Abstract
In this paper, the author discusses the existence of two positive solutions for an infinite boundary value problem of second order impulsive singular integro-differential equations on the half line by means of the fixed point theorem of cone expansion and compression with norm type.
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1 Introduction
The theory of impulsive differential equations has been emerging as an important area of investigation in recent years (see [1–3]). Many problems have been investigated for impulsive differential equations, impulsive functional differential equations and impulsive differential inclusions. These problems include existence of solutions, stability theory, geometric properties, applications, etc. There is a vast literature on existence of solutions: by using upper and lower solutions together with the monotone iterative technique to obtain the extremal solutions [4–8]; by using fixed point theorems to obtain the existence of solution and multiple solutions [9–14]; by using the Leray-Schauder degree theory or fixed point index theory to obtain multiple solutions [15–19]; by using the variational method to obtain the existence of solution and existence of infinite many solutions [20–25]. In recent article [14], the author discussed the existence of two positive solutions for an infinite boundary value problem of first order impulsive singular integro-differential equations on the half line by means of the fixed point theorem of cone expansion and compression with norm type, which was established by the author in [26] (see also [27–30]). Now, in this article, we shall discuss such problem for a class of second order equations. The discussion for second order equations is more complicated than the first order case. We must introduce a new Banach space and a new cone in it to control both the unknown function and its derivative so that we can still use the fixed point theorem of cone expansion and compression with norm type.
Consider the infinite boundary value problem (IBVP) for second order impulsive singular integro-differential equation of mixed type on the half line:
where R denotes the set of all real numbers, \(R_{+}=\{x\in R: x\geq 0\}\), \(R_{++}=\{x\in R: x>0\}\), \(0< t_{1}<\cdots<t_{k}<\cdots\), \(t_{k}\rightarrow\infty\), \(R'_{++}=R_{++}\backslash\{t_{1},\ldots ,t_{k},\ldots\}\), \(f\in C[R_{++}\times R_{++}\times R_{++}\times R_{+}\times R_{+},R_{+}]\), \(I_{k}, \bar{I}_{k}\in C[R_{++},R_{+}]\) (\(k=1,2,3,\ldots\)), \(\beta>1\), \(u'(\infty)=\lim_{t\rightarrow \infty}u'(t)\) and
\(K\in C[D,R_{+}]\), \(D=\{(t,s)\in R_{+}\times R_{+}: t\geq s\}\), \(H\in C[R_{+}\times R_{+},R_{+}]\). \(\Delta u |_{t=t_{k}}\) and \(\Delta u' |_{t=t_{k}}\) denote the jumps of \(u(t)\) and \(u'(t)\) at \(t=t_{k}\), respectively, i.e.
where \(u(t_{k}^{+})\) and \(u(t_{k}^{-})\) represent the right and left limits of \(u(t)\) at \(t=t_{k}\), respectively, and \(u'(t_{k}^{+})\) and \(u'(t_{k}^{-})\) represent the right and left limits of \(u'(t)\) at \(t=t_{k}\), respectively. In what follows, we always assume that
and
i.e. \(f(t,u,v,w,z)\) is singular at \(t=0, u=0\) and \(v=0\). We also assume that
and
i.e. \(I_{k}(v)\) and \(\bar{I}_{k}(v)\) (\(k=1,2,3,\ldots\)) are singular at \(v=0\). Let \(\mathit{PC}[R_{+},R]\) = {\(u: u\) is a real function on \(R_{+}\) such that \(u(t)\) is continuous at \(t\neq t_{k}\), left continuous at \(t=t_{k}\), and \(u(t^{+}_{k})\) exists, \(k=1,2,3,\ldots\)} and \(\mathit{PC}^{1}[R_{+},R]\) = {\(u\in \mathit{PC}[R_{+},R]: u'(t)\) is continuous at \(t\neq t_{k}\), and \(u'(t_{k}^{+})\) and \(u'(t_{k}^{-})\) exist for \(k=1,2,3,\ldots\)}. Let \(u\in \mathit{PC}^{1}[R_{+},R]\). For \(0< h<t_{k}-t_{k-1}\), by the mean value theorem, there exists \(t_{k}-h<\xi_{k} <t_{k}\) such that
hence the left derivative of \(u(t)\) at \(t=t_{k}\), which is denoted by \(u'_{-}(t_{k})\), exists, and
In what follows, it is understood that \(u'(t_{k})=u'_{-}(t_{k})\). So, for \(u\in \mathit{PC}^{1}[R_{+},R]\), we have \(u'\in \mathit{PC}[R_{+},R]\).
