- Research
- Open access
- Published:

# Existence of strong solutions for a class of semilinear evolution equations with nonlocal initial conditions

*Advances in Difference Equations*
**volume 2012**, Article number: 79 (2012)

## Abstract

This paper discusses the existence of strong solutions for a class of semilinear evolution equations with nonlocal initial conditions in Hilbert spaces. The discussion is based on analytic semigroups theory and fixed point theorem. An application to a partial differential equation with nonlocal condition is also considered.

**Mathematics Subject Classification(2010)**: 34G20; 34K30; 35D35; 47D06.

## 1 Introduction

In this paper, we discuss the existence of strong solutions for a class of semilinear evolution equations with nonlocal initial conditions in a Hilbert space *H*

where *A*: *D*(*A*) ⊂ *H* → *H* is a positive definite self-adjoint operator, *f*: *J × H* → *H* is given function satisfying some assumptions, *J* denote the real compact interval [0, *a*], *a >* 0 is a constant, 0 *< t*_{1} *< t*_{2} *< · · · < t*_{
m
}≤ *a*, *m* ∈ ℕ, *γ*_{
i
}are real numbers, *γ*_{
i
}≠ 0, *i* = 1, 2, ..., *m*.

In 1990, Byszewski and Lakshmikantham [1] first investigated the nonlocal problems. They studied and obtained the existence and uniqueness of mild solutions for nonlocal differential equations. Since it is demonstrated that the nonlocal problems have better effects in applications than the classical Cauchy problems, differential equations with nonlocal conditions were studied by many authors and some basic results on nonlocal problems have been obtained, see [2–10] and the references therein. The importance of nonlocal conditions have also been discussed in [11–15]. For example, Deng [11] used the nonlocal condition of type (2) to describe the diffusion phenomenon of a small amount of gas in a transparent tube. In this case, condition (2) allows the additional measurements at *t*_{
i
}, *i* = 1, 2, ..., *m*, which is more precise than the measurement just at *t* = 0. In [12], Byszewski pointed out that if *γ*_{
i
}≠ 0, *i* = 1, 2, ..., *m*, then the results can be applied to kinematics to determine the location evolution *t* → *u*(*t*) of a physical object for which we do not know the positions *u*(0), *u*(*t*_{1}), ..., *u*(*t*_{
m
}), but we know that the nonlocal condition (2) holds. Consequently, to describe some physical phenomena, the nonlocal condition can be more useful than the standard initial condition.

However, most of the existing articles only studied the existence and uniqueness of mild solutions for evolution equations with nonlocal conditions, there are very few papers considered the regularity results for nonlocal evolution equations. In [2, 3], Byszewski discussed the existence of strong and classical solutions for the evolution equation

with the nonlocal condition

or

in a reflexive Banach space *X*, but the conditions in [2, 3] are very strong and some of them can not be satisfied in applications. In this paper, we obtained the existence of strong solutions for the nonlocal problem (1)-(2) in a frame of abstract Hilbert spaces. Furthermore, an optimal condition (see condition (H1)) on the coefficients *γ*_{
i
}(*i* = 1, 2, ..., *m*) to guarantee that the nonlocal problem (1)-(2) has solutions has been obtained. At last, we demonstrated that the abstract results obtained can be applied to the parabolic partial differential equation with nonlocal conditions. Our discussions are based on analytic semigroups theory and the famous Schauder's fixed point theorem.

## 2 Preliminaries

Let *H* be a Hilbert space with inner product (*·,·*), then \u2225\cdot \u2225=\sqrt{\left(\cdot ,\cdot \right)} is the norm on *H* induced by inner product (·,·). We denote by *C*(*J*, *H*) the Banach space of all continuous *H*-value functions on interval *J* with the maximum norm {\u2225u\u2225}_{C}=\underset{t\in J}{\mathsf{\text{max}}}\u2225u\left(t\right)\u2225 and by \mathcal{L}\left(H\right) the Banach space of all linear and bounded operators on *H*.

