Abstract
In this paper, we investigate nonlinear second-order double impulsive differential equations integral boundary value problem with p-Laplacian on an infinite interval with the infinite number of impulsive times. Based on the cone theory and monotone iterative technique, we establish the existence of minimal nonnegative solution and iteration of positive solutions for such a boundary value problem. The main results are new and extend the existing results. At last, some examples are worked out to demonstrate the use of the main results.
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1 Introduction
Boundary value problems on infinite intervals appear often in applied mathematics and physics, for example, in the study of the unsteady flow of a gas through semi-infinite porous medium, in analyzing the heat transfer in radial flow between circular disks, in the study of plasma physics, and in an analysis of the mass transfer on a rotating disk in non-Newtonian fluid, see [1, 2] and the references therein. For extensive applications, this kind of BVPs attract lots of scholars to devote themselves to developing them. Scholars do some work and apply many techniques to deal with such problems, see [3–10] and the references therein. While boundary value problems with integral boundary conditions for ordinary differential equations on an infinite interval also arise in different fields such as heat conduction, chemical engineering, underground water flow, thermoelasticity, and plasma physics. In the past few years, many people have started to be active in studying the existence of the solutions to nonlinear integral boundary value problems (IBVPs) on infinite intervals. Some conclusions appeared in the meantime, see [11–18] and the references therein.
Considering the theory of impulsive differential equations, it has been emerging as an important area of investigation in recent years and has been extensively applied in chemical technology, population dynamics, and so on. It is much wider because all the structure of its emergence has deep physical background and realistic mathematical model and coincides with many phenomena in nature. For an introduction of the basic theory of impulsive differential equations in \(R^{n}\), see [19–21] and the references therein.
We notice that there has been increasing interest in studying nonlinear differential equation and impulsive integro differential equation on an infinite interval with an infinite number of impulsive times to identify a few, see [22–26] and the references therein. There are relatively few papers available for integral boundary value problems for impulsive differential equations on an infinite interval with an infinite number of impulsive times up to now, see [27–31] and the references therein.
Recently, in [32], Zhang et al. investigated the existence of minimal nonnegative solution for the following second-order impulsive differential equation IBVP:
where \(f\in C(J\times J\times J,J)\), \(I_{k}, \overline{I}_{k} \in C(R,R)\), \(J=[0,+\infty)\), \(0=t_{0}< t_{1}<\cdots<t_{k}<\cdots\), \(t_{k}\rightarrow\infty\) for \(k=1,2,\ldots\) , and \(g(t)\in L[J,J]\) with \(0< \int_{0}^{+\infty}g(t)\,dt<1\). \(\Delta x|_{t_{k}}\) denotes the jump of \(x(t)\) at \(t=t_{k}\), that is,
where \(x(t_{k}^{+})\) and \(x(t_{k}^{-})\) represent the right-hand limit and left-hand limit of \(x(t)\) at \(t=t_{k}\), respectively, \(\Delta x'|_{t_{k}}\) has a similar meaning to \(x'(t)\).
More recently, in [33], Zhang studied the existence and iteration of positive solutions for nonlinear second-order impulsive IBVP with p-Laplacian on infinite intervals
where \(\phi_{p}(s)=|s|^{p-2}s\), \(p>1\), \(J=[0,+\infty)\), \(J_{+}=(0,+\infty)\), \(J'_{+}=J_{+}\backslash\{t_{1},t_{2},\ldots,t_{k},\ldots \}\), \(0< t_{1}< t_{2}<\cdots<t_{k}<\cdots\), \(t_{k}\rightarrow\infty\) for \(k=1,2,\ldots \) , and \(g(t)\in L[J,J]\) with \(\int_{0}^{+\infty}g(t)\,dt<1\), \(\int_{0}^{+\infty}tg(t)\,dt<\infty\), and \(0 \leq x'(\infty)=\lim_{t\rightarrow+\infty}x'(t)\).