A function \(u\in \mathit{PC}^{1}[R_{+},R]\cap C^{2}[R'_{++},R]\) is called a positive solution of IBVP (1) if \(u(t)>0\) for \(t\in R_{++}\) and \(u(t)\) satisfies (1). Now, we need to introduce a new space \(\mathit{DPC}^{1}[R_{+},R]\) and a new cone Q in it. Let
It is easy to see that \(\mathit{DPC}^{1}[R_{+},R]\) is a Banach space with the norm
where
Let \(W=\{u\in \mathit{DPC}^{1}[R_{+},R]: u(t)\geq0, u'(t)\geq0, \forall t\in R_{+}\} \) and
Obviously, W and Q are two cones in the space \(\mathit{DPC}^{1}[R_{+},R]\) and \(Q\subset W\) (for details on cone theory, see [28]). Let \(Q_{+}=\{u\in Q: \|u\|_{D}>0\}\) and \(Q_{pq}=\{u\in Q: p\leq\|u\|_{D}\leq q\}\) for \(q>p>0\).
2 Several lemmas
Remark 1
(a) For \(u\in \mathit{DPC}^{1}[R_{+},R]\), we have \(u(0)=0\). This is clear since \(u(0)\neq0\) implies
(b) For \(u\in Q_{+}\), we have \(u(t)>0\) for \(t\in R_{++}\) and \(u'(t)>0\) for \(t\in R_{+}\).
Lemma 1
For \(u\in Q\), we have
and
Proof
Since (8) implies (9) and (8) and (9) imply (10), we need only to show (8).
For fixed \(0< t<t_{1}\), observing \(u(0)=0\) and by the mean value theorem, there exists \(0<\xi<t\) such that
So,
On the other hand, for any \(0< t<t_{1}\), we have
so,
hence,
□
Let us list some conditions.
(H1) \(\sup_{t\in J}\int_{0}^{t}K(t,s)s\, ds<\infty\), \(\sup_{t\in J}\int_{0}^{\infty}H(t,s)s\, ds<\infty\) and
In this case, let
(H2) There exist \(a,b\in C[R_{++},R_{+}]\), \(g\in C[R_{++},R_{+}]\) and \(G\in C[R_{++}\times R_{+}\times R_{+},R_{+}]\) such that
and
for any \(r>0\), where
and
(H3) \(I_{k}(v)\leq t_{k}\bar{I}_{k}(v)\), \(\forall v\in R_{++}\) (\(k=1,2,3,\ldots\)), and there exist \(\gamma_{k}\in R_{+}\) (\(k=1,2,3,\ldots\)) and \(F\in C[R_{++},R_{+}]\) such that
and
and, consequently,
It is clear: if condition (H3) is satisfied, then (6) implies (7).
