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Infinitely Many Solutions for a Class of Fractional Impulsive Coupled Systems with (p, q)-Laplacian.

1. Introduction and Main Results

In this paper, we are concerned with existence of infinitely many solutions for the following fractional impulsive differential system with (p, q)-Laplacian:

[mathematical expression not reproducible], (1)

where T > 0, [alpha] [member of] (1/p,1] with p > 1, [beta] [member of] (1/q,1] with q > 1, [[PHI].sub.s](x) = [[absolute value of x].sup.s-2]x (s > 1 and s = p,q), and (sub.t][D.sup.[alpha].sub.T](or [sub.t][D.sup.[beta].sub.T]) denotes the right Riemann-Liouville fractional derivative of order [alpha] (or [beta]), [mathematical expression not reproducible] is the left Caputo fractional derivative of order a (or [beta]), [rho], [gamma] [member of] [L.sup.[infinity]] ([0, T], [R.sup.+]), [I.sub.i],[H.sub.j] : [R.sup.N] (N [greater than or equal to] 1) [right arrow] R are continuously differentiable, 0 = [t.sub.0] < [t.sub.1] < ... < [t.sub.l+1] = T, 0 = [s.sub.0] < [s.sub.1] < ... < [s.sub.m+1] = T, and

[mathematical expression not reproducible] (2)

where

[mathematical expression not reproducible], (3)

i = 1,2, ..., l, j = 1,2, ..., m, and W: [0, r] x [R.sup.N] x [R.sup.N] [right arrow] R satisfies W(i, x, y) = -K(t, x, y) + F(t, x, y) and the following assumptions.

(W0) W(t, x, y) is measurable in t for each (x, y) [member of] [R.sup.N] x [R.sup.N], continuously differentiable in (x, y) [member of] [R.sup.N] x [R.sup.N] for a.e. t [member of] [0,T], and there exist [a.sub.1],[a.sub.2] [member of] C([R.sup.+], [R.sup.+]) and b [member of] [L.sup.[infinity]]([0,T]; [R.sup.+]) such that

[mathematical expression not reproducible], (4)

for all (x, y) [member of] [R.sup.N] x [R.sup.N] and a.e. t [member of] [0,T].

It is well known that critical point theory is a very important and effective tool to investigate the existence and multiplicity of various solutions for partial differential equations, ordinary differential equations, Hamiltonian systems, difference equations, and so on. Lots of important and interesting results have been established (see, e.g., [1-11] and reference therein). In 2011, Jiao and Zhou [12] first used critical point theory to investigate the existence of solutions for a class of fractional boundary value problems. Since then, critical point theory has also become an effective tool to obtain the existence and multiplicity results of solutions for various fractional differential equations (see, e.g., [13-17] and reference therein). Particularly, in [14], Zhao et al. considered existence of solutions for the following fractional coupled differential system with a parameter:

[mathematical expression not reproducible], (5)

where [lambda] > 0 is a parameter, a,b [member of] [L.sup.[infinity]]([0, T]; [R.sup.+]), and F : [0, T] x [R.sup.2] [right arrow] R. By using a critical point theorem in [18], they obtained system (5) which has at least three weak solutions. In [13], Li et al. investigated a class of fractional coupled differential systems with a parameter:

[mathematical expression not reproducible], (6)

where [lambda] > 0 is a parameter, a,b [member of] [L.sup.[infinity]]([0, T]; [R.sup.+]), and F : [0, T] x [R.sup.2] [right arrow] R. By using the least action principle and symmetric mountain pass theorem, they obtained system (6) which has at least one solution under asymptotically quadratic case and has infinitely many solutions under superquadratic case. For the superquadratic case, they assumed the following well-known Ambrosetti-Rabinowitz (AR) condition.

(AR) There are constants [mu] > 2, M > 0 such that

0 < [mu]F (t, x, y) [less than or equal to] x [F.sub.x] (t, x, y) + y[F.sub.y] (t, x, y) (7)

for all t [member of] [0, T] and [absolute value of x] + [absolute value of y] [greater than or equal to] M.

Over the past ten years, integer order impulsive differential equations with different boundary value conditions have been investigated deeply via variational methods (e.g., see [19-26] and reference therein). Recently, Bonanno et al. [27] and Rodriguez-Lopez and Tersian [28] were concerned with the following second-order impulsive fractional differential equation:

[mathematical expression not reproducible], (8)

where [alpha] [member of] (1/2,1], [lambda] [member of] (0,+[infinity]) and [mu] [member of] (0,+[infinity]) are two parameters, f [member of] C([0,T] x R, R), [Q.sub.i] [member of] C(R, R), and a [member of] C([0, T], R). By using variational methods, they obtained some existence results about one or three solutions of (8). Subsequently, in [29], Nyamoradi and RodriguezLopez investigated the existence and multiplicity of solutions for (8) with [lambda] = [mu] = 1. They obtained some existence results about one or infinitely many solutions of (8) by using the least action principle, the mountain pass theorem, and the symmetric mountain pass theorem. In [30],Y. Zhao and Y. Zhao investigated the existence and multiplicity of solutions for a class of perturbed fractional differential system with impulsive effects and one parameter, and they obtained system that has at least one or two nontrivial solutions by using two abstract critical point theorems due to [31]. In [32], Heidarkhani et al. investigated the multiplicity of solutions for a class of perturbed fractional differential system with impulsive effects and two parameters, and they obtained that system has infinitely many solutions by using the smooth version of an abstract critical point theorem due to [33]. In [34], Zhao et al. investigated the existence of solution for (8) with [lambda] = [mu] = 1. By using the Morse theory and local linking argument, they obtained that equation has at least one nontrivial solution.

