AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 10.1155/2016/5784273 5784273 Research Article Existence of Infinitely Many Periodic Solutions for Perturbed Semilinear Fourth-Order Impulsive Differential Inclusions Ferrara Massimiliano 1 http://orcid.org/0000-0003-1953-5198 Caristi Giuseppe 2 http://orcid.org/0000-0002-4558-1860 Salari Amjad 3 Zafer Agacik 1 Department of Law and Economics Mediterranea University of Reggio Calabria Via dei Bianchi 2 89131 Reggio Calabria Italy unirc.it 2 Department of Economics University of Messina Via dei Verdi 75 98122 Messina Italy unime.it 3 Department of Mathematics Faculty of Sciences Razi University Kermanshah 67149 Iran razi.ac.ir 2016 632016 2016 28 10 2015 11 02 2016 2016 Copyright © 2016 Massimiliano Ferrara et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper discusses the existence of infinitely many periodic solutions for a semilinear fourth-order impulsive differential inclusion with a perturbed nonlinearity and two parameters. The approach is based on a critical point theorem for nonsmooth functionals.

1. Introduction

The goal of this paper is to establish the existence of infinitely many periodic solutions for the following perturbed semilinear fourth-order impulsive differential inclusion: (1) u i v x - p u x + a x u x λ F u x + μ G u x , u x , x 0 , T Q , Δ u x k = I 1 k u x k , - Δ u x k = I 2 k u x k , k = 1,2 , , m , u 0 = u 0 = u T = u T = 0 , where p is a positive constant, a is continuous positive even 2 T -periodic function on R , T > 0 , 0 = x 0 < x 1 < < x m < x m + 1 = T , Q = { x 1 , x 2 , , x m } , F ( u ) and G u ( x , u ) are generalized gradients of F and G u , respectively, the operator Δ is defined as Δ u x k u ( x k + ) - u ( x k - ) , with u ( x k + ) and u ( x k - ) denoting the right and left limits, respectively, of u ( x ) at x = x k , I 1 k , I 2 k C ( R , R ) , k = 1,2 , , m , λ > 0 , μ 0 , and F : R R is a locally Lipschitz function satisfying the following:

F ( 0 ) = 0 and F ( s ) = - F ( - s ) for all s R ;

there exist two constants c > 0 and r [ 1,2 ) , such that (2) ξ c 1 + s r - 1 , ξ F s , s R .

Also, G is a function defined on [ 0 , T ] × R , satisfying the following:

G ( · , s ) : R R is measurable for each s R , G ( x , · ) : R R is locally Lipschitz for x [ 0 , T ] Q , G ( x , 0 ) = 0 , and - G s ( - x , - s ) = G s ( x , s ) = G s ( x + 2 T , s ) for a.e. x [ 0 , T ] Q and s R ;

there exists a constant b > 0 , such that (3) ξ b 1 + s r - 1 s R , ξ G s x , s ,

where r is defined in ( F 2 ) .

We study the existence of solutions, that is, absolutely continuous on every ( x k , x k + 1 ) and left continuous at x k functions which satisfy (1) for a.e. x with (possible) jumps (impulses) at x k .

Fourth-order ordinary differential equations act as models for the bending or deforming of elastic beams, and, therefore, they have important applications in engineering and physical sciences. Boundary value problems for fourth-order ordinary differential equations have been of great concern in recent years (e.g., see ). On the other hand, impulsive differential equations occur in many applications such as various mathematical models including population dynamics, ecology, biological systems, biotechnology, industrial robotic, pharmacokinetics, and optimal control. For the general aspects of impulsive differential equations, we refer the reader to . In association with this development, a theory of impulsive differential equations has been given extensive attention. Very recently, some researchers have studied the existence and multiplicity of solutions for impulsive fourth-order two-point boundary value problems; we refer the reader to  and references therein. Differential inclusions arise in models for control systems, mechanical systems, economical systems, game theory, and biological systems to name a few.

Recently, multiplicity of solutions for differential inclusions via nonsmooth variational methods and critical point theory has been considered and here we cite the papers . For example, in  the existence of infinitely many antiperiodic solutions for second-order impulsive differential inclusions has been discussed. In , Kristály employing a nonsmooth Ricceri-type variational principle , developed by Marano and Motreanu , has established the existence of infinitely many radially symmetric solutions for a differential inclusion problem in R N . Also, in , the authors extended a recent result of Ricceri concerning the existence of three critical points of certain nonsmooth functionals. Two applications have been given, both in the theory of differential inclusions; the first one concerns a nonhomogeneous Neumann boundary value problem and the second one treats a quasilinear elliptic inclusion problem in the whole R N . Tian and Henderson in , based on a nonsmooth version of critical point theory of Ricceri due to Iannizzotto , have established the existence of at least three solutions for the a second-order impulsive differential inclusion with a perturbed nonlinearity and two parameters. In , three periodic solutions with prescribed wavelength for a class of semilinear fourth-order differential inclusions are obtained by using a nonsmooth version critical point theorem.

In the present paper, motivated by [13, 18, 19], employing an abstract critical point result (see Theorem 7 below), we are interested in ensuring the existence of infinitely many periodic solutions for problem (1); see Theorem 12. We refer to , in which related variational methods are used for nonhomogeneous problems.

To the best of our knowledge, no investigation has been devoted to establishing the existence of infinitely many solutions to a problem such as (1). As one reference on impulsive differential inclusions, we can refer to .

A special case of our main result is the following theorem.

Theorem 1.

