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On mixed boundary value problem of impulsive semilinear evolution equations of fractional order

Lihong Zhang1, Guotao Wang1* and Guangxing Song2

Author affiliations

1 School of Mathematics and Computer Science, Shanxi Normal University, Linfen, Shanxi 041004, P. R. China

2 Department of Mathematics, China University of Petroleum, Qingdao, Shandong 266555, P. R. China

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Citation and License

Boundary Value Problems 2012, 2012:17  doi:10.1186/1687-2770-2012-17

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/17

Received:15 November 2011
Accepted:14 February 2012
Published:14 February 2012

© 2012 Zhang et al; licensee Springer.

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


This article studies the existence and uniqueness of solutions for impulsive semi-linear evolution equations of fractional order α ∈ (1, 2] with mixed boundary conditions. Some standard fixed point theorems are applied to prove the main results. An illustrative example is also presented.

Mathematics Subject Classification: 26A33; 34K30; 34K45.

evolution equations of fractional order; impulse; mixed boundary conditions; fixed point theorem.

1 Introduction and preliminaries

Fractional differential equations arise in many engineering and scientific disciplines as the mathematical modeling of systems and processes in the fields of physics, chemistry, aerodynamics, control theory, signal, and image processing, biophysics, electrodynamics of complex medium, polymer rheology, fitting of experimental data, etc. [1-6]. For example, one could mention the problem of anomalous diffusion [7-9], the nonlinear oscillation of earthquake can be modeled with fractional derivative [10], and fluid-dynamic traffic model with fractional derivatives [11] can eliminate the deficiency arising from the assumption to continuum traffic flow and many other [12,13] recent developments in the description of anomalous transport by fractional dynamics. For some recent development on nonlinear fractional differential equations, see [14-24] and the references therein.

Impulsive differential equations, which provide a natural description of observed evolution processes, are regarded as important mathematical tools for the better understanding of several real world problems in applied sciences. The theory of impulsive differential equations of integer order has found its extensive applications in realistic mathematical modelling of a wide variety of practical situations and has emerged as an important area of investigation. The impulsive differential equations of fractional order have also attracted a considerable attention and a variety of results can be found in the articles [25-36].

Motivated by Agarwal and Ahmad's work [33], in this article, we study a mixed boundary value problem for impulsive evolution equations of fractional order given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/17/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/17/mathml/M1">View MathML</a>


where CDα is the Caputo fractional derivative, A(t) is a bounded linear operator on J (the function t A(t) is continuous in the uniform operator topology), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/17/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/17/mathml/M2">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/17/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/17/mathml/M3">View MathML</a> denote the right and the left limits of u(t) at t = tk(k = 1, 2, ..., p), respectively and Δu'(tk) have a similar meaning for u'(t).

It is worthwhile pointing out that the boundary conditions in (1.1) interpolate between Neumann (a = b = c = d = 0) and Dirichlet (a, d → ∞ with finite values of b and c) boundary conditions. Note that Zaremba boundary conditions (u(0) = 0, u'(T) = 0) can be considered as mixed boundary conditions with a → ∞, c = d = 0. For more details on Zaremba boundary conditions, see ( [37-39]).

Let J0 = [0, t1], J1 = (t1, t2], ..., Jp-1 = (tp-1, tp], Jp = (tp, T], and we introduce the spaces: PC(J, ℝ) = {u : J → ℝ|u C(Jk), k = 0, 1, ..., p, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/17/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/17/mathml/M4">View MathML</a> exist,k = 1, 2, ..., p,} with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/17/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/17/mathml/M5">View MathML</a>, and PC1(J, ℝ) = {u : J → ℝ|u C1(Jk), k = 0, 1, ..., p, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/17/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/17/mathml/M6">View MathML</a> exist,k = 1, 2, ..., p,} with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/17/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/17/mathml/M7">View MathML</a>. Obviously, PC(J, ℝ) and PC1(J, ℝ) are Banach spaces.

Definition 1.1 A function u PC1(J, ℝ) with its Caputo derivative of order α existing on J is a solution of (1.1) if it satisfies (1.1).

For convenience, we give some notations:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/17/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/17/mathml/M8">View MathML</a>

where Λ = (b + 1)(c + d) - (a + b)(d - 1) ≠ 0.

