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

In this paper, we discuss the existence of positive solutions for second-order differential equations subject to nonlinear impulsive conditions and non-separated periodic boundary value conditions. Our criteria for the existence of positive solutions will be expressed in terms of the first eigenvalue of the corresponding nonimpulsive problem. The main tool of study is a fixed point theorem in a cone.

MSC:34B37, 34B18.

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.

(H1) Let J=[0,\omega ], and , f\in C(J\times R^{+},R^{+}), I_i\in C(R^{+},R^{+}), R^{+}=[0,+\mathrm{\infty }). \mathrm{\Delta }(u^{[1]}(t_i))=u^{[1]}(t_i^{+})-u^{[1]}(t_i^{-}), where u^{[1]}(t_i^{+}) (respectively u^{[1]}(t_i^{-})) denotes the right limit (respectively, the left limit) of u^{[1]}(t) at t=t_i.

1. (H)

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^{-}.

For the case of I_i=0 (i=1,2,\dots ,m), the problem (1.1) is related to a non-separated periodic boundary value problem of ODE. Atici and Guseinov [1] have proved the existence of a positive and twin positive solutions to BVP (1.1) by applying a fixed point theorem for the completely continuous operators in cones. In [2], Graef and Kong studied the following periodic boundary value problem:

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].

In this section, we collect some preliminary results that will be used in the subsequent section. We denote by \phi (t) and \psi (t) the unique solutions of the corresponding homogeneous equation

under the initial boundary conditions

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

Definition 2.1 For two differential functions y and z, we define their Wronskian by

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

Proof Suppose that y and z are two solutions of (2.1), then

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

Consider the following equation:

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.

(G_3)

Combining with Theorem 2.1, we can also prove that

(G_4)

Remark 1 From paper [1], we can get G(t,s) when q(t)=\frac{c^2}{p(t)} (c>0) and p(t)>0,

Especially, in the case of p(t)\equiv 1, q(t)\equiv c^2 (c>0), Green’s function G(t,s) has the form

Define an operator

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}.

In what follows, we denote the positive eigenfunction corresponding to {\lambda }_1 by ϕ and {max}_{t\in J}\varphi (t)=1. Define a mapping Φ and a cone K in a Banach space C(J) by

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).

Proof Clearly, Φu is continuous in t. For t\ne t_k,

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]

Let X be a Banach space and K be a cone in X. Suppose {\mathrm{\Omega }}_1 and {\mathrm{\Omega }}_2 are open subsets of X such that 0\in {\mathrm{\Omega }}_1\subset {\overline{\mathrm{\Omega }}}_1\subset {\mathrm{\Omega }}_2, and suppose that

is a completely continuous operator such that

• {inf}_{u\in K\cap \partial {\mathrm{\Omega }}_1}\parallel \mathrm{\Phi }u\parallel >0, u\ne \mu \mathrm{\Phi }uforu\in K\cap \partial {\mathrm{\Omega }}_1and\mu \ge 1, andu\ne \mu \mathrm{\Phi }uforu\in K\cap \partial {\mathrm{\Omega }}_2and, or

• {inf}_{u\in K\cap \partial {\mathrm{\Omega }}_2}\parallel \mathrm{\Phi }u\parallel >0, u\ne \mu \mathrm{\Phi }uforu\in K\cap \partial {\mathrm{\Omega }}_2and\mu \ge 1, andu\ne \mu \mathrm{\Phi }uforu\in K\cap \partial {\mathrm{\Omega }}_1and.

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

Lemma 2.4 [18]

Let X be a Banach space and K be a cone in X. Suppose {\mathrm{\Omega }}_1 and {\mathrm{\Omega }}_2 are open subsets of X such that 0\in {\mathrm{\Omega }}_1\subset {\overline{\mathrm{\Omega }}}_1\subset {\mathrm{\Omega }}_2, and suppose that

is a completely continuous operator such that

• There existsu_0\in K\mathrm{\setminus }\{0\}such thatu\ne \mathrm{\Phi }u+\mu u_0foru\in K\cap \partial {\mathrm{\Omega }}_1and\mu >0, \parallel \mathrm{\Phi }u\parallel \le \parallel u\parallelforu\in K\cap \partial {\mathrm{\Omega }}_2, or

• There existsu_0\in K\mathrm{\setminus }\{0\}such thatu\ne \mathrm{\Phi }u+\mu u_0foru\in K\cap \partial {\mathrm{\Omega }}_2and\mu >0, \parallel \mathrm{\Phi }u\parallel \le \parallel u\parallelforu\in K\cap \partial {\mathrm{\Omega }}_1.

