### Abstract

This paper investigates the existence of solutions for weighted *p*(*r*)-Laplacian impulsive system mixed type boundary value problems. The proof of our
main result is based upon Gaines and Mawhin's coincidence degree theory. Moreover,
we obtain the existence of nonnegative solutions.

##### Keywords:

Weighted*p*(

*r*)-Laplacian; impulsive system; coincidence degree

### 1 Introduction

In this paper, we mainly consider the existence of solutions for the weighted *p*(*r*)-Laplacian system

where *u*: [0, *T*] → ℝ* ^{N}*, with the following impulsive boundary conditions

where *p *∈ *C *([0, *T*], ℝ) and *p*(*r*) > 1, -Δ_{p(r) }*u*:= -(*w*(*r*) |*u*'|^{p(r)-2 }*u*'(*r*))' is called weighted *p*(*r*)-Laplacian; 0 < *r*_{1 }< *r*_{2 }< ⋯ < *r _{k }*<

*T*;

*A*,

_{i}*B*∈

_{i }*C*(ℝ

*× ℝ*

^{N }*, ℝ*

^{N}*);*

^{N}*a*,

*b*,

*c*,

*d*∈ [0, +∞),

*ad*+

*bc*> 0.

Throughout the paper, *o*(1) means functions which uniformly convergent to 0 (as *n *→ +∞); for any *v *∈ ℝ* ^{N}*,

*v*will denote the

^{j }*j*-th component of

*v*; the inner product in ℝ

*will be denoted by 〈·,·〉; |·| will denote the absolute value and the Euclidean norm on ℝ*

^{N }*. Denote*

^{N}*J*= [0,

*T*],

*J*' = [0,

*T*]\{

*r*

_{0},

*r*

_{1},...,

*r*

_{k+1}},

*J*

_{0 }= [

*r*

_{0},

*r*

_{1}],

*J*= (

_{i }*r*,

_{i}*r*

_{i+1}],

*i*= 1, ...,

*k*, where

*r*

_{0 }= 0,

*r*

_{k+1 }=

*T*. Denote

*J*,

_{i}*i*= 0, 1,...,

*k*. Let

*PC*(

*J*, ℝ

*) = {*

^{N}*x*:

*J*→ ℝ

*|*

^{N }*x*∈

*C*(

*J*, ℝ

_{i}*),*

^{N}*i*= 0, 1,...,

*k*, and

*i*= 1,...,

*k*};

*w*∈

*PC*(

*J*, ℝ) satisfies 0 <

*w*(

*r*), ∀

*r*∈

*J*', and

*i*= 0, 1,...,

*k*}. For any

*x*= (

*x*

^{1},...,

*x*) ∈

^{N}*PC*(

*J*, ℝ

*), denote |*

^{N}*x*|

^{i}_{0 }= sup

_{r∈J' }|

*x*(

^{i}*r*)|. Obviously,

*PC*(

*J*, ℝ

*) is a Banach space with the norm*

^{N}*PC*

^{1}(

*J*, ℝ

*) is a Banach space with the norm*

^{N}*PC*(

*J*, ℝ

*) and*

^{N}*PC*

^{1}(

*J*, ℝ

*) will be simply denoted by*

^{N}*PC*and

*PC*

^{1}, respectively. Let

*L*

^{1 }=

*L*

^{1}(

*J*, ℝ

*) with the norm*

^{N}*x*∈

*L*

^{1}, where

The study of differential equations and variational problems with nonstandard *p*(*r*)-growth conditions is a new and interesting topic. It arises from nonlinear elasticity
theory, electro-rheological fluids, image processing, etc. (see [1-4]). Many results have been obtained on this problems, for example [1-25]. If *p*(*r*) ≡ *p *(a constant), (1) is the well-known *p*-Laplacian system. If *p*(*r*) is a general function, -Δ_{p(r) }represents a nonhomogeneity and possesses more nonlinearity, thus -Δ_{p(r) }is more complicated than -Δ* _{p}*; for example, if Ω ⊂ ℝ

*is a bounded domain, the Rayleigh quotient*

^{N }

is zero in general, and only under some special conditions *λ*_{p(·) }> 0 (see [8,17-19]), but the property of *λ _{p }*> 0 is very important in the study of

*p*-Laplacian problems.

