\bigl(\tau,x(\tau ),x'( \tau),(Tx) (\tau),(Sx) (\tau)\bigr)\,d\tau \biggr)\,ds \\ &\quad{} -\sum_{0< t_{k}< t}\overline{I}_{k} \bigl(x(t_{k}),x'(t_{k})\bigr). \end{aligned}$$
(2.7)
Taking the integral of (2.7) on \([0,t]\), we have
$$\begin{aligned} x(t)&=x(0)- \int_{0}^{t}(t-s)\varphi^{-1} \biggl( \int_{0}^{s}f\bigl(\tau ,x(\tau),x'( \tau),(Tx) (\tau),(Sx) (\tau)\bigr)\,d\tau \biggr)\,ds \\ &\quad{} -\sum_{0< t_{k}< t}I_{k}(x(t_{k})- \sum_{0< t_{k}< t}(t-t_{k})\overline{I}_{k} \bigl(x(t_{k}),x'(t_{k})\bigr). \end{aligned}$$
(2.8)
Letting \(t=1\), (2.8) shows that
$$\begin{aligned} x(1)&=x(0)- \int_{0}^{1}(1-s)\varphi^{-1} \biggl( \int_{0}^{s}f\bigl(\tau ,x(\tau),x'( \tau),(Tx) (\tau),(Sx) (\tau)\bigr)\,d\tau \biggr)\,ds \\ &\quad{} -\sum_{k=1}^{m}I_{k}(x(t_{k})- \sum_{k=1}^{m}(1-t_{k}) \overline{I}_{k}\bigl(x(t_{k}),x'(t_{k}) \bigr), \end{aligned}$$
that is,
$$\begin{aligned} x(0)&=x(1)+ \int_{0}^{1}(1-s)\varphi^{-1} \biggl( \int_{0}^{s}f\bigl(\tau ,x(\tau),x'( \tau),(Tx) (\tau),(Sx) (\tau)\bigr)\,d\tau \biggr)\,ds \\ &\quad{} +\sum_{k=1}^{m}I_{k}(x(t_{k})+ \sum_{k=1}^{m}(1-t_{k}) \overline{I}_{k}\bigl(x(t_{k}),x'(t_{k}) \bigr). \end{aligned}$$
(2.9)
Substituting \(x(1)=\int_{0}^{1}g(t)x(t)\,dt\) into (2.9), we obtain
$$\begin{aligned} x(0)&= \int_{0}^{1}g(s)x(s)\,ds + \int_{0}^{1}(1-s)\varphi^{-1} \biggl( \int_{0}^{s}f\bigl(\tau,x(\tau ),x'( \tau),(Tx) (\tau),(Sx) (\tau)\bigr)\,d\tau \biggr)\,ds \\ &\quad{} +\sum_{k=1}^{m}I_{k}(x(t_{k})+ \sum_{k=1}^{m}(1-t_{k}) \overline{I}_{k}\bigl(x(t_{k}),x'(t_{k}) \bigr). \end{aligned}$$
(2.10)
Substituting (2.10) into (2.8), we have
$$\begin{aligned} x(t)&= \int_{0}^{1}g(s)x(s)\,ds+ \int_{0}^{1}(1-s)\varphi^{-1} \biggl( \int_{0}^{s} f\bigl(\tau,x(\tau),x'( \tau), (Tx) (\tau),(Sx) (\tau)\bigr)\,d\tau \biggr)\,ds \\ &\quad{} +\sum_{k=1}^{m}I_{k}(x(t_{k})+ \sum_{k=1}^{m}(1-t_{k}) \overline{I}_{k}\bigl(x(t_{k}),x'(t_{k}) \bigr) \\ &\quad{} - \int_{0}^{t}(t-s)\varphi^{-1} \biggl( \int_{0}^{s}f\bigl(\tau,x(\tau ),x'( \tau),(Tx) (\tau),(Sx) (\tau)\bigr)\,d\tau \biggr)\,ds \\ &\quad{} -\sum_{0< t_{k}< t}I_{k}(x(t_{k})- \sum_{0< t_{k}< t}(t-t_{k})\overline{I}_{k} \bigl(x(t_{k}),x'(t_{k})\bigr) \\ &= \biggl( \int_{0}^{t}+ \int_{t}^{1} \biggr) \biggl[(1-s)\varphi ^{-1} \biggl( \int_{0}^{s}f\bigl(\tau,x(\tau),x'( \tau),(Tx) (\tau ),(Sx) (\tau)\bigr)\,d\tau \biggr) \biggr]\,ds \\ &\quad{} - \int_{0}^{t}(t-s)\varphi^{-1} \biggl( \int_{0}^{s}f\bigl(\tau,x(\tau ),x'( \tau),(Tx) (\tau),(Sx) (\tau)\bigr)\,d\tau \biggr)\,ds+ \int _{0}^{1}g(s)x(s)\,ds \\ &\quad{} +\sum_{t\leq t_{k}< 1}I_{k}\bigl(x(t_{k}) \bigr)+\sum_{t\leq t_{k}< 1}(1-t_{k}) \overline{I}_{k}\bigl(x(t_{k}),x'(t_{k}) \bigr) +(1-t)\sum_{0< t_{k}< t}\overline{I}_{k} \bigl(x(t_{k}),x'(t_{k})\bigr) \\ &= \biggl( \int_{0}^{t}(1-t)+ \int_{t}^{1}(1-s) \biggr)\varphi^{-1} \biggl( \int_{0}^{s}f\bigl(\tau,x(\tau),x'( \tau),(Tx) (\tau),(Sx) (\tau )\bigr)\,d\tau \biggr)\,ds \\ &\quad{} + \int_{0}^{1}g(s)x(s)\,ds+(1-t)\sum _{0< t_{k}< t}\overline {I}_{k}\bigl(x(t_{k}),x'(t_{k}) \bigr)+\sum_{t< t_{k}< 1} \bigl[I_{k} \bigl(x(t_{k})\bigr)+\overline{I}_{k}\bigl(x(t_{k}),x'(t_{k}) \bigr) \bigr] \\ &= \int_{0}^{1}G(t,s)\varphi^{-1} \biggl( \int_{0}^{s}f\bigl(\tau,x(\tau),x'( \tau),(Tx) (\tau),(Sx) (\tau )\bigr)\,d\tau \biggr)\,ds+ \int_{0}^{1}g(s)x(s)\,ds \\ &\quad{} +\sum_{t\leq t_{k}< 1}I_{k}\bigl(x(t_{k}) \bigr)+\sum_{t\leq t_{k}< 1}(1-t_{k}) \overline{I}_{k}\bigl(x(t_{k}),x'(t_{k}) \bigr) \\ &\quad{} +(1-t)\sum_{0< t_{k}< t}\overline{I}_{k} \bigl(x(t_{k}),x'(t_{k})\bigr), \end{aligned}$$
(2.11)
where \(G(t,s)\) is defined as (2.3).
Integrating both sides of (2.11) from 0 to 1, we have
$$\begin{aligned}& \int_{0}^{1}g(s)x(s)\,ds \\& \quad = \int_{0}^{1}g(s) \biggl[ \int_{0}^{1}G(s,\tau )\varphi^{-1} \biggl( \int_{0}^{\tau} f\bigl(\eta,x(\eta),x'( \eta),(Tx) (\eta),(Sx) (\eta)\bigr)\,d\eta \biggr) \,d\tau \biggr] \,ds \\& \quad \quad{} + \int_{0}^{1}g(s)\,ds\times \int_{0}^{1}g(s)x(s)\,ds+ \int _{0}^{1}(1-s)g(s) \biggl(\sum _{0< t_{k}< s}\overline {I}_{k}\bigl(x(t_{k}),x'(t_{k}) \bigr) \biggr)\,ds \\& \quad \quad{} + \int_{0}^{1}g(s) \biggl(\sum _{s\leq t_{k}< 1} \bigl[I_{k}\bigl(x(t_{k}) \bigr)+(1-t_{k})\overline{I}_{k}\bigl(x(t_{k}),x'(t_{k}) \bigr) \bigr] \biggr)\,ds, \end{aligned}$$
which implies that
$$\begin{aligned} & \int_{0}^{1}g(s)x(s)\,ds \\ &\quad =\frac{1}{1-\sigma} \int _{0}^{1}g(s) \biggl[ \int_{0}^{1}G(s,\tau)\varphi^{-1} \biggl( \int _{0}^{\tau} f\bigl(\eta,x(\eta),x'( \eta),(Tx) (\eta),(Sx) (\eta)\bigr)\,d\eta \biggr)\,d\tau \\ &\quad\quad{} +(1-s)\sum_{0< t_{k}< s}\overline{I}_{k} \bigl(x(t_{k}),x'(t_{k})\bigr) \\ &\quad\quad{} +\sum _{s\leq t_{k}< 1} \bigl[I_{k}\bigl(x(t_{k}) \bigr)+(1-t_{k})\overline{I}_{k}\bigl(x(t_{k}),x'(t_{k}) \bigr) \bigr] \biggr] \,ds. \end{aligned}$$
(2.12)
According to (2.11) and (2.12), we derive
