PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 99(113) (2016), 227–235 DOI: 10.2298/PIM1613227Z

EXISTENCE RESULTS FOR NONLINEAR IMPULSIVE q_{k}-INTEGRAL BOUNDARY VALUE PROBLEMS

Lihong Zhang, Bashir Ahmad, and Guotao Wang

Abstract. We investigate a nonlinear impulsive q_{k}-integral boundary value problem by means of Leray–Schauder degree theory and contraction mapping principle. The conditions ensuring the existence and uniqueness of solutions for the problem are presented. An illustrative example is discussed.

1. Introduction

We investigate the existence and uniqueness of solutions for a nonlinear impulsive q_{k}-integral boundary value problem

(1.1)  \begin{aligned}D_{q_{k}} u(t) & = f(t, u(t)), 0 < q_{k} < 1, t ∈ J′, \\ Δu(t_{k}) & = I_{k}(u(t_{k})), k = 1, 2, …, m, \\ u(T) & = ∑ _{i=0}^{m} ∫_{t_{i}}^{t_{i+1}} g(s, u(s)) d_{q_{i}} s, \end{aligned}

where D_{q_{k}} are q_{k}-derivatives (k = 0, 1, 2, …, m), f, g ∈ C(J × ℝ, ℝ), I_{k} ∈ C(ℝ, ℝ), J = [0, T](T > 0), 0 = t_{0} < t_{1} < ⋯< t_{k} < ⋯< t_{m} < t_{m+1} = T, J′ = J\{t_{1}, t_{2}, …, t_{m}}, and Δu(t_{k}) = u(t_{k}^{+} ) − u(t_{k}^{−} ), u(t_{k}^{+} ) and u(t_{k}^{−} ) denote the right and the left limits of u(t) at t = t_{k} (k = 1, 2, …, m) respectively.

The study of q-difference equations, initiated with the pioneer work of Jackson [1], has been developed over the years. The concept of q-calculus corresponds to the classical calculus without the idea of limit. This subject is also known as quantum calculus and finds its applications in a variety of disciplines such as special functions, super-symmetry, control theory, operator theory, combinatorics, initial and boundary value problems of q-difference equations, etc. For the systematic development of q-calculus, we refer the reader to the books [2–4] and papers [5–10].

2010 Mathematics Subject Classification: 39A13; 05A13.

Key words and phrases: q_{k}-difference equation, impulse, q_{k}-integral boundary conditions, Leray–Schauder degree theory, contraction mapping principle.

Partially supported by National Natural Science Foundation of China (No. 11501342) and the Scientific and Technologial Innovation Programs of Higher Education Institutions in Shanxi (Nos. 2014135 and 2014136).

Communicated by Stevan Pilipović.

The importance of q-difference equations lies in the fact that these equations are always completely controllable and appear in the q-optimal control problems [11]. The variational q-calculus is regarded as a generalization of the continuous variational calculus due to the presence of an extra-parameter q that may be physical or economical in its nature. The variational calculus on the q-uniform lattice includes the study of the q-Euler equations and its applications to the isoperimetric and Lagrange problems and commutation equations. In other words, it suffices to solve the q-Euler-Lagrange equation for finding the extremum of the functional involved instead of solving the Euler–Lagrange equation [12]. Further details can be found in [13–16].

The initial and boundary value problems of impulsive fractional differential equations have been extensively investigated by many researchers, for instance, see [17–25]) and references therein. In a recent paper [26], the authors discussed the existence and uniqueness of solutions for impulsive q_{k}-difference equations.

Motivated by [26], the present work is devoted to the study of impulsive q_{k}- difference equations with integral boundary condition. The paper is organized as follows. In Section 2, we present some basic concepts of the topic and an auxiliary lemma. Section 3 contains the main results, while an illustrative example is discussed in Section 4.

2. Preliminaries

Let us set J_{0} = [0, t_{1}], J_{1} = (t_{1}, t_{2}], ⋯, J_{m−1} = (t_{m−1} , t_{m}], J_{m} = (t_{m}, T] and introduce the space: P C(J, ℝ) = {u : J → ℝ | u ∈ C(J_{k}), k = 0, 1, ⋯, m, and u(t_{k}^{+} ) exist for k = 1, 2, ⋯, m} with the norm ∥u∥ = sup_{t∈J} |u(t)|. Obviously P C(J, ℝ) is a Banach space.

