Existence of three solutions for impulsive multi-point boundary value problems

http://dx.doi.org/10.7494/OpMath.2017.37.3.353

EXISTENCE OF THREE SOLUTIONS

FOR IMPULSIVE MULTI-POINT

BOUNDARY VALUE PROBLEMS

Martin Bohner, Shapour Heidarkhani, Amjad Salari,

and Giuseppe Caristi

Communicated by Marek Galewski

Abstract. This paper is devoted to the study of the existence of at least three classical solutions for a second-order multi-point boundary value problem with impulsive effects. We use variational methods for smooth functionals defined on reflexive Banach spaces in order to achieve our results. Also by presenting an example, we ensure the applicability of our results.

Keywords: multi-point boundary value problem, impulsive condition, classical solution, variational method, three critical points theorem.

Mathematics Subject Classification:34B10, 34B15, 34A37.

1. INTRODUCTION

In this paper, we consider the second-order ℓ-point boundary value problem with m-impulsive effects

    

−(φpi(u

′

i))′ =λFui(t, u1, . . . , un) +µGui(t, u1, . . . , un), t∈(0,1)\Q,

∆φpi(u

′

i(tj)) =Iij(ui(tj)), j= 1, . . . , m, ui(0) =Pℓ_k=1akui(sk), ui(1) =Pℓ_k=1bkui(sk)

(φλ,µ)

for i = 1, . . . , n, where Q = {t1, . . . , tm}, pi ∈ (1,∞), φpi(x) = |x|

pi−2_{x for i =}

1, . . . , n, λ > 0, µ ≥ 0 are parameters, m, n, ℓ ∈ N_{, 0 = t0 < t1 < t2 < . . . <}

tm < tm+1 = 1, 0 < s1 ≤s2 ≤ . . . ≤ sℓ <1, tj 6=sk, j = 1, . . . , m, k = 1, . . . , ℓ,

F, G: [0,1]×Rn _→R_{are measurable with respect to t, for every (x1, . . . , xn)∈}Rn_,

c

continuously differentiable in (x1, . . . , xn), for almost everyt ∈[0,1] and satisfy the

standard summability condition

sup

|ξ|≤̺1

max{|F(·, ξ)|, |G(·, ξ)|, |Fξi(·, ξ)|, |Gξi(·, ξ)|, i= 1, . . . , n}

∈L1([0,1])

(1.1)

for any̺1>0 withξ= (ξ1, . . . , ξn) and|ξ|= q_P

n

i=1ξi2,Fui andGui are the partial

derivatives of F andGwith respect toui, respectively,

∆φpi(u

′

i(tj)) =φpi(u

′

i(t+j))−φpi(u

′

i(t−j)),

where u′_i(t+_j) and u′_i(t−_j) represent the right-hand limit and left-hand limit of u′_i(t) at t = tj, respectively, Iij ∈ C(R_,R_{), i = 1, . . . , n, j = 1, . . . , m and ak, bk ∈}R _for

k= 1, . . . , ℓ.

The theory of multi-point boundary value problems for ordinary differential equa-tions arise in different areas of applied mathematics and physics, especially in heat conduction [4, 5, 27, 29], the vibration of cables with nonuniform weights [35], and other problems in nonlinear elasticity [45]. The study of multi-point BVPs for linear second-order ordinary differential equations was initiated by Il’in and Moiseev [26]. From then on, many authors studied more general nonlinear multi-point boundary value problems. Recently, the existence and multiplicity of positive solutions for non-linear multi-point BVPs have received a great deal of attention, we refer the reader to [11, 12, 15, 16, 18–20, 23–25, 34] and the references therein.

On the other hand, impulsive differential equations serve as basic models to study the dynamics of processes that are subjected to sudden changes in their states. These kinds of processes naturally occur in control theory, biology, optimization theory, medicine, and so on (see [6, 10, 28, 36]). The theory of impulsive differential equa-tions has recently received considerable attention, see [2, 3, 6, 30, 33, 39]. There has been increasing interest in the investigation for boundary value problems of nonlin-ear impulsive differential equations during the past few ynonlin-ears, and many works have been published about the existence of solutions for second-order impulsive differential equations. There are some common techniques to approach these problems: Fixed point theorems [8, 9, 31], the method of upper and lower solutions [7], and topological degree theory [37]. In the last few years, variational methods and critical point theory have been used to determine the existence of solutions for impulsive differential equa-tions under certain boundary condiequa-tions, see [1, 21, 43, 44, 46, 48] and the references therein. We note that the difficulties dealing with such problems are that their states are discontinuous. Therefore, the results of impulsive differential equations, especially for higher-order impulsive differential equations, are fewer in number than those for differential equations without impulses.

and multiplicity of positive solutions of a boundary value problem for second-order three-point nonlinear impulsive integrodifferential equation of mixed type in a real Banach space. In [32], Liu and Yu with the help of the coincidence degree continuation theorem, achieved a general result concerning the existence of solutions of m-point boundary value problems for second-order differential systems with impulses. They also give a definition of autonomous curvature bound set relative to this m-point boundary value problems, and by using this definition and the above existence theo-rem, obtained some simple existence conditions for solutions of these boundary value problems. Meiqiang and Dongxiu in [14], based on fixed point theory in a cone, dis-cussed the existence of solutions for the m-point BVPs for second-order impulsive differential systems, and Thaiprayoon et al. in [42], introduced a new definition of impulsive conditions for boundary value problems of first-order impulsive differential equations with multi-point boundary conditions was introduced, and used the method of lower and upper solutions in reversed order coupled with the monotone iterative technique to obtain the extremal solutions of the boundary value problem.

Motivated by the above works, in this paper we are interested to investigate the ex-istence of at least three nontrivial classical solutions for second-orderℓ-point boundary value problems withm-impulsive effects (φλ,µ) for appropriate values of the parame-tersλand µbelonging to real intervals. Our approach uses variational methods and a three critical points theorem due to Ricceri [38].

Here, we state a special case of our main result.