$$\begin{aligned} x(t)&= \int_{0}^{1}G(t,s)\varphi^{-1} \biggl( \int_{0}^{s}f\bigl(\tau,x(\tau),x'( \tau),(Tx) (\tau),(Sx) (\tau )\bigr)\,d\tau \biggr)\,ds \\ &\quad{} +(1-t)\sum_{0< t_{k}< t}\overline{I}_{k} \bigl(x(t_{k}),x'(t_{k})\bigr)+\sum _{t\leq t_{k}< 1} \bigl[I_{k}\bigl(x(t_{k}) \bigr)+(1-t_{k})\overline {I}_{k}\bigl(x(t_{k}),x'(t_{k}) \bigr) \bigr] \\ &\quad{} +\frac{1}{1-\sigma} \int_{0}^{1}g(s) \biggl[ \int_{0}^{1}G(s,\tau )\varphi^{-1} \biggl( \int_{0}^{\tau} f\bigl(\eta,x(\eta),x'( \eta),(Tx) (\eta),(Sx) (\eta)\bigr)\,d\eta \biggr)\,d\tau \\ &\quad{} +(1-s)\sum_{0< t_{k}< s}\overline{I}_{k} \bigl(x(t_{k}),x'(t_{k})\bigr)+\sum _{s\leq t_{k}< 1} \bigl[I_{k}\bigl(x(t_{k}) \bigr)+(1-t_{k})\overline{I}_{k}\bigl(x(t_{k}),x'(t_{k}) \bigr) \bigr] \biggr] \,ds \\ &= \int_{0}^{1}G(t,s)\varphi^{-1} \biggl( \int_{0}^{s}f\bigl(\tau,x(\tau),x'( \tau),(Tx) (\tau),(Sx) (\tau )\bigr)\,d\tau \biggr)\,ds \\ &\quad{} +\frac{1}{1-\sigma} \int_{0}^{1} \biggl[ \int_{0}^{1}g(\tau)G(\tau ,s)\,d\tau \biggr] \varphi^{-1} \biggl( \int_{0}^{s} f\bigl(\eta,x(\eta),x'( \eta),(Tx) (\eta),(Sx) (\eta)\bigr)\,d\eta \biggr)\,ds \\ &\quad{} +(1-t)\sum_{0< t_{k}< t}\overline{I}_{k} \bigl(x(t_{k}),x'(t_{k})\bigr)+\sum _{t\leq t_{k}< 1} \bigl[I_{k}\bigl(x(t_{k}) \bigr)+(1-t_{k})\overline {I}_{k}\bigl(x(t_{k}),x'(t_{k}) \bigr) \bigr] \\ & \quad{} +\frac{1}{1-\sigma} \int_{0}^{1}g(s) \biggl[(1-s)\sum _{0< t_{k}< s}\overline{I}_{k}\bigl(x(t_{k}),x'(t_{k}) \bigr) \\ &\quad{} +\sum_{s\leq t_{k}< 1} \bigl[I_{k} \bigl(x(t_{k})\bigr)+(1-t_{k})\overline{I}_{k} \bigl(x(t_{k}),x'(t_{k})\bigr) \bigr] \biggr] \,ds \\ &= \int_{0}^{1}H(t,s)\varphi^{-1} \biggl( \int_{0}^{s}f\bigl(\tau,x(\tau ),x'( \tau),(Tx) (\tau),(Sx) (\tau)\bigr)\,d\tau \biggr)\,ds+\phi(t)+\psi (t) \\ &\quad{} +\frac{1}{1-\sigma} \int_{0}^{1}g(s)\phi(s)\,ds+\frac{1}{1-\sigma } \int_{0}^{1}g(s)\psi(s)\,ds, \end{aligned}$$
(2.13)
where \(H(t,s)\), \(\phi(s)\) and \(\psi(s)\) are defined as (2.2), (2.4), and (2.5), respectively. To sum up, we know that x is a solution of the impulsive integral equation (2.1).
Next, we show the sufficiency of Lemma 2.2. Let \(x\in C^{3}[\overline{J},\mathbb{E}]\) is a solution of impulsive integral equation (2.1). Taking the derivative at both sides of (2.1), we have
$$\begin{aligned} x^{\prime}(t)&= \int_{0}^{1}\frac{\partial G(t,s)}{\partial t}\varphi^{-1} \biggl( \int_{0}^{s}f\bigl(\tau,x(\tau),x'( \tau),(Tx) (\tau ),(Sx) (\tau)\bigr)\,d\tau \biggr)\,ds+\phi'(t)+ \psi'(t) \\ &=- \int_{0}^{t}\varphi^{-1} \biggl( \int_{0}^{s}f\bigl(\tau ,x(\tau),x'( \tau),(Tx) (\tau),(Sx) (\tau)\bigr)\,d\tau \biggr)\,ds \\ &\quad{} -\sum_{0< t_{k}< t}\overline{I}_{k} \bigl(x(t_{k}),x'(t_{k})\bigr). \end{aligned}$$
(2.14)
Equation (2.14) gives \(x'(0)=\theta\). Taking the derivative at both sides of (2.14), we get
$$\begin{aligned} x^{\prime\prime}(t)=-\varphi^{-1} \biggl( \int_{0}^{t}f\bigl(\tau,x(\tau ),x'( \tau),(Tx) (\tau),(Sx) (\tau)\bigr)\,d\tau \biggr), \end{aligned}$$
which implies that \(x''(0)=\theta\) and
$$\begin{aligned} \bigl(\varphi\bigl(-x^{\prime\prime}(t)\bigr)\bigr)'=f \bigl(t,x(t),x'(t),(Tx) (t),(Sx) (t)\bigr). \end{aligned}$$
From (2.11) and noting that \(G(0,s)=G(1,s)=0\), we obtain
$$\begin{aligned} x(1)= \int_{0}^{1}g(t)x(t)\,dt. \end{aligned}$$
In addition, it follows from (2.1) that
$$\begin{aligned} x\bigl(t_{i}^{-}\bigr)&=x(t_{i}) \\ &= \int_{0}^{1}H(t_{i},s) \varphi^{-1} \biggl( \int _{0}^{s}f\bigl(\tau,x(\tau),x'( \tau),(Tx) (\tau),(Sx) (\tau)\bigr)\,d\tau \biggr)\,ds+\phi(t_{i})+ \psi(t_{i}) \\ &\quad{} +\frac{1}{1-\sigma} \int_{0}^{1}(1-s)g(s)\phi(s)\,ds+\frac{1}{1-\sigma } \int_{0}^{1}g(s)\psi(s)\,ds \\ &= \int_{0}^{1}H(t_{i},s) \varphi^{-1} \biggl( \int_{0}^{s}f\bigl(\tau ,x(\tau),x'( \tau),(Tx) (\tau),(Sx) (\tau)\bigr)\,d\tau \biggr)\,ds \\ &\quad{} +(1-t_{i})\sum_{k=1}^{i-1} \overline {I}_{k}\bigl(x(t_{k}),x'(t_{k}) \bigr)+\sum_{k=i}^{m} \bigl[I_{k} \bigl(x(t_{k})\bigr)+(1-t_{k})\overline{I}_{k} \bigl(x(t_{k}),x'(t_{k})\bigr) \bigr] \\ &\quad{} +\frac{1}{1-\sigma} \int_{0}^{1}(1-s)g(s)\phi(s)\,ds+\frac{1}{1-\sigma } \int_{0}^{1}g(s)\psi(s)\,ds \end{aligned}$$
and
$$\begin{aligned} x\bigl(t_{i}^{+}\bigr)&= \int_{0}^{1}H(t_{i},s) \varphi^{-1} \biggl( \int_{0}^{s}f\bigl(\tau ,x(\tau),x'( \tau),(Tx) (\tau),(Sx) (\tau)\bigr)\,d\tau \biggr)\,ds+\phi \bigl(t_{i}^{+} \bigr)+\psi\bigl(t_{i}^{+}\bigr) \\ &\quad{} +\frac{1}{1-\sigma} \int_{0}^{1}(1-s)g(s)\phi(s)\,ds+\frac{1}{1-\sigma } \int_{0}^{1}g(s)\psi(s)\,ds \\ &= \int_{0}^{1}H(t_{i},s) \varphi^{-1} \biggl( \int_{0}^{s}f\bigl(\tau ,x(\tau),x'( \tau),(Tx) (\tau),(Sx) (\tau)\bigr)\,d\tau \biggr)\,ds \\ &\quad{} +(1-t_{i})\sum_{k=1}^{i} \overline {I}_{k}\bigl(x(t_{k}),x'(t_{k}) \bigr)+\sum_{k=i+1}^{m} \bigl[I_{k} \bigl(x(t_{k})\bigr)+(1-t_{k})\overline{I}_{k} \bigl(x(t_{k}),x'(t_{k})\bigr) \bigr] \\ &\quad{} +\frac{1}{1-\sigma} \int_{0}^{1}(1-s)g(s)\phi(s)\,ds+\frac{1}{1-\sigma } \int_{0}^{1}g(s)\psi(s)\,ds. \end{aligned}$$
Thus we derive
$$\begin{aligned} \Delta x|_{t=t_{i}}=x\bigl(t_{i}^{+}\bigr)-x\bigl(t_{i}^{-} \bigr)=-I_{k}\bigl(x(t_{k})\bigr),\quad k=1,\ldots,m. \end{aligned}$$
Similarly, by means of (2.14), we have
$$\begin{aligned} \Delta x'|_{t=t_{k}}=x'\bigl(t_{i}^{+} \bigr)-x'\bigl(t_{i}^{-}\bigr)=-\overline{I}_{k} \bigl(x(t_{k}),x'(t_{k})\bigr), \quad k=1,\ldots,m. \end{aligned}$$
From the above discussions, we know that \(x(t)\) is a solution of (1.1). Thus the proof of Lemma 2.2 is complete. □