Next we recall some basic concepts of q_{k}-calculus [26]. For 0 < q_{k} < 1 and t ∈ J_{k}, we define the q_{k}-derivatives of a real valued continuous function f as

(2.1)  D_{q_{k}} f(t) = \frac{f(t) − f(q_{k}t + (1 − q_{k})t_{k})}{(1 − q_{k})(t − t_{k})} , D_{q_{k}} f(t_{k}) = lim_{t→t_{k}} D_{q_{k}} f(t).

Higher order q_{k}-derivatives are given by

D_{q_{k}}^{0} f(t) = f(t), D_{q_{k}}^{n} f(t) = D_{q_{k}} D_{q_{k}}^{n−1} f(t), n ∈ ℕ, t ∈ J_{k}.

The q_{k}-integral of a function f is defined by

(2.2) _{t_{k}} I_{q_{k}} f(t) := ∫_{t_{k}}^{t} f(s) d_{q_{k}} s = (1−q_{k})(t−t_{k}) ∑_{n=0}^{∞} q_{k}^{n}f(q_{k}^{n}t+(1−q_{k}^{n})t_{k}), t ∈ J_{k},

provided the series converges. If a ∈ (t_{k}, t) and f is defined on the interval (t_{k}, t), then

∫_{a}^{t} f(s) d_{q_{k}} s = ∫_{t_{k}}^{t} f(s) d_{q_{k}} s − ∫_{t_{k}}^{a} f(s) d_{q_{k}} s.

Observe that

D_{q_{k}} (_{t_{k}} I_{q_{k}} f(t)) = D_{q_{k}} ∫_{t_{k}}^{t} f(s) d_{q_{k}} s = f(t),

EXISTENCE RESULTS FOR NONLINEAR IMPULSIVE q_{k} -INTEGRAL... 229

t_{k} I_{q_{k}} (D_{q_{k}} f(t)) = ∫_{t_{k}}^{t} D_{q_{k}} f(s) d_{q_{k}} s = f(t),
aI_{q_{k}} (D_{q_{k}} f(t)) = ∫_{a}^{t} D_{q_{k}} f(s) d_{q_{k}} s = f(t) − f(a), a ∈ (t_{k}, t).

In the case t_{k} = 0 and q_{k} = q in (2.1) and (2.2), then D_{q_{k}} f = D_{q}f, _{t_{k}} I_{q_{k}} f = 0I_{q}f, where D_{q} and _{0}I_{q} are the well-known q-derivative and q-integral of the function f(t) and are defined by

D_{q}f(t) = \frac{f(t) − f(qt)}{(1 − q)t} , _{0}I_{q}f(t) = ∫_{0}^{t} f(s) d_{q}s = ∑_{n=0}^{∞} t(1 − q)q^{n}f(tq^{n}).

Lemma 2.1. For a given σ(t) ∈ C(J, ℝ), a function u ∈ P C(J, ℝ) is a solution of the following impulsive q_{k}-integral boundary value problem

(2.3)  \{ \begin{matrix}D_{q_{k}} u(t) = σ(t), 0 < q_{k} < 1, t ∈ J′, \\ Δu(t_{k}) = I_{k}(u(t_{k})), k = 1, 2, …, m, \\ u(T) = ∑_{i=0}^{m} ∫_{t_{i}}^{t_{i+1}} g(s, u(s)) d_{q_{i}} s, \end{matrix}

if and only if u satisfies the q_{k}-integral equation (2.4)

u(t) = \{ \begin{matrix}∫_{0}^{t} σ(s) d_{q_{0}}s + ∑_{i=0}^{m} ∫_{t_{i}}^{t_{i+1}} [g(s, u(s)) − σ(s)]d_{q_{i}} s − ∑_{i=1}^{m} I_{i}(u(t_{i})), t ∈ J_{0}; \\ ∫_{t_{k}}^{t} σ(s) d_{q_{k}} s + ∑_{i=0}^{k−1} ∫_{t_{i}}^{t_{i+1}} σ(s) d_{q_{i}} s + ∑_{i=0}^{m} ∫_{t_{i}}^{t_{i+1}} [g(s, u(s)) − σ(s)]d_{q_{i}} s \\ − ∑_{i=k+1}^{m} I_{i}(u(t_{i})), t ∈ J_{k}. \end{matrix}

Proof. Let u be a solution of q_{k}-integral boundary value problem (2.3). For t ∈ J_{0}, applying the operator _{0}I_{q_{0}} on both sides of D_{q_{0}}u(t) = σ(t), we get

u(t) = u(0) + _{0}I_{q_{0}}σ(t) = u(0) + ∫_{0}^{t} σ(s) d_{q_{0}}s.