Theorem 1.1. Let p1, p2>1 such that eithermin{p1, p2} ≥2 ormax{p1, p2}<2,

F :R2_→Rbe aC1_{-function satisfying the condition}

sup

√

ξ2 1+ξ

2 2≤̺2

max{|F(ξ1, ξ2)|, |Fξ1(ξ1, ξ2)|, |Fξ2(ξ1, ξ2)|} ∈L 1_([0,1])

for any ̺2 > 0, F(0,0) = 0, I11, I12, I21, I22 ∈ C(R,R) be nondecreasing functions such that Iij(0) = 0 and Iij(s)s > 0,s ₆= 0 for i = 1,2, j = 1,2, a1, a2, b1, b2 ∈ R such thata1+a26= 1andb1+b26= 1,0< t1< t2<1,0< s1≤s2<1, andtj6=sk,

j= 1,2,k= 1,2. Assume that

max

(

lim sup

(u1,u2)→(0,0)

F(u1, u2) |u1|p1+|u

2|p2, lim sup |u|→∞

F(u1, u2) |u1|p1+|u

2|p2 )

<0

and

sup u∈Eb1×Eb2

1 R

0

F(u1(t), u2(t))dt

P2

i=1 kuikpi_pi

pi +

P2

i=1 P2

j=1

uiR(tj)

0

Iij(ζ)dζ >0,

where

b

Ei=

(

ξ_∈W1,pi_{([0,1]) : ξ(0) =a}

1ξ(s1)+a2ξ(s2), ξ(1) =b1ξ(s1)+b2ξ(s2) )

Then, for each compact interval [c, d]⊂(˜λ,∞), where

˜

λ= inf u∈Eb1×Eb2

( P2

i=1 kuikpi_pi

pi +

P2

i=1 P2

j=1

ui_R(tj)

0

Iij(ζ)dζ

1 R

0

F(u1(t), u2(t))dt

:

1 Z

0

F(u1(t), u2(t))dt >0 )

,

there exists R > 0 with the following property: for every λ _∈ [c, d] and for every

G_∈C1_(R2_{,R)such that}

sup

√

ξ2 1+ξ

2 2≤̺3

max{|G(ξ1, ξ2)|, |Gξ1(ξ1, ξ2)|, |Gξ2(ξ1, ξ2)|} ∈L 1_([0,1])

for any ̺3 > 0 with G(0,0) = 0, there exists γ > 0 such that, for each µ ∈ [0, γ], the system

    

−|u′

i(t)|pi−2ui′′(t) =λFui(u1, u2) +µGui(u1, u2), t∈(0,1)\ {t1, t2},

|u′i(t+j)|pi−2u′′i(t+j)− |u′i(t−j)|pi−2u′′i(t−j) =Iij(ui(tj)), j= 1,2, ui(0) =a1ui(s1) +a2ui(s2), ui(1) =b1ui(s1) +b2ui(s2)

for i= 1,2, has at least three classical solutions whose norms in the space Eb1×Eb2 are less than R.

2. PRELIMINARIES

In this section, we will introduce some notations, definitions and preliminary facts which are used throughout this paper.

To construct appropriate function spaces and apply critical point theory in order to investigate the existence of solutions for system (φλ,µ), we introduce the following basic notations and results which will be used in the proofs of our main results.

Throughout this article, we letE be the Cartesian product ofnspaces

Ei=

(

ξ_∈W1,pi_{([0,1]) : ξ(0) =}

ℓ

X

k=1

akξ(sk), ξ(1) = ℓ

X

k=1

bkξ(sk)

)

fori= 1, . . . , n, i.e.,E=E1×. . .×En, endowed with the norm

kuk=k(u1, . . . , un)k=

n

X

i=1 kuikpi,

where

kui_kpi=

1 Z

0

|u′_i(t)|pi_dt

!1

pi

In this paper, we assume throughout, and without further mention, that the following conditions hold:

(H1) eitherp≥2 orp <2, wherep= min{p1, . . . , pn}andp= max{p1, . . . , pn};

(H2) Pℓk=1ak6= 1 and Pℓ

k=1bk 6= 1.

Let

c= max

1≤i≤n

(

sup ui∈Ei\{0}

maxt∈[0,1]|ui(t)|pi kui_kpi

pi

)

. (2.1)

Since pi > 1 for i = 1, . . . , n, the embedding E = E1×. . .×En ֒→ C0([0,1])

n

is compact, and soc <_∞. Moreover, by (H2), from [11, Lemma 3.1], we have

sup v∈Ei\{0}

maxt∈[0,1]|v(t)|pi kvkpi

pi

≤ 1₂ 1 +

Pℓ k=1|ak| |1−Pℓk=1ak|

+

Pℓ k=1|bk| |1−Pℓk=1bk|

!

(2.2)

fori= 1, . . . , n.

By a classical solution of (φλ,µ), we mean a functionu= (u1, . . . , un) such that,

fori= 1, . . . , n, the left-hand side limit and the right-hand side limit of the derivative uiintj must exist and must be finite,ui_∈C1_{([0,1]\Q),φp}

i(u

′

i)∈C1([0,1]\Q), and ui satisfies (φλ,µ). We say that a function u = (u1, . . . , un) ∈E is a weak solution

of (φλ,µ) if

1 Z

0 _Xn

i=1

φpi(u

′

i(t))vi′(t)

dt+ n

X

i=1

m

X

j=1

Iij(ui(tj))vi(tj)

−λ

1 Z

0

n

X

i=1

Fui(t, u1(t), . . . , un(t))vi(t)dt−µ

1 Z

0

n

X

i=1

Gui(t, u1(t), . . . , un(t))vi(t)dt= 0

for anyv= (v1, . . . , vn)∈W10,p1([0,1])×W 1,p2

0 ([0,1])×. . .×W 1,pn

0 ([0,1]).

Let φ−pi1 denote the inverse of φpi for everyi = 1, . . . , n. Thenφ

−1

pi =φqi, where

1

pi +

1

qi = 1. It is clear thatφpi is increasing on R_,

lim

x→−∞φpi(x) =−∞ and xlim→∞φpi(x) =∞. (2.3) Lemma 2.1. For fixedλ, µ _∈R, u= (u1, . . . , un)∈ (C(tj, tj+1))n,j = 0,1, . . . , m, define αij(x;u) :R_→R,i= 1, . . . , n,j= 0,1, . . . , m by

αij(x;u)

= tZj+1

tj

φ−pi1 x−λ

δ

Z

0

Fui(t, u1(t), . . . , un(t))dt−µ

δ

Z

0

Gui(t, u1(t), . . . , un(t))dt

!

dδ

+ ℓ

X

k=1

akuk(sk)−

ℓ

X

k=1

Then, the equation

αij(x;u) = 0 (2.4)

has a unique solutionxu,i,j.

