In this paper, we consider the existence of solutions and nonnegative solutions for the following weighted
-Laplacian integro-differential system:
, with the following impulsive boundary value conditions:
where and , is called the weighted -Laplacian; , ; (); is nonnegative, ; , ; ; T and S are linear operators defined by , , , where .
If and , we say the problem is nonresonant, but if or , we say the problem is resonant.
Throughout the paper,
means functions which are uniformly convergent to 0 (as
); for any
will denote the j
th component of v
; the inner product in
will be denoted by
will denote the absolute value and the Euclidean norm on
. Denote by
the interior of
For any , denote .
is a Banach space with the norm
is a Banach space with the norm
with the norm
In the following,
will be simply denoted by PC
, respectively. We denote
The study of differential equations and variational problems with nonstandard
-growth conditions has attracted more and more interest in recent years (see [1
]). The applied background of these kinds of problems includes nonlinear elasticity theory [4
], electro-rheological fluids [1
], and image processing [2
]. Many results have been obtained on these kinds of problems; see, for example, [5
]. Recently, the applications of variable exponent analysis in image restoration have attracted more and more attention [16
(a constant), (1)-(4) becomes the well-known p
-Laplacian problem. If
is a general function, one can see easily
in general, but
represents a non-homogeneity and possesses more nonlinearity, thus
is more complicated than
. For example:
is a bounded domain, the Rayleigh quotient
is zero in general, and only under some special conditions
) is an interval, the results show that
if and only if
is monotone. But the property of
is very important in the study of p
-Laplacian problems, for example, in [20
], the authors use this property to deal with the existence of solutions.
In recent years, many results have been devoted to the existence of solutions for the Laplacian impulsive differential equation boundary value problems; see, for example, [22–29]. There are some methods to deal with these problems, for example, sub-super-solution method, fixed point theorem, monotone iterative method, coincidence degree. Because of the nonlinear property of , results on the existence of solutions for p-Laplacian impulsive differential equation boundary value problems are rare (see [30–33]). In , using the coincidence degree method, the present author investigates the existence of solutions for -Laplacian impulsive differential equation with multi-point boundary value conditions, when the problem is nonresonant. Integral boundary conditions for evolution problems have various applications in chemical engineering, thermo-elasticity, underground water flow and population dynamics. There are many papers on the differential equations with integral boundary value problems; see, for example, [35–38].
In this paper, when
is a general function, we investigate the existence of solutions and nonnegative solutions for the weighted
-Laplacian impulsive integro-differential system with integral and multi-point boundary value conditions. Results on these kinds of problems are rare. Our results contain both of the cases of resonance and nonresonance. Our method is based upon Leray-Schauder’s degree. The homotopy transformation used in [34
] is unsuitable for this paper. Moreover, this paper will consider the existence of (1) with (2), (4) and the following impulsive condition:
where , the impulsive condition (5) is called a linear impulsive condition (LI for short), and (3) is called a nonlinear impulsive condition (NLI for short). In general, p-Laplacian impulsive problems have two kinds of impulsive conditions, including LI and NLI; but Laplacian impulsive problems only have LI in general. It is another difference between p-Laplacian impulsive problems and Laplacian impulsive problems. Moreover, since the Rayleigh quotient in general and the -Laplacian is non-homogeneity, when we deal with the existence of solutions of variable exponent impulsive problems like (1)-(4), we usually need the nonlinear term that satisfies the sub- growth condition, but for the p-Laplacian impulsive problems, the nonlinear term only needs to satisfy the sub- growth condition.
, the function
is assumed to be Caratheodory, by which we mean:
For almost every , the function is continuous;
For each , the function is measurable on J;
, there is a
such that, for almost every
, one has
We say a function is a solution of (1) if with absolutely continuous on , , which satisfies (1) a.e. on J.
In this paper, we always use
to denote positive constants, if it cannot lead to confusion. Denote
We say f
satisfies the sub-
growth condition if f
where and .
We will discuss the existence of solutions for system (1)-(4) or (1) with (2), (4) and (5) in the following three cases:
Case (i): , ;
Case (ii): , ;
Case (iii): , .
This paper is organized as five sections. In Section 2, we present some preliminaries and give the operator equation which has the same solutions of (1)-(4) in the three cases, respectively. In Section 3, we give the existence of solutions for system (1)-(4) or (1) with (2), (4) and (5) when , . In Section 4, we give the existence of solutions for system (1)-(4) or (1) with (2), (4) and (5) when , . Finally, in Section 5, we give the existence of solutions and nonnegative solutions for system (1)-(4) or (1) with (2), (4) and (5) when , .