Lemma 2.3

Denote \(\sigma:=\int_{0}^{1}g(s)\,ds\). If \(0\leq\sigma<1\) and \(g\in L^{1}[0,1]\) is nonnegative, for any \(\delta\in(0,\frac{1}{2})\), then the functions \(H(t,s)\) and \(G(t,s)\) defined by (2.2) and (2.3) have the following properties:
  1. (i)

    \(0\leq G(t,s)\leq G(s,s)=1-s\) and \(0\leq H(t,s)\leq\frac{1}{1-\sigma}G(s,s)\), for all \(t,s\in[0,1]\);

     
  2. (ii)

    \(0\leq H'_{t}(t,s)\leq 1-s=G(s,s)\), for all \(t,s\in[0,1]\);

     
  3. (iii)

    \(G(t,s)\geq\delta G(s,s)\) and \(H(t,s)\geq\frac{\delta}{1-\sigma}G(s,s)\), for all \(t,s\in[\delta,1-\delta]\subset[0,1]\).

     

Proof

(i) When \(0\leq t\leq s\leq1\), \(G(t,s)=G(s,s)\equiv1-s\). When \(0\leq s\leq t\leq1\), \(G'_{t}(t,s)=-1<0\) implies \(0\leq G(t,s)=1-t\leq1-s=G(s,s)\). Thus we have \(0\leq G(t,s)\leq G(s,s)=1-s\), \(\forall s,t\in[0,1]\) and for all \(s,t\in[0,1]\)
$$\begin{aligned} 0&\leq H(t,s)=G(s,t)+\frac{1}{1-\sigma} \int_{0}^{1}g(\tau)G(\tau,s)\,d\tau \\ &\leq G(s,s)+\frac{1}{1-\sigma} \int_{0}^{1}g(\tau)G(s,s)\,d\tau \\ &\leq G(s,s)+\frac{1}{1-\sigma} \int_{0}^{1}g(\tau)G(s,s)\,d\tau=\frac {1}{1-\sigma} G(s,s). \end{aligned}$$

(ii) When \(0\leq s\leq t\leq1\), \(H'_{t}(t,s)=-1<1-s\). When \(0\leq t\leq s\leq1\), \(H'_{t}(t,s)=0<1-s\).

(iii) When \(0<\delta\leq t\leq s\leq1-\delta\leq1\), \(G(t,s)\equiv1-s\geq\delta(1-s)=\delta G(s,s)\). When \(0\leq\delta\leq s\leq t\leq1-\delta\leq1\), \(G'_{t}(t,s)=-1\leq0\) implies \(G(t,s)=1-t\geq1-(1-\delta)=\delta\geq\delta(1-s)=\delta G(s,s)\). Thus we have \(G(t,s)\geq\delta G(s,s)\), \(\forall s,t\in[\delta,1-\delta]\subset[0,1]\) and for all \(s,t\in[\delta,1-\delta]\subset[0,1]\)
$$\begin{aligned} H(t,s)&=G(s,t)+\frac{1}{1-\sigma} \int_{0}^{1}g(\tau)G(\tau,s)\,d\tau \\ &\geq \delta G(s,s)+\frac{1}{1-\sigma} \int_{0}^{1}g(\tau)\delta G(s,s)\,d\tau= \frac{\delta}{1-\sigma}G(s,s). \end{aligned}$$
Thus the proof of Lemma 2.3 is complete. □

Denote \(K:=\{x\in \operatorname {PC}[\overline{J},\mathbb{P}]:x\geq\delta x,t\in[0,1]\}\), \(0<\delta<\frac{1}{2}\). For any \(0< r<+\infty\), let \(K_{r}:=\{x\in K:\Vert x\Vert < r\}\), \(\partial K_{r}:=\{x\in K:\Vert x\Vert =r\}\), \(\overline{K}_{r}:=\{x\in K:\Vert x\Vert \leq r\}\).

According to Lemma 2.2, we define an operator \(A: K\rightarrow K\) as
$$\begin{aligned} (Ax) (t)&= \int_{0}^{1}H(t,s)\varphi^{-1} \biggl( \int_{0}^{s}f\bigl(\tau ,x(\tau),x'( \tau),(Tx) (\tau),(Sx) (\tau)\bigr)\,d\tau \biggr)\,ds+\phi (t)+\varphi(t) \\ &\quad{} +\frac{1}{1-\sigma} \int_{0}^{1}g(s)\phi(s)\,ds+\frac{1}{1-\sigma } \int_{0}^{1}g(s)\varphi(s)\,ds. \end{aligned}$$
(2.15)
It is easy to derive that x is a positive solution of BVP (1.1) if x is a nontrivial fixed point of operator A defined as (2.15).
For convenience and simplicity, we introduce some notations and assumptions as follows:
$$\begin{aligned} &f_{v}=\liminf_{\sum_{i=1}^{4}\Vert x_{i}\Vert \rightarrow v}\min_{t\in[0,1]} \frac{\Vert f(t,x_{1},x_{2},x_{3},x_{4})\Vert }{\varphi (\sum_{i=1}^{4}\Vert x_{i}\Vert )}, \\ &f^{v}=\limsup_{\sum_{i=1}^{4}\Vert x_{i}\Vert \rightarrow v}\max_{t\in[0,1]} \frac{\Vert f(t,x_{1},x_{2},x_{3},x_{4})\Vert }{\varphi (\sum_{i=1}^{4}\Vert x_{i}\Vert )} . \end{aligned}$$
Here v denotes 0 or ∞.
\((\mathrm{H}_{1})\)

\(0\leq\sigma=\int_{0}^{1}g(s)\,ds<1\) and \(g\in L^{1}[0,1]\) is nonnegative.

\((\mathrm{H}_{2})\)

For any bounded set \(B_{i}\subset\mathbb{E}\) (\(i=1,2,3,4\)), \(f(t,B_{1},B_{2},B_{3},B_{4})\) and \(I_{k}(B_{1})\) together with \(\overline{I}_{k}(B_{1},B_{2})\) (\(k=1,2,\ldots,m\)) are relatively compact sets, where \(f(t,B_{1},B_{2},B_{3},B_{4}):=\{f(t,w_{1},w_{2},w_{3},w_{4}):w_{i}\in B_{i},i=1,2,3,4\}\), \(I_{k}(B_{1}):=\{I_{k}(w_{1}):w_{i}\in B_{1}\}\) and \(\overline{I}_{k}(B_{1},B_{2}):=\{\overline{I}_{k}(w_{1},w_{2}):w_{i}\in B_{i},i=1,2\}\).

\((\mathrm{H}_{3})\)
For \(I_{k}\in C(\mathbb{P},\mathbb {P}^{+})\), \(\overline{I}_{k}\in C(\mathbb{P}^{2},\mathbb{P}^{+})\) (\(\mathbb{P}^{+}\) represents a positive cone in a real Banach space \(\mathbb{E}\), there exist some positive constants \(v_{k}\), \(v^{\ast}_{k}\) and \(\overline{v}_{k}\) (\(k=1,\ldots,m\)) such that
$$\begin{aligned} \bigl\Vert I_{k}(x)\bigr\Vert \leq v_{k}\Vert x \Vert , \quad\quad \bigl\Vert \overline{I}_{k}(x_{1},x_{2}) \bigr\Vert \leq v^{\ast}_{k}\Vert x_{1}\Vert + \overline{v}_{k}\Vert x_{2}\Vert , \quad \forall t\in J. \end{aligned}$$

Lemma 2.4

Suppose that \((\mathrm{H}_{1})\)-\((\mathrm{H}_{3})\) hold. \(A:K\rightarrow K\) defined by (2.15) is completely continuous.

Proof

(1) Now we show that \(A(K)\subset K\), that is, \(A:K\rightarrow K\) is well defined. Clearly, \((Ax)(t)\geq\delta(Ax)(t)\). Hence, \(A(K)\subset K\).