Proof. Taking (2.3) into account, we have

lim

x→−∞αij(x, u) =−∞ and xlim→∞αij(x, u) =∞.

Since αij(·, u) is continuous and strictly increasing on R_{, the conclusion follows}

fori= 1, . . . , n andj= 1, . . . , m.

Lemma 2.2. The functionu= (u1, . . . , un)is a solution of the system (φλ,µ)if and only ifui is a solution of the equation

ui(t) = ℓ

X

k=1

akuk(sk) + t

Z

0

φ−_pi1 xu,i,j₋λ δ

Z

0

Fui(t, u1(t), . . . , un(t))dt

−µ δ

Z

0

Gui(t, u1(t), . . . , un(t))dt

!

dδ

for i = 1, . . . , n, where xu,i,j is the unique solution of (2.4) for j = 0,1, . . . , m, and∆φpi(u

′

i(tj)) =Iij(ui(tj))fori= 1, . . . , nandj = 0,1, . . . , m.

Proof. This can be verified by direct computations (see [17, Lemma 2.4]).

Lemma 2.3. If a function u_∈E is a weak solution of (φλ,µ), then uis a classical solution of (φλ,µ).

Proof. Letu= (u1, . . . , un)∈E be a weak solution of (φλ,µ). Then

1 Z

0 _Xn

i=1

φpi(u

′

i(t))vi′(t)

dt+ n

X

i=1

m

X

j=1

Iij(ui(tj))vi(tj)

−λ

1 Z

0

n

X

i=1

Fui(t, u1(t), . . . , un(t))vi(t)dt

−µ

1 Z

0

n

X

i=1

Gui(t, u1(t), . . . , un(t))vi(t)dt= 0

for anyv = (v1, . . . , vn)∈W10,p1([0,1])×W 1,p2

0 ([0,1])×. . .×W 1,pn

0 ([0,1]). So (2.5)

holds for allv∈W1,p1

0 (tj, tj+1)×W10,p2(tj, tj+1)×. . .×W10,pn(tj, tj+1),v(t) = 0 for

t∈(0,1)\(tj, tj+1) forj= 0,1, . . . , m. Then (2.5) becomes

tZj+1

tj

_Xn

i=1

φp_i(u′i(t))vi′(t)

dt−λ tZj+1

tj

n

X

i=1

Fu_i(t, u1(t), . . . , un(t))vi(t)dt

−µ tj+1 Z

tj

n

X

i=1

Gui(t, u1(t), . . . , un(t))vi(t)dt= 0

(2.6)

forj = 0,1, . . . , m. Recall that, in one dimension, any weakly differentiable function is absolutely continuous, so that its classical derivative exists almost everywhere, and the classical derivative coincides with the weak derivative. Integrating (2.6) by parts gives

n

X

i=1

tZj+1

tj

[−(φpi◦u

′

i)′(t)−λFui(t, u1(t), . . . , un(t))−µGui(t, u1(t), . . . , un(t))]vi(t)dt= 0

forj= 0,1, . . . , m. Thus fori= 1, . . . , n,

−(φpi◦u

′

i)′(t)−λFui(t, u1(t), . . . , un(t))−µGui(t, u1(t), . . . , un(t)) = 0 (2.7)

for almost everyt_∈(tj, tj+1),j = 0,1, . . . , m. Then, by Lemmas 2.1 and 2.2, we see

that

ui(t) = ℓ

X

k=1

akuk(sk) + t

Z

0

φ−pi1 xu,i,j−λ

δ

Z

0

Fui(t, u1(t), . . . , un(t))dt

−µ δ

Z

0

Gu_i(t, u1(t), . . . , un(t))dt !

dδ

fori= 1, . . . , n, wherexu,i,j is the unique solution of (2.4) forj= 0,1, . . . , m. Hence, ui ∈ C1_{(tj, tj}

+1) and φpi ◦u

′

i ∈ C1(tj, tj+1) for i = 1, . . . , n and j = 0,1, . . . , m.

Therefore (2.7) holds fort∈(0,1)\Q. Now we shall show that the impulsive conditions are satisfied. For allv= (v1, . . . , vn)∈W01,p1([0,1])×W

1,p2

0 ([0,1])×. . .×W 1,pn

0 ([0,1]),

from the equality

(φpi◦u

′

i)′(t)vi(t) = d dt

 

t

Z

0

(φpi◦u

′

i)′(ζ)dζvi(t)

 ₋

 

t

Z

0

(φpi◦u

′

i)′(ζ)dζ

we have

1 Z

0

(φpi◦u

′

i)′(t)vi(t)dt

= 1 Z 0  d dt   t Z 0

(φpi◦u

′

i)′(ζ)dζ vi(t)

 ₋   t Z 0

(φpi◦u

′

i)′(ζ)dζ

 vi′(t)

 dt =vi(1) 1 Z 0

(φpi◦u

′

i)′(t)dt

− 1 Z

0  φpi(u

′

i(t))−φpi(u

′

i(0))−

X

0≤tj<t

∆φpi(u

′

i(tj))

 vi′(t)dt

=

 φpi(u

′

i(1))−φpi(u

′

i(0))− m

X

j=1

∆φpi(u

′

i(tj))

 vi(1) − 1 Z 0

φpi(u

′

i(t))vi′(t)dt+φpi(u

′

i(0))[vi(1)−vi(0)]

+ m

X

j=1

tZj+1

tj

X

0≤tj<t

∆φpi(u

′

i(tj))v′i(t)dt

=φpi(u

′

i(1))vi(1)−φpi(u

′

i(0))vi(0)− m

X

j=1

∆φpi(u

′ i(tj))vi(1) − 1 Z 0

φpi(u

′

i(t))vi′(t)dt− m

X

j=1

∆φpi(u

′

i(tj))vi(tj) + m

X

j=1

∆φpi(u

′

i(tj))vi(1)

=−

m

X

j=1

∆φpi(u

′

i(tj))vi(tj)−

1 Z

0

φpi(u

′

i(t))vi′(t)dt

(2.8)

fori= 1, . . . , n. Substituting (2.8) into (2.5), we have

n X i=1 1 Z 0

(−(φpi◦u

′

i)′(t)−λFui(t, u1(t), . . . , un(t))−µGui(t, u1(t), . . . , un(t)))vi(t)dt

+ n X i=1 m X j=1

[−∆φpi(u

fori= 1, . . . , n. Sinceusatisfies (2.7), we have

∆φpi(u

′_{(tj))vi(tj) =Iij(ui(tj))vi(tj), j = 0,1, . . . , m}

fori= 1, . . . , n. Souis a classical solution of (φλ,µ).