Impulsive differential equations have been studied extensively in recent years. Such
equations arise in many applications such as spacecraft control, impact mechanics,
chemical engineering and inspection process in operations research (see [26-28] and the references therein). It is interesting to note that *p*(*r*)-Laplacian impulsive boundary problems are about comparatively new applications like
ecological competition, respiratory dynamics and vaccination strategies. On the Laplacian
impulsive differential equation boundary value problems, there are many results (see
[29-37]). There are many methods to deal with this problem, e.g., subsupersolution method,
fixed point theorem, monotone iterative method and coincidence degree. Because of
the nonlinearity of -Δ* _{p}*, results on the existence of solutions for

*p*-Laplacian impulsive differential equation boundary value problems are rare (see [38,39]). On the Laplacian (

*p*(

*x*) ≡ 2) impulsive differential equations mixed type boundary value problems, we refer to [30,32,34].

In [39], Tian and Ge have studied nonlinear IBVP

where Φ* _{p}*(

*x*) = |

*x*|

^{p-2 }

*x*,

*p*> 1,

*ρ*,

*s*∈

*L*

^{∞ }[

*a*,

*b*] with

*essin f*

_{[a, b] }

*ρ*> 0, and

*essin f*

_{[a,b] }

*s*> 0, 0 <

*ρ*(

*a*),

*p*(

*b*) <∞,

*σ*

_{1 }≤ 0,

*σ*

_{2 }≥ 0,

*α*,

*β*,

*γ*,

*σ*> 0,

*a*=

*t*

_{0 }<

*t*

_{1 }< ⋯ <

*t*

_{l }<

*t*

_{l+1 }=

*b*,

*I*∈

_{i }*C*([0, +∞), [0, ∞)),

*i*= 1,...,

*l*,

*f*∈

*C*([

*a*,

*b*] × [0, +∞), [0, ∞)),

*f*(·, 0) is nontrivial. By using variational methods, the existence of at least two positive solutions was obtained.

In [24,25], the present author investigates the existence of solutions of *p*(*r*)-Laplacian impulsive differential equation (1-3) with periodic-like or multi-point
boundary value conditions.

In this paper, we consider the existence of solutions for the weighted *p*(*r*)-Laplacian impulsive differential system mixed type boundary value condition problems,
when *p*(*r*) is a general function. The proof of our main result is based upon Gaines and Mawhin's
coincidence degree theory. Since the nonlinear term *f *in (5) is independent on the first-order derivative, and the impulsive conditions
are simpler than (2), our main results partly generalized the results of [30,32,34,39]. Since the mixed type boundary value problems are different from periodic-like or
multi-point boundary value conditions, and this paper gives two kinds of mixed type
boundary value conditions (linear and nonlinear), our discussions are different from
[24,25] and have more difficulties. Moreover, we obtain the existence of nonnegative solutions.
This paper was motivated by [24-26,38,40].

Let *N *≥ 1, the function *f*: *J *× ℝ* ^{N }*× ℝ

*→ ℝ*

^{N }*is assumed to be Caratheodory; by this, we mean:*

^{N }(i) for almost every *t *∈ *J*, the function *f*(*t*, ·, ·) is continuous;

(ii) for each (*x*, *y*) ∈ ℝ* ^{N }*× ℝ

*, the function*

^{N}*f*(·,

*x*,

*y*) is measurable on

*J*;

(iii) for each *R *> 0, there is a *α _{R }*∈

*L*

^{1 }(

*J*, ℝ), such that, for almost every

*t*∈

*J*and every (

*x*,

*y*) ∈ ℝ

*× ℝ*

^{N }*with |*

^{N }*x*| ≤

*R*, |

*y*| ≤

*R*, one has

We say a function *u*: *J *→ ℝ* ^{N }*is a solution of (1) if

*u*∈

*PC*

^{1 }with

*w*(·) |

*u*'|

^{p(·)-2 }

*u*'(·) absolutely continuous on every

*i*= 0, 1,...,

*k*, which satisfies (1)

*a*.

*e*. on

*J*.