In [35], Zhao and Tang investigated the following impulsive fractional differential equations with p-Laplacian:

[mathematical expression not reproducible], (9)

where f [member of] C([0,T] x R, R) and [Q.sub.i] [member of] C(R, R). By using the mountain pass theorem, a critical point theorem in [36], and symmetric mountain pass theorem, they obtained two multiplicity results of solutions for (9). In detail, they obtained the following theorems.

Theorem A (see [35]). Suppose the following conditions hold.

(A1) There exists a constant [mu] > p such that [Q.sub.i](u)u [less than or equal to] [mu][[integral].sup.u.sub.0][Q.sub.i](s)ds < 0 for any u [member of] [E.sup.[alpha],p]\{0}, i = 1,2, ..., I, where [E.sup.[alpha],p.sub.0] is defined in Section 2.

(A2) There exists a constant [theta] [member of] (p, p] such that [theta]F (t, u) [less than or equal to] f(t,u)u for all u [member of] [E.sup.[alpha],p] and t [member of] [0, T], where F(t,u) = [[integral].sup.u.sub.0] f(t,s)ds.

(A3) There exist constants [delta],[gamma] > 0 such that [F.sup.0] [less than or equal to] [delta] and [F.sub.[infinity]][greater than or equal to] [gamma], where

[mathematical expression not reproducible] (10)

(A4) There exist constants [[delta].sub.i] > 0 such that [[infinity].sup.u.sub.0] [Q.sub.i](s)ds [greater than or equal to] -[[delta].sub.i][[absolute value of u].sup.[mu]] for all u [E.sup.[alpha],p]\{0}, i =1,2, ..., l.

Then (9) has at least two weak solutions.

Theorem B (see [35]). Suppose (A1)-(A4) hold and f(t,u) and [I.sub.i](u) are odd about u, where i = 1, ..., l. Then (9) has infinitely many weak solutions.

Motivated by [12-14, 35], in this paper, we investigate the existence of infinitely many solutions for system (1). Obviously, system (1) is more general and complex than system (5), system (6), and (9). We present some techniques in [35], which were applied to fractional p-Laplacian impulsive differential equation and can also be applied to fractional (p, q)-Laplacian impulsive differential system, and present some more relaxed superquadratic conditions for nonlinearities than those in [35]. It is remarkable that the fractional coupled (p, q)-Laplacian differential systems are different from the fractional p-Laplacian differential equations. One stark difference is that the solutions of system (1) are the combination of [u.sub.1] and [u.sub.2] but not of (5), which causes the fact that system (1) number is possibly more than that of (9) and, hence, it is impossible that system (1) reduces to system (9). Moreover, since, in general, p [not equal to] q and we present more relaxed superquadratic conditions, it is difficult to prove the boundness of Cerami sequence (see the definition in Section 2 below) and we have to develop some techniques on inequalities. When [alpha] = [beta] = 1, system (1) becomes the following integer order (p, q)-Laplacian impulsive differential system:

[mathematical expression not reproducible], (11)

There have been some results on existence and multiplicity of solutions for integer order (p, q)-Laplacian impulsive differential systems with different boundary value conditions (see, e.g., [4, 37, 38]). However, system (11) which has Dirichlet boundary value is different from those systems in [4, 37, 38] and our assumptions on W are more relaxed than the well-known (AR) condition. Hence, our results are still new for integer order (p, q)-Laplacian impulsive differential systems. Next, we state our results.

Theorem 1. Suppose that (W0) the following conditions hold.

(A) [[rho].sup.-] := [essinf.sub.[0,T]][rho](t) > 0, [[gamma].sup.-] := [essinf.sub.[0,T]][gamma](t) > 0.

(W1) W(t,x,y) is even in (x,y) [member of] [R.sup.N] x [R.sup.N] and W(t,0,0) [equivalent to] 0 for a.e. t [member of] [0,T].

(K1) There exist constants [[theta].sub.1] [member of] [0,p) [[theta].sub.2] [member of] [0,q), [d.sub.1], [d.sub.2] > 0, [M.sub.1] > 0 such that

[mathematical expression not reproducible] (12)

for a.e. t [member of] [0,T] and all (x,y) [member of] [R.sub.N] x [R.sub.N] with [absolute value of x] + [absolute value of y] [greater than or equal to] [M.sub.1].

(K2) There exists a positive constant [M.sub.2] such that

([[nabla].sub.x]K(t,x,y),x) + ([[nabla].sub.y]K(t,x,y),y) [less than or equal to] max {p,q} K (t, x, y) (13)

For a.e. t [member of] [0,] and all (x,y) [member of] [R.sup.N] x [R.sup.N] with [absolute value of x] + [absolute value of y] [greater than or equal to] [M.sub.2].

(K3) There exists positive constants [c.sub.1], [c.sub.2], [M.sub.3] such that

K(t,x,y) [greater than or equal to] [c.sub.1] [[absolute value of x].sup.p] + [c.sub.2] [[absolute value of y].sup.q] (14)

for a.e. t [member of] [0,T] and all (x,y) [member of] [R.sup.N] x [R.sup.N] with [absolute value of x] + [absolute value of y] [less than or equal to] [M.sub.3].

(F1) There exist [L.sub.1] [member of] (0,1], [b.sub.1] [member of] (0, min{[[rho].sup.-]/p,[c.sub.1]}), [b.sub.2] [member of] (0, min{[[gamma].sup.-]/q, [c.sub.2]}) such that

F(t,x,y) [less than or equal to] [b.sub.1][[absolute value of ].sup.p] + [b.sub.2] [[absolute value of y].sup.q] (15)

for a.e. t [member of] [0, T] and all (x,y) [member of] [R.sup.N] x [R.sup.N] with [absolute value of x] + [absolute value of y] [less than or equal to] [L.sub.1]. (F2)

[mathematical expression not reproducible] (16)

uniformly for a.e. t [member of] [0, T].

