Abstract
This article studies the existence and uniqueness of solutions for impulsive semilinear 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.
Keywords:
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. [16]. For example, one could mention the problem of anomalous diffusion [79], the nonlinear oscillation of earthquake can be modeled with fractional derivative [10], and fluiddynamic 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 [1424] 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 [2536].
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
where ^{C}D^{α }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), and denote the right and the left limits of u(t) at t = t_{k}(k = 1, 2, ..., p), respectively and Δu'(t_{k}) 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 ( [3739]).
Let J_{0 }= [0, t_{1}], J_{1 }= (t_{1}, t_{2}], ..., J_{p1 }= (t_{p1}, t_{p}], J_{p }= (t_{p}, T], and we introduce the spaces: PC(J, ℝ) = {u : J → ℝu ∈ C(J_{k}), k = 0, 1, ..., p, and exist,k = 1, 2, ..., p,} with the norm , and PC^{1}(J, ℝ) = {u : J → ℝu ∈ C^{1}(J_{k}), k = 0, 1, ..., p, and exist,k = 1, 2, ..., p,} with the norm . Obviously, PC(J, ℝ) and PC^{1}(J, ℝ) are Banach spaces.
Definition 1.1 A function u ∈ PC^{1}(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:
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
if and only if u is a solution of the impulsive fractional integral equation
where
2 Uniqueness and existence results
Define an operator T : PC(J, ℝ) → PC(J, ℝ) as
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, I_{k }and . Let Ω ⊂ PC(J, ℝ) be bounded, where Ω = {u ∈ PC(J, ℝ): ∥u∥ ≤ r}. Then, there exist positive constants L_{i }> 0(i = 1, 2, 3) such that f(t, u(t)) ≤ L_{1}, I_{k}(u) ≤ L_{2 }and ∀u ∈ Ω. Thus, ∀u ∈ Ω, we have
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 ∈ J_{k}, 0 ≤ k ≤ p, we have
Hence, for t_{1}, t_{2 }∈ J_{k}, t_{1 }< t_{2}, 0 ≤ k ≤ p, we have
which implies that T is equicontinuous on all J_{k}, k = 0, 1, 2, ..., p. Thus, by the ArzelaAscoli 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 Ω is an open bounded subset of E with θ ∈ Ω and let be a completely continuous operator such that
Theorem 2.2 Let and , then the problem (1.1) has at least one solution.
Proof. In view of and , then there exists a constant r > 0 such that f(t, u) ≤ δ_{1}u, I_{k}(u) ≤ δ_{2}u and for 0 < u < r,
where δ_{i }> 0(i = 1, 2, 3) satisfy the inequality
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
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 .
Theorem 2.3 Assume that there exist positive constants K_{i}(i = 1, 2, 3) such that
for t ∈ J, u, v ∈ ℝ and k = 1, 2, ..., p.
Then the problem (1.1) has a unique solution if , where
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
Thus, we obtain , where is given by (2.5). As , therefore, T is a contraction. Thus, the conclusion of the theorem follows by the contraction mapping principle. This completes the proof.
3 Examples
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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