Let *A*: *D*(*A*) ⊂ *H* → *H* be a positive definite self-adjoint operator in Hilbert space *H* and it have compact resolvent. By the spectral resolution theorem of selfadjoint operator, the spectrum *σ*(*A*) only consists of real eigenvalues and it can be arrayed in sequences as

By the positive definite property of *A*, the first eigenvalue *λ*_{1} *>* 0. From [16, 17], we know that -*A* generates an analytic operator semigroup *T* (*t*)(*t* ≥ 0) on *H*, which is exponentially stable and satisfies

Since the positive definite self-adjoint operator *A* has compact resolvent, the embedding *D*(*A*) ↪ *H* is compact, and therefore *T* (*t*)(*t* ≥ 0) is also a compact semigroup.

We recall some concepts and conclusions on the fractional powers of *A*. For *α >* 0, *A*^{-α}is defined by

where Γ(·) is the Euler gamma function. {A}^{-\alpha}\in \mathcal{L}\left(H\right) is injective, and *A*^{α}can be defined by *A*^{α}= (*A*^{-α})^{-1} with the domain *D*(*A*^{α}) = *A*^{-α}(*H*). For *α* = 0, let *A*^{α}= *I*.

We endow an inner product (·,·)_{
α
}= (*A*^{α}·, *A*^{α}·) to *D*(*A*^{α}). Since *A*^{α}is a closed linear operator, it follows that (*D*(*A*^{α}), (·,·)_{
α
}) is a Hilbert space. We denote by *H*_{
α
}the Hilbert space (*D*(*A*^{α}), (·,·)_{
α
}). Especially, *H*_{0} = *H* and *H*_{1} = *D*(*A*). For 0 ≤ *α < β*, *H*_{
β
}is densely embedded into *H*_{
α
}and the embedding *H*_{
β
}↪ *H*_{
α
}is compact. For the details of the properties of the fractional powers, we refer to [17] and [18].

It is well known [[16], Chapter 4, Theorem 2.9] that for any *u*_{0} ∈ *D*(*A*) and *h* ∈ *C*^{1}(*J*, *H*), the initial value problem of linear evolution equation (LIVP)

has a unique classical solution *u* ∈ *C*^{1}(*J*, *H*) ∩ *C*(*J*, *H*_{1}) expressed by

If *u*_{0} ∈ *H* and *h* ∈ *L*^{1}(*J*, *H*), the function *u* given by (8) belongs to *C*(*J*, *H*), which is known as a mild solution of the LIVP(7). If a mild solution *u* of the LIVP(7) belongs to *W*^{1,1}(*J*, *H*) ∩ *L*^{1}(*J*, *H*_{1}) and satisfies the equation for a.e. *t* ∈ *J*, we call it a strong solution.

Throughout this paper, we assume that

By assumption (H1), we have \u2225\sum _{i=1}^{m}{\gamma}_{i}T\left({t}_{i}\right)\u2225\le \sum _{i=1}^{m}\left|{\gamma}_{i}\right|\phantom{\rule{0.3em}{0ex}}{e}^{-{\lambda}_{1}{t}_{1}}<1. By operator spectrum theorem, we know that the operator

exists, bounded and D\left(\mathcal{B}\right)=H. Furthermore, by Neumann expression, can be expressed by

Therefore

To prove our main results, for any *h* ∈ *C*(*J*, *H*), we consider the linear evolution equation nonlocal problem (LNP)

**Lemma 1** *If the condition (H1) holds, then the LNP (12)-(13) has a unique mild solution u* ∈ *C*(*J*, *H*) *given by*

*Moreover*, *u*∈ *W* ^{1,2}(*J*, *H*) ∩ *L*^{2}(*J*, *H*_{1}) *is a strong solution of the LNP (12)-(13).*

**Proof**. By (7) and (8), we know that Eq. (12) has a unique mild solution *u* ∈ *C*(*J, H*) which can be expressed by

From (15),

By (13) and (16),

Since I-{\sum}_{i=1}^{m}{\gamma}_{i}T\left({t}_{i}\right) has a bounded inverse operator ,

From (15) and (18), we know that *u* satisfies (14).

Inversely, we can verify directly that the function *u* ∈ *C*(*J, H*) given by (14) is a mild solution of the LNP (12)-(13).

By the maximal regularity of linear evolution equations with positive definite operator in Hilbert spaces (see [19], Chapter II, Theorem 3.3), when *u*(0) = *u*_{0} ∈ *H*_{1}_{
/
}_{2}, the mild solution of the LIVP (7) has the regularity

and it is a strong solution.