This paper is divided into three sections; in the second section, we present some preliminary. Finally, in the third section, we give the existence of solutions and nonnegative solutions of system (1)-(4).

### 2 Preliminary

Let *X *and *Y *be two Banach spaces and *L*: *D*(*L*) ⊂ *X *→ *Y *be a linear operator, where *D*(*L*) denotes the domain of *L*. *L *will be a Fredholm operator of index 0, *i*.*e*., *ImL *is closed in *Y *and the linear spaces *KerL *and *coImL *have the same dimension which is finite. We define *X*_{1 }= *KerL *and *Y*_{1 }= *coImL*, so we have the decompositions *X *= *X*_{1 }⊕ *coKerL *and *Y *= *Y*_{1 }⊕ *ImL*. Now, we have the linear isomorphism Λ: *X*_{1 }→ *Y*_{1 }and the continuous linear projectors *P*: *X *→ *X*_{1 }and *Q*: *Y *→ *Y*_{1 }with *KerQ *= *ImL *and *ImP *= *X*_{1}.

Let Ω be an open bounded subset of *X *with Ω ∩ *D*(*L*) ≠ ∅. Operator
*L*, *S*) in Ω, as in [40,41], denoted by *d*(*L *- *S*, Ω), we assume that

It is easy to see that the operator
*M *= (*L *+ Λ*P*)^{-1 }(*S *+ Λ*P*) is well defined, and

If *M *is continuous and compact, then *S *is called *L*-compact, and the Leray-Schauder degree of *I _{X }*-

*M*(where

*I*is the identity mapping of

_{X }*X*) is well defined in Ω, and we will denote it by

*d*(

_{LS }*I*-

_{X }*M*, Ω, 0). This number is independent of the choice of

*P*,

*Q*and Λ (up to a sign) and we can define

**Definition 2.1**. (see [40,41]) The coincidence degree of (*L*, *S*) in Ω, denoted by *d*(*L *- *S*, Ω), is defined as *d*(*L *- *S*, Ω) = *d _{LS }*(

*I*-

_{X }*M*, Ω, 0).

There are many papers about coincidence degree and its applications (see [40-43]).

**Proposition 2.2**. (see [40]) (i) (Existence property). If *d*(*L *- *S*, Ω) ≠ 0, then there exists *x *∈ Ω such that *Lx *= *Sx*.

(ii) (Homotopy invariant property). If
*L*-compact and *Lx *≠ *H*(*x*, *λ*) for all *x *∈ ∂Ω and *λ *∈ [0, 1], then *d*(*L *- *H *(·, *λ*), Ω) is independent of *λ*.

The effect of small perturbations is negligible, as is proved in the next Proposition (see [41] Theorem III.3, page 24).

**Proposition 2.3**. Assume that *Lx *≠ *Sx *for each *x *∈ ∂Ω. If *S _{ε }*is such that sup

_{x∈∂Ω}||

*S*||

_{ε}x*is sufficiently small, then*

_{Y }*Lx*≠

*Sx*+

*S*for all

_{ε}x*x*∈ ∂Ω and

*d*(

*L*-

*S - S*, Ω) =

_{ε}*d*(

*L*-

*S*, Ω).

For any (*r*, *x*) ∈ (*J *× ℝ* ^{N}*), denote

*φ*

_{p(r)}(

*x*) = |

*x*|

^{p(r)-2}

*x*. Obviously,

*φ*has the following properties

**Proposition 2.4 **(see [41]) *φ *is a continuous function and satisfies

(i) For any *r *∈ [0, *T*], *φ*_{p(r)}(·) is strictly monotone, i.e.,

(ii) There exists a function *η*: [0, +∞) → [0, +∞), *η*(*s*) → +∞ as *s *→ +∞, such that

It is well known that *φ*_{p(r)}(·) is a homeomorphism from ℝ* ^{N }*to ℝ

*for any fixed*

^{N }*r*∈

*J*. Denote

It is clear that
*X *= {(*x*_{1}, *x*_{2}) | *x*_{1 }∈ *PC*, *x*_{2 }∈ *PC*} with the norm ||(*x*_{1}, *x*_{2})||* _{X }*= ||

*x*

_{1}||

_{0 }+ ||

*x*

_{2}||

_{0},

*Y*=

*L*

^{1 }×

*L*

^{1 }× ℝ

^{2(k + 1)N}, and we define the norm on

*Y*as

where *y*_{1}, *y*_{2 }∈ *L*^{1}, *z _{m }*∈ ℝ

*,*

^{N}*m*= 1,..., 2(

*k +*1), then

*X*and

*Y*are Banach spaces.