(F3) There exist constants [xi],[[eta].sub.1],[[eta].sub.2],[L.sub.2] [member of] (0,+[infinity]) and [v.sub.1], [v.sub.2] [member of] [0, min{p,q}) such that

[mathematical expression not reproducible] (17)

for a.e. t [member of] [0, T] and all (x, y) [member of] [R.sup.N] x [R.sup.N] with [absolute value of x] + [absolute value of y] > [L.sub.2].

(I1) There exist constants [d.sub.3] > 0, [d.sub.4] > 0, [[theta].sub.3] [member of] [0, p), [G.sub.1] > 0 such that

[mathematical expression not reproducible], (18)

for all x [member of] [R.sup.N] with [absolute value of x] [greater than or equal to] [G.sub.1], i=1, ..., l.

(Hl) There exist constants [d.sub.5] > 0, [d.sub.6] > 0, [[theta].sub.4] [member of] [0,q), [G.sub.2] > 0 such that

[mathematical expression not reproducible] (19)

for all y [member of] [R.sup.N] with [absolute value of y] [greater than or equal to] [G.sub.2], j = 1, ..., m.

(I2) There exist constants [d.sub.7] > 0, [[theta].sub.5] [member of] [0, min{p,q}), [G.sub.3] > 0 such that

[mathematical expression not reproducible] (20)

for all x [member of] [R.sup.N] with [absolute value of x] [less than or equal to] [G.sub.3], i=1, ..., l.

(H2) There exist constants [d.sub.8] >0, [[theta].sub.6] [member of] [0,min{p,q}), [G.sub.4] > 0 such that

[mathematical expression not reproducible] (21)

for all y [member of] [R.sup.N] with [absolute value of y] < [G.sub.4], j = 1, ..., m.

(13) [I.sub.i](0) = 0, [I.sub.i](x) is even in x [member of] [R.sup.N], i=1, ..., l.

(H3) [H.sub.j](0) = 0, [H.sub.j](y) is even in y [member of] [R.sup.N], j = 1, ..., m.

Then system (1) has an unbounded sequence of weak solutions.

Theorem 2. Suppose that (W0), (A), (W1),(K1)-(K3), (F1)-(F3), (11), (H1), (13), (H3), and the following conditions hold.

(IH2)' There exist positive constants [d'.sub.7], [d'.sub.8] with

[mathematical expression not reproducible], (22)

and [G'.sub.3] > 0 such that

[I.sub.i](x) [greater than or equal to] -[d'.sub.7][[absolute value of x].sup.min{p,q}] (23)

for all x [member of] [R.sup.N] with [absolute value of x] [less than or equal to] [G'.sub.3], i = 1, ..., l, and [G'.sub.4] >0 such that

[H.sub.j](y) [greater than or equal to] [-d'.sub.8][[absolute value of y].sup.min{p,q]}] (24)

for all y [member of] [R.sup.N] with [absolute value of y] [less than or equal to] [G'.sub.4], j = 1, ..., m, where

[mathematical expression not reproducible]. (25)

Then system (1) has an unbounded sequence of weak solutions.

It is easy to prove that the following (AR)' condition implies that (F2) and (F3) hold.

(AR)' there are constants [mu] > [max.sub.{p, q}, [L.sub.2] > 0 such that

0 < [mu]F (t,x,y) [less than or equal to] ([[nabla].sub.x]F(t,x,y),x) + ([[nabla].sub.y]F(t,x,y),y) (26)

for a.e. t [member of] [0, T] and all (x, y) [member of] [R.sup.N] x [R.sup.N] with [absolute value of x] + [absolute value of y] [greater than or equal to] [L.sub.2].

Indeed, obviously, (AR)' implies that (F3) holds with [v.sub.1] = [v.sub.2] = 0 and 1/[xi] = [mu] - max{p,q}. Moreover, by the proof of Theorem 1.2 in [4], (AR)' and (W0) imply that there exist positive constants [B.sub.i], i = 1,2,3,4 such that

F (t, x, y) [greater than or equal to] [B.sub.1] [[absolute value of x].sup.[mu]] + [B.sub.2][[absolute value of y].sup.[mu]] - [B.sub.3] - [B.sub.4]b(t) (27)

for all (x, y) [member of] [R.sup.N] x [R.sup.N] and a.e. t [member of] [0, T], and so it is easy to see that (F2) holds. Then by Theorems 1 and 2, we have the following corollaries.

Corollary 3. Suppose that (W0), (A), (W1), (K1)-(K3), (F1), (11)-(13), (H1)-(H3), and (AR)' hold. Then system (1) has an unbounded sequence of weak solutions.

Corollary 4. Assume that (W0), (A), (W1), (K1)-(K3), (F1), (11), (13), (H1), (H3), (IH2)', and (AR)' hold. Then system (1) has an unbounded sequence of weak solutions.

Remark 5. There exist examples satisfying Theorems 1 and 2. For example, let p > 1, q > 1, [rho](t) = [gamma](t) = [t.sup.2] + 1, [I.sub.i](x) = [[absolute value of x].sup.p], i=1, ..., l, [H.sub.j](y) = [[absolute value of y].sup.p], j=1, ..., m, and

[mathematical expression not reproducible], (28)

for all (x, y) [member of] [R.sup.N] x [R.sup.N] and a.e. t [member of] [0, T], where [[theta].sub.1] [member of] (0, p] and [[theta].sub.2] [member of] (0, q].

With similar proofs of Theorems 1 and 2, we can obtain the corresponding theorems for the following p-Laplacian system:

[mathematical expression not reproducible], (29)

where W(t,x) = -K(t,x) + F(t,x) for all x [member of] [R.sup.N] (N [greater than or equal to] 1) and a.e. t [member of] [0,T].