Assume that ( F 1 ) and ( F 2 ) hold, and I i ( 0 ) = 0 and I i ( s ) s > 0 , s R , i = 1,2 , , m . Furthermore, suppose that (4) liminf ξ + s u p t ξ F t ξ 2 = 0 , limsup ξ + 0 1 F ξ x 2 - x d x 131 / 60 ξ 2 + k = 1 m 0 ξ 2 x k - 1 I 1 k s d s + k = 1 m 0 ξ x k 2 - x k I 2 k s d s = + . Then, the problem (5) u i v x - u x + u x F u x , x 0,1 Q , Δ u x k = I 1 k u x k , - Δ u x k = I 2 k u x k , k = 1,2 , , m , u 0 = u 0 = u 1 = u 1 = 0 admits a sequence of classical solutions.

2. Basic Definitions and Preliminary Results

Let ( X , · X ) be a real Banach space. We denote by X the dual space of X , while · , · stands for the duality pairing between X and X . A function φ : X R is called locally Lipschitz if, for all u X , there exist a neighborhood U of u and a real number L > 0 such that (6) φ v - φ w L v - w X v , w U . If φ is locally Lipschitz and u X , the generalized directional derivative of φ at u along the direction v X is (7) φ u ; v limsup w u , τ 0 + φ w + τ v - φ w τ . The generalized gradient of φ at u is the set (8) φ u u X : u , v φ u ; v v X . So φ : X 2 X is a multifunction. We say that φ has compact gradient if φ maps bounded subsets of X into relatively compact subsets of X .

Lemma 2 (see [<xref ref-type="bibr" rid="B21">24</xref>, Proposition 1.1]).

Let φ C 1 ( X ) be a functional. Then φ is locally Lipschitz and (9) φ u ; v = φ u , v u , v X ; φ u = φ u u X .

Lemma 3 (see [<xref ref-type="bibr" rid="B21">24</xref>, Proposition 1.3]).

Let φ : X R be a locally Lipschitz functional. Then φ ( u ; · ) is subadditive and positively homogeneous for all u X , and (10) φ u ; v L v u , v X , with L > 0 being a Lipschitz constant for φ around u .

Lemma 4 (see [<xref ref-type="bibr" rid="B9">25</xref>]).

Let φ : X R be a locally Lipschitz functional. Then φ : X × X R is upper semicontinuous and, for all λ 0 , u , v X , (11) λ φ u ; v = λ φ u ; v . Moreover, if φ , ψ : X R are locally Lipschitz functionals, then (12) φ + ψ u ; v φ u ; v + ψ u ; v u , v X .

Lemma 5 (see [<xref ref-type="bibr" rid="B21">24</xref>, Proposition 1.6]).

Let φ , ψ : X R be locally Lipschitz functionals. Then (13) λ φ u = λ φ u u X , λ R , φ + ψ u φ u + ψ u u X .

Lemma 6 (see [<xref ref-type="bibr" rid="B14">14</xref>, Proposition 1.6]).

Let φ : X R be a locally Lipschitz functional with a compact gradient. Then φ is sequentially weakly continuous.

We say that u X is a (generalized) critical point of a locally Lipschitz functional φ if 0 φ ( u ) ; that is, (14) φ u ; v 0 v X . When a nonsmooth functional, g : X ( - , + ) , is expressed as a sum of a locally Lipschitz function, φ : X R , and a convex, proper, and lower semicontinuous function, j : X ( - , + ) , that is, g φ + j , a (generalized) critical point of g is every u X such that (15) φ u ; v - u + j v - j u 0 for all v X (see [24, Chapter 3]).

Henceforth, we assume that X is a reflexive real Banach space, N : X R is a sequentially weakly lower semicontinuous functional, Υ : X R is a sequentially weakly upper semicontinuous functional, λ is a positive parameter, j : X ( - , + ) is a convex, proper, and lower semicontinuous functional, and D ( j ) is the effective domain of j . Write (16) M Υ - j , I λ N - λ M = N - λ Υ + λ j . We also assume that N is coercive and (17) D j N - 1 - , r for all r > i n f X N . Moreover, owing to (17) and provided that r > i n f X N , we can define (18) φ r inf u N - 1 - , r sup v N - 1 - , r M v - M u r - N u , γ liminf r + φ r , δ liminf r inf X N + φ r . If N and Υ are locally Lipschitz functionals, in [22, Theorem 2.1] the following result is proved; it is a more precise version of [21, Theorem 1.1] (see also ).

Theorem 7.

Under the above assumption on X , N , and M , one has the following:

For every r > i n f X N and every λ ( 0,1 / φ ( r ) ) , the restriction of the functional I λ = N - λ M to N - 1 ( ( - , r ) ) admits a global minimum, which is a critical point (local minimum) of I λ in X .

If γ < + , then, for each λ ( 0,1 / γ ) , the following alternative holds: either

I λ possesses a global minimum or

there is a sequence { u n } of critical points (local minima) of I λ such that l i m n + N ( u n ) = + .

If δ < + , then, for each λ ( 0,1 / δ ) , the following alternative holds: either

there is a global minimum of N which is a local minimum of I λ or

there is a sequence { u n } of pairwise distinct critical points (local minima) of I λ , with l i m n + N ( u n ) = i n f X N , which converges weakly to a global minimum of N .