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

Recalling that δ was defined after Lemma 2.1, for convenience, we introduce the following notations. Assume that the constant r>0 and γ is some positive function on J,

Theorem 3.1 Assume that there exist positive constants α, β such that f_q^{\alpha }\ge 0, {\overline{f}}_q^{\beta }\ge 0, I_i^{\alpha }\ge 0, I_i^{\beta }\ge 0 and

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

Proof Clearly, \alpha \ne \beta, let \alpha =min\{\alpha ,\beta \}, \beta =max\{\alpha ,\beta \}. Define the open sets

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)

and

Let u_0\equiv 1. We show that

If not, there exist u_1\in K\cap \partial {\mathrm{\Omega }}_{\alpha } and {\mu }_1>0 such that u_1=\mathrm{\Phi }{u}_1+{\mu }_1. Let u_1(\rho )={min}_{t\in J}{u}_1(t). Noting that \delta \alpha \le u_1\le \alpha for any t\in J, we obtain that for t\in J,

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

On the other hand, for \mathrm{\forall }u\in K\cap \partial {\mathrm{\Omega }}_{\beta }, \delta \beta \le u(t)\le \beta, we have

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.

Theorem 3.2 Assume that there exist positive constants α, β such that f_{\gamma }^{\alpha }\ge 0, f_{\gamma }^{\beta }\ge 0, I_i^{\alpha }\ge 0, I_i^{\beta }\ge 0 and

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 \}.

Proof Obviously, \alpha \ne \beta, put \alpha =min\{\alpha ,\beta \}, \beta =max\{\alpha ,\beta \}. Define the open sets

At first, we show that \mathrm{\Phi }:K\cap ({\overline{\mathrm{\Omega }}}_{\beta }\mathrm{\setminus }{\mathrm{\Omega }}_{\alpha })\to K. For any u\in K\cap ({\overline{\mathrm{\Omega }}}_{\beta }\mathrm{\setminus }{\mathrm{\Omega }}_{\alpha }), from (G_2), we have

On the other hand,

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

Next, we show that

If not, there exist {\mu }_0\in (0,1] and u_0\in K\cap \partial {\mathrm{\Omega }}_{\beta } such that {\mu }_0\mathrm{\Phi }{u}_0=u_0. Hence,

Multiplying the first equation of (3.8) by ϕ and integrating from 0 to ω, we obtain that

One can find that

Substituting (3.10) into (3.9), we get

Noting that \delta \parallel u_0\parallel \le u_0\le \parallel u_0\parallel, therefore,

which implies that

a contradiction.

Finally, we show that

Since f(t,u) and I_i(u) are negative for u\in [\delta \alpha ,\alpha ] and t\in J, the condition (3.6) implies that f_{\gamma }^{\alpha }>0. Hence, {\int }_0^{\omega }f(s,u(s))ds>0 for u\in K\cap \partial {\mathrm{\Omega }}_{\alpha } and for any u\in K\cap \partial {\mathrm{\Omega }}_{\alpha },

Suppose that there exist {\mu }_0\ge 1 and u_0\in K\cap \partial {\mathrm{\Omega }}_{\alpha } such that {\mu }_0\mathrm{\Phi }{u}_0=u_0, that is,

Multiplying the first equation of (3.11) by ϕ and integrating from 0 to ω, we obtain that

One can get that

Substituting (3.13) into (3.12), we get

Noting that \delta \parallel u_0\parallel \le u_0\le \parallel u_0\parallel, therefore,

It is impossible for {\lambda }_1{\int }_0^{\omega }\varphi (s)ds-\delta {\sum }_{i=1}^m{I}_i^{\alpha }\varphi (t_i)\le 0. When {\lambda }_1{\int }_0^{\omega }\varphi (s)ds-\delta {\sum }_{i=1}^m{I}_i^{\alpha }\varphi (t_i)>0,

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. □

Corollary 3.1 Assume that f_{\gamma }^0>0, f_{\gamma }^{\mathrm{\infty }}>0, I_i^0>0, I_i^{\mathrm{\infty }}>0 and

or

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

Corollary 3.2 Assume that there exists a constant α such that f_{\gamma }^{\rho }>0, I_i^{\rho }>0 (\rho =0, α and ∞) and

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 }.

Example 1 Consider the equation

where p(t)=1, q(t)=3 and

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.

Example 2 Consider the equation

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

It is well known that, for the problem consisting of the equation -u^{″}=\lambda u, t\in (0,1), and the boundary condition

the first eigenvalue is 0 (see, for example, [[19], p.428]). It follows that the first eigenvalue is {\lambda }_1=20 for the problem consisting of the equation

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 }.

The authors would like to thank anonymous referees very much for helpful comments and suggestions which led to the improvement of presentation and quality of work. This research was partially supported by the NNSF of China (No. 11001274, 11171085) and the Postdoctoral Science Foundation of Central South University and China (No. 2011M501280).

Affiliations

1. Department of Mathematics and Statistics, Central South University, Changsha, Hunan, 410075, P.R. China

   • Ruixi Liang

2. Department of Mathematics, Hangzhou Normal University, Hangzhou, Zhejiang, 310036, P.R. China

   • Jianhua Shen

Corresponding author

Correspondence to Ruixi Liang.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the manuscript and read and approved the final manuscript.

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