The method of upper and lower solutions to impulsive differential equations with integral boundary conditions

The theory of impulsive differential equations describes processes which experience a sudden change of their state at certain moments. Processes with such a character arise naturally and often, especially in phenomena studied in physics, chemical technology, population dynamics, biotechnology, and economics. For an introduction of the basic theory of impulsive differential equations in $R^n$, see Lakshmikantham et al. [1], Bainov and Simeonov [2], Samoilenko and Perestyuk [3], and the references therein. The theory of impulsive differential equations has become an important area of investigation in recent years, and it is much richer than the corresponding theory of differential equations (see, for instance, [4–6] and the references therein).

On the other hand, the theory of boundary value problems with integral boundary conditions for ordinary differential equations arises in different areas of applied mathematics and physics. For example, heat conduction, chemical engineering, underground water flow, thermo-elasticity, and plasma physics can be reduced to nonlocal problems with integral boundary conditions. For boundary value problems with integral boundary conditions and comments on their importance, we refer the reader to the papers in Gallardo [7–9] and the references therein. For more information about the general theory of integral equations and their relation with boundary value problems, we refer to the book of Corduneanu [10] and Agarwal and O’Regan [11].

Recently, some well-known works, such as Hao et al. [12], Zhang et al. [13] and Ding and Wang [14], deal with impulsive differential equations with integral boundary conditions. However, most of these results are obtained by using the fixed point theorem in cones. It is well known that the method of upper and lower solutions is a powerful tool for proving the existence results for a large class of boundary value problems, see [15–17].

where $J=[0,1]$, $J^{\ast }=J\mathrm{\setminus }\{t_1,t_2,\dots ,t_p\}$, $f:J\times R^2\to R$ is continuous, $I_k,J_k\in C(R)$ for $1⩽k⩽p$, $g\in L^1[0,1]$ is nonnegative, $0=t_0<t_1<t_2<\cdots <t_p<t_{p+1}=1$, $\mathrm{\Delta }x(t_k)=x(t_k^{+})-x(t_k^{-})$ denotes the jump of $x(t)$ at $t=t_k$, $x(t_k^{+})$ and $x(t_k^{-})$ represent the right and left limits of $x(t)$ at $t=t_k$, respectively, $\mathrm{\Delta }{x}^{\prime }(t_k)$ has a similar meaning for $x^{\prime }(t)$.

the impulsive conditions $\mathrm{\Delta }x(t_k)=I_k(x(t_k))$, $\mathrm{\Delta }{x}^{\prime }(t_k)=J_k(x(t_k),x^{\prime }(t_k))$ and the boundary conditions $x(0)=x(1)={\int }_0^{1}g(s)x(s)ds$.

For $\alpha ,\beta \in E$, we write $\alpha ⩽\beta$ if $\alpha (t)⩽\beta (t)$ for all $t\in J$.

In addition, we assume that the following conditions hold:

(H₁) f satisfies the Nagumo condition relative to α, β;

(H₂) $I_k,J_k\in C(R,R)$. $J_k(y,z)$ is nondecreasing in $z\in [-D,D]$ for all $1⩽k⩽p$;

(H₃) $0<{\int }_0^{1}g(s)ds<1$.

One can find the next result, with its proof, in [18].

Proof Denote $y(t)=x(t)-\beta (t)$, we will only see that $x(t)⩽\beta (t)$ for every $t\in J$. An analogous reasoning shows that $x(t)⩾\alpha (t)$ for all $t\in J$. Otherwise, if $x(t)⩽\beta (t)$, $t\in [0,1]$, does not hold, then ${sup}_{0⩽t⩽1}(x(t)-\beta (t))>0$, there are three cases.

which is a contradiction.

which is a contradiction. Hence the function y cannot have any positive maximum interior to the interval $(t_k,t_k+1)$ for $k=1,2,\dots ,p$.

Case 3. According to Case 2, if ${sup}_{0\le t\le 1}y(t)>0$, then ${sup}_{0⩽t⩽1}y(t)=y(t_k^{+})$, or ${sup}_{0⩽t⩽1}y(t)=y(t_k^{-})=y(t_k)$, we only prove that ${sup}_{0⩽t⩽1}y(t)=y(t_k^{-})$, $k=1,2,\dots ,p$. Suppose that ${sup}_{0⩽t⩽1}y(t)=y(t_1^{-})$, easily, $y^{\prime }(t_1)=y^{\prime }(t_1^{-})\ge 0$.

which contradicts that $y(1)$ cannot be the maximum point. By the same analysis, we can get that $y(t_k)>0$, $k=2,3,\dots ,p$, cannot hold. □

Lemma 2.4 $-D⩽x^{\prime }(t)⩽D$ on J, where $x(t)$ is the solution of (2.1).

This, obviously, contradicts the choice of D. The proof is complete. □

Lemma 2.5 (Schauder’s fixed point theorem)

Let K be a convex subset of a normed linear space E. Each continuous, compact map $L:K\to K$ has a fixed point.

where $I_k^{\ast }(x(t_k))=I_k(m_{\alpha ,\beta }(t_k,x(t_k)))$, $J_k^{\ast }(x(t_k),x^{\prime }(t_k))=J_k(m_{\alpha ,\beta }(t_k,x(t_k)),n(t_k,x^{\prime }(t_k)))$.

It is obvious that $T:E\to E$ is completely continuous.

By the Schauder fixed point theorem, we can easily obtain that T has a fixed point $x\in E$, which is a solution of BVP (2.1). And by Lemma 2.3 and Lemma 2.4, we know that $\alpha (t)⩽x(t)⩽\beta (t)$, $-D⩽x^{\prime }(t)⩽D$, then BVP (2.1) becomes BVP (1.1), therefore $x(t)$ is a solution of BVP (1.1). The proof is complete. □

where ${\overline{I}}_k(x(t_k))=I_k(m_{{\alpha }_1,{\beta }_2}(t_k,x(t_k)))$, ${\overline{J}}_k(x(t_k),x^{\prime }(t_k))=J_k(m_{{\alpha }_1,{\beta }_2}(t_k,x(t_k)),n(t_k,x^{\prime }(t_k)))$.

It is standard that $\overline{T}:E\to E$ is completely continuous. It is immediate from the argument above that $\overline{T}(\overline{\mathrm{\Omega }})\subset \mathrm{\Omega }$.

and there are solutions in $\overline{{\mathrm{\Omega }}^{{\beta }_1}\cup {\mathrm{\Omega }}_{{\alpha }_2}}$, ${\mathrm{\Omega }}^{{\beta }_1}$, ${\mathrm{\Omega }}_{{\alpha }_2}$, as required.

We now show $deg(I-\overline{T},{\mathrm{\Omega }}_{{\alpha }_2},0)=1$. The proof that $deg(I-\overline{T},{\mathrm{\Omega }}^{{\beta }_1},0)=1$ is similar and hence omitted. We define $I-W$, the extension to $\overline{\mathrm{\Omega }}$ of the restriction of $I-\overline{T}$ to ${\overline{\mathrm{\Omega }}}_{{\alpha }_2}$ as follows.

for some constants $M_1$. Moreover, we may choose $M_1$ so that ${\parallel {\alpha }_2\parallel }_{\mathrm{\infty }},{\parallel {\beta }_1\parallel }_{\mathrm{\infty }}<M_1$.