Our main tool is Theorem 2.4 which has been obtained by Ricceri ([38, Theo-rem 2]). It is as follows:

If X is a real Banach space, then denote by WX the class of all functionals Φ :X _→R _{possessing the following property: If {}un_} is a sequence in X converg-ing weakly to u_∈ X with lim infn→∞Φ(un)≤ Φ(u), then {un} has a subsequence

converging strongly to u.

For example, if X is uniformly convex and g : [0,∞) → R _{is a continuous and}

strictly increasing function, then, by a classical result, the functional u _→ g(ku_k) belongs to the classWX.

Theorem 2.4. LetX be a separable and reflexive real Banach space, letΦ :X →R_be

a coercive, sequentially weakly lower semicontinuousC1_{-functional, belonging toW}

X,

bounded on each bounded subset of X and whose derivative admits a continuous in-verse on X∗, and letJ :X →R_{be a C}1_{-functional with compact derivative. Assume}

that Φhas a strict local minimumu0 with Φ(u0) =J(u0) = 0. Finally, setting

ρ= max

(

0,lim sup

kuk→∞

J(u)

Φ(u),lim supu→u0

J(u) Φ(u)

)

,

σ= sup

u∈Φ−1_((0,∞))

J(u) Φ(u),

assume that ρ < σ. Then for each compact interval [c, d] ⊂ (1

σ,

1

ρ) (with the

con-ventions 1₀ =∞,_∞1 = 0), there exists R >0 with the following property: for every

λ∈ [c, d] and every C1_{-functional Ψ : X →R with compact derivative, there exists}

γ >0such that, for each µ∈[0, γ],

Φ′(u) =λJ′(u) +µΨ′(u)

has at least three solutions inX whose norms are less thanR.

We refer the reader to the papers [22, 41] in which Theorem 2.4 was successfully employed to ensure the existence of at least three solutions for boundary value prob-lems.

Now for every u_∈E, we define

Φ(u) := n

X

i=1 kui_kpi

pi

pi +

n

X

i=1

m

X

j=1

uZi(tj)

0

Iij(ζ)dζ, (2.9)

J(u) = T

Z

0

and

Ψ(u) = T

Z

0

G(t, u1(t), . . . , un(t))dt. (2.11)

Standard arguments show that Φ−µΨ−λJ is a Gâteaux differentiable functional whose Gâteaux derivative at the pointu_∈E is given by

(Φ′−µΨ′−λJ′)(u)(v)

=

1 Z

0 " _n

X

i=1

φpi(u

′

i(t))v′i(t)

#

dt−λ

1 Z

0

n

X

i=1

Fui(t, u1(t), . . . , un(t))vi(t)dt

−µ

1 Z

0

n

X

i=1

Gui(t, u1(t), . . . , un(t))vi(t)dt+

n

X

i=1

m

X

j=1

Iij(ui(tj))vi(tj)

for all v = (v1, . . . , vn) ∈ W10,p1([0,1])×W 1,p2

0 ([0,1])×. . .×W 1,pn

0 ([0,1]). Hence,

a critical point of the functional Φ−µΨ−λJ gives us a weak solution of (φλ,µ), and in view of Lemma 2.3, every weak solution of the problem (φλ,µ) is a classical one.

We suppose that the impulsive terms satisfy the condition

(I) Iij : R→R is nondecreasing, Iij(0) = 0 and Iij(s)s > 0,s 6= 0 for i= 1, . . . , n and j= 1, . . . , m.

We need the following proposition in the proof of our main result.

Proposition 2.5. Assume that (H1) holds. Let S :E→E∗ be the operator defined by

S(u)(v) =

1 Z

0 " _n

X

i=1

φpi(u

′

i(t))vi′(t)

#

dt+ n

X

i=1

m

X

j=1

Iij(ui(tj))vi(tj)

Proof. In the proof, we useC1, C2 and C3 to denote appropriate positive constants.

By (I), we have

S(u)(u) = n X i=1 1 Z 0

φpi(u

′

i(t))u′i(t)dt+ n X i=1 m X j=1

Iij(ui(tj))ui(tj)

= n X i=1 1 Z 0

|u′i(t)|pidt+ n X i=1 m X j=1 Iij(ui(tj))ui(tj) ≥ n X i=1      1 Z 0

|u′i(t)|pidt

 

1/pi

  pi ≥ n X i=1      1 Z 0

|u′_i(t)|pi_dt

 

1/pi

 

p

≥C1    n X i=1   1 Z 0

|u′_i(t)|pi_dt

 

1/pi

 

p

=C1kukp.

This implies that S is coercive. Now, for any u ∈ (u1, . . . , un) ∈ E and v ∈

(v1, . . . , vn)∈E, we have

hS(u)−S(v), u−v_i=

1 Z 0 " _n X i=1

(φpi(u

′

i(t))−φpi(v

′

i(t))) (u′i(t)−v′i(t))

# dt + n X i=1 m X j=1

(Iij(ui(t))−Iij(vi(t))) (ui(t)−vi(t)).

Then, by [40, Relation (2.2)] and (I), we see that

hS(u)−S(v), u−vi ≥             

C2Pni=1 1 R

0

|u′_i(t)−v_i′(t)|pi_{dt, p≥2,}

C3Pni=1 1 R

0 |u′

i(t)−v

′

i(t)|

2 (|u′

i(t)|+|v

′

i(t)|)

2−_pidt, p <2.

(2.12)

S−1 exists and it is bounded. So by [17, Relation (2.8)], S−1 is locally Lipschitz continuous and hence continuous. This completes the proof.

Remark 2.6. Suppose that the condition (I) is replaced by the condition

(I’) Iij :R_→R_{is odd and nondecreasing in}R_{for anyi= 1, . . . , nandj= 1, . . . , m.}

Then the conclusion of Proposition 2.5 holds again. It is easy to see that the condition (I’) implies the condition (I).