We note that *u*(*t*) defined by (14) is the mild solution of the LIVP (7) for u\left(0\right)={\sum}_{i=1}^{m}{\gamma}_{i}\mathcal{B}{\int}_{0}^{{t}_{i}}T\left({t}_{i}-s\right)h\left(s\right)ds. By the representation (8) of mild solution, *u*(*t*) = *T* (*t*)*u*(0) + *v*(*t*), where v\left(t\right)={\int}_{0}^{t}T\left(t-\phantom{\rule{2.77695pt}{0ex}}s\right)h\left(s\right)ds Since the function *v*(*t*) is a mild solution of the LIVP (7) with the null initial value *u*(0) = *θ*, *v* has the regularity (19). By the analytic property of the semigroup *T* (*t*), *T* (*t*_{
i
})*u*(0) ∈ *D*(*A*) *H*_{1}_{
/
}_{2}. Hence, u\left(\mathsf{\text{0}}\right)={\sum}_{i=1}^{m}{\gamma}_{i}T\left({t}_{i}\right)u\left(0\right)+{\sum}_{i=1}^{m}{\gamma}_{i}v\left({t}_{i}\right)\in {H}_{1/2}. Using the regularity (19) again, we obtain that *u* ∈ *W* ^{1,2}(*J*, *H*) ∩ *L*^{2}(*J*, *H*_{1}) and it is a strong solution of the LNP (12)-(13). □

For any *r >* 0, let

then Ω_{
r
}is a bounded closed and convex set on *C*(*J*, *H*).

## 3 Main results

**Theorem 1** *Let A be a positive definite self-adjoint operator in Hilbert space H and it have compact resolvent, f: J × H* → *H be continuous. If condition (H1) and the following condition*

*(H2) For some r >* 0, *there exists a function φ* ∈ *L*(*J*, ℝ^{+}) *such that for all t* ∈ *J and u* ∈ *H satisfying* ||*u*|| ≤ *r*, ||*f*(*t*, *u*) || ≤ *φ*(*t*), *hold, then the problem (1)-(2) has at least one strong solution u* ∈ *W* ^{1,2}(*J*, *H*) ∩ *L*^{2}(*J*, *H*_{1}).

**Proof**. We consider the operator *Q* on *C*(*J*, *H*) defined by

By assumption (H1) and Lemma 1, it is easy to see that the mild solution of problem (1)-(2) is equivalent to the fixed point of the operator *Q*. In the following, we will prove that *Q* has a fixed point by using the famous Schauder Fixed Point Theorem.

At first, we prove that *Q* is continuous on *C*(*J*, *H*). To this end, let {\left\{{u}_{n}\right\}}_{n=1}^{\mathrm{\infty}}\subset C\left(J,H\right) be a sequence such that \underset{n\to +\mathrm{\infty}}{\mathsf{\text{lim}}}{u}_{n}=u on *C*(*J*, *H*). By the continuity of the nonlinear term *f*, for each *s* ∈ *J*, \underset{n\to +\mathrm{\infty}}{\mathsf{\text{lim}}}f\left(s,{u}_{n}\left(s\right)\right)=f\left(s,u\left(s\right)\right). Therefore, we can conclude that

From (6) and (20), for *t* ∈ *J*, we have

which implies that

From (21), we know that

That is, *Q* is continuous on *C*(*J*, *H*).

Subsequently, we prove that *Q*: *C*(*J*, *H*) → *C*(*J*, *H*) is a compact operator. Let 0\le \alpha <\frac{1}{2},0<\nu <\frac{1}{2}-\phantom{\rule{0.3em}{0ex}}\alpha. By [20], we can prove that the operator *Q* defined by (20) maps *C*(*J*, *H*) into *C*^{ν}(*J*, *H*_{
α
}). By Arzela-Ascoli's theorem, the embedding *C*^{ν}(*J*, *H*_{
α
}) ↪ *C*(*J*, *H*) is compact. This implies that *Q*: *C*(*J*, *H*) → *C*(*J*, *H*) is a compact operator. Combining this with the continuity of *Q* on *C*(*J*, *H*), we know that *Q*: *C*(*J*, *H*) → *C*(*J*, *H*) is a completely continuous operator.