(2) We need to show that \(A:K\rightarrow K\) is continuous. In fact, assume that \(x_{n},x_{0}\in K\) and \(\Vert x_{n}-x_{0}\Vert \rightarrow0\), \(\Vert x'_{n}-x'_{0}\Vert \rightarrow0\) (\(n\rightarrow\infty\)). Since \(f\in C(\overline{J}\times\mathbb{P}^{4},\mathbb{P}^{+})\), \(I_{k}\in C(\mathbb{P},\mathbb{P})\), \(\overline{I}_{k}\in C(\mathbb{P},\mathbb{P}^{+})\) and \(\varphi:\mathbb{P}\rightarrow \mathbb{E}\) is an increasing and positive homomorphism, then
$$\begin{aligned} &\lim_{n\rightarrow\infty}\bigl\Vert f\bigl(t,x_{n},x'_{n},(Tx_{n}) (t),(Sx_{n}) (t)\bigr)-f\bigl(t,x_{0},x'_{0},(Tx_{0}) (t),(Sx_{0}) (t)\bigr)\bigr\Vert =0, \\ &\lim_{n\rightarrow\infty}\bigl\Vert \varphi ^{-1}\bigl(f \bigl(t,x_{n},x'_{n},(Tx_{n}) (t),(Sx_{n}) (t)\bigr)\bigr)-\varphi ^{-1}\bigl(f \bigl(t,x_{0},x'_{0},(Tx_{0}) (t),(Sx_{0}) (t)\bigr)\bigr)\bigr\Vert =0, \\ &\lim_{n\rightarrow\infty}\bigl\Vert I_{k}\bigl(x_{n}(t_{k}) \bigr)-I_{k}\bigl(x_{0}(t_{k})\bigr)\bigr\Vert =0, \quad k=1,\ldots,m, \\ &\lim_{n\rightarrow\infty}\bigl\Vert \overline {I}_{k} \bigl(x_{n}(t_{k}),x'_{n}(t_{k}) \bigr)-\overline{I}_{k}\bigl(x_{0}(t_{k}),x'_{0}(t_{k}) \bigr)\bigr\Vert =0,\quad k=1,\ldots,m. \end{aligned}$$
Thus, for any \(t\in J\), from the Lebesgue dominate convergence theorem and Lemma 2.3 together with (2.15), we have
$$\begin{aligned}& \bigl\Vert (Ax_{n}) (t)-(Ax_{0}) (t)\bigr\Vert _{\operatorname {PC}} \\& \quad \leq \biggl\Vert \frac{1}{1-\sigma} \int_{0}^{1}G(s,s) \biggl[\varphi ^{-1} \biggl( \int_{0}^{s}f\bigl(\tau,x_{n}( \tau),x_{n}'(\tau),(Tx_{n}) (\tau ),(Sx_{n}) (\tau)\bigr)\,d\tau \biggr) \\& \quad\quad{} -\varphi^{-1} \biggl( \int_{0}^{s}f\bigl(\tau,x_{0}(\tau ),x_{0}'(\tau),(Tx_{0}) (\tau),(Sx_{0}) (\tau)\bigr)\,d\tau \biggr) \biggr]\,ds \\& \quad\quad{}+(1-t)\sum _{0< t_{k}< t} \bigl[\overline {I}_{k}\bigl(x_{n}(t_{k}),x'_{n}(t_{k}) \bigr)-\overline{I}_{k}\bigl(x_{0}(t_{k}),x'_{0}(t_{k}) \bigr) \bigr] \\& \quad\quad{}+\sum_{t\leq t_{k}< 1} \bigl[ \bigl(I_{k}\bigl(x_{n}(t_{k}) \bigr)-I_{k}\bigl(x_{0}(t_{k})\bigr) \bigr)+(1-t_{k}) \bigl(\overline{I}_{k}\bigl(x_{n}(t_{k}),x'_{n}(t_{k}) \bigr)-\overline {I}_{k}\bigl(x_{0}(t_{k}),x'_{0}(t_{k}) \bigr) \bigr) \bigr] \\& \quad\quad{}+\frac{1}{1-\sigma} \int_{0}^{1}(1-s)g(s)\sum _{0< t_{k}< s} \bigl[\overline{I}_{k}\bigl(x_{n}(t_{k}),x'_{n}(t_{k}) \bigr)-\overline {I}_{k}\bigl(x_{0}(t_{k}),x'_{0}(t_{k}) \bigr) \bigr]\,ds \\& \quad\quad{}+\frac{1}{1-\sigma} \int_{0}^{1}g(s)\sum_{t\leq t_{k}< 1} \bigl[ \bigl(I_{k}\bigl(x_{n}(t_{k}) \bigr)-I_{k}\bigl(x_{0}(t_{k})\bigr) \bigr) \\& \quad\quad{}+(1-t_{k}) \bigl(\overline {I}_{k} \bigl(x_{n}(t_{k}),x'_{n}(t_{k}) \bigr)-\overline{I}_{k}\bigl(x_{0}(t_{k}),x'_{0}(t_{k}) \bigr) \bigr) \bigr]\,ds\biggr\Vert \\& \quad \leq \frac{1}{1-\sigma} \int_{0}^{1}G(s,s)\biggl\Vert \varphi^{-1} \biggl( \int_{0}^{s}f\bigl(\tau,x_{n}( \tau),x_{n}'(\tau),(Tx_{n}) ( \tau),(Sx_{n}) (\tau )\bigr)\,d\tau \biggr) \\& \quad\quad{}- \varphi^{-1} \biggl( \int_{0}^{s}f\bigl(\tau,x_{0}(\tau ),x_{0}'(\tau),(Tx_{0}) (\tau),(Sx_{0}) (\tau)\bigr)\,d\tau \biggr)\biggr\Vert \,ds \\& \quad\quad{} +(1-t)\sum_{0< t_{k}< t}\bigl\Vert \overline {I}_{k}\bigl(x_{n}(t_{k}),x'_{n}(t_{k}) \bigr)-\overline{I}_{k}\bigl(x_{0}(t_{k}),x'_{0}(t_{k}) \bigr)\bigr\Vert \\& \quad\quad{} +\sum_{t\leq t_{k}< 1} \bigl[\bigl\Vert I_{k} \bigl(x_{n}(t_{k})\bigr)-I_{k} \bigl(x_{0}(t_{k})\bigr)\bigr\Vert +(1-t_{k}) \bigl\Vert \overline{I}_{k}\bigl(x_{n}(t_{k}),x'_{n}(t_{k}) \bigr)-\overline {I}_{k}\bigl(x_{0}(t_{k}),x'_{0}(t_{k}) \bigr)\bigr\Vert \bigr] \\& \quad\quad{} +\frac{1}{1-\sigma} \int_{0}^{1}(1-s)g(s)\sum _{0< t_{k}< s}\bigl\Vert \overline{I}_{k} \bigl(x_{n}(t_{k}),x'_{n}(t_{k}) \bigr)-\overline {I}_{k}\bigl(x_{0}(t_{k}),x'_{0}(t_{k}) \bigr)\bigr\Vert \,ds \\& \quad\quad{} +\frac{1}{1-\sigma} \int_{0}^{1}g(s)\sum_{t\leq t_{k}< 1} \bigl[\bigl\Vert I_{k}\bigl(x_{n}(t_{k}) \bigr)-I_{k}\bigl(x_{0}(t_{k})\bigr)\bigr\Vert \\& \quad\quad{} +(1-t_{k})\bigl\Vert \overline {I}_{k} \bigl(x_{n}(t_{k}),x'_{n}(t_{k}) \bigr)-\overline{I}_{k}\bigl(x_{0}(t_{k}),x'_{0}(t_{k}) \bigr)\bigr\Vert \bigr]\,ds \rightarrow0, \quad \mbox{as } n\rightarrow\infty \end{aligned}$$
and
$$\begin{aligned}& \bigl\Vert (Ax_{n})'(t)-(Ax_{0})'(t) \bigr\Vert _{\operatorname {PC}} \\& \quad \leq \biggl\Vert \frac{1}{1-\sigma} \int_{0}^{1}H'_{t}(t,s) \biggl[\varphi ^{-1} \biggl( \int_{0}^{s}f\bigl(\tau,x_{n}( \tau),x_{n}'(\tau),(Tx_{n}) (\tau ),(Sx_{n}) (\tau)\bigr)\,d\tau \biggr) \\& \quad\quad{} -\varphi^{-1} \biggl( \int_{0}^{s}f\bigl(\tau,x_{0}(\tau ),x_{0}'(\tau),(Tx_{0}) (\tau),(Sx_{0}) (\tau)\bigr)\,d\tau \biggr) \biggr]\,ds \\& \quad\quad{} -\sum _{0< t_{k}< t} \bigl[\overline {I}_{k}\bigl(x_{n}(t_{k}),x'_{n}(t_{k}) \bigr)-\overline{I}_{k}\bigl(x_{0}(t_{k}),x'_{0}(t_{k}) \bigr) \bigr]\biggr\Vert \\& \quad \leq \frac{1}{1-\sigma} \int_{0}^{1}G(s,s)\biggl\Vert \varphi^{-1} \biggl( \int_{0}^{s}f\bigl(\tau,x_{n}( \tau),x_{n}'(\tau),(Tx_{n}) ( \tau),(Sx_{n}) (\tau )\bigr)\,d\tau \biggr) \\& \quad\quad{}- \varphi^{-1} \biggl( \int_{0}^{s}f\bigl(\tau,x_{0}(\tau ),x_{0}'(\tau),(Tx_{0}) (\tau),(Sx_{0}) (\tau)\bigr)\,d\tau \biggr)\biggr\Vert \,ds \\& \quad\quad{}+\sum _{0< t_{k}< t}\bigl\Vert \overline {I}_{k} \bigl(x_{n}(t_{k}),x'_{n}(t_{k}) \bigr)-\overline{I}_{k}\bigl(x_{0}(t_{k}),x'_{0}(t_{k}) \bigr)\bigr\Vert \rightarrow0, \quad \mbox{as } n\rightarrow\infty. \end{aligned}$$
Hence, \(A:K\rightarrow K\) is continuous.
(3) Now we are going to prove that A is compact by the Kuratowski’s measure of noncompactness. Let \(\Omega\subset K\) be any bounded subset in \(\operatorname {PC}(J,\mathbb{E})\). For any \(x\in\Omega\), \(t,s\in[0,1]\), there exist some constants \(L_{i}>0\) (\(i=1,2,3,4\)) such that
$$\begin{aligned} &\max_{t,s\in[0,1]}\bigl\vert H(t,s)\bigr\vert \leq L_{1}, \quad\quad \max_{x\in\Omega,t\in[0,1]}\bigl\Vert f\bigl(t,x(t),x'(t),(Sx) (t),(Tx) (t)\bigr)\bigr\Vert \leq L_{2}, \\ &\max_{1\leq k\leq m}\max_{x\in\Omega}\bigl\Vert I_{k}\bigl(x(t_{k})\bigr)\bigr\Vert \leq L_{3},\quad\quad \max_{1\leq k\leq m}\max_{x\in\Omega}\bigl\Vert \overline{I}_{k}\bigl(x(t_{k}),x'(t_{k}) \bigr)\bigr\Vert \leq L_{4}. \end{aligned}$$
Then we have
$$\begin{aligned}& \bigl\Vert \phi(t)\bigr\Vert _{\operatorname {PC}}\leq \vert 1-t\vert \sum _{0< t_{k}< t}\bigl\Vert \overline {I}_{k} \bigl(x(t_{k}),x'(t_{k})\bigr)\bigr\Vert \leq \sum_{k=0}^{m}\bigl\Vert \overline {I}_{k}\bigl(x(t_{k}),x'(t_{k}) \bigr)\bigr\Vert \leq mL_{4}, \\& \begin{aligned} \bigl\Vert \psi(t)\bigr\Vert _{\operatorname {PC}}&\leq\sum _{t\leq t_{k}< 1} \bigl[\bigl\Vert I_{k}\bigl(x(t_{k}) \bigr)\bigr\Vert +\vert 1-t_{k}\vert \bigl\Vert \overline{I}_{k}\bigl(x(t_{k}),x'(t_{k}) \bigr)\bigr\Vert \bigr] \\ &\leq\sum_{k=0}^{m} \bigl[\bigl\Vert I_{k}\bigl(x(t_{k})\bigr)\bigr\Vert +\vert 1-t_{k}\vert \bigl\Vert \overline{I}_{k} \bigl(x(t_{k}),x'(t_{k})\bigr)\bigr\Vert \bigr]\leq m(L_{3}+L_{4}), \end{aligned} \\& \begin{aligned} \bigl\Vert (Ax) (t)\bigr\Vert _{\operatorname {PC}}&\leq \int_{0}^{1}\bigl\vert H(t,s)\bigr\vert \varphi^{-1} \biggl( \int _{0}^{s}\bigl\Vert f\bigl(\tau,x( \tau),x'(\tau),(Tx) (\tau),(Sx) (\tau)\bigr)\bigr\Vert \,d\tau \biggr)\,ds \\ &\quad {}+\bigl\Vert \phi(t)\bigr\Vert _{\operatorname {PC}}+\bigl\Vert \psi(t)\bigr\Vert _{\operatorname {PC}} \\ &\quad{} +\frac{1}{1-\sigma} \int _{0}^{1}g(s)\bigl\Vert \phi(s)\bigr\Vert \,ds+ \frac{1}{1-\sigma} \int_{0}^{1}g(s)\bigl\Vert \varphi(s)\bigr\Vert \,ds \\ &\leq L_{1}\varphi^{-1}(L_{2})+mL_{4}+m(L_{3}+L_{4})+ \frac{mL_{4}\sigma }{1-\sigma}+\frac{m\sigma(L_{3}+L4)}{1-\sigma} \\ &=L_{1}\varphi^{-1}(L_{2})+mL_{3}+2mL_{4}+ \frac{m\sigma (L_{3}+2L_{4})}{1-\sigma}:= M_{1}, \end{aligned} \end{aligned}$$