Define *L*: *D*(*L*) ⊂ *X *→ *Y *and *S*: *X *→ *Y *as the following

where

Obviously, the problem (1)-(4) can be written as *Lx *= *Sx*, where *L*: *X *→ *Y *is a linear operator, *S*: *X *→ *Y *is a nonlinear operator, and *X *and *Y *are Banach spaces.

Since

we have dim*KerL *= dim(*Y*/*ImL*) = 2*N *< +∞ is even and *ImL *is closed in *Y*, then *L *is a Fredholm operator of index zero. Define

at the same time the projectors *P*: *X *→ *X *and *Q*: *Y *→ *Y *satisfy

Since *ImQ *is isomorphic to *KerL*, there exists an isomorphism Λ: *KerL *→ *ImQ*. It is easy to see that *L *|_{D(L)∩KerP }: *D*(*L*) ∩ *KerP *→ *ImL *is invertible. We denote the inverse of that mapping by *K _{p}*, then

*K*:

_{p }*ImL*→

*D*(

*L*) ∩

*KerP*as

then

**Proposition 2.5 **(i) *K _{p}*(·) is continuous;

(ii) *K _{p }*(

*I*-

*Q*)

*S*is continuous and compact.

**Proof**. (i) It is easy to see that *K _{p}*(·) is continuous. Moreover, the operator

*L*

^{1 }to relatively compact set of

*PC*.

(ii) It is easy to see that *K _{p}*(

*I*-

*Q*)

*Sx*∈

*X*, ∀

*x*∈

*X*. Since

*f*is Caratheodory, it is easy to check that

*S*is a continuous operator from

*X*to

*Y*, and the operators (

*x*

_{1},

*x*

_{2}) →

*φ*

_{q(r) }((

*w*(

*r*))

^{-1 }

*x*

_{2}) and (

*x*

_{1},

*x*

_{2}) →

*f*(

*r*,

*x*

_{1},

*φ*

_{q(r) }((

*w*(

*r*))

^{-1 }

*x*

_{2})) both send bounded sets of

*X*to equi-integrable set of

*L*

^{1}. Obviously,

*A*,

_{i}*B*and

_{i }*QS*are compact continuous. Since

*f*is Caratheodory, by using the Ascoli-Arzela theorem, we can show that the operator

Denote

where *A _{i}*,

*B*are defined in (6),

_{i }*i*= 1,...,

*k*.

Consider

Define
*M*(·, ·) = (*L *+ Λ*P*)^{-1 }(*S*(·, ·) + Λ*P*), then

Since (*I *- *Q*)*S*(·, 0) = 0 and *K _{p }*(0) = 0, we have

It is easy to see that all the solutions of *Lx *= *S*(*x*, 0) belong to *KerL*, then

Notice that *P *|* _{KerL }*=

*I*, then

_{KerL}

**Proposition 2.6 **(continuation theorem) (see [40]). Suppose that *L *is a Fredholm operator of index zero and *S *is *L*-compact on
*X*. If the following conditions are satisfied,

(i) for each *λ *∈ (0, 1), every solution *x *of

is such that *x *∉ ∂Ω;

(ii) *QS*(*x*, 0) ≠ 0 for *x *∈ ∂Ω ∩ *KerL *and *d _{B}*(Λ

^{-1 }

*QS*(·,0), Ω ∩

*KerL*, 0) ≠ 0, then the operator equation

*Lx*=

*S*(

*x*, 1) has one solution lying in

The importance of the above result is that it gives sufficient conditions for being
able to calculate the coincidence degree as the Brouwer degree (denoted with *d _{B}*) of a related finite dimensional mapping. It is known that the degree of finite dimensional
mappings is easier to calculate. The idea of the proof is the use of the homotopy
of the problem

*Lx*=

*S*(

*x*, 1) with the finite dimensional one

*Lx*=

*S*(

*x*, 0).