Theorem 6. Suppose that the following conditions hold.

(W0)' W(t, x) is measurable in t for each x [member of] [R.sup.N], continuously differential in x [member of] [R.sup.N] for a.e. t [member of] [0,T], and there exist a [member of] C([R.sup.+], [R.sup.+]) and b [member of] [L.sup.1]([0, T]; [R.sup.+]) such that

[mathematical expression not reproducible], (30)

for all x [member of] [R.sup.N] and a.e. t [member of] [0, T].

(A)' [[rho].sup.-] := [essinf.sub.[0,T]][rho](t) > 0.

(W1)' W(t, x) is even in x [member of] [R.sup.N] and W(t, 0) [equivalent to] 0 for a.e. t [member of] [0, T].

(K1)' There exist [[theta].sub.1] [member of] [0,p), [d.sub.1] > 0, [M.sub.1] > 0 such that

[mathematical expression not reproducible] (31)

for a.e. t [member of] [0, T] and all x [member of] [R.sup.N] with [absolute value of x] [greater than or equal to] [M.sub.1].

(K2)' There exists a positive constant [M.sub.2] such that

([[nabla].sub.x]K(t, x), x) [less than or equal to] pK(t, x), (32)

for a.e. t [member of] [0, T] and all x [member of] [R.sup.N] with [absolute value of x] [greater than or equal to] [M.sub.2].

(K3)' There exist positive constants c, [M.sub.3] such that

K(t,x) [greater than or equal to] c[[absolute value of x].sup.p] (33)

for a.e. t [member of] [0, T] and all x [member of] [R.sup.N] with [absolute value of x] [less than or equal to] [M.sub.3].

(F1)' There exist [L.sub.1] [member of] (0, 1], b [member of] (0, min[[[rho].sup.-]/p, c}) such that

F(t,x) [less than or equal to] b [[absolute value of x].sup.p] (34)

for a.e. t [member of] [0, T] and all x [member of] [R.sup.N] with [absolute value of x] [less than or equal to] [L.sub.1].

(F2)'

[mathematical expression not reproducible]. (35)

(F3)' There exist constants [xi], [eta], [L.sub.2] [member of] (0, +[infinity]) and v [member of] [0, p) such that

(p + 1/[xi]+[eta][[absolute value of x].sup.v])F(t, x, y) [less than or equal to] ([[nabla].sub.x]F(t, x), x) (36)

for a.e. t [member of] [0, T] and all x [member of] [R.sup.N] with [absolute value of x] > [L.sub.2].

(I1)' There exist constants [d.sub.2] > 0, [[theta].sub.2] [member of] [0,p), [G.sub.1] > 0 such that

[mathematical expression not reproducible] (37)

for all x [member of] [R.sup.N] with [absolute value of x][greater than or equal to][G.sub.1], i=1, ..., l.

(I2)' There exist constants [d.sub.3] >0, [[theta].sub.3] [member of] [0, p), [G.sub.2] > 0 such that

[mathematical expression not reproducible] (38)

for all x [member of] [R.sup.N] with [absolute value of x] [less than or equal to] [G.sub.2], i=1, ..., l.

(I3)' [I.sub.i](0) = 0, [I.sub.i](x) is even in x [member of] [R.sup.N], i=1, ..., l.

Then system (29) has an unbounded sequence of weak solutions.

Theorem 7. Suppose that (W0)', (A)', (W1)', (K1)'-(K3)', (F1)'-(F3)', (11)', (13)', and the following condition holds.

(I2)" There exist positive constants [d'.sub.2] with

[d'.sub.2]l[C.sup.p.sub.[alpha],p,[infinity]] [less than or equal to] min {[[rho].sup.-]/p, [c.sub.1]}-[b.sub.1], (39)

and [G'.sub.2] > 0 such that

[I.sub.i](x) [greater than or equal to] -[d'.sub.2][[absolute value of x].sup.p] (40)

for all x [member of] [R.sup.N] with [absolute value of x][less than or equal to][G'.sub.2], i=1, ..., l.

Then system (29) has an unbounded sequence of weak solutions.

Corollary 8. Suppose that (W0)', (A)', (W1)', (K1)'-(K3)', (F1)', and (11)'-(13)' and the following condition holds.

(AR)" There are constants [mu] > p, [L.sub.2] > 0 such that

0 < [mu]F (t, x) [less than or equal to] ([[nabla].sub.x]F (t, x), x) (41)

for a.e. t [member of] [0, T] and all x [member of] [R.sup.N] with [absolute value of x] [greater than or equal to] [L.sub.2].

Then system (29) has an unbounded sequence of weak solutions.

Corollary 9. Assume that (W0)', (A)', (W1)', (K1)'-(K3)', (F1)', (11)', (12)", (13)', and (AR)" hold. Thensystem (29) has an unbounded sequence of weak solutions.

Remark 10. Corollaries 8 and 9 are still different from Theorem B. Indeed, if N = 1, [rho](t) [equivalent to] 1, and K(t,x) [equivalent to] (1/p)[[absolute value of x].sup.p] for a.e. t [member of] [0, T] and all x [member of] R, system (29) reduces to (9). However, it is easy to see that (I1)' and (I2)' (or (I2)") are different from (A1). There exist examples satisfying (I1)' and (I2)' but not satisfying (A1) and (A4). For example, let N = 1 and [I.sub.i](x) = [[absolute value of x].sup.p] for all % [member of] R, i = 1, ..., l. Then [I'.sub.i](x) = p[[absolute value of x].sup.p-2]x. It is easy to see that [I.sub.i], i = 1, ..., l satisfy (I1)' and (I2)'. Set [Q.sub.i](x) = [I'.sub.i](x), i = 1, ..., l. Obviously, [Q.sub.i](x), i = 1, ..., l do not satisfy (A1). Moreover, there exist examples satisfying (F1)'-(F3)' but not satisfying (A2). For example, let

F(t, x) [equivalent to]F(x) = [[absolute value of x].sup.p]ln (1 + [[absolute value of x].sup.p]) (42)

for a.e. t [member of] [0, T]. Finally, one can also establish some results which are similar to Theorem A for system (1) and system (29) by combining those assumptions and arguments of Theorems 1 and 2 with those ideas proving Theorem A.