and
$$\begin{aligned}& \bigl\Vert (Ax)'(t)\bigr\Vert _{\operatorname {PC}} \\& \quad \leq \int_{0}^{1}\bigl\vert H'_{t}(t,s) \bigr\vert \varphi^{-1} \biggl( \int _{0}^{s}\bigl\Vert f\bigl(\tau,x( \tau),x'(\tau),(Tx) (\tau),(Sx) (\tau)\bigr)\bigr\Vert \,d\tau \biggr)\,ds+\bigl\Vert \phi'(t)\bigr\Vert \\& \quad \leq\varphi^{-1}(L_{1})+mL_{4}:=M_{2}. \end{aligned}$$
Therefore, \(A(\Omega)\) is uniformly bounded in \(\operatorname {PC}(J,\mathbb{E})\).
Next we verify that \(A:K\rightarrow K\) is equicontinuous. In fact, since \(H(t,s)\) are continuous on \([0,1]\times[0,1]\), it is uniformly continuous on \([0,1]\times[0,1]\). Thus, for fixed \(s\in[0,1]\) and for any \(\epsilon>0\), there exists a constant \(\delta_{1}>0\) such that for any \(\overline{t}_{1},\overline{t}_{2}\in[0,1]\) with \(\vert \overline{t}_{2}-\overline{t}_{1}\vert <\delta_{1}\), we have \(\vert H(\overline{t}_{2},s)-H(\overline{t}_{1},s)\vert <\epsilon\). Without loss of the generality, assume that \(\overline{t}_{1},\overline{t}_{2}\in[0,1]\) with \(\overline{t}_{1}\leq\overline{t}_{1}\), then
$$\begin{aligned}& \bigl\Vert (Ax) (\overline{t}_{2})-(Ax) (\overline{t}_{1}) \bigr\Vert _{\operatorname {PC}} \\& \quad \leq \int_{0}^{1}\bigl\vert H(\overline{t}_{2},s)-H( \overline {t}_{1},s)\bigr\vert \varphi^{-1} \biggl( \int_{0}^{s}\bigl\Vert f\bigl(\tau,x( \tau),x'(\tau ),(Tx) (\tau),(Sx) (\tau)\bigr)\bigr\Vert \,d\tau \biggr)\,ds \\& \quad\quad{} +\sum_{\overline{t}_{1}\leq t_{k}< \overline{t}_{2}}\bigl\Vert \overline{I}_{k} \bigl(x(t_{k}),x'(t_{k})\bigr)\bigr\Vert + \vert t_{2}-t_{1}\vert \sum_{0< t_{k}< \overline{t}_{1}} \bigl\Vert \overline{I}_{k}\bigl(x(t_{k}),x'(t_{k}) \bigr)\bigr\Vert \\& \quad\quad{} +\sum_{\overline{t}_{1}\leq t_{k}< \overline{t}_{2}} \bigl[\bigl\Vert I_{k} \bigl(x(t_{k})\bigr)\bigr\Vert +\vert 1-t_{k}\vert \bigl\Vert \overline {I}_{k}\bigl(x(t_{k}),x'(t_{k}) \bigr)\bigr\Vert \bigr] \\& \quad \leq \vert \bar{t}_{2}-\bar{t}_{1}\vert \varphi^{-1}(L_{2})+\vert \bar{t}_{2}-\bar {t}_{1}\vert L_{3}+\vert \bar{t}_{2}- \bar{t}_{1}\vert mL_{4}+\vert \bar{t}_{2}- \bar{t}_{1}\vert (L_{3}+L_{4}) \\& \quad = \bigl[\varphi^{-1}(L_{2})+2L_{3}+(m+1)L_{4} \bigr]\vert \bar{t}_{2}-\bar{t}_{1}\vert := N\vert \bar{t}_{2}-\bar{t}_{1}\vert , \end{aligned}$$
and noting that
$$H'_{t}(t,s)=\left \{ \textstyle\begin{array}{l@{\quad}l} -1, & 0\leq t\leq s\leq1, \\ 0, & 0\leq s\leq t\leq1, \end{array}\displaystyle \right .$$
we have
$$\begin{aligned} &\bigl\Vert (Ax)'(\overline{t}_{2})-(Ax)'( \overline{t}_{1})\bigr\Vert _{\operatorname {PC}} \\ &\quad \leq \int_{0}^{1}\bigl\vert H'_{t}( \overline{t}_{2},s)-H'_{t}(\overline {t}_{1},s)\bigr\vert \varphi^{-1} \biggl( \int_{0}^{s}\bigl\Vert f\bigl(\tau,x( \tau),x'(\tau ),(Tx) (\tau),(Sx) (\tau)\bigr)\bigr\Vert \,d\tau \biggr)\,ds \\ &\quad\quad{} +\sum_{\overline{t}_{1}\leq t_{k}< \overline{t}_{2}}\bigl\Vert \overline{I}_{k} \bigl(x(t_{k}),x'(t_{k})\bigr)\bigr\Vert \\ &\quad \leq L_{4}\vert \bar{t}_{2}-\bar{t}_{1}\vert , \end{aligned}$$
where \(N:=\varphi^{-1}(L_{2})+2L_{3}+(m+1)L_{4}\). Take \(\delta=\min\{\frac{\epsilon}{N},\frac{\epsilon}{L_{4}},\}=\frac {\epsilon}{N}\). Thus, for fixed \(s\in[0,1]\) and for any \(\epsilon>0\), there exists a constant \(\delta>0\) such that, for any \(\overline{t}_{1},\overline{t}_{2}\in[0,1]\) with \(\vert \overline{t}_{2}-\overline{t}_{1}\vert <\delta\), we have \(\Vert (Ax)(\overline{t}_{2})-(Ax)(\overline{t}_{1})\Vert _{\operatorname {PC}^{1}}<\epsilon\), which means that \(A:K\rightarrow K\) is equicontinuous. Hence, by [11], Lemma 7, we get
$$\begin{aligned} \alpha_{\operatorname {PC}^{1}}\bigl(A(\Omega)\bigr)=\max \Bigl\{ \sup _{t\in J}\alpha\bigl(A(\Omega)\bigr) (t), \sup_{t\in J} \alpha\bigl(A'(\Omega)\bigr) (t) \Bigr\} , \end{aligned}$$
(2.16)
where \(A(\Omega):=\{(Ax)(t):x\in\Omega\}\) and \(A'(\Omega):=\{(Ax)'(t):x\in\Omega\}\), α, \(\alpha_{\operatorname {PC}^{1}}\) denote the Kuratowski measures of noncompactness of bounded sets in \(\mathbb{E}\) and \(\operatorname {PC}^{1}(J, \mathbb{E})\), respectively (see [26], Section 1.2).
It follows from (2.15) and [26], Theorem 1.2.3, that
$$\begin{aligned} &\alpha\bigl((A\Omega)\bigr) (t) \\ &\quad \leq\frac{1}{1-\sigma} \int _{0}^{1}(1-s)\varphi^{-1} \biggl(2 \int_{0}^{s}\alpha \bigl(f\bigl(\tau ,\Omega(\tau), \Omega'(\tau),(T\Omega) (\tau),(S\Omega) (\tau )\bigr)\,d\tau \bigr) \biggr)\,ds \\ &\quad\quad{} +2\sum_{0< t_{k}< 1} \bigl[\alpha\bigl(I_{k} \bigl(\Omega (t_{k})\bigr)\bigr)+(1-t_{k})\alpha\bigl( \overline{I}_{k}\bigl(\Omega(t_{k}),x'(t_{k}) \bigr)\bigr) \bigr] \\ &\quad\quad{} +2(1-t)\sum_{0< t_{k}< 1}\alpha\bigl( \overline{I}_{k}\bigl(\Omega (t_{k}),\Omega'(t_{k}) \bigr)\bigr) \\ &\quad\quad{} +\frac{2 }{1-\sigma} \int_{0}^{1}g(s)\sum_{0< t_{k}< 1} \bigl[\alpha \bigl(I_{k}\bigl(\Omega(t_{k})\bigr) \bigr)+(1-t_{k})\alpha\bigl(\overline{I}_{k}\bigl(\Omega (t_{k}),x'(t_{k})\bigr)\bigr) \bigr]\,ds \\ &\quad\quad{} +\frac{2 }{1-\sigma} \int_{0}^{1}(1-s)g(s)\sum _{0< t_{k}< 1}\alpha\bigl(\overline {I}_{k}\bigl( \Omega(t_{k}),\Omega'(t_{k})\bigr)\bigr)\,ds \end{aligned}$$
(2.17)
and
$$\begin{aligned} \alpha\bigl((A\Omega)\bigr)'(t)&\leq\frac{1}{1-\sigma} \int _{0}^{1}(1-s)\varphi^{-1} \biggl(2 \int_{0}^{s}\alpha \bigl(f\bigl(\tau ,\Omega(\tau), \Omega'(\tau),(T\Omega) (\tau),(S\Omega) (\tau )\bigr)\,d\tau \bigr) \biggr)\,ds \\ & \quad{} +2\sum_{0< t_{k}< 1}\alpha\bigl(\overline{I}_{k} \bigl(\Omega(t_{k}),\Omega'(t_{k})\bigr)\bigr). \end{aligned}$$
(2.18)
According to condition \((\mathrm{H}_{2})\), we have
$$\begin{aligned} \alpha \bigl(f\bigl(\tau,\Omega(\tau),\Omega'( \tau),(T\Omega) (\tau ),(S\Omega) (\tau)\bigr)\,d\tau \bigr)=0, \quad \forall \tau\in J, \end{aligned}$$
(2.19)
and
$$\begin{aligned} \alpha\bigl(I_{k}\bigl(\Omega(t_{k})\bigr) \bigr)=0, \quad\quad \alpha\bigl(\overline{I}_{k}\bigl(\Omega (t_{k}), \Omega'(t_{k})\bigr)\bigr)=0, \quad k=1,2,\ldots,m. \end{aligned}$$
(2.20)
It follows from (2.16)-(2.20) that
$$\begin{aligned} \alpha_{\operatorname {PC}^{1}}\bigl(A(\Omega)\bigr)=\max \Bigl\{ \sup_{t\in J} \alpha\bigl(A(\Omega)\bigr) (t), \sup_{t\in J}\alpha \bigl(A'(\Omega)\bigr) (t) \Bigr\} =0. \end{aligned}$$
Thus the compactness of A is proved. Therefore, \(A:K\rightarrow K\) defined by (2.15) is completely continuous. The proof is complete. □