Now we recall some basic definitions and notations. We consider the reflexive Banach space X H 2 ( 0 , T ) H 0 1 ( 0 , T ) endowed with the norm (19) u = 0 T u x 2 + u x 2 + u x 2 d x 1 / 2 . Obviously, X is a reflexive Banach space and completely embedded in C ( [ 0 , T ] ) . So there exists a constant C 0 , such that C 0 = s u p u X , u 0 u / u . From the positivity of p and a , it is easy to see that (20) u X = 0 T u x 2 + p u x 2 + a x u x 2 d x 1 / 2 is also a norm of X , which is equivalent to the usual norm. Therefore, there exist two constants C 1 and C 2 such that (21) C 1 u u X C 2 u . Thus, (22) u C 0 C 1 u X .

Definition 8.

A function u X is said to be a weak solution of problem (1) such that, corresponding to it, there exists a mapping [ 0 , T ] x u ( x ) with u ( x ) λ F ( u ( x ) ) + μ G u ( x , u ( x ) ) , for a.e. x [ 0 , T ] , and having the property that, for every v X , u v L 1 [ 0 , T ] and (23) 0 T u x v x + p u x v x + a x u x v x - u x v x d x + k = 1 m I 2 k u x k v x k + k = 1 m I 1 k u x k v x k = 0 .

Definition 9.

A solution u is called a classical solution of the impulsive differential inclusion (1) if u A C 3 ( x k , x k + 1 ) and u ( x k - ) = u ( x k ) , for k = 1 , , n , and (24) u i v x + p u x + a x u x = u x , x 0 , T Q , Δ u x k = I 1 k u x k , - Δ u x k = I 2 k u x k , k = 1,2 , , m , u 0 = u 0 = u T = u T = 0 , where u ( x ) λ F ( u ( x ) ) + μ G u ( x , u ( x ) ) for a.e. x [ 0 , T ] .

Lemma 10.

If u X is a weak solution of (1), then u is a classical solution of (1).

Proof.

Let u X be a weak solution of (1). Then there exists u with u ( x ) λ F ( u ( x ) ) + μ G u ( x , u ( x ) ) for a.e. x [ 0 , T ] , satisfying (23). Using integration by parts (23) becomes (25) 0 T u x v x d x = 0 T p u x - a x u x + u x v x d x - k = 1 m I 2 k u x k v x k - k = 1 m I 1 k u x k v x k , so u H 4 ( ( 0 , T ) Q ) , and (25) holds for each v X with v ( x k ) = v ( x k ) = 0 , k = 1 , , m . Through integration by parts, we obtain (26) u i v x - p u x + a x u x - u x = 0 , a.e.   x 0 , T Q . Due to u X , similar to [26, Section 2], we can get u A C 3 ( x k , x k + 1 ) and u ( x k - ) = u ( x k ) for k = 1 , , n . Now we show that the boundary conditions are satisfied. Choose any j { 0,1 , 2 , , m - 1 } and v X such that v ( x ) = 0 if x [ x k , x k + 1 ] for k j . Then, from (25) and by means of integration by parts, we can get (27) u x j + 1 v x j + 1 - u x j v x j + x j x j + 1 u i v - p u + a x u - u v d x = 0 . Thus, (28) j = 0 m - 1 u x j + 1 v x j + 1 - u x j v x j + j = 0 m - 1 x j x j + 1 u i v - p u + a x u - u v d x = 0 , which implies that (29) u T v T - u 0 v 0 + 0 T u i v x - p u x + a x u x - u x v x d x = 0 . Since u satisfies (26), we have that u T v T - u 0 v 0 = 0 holds for all v X , which implies that u 0 = u T = 0 and in view of u X we get u ( 0 ) = u ( T ) = 0 . Now using a technique similar to the technique of [18, Lemma 3.5] shows that the impulsive conditions are satisfied. From the equality (30) u i v x v x = 0 x u i v s d s v x - 0 x u i v s d s v x ,

we have (31) 0 T u i v x v x d x = 0 T 0 x u i v s d s v x - 0 x u i v s d s v x d x = v T 0 T u i v x d x - 0 T u x - u 0 - 0 x k < x Δ u x k v x d x = u T - u 0 - k = 1 m Δ u x k v T - 0 T u x v x d x + u 0 v T - v 0 + k = 1 m x k x k + 1 0 x k < x Δ u x k v x d x = u T v T - u 0 v 0 - k = 1 m Δ u x k v T - 0 T u x v x d x - k = 1 m Δ u x k v x k + k = 1 m Δ u x k v T = u T v T - u 0 v 0 - k = 1 m Δ u x k v x k - 0 T u x v x d x . Substituting (31) into (25), we have (32) 0 T u i v x - p u x + a x u x - u x v x d x + u T v T - u 0 v 0 + k = 1 m - Δ u x k - I 2 k u x k v x k = 0 . Since u satisfies (26), we have (33) u T v T - u 0 v 0 - k = 1 m Δ u x k v x k - k = 1 m I 2 k u x k v x k = 0 for all v X . Thus, - Δ u ( x k ) = I 2 k ( u ( x k ) ) , k = 1 , , m . Similarly from the equality (34) u x v x = 0 x u s d s v x - 0 x u s d s v x , we have Δ u x k = I 1 k u x k . So u is a classical solution of (1).

Now we introduce the functionals N , F , G , and I λ by (35) N u = 1 2 u X + k = 1 m 0 u t k I 1 k s d s + k = 1 m 0 u t k I 2 k s d s , u X , F u = 0 T F u x d x , G u = 0 T G x , u x d x , u X , I λ , μ u = N u - λ F u - μ G u u X . Thus, N C 1 ( X , R ) and (36) N u , v = 0 T u x v x + p u x v x + a x u x v x d x + k = 1 m I 2 k u t k v t k + k = 1 m I 1 k u t k v t k for all u , v X .