(H4) There exists \(c\in C[R_{++},R_{++}]\) such that
uniformly for \(t,u\in R_{++}\), \(w,z\in R_{+}\), and
(H5) There exists \(d\in C[R_{++},R_{++}]\) such that
uniformly for \(t,u\in R_{++}\), \(w,z\in R_{+}\), and
Remark 2
It is clear: if condition (H1) is satisfied, then the operators T and S defined by (2) are bounded linear operators from \(\mathit{DPC}^{1}[R_{+},R]\) into \(\mathit{BC}[R_{+},R]\) (the Banach space of all bounded continuous functions \(u(t)\) on \(R_{+}\) with the norm \(\|u\|_{B}=\sup_{t\in R_{+}}|u(t)|\)) and \(\|T\|\leq k^{*}\), \(\|S\|\leq h^{*}\); moreover, we have \(T(\mathit{DPC}^{1}[R_{+},R_{+}])\subset \mathit{BC}[R_{+},R_{+}]\) (\(\mathit{BC}[R_{+},R_{+}]=\{u\in \mathit{BC}[R_{+},R]: u(t)\geq0, \forall t\in R_{+}\}\)) and \(S(\mathit{DPC}^{1}[R_{+},R_{+}])\subset \mathit{BC}[R_{+},R_{+}]\).
Remark 3
Condition (H4) means that the function \(f(t,u,v,w,z)\) is superlinear with respect to v.
Remark 4
Condition (H5) means that the function \(f(t,u,v,w,z)\) is singular at \(v=0\) and it is stronger than (5).
Remark 5
In what follows, we need the following two formulas (see [6], Lemma 1):
-
(a)
If \(u\in \mathit{PC}[R_{+},R]\cap C^{1}[R'_{++},R]\), then
$$ u(t)=u(0)+\int_{0}^{t}u'(s)\,ds+\sum _{0< t_{k}<t}\bigl[u\bigl(t_{k}^{+}\bigr)-u \bigl(t_{k}^{-}\bigr)\bigr], \quad \forall t\in R_{+}. $$(11) -
(b)
If \(u\in \mathit{PC}^{1}[R_{+},R]\cap C^{2}[R'_{++},R]\), then
$$\begin{aligned} u(t) =&u(0)+tu'(0)+\int_{0}^{t}(t-s)u''(s) \,ds \\ &{}+\sum_{0< t_{k}<t} \bigl\{ \bigl[u\bigl(t_{k}^{+} \bigr)-u\bigl(t_{k}^{-}\bigr)\bigr]+(t-t_{k}) \bigl[u'\bigl(t_{k}^{+}\bigr)-u' \bigl(t_{k}^{-}\bigr)\bigr] \bigr\} ,\quad \forall t\in R_{+}. \end{aligned}$$(12)
We shall reduce IBVP (1) to an impulsive integral equation. To this end, we first consider operator A defined by
In what follows, we write \(J_{1}=[0,t_{1}]\), \(J_{k}=(t_{k-1},t_{k}]\) (\(k=2,3,4,\ldots\)).
Lemma 2
If conditions (H1)-(H3) are satisfied, then operator A defined by (13) is a continuous operator from \(Q_{+}\) into Q; moreover, for any \(q>p>0\), \(A(Q_{pq})\) is relatively compact.
Proof
Let \(u\in Q_{+}\) and \(\|u\|_{B}=r\). Then \(r>0\) and, by (10) and Remark 1(a),
By conditions (H1), (H2) and (14), we have (for \(k^{*}\), \(h^{*}\), \(a(t)\), \(g(u)\), \(b(t)\), \(G(v,w,z)\), \(g_{r}(t)\) and \(a_{r}^{*}\), \(b^{*}\), see conditions (H1) and (H2))
where
which implies the convergence of the infinite integral
and
On the other hand, by condition (H3) and (14), we have
where
which implies the convergence of the infinite series
and
In addition, from (13) we get
Moreover, by condition (H3), we have
so, (13) gives
On the other hand, by (13), we have
so,
and
It follows from (13), (21)-(25) that \(Au\in Q\), i.e. \(Au\in \mathit{DPC}^{1}[R_{+},R]\) and
and, by (17), (20), (22) and (25),
Thus, we have proved that A maps \(Q_{+}\) into Q.