However, to the authors’ knowledge, the corresponding theory for double impulsive integral boundary value problems with p-Laplacian operator and infinite impulsive times on infinite intervals has not been considered till now. Motivated by the above mentioned works, in this paper, we study the existence of solutions for nonlinear double impulsive IBVPs of second-order differential equations with p-Laplacian on an infinite interval
where \(\phi_{p}(s)=|s|^{p-2}s\), \(p>1\), \(\eta\geq0\) is a constant, \(f\in C(J^{3},J)\), \(J=[0,+\infty)\), \(0=t_{0}< t_{1}<\cdots<t_{k}<\cdots\), \(t_{k}\rightarrow\infty\) for \(k=1,2,\ldots \) and note \(J_{0}=[0,t_{1}]\) and \(J_{i}=(t_{i},t_{i+1}] \) (\(i=1,2,\ldots\)), \(g(t)\in L^{1}[J,J]\) with \(\int_{\eta}^{+\infty}g(t)\,dt<1\), \(\int_{\eta }^{+\infty}tg(t)\,dt<+\infty\), \(I_{k}, \overline{I}_{k} \in C(R,R)\), and
It is clear that
Throughout this paper, we adopt the following assumptions:
- (H1):
-
Suppose that \(f\in C[J\times J\times J,J]\), and there exist \(p,q,r\in C(J,J)\) such that
$$f(t,u,v)\leq p(t)\phi_{p}(u)+q(t)\phi_{p}(v)+r(t), \quad \forall t\in J\mbox{ and } \forall u,v\in J, $$and note
$$\begin{aligned}& p^{*}= \int_{0}^{+\infty }a(t)p(t) (1+t)^{p-1}\,dt< +\infty,\qquad q^{*}= \int _{0}^{+\infty}a(t)q(t)\,dt< +\infty, \\& r^{*}= \int_{0}^{+\infty}a(t)r(t)\,dt< + \infty. \end{aligned}$$ - (H2):
-
\(I_{k},\overline{I}_{k}\in C(J,J)\) and there exist nonnegative constants \(a_{k}\geq0\), \(b_{k}\geq0\), \(c_{k}\geq0\), \(d_{k}\geq0\) such that
$$\begin{aligned}& 0\leq I_{k}(u)\leq a_{k}u+b_{k}, \quad\forall u \in J\ (k=1,2,3,\ldots), \\& 0\leq\overline{I}_{k}(u)\leq c_{k}\phi _{p}(u)+d_{k}, \quad \forall u\in J\ (k=1,2,3,\ldots) \end{aligned}$$and note
$$\begin{aligned}& a^{*}=\sum_{k=1}^{\infty}(t_{k}+1)a_{k}< + \infty,\qquad b^{*}=\sum_{k=1}^{\infty}b_{k}< + \infty, \\& c^{*}=\sum_{k=1}^{\infty }(t_{k}+1)^{p-1}c_{k}< + \infty,\qquad d^{*}=\sum_{k=1}^{\infty}d_{k}< + \infty, \end{aligned}$$with \(a^{*}<(1/3)(1-\int_{\eta}^{\infty}g(t)\,dt)\) and \(c^{*}<\phi_{p}(1/(3n))\), where n is a constant and it firstly appears in (3.12).
- (H3):
-
\(f(t,u_{1},v_{1})\leq f(t,u_{2},v_{2})\), \(I_{k}(u_{1})\leq I_{k}(u_{2})\), \(\overline{I}_{k}(u_{1})\leq \overline{I}_{k}(u_{2})\) for \(t\in J\), \(u_{1}\leq u_{2}\), \(v_{1}\leq v_{2}\) (\(k=1,2,3,\ldots\)).
- (H4):
-
Suppose that \(f\in C[J\times J\times J,J]\), \(f(t,0,0)\neq0\) on any subinterval of J, and u, v are bounded, \(f(t,(1+t)u,v)\) is bounded on J.
- (H5):
-
\(a(t)\) is a nonnegative measurable function defined in \(J\backslash\{0\}\), and \(a(t)\) does not identically vanish on any subinterval of \(J\backslash\{0\}\), and
$$\begin{aligned}& 0< \int_{0}^{+\infty}a(t)\,dt< +\infty,\qquad 0< \int _{0}^{+\infty}\phi_{p}^{-1}\biggl( \int_{t}^{+\infty}a(s)\,ds\biggr)\,dt< +\infty. \end{aligned}$$
2 Preliminary results
In this section, we firstly present some definitions and lemmas, which will be needed in the proof of the main results.
Definition 2.1
Let E be a real Banach space. A nonempty closed set \(P\subset E\) is said to be a cone provided that
-
(1)
\(au+bv\in P\) for all \(u,v\in P\) and all \(a\geq0\), \(b\geq0\),
-
(2)
\(u,-u\in P\) implies that \(u=0\).