Lemma 11.

Assume that ( F 1 ) , ( F 2 ) , ( G 1 ) , and ( G 2 ) hold. Then the functional I λ , μ : X R is locally Lipschitz. Moreover, each critical point u X of I λ , μ is a weak solution of (1).

Proof.

Let I λ , μ = N ( u ) + I 1 ( u ) , where I 1 ( u ) = - λ F ( u ) - μ G ( u ) . Since N C 1 ( X , R ) by Lemma 2, N is locally Lipschitz on X . From ( F 2 ) and ( G 2 ) , we know that I 1 is locally Lipschitz on L r ( [ 0 , T ] ) . Moreover, X is compactly embedded into L r ( [ 0 , T ] ) . Thus, I 1 is locally Lipschitz on X [27, Theorem 2.2]. According to Lemma 5, we get (37) I 1 u - λ 0 T F u x d x - μ 0 T G u x , u x d x . The explanation of (37) is as follows: for every u I 1 ( u ) , there is a corresponding mapping u ( x ) - λ F ( u ( x ) ) - μ G u ( x , u ( x ) ) for a.e. x [ 0 , T ] having the property that, for every v X , the function u v L 1 [ 0 , T ] and u , v = 0 T u ( x ) v ( x ) d x . Therefore, I λ , μ is locally Lipschitz on X .

Now we show that each critical point of I λ , μ is a weak solution of (1). Assume that u X is a critical point of I λ , μ . So (38) 0 I λ , μ u = u X : u , v I λ , μ u ; v   for   v X . So, by Lemma 2 and (38), we have (39) N u + u = 0 w i t h u I 1 u and hence u ( x ) - λ F ( u ( x ) ) - μ G u ( x , u ( x ) ) a.e. on [ 0 , T ] . It follows from (37) and (39) that for every v X we have (40) 0 T u x v x + p u x v x + a x u x v x + u x v x d x + k = 1 m I 2 k u t k v t k + k = 1 m I 1 k u t k v t k = 0 for all v X . Thus, by Definition 8, u is a weak solution of (1).

3. Main Results

First for every ξ R + we set (41) D ξ = C 2 2 2 ξ 2 4 T + T 3 3 + T 5 30 + k = 1 m 0 ξ 2 x k - T I 1 k s d s + k = 1 m 0 ξ x k 2 - T x k I 2 k s d s .

Now we formulate our main result using the following assumptions:

l i m i n f ξ + s u p t ξ F ( t ) / ξ 2 < 1 / 2 T C 1 / C 0 2 l i m s u p ξ + 0 T F ( ξ ( x 2 - T x ) ) d x / D ( ξ ) ;

I j k ( 0 ) = 0 , I j k ( s ) s > 0 , s R , j = 1,2 and k = 1,2 , , m .

Theorem 12.

Assume that ( F 1 ) ( F 3 ) and ( I 1 ) hold. Let (42) λ 1 1 limsup ξ + 0 T F ξ x 2 - T x d x / D ξ , λ 2 C 1 2 2 T C 0 2 liminf ξ + sup t ξ F t / ξ 2 . Then, for every λ ( λ 1 , λ 2 ) and every nonnegative function G satisfying ( G 1 ) , ( G 2 ) , and the assumption

G l i m ξ + 0 T s u p t ξ G ( x , t ) d x / ξ 2 < + ,

putting (43) μ G , λ 1 2 G C 1 C 0 2 1 - 2 T λ C 0 C 1 2 liminf ξ + sup t ξ F t ξ 2 , where μ G , λ = + when G = 0 , for every μ [ 0 , μ G , λ ) problem (1) admits an unbounded sequence of classical solutions in X .

Proof.