3. MAIN RESULTS

In this section, we formulate our main results. Let us denote by F the class of all functions F : [0,1]×Rn _→R_{that are measurable with respect to t, for allξ∈}Rn_,

continuously differentiable inξ, for almost everyt_∈[0,1], satisfy the standard summa-bility condition (1.1). Set

A:= 1

2 1 +

Pℓ k=1|ak| |1−Pℓk=1ak|

+

Pℓ k=1|bk| |1−Pℓk=1bk|

!

.

Let

λ1= inf

u∈E

( Pn i=1

kuikpi_pi

pi +

Pn i=1

Pm j=1

uiR(tj)

0

Iij(ζ)dζ

1 R

0

F(t, u(t))dt

:

1 Z

0

F(t, u(t))dt >0

)

andλ2=_max{0,λ1

0,λ∞}, where

λ0= lim sup |u|→0

1 R

0

F(t, u(t))dt

Pn i=1

kuikpipi

pi +

Pn i=1

Pm j=1

uiR(tj)

0

Iij(ζ)dζ

and

λ∞= lim sup kuk→∞

1 R

0

F(t, u(t))dt

Pn i=1

kuikpi_pi

pi +

Pn i=1

Pm j=1

uiR(tj)

0

Iij(ζ)dζ

Theorem 3.1. Suppose that F ∈ F. Assume that

(A1) there exists a constant ε >0 such that

max

(

lim sup u→(0,...,0)

maxt∈[0,1]F(t, u) Pn

i=1|ui|pi

,lim sup

|u|→∞

maxt∈[0,1]F(t, u) Pn

i=1|ui|pi )

< ε,

whereu= (u1, . . . , un) with|u|=pPni=1u2i; (A2) there exists a functionw∈E such that

n

X

i=1 kwikpi

pi

pi +

n

X

i=1

m

X

j=1

w_Zi(tj)

0

Iij(ζ)dζ6= 0

and

Apε <

1 R

0

F(t, w(t))dt

Pn i=1

kwikpipi

pi +

Pn i=1

Pm j=1

wiR(tj)

0

Iij(ζ)dζ .

Then, for each compact interval[c, d]⊂(λ1, λ2), there existsR >0with the following property: for every λ_∈[c, d]and every G_{∈ F}, there existsγ >0 such that, for each

µ_∈ [0, γ], the system (φλ,µ) has at least three classical solutions whose norms in E

are less than R.

Proof. Take X =E. Clearly, X is a separable and uniformly convex Banach space. Let the functionals Φ, J and Ψ be as given in (2.9), (2.10) and (2.11), respectively. The functional Φ is C1, and due to Proposition 2.5 its derivative admits a contin-uous inverse on X∗. Moreover, by the sequentially weakly lower semicontinuity of

Pn i=1

kuikpi_pi

pi and the continuity of Iij, i = 1, . . . , n, j = 1, . . . , m, Φ is sequentially

weakly lower semicontinuous inX. On the other hand, we letkui_kpi>1,i= 1, . . . , n,

by (I),

Φ(u) = n

X

i=1 kuikpi

pi

pi + n

X

i=1

m

X

j=1

uZi(tj)

0

Iij(ζ)dζ≥

n

X

i=1 kuikpi

pi

pi ≥

1 pkuk

p _(3.1)

for every u= (u1, . . . , un)∈X. Thus Φ is coercive. Moreover, let M be a bounded

subset of X. That is, there exist constantsci >0,i= 1, . . . , n such thatkuiki ≤ci for eachu∈M. Then, we have

|Φ(u)| ≤ 1_p

n

X

i=1 (

kcikp, ifkuiki≤1,

kcikp, if kuiki>1.

Hence Φ is bounded on each bounded subset of X. Furthermore, Φ ∈ WX. Indeed, let the sequence

and lim infk→∞Φ(uk) ≤ Φ(u). Since the function Iij is continuous, i = 1, . . . , n,

j= 1, . . . , m, one has

lim inf n→∞

n

X

i=1 kuki_kpi

pi

pi ≤

n

X

i=1 kui_kpi

pi

pi .

Thus,{uk_}∞

k=1 has a subsequence converging strongly to u. Therefore, Φ∈ WX. The

functionalsJand Ψ are two C1_{-functionals with compact derivatives. Moreover, Φ has}

a strict local minimum 0 with Φ(0) = J(0) = 0. In view of (A1), there exist τ1, τ2

with 0< τ1< τ2 such that

F(t, u)≤ε n

X

i=1

|ui|pi _(3.2)

for every t ∈[0,1] and everyu= (u1, . . . , un) with |u| ∈[0, τ1)∪(τ2,∞). By (1.1),

F(t, u) is bounded on [0,1]×[τ1, τ2]. Thus we can chooseη >0 andυ > psuch that

F(t, u)≤ε n

X

i=1

|ui|pi_+η

n

X

i=1 |ui|υ

for all (t, u)∈[0,1]×Rn_{. So, by (2.2), we have}

J(u)≤Aε n

X

i=1 kuikpi

pi+η

n

X

i=1

Aυ/pi_kuikυ

pi (3.3)

for allu_∈X. Hence, from (3.3), we have

lim sup

|u|→0

J(u)

Φ(u) ≤Apε. (3.4)

Moreover, by using (3.2), for eachu_∈X_{\ {}0}, we obtain

J(u) Φ(u) =

R

|u|≤τ2

F(t, u(t))dt

Φ(u) +

R

|u|>τ2

F(t, u(t))dt

Φ(u)

≤supt∈[0,1],|u|∈[0,τ2]F(t, u)

Φ(u) +

AεPn_i=1kui_kpi

pi

Φ(u)

≤supt∈[0,1],|u|∈[0,τ2]F(t, u) Pn

i=1 kuikpipi

pi

+Apε.

So, we get

lim sup

kuk→∞

J(u)

Φ(u) ≤Apε. (3.5)

In view of (3.4) and (3.5), we have

ρ= max

(

0,lim sup

kuk→∞

J(u)

Φ(u),ulim sup→(0,...,0)

J(u) Φ(u)

)

Assumption (A2) in conjunction with (3.6) yields

σ= sup

u∈Φ−1_((0,∞))

J(u)

Φ(u)= supX\{0}

J(u) Φ(u)

≥ 1 R

0

F(t, w(t))dt

Φ(w(t)) =

1 R

0

F(t, w(t))dt

Pn i=1

kwikpipi

pi +

Pn i=1

Pm j=1

wiR(tj)

0

Iij(ζ)dζ

> Apε_≥ρ.