Definition 2.2
A map \(\alpha:P\rightarrow[0,+\infty)\) is said to be concave on P if
Definition 2.3
(see [8])
Let \(V=\{x\in X:\|x\|< l\}\) (\(l>0\)), \(V_{1}:=\{\frac{x(t)}{1+t},x\in V\}\cup\{x'(t),x\in V\} \) is called equiconvergent at infinity if and only if for all \(\varepsilon>0\), there exists \(T=T(\varepsilon)>0 \) such that for all \(x\in V_{1}\), the following holds:
Lemma 2.4
If \(\{\frac{x(t)}{1+t},x\in V\}\) and \(\{x'(t),x\in V\}\) are both equicontinuous on any compact intervals of \([0,+\infty)\) and equiconvergent at infinity, then V is relatively compact on X.
Definition 2.5
Let \(J'=J\setminus\{t_{1},t_{2},\ldots t_{k},\ldots\}\), \(x\in E\cap C^{2}[J',R]\) is called a nonnegative solution of IBVP (1.1) if \(x(t)\geq0\), \(x'(t)\geq0\) and \(x(t)\) satisfies IBVP (1.1). Moreover, \(\overline{x}(t)\) is called a minimal nonnegative solution if x is an arbitrary nonnegative solution of (1.1), then \(x(t)\geq \overline{x}(t)\), \(x'(t)\geq\overline{x}'(t)\) for all \(t\in J'\).
Let
with the norm \(\|x\|=\max\{\|x\|_{1},\|x'\|_{\infty}\}\), where \(\|x\|_{1}=\sup_{t\in J}\frac{|x(t)|}{1+t}\), \(\|x'\|_{\infty}=\sup_{t\in J}|x'(t)|\). At the same time, define a cone \(P\subset E\) by
Lemma 2.6
Suppose that (H1), (H2) hold. Then, for all \(x\in P\), \(\int_{0}^{+\infty}a(t)f(t,x(t),x'(t))\,dt\), \(\sum_{k=1}^{\infty}I_{k}(x(t_{x}))\) and \(\sum_{k=1}^{\infty}\overline{I}_{k}(x(t_{x}))\) are convergent.
Proof
By (H1) and (H2), we have
Thus
The proof is complete. □
Lemma 2.7
Let \(y(t)\in L^{1}[0,+\infty)\) and \(\int_{\eta}^{+\infty}g(t)\,dt<1\), then the IBVP
has a unique solution
Proof
For \(t\in[0,t_{1}]\), integrating (2.1) from 0 to t, we have
That is,
which implies that
For \(t\in[t_{1},t_{2}]\), integrating (2.1) from \(t_{1}\) to t, we have
That is,
Adding (2.4) and (2.5) together, we have
Repeating the previous process, for any \(t\in[0,+\infty)\), we get that
Taking limit for \(t\rightarrow+\infty\), by the boundary condition, we have
For \(t\in[0,t_{1}]\), integrating (2.8) from 0 to t, we have
which implies that
For \(t\in[t_{1},t_{2}]\), integrating (2.8) from \(t_{1}\) to t, we have
Adding (2.10) and (2.11) together, we have
Repeating the previous process, for any \(t\in[0,+\infty)\), we get that
By (2.13) and the boundary condition, for any \(t\in[0,+\infty)\), we have
This completes the proof. □
Define an integral operator \(T:P\rightarrow E\) by
Obviously, T is well defined and \(x\in PC(J,R)\) is a solution of BVP (1.1) if and only if x is a fixed point of T.
Lemma 2.8
Assume that (H1)-(H3) hold. Then the operator T maps P into P, and
Moreover, for \(x,y\in P\) with \(x(t)\leq y(t)\), \(x'(t)\leq y'(t)\), for all \(t\in J\), and one has
where
Proof
Let \(x\in P\). From the definition of T, (H1)-(H3) and (2.15), we can obtain that T is an operator from P to P, and
Therefore,
Direct differentiation of T implies, for \(t\neq t_{k}\),
Thus, we have
It follows that (2.15) is satisfied and equation (2.16) is easily obtained by (H3). □
Lemma 2.9
Let (H2), (H4), and (H5) hold. Then \(T:P\rightarrow P\) is completely continuous.