Let us now consider the following simple impulsive problem

where *J*' = [0, *T*]\{*r*_{0}, *r*_{1}, ..., *r*_{k+1}}, *a _{i}*,

*b*∈ ℝ

_{i }*;*

^{N}*g*∈

*L*

^{1}.

If *u *is a solution of (7), then we have

Denote *ρ*_{0 }= *w*(0)_{φp(0) }(*u*'(0)). Obviously, *ρ*_{0 }is dependent on *g*, *a _{i}*,

*b*. Define

_{i}*F*:

*L*

^{1 }→

*PC*as

By (8), we have

If *a *≠ 0, then the boundary condition

The boundary condition

Denote *H *= *L*^{1 }× ℝ^{2kN }with the norm

then *H *is a Banach space. For fixed *h *∈ *H*, we denote

**Lemma 2.7 **The mapping Θ* _{h}*(·) has the following properties

(i) For any fixed *h *∈ *H*, the equation

has a unique solution *ρ*(*h*) ∈ ℝ* ^{N}*.

(ii) The mapping *ρ*: *H *→ ℝ* ^{N}*, defined in (i), is continuous and sends bounded sets to bounded sets. Moreover,

*h*= (

*g*,

*a*,

_{i}*b*) ∈

_{i}*H*,

*p*means

^{# }**Proof**. (i) From Proposition 2.4, it is immediate that

and hence, if (10) has a solution, then it is unique.

Let
*F*(*g*) ∈ *PC*, if |*ρ*| > *R*_{0}, it is easy to see that there exists a *j*_{0 }such that, the *j*_{0}-th component
*ρ *satisfies

Obviously,

then

and

By (11) and (12), the *j*_{0}-th component of
*J *and

Combining (13) and (14), the *j*_{0}-th component

From the definition

Without loss of generality, we may assume that

Therefore, the *j*_{0}-th component of
*j*_{0}-th component of
*a*, *b*, *c*, *d *∈ [0, +∞) and *ad *+ *bc *> 0, we can easily see that the *j*_{0}-th component of Θ* _{h}*(

*ρ*) keeps the same sign of

Let us consider the equation

According to the above discussion, all the solutions of (15) belong to *b*(*R*_{0 }+ 1) = {*x *∈ ℝ* ^{N}*| |

*x*| <

*R*

_{0 }+ 1}. So, we have

It means the existence of solutions of Θ* _{h}*(

*ρ*) = 0.

In this way, we define a mapping *ρ*(*h*): *H *→ ℝ* ^{N}*, which satisfies

(ii) By the proof of (i), we also obtain *ρ *sends bounded set to bounded set, and

It only remains to prove the continuity of *ρ*. Let {*u _{n}*} is a convergent sequence in

*H*and

*u*→

_{n }*u*, as

*n*→ +∞. Since {

*ρ*(

*u*)} is a bounded sequence, it contains a convergent subsequence

_{n}*j*→ +∞. Since Θ

*(*

_{h}*ρ*) consists of continuous functions, and

Letting *j *→ +∞, we have

from (i) we get *ρ*_{* }= *ρ*(*u*), it means that *ρ *is continuous.

This completes the proof.

If *a *= 0, the boundary condition

Since *ad *+ *bc *> 0, we have *c *> 0. Thus,

the boundary condition

Denote *G*: *H *→ ℝ* ^{N }*as

It is easy to see that

**Lemma 2.8 **The function *G*(·) is continuous and sends bounded sets to bounded sets. Moreover,
*p** means

### 3 Main results and proofs

In this section, we will apply coincidence degree to deal with the existence of solutions
for (1)-(4). In the following, we always use *C *and *C _{i }*to denote positive constants, if it cannot lead to confusion.

**Theorem 3.1 **Assume that Ω is an open bounded set in *X *such that the following conditions hold.

(1^{0}) For each *λ *∈ (0, 1) the problem

has no solution on ∂Ω.

(2^{0}) (0, 0) ∈ Ω.