Lemma 1.1 [26]For a given y C[0, T], a function u is a solution of the impulsive mixed boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/17/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/17/mathml/M9">View MathML</a>


if and only if u is a solution of the impulsive fractional integral equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/17/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/17/mathml/M10">View MathML</a>



<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/17/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/17/mathml/M11">View MathML</a>

2 Uniqueness and existence results

Define an operator T : PC(J, ℝ) → PC(J, ℝ) as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/17/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/17/mathml/M12">View MathML</a>


Lemma 2.1 The operator T : PC(J, ℝ) → PC(J, ℝ) is completely continuous.

Proof. Observe that T is continuous in view of the continuity of f, Ik and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/17/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/17/mathml/M13">View MathML</a>. Let Ω ⊂ PC(J, ℝ) be bounded, where Ω = {u PC(J, ℝ): ∥u∥ ≤ r}. Then, there exist positive constants Li > 0(i = 1, 2, 3) such that |f(t, u(t))| ≤ L1, |Ik(u)| ≤ L2 and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/17/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/17/mathml/M14">View MathML</a>u ∈ Ω. Thus, ∀u ∈ Ω, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/17/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/17/mathml/M15">View MathML</a>


Since t ∈ [0, T], therefore there exists a positive constant L, such that ∥Tu∥ ≤ L, which implies that the operator T is uniformly bounded.

On the other hand, for any t Jk, 0 ≤ k p, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/17/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/17/mathml/M16">View MathML</a>

Hence, for t1, t2 Jk, t1 < t2, 0 ≤ k p, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/17/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/17/mathml/M17">View MathML</a>

which implies that T is equicontinuous on all Jk, k = 0, 1, 2, ..., p. Thus, by the Arzela-Ascoli Theorem, the operator T : PC(J, ℝ) → PC(J, ℝ) is completely continuous.

We need the following known results to prove the existence of solutions for (1.1).

Theorem 2.1 [40]Let E be a Banach space. Assume that is an open bounded subset of E with θ ∈ Ω and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/17/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/17/mathml/M18">View MathML</a>be a completely continuous operator such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/17/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/17/mathml/M19">View MathML</a>

Then T has a fixed point in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/17/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/17/mathml/M20">View MathML</a>.

Theorem 2.2 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/17/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/17/mathml/M21">View MathML</a>and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/17/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/17/mathml/M22">View MathML</a>, then the problem (1.1) has at least one solution.

Proof. In view of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/17/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/17/mathml/M21">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/17/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/17/mathml/M22">View MathML</a>, then there exists a constant r > 0 such that |f(t, u)| ≤ δ1|u|, |Ik(u)| ≤ δ2|u| and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/17/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/17/mathml/M23">View MathML</a> for 0 < |u| < r,

where δi > 0(i = 1, 2, 3) satisfy the inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/17/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/17/mathml/M24">View MathML</a>


Let us set Ω = {u ∈ PC(J, ℝ) | ∥u∥ < r} and take u ∈ PC(J, ℝ) such that ∥u∥ = r, that is, u Ω. Then, by the process used to obtain (2.2), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/17/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/17/mathml/M25">View MathML</a>


Thus, it follows that ∥Tu∥ ≤ ∥u∥, u Ω. Therefore, by Theorem 2.1, the operator T has at least one fixed point, which in turn implies that the problem (1.1) has at least one solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/17/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/17/mathml/M26">View MathML</a>.

Theorem 2.3 Assume that there exist positive constants Ki(i = 1, 2, 3) such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/17/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/17/mathml/M27">View MathML</a>

for t J, u, v ∈ ℝ and k = 1, 2, ..., p.

Then the problem (1.1) has a unique solution if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/17/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/17/mathml/M28">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/17/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/17/mathml/M29">View MathML</a>


Proof. Denote F(s) = |f(s, u(s)) - f(s,v(s))| + |A(s)u(s) - A(s)v(s)|.

For u, v PC(J, ℝ), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/17/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/17/mathml/M30">View MathML</a>

Thus, we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/17/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/17/mathml/M31">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/17/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/17/mathml/M32">View MathML</a> is given by (2.5). As <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/17/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/17/mathml/M28">View MathML</a>, therefore, T is a contraction. Thus, the conclusion of the theorem follows by the contraction mapping principle. This completes the proof.

3 Examples

Example 3.1 Consider the following fractional order impulsive mixed boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/17/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/17/mathml/M33">View MathML</a>


where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/17/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/17/mathml/M34">View MathML</a>and p = 1.

Clearly all the assumptions of Theorem 2.2 hold. Thus, the conclusion of Theorem 2.2 applies and the impulsive fractional mixed boundary value problem (3.1) has at least one solution.

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

GW and LZ completed the main part of this paper, GS corrected the main theorems and gave an example. All authors read and approved the final manuscript.


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