Proof
For any \(x\in P\), by (2.14), we have
It follows from (2.14), (2.19) and (H4) that
that is, \(T(P)\subset P\). Next, we divide the proof into two steps.
Step 1. We prove that T is continuous.
Let \(x_{n}\rightarrow x\) as \(n\rightarrow\infty\) in P, then there exists \(r_{0}\) such that \(\sup_{n\in N\setminus\{0\}}\|x\|< r_{0}\). Set \(B_{r_{0}}=\sup\{f(t,(1+t)u,v), (t,u,v)\in J\times [0,r_{0}]^{2}\}\), and we have
Therefore, by the Lebesgue dominated convergence theorem, we have
From above and (H4), (H5), we get
and
We can easily get \(\|Tx_{n}-Tx\|\rightarrow 0\) (\(n\rightarrow\infty\)). Hence, T is continuous.
Step 2. We prove that T is compact provided that it maps bounded sets into relatively compact sets.
First, let Ω be a bounded subset of P, then there exists \(r>0\) such that \(\|x\|< r\) for all \(x\in\Omega\). By (2.18), we have
and
where \(R_{1}=\frac{1}{1-\int_{\eta}^{+\infty}g(t)\,dt} [(2+\alpha\int_{\eta}^{+\infty}g(t)\,dt)R_{0}+a^{*}r+b^{*} ]\). Hence, \(\|Tx\|\leq\max\{R_{0},R_{1}\}\). So TΩ is bounded.
Second, for any \(L\in(0,+\infty)\) and \(t', t''\in J_{k}\cap[0,L]\) with \(t'< t''\), we have
and
for all \(x\in\Omega\). So TΩ is equicontinuous on any compact interval of \(J_{k}\) (\(k=1,2,\ldots\)).
Third, we prove that for any \(\varepsilon>0\), \(x\in\Omega\), there exists sufficiently large \(N>0\) such that
For any \(x\in\Omega\), we have
and
Hence, we obtain that
and
So TΩ is equiconvergent at infinity. By Lemma 2.4, we obtain TΩ is relatively compact, that is, T is a compact operator.
Therefore, \(T:P\rightarrow P\) is completely continuous. The proof is complete. □
Remark 2.10
Similarly, we may prove that when (H1) and (H2) hold, then \(T:P\rightarrow P\) is completely continuous.
3 Main result
Theorem 3.1
Let conditions (H1)-(H3) be satisfied. Suppose further that \(A<1\). Then IBVP (1.1) has the minimal nonnegative solution x̅ with \(\|\overline{x}\|\leq\frac{B}{1-A}\), where A and B are defined as in Lemma 2.8. Moreover, if we let \(x_{0}(t)=0\), \(x_{n}(t)=(Tx_{n-1}(t))\) for all \(t\in J\) (\(n=1,2,\ldots\)), then \(x_{n}(t)\in P\) with
and \(\{x_{n}(t)\}\) and \(\{x_{n}'(t)\}\) converge uniformly to \(\overline{x}(t)\) and \(\overline{x}'(t)\) on \(J_{i} \) (\(i=1,2,\ldots\)), respectively.