3 Main results

In this section, we show that there exists at least one positive solution for BVP (1.1).

Theorem 3.1

Assume that \((\mathrm{H}_{1})\)-\((\mathrm{H}_{3})\) hold. If \(0< f^{\infty}<\varphi (1-\sigma )\) and \(\lambda<\frac{1}{2}\), then BVP (1.1) has at least one positive solution, where \(\lambda=\frac{1}{1-\sigma}\sum_{k=1}^{m} [v_{k}+2v^{*}_{k}+2\overline {v}_{k} ]\).

Proof

According to \(0< f^{\infty}<\varphi (1-\sigma )\), there exists \(\varepsilon_{2}>0\) such that \(f^{\infty}< M-\varepsilon_{2}\). From the definition of \(f^{\infty}\), there exists \(R_{1}>1\) such that \(\Vert f(t,x_{1},x_{2},x_{3},x_{4})\Vert \leq f^{\infty}\varphi (\sum_{i=1}^{4}\Vert x_{i}\Vert )\) for all \(\sum_{i=1}^{4}\Vert x_{i}\Vert \geq R_{1}\), \(t\in[0,1]\). Let \(K_{R_{1}}:=\{x\in K:\Vert x\Vert _{\operatorname {PC}}< R_{1}\}\), \(\partial K_{R_{1}}:=\{x\in K:\Vert x\Vert _{\operatorname {PC}}=R_{1}\}\). Define an operator \(A:K_{R_{1}}\rightarrow K_{R_{1}}\) by (2.15). By Lemma 2.4, we know that \(A:K_{R_{1}}\rightarrow K_{R_{1}}\) is completely continuous. For any \(x\in\partial K_{R_{1}}\), that is, \(\Vert x\Vert _{\operatorname {PC}}=R_{1}\), it follows from \((\mathrm{H}_{1})\) and \((\mathrm{H}_{3})\) that
$$\begin{aligned} &\bigl\Vert (Ax) (t)\bigr\Vert _{\operatorname {PC}} \\ &\quad \leq\frac{1}{1-\sigma} \int_{0}^{1}G(s,s) \\ &\quad\quad{}\times \varphi ^{-1} \biggl( \int_{0}^{s} \bigl(\varphi (1-\sigma ) - \varepsilon_{2} \bigr)\varphi \bigl(\Vert x\Vert _{\operatorname {PC}}+ \bigl\Vert x'\bigr\Vert _{\operatorname {PC}}+\Vert Tx\Vert _{\operatorname {PC}}+\Vert Sx\Vert _{\operatorname {PC}} \bigr)\,d\tau \biggr)\,ds \\ &\quad\quad{} +\sum_{0< t_{k}< t} \bigl[v^{*}_{k}\Vert x \Vert _{\operatorname {PC}}+\overline{v}_{k}\bigl\Vert x' \bigr\Vert _{\operatorname {PC}} \bigr] +\sum_{t< t_{k}< 1} \bigl[v_{k}\Vert x\Vert _{\operatorname {PC}}+v^{*}_{k}\Vert x \Vert _{\operatorname {PC}}+\overline {v}_{k}\bigl\Vert x' \bigr\Vert _{\operatorname {PC}} \bigr] \\ &\quad\quad{} +\frac{1 }{1-\sigma} \int_{0}^{1}(1-s)g(s)\sum _{0< t_{k}< s} \bigl[v^{*}_{k}\Vert x\Vert _{\operatorname {PC}}+\overline{v}_{k}\bigl\Vert x'\bigr\Vert _{\operatorname {PC}} \bigr]\,ds \\ &\quad\quad{} +\frac{1 }{1-\sigma} \int_{0}^{1}g(s)\sum_{s< t_{k}< 1} \bigl[v_{k}\Vert x\Vert _{\operatorname {PC}}+v^{*}_{k}\Vert x\Vert _{\operatorname {PC}}+\overline{v}_{k}\bigl\Vert x' \bigr\Vert _{\operatorname {PC}} \bigr]\,ds \\ &\quad \leq\frac{1}{1-\sigma}\varphi^{-1} \bigl(\varphi (1-\sigma )- \varepsilon_{2} \bigr) \int_{0}^{1}G(s,s)\,ds \\ &\quad\quad{} \times \varphi^{-1} \bigl(\varphi \bigl(\Vert x\Vert _{\operatorname {PC}}+\bigl\Vert x' \bigr\Vert _{\operatorname {PC}}+\Vert Tx\Vert _{\operatorname {PC}}+\Vert Sx\Vert _{\operatorname {PC}} \bigr) \bigr) \\ &\quad\quad{} +\sum_{k=1}^{m} \bigl[v^{*}_{k} \Vert x\Vert _{\operatorname {PC}}+\overline{v}_{k}\bigl\Vert x'\bigr\Vert _{\operatorname {PC}} \bigr] +\sum _{k=1}^{m} \bigl[v_{k}\Vert x\Vert _{\operatorname {PC}}+v^{*}_{k}\Vert x\Vert _{\operatorname {PC}}+\overline {v}_{k}\bigl\Vert x'\bigr\Vert _{\operatorname {PC}} \bigr] \\ & \quad\quad{} +\frac{1 }{1-\sigma} \int_{0}^{1}(1-s)g(s)\sum _{k=1}^{m} \bigl[v^{*}_{k}\Vert x\Vert _{\operatorname {PC}}+\overline{v}_{k}\bigl\Vert x'\bigr\Vert _{\operatorname {PC}} \bigr]\,ds \\ &\quad\quad{} +\frac{1 }{1-\sigma} \int_{0}^{1}g(s)\sum_{k=1}^{m} \bigl[v_{k}\Vert x\Vert _{\operatorname {PC}}+v^{*}_{k}\Vert x\Vert _{\operatorname {PC}}+\overline{v}_{k}\bigl\Vert x' \bigr\Vert _{\operatorname {PC}} \bigr]\,ds \\ &\quad \leq\frac{1}{1-\sigma}\varphi^{-1} \bigl(\varphi (1-\sigma ) \bigr) \int_{0}^{1}(1-s)\,ds\times\varphi^{-1} \bigl(\varphi \bigl(\Vert x\Vert _{\operatorname {PC}}+\bigl\Vert x' \bigr\Vert _{\operatorname {PC}}+\Vert Tx\Vert _{\operatorname {PC}}+\Vert Sx\Vert _{\operatorname {PC}} \bigr) \bigr) \\ &\quad\quad{} +\sum_{k=1}^{m} \bigl[v^{*}_{k} \Vert x\Vert _{\operatorname {PC}}+\overline{v}_{k}\bigl\Vert x'\bigr\Vert _{\operatorname {PC}} \bigr] +\sum _{k=1}^{m} \bigl[v_{k}\Vert x\Vert _{\operatorname {PC}}+v^{*}_{k}\Vert x\Vert _{\operatorname {PC}}+\overline {v}_{k}\bigl\Vert x'\bigr\Vert _{\operatorname {PC}} \bigr] \\ & \quad\quad{} +\frac{1 }{1-\sigma} \int_{0}^{1}(1-s)g(s)\sum _{k=1}^{m} \bigl[v^{*}_{k}\Vert x\Vert _{\operatorname {PC}}+\overline{v}_{k}\bigl\Vert x'\bigr\Vert _{\operatorname {PC}} \bigr]\,ds \\ & \quad\quad{}+\frac{1 }{1-\sigma} \int_{0}^{1}g(s)\sum_{k=1}^{m} \bigl[v_{k}\Vert x\Vert _{\operatorname {PC}}+v^{*}_{k}\Vert x\Vert _{\operatorname {PC}}+\overline{v}_{k}\bigl\Vert x' \bigr\Vert _{\operatorname {PC}} \bigr]\,ds \\ &\quad \leq\frac{1 }{1-\sigma}\times(1-\sigma)\times\frac{1}{2}\times \Vert x \Vert _{\operatorname {PC}} +\Vert x\Vert _{\operatorname {PC}}\sum _{k=1}^{m} \bigl[v^{*}_{k}+ \overline{v}_{k} \bigr] +\Vert x\Vert _{\operatorname {PC}}\sum _{k=1}^{m} \bigl[v_{k}+ v^{*}_{k}+ \overline{v}_{k} \bigr] \\ & \quad\quad{}+\frac{\sigma \Vert x\Vert _{\operatorname {PC}} }{1-\sigma}\sum_{k=1}^{m} \bigl[v^{*}_{k}+\overline{v}_{k} \bigr] +\frac{\sigma \Vert x\Vert _{\operatorname {PC}} }{1-\sigma} \sum_{k=1}^{m} \bigl[v_{k}+ v^{*}_{k}+\overline{v}_{k} \bigr] \\ &\quad \leq\frac{1}{2}\Vert x\Vert _{\operatorname {PC}} +\lambda \Vert x\Vert _{\operatorname {PC}}< \frac{1}{2}\Vert x\Vert _{\operatorname {PC}}+ \frac{1}{2}\Vert x\Vert _{\operatorname {PC}}=\Vert x\Vert _{\operatorname {PC}}=R_{1}. \end{aligned}$$
(3.1)
It is worth noticing that the inequality of (3.1) is strict. So \(Ax\neq x\) for all \(\Vert x\Vert _{\operatorname {PC}}=R_{1}\). By way of (2) of Lemma 2.1, we obtain
$$\begin{aligned} i(A,K_{R_{1}}\cap K,K)=1. \end{aligned}$$
(3.2)
Therefore, A has at least one positive fixed point on \(x\in K_{R_{1}}\cap K\), namely, \(\Vert x\Vert _{\operatorname {PC}}\leq R_{1}\) and \(\Vert Ax\Vert _{\operatorname {PC}}>\delta \Vert A\Vert _{\operatorname {PC}}\). This positive fixed point x of A is a solution of BVP (1.1). □