Our goal is to apply Theorem 7(b) to (1). For this purpose, we fix λ ¯ ( λ 1 , λ 2 ) and let G be a nonnegative function satisfying ( G 1 )–( G 3 ). Since λ ¯ < λ 2 , we have (44) μ G , λ ¯ = 1 2 G C 1 C 0 2 1 - 2 T λ ¯ C 0 C 1 2 liminf ξ + sup t ξ F t ξ 2 > 0 . Now fix μ ¯ ( 0 , μ g , λ ¯ ) , put ν 1 λ 1 , and (45) ν 2 λ 2 1 + 2 C 0 / C 1 2 μ ¯ / λ ¯ λ 2 G . If G = 0 , then ν 1 = λ 1 , ν 2 = λ 2 , and λ ¯ ( ν 1 , ν 2 ) . If G 0 , since μ ¯ < μ G , λ ¯ , we have (46) λ ¯ λ 2 + 2 C 0 C 1 2 μ ¯ G < 1 , and so (47) λ 2 1 + 2 C 0 / C 1 2 μ ¯ / λ ¯ λ 2 G > λ ¯ ; to wit, λ ¯ < ν 2 . Hence, taking into account that λ ¯ > λ 1 = ν 1 , one has λ ¯ ( ν 1 , ν 2 ) . Now, set (48) J x , s F s + μ ¯ λ ¯ G x , s for all ( x , s ) [ 0 , T ] × R . Assume that j is exactly zero in X and for each u X and put (49) Υ u 0 T J x , u x d x , M u Υ u - j u = Υ u , I λ ¯ u N u - λ ¯ M u = N u - λ ¯ Υ u . It is easy to prove that N is sequentially weakly lower semicontinuous on X . Obviously, N C 1 ( X ) . By Lemma 2, N is locally Lipschitz on X . By Lemma 11, F and G are locally Lipschitz on L 2 ( [ 0 , T ] ) . So, Υ is locally Lipschitz on L 2 ( [ 0 , T ] ) , and since X is compactly embedded into L 2 ( [ 0 , T ] ) , Υ is locally Lipschitz on X . In addition, Υ is sequentially weakly upper semicontinuous. For all u X , by ( I 1 ) , (50) 0 u x k I 1 k s d s > 0 , k = 1,2 , , m , 0 u x k I 2 k s d s > 0 , k = 1,2 , , m . So, we have (51) N u = 1 2 u X 2 + k = 1 m 0 u x k I 1 k s d s + k = 1 m 0 u x k I 2 k s d s > 1 2 u X 2 for all u X . Hence, N is coercive and i n f X N = N ( 0 ) = 0 . Under our hypotheses, we want to show that there exists a sequence { u ¯ n } X of critical points for the functional I λ ¯ ; that is, every element u ¯ n satisfies (52) I λ ¯ u ¯ n , v - u ¯ n 0 , for  every   v X . Now, we prove that γ < + . For this, let { ξ n } be a sequence of positive numbers such that l i m n + ξ n = + and (53) lim n + 0 T s u p t ξ n J x , t d x ξ n 2 = liminf ξ + 0 T sup t ξ J x , t d x ξ 2 . Put (54) r n 1 2 C 1 ξ n C 0 2 , n N . Then, for all v X with N ( v ) < r n , taking into account that v X 2 < 2 r n and v C 0 / C 1 v X , one has v x ξ n for every x [ 0 , T ] . Therefore, for all n N , (55) φ r n = inf u N - 1 - , r sup v N - 1 - , r M v - M u r - N u s u p v X 2 < 2 r n F v + μ ¯ / λ ¯ G v r n sup t ξ n T F t + μ ¯ / λ ¯ 0 T G x , t d x r n 2 C 0 C 1 2 T sup t ξ n F t ξ n 2 + μ ¯ λ ¯ 0 T sup t ξ n G x , t d x ξ n 2 . Moreover, from assumptions ( F 3 ) and ( G 3 ) , we have (56) lim n + T sup t ξ n F t ξ n 2 + μ ¯ λ ¯ lim n + 0 T sup t ξ n G x , t d x ξ n 2 < + , which follows (57) lim n + 0 T sup t ξ n J x , t d x ξ n 2 < + . Therefore, (58) γ liminf n + φ r n 2 C 0 C 1 2 liminf ξ + 0 T sup t ξ J x , t d x ξ 2 < + . Since (59) 0 T sup t ξ J x , s d x ξ 2 T sup t ξ F t ξ 2 + μ ¯ λ ¯ 0 T sup t ξ G x , t d x ξ 2 , and taking ( G 3 ) into account, we get (60) liminf ξ + 0 T sup t ξ J x , t d x ξ 2 liminf ξ + T sup t ξ F t ξ 2 + μ ¯ λ ¯ G . Moreover, by assumption ( G 3 ) we have (61) limsup ξ + 0 T J x , ξ x 2 - T x d x D ξ limsup ξ + 0 T F ξ x 2 - T x d x D ξ . Therefore, from (60) and (61), we observe that (62) λ ¯ ν 1 , ν 2 1 limsup ξ + 0 T J x , ξ x 2 - T x d x / D ξ , 1 2 C 0 / C 1 2 liminf ξ + 0 T sup t ξ J x , t d x / ξ 2 0 , 1 γ . For the fixed λ ¯ , inequality (58) ensures that condition (b) of Theorem 7 can be applied and either I λ ¯ has a global minimum or there exists a sequence { u n } of weak solutions of problem (1) such that l i m n u n = + . Now we prove that for the fixed λ ¯ the functional I λ ¯ has no global minimum. Let us verify that the functional I λ ¯ is unbounded from below. Since (63) 1 λ ¯ < limsup ξ + 0 T F ξ x 2 - T x d x D ξ , there exist a sequence { η n } of positive numbers and a constant τ such that l i m n + η n = + and (64) 1 λ ¯ < τ < 0 T F η n x 2 - T x d x D η n for each n N large enough. For all n N , put (65) w n x η n x 2 - T x for  every   x 0 , T . For any fixed n N , clearly w n X and one has (66) w n X 2 C 2 2 w n 2 = η n 2 C 2 2 4 T + T 3 3 + T 5 30 , and so (67) N w n C 2 2 2 η n 2 4 T + T 3 3 + T 5 30 + k = 1 m 0 η n 2 x k - T I 1 k s d s + k = 1 m 0 η n x k 2 - T x k I 2 k s d s = D η n . By (64) and (67) and since G is nonnegative we observe that (68) I λ ¯ w n = N w n - λ ¯ M w n D η n - λ ¯ 0 T F η n x 2 - T x d x < D η n 1 - λ ¯ τ for every n N large enough; since λ ¯ τ > 1 and l i m n + η n = + , it follows that (69) lim n + I λ ¯ w n = - . So, the functional I λ ¯ is unbounded from below, and it shows that I λ ¯ has no global minimum. Therefore, from part (b) of Theorem 7, the functional I λ ¯ admits a sequence of critical points { u ¯ n } X such that l i m n + N ( u ¯ n ) = + . Since N is bounded on bounded sets and taking into account that l i m n + N ( u ¯ n ) = + , then { u ¯ n } has to be unbounded; that is, (70) lim n + u ¯ n X = + . Also, if u ¯ n X is a critical point of I λ ¯ , clearly, by definition, one has (71) I λ ¯ u ¯ n , v - u ¯ n 0 , for  every   v X . Finally, by Lemma 11, the critical points of I λ ¯ are weak solutions for problem (1), and, by Lemma 10, every weak solution of (1) is a classical solution of (1). Hence, we have the result.