Proof
By Lemma 2.8 and the definition of operator T, we have \(x_{n}(t)\in P\) and
By (3.3), we can get
From (3.4)-(3.6), we know that \(\lim_{n\rightarrow+\infty}x_{n}(t)\) and \(\lim_{n\rightarrow+\infty}x'_{n}(t)\) exist. Suppose that
According to the definition of \(x_{n}(t)\), we have
From (3.6), we obtain
It follows that \({x_{n}(t)}\) is equicontinuous on every \(J_{i}\) (\(i=0,1,2,\ldots\)). Combining this with the Ascoli-Arzela theorem and diagonal process, there exists a subsequence which converges uniformly to x̅ on \(J_{i}\) (\(i=0,1,2,\ldots\)), which together with (3.4) imply that \({x_{n}(t)}\) converges uniformly to \(\overline{x}(t)\) on \(J_{i} \) (\(i=0,1,2,\ldots\)), and \(\overline{x}(t)\in PC[J,R]\), \(\|\overline{x}\|_{1}\leq \frac{B}{1-A}\). On the other hand, by (H1), (3.6) and (3.9), we have
Since \(s(t)\) is bounded on \([0,M]\) (M is a finite positive number), \({x'_{n}(t)}\) is equicontinuous on every \(J_{i} \) (\(i=0,1,2,\ldots\)). Combining this with the Ascoli-Arzela theorem and diagonal process, there exists a subsequence which converges uniformly to \(h(t)\) on \(J_{i}\) (\(i=0,1,2,\ldots\)), which together with (3.5) imply that \({x'_{n}(t)}\) converges uniformly to \(h(t)\) on \(J_{i}\) (\(i=0,1,2,\ldots\)), and \(h(t)\in PC[J,R]\), \(\|h\|_{\infty}\leq \frac{B}{1-A}\). From above, we know that \(\overline{x}'(t)\) exists and \(\overline{x}'(t)=h(t)\) for all \(t\in J\). It follows that \(\overline{x}\in P\) and \(\|\overline{x}\|\leq\frac{B}{1-A}\). Now taking limits from two sides of \(x_{n}(t)=(Tx_{n-1})(t)\), we have \(\overline{x}(t)=(T\overline{x})(t)\), that is, T has a fixed point. By Lemma 2.7, \(\overline{x}(t)\) is a nonnegative solution of IBVP (1.1).
Suppose that \(x\in P\cap C^{2}[J, R]\) is an arbitrary nonnegative solution of IBVP (1.1). Then \(x(t)=(Tx)(t)\). It is clear that \(x(t)\geq0\), \(x'(t)\geq0\), \(\forall t\in J\). Suppose that \(x(t)\geq x_{n-1}(t)\), \(x'(t)\geq x'_{n-1}(t)\) for \(t\in J\). By (2.14), we have \((Tx)(t)\geq(Tx_{n-1})(t)\), \((Tx)'(t)\geq(Tx_{n-1})'(t)\) for all \(t\in J\). This means that \(x(t)\geq x_{n}(t)\), \(x'(t)\geq x'_{n}(t)\) for all \(t\in J\) (\(n=1,2,\ldots\)). Taking limit, we have \(x(t)\geq \overline{x}(t)\), \(x'(t)\geq\overline{x}'(t)\) for all \(t\in J\). The proof of Theorem 3.1 is complete. □
Next, for notational convenience, we denote that
Theorem 3.2
Assume that (H2)-(H5) hold, and there exists
such that
- (A1):
-
$$\begin{aligned} f\bigl(t,(1+t)u,v\bigr)\leq \left \{ \textstyle\begin{array}{@{}l@{\quad}l} \phi_{p}(\frac{d}{3m}), & p\geq2, \\ \phi_{p}(\frac{d}{3m'}),& 1< p< 2, \end{array}\displaystyle \displaystyle \displaystyle \right . \quad\textit{for }(t,u,v)\in[0,+\infty)\times[0,d]\times[0,d]. \end{aligned}$$
Then IBVP (1.1) admits positive, nondecreasing on \([0,+\infty)\) and concave solutions \(w^{*}\) and \(v^{*}\) such that \(0<\|w^{*}\|\leq d\), and \(\lim_{n\rightarrow+\infty}w_{n}=\lim_{n\rightarrow+\infty}A^{n}w_{0}=w^{*}\), where
and \(0<\|v^{*}\|\leq d\), and \(\lim_{n\rightarrow+\infty}v_{n}=\lim_{n\rightarrow+\infty}A^{n}v_{0}=v^{*}\), where \(v_{0}(t)=0\), \(t\in J\).
Proof
We only prove the case that \(p\geq2\), another case can be proved in a similar way. By Lemma 2.9, we know that \(T:P\rightarrow P\) is completely continuous. From the definition of T and (H3), we can easily get that \(Tx_{1}\leq Tx_{2}\) for any \(x_{1},x_{2}\in P\) with \(x_{1}\leq x_{2}\), \(x'_{1}\leq x'_{2}\). Denote that
In what follows, we first prove that \(T:\overline{P}_{d}\rightarrow \overline{P}_{d}\). If \(x\in\overline{P}_{d}\), then \(\|x\|\leq d\). By (1.3), (2.18), (3.10), (3.12) and (3.14), we get that
and
Thus, we get that \(\|Tx\|\leq d\). Hence, we have proved that \(T:\overline{P}_{d}\rightarrow\overline{P}_{d}\).