Similar to the proof of 3.1, we have Theorem 3.2.

Theorem 3.2

Assume that \((\mathrm{H}_{1})\)-\((\mathrm{H}_{3})\) hold. If \(0<\lambda<1\) and \(0< f^{0}<\varphi (\frac{2(1-\lambda)}{\rho} )\), for all \(\rho>\frac{1}{1-\lambda}\), then BVP (1.1) has at least one positive solution, where \(\lambda=\frac{1}{1-\sigma}\sum_{k=1}^{m} [v_{k}+2v^{*}_{k}+2\overline {v}_{k} ]\).

Next similarly we prove some corollaries for BVP (1.3) and (1.4).

Corollary 3.1

Assume that \((\mathrm{H}'_{1})\)-\((\mathrm{H}'_{3})\) hold.
\((\mathrm{H}'_{1})\)

\(0\leq\sigma=\int_{0}^{1}g(s)\,ds<1\) and \(g\in L^{1}[0,1]\) is nonnegative.

\((\mathrm{H}'_{2})\)

For any bounded set \(B_{i}\subset\mathbb{E}\) (\(i=1,2,3,4\)), \(f(t,B_{1},B_{2},B_{3},B_{4})\) and \(I_{k}(B_{1})\) together with \(\overline{I}_{k}(B_{1},B_{2})\) (\(k=1,2,\ldots,m\)) are relatively compact sets, where \(f(t,B_{1},B_{2},B_{3},B_{4}):=\{f(t,w_{1},w_{2},w_{3},w_{4}):w_{i}\in B_{i},i=1,2,3,4\}\), \(I_{k}(B_{1}):=\{I_{k}(w_{1}):w_{i}\in B_{1}\}\) and \(\overline{I}_{k}(B_{1},B_{2}):=\{\overline{I}_{k}(w_{1},w_{2}):w_{i}\in B_{i},i=1,2\}\).

\((\mathrm{H}'_{3})\)
For \(I_{k}\in C(\mathbb{P},\mathbb {P}^{+})\), \(\overline{I}_{k}\in C(\mathbb{P}^{2},\mathbb{P}^{+})\) (\(\mathbb{P}^{+}\) represents a positive cone in a real Banach space \(\mathbb{E}\), there exist some positive constants \(v_{k}\), \(v^{\ast}_{k}\) and \(\overline{v}_{k}\) (\(k=1,\ldots,m\)) such that
$$\begin{aligned} \bigl\Vert I_{k}(x)\bigr\Vert \leq v_{k}\Vert x \Vert , \quad\quad \bigl\Vert \overline{I}_{k}(x_{1},x_{2}) \bigr\Vert \leq v^{\ast}_{k}\Vert x_{1}\Vert + \overline{v}_{k}\Vert x_{2}\Vert , \quad \forall t\in J. \end{aligned}$$
If \(0< f^{\infty}<1-\sigma\) and \(\lambda<\frac{1}{2}\), then BVP (1.3) has at least one positive solution, where \(\lambda=\frac{1}{1-\sigma}\sum_{k=1}^{m} [v_{k}+2v^{*}_{k}+2\overline {v}_{k} ]\).

Corollary 3.2

Assume that \((\mathrm{H}'_{1})\)-\((\mathrm{H}'_{3})\) hold. If \(0<\lambda<1\) and \(0< f^{0}<\frac{2(1-\lambda)}{\rho}\), for all \(\rho>\frac{1}{1-\lambda}\), then BVP (1.3) has at least one positive solution, where \(\lambda=\frac{1}{1-\sigma}\sum_{k=1}^{m} [v_{k}+2v^{*}_{k}+2\overline {v}_{k} ]\).

Corollary 3.3

Assume that \((\mathrm{H}''_{1})\)-\((\mathrm{H}''_{3})\) hold.
\((\mathrm{H}''_{1})\)

\(0\leq\sigma=\int_{0}^{1}g(s)\,ds<1\) and \(g\in L^{1}[0,1]\) is nonnegative.

\((\mathrm{H}''_{2})\)

For any bounded set \(B_{i}\subset\mathbb{E}\) (\(i=1,2,3,4\)), \(f(t,B_{1},B_{2},B_{3},B_{4})\) and \(I_{k}(B_{1})\) together with \(\overline{I}_{k}(B_{1},B_{2})\) (\(k=1,2,\ldots,m\)) are relatively compact sets, where \(f(t,B_{1},B_{2},B_{3},B_{4})\triangleq\{f(t,w_{1},w_{2},w_{3},w_{4}):w_{i}\in B_{i},i=1,2,3,4\}\), \(I_{k}(B_{1}):=\{I_{k}(w_{1}):w_{i}\in B_{1}\}\) and \(\overline{I}_{k}(B_{1},B_{2}):=\{\overline{I}_{k}(w_{1},w_{2}):w_{i}\in B_{i},i=1,2\}\).

\((\mathrm{H}''_{3})\)
For \(I_{k}\in C(\mathbb{P},\mathbb {P}^{+})\), \(\overline{I}_{k}\in C(\mathbb{P}^{2},\mathbb{P}^{+})\) (\(\mathbb{P}^{+}\) represents a positive cone in a real Banach space \(\mathbb{E}\), there exist some positive constants \(v_{k}\), \(v^{\ast}_{k}\) and \(\overline{v}_{k}\) (\(k=1,\ldots,m\)) such that
$$\begin{aligned} \bigl\Vert I_{k}(x)\bigr\Vert \leq v_{k}\Vert x \Vert , \quad\quad \bigl\Vert \overline{I}_{k}(x_{1},x_{2}) \bigr\Vert \leq v^{\ast}_{k}\Vert x_{1}\Vert + \overline{v}_{k}\Vert x_{2}\Vert ,\quad \forall t\in J. \end{aligned}$$
If \(0< f^{\infty}< (1-\sigma )^{p-1}\) and \(\lambda<\frac{1}{2}\), then BVP (1.4) has at least one positive solution, where \(\lambda=\frac{1}{1-\sigma}\sum_{k=1}^{m} [v_{k}+2v^{*}_{k}+2\overline {v}_{k} ]\).

Corollary 3.4

Assume that \((\mathrm{H}''_{1})\)-\((\mathrm{H}''_{3})\) hold. If \(0<\lambda<1\) and \(0< f^{0}< (\frac{2(1-\lambda)}{\rho} )^{p-1}\), for all \(\rho>\frac{1}{1-\lambda}\), then BVP (1.4) has at least one positive solution, where \(\lambda=\frac{1}{1-\sigma}\sum_{k=1}^{m} [v_{k}+2v^{*}_{k}+2\overline {v}_{k} ]\).