Remark 13.

Under the conditions (72) liminf ξ + sup t ξ F t ξ 2 = 0 , limsup ξ + 0 T F ξ x 2 - T x d x D ξ = + , from Theorem 12, we see that, for every λ > 0 and for each μ [ 0 , C 1 2 / 2 C 0 2 G ) , problem (1) admits a sequence of solutions which is unbounded in X . Moreover, if G = 0 , the result holds for every λ > 0 and μ 0 .

The following result is a special case of Theorem 12 with μ = 0 .

Theorem 14.

Assume that ( F 1 ) ( F 3 ) and ( I 1 ) hold. Then, for each (73) λ 1 limsup ξ + 0 T F ξ x 2 - T x d x / D ξ , 1 2 T C 0 / C 1 2 liminf ξ + sup t ξ F t / ξ 2 , the problem (74) u i v x - p u x + a x u x λ F u x , x 0 , T Q , Δ u x k = I 1 k u x k , - Δ u x k = I 2 k u x k , k = 1,2 , , m , u 0 = u 0 = u T = u T = 0 has an unbounded sequence of classical solutions in X .

Now, we present the following example to illustrate Theorem 14.

Example 15.

Let T = 1 , p = 1 , and a ( x ) = 1 + cos x for every x R . Thus, a is a continuous positive even 2 π -periodic function on R . Define the function (75) F s = 0 , if   s - 1,1 , s 2 - 1 , if   s - 2 , - 1 1,2 , 3 , if   s - , - 2 2 , + . Clearly, F is a continuous convex function with F ( 0 ) = 0 . An easy calculation shows that (76) F s = 0 , if   s - , - 2 , - 4,0 , if   s = - 2 , - 4 , - 2 , if   s - 2 , - 1 , - 2,0 , if   s = - 1 , 0 , if   s - 1,1 , 0,2 , if   s = 1 , 2,4 , if   s 1,2 , 0,4 , if   s = 2 , 0 , if   s 2 , + . Hence, assumptions ( F 1 ) and ( F 2 ) hold. Moreover, let Q = 1 / 3,2 / 3 , I 11 ( x ) = I 12 ( x ) = 1 / 8 x , and I 21 ( x ) = I 22 ( x ) = 2 / 5 x for every x R . Thus, ( I 1 ) is satisfied and (77) D ξ = 131 C 2 2 30 ξ 2 + 2 k = 1 2 0 ξ 2 x k - 1 I 1 k s d s + 2 k = 1 2 0 ξ x k 2 - x k I 2 k s d s = 131 C 2 2 30 ξ 2 + 17 324 . So, we have (78) liminf ξ + sup t ξ F t ξ 2 = 0 , limsup ξ + 0 1 F ξ x 2 - ξ x d x D ξ lim ξ ξ 2 - 1 131 C 2 2 / 30 ξ 2 + 17 / 324 = 30 131 C 2 2 . Therefore, by applying Theorem 14, the problem (79) u iv x - u x + a x u x λ F u x , x 0,1 1 3 , 2 3 , Δ u 1 3 = 1 8 u 1 3 , - Δ u 1 3 = 2 5 u 1 3 , Δ u 2 3 = 1 8 u 2 3 , - Δ u 2 3 = 2 5 u 2 3 , u 0 = u 0 = u 1 = u 1 = 0 for λ ( 0 , 30 / 131 C 2 2 ) has an unbounded sequence of classical solutions in H 2 ( 0,1 ) H 0 1 ( 0,1 ) .

Now we state the following consequence of Theorem 14, using the following assumptions:

liminf ξ + s u p t ξ F ( t ) / ξ 2 < C 1 2 / 2 T C 0 2 ;

limsup ξ + 0 T F ( ξ ( x 2 - T x ) ) d x / D ( ξ ) > 1 .

Corollary 16.

Assume that ( F 1 ) , ( F 2 ) , ( F 4 ) , ( F 5 ) , and ( I 1 ) hold. Then, the problem (80) u i v x - p u x + a x u x F u x , x 0 , T Q , Δ u x k = I 1 k u x k , - Δ u x k = I 2 k u x k , k = 1,2 , , m , u 0 = u 0 = u T = u T = 0 has an unbounded sequence of classical solutions in X .

Remark 17.

Theorem 1 in Introduction is an immediate consequence of Corollary 16.

Here, we give a consequence of the main result.

Corollary 18.

Let F 1 : R R be a locally Lipschitz function such that F 1 ( 0 ) = 0 , - F 1 ( - s ) = F 1 ( s ) , ξ a ( 1 + s r 1 - 1 ) for all s R , ξ F 1 ( s ) , r 1 [ 1,2 ) ( a > 0 ) . Furthermore, suppose that

liminf ξ + s u p t ξ F 1 ( t ) / ξ 2 < + ,

limsup ξ + 0 T F 1 ( ξ ( x 2 - T x ) ) d x / D ( ξ ) = + .