Let \(w_{0}(t)=d+dt\), \(0\leq t<+\infty\), then \(w_{0}(t)\in \overline{P}_{d}\). Let \(w_{1}(t)=Tw_{0}(t)\), \(w_{2}(t)=T^{2}w_{0}(t)\), then by Lemma 2.9, we have \(w_{1}(t)\in\overline{P}_{d}\) and \(w_{2}(t)\in \overline{P}_{d}\). Denote that
Since \(T:\overline{P}_{d}\rightarrow\overline{P}_{d}\), we have that
It follows from the complete continuity of T that \(\{w_{n}\}_{n=1}^{\infty}\) has a convergent subsequence \(\{w_{n_{k}}\}_{k=1}^{\infty}\), and there exists \(w^{*}\in \overline{P}_{d}\) such that \(w_{n_{k}}\rightarrow w^{*}\).
By (3.18), (H3) and (A1), we get that
and
So, by (3.18), (H3) and (A1), we have
By induction, we see
Hence, we claim that \(w_{n}\rightarrow w^{*}\) as \(n\rightarrow \infty\). Applying the continuity of T and \(w_{n+1}(t)=Tw_{n}(t)\), we know \(Tw^{*}=w^{*}\). Let \(v_{0}=0\), \(0\leq t<+\infty\), then \(v_{0}(t)\in\overline{P}_{d}\). Let \(v_{1}=Tv_{0}\), \(v_{2}=T^{2}v_{0}\). By Lemma 2.9, we get \(v_{1}\in \overline{P}_{d}\) and \(v_{2}\in\overline{P}_{d}\). Denote
Since \(T:\overline{P}_{d}\rightarrow\overline{P}_{d}\), we have \(v_{n}\in T(\overline{P}_{d})\subset \overline{P}_{d}\), \(n=1,2,\ldots \) . It follows from the complete continuity of T that \(\{v_{n}\}_{n=1}^{\infty}\) is a sequentially compact set. Furthermore, we assert that \(\{v_{n}\}_{n=1}^{\infty}\) has a convergent subsequence \(\{v_{n_{k}}\}_{k=1}^{\infty}\), and there exists \(v^{*}\in\overline{P}_{d}\) such that \(v_{n_{k}}\rightarrow v^{*}\).
For \(v_{1}=Tv_{0}\in\overline{P}_{d}\), we obtain
By (H3) and (A1), we have
By induction, we see
Hence, we claim that \(v_{n}\rightarrow v^{*}\) as \(n\rightarrow \infty\). Applying the continuity of T and \(v_{n+1}(t)=Tv_{n}(t)\), we know \(Tv^{*}=v^{*}\).
Since \(f(t,0,0)\neq0\), \(0\leq t<+\infty\), then the zero function is not the solution of IBVP (1.1). Thus, \(v^{*}\) is a positive solution of IBVP (1.1). By Lemma 2.7, we know that \(w^{*}\) and \(v^{*}\) are positive, nondecreasing on \([0,\infty)\) and concave solutions of IBVP (1.1).
We can easily get that the theorem holds for \(1< p<2\) in a similar way. □
Theorem 3.3
Assume that (H2)-(H5) hold, and there exists
such that
- (A2):
-
$$\begin{aligned} &f\bigl(t,(1+t)u,v\bigr)\leq \left \{ \textstyle\begin{array}{@{}l@{\quad}l} \phi_{p}(\frac{d_{k}}{3m}), & p\geq2, \\ \phi_{p}(\frac{d_{k}}{3m'}), & 1< p< 2, \end{array}\displaystyle \displaystyle \displaystyle \right .\\ &\quad\textit{for }(t,u,v)\in[0,+\infty)\times[0,d_{k}]\times [0,d_{k}]. \end{aligned}$$
Then the boundary value problem (1.3) admits positive, nondecreasing on \([0,+\infty)\) and concave solutions \(w_{k}^{*}\) and \(v_{k}^{*}\) such that \(0<\|w_{k}^{*}\|\leq d_{k}\), and \(\lim_{n\rightarrow+\infty}w_{kn}=\lim_{n\rightarrow+\infty }A^{n}w_{k0}=w_{k}^{*}\), where
and \(0<\|v_{k}^{*}\|\leq d\), and \(\lim_{n\rightarrow+\infty}v_{kn}=\lim_{n\rightarrow+\infty }A^{n}v_{k0}=v_{k}^{*}\), where \(v_{0}(t)=0\), \(t\in J\).