Thus, u(t_{1}^{−} ) = u(0)+∫_{0}^{t_{1}} σ(s) d_{q_{0}}s. For t ∈ J_{1}, applying the operator _{t_{1}^{+}} I_{q_{1}} on both sides of D_{q_{1}}u(t) = σ(t) yields

u(t) = u(t_{1}^{+} ) + ∫_{t_{1}}^{t} σ(s) d_{q_{1}}s.

Taking into account the condition: Δu(t_{1}) = u(t_{1}^{+} ) − u(t_{1}^{−} ) = I_{1}(u(t_{1})), we obtain

u(t) = u(0) + ∫_{t_{1}}^{t} σ(s) d_{q_{1}}s + ∫_{0}^{t_{1}} σ(s) d_{q_{0}}s + I_{1}(u(t_{1})), ∀t ∈ J_{1}.

Repeating the above process, it is found that

(2.5) u(t) = u(0) + ∫_{t_{k}}^{t} σ(s) d_{q_{k}} s + ∑_{i=0}^{k−1} ∫_{t_{i}}^{t_{i+1}} σ(s) d_{q_{i}} s + ∑_{i=1}^{k} I_{i}(u(t_{i})), t ∈ J_{k}.

Substituting t = T in (2.5), we have

(2.6)  u(T) = u(0) + ∑_{i=0}^{m} ∫_{t_{i}}^{t_{i+1}} σ(s) d_{q_{i}} s + ∑_{i=1}^{m} I_{i}(u(t_{i})).

Using the boundary condition given by (2.3) in (2.6), we obtain

u(t) = ∫_{t_{k}}^{t} σ(s) d_{q_{k}} s + ∑_{i=0}^{k−1} ∫_{t_{i}}^{t_{i+1}} σ(s) d_{q_{i}} s + ∑_{i=0}^{m} ∫_{t_{i}}^{t_{i+1}} [g(s, u(s)) − σ(s)]d_{q_{i}} s

− ∑_{i=k+1}^{m} I_{i}(u(t_{i})), t ∈ J_{k}.

Conversely, assume that u satisfies q_{k}-integral equation (2.4). Then, by applying the operator D_{q_{k}} on both sides of (2.4) and using t = T, we obtain (2.3). □

3. Main results

By Lemma 2.1, the nonlinear impulsive q_{k}-integral boundary value problem (1.1) can be transformed into an equivalent fixed point problem: u = Gu, where the operator G : P C(J, ℝ) → P C(J, ℝ) is defined by

(Gu)(t)= ∫_{t_{k}}^{t} f(s, u(s)) d_{q_{k}} s + ∑_{i=0}^{k−1} ∫_{t_{i}}^{t_{i+1}} f(s, u(s)) d_{q_{i}} s

+ ∑_{i=0}^{m} ∫_{t_{i}}^{t_{i+1}} [g(s, u(s)) − f(s, u(s))]d_{q_{i}} s − ∑_{i=k+1}^{m} I_{i}(u(t_{i})).

One can notice that the existence of a fixed point of the operator G implies the existence of a solution of problem (1.1).

To show the existence of solutions for problem (1.1), we rely on Leray–Schauder degree theory and Banach fixed point theorem.

Theorem 3.1. Assume that (H_{1}) there exist nonnegative constants a, b, c, d and e such that \frac{(2a+c)T +me}{1−(2b+d)T} > 0 and

|f(t, u)| ⩽ a + b|u|, |g(t, u)| ⩽ c + d|u|, |I_{k}(u)| ⩽ e, k = 1, 2, …, m,

for all t ∈ J, u ∈ ℝ. Then impulsive q_{k}-integral boundary value problem (1.1) has at least one solution.