4 Illustrative example

Consider the following boundary value problem:
$$\begin{aligned} \left \{ \textstyle\begin{array}{l} (\varphi(-x''(t)))'=f(t,x(t),x'(t),(Tx)(t),(Sx)(t)), \quad t\in J, t\neq\frac{1}{4},\\ \Delta x|_{t_{1}=\frac{1}{4}}=I_{k}(x(\frac{1}{4})), \quad k=1,\ldots,m,\\ -\Delta x'|_{t_{1}=\frac{1}{4}}=\overline{I}_{k}(x(\frac {1}{4}),x'(\frac{1}{4})), \quad k=1,\ldots,m,\\ x(0)=x''(0)=\theta, \quad\quad x(1)=\int_{0}^{1}g(t)x(t)\,dt, \end{array}\displaystyle \right . \end{aligned}$$
(4.1)
where \(m=1\), \(t_{1}=\frac{1}{4}\), \(I_{1}(u)=\frac{u}{120}\), \(\overline{I}_{1}(u,v)=\frac{(u^{2}e^{-v}+v^{2}e^{-u})}{120(1+\vert uv\vert )}\), \(g(t)=t\), \(K(t,s)=e^{-ts}\), \(F(t,s)=e^{-2s}\), \(f(t,x,u,v,w)=\frac{\ln(3+t^{2})+(t^{3}\vert x\vert +t\vert u\vert +\vert v\vert +\vert w\vert )^{3}}{ 1+26(\vert x\vert +\vert u\vert +\vert v\vert +\vert w\vert )}\),
$$\varphi(u)=\left \{ \textstyle\begin{array}{l@{\quad}l} u^{3}, & u< 0, \\ u^{2}, & u>0. \end{array}\displaystyle \right . $$

Let \(\mathbb{E}=\mathbb{R}\) with the norm \(\Vert x\Vert =\vert x\vert \) and \(\mathbb{P}=\{x\in\mathbb{R}:x\geq0\}\).

Choose \(\delta=\frac{1}{5}\in(0,\frac{1}{2})\). By a simple computation, we get \(0<\sigma=\int_{0}^{1}g(s)\,ds=\frac{1}{2}<1\), \(\vert I_{1}(u)\vert =\frac{\vert u\vert }{120}\), \(\vert \overline{I}_{1}(u,v)\vert \leq\frac{\vert u\vert +\vert v\vert }{120}\), \(v_{1}=v_{1}^{*}=\overline{v}_{1}=\frac{1}{120}\). For all \(t,s\in[0,1]\), \(0< K(s,t)<1\), \(0< F(t,s)<1\),
$$\begin{aligned}& \varphi^{-1}(u)=\left \{ \textstyle\begin{array}{l@{\quad }l} \sqrt[3]{u}, & u< 0, \\ \sqrt{u}, & u>0, \end{array}\displaystyle \right . \\& \begin{aligned} f^{\infty}&=\limsup_{\vert x\vert +\vert u\vert +\vert v\vert +\vert w\vert \rightarrow v}\max_{t\in[0,1]} \frac{\vert f(t,x,u,v,w)\vert }{\varphi(\vert x\vert +\vert u\vert +\vert v\vert +\vert w\vert )} \\ &=\limsup_{\vert x\vert +\vert u\vert +\vert v\vert +\vert w\vert \rightarrow \infty}\max_{t\in[0,1]}\frac{\ln (3+t^{2})+(t^{3}\vert x\vert +t\vert u\vert +\vert v\vert +\vert w\vert )^{3}}{[1+26(\vert x\vert +\vert u\vert +\vert v\vert +\vert w\vert )](\vert x\vert +\vert u\vert +\vert v\vert +\vert w\vert )^{2}} \\ &=\frac{1}{26}< \frac{1}{4}=\varphi (1-\sigma ), \end{aligned} \\& \lambda=\frac{1}{1-\sigma}\sum_{k=1}^{m} \bigl[v_{k}+2v^{*}_{k}+2\overline {v}_{k} \bigr]= \frac{1}{1-\sigma} \bigl[v_{1}+2v^{*}_{1}+2\overline{v}_{1} \bigr]=\frac{1}{12}< \frac{1}{2}. \end{aligned}$$
Thus all conditions of Theorem 3.1 are satisfied. It follows from Theorem 3.1 that for BVP (4.1) there exists at least one positive solution.

Declarations

Acknowledgements

The authors would like to thank the anonymous referees for their useful and valuable suggestions. This work was supported by the National Natural Sciences Foundation of the People’s Republic of China under Grant (No. 11161025, No. 11661047), Yunnan Province natural scientific research fund project (No. 2011FZ058).

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Department of Applied Mathematics, Kunming University of Science and Technology

References

  1. Liang, SH, Zhang, JH: Existence of three positive solutions of three-order with m-point impulsive boundary value problems. Acta Appl. Math. 110, 1265-1292 (2010) MathSciNetView ArticleGoogle Scholar
  2. Wu, CZ, Teo, KL, Zhou, Y, Yan, WY: An optimal control problem involving impulsive integro-differential systems. Optim. Methods Softw. 22, 531-549 (2007) MathSciNetView ArticleMATHGoogle Scholar
  3. Wu, CZ, Teo, KL, Zhou, Y, Yan, WY: Solving an identification problem as an impulsive optimal parameter selection problem. Comput. Math. Appl. 50, 217-229 (2005) MathSciNetView ArticleMATHGoogle Scholar
  4. Sun, Y: Positive solutions of Sturm-Liouville boundary value problems for singular nonlinear second-order impulsive integro-differential equation in Banach spaces. Bound. Value Probl. 2012, 86 (2012) MathSciNetView ArticleMATHGoogle Scholar
  5. Guo, DJ: Boundary value problems for impulsive integro-differential equations on unbounded domains in a Banach space. Appl. Math. Comput. 99, 1-15 (1999) MathSciNetMATHGoogle Scholar
  6. Guo, DJ: Extremal solutions for nth order impulsive integro-differential equations on the half-line in Banach space. Nonlinear Anal. 65, 677-696 (2006) MathSciNetView ArticleMATHGoogle Scholar
  7. Guo, DJ: Existence of positive solutions for nth order nonlinear impulsive singular integro-differential equations in a Banach space. Nonlinear Anal. 68, 2727-2740 (2008) MathSciNetView ArticleMATHGoogle Scholar
  8. Guo, DJ: Existence of solutions for nth order impulsive integro-differential equations in Banach space. Nonlinear Anal. 44, 1339-1356 (2002) Google Scholar
  9. Guo, DJ: Multiple positive solutions for nth order impulsive integro-differential equations in Banach space. Nonlinear Anal. 60, 955-976 (2005) MathSciNetView ArticleMATHGoogle Scholar
  10. Guo, DJ: Periodic boundary value problem for second order impulsive integro-differential equations in Banach spaces. Nonlinear Anal. 28, 938-997 (1997) View ArticleGoogle Scholar
  11. Guo, D: Existence of solutions for nth-order impulsive integro-differential equations in a Banach space. Nonlinear Anal. 47, 741-752 (2001) MathSciNetView ArticleMATHGoogle Scholar
  12. Xu, Y: A global solutions of initial value problems for second order impulsive integro differential equations in Banach spaces. Acta Math. Sin. 25(1), 47-56 (2005) View ArticleGoogle Scholar
  13. Liu, L, Xu, Y, Wu, Y: On unique solution of an initial value problem for nonlinear first-order impulsive integro-differential equations of Volterra type in Banach spaces. Dyn. Contin. Discrete Impuls. Syst., Ser. A Math. Anal. 13, 641-652 (2006) MathSciNetMATHGoogle Scholar
  14. Liu, L, Wu, Y, Zhang, X: On well-posedness of an initial value problem for nonlinear second order impulsive integro differential equations of Volterra type in Banach spaces. J. Math. Anal. Appl. 317, 634-649 (2006) MathSciNetView ArticleMATHGoogle Scholar
  15. Zhang, X, Liu, L: Initial value problem for nonlinear second order impulsive integro differential equations of mixed type in Banach spaces. Nonlinear Anal. 64, 2562-2574 (2006) MathSciNetView ArticleMATHGoogle Scholar
  16. Liu, L, Wu, C, Guo, F: A unique solution of initial value problems for first order impulsive integro-differential equations of mixed type in Banach spaces. J. Math. Anal. Appl. 275, 369-385 (2002) MathSciNetView ArticleMATHGoogle Scholar
  17. Zhang, X, Feng, M, Ge, W: Existence of solutions of boundary value problems with integral boundary conditions for second-order impulsive integro-differential equations in Banach spaces. J. Comput. Appl. Math. 233, 1915-1926 (2010) MathSciNetView ArticleMATHGoogle Scholar
  18. Feng, M, Pang, H: A class of three-point boundary-value problems for second-order impulsive integro-differential equations in Banach spaces. Nonlinear Anal. 70, 64-82 (2009) MathSciNetView ArticleMATHGoogle Scholar
  19. Zhao, KH, Gong, P: Positive solutions for impulsive fractional differential equations with generalized periodic boundary value conditions. Adv. Differ. Equ. 2014, 255 (2014) MathSciNetView ArticleMATHGoogle Scholar
  20. Zhao, KH: Multiple positive solutions of integral BVPs for high-order nonlinear fractional differential equations with impulses and distributed delays. Dyn. Syst. 30(2), 208-223 (2015) MathSciNetView ArticleMATHGoogle Scholar
  21. Zhao, KH: Impulsive boundary value problems for two classes of fractional differential equation with two different Caputo fractional derivatives. Mediterr. J. Math. 13, 1033-1050 (2016) MathSciNetView ArticleMATHGoogle Scholar
  22. Zhao, KH: Impulsive integral boundary value problems of the higher-order fractional differential equation with eigenvalue arguments. Adv. Differ. Equ. 2015, 382 (2015) MathSciNetView ArticleGoogle Scholar
  23. Fu, D, Ding, W: Existence of positive solutions of the fourth-order boundary value problems with integral boundary conditions in Banach spaces. Adv. Differ. Equ. 2013, 65 (2013) MathSciNetView ArticleGoogle Scholar
  24. Webb, JRL, Infante, G: Positive solutions of nonlocal boundary value problems involving integral conditions. Nonlinear Differ. Equ. Appl. 15(1-2), 45-67 (2008) MathSciNetView ArticleMATHGoogle Scholar
  25. Deimling, K: Nonlinear Functional Analysis. Springer, Berlin (1985) View ArticleMATHGoogle Scholar
  26. Guo, D, Lakshmikantham, V, Liu, XZ: Nonlinear Integral Equations in Abstract Spaces. Kluwer Academic, Dordrecht (1996) View ArticleMATHGoogle Scholar

Copyright

© Zhao and Wang 2016