Proof
It is similar to the proof of Theorem 3.2. □
4 Example
Example 4.1
Consider the following IBVP for double impulsive differential equation with p-Laplacian on an infinite interval:
Here, \(p=3\), \(a(t)=e^{-t}\), \(f(t,x(t),x'(t))=\frac{\ln(1+\phi_{p}(x))}{100(1+t)^{2}}+\frac {e^{t}\arctan(\phi_{p}(x'))}{100(1+t^{2})}+1\), \(I_{k}(x(t_{k}))=\frac{1}{4^{k}}(x(t_{k})+1)^{\frac{3}{16}}\), \(\overline{I}_{k}(x(t_{k}))=\frac{1}{5^{k}}(\frac{\ln(\phi _{p}(x(t_{k})))}{25(1+2^{k})^{2}}+1)\), \(g(t)=\frac{1}{2}e^{-2t}\), \(\eta=1\). Evidently, \(x(t)=0\) is not the solution of IBVP (4.1).
It is clear that \(\int_{1}^{+\infty}\frac{1}{2}e^{-2t}\,dt<1\) and (H3) is satisfied. Since
So we have
Then we easily obtain that
Thus, (H1) and (H2) are satisfied. Clearly, \(A\approx0.8166< 1\). By Theorem 3.1, we obtain that BVP (4.1) has a minimal positive solution x̅ and \(\|\overline{x}\|\leq19.9189\).
Example 4.2
Consider the following IBVP for double impulsive differential equation with p-Laplacian on an infinite interval:
Here,
It is clear that (H2) and (H4) hold for \(p=3\), \(a(t)=e^{-4t}\), \(g(t)=4/(1+t)^{4}\). By direct computation, we obtain that
which implies that (H5) holds.
Obviously,
Hence, we can obtain that
Take \(d=13\). In this case, we have
On the other hand, nonlinear term f satisfies
which means that (A2) holds. Thus, we have checked that all the conditions of Theorem 3.2 are satisfied. Therefore, we obtain that IBVP (4.2) has two iteration positive solutions.
References
Agarwal, RP, O’Regan, D: Infinite Interval Problems for Differential, Difference and Integral Equations. Kluwer Academic, Dordrecht (2001)
O’Regan, D: Theory of Singular Boundary Value Problems. World Scientific, River Edge (1994)
Agarwal, RP, O’Regan, D: Nonlinear boundary value problems on the semi-infinite interval: an upper and lower solution approach. Mathematica 49, 129-140 (2002)
Baxley, JV: Existence and uniqueness for nonlinear boundary value problems on infinite interval. J. Math. Anal. Appl. 147, 127-133 (1990)
Chen, SZ, Zhang, Y: Singular boundary value problems on a half-line. J. Math. Anal. Appl. 195, 449-468 (1995)
Ma, R: Existence of positive solution for second-order boundary value problems on infinite interval. Appl. Math. Lett. 16, 33-39 (2003)
Lian, H, Pang, H, Ge, W: Triple positive solutions for boundary value problems on infinite intervals. Nonlinear Anal., Theory Methods Appl. 67, 2199-2207 (2007)
Guo, Y, Yu, C, Wang, J: Existence of three positive solutions for m-point boundary value problems on infinite intervals. Nonlinear Anal., Theory Methods Appl. 71, 717-722 (2009)
Rachunková, I, Tomeček, J: Superlinear singular problems on the half line. Bound. Value Probl. 2010, 429813 (2010)
Yu, C, Li, Z, Wei, H, Wang, J: Existence and uniqueness of solutions for second-order m-point boundary value problems at resonance on infinite interval. J. Hebei Univ. Sci. Technol. 34(1), 7-14 (2013)
Karakostas, GL, Tsamatos, PC: Multiple positive solutions of some Fredholm integral equations arisen from nonlocal boundary-value problems. Electron. J. Differ. Equ. 2002, 30 (2002)
Webb, JRL, Infante, G: Positive solutions of nonlocal boundary value problems involving integral conditions. Nonlinear Differ. Equ. Appl. 15, 45-67 (2008)
Yang, Z: Existence and nonexistence results for positive solutions of an integral boundary value problem. Nonlinear Anal., Theory Methods Appl. 65, 1489-1511 (2006)
Yang, Z: Positive solutions of a second-order integral boundary value problem. J. Math. Anal. Appl. 321, 751-765 (2006)
Feng, M: Existence of symmetric positive solutions for a boundary value problem with integral boundary conditions. Appl. Math. Lett. 24, 1419-1427 (2011)