Next, we prove that there exists a positive constant *R* big enough, such that *Q*(Ω_{
R
}) ⊂ Ω_{
R
}. In fact, choosing

For any *u* ∈ Ω_{
R
}, we have

Therefore, *Q*(Ω_{
R
}) ⊂ Ω_{
R
}. Thus, *Q*: Ω_{
R
}→ Ω_{
R
}is a completely continuous operator.

By Schauder Fixed Point Theorem, we know that *Q* has at least one fixed point *u* ∈ Ω_{
R
}. Since *u* is mild solution of the LNP (12)-(13) for *h*(·) = *f*(·, *u*(·)), by Lemma 1, *u* ∈ *W* ^{1,2}(*J, H*) ∩ *L*^{2}(*J, H*_{1}) is a strong solution of the problem (1)-(2).

□

If we replace the assumption (H2) by the following assumption (H2)* For some *r >* 0, there exist a function *φ* ∈ *L*(*J*, ℝ^{+}) and a non-decreasing continuous function *ψ*: ℝ^{+} → ℝ^{+} such that for all *t* ∈ *J* and *u* ∈ *H* satisfying ||*u*|| ≤ *r*,

We have the following existence result.

**Theorem 2** *Let A be a positive definite self-adjoint operator in Hilbert space H and it have compact resolvent, f: J × H* → *H be continuous. If the conditions (H1) and (H2)* are satisfied, then the problem (1)*-*(2) has at least one strong solution u* ∈ *W* ^{1,2}(*J*, *H*) ∩ *L*^{2}(*J*, *H*_{1}) *provided that there exists a constant R with*

where

**Proof**. By the proof of Theorem 1, we know that the operator *Q*: *C*(*J*, *H*) → *C*(*J*, *H*) is completely continuous. For any *u* ∈ Ω_{
R
}, from the assumption (H2)* and (22), we have

which implies *Q*(Ω_{
R
}) ⊂ Ω_{
R
}. Thus, *Q*: Ω_{
R
}→ Ω_{
R
}is a completely continuous operator. By Schauder Fixed Point Theorem, we know that *Q* has at least one fixed point *u* ∈ Ω_{
R
}. Since *u* is mild solution of the LNP (12)-(13) for *h*(·) = *f*(·, *u*(·)), by Lemma 1, *u* ∈ *W* ^{1,2}(*J*, *H*) ∩ *L*^{2}(*J, H*_{1}) is a strong solution of the problem (1)-(2).

□

**Corollary 1** *Let A be a positive definite self-adjoint operator in Hilbert space H and it have compact resolvent, f: J × H* → *H be continuous. If the conditions (H1) and (H2)** *are satisfied, then the problem (1)*-*(2) has at least one strong solution u* ∈ *W*^{1,2}(*J, H*) ∩ *L*^{2}(*J, H*_{1}) *provided that*

*where M is defined by (23)*.

## 4 An example

In order to illustrate our main results, we consider the parabolic partial differential equation with nonlocal condition

where *J* = [0, *a*], 0 *< t*_{1} *< t*_{2} *<* ⋯ < *t*_{
m
}≤ *a*, *γ*_{
i
}are real numbers, *γ*_{
i
}≠ 0, *i* = 1, 2, ..., *m*, *f*: [0, 1] × *J ×* ℝ → ℝ is continuous.

Let *H* = *L*^{2}(0, 1; ℝ) with the norm || · ||_{2}. We define the linear operator *A* in Hilbert space *H* by

where *H*^{2}(0, 1) = *W*^{2,2}(0, 1), {H}_{0}^{1}\left(0,1\right)={W}_{0}^{1,2}\left(0,1\right). It is well know from [16, 17] that *A* is a positive definite self-adjoint operator on *H* and -*A* is the infinitesimal generator of an analytic, compact semigroup *T*(*t*)(*t* ≥ 0). Moreover, *A* has discrete spectrum with eigenvalues *λ*_{
n
}= *n*^{2}*π*^{2}, *n* ∈ ℕ, associated normalized eigenvectors {v}_{n}\left(x\right)=\sqrt{2}\mathsf{\text{sin}}n\pi x, the set {*v*_{
n
}: *n* ∈ ℕ} is an orthonormal basis of *H* and

Let *f*(*t*, *u*(*t*)) = *f*(·, *t*, *u*(·, *t*)), then the problem (24) can be rewritten into the abstract form of problem (1)-(2).