Proof. In the first step, it will be shown that the operator G : P C(J, ℝ) → P C(J, ℝ) is completely continuous. Let H ⊂ P C(J, ℝ) be bounded. Then, for ∀t ∈ J, u ∈ H, we have |f(t, u)| ⩽ L_{1}, |g(t, u)| ⩽ L_{2}, |I_{k}(u)| ⩽ L_{3}, where L_{i} (i = 1, 2, 3) are constants and k = 1, 2, …, m. Hence, for (t, u) ∈ J × H, the following inequality holds

|(Gu)(t)|⩽ ∫_{t_{k}}^{t} |f(s, u(s))|d_{q_{k}} s + ∑_{i=0}^{k−1} ∫_{t_{i}}^{t_{i+1}} |f(s, u(s))|d_{q_{i}} s

+ ∑_{i=0}^{m} ∫_{t_{i}}^{t_{i+1}} [|g(s, u(s))| + |f(s, u(s))|]d_{q_{i}} s + ∑_{i=k+1}^{m} |I_{i}(u(t_{i}))|

⩽ L_{1}(t − t_{k}) + L_{2} ∑_{i=0}^{k−1} (t_{i+1} − t_{i}) + (L_{1} + L_{2}) ∑_{i=0}^{m} (t_{i+1} − t_{i}) + (m − k)L_{3}

⩽ T L_{1} + T (L_{1} + L_{2}) + (m − k)L_{3}

⩽ T (2L_{1} + L_{2}) + mL_{3} =: L (constant).

This implies that ∥Gu∥ ⩽ L. Furthermore, for any t′, t′′∈ J_{k} (k = 0, 1, 2, …, m) such that t′ < t′′< T, we have

(3.1)  |(Gu)(t′′) − (Gu)(t′)|⩽ | ∫_{t_{k}}^{t′′} f(s, u(s)) d_{q_{k}} s − ∫_{t_{k}}^{t′} f(s, u(s)) d_{q_{k}} s|

⩽ ∫_{t′}^{t′′} |f(s, u(s))|d_{q_{k}} s ⩽ L_{1}(t′′− t′).

As t′ → t′′, the right-hand side of (3.1) tends to zero. Thus, G(H) is a relatively compact set. Therefore, by the Arzelá–Ascoli theorem, the operator G is compact. Also, continuity of functions f, g and I_{k} imply that G is a continuous operator. In consequence, it follows that the operator G is completely continuous.

Now let us define H(λ, u) = λGu, u ∈ P C(J, ℝ), λ ∈ [0, 1] and note that h_{λ}(u) = u − H(λ, u) = u − λGu is completely continuous.

Next, we fix R = \frac{(2a+c)T +me}{1−(2b+d)T} + 1 and define a set B_{R} = \{u ∈ P C(J, ℝ) | ∥u∥ < R}. To arrive at the desired conclusion, it is sufficient to show that G : B̄_{R} → P C(J, ℝ) satisfies

(3.2)  u ≠ λGu, ∀u ∈ ∂B_{R} ∀λ ∈ [0, 1].

Suppose that (3.2) is not true. Then, there exists some λ ∈ [0, 1] such that u = λGu for any u ∈ ∂B_{R} and t ∈ J. Thus, we have

|u(t)| = |λ(Gu)(t)| ⩽ ∫_{t_{k}}^{t} |f(s, u(s))|d_{q_{k}} s + ∑_{i=0}^{k−1} ∫_{t_{i}}^{t_{i+1}} |f(s, u(s))|d_{q_{i}} s

+ ∑_{i=0}^{m} ∫_{t_{i}}^{t_{i+1}} [|g(s, u(s))| + |f(s, u(s))|]d_{q_{i}} s + ∑_{i=k+1}^{m} |I_{i}(u(t_{i}))|

⩽ ∫_{t_{k}}^{t}(a + b|u(s)|) d_{q_{k}} s + ∑_{i=0}^{k−1} ∫_{t_{i}}^{t_{i+1}} (a + b|u(s)|) d_{q_{i}} s

+ ∑_{i=0}^{m} ∫_{t_{i}}^{t_{i+1}} (c + d|u(s)| + a + b|u(s)|) d_{q_{i}} s + ∑_{i=k+1}^{m} e

⩽ (a + b∥u∥)[(t − t_{k}) + ∑_{i=0}^{k−1} (t_{i+1} − t_{i})]