2. Preliminaries

In this section, we recall some known definitions and lemmas about fractional derivatives. For more details, the readers can see [12, 39-42].

Let a,b [member of] (-[infinity], +[infinity]) and

AC([a,b]) := {u : [a,b] [right arrow] [R.sup.N] | u is absolutely continuous on [a, b]}. (43)

Definition 11 (see [40, 42]). Let f [member of] AC[a,b] and [??] [member of] (0,1). [sub.a][D.sup.[??].sub.t] and [sub.t][D.sup.[??].sub.b] denote the left and right Riemann-Liouville fractional derivatives of order a for function f, respectively, which are defined by

[mathematical expression not reproducible]. (44)

Definition 12 (see [40, 42]). Let f [member of] AC[a,b] and [??] [member of] (0,1). [mathematical expression not reproducible] denote the left and right Caputo fractional derivatives of order [??] for function f, respectively, which are defined by

[mathematical expression not reproducible]. (45)

Remark 13 (see [40, 42]). When [mathematical expression not reproducible].

Let

[C.sup.[infinity]]([0, T], [R.sup.N]) := {u|u [member of] [C.sup.[infinity]]([0, T], [R.sup.N]), u(0) = u(T) = 0} (46)

with the norm [[parallel]w[parallel].sub.[infinity]] = [max.sub.[0)T]][absolute value of u(t)], and, for s > 1,

[L.sup.s]([0, T], [R.sup.N]) := {u|u: [0,T] [right arrow] [R.sup.N], [[integral].sup.T.sub.0][[absolute value of u(t)].sup.s] dt < [infinity]} (47)

with the norm [mathematical expression not reproducible].

For [??] [member of] (0,1] and s > 1, we define [E.sup.[??],s.sub.0](0, T) as the closure of [C.sup.[infinity].sub.0]([0, T], [R.sup.N]), with respect to the norm:

[mathematical expression not reproducible] (48)

Then by Proposition 3.1 in [12], E is separable and reflexive Banach space, and if u [member of] [E.sup.[??],s.sub.0](0, T), then u, [mathematical expression not reproducible] and u(0) = u(T) = 0. Moreover, by Remark 3.1 in [12], [mathematical expression not reproducible].

Proposition 14 (see [12]). Assume that [??] [member of] (0,1] and s > 1. For all u [member of] [E.sup.[??],s.sub.0](0, T),

[mathematical expression not reproducible], (49)

where [C.sub.[??]] = [T.sup.[??]]/[GAMMA]([??] + 1). Moreover, if [??] > 1/s, then

[mathematical expression not reproducible], (50)

where [C.sub.[??],s,[infinity]] := [T.sup.[??]-1/s]/[GAMMA]([??])[([??]s - s' + 1).sup.1/s'] and s' = s/(s - 1).

By Proposition 14, it is easy to obtain that

[mathematical expression not reproducible]. (51)

Proposition 15 (see [12]). Assume that 1/p < [alpha] [less than or equal to] 1 and 1 < p < [infinity], and the sequence {[u.sub.k]} converges weakly to u in [E.sup.[alpha],p.sub.0]. Then [u.sub.k] [right arrow] u in C([0,T], [R.sup.N]).

Assume that [alpha], [beta] [member of] (0,1]. Let E = [E.sup.[alpha],p.sub.0](0, T) x [E.sup.[beta],q.sub.0](0, T).

On E, define the norm:

[[parallel](u, V)[parallel].sub.E] = [[parallel]u[parallel].sub.p] + [[parallel]v[parallel].sub.q] (52)

for all (u, v) [member of] [E.sup.[alpha],p.sub.0](0, T) x [E.sup.[beta],q.sub.0](0, T) = E.

Similar to Definitions 2.4 and 2.5 in [35], we also present the following two definitions.

Definition 16. Let

[mathematical expression not reproducible]. (53)

If (u, v) satisfies the first equation of (1) for a.e. t [member of] [0, T] \ [[t.sub.1], ..., [t.sub.l]] and the second equation of (1) for a.e. t [member of] [0, T]\ [[s.sub.1] ..., [mathematical expression not reproducible], and [mathematical expression not reproducible] exist and satisfy the impulsive conditions of (1), and boundary conditions u(0) = u(T) = 0 and v(0) = v(T) = 0, then we call (u, v) a classical solution of (1).

Definition 17. For any (h, w) [member of] E, if the following two equalities

[mathematical expression not reproducible] (54)

hold then the vector function (u, v) [member of] E is called a weak solution of (1).

For (u, v) [member of] E, we define the functional J : E [right arrow] R by

[mathematical expression not reproducible] (55)

where

[mathematical expression not reproducible] (56)

It follows from (W0), the continuity of [nabla][I.sub.i], and [nabla][H.sub.j] and Theorem 5.41 in [42], that [PHI] and [PSI] are continuously differentiable and so J [member of] [C.sup.1](E, R) and

[mathematical expression not reproducible] (57)

Hence, the critical point of J is a weak solution of (1). Similar to the arguments of Propositions 2.5 and 2.6 in [35], it is easy to obtain that (u, v) is a classical solution of (1) if (u, v) [member of] E is a weak solution of (1).