Ahmad, B, Alsaedi, A, Alghamdi, BS: Analytic approximation of solutions of the forced Duffing equation with integral boundary conditions. Nonlinear Anal., Real World Appl. 9, 1727-1740 (2008)
Yu, C, Wang, J, Guo, Y: Positive solutions for boundary-value problems with integral boundary conditions on infinite interval. Electron. J. Differ. Equ. 2012, 158 (2012)
Zhang, X, Ge, W: Symmetric positive solutions of boundary value problems with integral boundary conditions. Appl. Math. Comput. 219, 3553-3564 (2012)
Lakshimkantham, V, Baĭnov, DD, Simeonov, PS: Theory of Impulsive Differential Equations. Series in Modern Applied Mathematics, vol. 6. World Scientific, Teaneck (1989)
Baĭnov, DD, Simeonov, PS: Systems with Impulse Effect: Stability, Theory and Application. Ellis Horwood Series: Mathematics and Its Applications. Ellis Horwood, Chichester (1989)
Samoĭlenko, BAM, Perestyuk, NA: Impulsive Differential Equations. World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises, vol. 14. World Scientific, River Edge (1995)
Guo, D, Liu, X: Impulsive integro-differential equations on unbounded domain in a Banach space. Nonlinear Stud. 3, 49-57 (1996)
Guo, D: Boundary value problems for impulsive integro-differential equations on unbounded domains in a Banach space. Appl. Math. Comput. 99, 1-15 (1999)
Guo, D: Second order impulsive integro-differential equations on unbounded domains in Banach spaces. Nonlinear Anal., Theory Methods Appl. 35, 413-423 (1999)
Guo, D: Multiple positive solutions for first order nonlinear impulsive integro-differential equations in a Banach space. Appl. Math. Comput. 143, 233-249 (2003)
Li, J, Shen, J: Existence of positive solution for second-order impulsive boundary value problems on infinity intervals. Bound. Value Probl. 2006, 14594 (2006)
Benchohra, M, Henderson, J, Ntouyas, SK: Impulsive Differential Equations and Inclusions. Hindawi Publishing Corporation, New York (2006)
Nieto, JJ, Rodríguez-López, R: New comparison results for impulsive integro-differential equations and applications. J. Math. Anal. Appl. 328, 1343-1368 (2007)
Liu, X: Solutions of impulsive boundary value problems on the half-line. J. Math. Anal. Appl. 222, 411-430 (1998)
Li, J, Nieto, JJ: Existence of positive solutions for multipoint boundary value problem on the half-line with impulses. Bound. Value Probl. 2009, 834158 (2009)
Jiang, W: The existence of solutions for impulsive p-Laplacian boundary value problems at resonance on the half-line. Bound. Value Probl. 2015, 39 (2015)
Zhang, X, Yang, X, Feng, M: Minimal nonnegative solution of nonlinear impulsive differential equations on infinite interval. Bound. Value Probl. 2011, 684542 (2011)
Zhang, X: Computation of positive solutions for nonlinear impulsive integral boundary value problems with p-Laplacian on infinite intervals. Abstr. Appl. Anal. 2013, 708281 (2013)
Liu, Y: A boundary value problem for a second-order differential equation on unbounded domain. Acta Anal. Funct. Appl. 4, 211-216 (2002)
Corduneanu, C: Integral Equations and Stability of Feedback Systems. Academic Press, New York (1973)
Acknowledgements
Supported by the Natural Science Foundation of China (11201112), the Natural Science Foundation of Hebei Province (A2013208147), (A2014208152) and (A2015208114), the Foundation of Hebei Education Department (Z2014062) and (QN2015175).
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Yu, C., Wang, J. & Guo, Y. Positive solutions for nonlinear double impulsive differential equations with p-Laplacian on infinite intervals. Bound Value Probl 2015, 147 (2015). https://doi.org/10.1186/s13661-015-0409-2
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DOI: https://doi.org/10.1186/s13661-015-0409-2