Eigenvalue criteria for existence of positive solutions of impulsive differential equations with non-separated boundary conditions

Let ω be a fixed positive number. In this paper, we are concerned with the existence of positive solutions for the following boundary value problem (BVP) with impulses:

(1.1a)

(1.1b)

(1.1c)

Here, $u^{[1]}(t)=p(t)u^{\prime }(t)$ denotes the quasi-derivative of $u(t)$. The condition (1.1c) is called a non-separated periodic boundary value condition for (1.1a).

We assume throughout, and with further mention, that the following conditions hold.

A function $u(t)$ defined on $J^{-}=J\mathrm{\setminus }\{t_1,t_2,\dots ,t_m\}$ is called a solution of BVP (1.1) ((1.1a)-(1.1c)) if its first derivative $u^{\prime }(t)$ exists for each $t\in J^{-}$, $p(t)u^{\prime }(t)$ is absolutely continuous on each close subinterval of $J^{-}$, there exist finite values $u^{[1]}(t_i^{\pm })$, the impulse conditions (1.1b) and the boundary conditions (1.1c) are satisfied, and the equation (1.1a) is satisfied almost everywhere on $J^{-}$.

where $h(t)>0$. Based upon the properties of Green’s function obtained in [1], the authors extended and improved the work of [1] by using topological degree theory. They derived new criteria for the existence of non-trivial solutions, positive solutions and negative solutions of the problem (1.2) when f is a sign-changing function and not necessarily bounded from below even over $[0,\omega ]\times R^{+}$. Very recently, He et al. [3] studied BVP (1.1) without impulses and generalized the results of [1, 4] via the fixed point index theory. The problem (1.2) in the case of $p\equiv 1$, the usual periodic boundary value problem, has been extensively investigated; see [4–7] for some results.

On the other hand, impulsive differential equations are a basic tool to study processes that are subjected to abrupt changes in their state. There has been a significant development in the last two decades. Boundary problems of second-order differential equations with impulse have received considerable attention and much literature has been published; see, for instance, [8–17] and their references. However, there are fewer results about positive solutions for second-order impulsive differential equations. To our best knowledge, there is no result about nonlinear impulsive differential equations with non-separated periodic boundary conditions.

Motivated by the work above, in this paper we study the existence of positive solutions for the boundary value problem (1.1). By using fixed point theorems in a cone, criteria are established under some conditions on $f(t,u)$ concerning the first eigenvalue corresponding to the relevant linear operator. More important, the impulsive terms are different from those of papers [8, 9].

Put $D=\phi (\omega )+{\psi }^{[1]}(\omega )-2$, then by [[1], Lemma 2.3], $D>0$.

Theorem 2.1 The Wronskian of any two solutions for equations (2.1) is constant. Especially, $W_t(\phi ,\psi )\equiv 1$.

therefore, the Wronskian is constant. Further, from the initial conditions (2.2), we have $W_t(\phi ,\psi )\equiv 1$. The proof is complete. □

From Theorem 2.5 in [1], equation (2.3) has a Green function $G(t,s)>0$ for all $s,t\in J$, which has the following properties:

($G_1$) $G(t,s)$ is continuous in t and s for all $t,s\in J$.

($G_2$) If $A={min}_{0\le t,s\le \omega }G(t,s)$ and $B={max}_{0\le t,s\le \omega }G(t,s)$, then $B>A>0$.

Combining with Theorem 2.1, we can also prove that

then it is easy to check that $T:C(J)\to C(J)$ is a completely continuous operator. By virtue of the Krein-Rutman theorem, the authors in [3] got the following result.

Lemma 2.1 The spectral radius $r(T)>0$ and T has a positive eigenfunction corresponding to its first eigenvalue ${\lambda }_1={(r(T))}^{-1}$.

where $\delta =\frac{A}{B}$, $\parallel u\parallel ={max}_{t\in J}|u(t)|$.

Lemma 2.2 The fixed point of the mapping Φ is a solution of (1.1).

Using ($G_3$) and ($G_4$), we have $(\mathrm{\Phi }u)(0)=(\mathrm{\Phi }u)(\omega )$, ${(\mathrm{\Phi }u)}^{[1]}(0)={(\mathrm{\Phi }u)}^{[1]}(\omega )$ and

which implies that the fixed point of Φ is the solution of (1.1). The proof is complete. □

The proofs of the main theorems of this paper are based on fixed point theory. The following two well-known lemmas in [18] are needed in our argument.