Thus, all the hypotheses of Theorem 2.4 are satisfied. Clearly,λ1 = 1β and λ2 = α1. Then, using Theorem 2.4, taking Lemma 2.3 into account, for each compact interval [c, d] ⊂(λ1, λ2), there exists R >0 with the following property: for everyλ∈ [c, d]

and everyG_{∈ F} there existsγ >0 such that, for eachµ_∈[0, γ], the system (φλ,µ) has at least three classical solutions whose norms in Eare less than R.

Another announced application of Theorem 2.4 reads as follows.

Theorem 3.2. Suppose that F ∈ F. Assume that

max

(

lim sup u→(0,...,0)

maxt∈[0,1]F(t, u) Pn

i=1|ui|pi

,lim sup

|u|→∞

maxt∈[0,1]F(t, u) Pn

i=1|ui|pi )

<0 (3.7)

whereu= (u1, . . . , un)with|u|=pPni=1u2i, and

sup u∈E

1 R

0

F(t, u(t))dt

Pn i=1

kuikpipi

pi +

Pn i=1

Pm j=1

uiR(tj)

0

Iij(ζ)dζ

>0. (3.8)

Then for each compact interval [c, d]⊂(λ1,∞) there existsR >0 with the following property: for everyλ_∈[c, d] and everyG_{∈ F}, there exists γ >0 such that for each

µ_∈ [0, γ], the system (φλ,µ) has at least three classical solutions whose norms in E

are less than R.

Proof. In view of (3.7), there exist an arbitrary ε > 0 andτ1, τ2 with 0 < τ1 < τ2

such that

F(t, u)≤ε n

X

i=1 |ui|pi

for every t ∈[0,1] and everyu= (u1, . . . , un) with |u| ∈[0, τ1)∪(τ2,∞). By (1.1),

F(t, u) is bounded on [0,1]×[τ1, τ2]. Thus we can chooseη >0 andυ > pin a manner

that

F(t, u)≤ε n

X

i=1

|ui|pi_+η

n

X

for all (t, u)∈ [0,1]×Rn_{. So, by the same process as in the proof of Theorem 3.1,}

we obtain (3.4) and (3.5). Sinceǫis arbitrary, (3.4) and (3.5) give

max

(

0, lim sup

kuk→+∞

J(u)

Φ(u),ulim sup→(0,...,0)

J(u) Φ(u)

) ≤0.

Then, with the notation of Theorem 2.4, we haveρ= 0. By (3.8), we also haveσ >0. In this case, clearly λ1 = _σ1 and λ2 = ∞. Thus, by using Theorem 2.4 and taking

Lemma 2.3 into account, the result is achieved.

Remark 3.3. Theorem 1.1 immediately follows from Theorem 3.2.

Remark 3.4. In Assumption (A2), if we choose

w(t) =w⋆(t) = (0, . . . ,0, wn(t)) (3.9)

with

wn(t) =

               δ Pℓ

k=1ak+

2 1−Pℓ

k=1

ak

s1 t

, if t∈[0,s1

2),

δ, if t_∈[s1

2, 1+sℓ

2 ],

δ

2−Pℓ

k=1

bk−sℓP ℓ k=1

bk

1−sℓ −

2(1−Pℓ

k=1

bk)

1−sℓ t

, if t∈(1+sℓ

2 ,1],

(3.10)

whereδ >0, then (A2) takes the following form:

(A′

2) there exists a positive constantδ such that

(ωnδ)pn_+pn

m

X

j=1

wZn(tj)

0

Inj(ζ)dζ6= 0

and Apε < pn 1 R 0

F(t, w⋆_(t))dt

(ωnδ)pn+pnPm

j=1

wnR(tj)

0

Inj(ζ)dζ .

Clearly,w⋆_{∈E and}

Φ(w⋆) =(ωnδ) pn

pn +

m

X

j=1

wZn(tj)

0

Inj(ζ)dζ,

where

ωn:=

"

2pn−1 _s1−pn

1 1− ℓ X k=1 ak pn

+ (1−sℓ)1−pn

! 1− ℓ X k=1 bk

pn#1/pn

Now, we point out some results in which the functionF has separated variables. To be precise, we consider the system

    

−(φpi(u

′

i))′ =λθ(t)Fui(u1, . . . , un) +µGui(t, u1, . . . , un), t∈(0,1)\Q,

∆φpi(u

′

i(tj)) =Iij(ui(tj)), j= 1, . . . , m, ui(0) =Pℓ_k=1akui(sk), ui(1) =Pℓ_k=1bkui(sk),

(φθ λ,µ)

where θ : [0,1]→ R_{is a nonzero function such that θ ∈L}1_{([0,1]) and F :} Rn _→R

is a C1_{-function and G : [0,1]×R}n _{→ R is as introduced for the system (φλ,µ) in} Section 1.

Set F(t, x1, . . . , xn) = θ(t)F(x1, . . . , xn) for every (t, x1, . . . , xn) ∈ [0,1]×Rn.

The following existence results are consequences of Theorem 3.1.

Theorem 3.5. Assume that

(A′

1) there exists a constant ε >0 such that

sup t∈[0,1]

θ(t).max

(

lim sup u→(0,...,0)

F(u)

Pn i=1|ui|pi

,lim sup

|u|→∞

F(u)

Pn i=1|ui|pi

)

< ε,

whereu= (u1, . . . , un) with|u|=pPni=1u2i; (A′′

2) there exists a positive constantδ such that

(ωnδ)pn_+pn

m

X

j=1

wZn(tj)

0

Inj(ζ)dζ6= 0

and

Apε <

pn

1 R

0

F(t, w⋆_(t))dt

(ωnδ)pn+pnPm

j=1

wnR(tj)

0

Inj(ζ)dζ

wherew⋆ _andω

n are given by (3.9)and (3.10), respectively.

Then, for each compact interval[c, d]⊂(λ3, λ4), where λ3 andλ4are the same asλ1 and λ2, but

R1

0 F(t, u(t))dt is replaced by R1

0 θ(t)F(u(t))dt, respectively, there exists

R > 0 with the following property: for every λ ∈ [c, d] and every G ∈ F, there existsγ >0such that for eachµ∈[0, γ], the system (φθ

λ,µ)has at least three classical

solutions whose norms in E are less thanR.