Assume that E is a real Banach space and [phi] [member of] [C.sup.1](E, R). For any sequence {[u.sub.k]} c E, if [phi]([u.sub.k]) is bounded and [phi]'([u.sub.k]) [right arrow] 0 as k [right arrow] [infinity], then we call {[u.sub.k]} a Palais-Smale sequence. If any Palais-Smale sequence {[u.sub.k]} has a convergent subsequence, then we call [phi] which satisfies Palais-Smale condition.

Similar to the proofs in [39], we will also use the following symmetric mountain pass theorem to prove our main results.

Lemma 18 (see [2]). Let E be an infinite dimensional Banach space and let [phi] [member of] [C.sup.1](E, R) be even and satisfy Palais-Smale condition, and [phi](0) = 0. If E = V [direct sum] X, where V is finite dimensional, and [phi] satisfies the following, then f possesses an unbounded sequence of critical values.

(i) There are constants [rho], [xi] > 0 such that [mathematical expression not reproducible].

(ii) For each finite dimensional subspace [??] [subset] E, there is R = R([??]) such that [phi] [less than or equal to] 0 on [??]\[B.sub.R([??)]

Remark 19. As shown in [43], a deformation lemma can be proved with replacing Palais-Smale condition with Cerami condition, which implies that Lemma 2.1 in [2] is true under Cerami condition. We say that [phi] satisfies Cerami condition; that is, for every sequence {[u.sub.k]} [subset] E, {[u.sub.k]} has a convergent subsequence if [phi]([u.sub.k]) is bounded and (1 + [[parallel][u.sub.k][parallel].sub.E][[parallel][phi]'(uk)[parallel].sub.E*] [right arrow] 0 as k [right arrow] [infinity], where E* with the norm [[parallel]*[parallel].sub.E*]. is the dual space of E.

3. Proofs of Theorems

Lemma 20. Assume that (A), (K1), (K2), (F2), (F3), (11), and (H1) hold. Then J satisfies Cerami condition.

Proof. For any sequence [{([u.sub.n], [v.sub.n])}.sup.[infinity].sub.n=1] [subset] E, suppose that there is a positive constant [C.sub.1] > 0 such that

[absolute value of F([u.sub.n],[v.sub.n])] [less than or equal to] [C.sub.1], (1 + [[parallel]([u.sub.n],[v.sub.n])[parallel].sub.E])[[parallel]J'([u.sub.n],[v.sub.n][parallel].sub.E*] [less than or equal to] [C.sub.1]. [nabla]n [member of] N, (58)

By (F2) and assumption (W0), there exist positive constants A, [C.sub.2], and [C.sub.3] such that

F (t,x,y) [greater than or equal to] A[[absolute value of x].sup.p] + A[[absolute value of y].sup.q] - [C.sub.2] - [C.sub.3]b(t), [nabla](x,y) [member of] [R.sup.N] x [R.sup.N], a.e. t [member of] [0,T]. (59)

It follows from (F3) that

[mathematical expression not reproducible] (60)

for all (x,y) [member of] [R.sup.N] x [R.sup.N] with [absolute value of x] + [absolute value of y] > [L.sub.2] and a.e. t [member of] [0,T]. Assumption (W0) and (60) imply that there exists a positive constant [C.sub.4] such that

[mathematical expression not reproducible] (61)

for all (x, y) [member of] [R.sup.N] x [R.sup.N] and a.e. t [member of] [0,T].

Assume that p [greater than or equal to] q. Then, for any x [member of] [R.sup.N] with [absolute value of x] [greater than or equal to] 1 and all y [member of] [R.sup.N],

[mathematical expression not reproducible] (62)

and, for any x [member of] [R.sup.N] with [absolute value of x] [less than or equal to] 1 and all y [member of] [R.sup.N],

[mathematical expression not reproducible]. (63)

Then, (62) and (63) imply that

[([absolute value of x] + [absolute value of y]).sup.min{p,q}] = [less than or equal to] [2.sup.q-1]([[absolute value of x].sup.p] + [[absolute value of y].sup.q] + 1). (64)

for all (x, y) [member of] [R.sup.N] x [R.sup.N]. Similarly, if q [greater than or equal to] p, we have

[([absolute value of x] + [absolute value of y]).sup.min{p,q}] = [less than or equal to] [2.sup.q-1]([[absolute value of x].sup.p] + [[absolute value of y].sup.q] + 1). (65)

for all (x, y) [member of] [R.sup.N] x [R.sup.N]. Combining (64) and (65), we have

[[absolute value of x].sup.p]+[[absolute value of y].sup.q] [greater than or equal to] [([absolute value of x]+[absolute value of y]).sup.min{p,q}]/max[2.sup.q-1],[2.sup.p-1]-1 (66)

for all (x,y) [member of] [R.sup.N] x [R.sup.N]. Moreover, for [v.sub.1], [v.sub.2] [member of] (1,[infinity]) and [[eta].sub.1], [[eta].sub.2] [member of] (0, +[infinity]), we have

[mathematical expression not reproducible] (67)

for all (x, y) [member of] [R.sup.N] x [R.sup.N], and

[mathematical expression not reproducible]. (68)

for all (x,y) [member of] [R.sup.N] x [R.sup.N] with [absolute value of x] + [absolute value of y] [greater than or equal to] 1. It follows from assumptions (59), (61), (66), (67), and (68) that there exist positive constants [C.sub.5], [C.sub.6], [C.sub.7] such that