Then, problem (1)-(4) has a solution *u *satisfies
*v *= *w*(*r*)*φ*_{p(r)}(*u*'(*r*)), ∀*r *∈ *J*'.

**Proof**. Let us consider the following operator equation

It is easy to see that *x *= (*x*_{1}, *x*_{2}) is a solution of *Lx *= *S*(*x*, 1) if and only if *x*_{1}(*r*) is a solution of (1)-(4) and
*r *∈ *J*'.

According to Proposition 2.5, we can conclude that *S*(·, ·) is *L*-compact from *X *× [0, 1] to *Y*. We assume that for *λ *= 1, (16) does not have a solution on ∂Ω, otherwise we complete the proof. Now from
hypothesis (1^{0}), it follows that (16) has no solutions for (*x*, *λ*) ∈ ∂Ω × (0, 1]. For *λ *= 0, (17) is equivalent to *Lx *= *S*(*x*, 0), namely the following usual problem

The problem (??) is a usual differential equation. Hence,

where *c*_{1}, *c*_{2 }∈ ℝ* ^{N }*are constants. The boundary value condition of (??) holds,

Since (*ad *+ *bc*) > 0, we have

which together with hypothesis (2^{0}), implies that (0, 0)∈ Ω. Thus, we have proved that (16) has no solution on ∂Ω ×
[0, 1]. It means that the coincidence degree *d*[*L *- *S*(·, *λ*), Ω] is well defined for each *λ *∈ [0, 1]. From the homotopy invariant property of that degree, we have

Now, it is clear that the following problem

is equivalent to problem (1)-(4), and (18) tells us that problem (19) will have a solution if we can show that

Since by hypothesis (2^{0}), this last degree

where *ω*_{*}(*c*_{1}, *c*_{2}) = (*ac*_{1 }- *bφ*_{q(0)}(*c*_{2}), *cc*_{1 }+ *dc*_{2}). This completes the proof.

Our next theorem is a consequence of Theorem 3.1. Denote

**Theorem 3.2 **Assume that the following conditions hold

(1^{0}) *a *> 0;

(2^{0}) lim_{|u| + |v| → +∞ }(*f*(*r*, *u*, *v*)/(|*u*| + |*v*|)^{β(r) -1}) = 0, for *r *∈ *J *uniformly, where *β*(*r*) ∈ *C*(*J*, ℝ), and 1<*β ^{- }*≤

*β*

^{+ }<

*p*

^{-};

(3^{0})
*u*| + |*v*| is large enough, where

(4^{0})
*u*| + |*v*| is large enough, where 0 ≤ *ε *< *β *^{+ }- 1.

Then, problem (1)-(4) has at least one solution.

**Proof**. Now, we consider the following operator equation

For any *λ *∈ (0, 1], *x *= (*x*_{1}, *x*_{2}) = (*u*, *v*) is a solution of (20) if and only if
*u*(*r*) is a solution of the following

We claim that all the solutions of (21) are uniformly bounded for *λ *∈ (0, 1]. In fact, if it is false, we can find a sequence (*u _{n}*,

*λ*) of solutions for (21), such that ||

_{n}*u*||

_{n}_{1 }> 1 and ||

*u*||

_{n}_{1 }→ +∞ when

*n*→ +∞,

*λ*∈ (0, 1]. Since (

_{n }*u*,

_{n}*λ*) are solutions of (21), we have

_{n}

for any *r *∈ *J*', where

By computation, we have

Denote

We claim that

If it is false, without loss of generality, we may assume that

then for any *n *= 1, 2, ..., there is a *j _{n }*∈ {1, ...,

*N*} such that the

*j*-th component

_{n}*ρ*satisfies

_{n }

Thus, when *n *is large enough, the *j _{n}*-th component

*(*

_{n}*r*) keeps the same sign as

When *n *is large enough, we can conclude that the *j _{n}*-th component

*F*{

*φ*

_{q(r)}[Γ

*(*

_{n}*r*)]} (

*T*) keeps the same sign as

Since

from (22) and (24), we can see that
*n *is large enough.

But the boundary value conditions (4) mean that

It is a contradiction. Thus (23) is valid. Therefore,

It means that

From (22), (23) and (25), for any *r *∈ *J*, we have