Theorem 3.6. Assume that there exists a positive constantδ such that

(ωnδ)pn_+pn

m

X

j=1

wZn(tj)

0

Inj(ζ)dζ >0and 1 Z

0

wherew⋆ andωn are given by (3.9)and (3.10), respectively. Moreover, suppose that

lim sup u→(0,...,0)

F(u)

Pn i=1|ui|pi

= lim sup

|u|→∞

F(u)

Pn i=1|ui|pi

= 0, (3.12)

whereu= (u1, . . . , un)with|u|=pPni=1u2i. Then, for each compact interval[c, d]⊂ (λ3,∞)whereλ3 is the same asλ1 butR₀1F(t, u(t))dtis replaced byR₀1θ(t)F(u(t))dt, there exists R >0 with the following property: for every λ_∈[c, d] and every G_{∈ F}, there exists γ >0 such that for each µ_∈[0, γ], the system (φθλ,µ) has at least three

classical solutions whose norms in E are less than R.

Proof. We easily observe that, from (3.12), the assumption (A′

1) is satisfied for every

ε > 0. Moreover, using (3.11), by choosing ε > 0 small enough, one can derive the assumption (A′′

2). Hence, the conclusion follows from Theorem 3.5.

Now, we exhibit an example in which the hypotheses of Theorem 3.6 are satisfied.

Example 3.7. Letn= 2,p1=p2= 2,m=l= 1,t1 = 1₃, s1= 1₂,a1=b1= 2 and

I11(x) =I21(x) =x3 for each x∈R. Let θ(t) = [t] + 1 for all t ∈[0,1], where [t] is

the integer part oftand

F(x1, x2) = (

(x2

1+x22)2, ifx21+x22<1,

1, ifx2₁+x2_2≥1.

By choosingδ= 1, we havew(t) =w⋆(t) = (0, w2(t)) with

w2(t) =     

2(1−2t) ift_∈[0,1₄),

1 ift∈[1

4, 3 4], −2(1−2t) ift_∈(3₄,1],

andω2= 2 √

2. Thus we have

(ω2δ)p2+p2

m

X

j=1

wZ2(tj)

0

I2j(ζ)dζ= (2 √

2)2+ 2

1 Z

0

ζ3dζ= 17 2 >0,

1 Z

0

θ(t)F(w⋆(t))dt=

1 4 Z

0

F(0,2(1−2t))dt+

3 4 Z

1 4

F(0,1)dt+

1 Z

3 4

F(0,−2(1−2t))dt

=

1 4 Z

0

16(1−2t)4dt+

3 4 Z

1 4

dt+

1 Z

3 4

dt= 23 10>0,

lim

(u1,u2)→(0,0)

F(u1, u2)

u21+u22

= lim

(u1,u2)→(0,0)

(u2 1+u22)2

u21+u22

= lim

(u1,u2)→(0,0)

and

lim

|u|→∞

F(u1, u2)

u2 1+u22

= lim

|u|→∞

1 u2

1+u22

= 0,

where u= (u1, u2) with|u| = p

u21+u22. Hence, by applying Theorem 3.6 for each

compact interval [c, d] ⊂(0,∞), there exists R >0 with the following property: for every λ _∈[c, d] and every G_{∈ F}, there exists γ > 0 such that, for eachµ _∈ [0, γ], the system

                  

−u′′

1(t) =λ([t] + 1)Fu1(u1, u2) +µGu1(t, u1, u2), t∈(0,1)\ { 1 3}, −u′′2(t) =λ([t] + 1)Fu2(u1, u2) +µGu2(t, u1, u2), t∈(0,1)\ {

1 3},

u′1(13 +

)−u′1(13 −

) = (u1(1₃))3,

u′ 2(13

+

)−u′ 2(13

−

) = (u2(1₃))3,

u1(0) = 2u1(1₂), u1(1) = 2u1(1₂),

u2(0) = 2u2(12), u2(1) = 2u2(12)

has at least three classical solutions whose norms in the space

e

E=

(

(ξ1, ξ2)∈W1,2([0,1])×W1,2([0,1]) :ξ1(0) =ξ1(1) = 2ξ1

1 2

,

ξ2(0) =ξ2(1) = 2ξ2

1 2

)

are less thanR.

4. SINGLE IMPULSE

As an application of the results from Section 3, we consider the problem

    

−(φp(u′₎₎′_{=λf(t, u) +µg(t, u), t∈(0,1)\ {t} 1},

∆φp(u′_(t

1)) =I(u(t1)),

u(0) =u(1) =u(s1),

(4.1)

where p > 1, 0 < t1, s1 < 1, t1 6= s1, φp(x) = |x|p−2x, λ > 0, µ ≥ 0, and f, g :

[0,1]×R_→R_{are two L}1_{-Carathéodory functions, ∆φpu}′_{i(t1) =}φpu′_(t+

1)−φpu′(t−1),

whereu′_(t+

1) andu′(t−1) represent the right-hand limit and left-hand limit ofu′(t) at

t=t1, respectively, and I:R→Ris a continuous and nondecreasing function such

thatI(0) = 0 andI(s)s >0 for alls∈R_{\ {0}.}

We introduce the functionsF : [0,1]×R_→R_{andG: [0,1]×}R_→R_{, respectively,}

as

F(t, x) = x

Z

0

and

G(t, x) = x

Z

0

g(t, ζ)dζ for all (t, x)∈[0,1]×R_.

The following two theorems are consequences of Theorems 3.1 and 3.2, respectively.

Theorem 4.1. Assume that

(B1) there exists a constantε >0 such that

max

(

lim sup u→0

maxt∈[0,1]F(t, u)

|u|p , lim sup

|u|→∞

maxt∈[0,1]F(t, u) |u|p

)

< ε;

(B2) there exists a functionw∈Eb, where

b

E=

(

ξ_∈W1,p([0,1]) : ξ(0) =ξ(1) =ξ(s1) )

,

such that

kw_kp_p+p wZ(t1)

0

I(ζ)dζ6= 0

and

p2ε < 2

1 R

0

F(t, w(t))dt

kukpp+p uR(t1)

0

I(ζ)dζ .