Then, for every function F 2 : R R which is locally Lipschitz function such that F 2 ( 0 ) = 0 , - F 2 ( - s ) = F 2 ( s ) , and ξ a ( 1 + s r 1 - 1 ) for all s R ,   ξ F 2 ( s ) ,   r 1 [ 1,2 )    ( a > 0 ) and satisfies the conditions (81) sup t R F 2 t 0 , liminf ξ + 0 T F 2 ξ x 2 - T x d x D ξ > - , for each (82) λ 0 , 1 2 T C 0 / C 1 2 liminf ξ + sup t ξ F 1 t / ξ 2 , the problem (83) u i v x - p u x + a x u x λ F 1 u x + F 2 u x , x 0 , T Q , Δ u x k = I 1 k u x k , - Δ u x k = I 2 k u x k , k = 1,2 , , m , u 0 = u 0 = u T = u T = 0 has an unbounded sequence of classical solutions in X .

Proof.

Set F ( t ) = F 1 ( t ) + F 2 ( t ) for all t R . Assumption ( C 2 ) along with the condition (84) liminf ξ + 0 T F 2 ξ x 2 - T x d x D ξ > - yields (85) limsup ξ + 0 T F ξ x 2 - T x d x D ξ = limsup ξ + 0 T F 1 ξ x 2 - T x d x + 0 T F 2 ξ x 2 - T x d x D ξ = + . Moreover, assumption ( C 1 ) together with the condition (86) sup t R F 2 t 0 ensures (87) liminf ξ + sup t ξ F t ξ 2 liminf ξ + sup t ξ F 1 t ξ 2 < + . Since (88) 1 liminf ξ + sup t ξ F t / ξ 2 1 liminf ξ + sup t ξ F 1 t / ξ 2 , by applying Theorem 14, we have the desired conclusion.

Remark 19.

We observe that in Theorem 12 we can replace ξ + with ξ 0 + , and then by the same argument as in the proof of Theorem 12, but using conclusion (c) of Theorem 7 instead of (b), problem (1) has a sequence of pairwise distinct classical solutions, which strongly converges to 0 in X .

We end this paper by presenting the following example.

Example 20.