[mathematical expression not reproducible] (69)

for all (x, y) [member of] [R.sup.N] x [R.sup.N] with [absolute value of x] + [absolute value of y] [greater than or equal to] 1 and a.e. t [member of] [0, T]. By (69) and assumption (W0), there exist positive constants [C.sub.8], [C.sub.9] such that

[mathematical expression not reproducible] (70)

for all (x, y) [member of] [R.sup.N] x [R.sup.N] and a.e. t [member of] [0, T]. By (K1), (K2), and (W0), there exists positive constant [C.sub.10] such that

max {p,q} K(t,x,y) - ([[nabla].sub.x]K(t, x, y), x) -([[nabla].sub.y]K(t,x,y),y) [greater than or equal to] -[C.sub.10]b(t), (71)

[mathematical expression not reproducible] (72)

for all (x,y) [member of] [R.sup.N] x [R.sup.N] and a.e. t [member of] [0,T]. Moreover, by (71), (H1), and (W0), there exist positive constants C11 and [C.sub.12] such that

[mathematical expression not reproducible] (73)

for all (x, y) [member of] [R.sup.N] x [R.sup.N]. By (58) and (70), we have (max {p,q} + l)[C.sub.1] [greater than or equal to] max {p,q} J ([u.sub.n],[v.sub.n])

[mathematical expression not reproducible] (74)

which implies that there exists a positive constant [C.sub.13] such that [mathematical expression not reproducible]. Then [mathematical expression not reproducible] and [mathematical expression not reproducible]. By (60) and (F2), there exists C[1.sub.4] > 0 such that

[mathematical expression not reproducible] (75)

for all (x,y) [member of] [R.sup.N] x [R.sup.N] with [absolute value of x] + [absolute value of y] > [L.sub.2] + [C.sub.14] and a.e. t [member of] [0,T]. Let[??](t,x,y) := ([[nabla].sub.x]F(t,x,y),x) + ([[nabla].sub.y]F(t,x,y),y)-max[p,q]F(t,x,y) for all (x,y) [member of] [R.sup.N] x [R.sup.N] and a.e. t [member of] [0,T], [S.sub.1] := [t [member of][0,T]][absolute value of [u.sub.n](t)]+[absolute value of [v.sub.n](t)] > [L.sub.2] + [C.sub.14]), [S.sub.2] := [0,T]\[S.sub.1], and [LAMBDA] := min[p,q] - max[[v.sub.1], [v.sub.2]]. By (58), (W0), (A), (61), (72), (73), (74), (75), and (51), there exist positive constants [C.sub.15] and [C.sub.16] such that

[mathematical expression not reproducible] (76)

Since max{[v.sub.1], [v.sub.2]} < min{p,q}, [[theta].sub.1] < p, [[theta].sub.2] < q, [[theta].sub.3] < p, [[theta].sub.d4] < q, and 0 < [LAMBDA] < min{p,q}, the boundness of [[integral].sup.T.sub.0][[absolute value of [u.sub.n](t)].sup.[LAMBDA]] and [[integral].sup.T.sub.0][[absolute value of [v.sub.n](t)].sup.[LAMBDA]]dt implies that [[parallel][u.sub.n][parallel].sub.p] and [[parallel][v.sub.n][parallel].sub.q] are bounded. Going if necessary to a subsequence, assume that [u.sub.n] [??] u in [E.sup.[alpha],p.sub.0](0, T) and [v.sub.n] [??] [E.sup.[beta],q.sub.0](0, T). Then, by Proposition 15, we obtain [[parallel][u.sub.n]-u[parallel].sub.[infinity]][right arrow]0 and [[parallel][v.sub.n]- v[parallel].sub.[infinity]] [right arrow] 0, and so [mathematical expression not reproducible]. Similar to the arguments of Lemma 3.1 in [35], we can obtain that [mathematical expression not reproducible]. Therefore, [[parallel][u.sub.n]-u[parallel].sub.p] [right arrow] 0 and [[parallel][v.sub.n]-v[parallel].sub.q] [right arrow] 0 as n [right arrow] [infinity], which shows that [[parallel]([u.sub.n], [v.sub.n]) - (u, v)[parallel].sub.E] [right arrow] 0 as n [right arrow] [infinity].

Lemma 21. Assume that (A), (K3), (F1), (12), and (H2) hold. There are constants [rho],[xi] > 0 such that [mathematical expression not reproducible].

Proof. Choosing [rho] = min{[G.sub.3],[G.sub.4], [M.sub.3], [L.sub.1]}/max{[C.sub.[alpha],p,[infinity]], [C.sub.[beta],q,[infinity]]}. Then, for all (u, v) [member of] E [intersection] [partial derivative][B.sub.[rho]], we have [[parallel]u[parallel].sub.p] + [[parallel]v[parallel].sub.q] = [rho] and then by (51) [[parallel]u[parallel].sub.[infinity]] [less than or equal to] [C.sub.[alpha],p,[infinity]][[parallel]u[parallel].sub.p]+ [C.sub.[alpha],p,[infinity]][rho] [less than or equal to] min{[G.sub.3],[G.sub.4],[M.sub.3],[L.sub.1]} and, similarly, [[parallel]v[parallel].sub.[infinity]] [less than or equal to] min{[G.sub.3], [G.sub.4], [M.sub.3],[L.sub.1]}. Note that [b.sub.1] < min{[[rho].sup.-]/p, [c.sub.1]} and [b.sub.2] < min{[[gamma].sup.-]/q,[c.sub.2]}. Hence, it follows from (K3), (F1), (12), (H2), and (66) that

[mathematical expression not reproducible] (77)

Since [[theta].sub.5], [[theta].sub.6] [member of] [0, min{p, q}), (77) implies that there exists a sufficiently large constant [rho] > 0 such that J(u, v) > 0 for all (u, v) [member of] E with [[parallel](u, v)[parallel].sub.E] = [rho].