Now, we are going to show that A is continuous. Let \(u_{n},\bar{u}\in Q_{+}\), \(\|u_{n}-\bar{u}\|_{D}\rightarrow0\) (\(n\rightarrow\infty\)). Write \(\|\bar{u}\|_{D}=2\bar{r}\) (\(\bar{r}>0\)) and we may assume that
and
By (13), we have
When \(0< t\leq t_{1}\), we have
so,
It follows from (30) and (31) that
It is clear that
and, similar to (15) and observing (28), we have
where
It is easy to see that condition (H2) implies
for any \(q>p>0\), where
So,
and therefore,
It follows from (33), (34), (37) and the dominated convergence theorem that
On the other hand, similar to (18) and observing (29), we have
where
For any given \(\epsilon>0\), by (39) and condition (H3), we can choose a positive integer \(k_{0}\) such that
and
so,
and
It is clear that
and
so, we can choose a positive integer \(n_{0}\) such that
and
and
hence
and
It follows from (32), (38), (46) and (47) that
On the other hand, from (23) it is easy to get
It follows from (48) and (50) that \(\|Au_{n}-A\bar{u}\|_{D}\rightarrow0\) as \(n\rightarrow\infty\), and the continuity of A is proved.
Finally, we prove that \(A(Q_{pq})\) is relatively compact, where \(q>p>0\) are arbitrarily given. Let \(\bar{u}_{n}\in Q_{pq}\) (\(n=1,2,3,\ldots\)). Then, by (10),
Similar to (15), (18), (26) and observing (51), we have
and
where \(g_{pq}(t)\) and \(a_{pq}^{*}\) are defined by (36) and (35), respectively, and
From (54) we see that functions \(\{(A\bar{u}_{n})(t)\}\) (\(n=1,2,3,\ldots\)) are uniformly bounded on \([0,r]\) for any \(r>0\). On the other hand, by (13) and (52)-(54), we have
which implies that functions \(\{w_{n}(t)\}\) (\(n=1,2,3,\ldots\)) defined by (for any fixed k)
(\((A\bar{u}_{n})(t_{k-1}^{+})\) denotes the right limit of \((A\bar{u}_{n})(t)\) at \(t=t_{k-1}\)) are equicontinuous on \(\bar{J}_{k}=[t_{k-1},t_{k}]\). Consequently, by the Ascoli-Arzela theorem, \(\{w_{n}(t)\}\) has a subsequence which is convergent uniformly on \(\bar{J}_{k}\). So, functions \(\{A\bar{u}_{n}(t)\}\) (\(n=1,2,3,\ldots\)) have a subsequence which is convergent uniformly on \(J_{k}\). Now, by the diagonal method, we can choose a subsequence \(\{(A\bar{u}_{n_{i}})(t)\}\) (\(i=1,2,3,\ldots\)) of \(\{ (A\bar{u}_{n})(t)\}\) (\(n=1,2,3,\ldots\)) such that \(\{(A\bar{u}_{n_{i}})(t)\}\) (\(i=1,2,3,\ldots\)) is convergent uniformly on each \(J_{k}\) (\(k=1,2,3,\ldots\)). Let
Similarly, we can discuss \(\{(A\bar{u}_{n})'(t)\}\) (\(n=1,2,3,\ldots\)). Similar to (27) and by (23), we have
and
and by a similar method, we can prove that \(\{(A\bar{u}_{n_{i}})'(t)\}\) (\(n=1,2,3,\ldots\)) has a subsequence which is convergent uniformly on each \(J_{k}\) (\(k-1,2,3,\ldots\)). For the sake of simplicity of notation, we may assume that \(\{(A\bar{u}_{n_{i}})'(t)\}\) (\(i=1,2,3,\ldots\)) itself converges uniformly on each \(J_{k}\) (\(k=1,2,3,\ldots\)). Let
By (55), (57) and the uniformity of convergence, we have
and so, \(\bar{w}\in \mathit{PC}^{1}[R_{+},R]\). From (54) and (56), we get
and
Consequently, \(\bar{w}\in \mathit{DPC}^{1}[R_{+},R]\) and
Let \(\epsilon>0\) be arbitrarily given. Choose a sufficiently large positive number η such that