+ [a + c + (b + d)∥u∥] ∑_{i=0}^{m} (t_{i+1} − t_{i}) + (m − k)e

⩽ (2b + d)T ∥u∥ + (2a + c)T + me,

which leads to a contradiction: ∥u∥ ⩽ \frac{(2a+c)T +me}{1−(2b+d)T} < R. Hence our supposition is false and (3.2) is true. Applying the homotopy invariance of topological degree, it follows that

deg(h_{λ}, B_{R}, 0) = deg(I − λG, B_{R}, 0) = deg(h_{1}, B_{R}, 0)
= deg(h_{0}, B_{R}, 0) = deg(I, B_{R}, 0) = 1 ≠ 0, 0 ∈ B_{R},

where I is the unit operator. Since deg(I −G, B_{R}, 0) = 1, the operator G has at least one fixed point in B_{R} by the solvability of topological degree. Thus, the impulsive q_{k}-integral boundary value problem (1.1) has at least one solution in B_{R}. □

To prove the uniqueness of solutions, we list the following assumptions: (H_{2}) there exist nonnegative continuous functions M(t) and N(t) such that

\begin{matrix}|f(t, u) − f(t, v)| & ⩽ M(t)|u − v|, \\ |g(t, u) − g(t, v)| & ⩽ N(t)|u − v|, \end{matrix} for all t ∈ J, u, v ∈ ℝ.

(H_{3}) there exists a positive constant K such that

|I_{k}(u) − I_{k}(v)| ⩽ K|u − v|, u, v ∈ ℝ, k = 1, 2, …, m.

In the sequel, we set

M^{∗} = max_{t∈J}|f(t, 0)|, N^{∗} = max_{t∈J}|g(t, 0)|, γ = ∑_{i=0}^{m} _{t_{i}} I_{q_{i}} (2M + N)(t_{i+1} ) + mK,

β = (2M^{∗} + N^{∗})T, B_{r} = \{u ∈ P C(J, ℝ) | ∥u∥ ⩽ r\}, r ⩾ \frac{β}{1 − γ} .

Theorem 3.2. Let γ < 1 and the conditions (H_{2}) − (H_{3}) hold. Then the impulsive q_{k}-integral boundary value problem (1.1) has a unique solution in B_{r}.

Proof. Firstly, we show that the operator G maps B_{r} into itself. For ∀t ∈ J_{k}, u ∈ B_{r}, by (H_{2}) and (H_{3}), we find that

|(Gu)(t)|⩽ ∫_{t_{k}}^{t} |f(s, u(s))|d_{q_{k}} s + ∑_{i=0}^{k−1} ∫_{t_{i}}^{t_{i+1}} |f(s, u(s))|d_{q_{i}} s

+ ∑_{i=0}^{m} ∫_{t_{i}}^{t_{i+1}} [|g(s, u(s))| + |f(s, u(s))|]d_{q_{i}} s + ∑_{i=k+1}^{m} |I_{i}(u(t_{i}))|

⩽ ∫_{t_{k}}^{t} [|f(s, u(s)) − f(s, 0)| + |f(s, 0)|]d_{q_{k}} s

+ ∑_{i=0}^{k−1} ∫_{t_{i}}^{t_{i+1}} [|f(s, u(s)) − f(s, 0)| + |f(s, 0)|]d_{q_{i}} s

EXISTENCE RESULTS FOR NONLINEAR IMPULSIVE q_{k} -INTEGRAL... 233

+ ∑_{i=0}^{m} ∫_{t_{i}}^{t_{i+1}} [|g(s, u(s)) − g(s, 0)| + |g(s, 0)|

+ |f(s, u(s)) − f(s, 0)| + |f(s, 0)|]d_{q_{i}} s + ∑_{i=k+1}^{m} |I_{i}(u(t_{i}))|

⩽ ∫_{t_{k}}^{t}(M(s)|u(s)| + M^{∗}) d_{q_{k}} s + ∑_{i=0}^{k−1} ∫_{t_{i}}^{t_{i+1}} (M(s)|u(s)| + M^{∗}) d_{q_{i}} s

+ ∑_{i=0}^{m} ∫_{t_{i}}^{t_{i+1}} [(M + N)(s)|u(s)| + (M^{∗} + N^{∗})]d_{q_{i}} s + mK∥u∥

⩽ [_{t_{k}} I_{q_{k}} M(t) + ∑_{i=0}^{k−1} _{t_{i}} I_{q_{i}} M(t_{i+1} ) + ∑_{i=0}^{m} _{t_{i}} I_{q_{i}} (M + N)(t_{i+1} ) + mK]∥u∥

+ M^{∗}t + (M^{∗} + N^{∗})T

⩽ [ ∑_{i=0}^{m} _{t_{i}} I_{q_{i}} (2M + N)(t_{i+1} ) + mK]∥u∥ + (2M^{∗} + N^{∗})T

⩽ γ∥u∥ + β ⩽ r,

which implies that G(B_{r}) ⊂ B_{r}.