Lemma 22. Assume that (A), (K3), (F1), and (IH2)' hold. There are constants [rho],[xi] > 0 such that [mathematical expression not reproducible].

Proof. By (77) and (IH2)', it is easy to obtain that

[mathematical expression not reproducible]. (78)

Then (22) and (78) imply that there exists a sufficiently large constant [rho] > 0 such that J(u, v) > 0 for all (u, v) [member of] E with [[parallel](u, v)[parallel].sub.E] = [rho].

Lemma 23. For each finite dimensional subspace [??] [subset] E, there is R = R([??]) such that J(u, v) [less than or equal to] 0 on [??]\[B.sub.([??])]

Proof. For each given finite dimensional space [??] [subset] E, we claim that there exists R > 0 such that [phi](u) [less than or equal to] 0 on [??]/[B.sub.R]. Indeed, obviously, for any (u, v) [member of] E, (u, v) can be rewritten by (u, v) = (u, 0) + (0, v), where u [member of] [??] [subset] [E.sup.[alpha],p.sub.0](0, T), v [member of] [??] [subset] [E.sup.[beta],q.sub.0](0, T), and [[??].sub.1] and [[??].sub.2] are finite dimensional ones. So there exist positive constants dg, [d.sub.10], [d.sub.11], [d.sub.12], [d.sub.13], [d.sub.14], [d.sub.15], [d.sub.16] such that

[mathematical expression not reproducible] (79)

Similar to the proof of Theorem 1.1 in [4], by (K2) and assumption (W0), there exists positive constant [D.sub.0] such that

K (t,x,y) [less than or equal to] [D.sub.0]b(t)[[absolute value of x].sup.max{p,q}] + [D.sub.0]b(t)[[absolute value of y].sup.max{p,q}]+[D.sub.0]b(t) (80)

for all (x, y) [member of] [R.sup.N] x [R.sup.N] and a.e. t [member of] [0, T]. By (F2), there exist

[mathematical expression not reproducible] (81)

and positive constant [M.sub.4] such that

F(t,x,y) [greater than or equal to] [D.sub.1][[absolute value of x].sup.p]+[D.sub.1][[absolute value of y].sup.q]

for all (x,y) [member of] [R.sup.N] x [R.sup.N] with [absolute value of x] + [absolute value of y] [greater than or equal to] [M.sub.4] and a.e. t [member of] [0, T]. Then by (82) and assumption (W0), there exist positive constants [D.sub.2] and [D.sub.3] such that

F (t,x,y) [greater than or equal to] [D.sub.1] [[absolute value of x].sup.p] + [D.sub.0][[absolute value of y].sup.q] - [D.sub.2] - [D.sub.3]b(t) (83)

for all (x,y) [member of] [R.sup.N] x [R.sup.N] and a.e. t [member of] [0,T]. By (11) and (H1), there exist positive constants [D.sub.4], [D.sub.5] such that

[I.sub.i](x) [less than or equal to] [d.sub.3][[absolute value of x].sup.p] + [D.sub.4]b(t), [H.sub.j](y) [less than or equal to] [d.sub.5][[absolute value of y].sup.q] + [D.sub.5]b(t) (84)

for all (x, y) [member of] [R.sup.N] x [R.sup.N] and a.e. t [member of] [0, T]. It follows from (83), (80), (84), and (79) that

[mathematical expression not reproducible] (85)

Thus, (81) and the above inequality imply that there exists a sufficiently large constant R > 0 such that J(u, v) < 0 if [[parallel]u[parallel].sub.p] = [[parallel]v[parallel].sub.q] = R/2.

Proof of Theorems 1 and 2. By (W1), (13), and (H3), we have J(0, 0) = 0 and J is even. Similar to the proof of Lemma 4.4 in [44], we can choose [e.sub.j] [member of] E \ {0}, j = 1,2, ..., and define V = span{[e.sub.1]} and X = [bar.span{[e.sub.j], j = 2, ...}]. Then E = V [direct sum] X. Note that [partial derivative][B.sub.[rho]][intersection]X [subset] [partial derivative][B.sub.[rho]][intersection]E. Thus by Lemma 21 (or Lemma 22) and Lemma 23 it is easy to see that (i) and (ii) of Lemma 18 hold. Lemma 20 implies that J satisfies Cerami condition. So, by Lemma 18 and Remark 19, there exists a sequence {([u.sub.n], [v.sub.n])} such that J([u.sub.n], [v.sub.n]) [right arrow] +[intersection] and then it is easy to see that [parallel]([u.sub.n], [v.sub.n])[parallel] [right arrow] [infinity] as n [right arrow] [infinity].

https://doi.org/10.1155/2018/9256192

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This project is supported by the National Natural Science Foundation of China (no. 11226135 and no. 11301235).

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Junping Xie (iD) (1,2) and Xingyong Zhang (iD) (3,4)

(1) School of Geosciences and Info-Physics, Central South University, Changsha, Hunan 410083, China

(2) Key Laboratory of Metallogenic Prediction of Nonferrous Metals and Geological Environment Monitoring, Central South University, Ministry of Education, Changsha, China

(3) Faculty of Science, Kunming University of Science and Technology, Kunming, Yunnan 650500, China

(4) School of Mathematics and Statistics, Central South University, Changsha, Hunan 410083, China

Correspondence should be addressed to Xingyong Zhang; zhangxingyong1@163.com

Received 23 January 2018; Accepted 28 March 2018; Published 8 May 2018

Academic Editor: Antonio Iannizzotto
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Title Annotation:Research Article
Author:Xie, Junping; Zhang, Xingyong
Publication:Discrete Dynamics in Nature and Society
Geographic Code:9CHIN
Date:Jan 1, 2018
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