For any \(\eta< t<\infty\), we have, by (23), (52) and (53),
which implies by virtue of (59) that
Letting \(i\rightarrow\infty\) in (60) and observing (57) and (58), we get
On the other hand, since \(\{(A\bar{u}_{n_{i}})'(t)\}\) converges uniformly to \(\bar{w}'(t)\) on \([0,\eta]\) as \(i\rightarrow\infty\), there exists a positive integer \(i_{0}\) such that
It follows from (60)-(62) that
hence
It is clear that (13) implies
By virtue of the uniformity of convergence of \(\{(A\bar{u}_{n_{i}})(t)\}\), we see that
so, (65) implies that
exist and
Let
Then \(\alpha_{k}\geq0\) (\(k=1,2,3,\ldots\)) and
By (53) and condition (H3), we have
so,
For any given \(\epsilon>0\), choose a sufficiently large positive integer \(k_{0}\) such that
and then, choose another sufficiently large integer \(i_{1}\) such that
It follows from (67)-(70) that
hence
By formula (11) and (65), (66), we have
and
which imply
Since
(72) implies
By (64), (71) and (73), we have
It follows from (64) and (74) that \(\|A\bar{u}_{n_{i}}-\bar{w}\|_{D}\rightarrow0\) as \(i\rightarrow\infty\), and the relative compactness of \(A(Q_{pq})\) is proved. □
Lemma 3
Let conditions (H1)-(H3) be satisfied. Then \(u\in Q_{+}\cap C^{2}[R'_{++},R]\) is a positive solution of IBVP (1) if and only if \(u\in Q_{+}\) is a solution of the following impulsive integral equation:
i.e. u is a fixed point of operator A defined by (13).
Proof
If \(u\in Q_{+}\cap C^{2}[R'_{++},R]\) is a positive solution of IBVP (1), then, by (1) and formula (12), we have
Differentiation of (76) gives
Under conditions (H1)-(H3), we have shown in the proof of Lemma 2 that the infinite integral (15) and the infinite series (19) are convergent. So, by taking limits as \(t\rightarrow\infty\) in both sides of (77) and using the relation \(u'(\infty)=\beta u'(0)\), we get
Now, substituting (78) into (76), we see that \(u(t)\) satisfies equation (75).
Conversely, if \(u\in Q_{+}\) is a solution of equation (75), then direct differentiation of (75) twice gives
and
So, \(u\in C^{2}[R'_{++},R]\) and
Moreover, taking limits as \(t\rightarrow\infty\) in (79), we see that \(u'(\infty)\) exists and
Hence, \(u(t)\) is a positive solution of IBVP (1). □
Lemma 4
(Fixed point theorem of cone expansion and compression with norm type, see Corollary 1 in [26] or Theorem 3 in [27] or Theorem 2.3.4 in [28], see also [29, 30])
Let P be a cone in a real Banach space E and \(\Omega_{1}\), \(\Omega_{2}\) be two bounded open sets in E such that \(\theta\in\Omega_{1}\), \(\bar{\Omega}_{1}\subset \Omega_{2}\), where θ denotes the zero element of E and \(\bar{\Omega}_{i} \) denotes the closure of \(\Omega_{i}\) (\(i=1,2\)). Let the operator \(A: P\cap(\bar{\Omega}_{2}\backslash\Omega_{1})\rightarrow P\) be completely continuous (i.e. continuous and compact). Suppose that one of the following two conditions is satisfied:
-
(a)
\(\|Ax\|\leq\|x\|\), \(\forall x\in P\cap\partial\Omega_{1} \); \(\|Ax\|\geq\|x\|\), \(\forall x\in P\cap\partial\Omega_{2}\), where \(\partial\Omega_{i} \) denotes the boundary of \(\Omega_{i}\) (\(i=1,2\)).