Lemma 2.3 [18]

is a completely continuous operator such that

Then Φ has a fixed point in $K\cap ({\overline{\mathrm{\Omega }}}_2\mathrm{\setminus }{\mathrm{\Omega }}_1)$.

Lemma 2.4 [18]

is a completely continuous operator such that

Then Φ has a fixed point in ${\overline{\mathrm{\Omega }}}_2\mathrm{\setminus }{\mathrm{\Omega }}_1$.

Then (1.1) has at least one positive solution u such that $min\{\alpha ,\beta \}\le \parallel u\parallel \le max\{\alpha ,\beta \}$.

Then $\mathrm{\Phi }:K\cap ({\overline{\mathrm{\Omega }}}_{\beta }\mathrm{\setminus }{\mathrm{\Omega }}_{\alpha })$ is completely continuous. By (3.1) and the definition of $f_q^{\alpha }$, $I_i^{\alpha }$, ${\overline{f}}_q^{\beta }$, ${\overline{I}}_i^{\beta }$, there exists $\epsilon >0$ such that

(3.2)

(3.3)

which implies that $u_1(\rho )>u_1(\rho )$, a contradiction.

From Lemma 2.4 it follows that Φ has a fixed point $u\in K\cap ({\overline{\mathrm{\Omega }}}_{\beta }\mathrm{\setminus }{\mathrm{\Omega }}_{\alpha })$. Furthermore, $\alpha \le \parallel u\parallel \le \beta$ and $u(t)\ge \delta \alpha >0$, which means that $u(t)$ is a positive solution of Eq. (1.1). The proof is complete. □

In the next theorem, we make use of the eigenvalue ${\lambda }_1$ and the corresponding eigenfunction ϕ introduced in Lemma 2.1.

here $\gamma \equiv 1$ on J. Then (1.1) has at least one positive solution u such that $min\{\alpha ,\beta \}\le \parallel u\parallel \le max\{\alpha ,\beta \}$.

It is easy to check that $\mathrm{\Phi }:K\cap ({\overline{\mathrm{\Omega }}}_{\beta }\mathrm{\setminus }{\mathrm{\Omega }}_{\alpha })\to K$ is completely continuous.

a contradiction.

a contradiction.

From Lemma 2.3 it follows that Φ has a fixed point $u\in K\cap ({\overline{\mathrm{\Omega }}}_{\beta }\mathrm{\setminus }{\mathrm{\Omega }}_{\alpha })$. Furthermore, $\alpha \le \parallel u\parallel \le \beta$ and $u\ge \delta \alpha >0$, which means that $u(t)$ is a positive solution of Eq. (1.1). The proof is complete. □

here $\gamma \equiv 1$ on J. Then (1.1) has at least one positive solution.

here $\gamma \equiv 1$ on J. Then there exists one open interval $\mathrm{\Theta }:1\in \mathrm{\Theta }$ such that (1.1) has at least two positive solutions for $\lambda \in \mathrm{\Theta }$.

here $\rho >1$ and $i=1,2,\dots ,m$. Since $I_i^0=+\mathrm{\infty }$, ${\overline{I}}_i^{\mathrm{\infty }}=0$ and ${\overline{f}}_q^{\mathrm{\infty }}=0$, by Theorem 3.1, (3.14) has at least one positive solution for any $\lambda >0$.

where $f(t,u)=\frac{{\mathrm{e}}^{-u}}{10}$, $I_i(u)=\frac{u^2}{100(m+i)}$.

and the boundary condition (3.16). Meanwhile, we can obtain the positive eigenfunction $\varphi (t)\equiv 1$ corresponding to ${\lambda }_1$. It is also easy to check that $\delta =\frac{2\sqrt{\mathrm{e}}}{1+\mathrm{e}}$, $f_{\gamma }^0=+\mathrm{\infty }$, $f_{\gamma }^{\mathrm{\infty }}=0$ and $I_i^{\mathrm{\infty }}=+\mathrm{\infty }$ (here $\gamma =1$). So, the right-hand side of the inequality in Corollary 3.2 is obviously satisfied. Considering the monotonicity of $f(t,u)$ and $I_i$, we can choose a sufficiently small positive constant α such that the left-hand side of the inequality is true. Therefore, by a direct application of Corollary 3.2, there exists one open interval $\mathrm{\Theta }:1\in \mathrm{\Theta }$ such that (3.15) has at least two positive solutions for $\lambda \in \mathrm{\Theta }$.