Then, for each compact interval [c, d]⊂(¯λ1,¯λ2), where

¯ λ1= inf

u∈Eb

(_kukp p+p

uR(t1)

0

I(ζ)dζ

p

1 R

0

F(t, u(t))dt :

1 Z

0

F(t, u(t))dt >0

)

and

¯

λ2= max         

0,lim sup

|u|→0

p

1 R

0

F(t, u(t))dt

ku_kpp+p uR(t1)

0

I(ζ)dζ

, lim sup

kukp→∞

p

1 R

0

F(t, u(t))dt

ku_kpp+p u(Rt1)

0

I(ζ)dζ

        

,

there existsR >0 with the following property: for everyλ_∈[c, d], there existsγ >0

Theorem 4.2. Assume that

max

(

lim sup u→0

supt∈[0,1]F(t, u(t))

|u_|p , lim sup

|u|→∞

supt∈[0,1]F(t, u(t)) |u_|p

) ≤0

and

sup u∈Eb

p

1 R

0

F(t, u(t))dt

kukpp+p u(Rt1)

0

I(ζ)dζ >0.

Then for each compact interval [c, d]⊂(¯λ1,∞), whereλ¯1 is defined in Theorem 4.1, there existsR >0 with the following property: for everyλ∈[c, d], there existsγ >0

such that for each µ∈ [0, γ], the problem (4.1) has at least three classical solutions whose norms inEb are less than R.

Now letf :R_→R_{be a continuous function. Put}

F(x) = x

Z

0

f(ζ)dζ for allx∈R_.

The following theorems are immediate consequences of Theorems 3.5 and 3.6, respec-tively.

Theorem 4.3. Assume that

(B′

1) there exists a constantε >0 such that

max

(

lim sup u→0

F(u)

|u|p , lim sup

|u|→∞

F(u)

|u|p

)

< ε;

(B′′

2) there exists a positive constantδ such that

2p(p−1)δps₁1−p+ (1−s1)1−p

+p δ

Z

0

I(ζ)dζ6= 0

and

pε <

2pn

1 R

0

F(w(t))dt

2p(p−1)_δp_s1−p

1 + (1−s1)1−p

+p δ

R

0

I(ζ)dζ ,

where

w(t) =

    

2δ

s1t, if t∈[0,

s1 2),

δ, if t∈[s1 2,

1+s1 2 ], 2δ

1−s1(1−t), if t∈( 1+s1

2 ,1].

Then, for each compact interval [c, d] ⊂(¯λ3,λ¯4), where λ¯3 and λ¯4 are the same as

¯

λ1 and λ¯2, but R₀1F(t, u(t))dt is replaced by R₀1F(u(t))dt, respectively, there exists

R >0 with the following property: for everyλ_∈[c, d]and every continuous function

g:R_→R_{, there existsγ >0such that for eachµ∈[0, γ], the problem}

    

−(φp(u′))′=λf(u) +µg(u), t_∈(0,1)\ {t1},

∆φp(u′_(t

1)) =I(u(t1)),

u(0) =u(1) =u(s1)

(4.3)

has at least three classical solutions whose norms in Eb are less than R. Theorem 4.4. Assume that there exists a positive constantδ such that

2p(p−1)δps₁1−p+ (1−s1)1−p

+p δ

Z

0

I(ζ)dζ >0 and 1 Z

0

F(w(t))dt >0,

wherew is given by (4.2). Moreover, suppose that

lim sup u→0

f(u)

|u|p−1 = lim sup |u|→∞

f(u)

|u|p−1 = 0.

Then, for each compact interval [c, d] ⊂(¯λ3,+∞), where ¯λ3 is the same as ¯λ1, but R1

0 F(t, u(t))dt is replaced by R1

0 F(u(t))dt, there existsR >0with the following prop-erty: for every λ∈[c, d] and every continuous functiong:R_→R_{, there existsγ >0}

such that for each µ∈ [0, γ], the problem (4.3) has at least three classical solutions whose norms inEb are less than R.

Finally, we present the following example in order to illustrate Corollary 4.4.

Example 4.5. Letp= 3,t1= 2₃,s1= 1₂,I(x) =√3xfor allx∈Randf(x) =−x3e−x

for allx∈R_{. By choosing δ= 1, we have}

w(t) =

    

4t, if t∈[0,1 4),

1, if t_∈[1₄,3₄], 4(1−t), if t∈(3

4,1].

Thus

2p(p−1)δps₁1−p+ (1−s1)1−p

+p δ

Z

0

I(ζ)dζ= 29+ 3

1 Z

0 3 p

1 Z

0

F(w(t))dt=− 1 4 Z

0 4t

Z

0

ζ3e−ζdζdt− 3 4 Z

1 4

1 Z

0

ζ3e−ζdζdt− 1 Z

3 4

4(1_Z−t)

0

ζ3e−ζdζdt

=

1 4 Z

0

e−4t(64t3+ 58t2+ 24t+ 6)−6dt+

16 e −6

3 4 Z

1 4

dt

+ 2 e4

1 Z

3 4

e4t(76−166t+ 125t2−32t3)−6dt

= 29

6 +

8 e−

1089 16e4 +

122 64e7 −

70 32e8 >0.

Also limu→0f_|u(u_|2) = lim|u|→∞

f(u)

|u|2 = 0. Hence, by applying Corollary 4.4 for each

compact interval [c, d] ⊂(0,∞), there exists R >0 with the following property: for every λ _∈ [c, d] and every continuous functiong : R _→ R_{, there exists} γ > 0 such that, for eachµ_∈[0, γ], the problem

      

−|u′1(t)|u′′1(t) =−λu(t)3e−u(t)+µg(u), t∈(0,1)\ {23}, |u′(2

3 +

)|u′(2 3 +

)− |u′(2 3 −

)|u′1(23 −

) = 3 q

u(2 3),

u(0) =u(1) =u(1 2)

has at least three classical solutions whose norms in the space

e

E1= (

ξ∈W1,3_{([0,1]) : ξ(0) =ξ(1) =ξ}

1 2

)

are less thanR.

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Martin Bohner bohner@mst.edu

Department of Mathematics and Statistics Missouri S&T, Rolla

MO 65409-0020, USA

Shapour Heidarkhani s.heidarkhani@razi.ac.ir

Razi University Faculty of Sciences

Amjad Salari

amjads45@yahoo.com

Razi University Faculty of Sciences

Department of Mathematics 67149 Kermanshah, Iran

Giuseppe Caristi gcaristi@unime.it

University of Messina Department of Economics Messina, Italy