-
(b)
\(\|Ax\|\geq\|x\|\), \(\forall x\in P\cap\partial\Omega_{1} \); \(\|Ax\|\leq\|x\|\), \(\forall x\in P\cap\partial\Omega_{2}\).
Then A has at least one fixed point in \(P\cap(\bar{\Omega} _{2}\backslash\Omega_{1})\).
3 Main theorem
Theorem
Let conditions (H1)-(H5) be satisfied. Assume that there exists \(r>0 \) such that
where \(a_{r}^{*}\), \(b^{*}\) and \(\gamma^{*}\) are defined in conditions (H1) and (H2), and, \(G_{r}\) and \(N_{r}\) are defined by two equalities below (15) and (18), respectively. Then IBVP (1) has at least two positive solutions \(u^{*},u^{**}\in Q_{+}\cap C^{2}[R'_{++},R]\) such that
and
Proof
By Lemma 2 and Lemma 3, operator A defined by (13) is continuous from \(Q_{+}\) into Q, and we need to prove that A has two fixed points \(u^{*}\) and \(u^{**}\) in \(Q_{+}\) such that \(0<\|u^{*}\|_{D}<r<\|u^{**}\|_{D}\).
By condition (H4), there exists \(r_{1}>0\) such that
Choose
For \(u\in Q\), \(\|u\|_{D}=r_{2}\), we have, by (10) and (82),
and consequently,
hence
By condition (H5), there exists \(r_{3}>0\) such that
Choose
For \(u\in Q, \|u\|_{D}=r_{4}\), we have, by (10) and (85),
hence
and consequently,
On the other hand, for \(u\in Q\), \(\|u\|_{D}=r\), (26) and (27) imply
Thus, from (80) and (87), we get
By (82) and (85) we know \(0< r_{4}<r<r_{2}\), and, by Lemma 2, operator A is completely continuous from \(Q_{r_{4}r_{2}}\) into Q. Hence, (83), (86), (88) and Lemma 4 imply that A has two fixed points \(u^{*}, u^{**}\in Q_{r_{4}r_{2}}\) such that \(r_{4}<\|u^{*}\|_{D}<r<\|u^{**}\|_{D}\leq r_{2}\). The proof is complete. □
Example
Consider the infinite boundary value problem for second order impulsive singular integro-differential equation of mixed type on the half line:
Conclusion
IBVP (89) has at least two positive solutions \(u^{*},u^{**}\in \mathit{PC}^{1}[R_{+},R]\cap C^{2}[R'_{++},R]\) such that
and
Proof
System (89) is an IBVP of form (1). In this situation, \(t_{k}=k\) (\(k=1,2,3,\ldots\)), \(K(t,s)=e^{-(t+2)s}\), \(H(t,s)=(1+t+s)^{-3}\), \(\beta=2\), and
It is clear that (3)-(7) are satisfied, so, (89) is a singular problem. It is easy to see that condition (H1) is satisfied and \(k^{*}\leq1\), \(h^{*}\leq1\). We have
so, condition (H2) is satisfied for
and
with
and
It is obvious that condition (H3) is satisfied for \(\gamma _{k}=k^{-1}3^{-k-4}\) (\(\gamma^{*}=\frac{1}{162}\)) and \(F(v)=v^{-\frac {1}{2}}\). From
and
we see that conditions (H4) and (H5) are satisfied for
and
respectively. Finally, we check that inequality (80) is satisfied for \(r=1\), i.e.
and
Moreover, it is easy to get
Hence
Consequently, (92) holds, and our conclusion follows from the theorem. □
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Guo, D. Existence of two positive solutions for a class of second order impulsive singular integro-differential equations on the half line. Bound Value Probl 2015, 76 (2015). https://doi.org/10.1186/s13661-015-0337-1
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DOI: https://doi.org/10.1186/s13661-015-0337-1