**Theorem 3** *If the nonlinear term f*(*x*, *t*, *u*(*x*, *t*)) = sin *u*(*x*, *t*)*/*(*t*^{1/2} + 1), *x* ∈ [0, 1], *t* ∈ *J and* {\sum}_{i=1}^{m}\left|{\gamma}_{i}\right|<{e}^{{\pi}^{2}{t}_{1}}, *then the problem (24) has at least one strong solution* u\in C\left(J,\phantom{\rule{2.77695pt}{0ex}}{H}_{0}^{1}\left(0,1\right)\right)\cap {L}^{2}\left(J,\phantom{\rule{2.77695pt}{0ex}}{H}^{2}\left(0,1\right)\right)\cap {W}^{1,2}\left(J,\phantom{\rule{2.77695pt}{0ex}}{L}^{2}\left(0,1;\mathbb{R}\right)\right).

**Proof**. Let *φ*(*t*) = *t*^{-1/2}, from the condition {\sum}_{i=1}^{m}\left|{\gamma}_{i}\right|<{e}^{{\pi}^{2}{t}_{1}}, we easily see that the conditions (H1) and (H2) hold. Hence by Theorem 1, the problem (24) has a strong solution u\in C\left(J,\phantom{\rule{2.77695pt}{0ex}}{H}_{0}^{1}\left(0,1\right)\right)\cap {L}^{2}\left(J,\phantom{\rule{2.77695pt}{0ex}}{H}^{2}\left(0,1\right)\right)\cap {W}^{1,2}\left(J,\phantom{\rule{2.77695pt}{0ex}}{L}^{2}\left(0,1;\mathbb{R}\right)\right) in the sense of *L*^{2}(0, 1; ℝ). □

**Theorem 4** *If* {\sum}_{i=1}^{m}\left|{\gamma}_{i}\right|<{e}^{{\pi}^{2}{t}_{1}}, *f*: [0, 1] *× J ×* ℝ → ℝ *is continuous and satisfies the following conditions*

*(P1) For some r >* 0, *there exists a function φ* ∈ *L*(*J*, ℝ^{+}) *such that for all t* ∈ *J*, *x* ∈ [0, 1] *and u* ∈ ℝ, | *u* |≤ *r*, | *f*(*x*, *t*, *u*(*x*, *t*)) |≤ *φ*(*t*),

*(P2) There exists a function c*: ℝ^{+} → ℝ^{+} *such that*

*for any r >* 0, *μ* ∈ (0, 1) *and* (*x*, *t*, *ξ*), (*y*, *s*, *η*) ∈ [0, 1] *× J ×* [-*r*, *r*], *then the problem (24) has at least one classical solution u* ∈ *C*^{2+μ,1+μ/2}([0, 1] *× J*).

**Proof**. From the condition {\sum}_{i=1}^{m}\left|{\gamma}_{i}\right|<{e}^{{\pi}^{2}{t}_{1}} and assumption (P1), it is easy to verify that the conditions (H1) and (H2) are satisfied. Hence by Theorem 1, the problem (24) has at least one strong solution u\in C\left(J,\phantom{\rule{2.77695pt}{0ex}}{H}_{0}^{1}\left(0,1\right)\right)\cap {L}^{2}\left(J,\phantom{\rule{2.77695pt}{0ex}}{H}^{2}\left(0,1\right)\right)\cap {W}^{1,2}\left(J,\phantom{\rule{2.77695pt}{0ex}}{L}^{2}\left(0,1;\mathbb{R}\right)\right) in the sense of *L*^{2}(0, 1; ℝ). Since the nonlinear term *f* satisfies the condition (P2), by using the similar regularization method in [20], we can prove that *u* ∈ *C*^{2+μ,1+μ/2}([0, 1] *× J*) is a classical solution of the problem (24). □

## References

Byszewski L, Lakshmikantham V: Theorem about the existence and uniqueness of solutions of a nonlocal Cauchy problem in a Banach space.

*Appl Anal*1990, 40: 11–19.Byszewski L: Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem.

*J Math Appl Anal*1991, 162: 494–505. 10.1016/0022-247X(91)90164-UByszewski L: Application of preperties of the right hand sides of evolution equations to an investigation of nonlocal evolution problems.