Let T = π , p = 1 , and a ( x ) = sin x + 1 for every x R . Thus, a is a continuous positive even 2 π -periodic function on R . Now let F ( s ) = - s for all s R and G ( x , s ) = x ( 1 - cos s ) for all ( x , s ) [ 0 , π ] × R . Hence, F is locally Lipschitz function and (89) F s = 1 , if   s < 0 , - 1,1 , if   s = 0 , - 1 , if   s > 0 . Therefore, ( F 1 ) and ( F 2 ) hold, and also we can simply see that ( G 1 ) and ( G 2 ) hold. Moreover, let Q = { π / 3,2 π / 3 } , I 11 ( x ) = I 12 ( x ) = x ( 1 + x ) , and I 21 ( x ) = I 22 ( x ) = x 5 for every x R . Thus, ( I 1 ) is satisfied. An easy computation shows that (90) 0 π F ξ x 2 - x d x D ξ = ξ 0 1 x - x 2 d x + 1 π x 2 - x d x C 2 2 / 2 ξ 2 4 π + π 3 / 3 + π 5 / 30 + 2 k = 1 2 0 ξ 2 x k - π I 1 k s + 0 ξ x k 2 - π x k I 2 k s d s = ξ π 3 / 3 - π 2 / 2 + 1 / 3 C 2 2 / 2 ξ 2 4 π + π 3 / 3 + π 5 / 30 + 2 7 / 9 6 π 12 ξ 6 for some ξ R . So, (91) liminf ξ 0 + sup t ξ F t ξ 2 = 0 , limsup ξ 0 + 0 π F ξ x 2 - ξ x d x D ξ = + . Also (92) 0 π x 1 - cos t d x = π 2 2 1 - cos t 0 t R , lim ξ + 0 + sup t ξ 0 π x 1 - cos t d x ξ 2 = π 2 2 lim ξ + 0 + sup t ξ 1 - cos t ξ 2 = π 2 4 . Hence, taking Remark 19 into account, the problem (93) u iv x - u x + 2 + sin x u x λ F u x + μ G u x , u , x 0 , π π 3 , 2 π 3 , Δ u π 3 = u π 3 1 + u π 3 , - Δ u π 3 = u 5 π 3 , Δ u 2 π 3 = u 2 π 3 1 + u 2 π 3 , - Δ u 2 π 3 = u 5 2 π 3 , u 0 = u 0 = u π = u π = 0 for λ ( 0 , ) and μ [ 0 , 2 C 1 2 / C 0 2 π 3 ) has an unbounded sequence of pairwise distinct classical solutions in H 2 ( 0 , π ) H 0 1 ( 0 , π ) .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Afrouzi G. A. Heidarkhani S. O'Regan D. Existence of three solutions for a doubly eigen-value fourth-order boundary value problem Taiwanese Journal of Mathematics 2011 15 1 201 210 Bonanno G. Di Bella B. A boundary value problem for fourth-order elastic beam equations Journal of Mathematical Analysis and Applications 2008 343 2 1166 1176 10.1016/j.jmaa.2008.01.049 MR2417133 2-s2.0-41949103735 Graef J. R. Yang B. Existnence and nonexistence of positive solutions of fourth order nonlinear boundary value problems Applicable Analysis 2000 74 1-2 201 214 10.1080/00036810008840810 Ma R. Multiple positive solutions for a semipositone fourth-order boundary value problem Hiroshima Mathematical Journal 2003 33 2 217 227 MR1997695 ZBL1048.34048 Bainov D. Simeonov P. Systems with Impulse Effect 1989 Chichester, UK Ellis Horwood Ellis Horwood Series: Mathematics and Its Applications Benchohra M. Henderson J. Ntouyas S. Theory of Impulsive Differential Equations 2006 2 New York, NY, USA Hindawi Publishing Corporation Contemporary Mathematics and Its Applications Carter T. E. Necessary and sufficient conditions for optimal impulsive rendezvous with linear equations of motion Dynamics and Control 2000 10 3 219 227 10.1023/a:1008376427023 MR1809484 2-s2.0-0034217074 Heidarkhani S. Ferrara M. Salari A. Infinitely many periodic solutions for a class of perturbed second-order differential equations with impulses Acta Applicandae Mathematicae 2015 139 81 94 10.1007/s10440-014-9970-4 MR3400583 Liu X. Willms A. R. Impulsive controllability of linear dynamical systems with applications to maneuvers of spacecraft Mathematical Problems in Engineering 1996 2 4 277 299 10.1155/s1024123x9600035x 2-s2.0-0000177831 Cabada A. Tersian S. Existence and multiplicity of solutions to boundary value problems for fourth-order impulsive differential equations Boundary Value Problems 2014 2014, article 105 10.1186/1687-2770-2014-105 MR3352642 2-s2.0-84901487610 Sun J. Chen H. Yang L. Variational methods to fourth-order impulsive differential equations Journal of Applied Mathematics and Computing 2011 35 1-2 323 340 10.1007/s12190-009-0359-x MR2748367 ZBL1218.34029 2-s2.0-78651377366 Xie J. Luo Z. Solutions to a boundary value problem of a fourth-order impulsive differential equation Boundary Value Problems 2013 2013, article 154 14 10.1186/1687-2770-2013-154 MR3083529 2-s2.0-84884272859 Heidarkhani S. Afrouzi G. A. Hadjian A. Henderson J. Existence of infinitely many anti-periodic solutions for second-order impulsive differential inclusions Electronic Journal of Differential Equations 2013 2013, article 97 13 MR3065050 Iannizzotto A. Three critical points for perturbed nonsmooth functionals and applications Nonlinear Analysis: Theory, Methods & Applications 2010 72 3-4 1319 1338 10.1016/j.na.2009.08.001 MR2577534 2-s2.0-71549158802 Iannizzotto A. Three periodic solutions for an ordinary differential inclusion with two parameters Annales Polonici Mathematici 2012 103 1 89 100 10.4064/ap103-1-7 MR2854994 2-s2.0-83655182900 Kristály A. Infinitely many solutions for a differential inclusion problem in R N Journal of Differential Equations 2006 220 2 511 530 10.1016/j.jde.2005.02.007 Kristály A. Marzantowicz W. Varga C. A non-smooth three critical points theorem with applications in differential inclusions Journal of Global Optimization 2010 46 1 49 62 10.1007/s10898-009-9408-0 MR2566135 2-s2.0-72449127177 Tian Y. Henderson J. Three anti-periodic solutions for second-order impulsive differential inclusions via nonsmooth critical point theory Nonlinear Analysis 2012 75 18 6496 6505 10.1016/j.na.2012.07.025 MR2965234 2-s2.0-84865688489 Yang B.-X. Sun H.-R. Periodic solutions for semilinear fourth-order differential inclusions via nonsmooth critical point theory Journal of Function Spaces 2014 2014 6 816490 10.1155/2014/816490 MR3302055 Ricceri B. A general variational principle and some of its applications Journal of Computational and Applied Mathematics 2000 113 1-2 401 410 10.1016/S0377-0427(99)00269-1 MR1735837 ZBL0946.49001 2-s2.0-0033890661 Marano S. A. Motreanu D. Infinitely many critical points of non-differentiable functions and applications to a Neumann-type problem involving the p-Laplacian Journal of Differential Equations 2002 182 1 108 120 10.1006/jdeq.2001.4092 MR1912071 2-s2.0-0037053782 Bonanno G. Bisci G. M. Infinitely many solutions for a boundary value problem with discontinuous nonlinearities Boundary Value Problems 2009 2009 670675 MR2487254 Erbe L. H. Krawcewicz W. Existence of solutions to boundary value problems for impulsive second order differential inclusions The Rocky Mountain Journal of Mathematics 1992 22 2 519 539 10.1216/rmjm/1181072746 MR1180717 2-s2.0-0002602635 Motreanu D. Panagiotopoulos P. D. Minimax Theorems and Qualitative Properties of the Solutions of Hemivariational Inequalities 1999 29 Dordrecht, The Netherlands Kluwer Academic Publishers Nonconvex Optimization and Its Applications 10.1007/978-1-4615-4064-9 MR1675895 Clarke F. H. Optimization and Nonsmooth Analysis 1983 New York, NY, USA John Wiley & Sons Canadian Mathematical Society Series of Monographs and Advanced Texts MR709590 Tersian S. Chaparova J. Periodic and homoclinic solutions of extended Fisher-Kolmogorov equations Journal of Mathematical Analysis and Applications 2001 260 2 490 506 10.1006/jmaa.2001.7470 MR1845566 ZBL0984.34031 2-s2.0-0035882834 Chang K. C. Variational methods for nondifferentiable functionals and their applications to partial differential equations Journal of Mathematical Analysis and Applications 1981 80 1 102 129 10.1016/0022-247x(81)90095-0 MR614246 2-s2.0-0000993645