Next, we show that G is a contractive map. For each u, v ∈ P C(J, ℝ), it follows by (H_{2}) and (H_{3}) that

|(Gu)(t) − (Gv)(t)|

⩽ ∫_{t_{k}}^{t} |f(s, u(s)) − f(s, v(s))|d_{q_{k}} s + ∑_{i=0}^{k−1} ∫_{t_{i}}^{t_{i+1}} |f(s, u(s)) − f(s, v(s))|d_{q_{i}} s

+ ∑_{i=0}^{m} ∫_{t_{i}}^{t_{i+1}} [|g(s, u(s)) − g(s, v(s))| + |f(s, u(s)) − f(s, v(s))|]d_{q_{i}} s

+ ∑_{i=k+1}^{m} |I_{i}(u(t_{i})) − I_{i}(v(t_{i}))|

⩽ ∫_{t_{k}}^{t} M(s)∥u − v∥d_{q_{k}} s + ∑_{i=0}^{k−1} ∫_{t_{i}}^{t_{i+1}} M(s)∥u − v∥d_{q_{i}} s

+ ∑_{i=0}^{m} ∫_{t_{i}}^{t_{i+1}} (M + N)(s)∥u − v∥d_{q_{i}} s + mK∥u − v∥

⩽ [_{t_{k}} I_{q_{k}} M(t) + ∑_{i=0}^{k−1} _{t_{i}} I_{q_{i}} M(t_{i+1} ) + ∑_{i=0}^{m} _{t_{i}} I_{q_{i}} (M + N)(t_{i+1} ) + mK]∥u − v∥

⩽ γ∥u − v∥.

This implies that ∥Gu − Gv∥ ⩽ γ∥u − v∥. Clearly G is a contraction in view of the assumption γ < 1. Hence, the conclusion of Theorem 3.2 follows by contraction mapping principle due to Banach. □

4. Example

Consider the following nonlinear impulsive q_{k}-integral boundary value problem

(4.1) \begin{matrix}D_{\frac{2}{3+k}} u(t) = 5 + \frac{u(t)}{3 + u^{2}(t)} , t ∈ [0, 1], t ≠ \frac{k}{1 + k} , \\ Δu( \frac{k}{1 + k} ) = 10 sinu( \frac{k}{1 + k} ) , k = 1, 2, \\ u(1) = ∫_{0}^{1/2} (3s + \frac{1}{5}u(s)e^{−u^{2} (s)} ) d_{2/3} s \\ + ∫_{1/2}^{2/3} (3s + \frac{1}{5}u(s)e^{−u^{2} (s)} ) d_{1/2} s + ∫_{2/3}^{1} (3s + \frac{1}{5}u(s)e^{−u^{2} (s)} ) d_{2/5} s. \end{matrix}

Here, q_{k} = \frac{2}{3+k} (k = 0, 1, 2), t_{k} = \frac{k}{1+k} (k = 1, 2), f(t, u) = 5+ \frac{u}{3+u^{2}} , I_{k}(u) = 10 sinu, g(t, u) = 3t + \frac{1}{5}ue^{−u^{2}} . Clearly |f(t, u)| ⩽ 5 + \frac{1}{3}|u|, |g(t, u)| ⩽ 3 + \frac{1}{5}|u|, |I_{k}(u)| ⩽ 10. Selecting a = 5, b = \frac{1}{3}, c = 3, d = \frac{1}{5} and e = 10, all the conditions of Theorem 3.1 hold. Hence, by the conclusion of Theorem 3.1, there exists at least one solution for the problem (4.1).

Acknowledgment. We thank the anonymous reviewer for his/her constructive comments that led to improvement of the original manuscript.

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School of Mathematics and Computer Science (Received 09 04 2014) Shanxi Normal University (Revised 24 07 2015) Linfen, Shanxi People’s Republic of China

 zhanglih149@126.com

NAAM-Research Group Department of Mathematics Faculty of Science King Abdulaziz University Jeddah Saudi Arabia

 bashirahmad_{−} qau@yahoo.com

School of Mathematics and Computer Science Shanxi Normal University Linfen, Shanxi People’s Republic of China

 wgt2512@163.com