*Nonlinear Anal*1998, 33: 413–426. 10.1016/S0362-546X(97)00594-4Lin Y, Liu JH: Semilinear integrodifferential equations with nonlocal Cauchy problem.

*Nonlinear Anal*1996, 26: 1023–1033. 10.1016/0362-546X(94)00141-0Benchohra M, Ntouyas SK: Existence of mild solutions of semilinear evolution inclusions with nonlocal conditions.

*Georgian Math J*2000, 7: 221–230.Wang J, Zhou Y, Wei W, Xu H: Nonlocal problems for fractional integrodifferential equations via fractional operators and optimal controls.

*Comput Math Appl*2011, 62: 1427–1441. 10.1016/j.camwa.2011.02.040Liang J, Casteren JV, Xiao TJ: Nonlocal Cauchy problems for semilinear evolution equations. Nonlinear Anal. 2002, 50: 173–189.

Xiao TJ, Liang J: Existence of classical solutions to nonautonomous nonlocal parabolic problems.

*Nonlinear Anal*2005, 63: 225–232. 10.1016/j.na.2005.05.008Wang J, Zhou Y, Medved M: Picard and weakly Picard operators technique for nonlinear differential equations in Banach spaces.

*J Math Anal Appl*2012, 389: 261–274. 10.1016/j.jmaa.2011.11.059Boucherif A: Semilinear evolution inclutions with nonlocal conditions.

*Appl Math Lett*2009, 22: 1145–1149. 10.1016/j.aml.2008.10.004Deng K: Exponential decay of solutions of semilinear parabolic equations with nonlocal initial conditions.

*J Math Anal Appl*1993, 179: 630–637. 10.1006/jmaa.1993.1373Byszewski L: Existence and uniqueness of a classical solutions to a functional-differential abstract nonlocal Cauchy problem.

*J Math Appl Stoch Anal*1999, 12: 91–97. 10.1155/S1048953399000088Benchohra M, Ntouyas SK: Nonlocal Cauchy problems for neutral functional differential and integrodifferential inclusions in Banach spaces.

*J Math Anal Appl*2001, 258: 573–590. 10.1006/jmaa.2000.7394Liang J, Liu JH, Xiao TJ: Nonlocal Cauchy problems governed by compact operator families.

*Nonlinear Anal*2004, 57: 183–189. 10.1016/j.na.2004.02.007Ezzinbi K, Fu X, Hilal K: Existence and regularity in the

*α*-norm for some neutral partial differential equations with nonlocal conditions.*Nonlinear Anal*2007, 67: 1613–1622. 10.1016/j.na.2006.08.003Pazy A:

*Semigroups of Linear Operators and Applications to Partial Differential Equations.*Springer, Berlin; 1983.Henry D: Geometric Theory of Semilinear Parabolic Equations. In

*Lecture Notes in Mathematics*.*Volume 840*. Springer, New York; 1981.Xiang X, Ahmed NU: Existence of periodic solutions of semilinear evilution equations with time lags.

*Nonlinear Anal*1992, 18: 1063–1070. 10.1016/0362-546X(92)90195-KTeman R:

*Infinite-Dimensional Dynamical Systems in Mechanics and Physics.*2nd edition. Springer, New York; 1997.Amann H: Periodic solutions of semilinear parabolic equations. In

*Nonlinear Analysis: A Collection of Papers in Honor of Erich H Rothe*. Edited by: Cesari L, Kannan R, Weinberger R. Academic Press, New York; 1978:1–29.

## Acknowledgements

This research was supported by NNSF of China (10871160) and NNSF of China (11061031).

## Author information

### Authors and Affiliations

### Corresponding author

## Additional information

### Competing interests

The authors declare that they have no competing interests.

### Authors' contributions

PC carried out the first draft of this manuscript, YL corrected this draft, HF prepared the final version of the manuscript. All authors read and approved the final version of the manuscript.

## Rights and permissions

**Open Access**
This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (
https://creativecommons.org/licenses/by/2.0
), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

## About this article

### Cite this article

Chen, P., Li, Y. & Fan, H. Existence of strong solutions for a class of semilinear evolution equations with nonlocal initial conditions.
*Adv Differ Equ* **2012**, 79 (2012). https://doi.org/10.1186/1687-1847-2012-79

Received:

Accepted:

Published:

DOI: https://doi.org/10.1186/1687-1847-2012-79