The theory of impulsive differential equations describes processes which experience a sudden change of their state at certain moments. Processes with such a character arise naturally and often, especially in phenomena studied in physics, chemical technology, population dynamics, biotechnology, and economics. For an introduction of the basic theory of impulsive differential equations, see Lakshmikantham et al. ; for an overview of existing results and of recent research areas of impulsive differential equations, see Benchohra et al. . The theory of impulsive differential equations has become an important area of investigation in recent years and is much richer than the corresponding theory of differential equations (see, e.g., [3–18] and references cited therein).
Moreover, the theory of boundary-value problems with integral boundary conditions for ordinary differential equations arises in different areas of applied mathematics and physics. For example, heat conduction, chemical engineering, underground water flow, thermo-elasticity, and plasma physics can be reduced to the nonlocal problems with integral boundary conditions. For boundary-value problems with integral boundary conditions and comments on their importance, we refer the reader to the papers by Gallardo , Karakostas and Tsamatos , Lomtatidze and Malaguti , and the references therein. For more information about the general theory of integral equations and their relation with boundary-value problems, we refer to the book of Corduneanu  and Agarwal and O'Regan .
On the other hand, boundary-value problems with integral boundary conditions constitute a very interesting and important class of problems. They include two, three, multipoint and nonlocal boundary-value problems as special cases. The existence and multiplicity of positive solutions for such problems have received a great deal of attention in the literature. To identify a few, we refer the reader to [24
] and references therein. In particular, we would like to mention some results of Zhang et al. [34
], Kang et al. [44
], and Webb et al. [45
]. In [34
], Zhang et al. studied the following fourth-order boundary value problem with integral boundary conditions
is a positive parameter,
is the zero element of
. The authors investigated the multiplicity of positive solutions to problem (1.1) by using the fixed point index theory in cone for strict set contraction operator.
], Kang et al. have improved and generalized the work of [34
] by applying the fixed point theory in cone for a strict set contraction operator; they proved that there exist various results on the existence of positive solutions to a class of fourth-order singular boundary value problems with integral boundary conditions
and may be singular at
are continuous and may be singular at
More recently, by using a unified approach, Webb et al. [45
] considered the widely studied boundary conditions corresponding to clamped and hinged ends and many nonlocal boundary conditions and established excellent existence results for multiple positive solutions of fourth-order nonlinear equations which model deflections of an elastic beam
subject to various boundary conditions
denotes a linear functional on
involving a Stieltjes integral, and
is a function of bounded variation.
At the same time, we notice that there has been a considerable attention on
-Laplacian BVPs [18
-Laplacian appears in the study of flow through porous media (
), nonlinear elasticity (
), glaciology (
), and so forth. Here, it is worth mentioning that Liu et al. [43
] considered the following fourth-order four-point boundary value problem:
. By using upper and lower solution method, fixed-point theorems, and the properties of Green's function
, the authors give sufficient conditions for the existence of one positive solution.
Motivated by works mentioned above, in this paper, we consider the existence of positive solutions for a class of boundary value problems with integral boundary conditions of fourth-order impulsive differential equations:
-Laplace operator, that is,
is fixed positive integer) are fixed points with
represent the right-hand limit and left-hand limit of
, respectively, and
For the case of
, problem (1.6) reduces to the problem studied by Zhang et al. in . By using the fixed point theorem in cone, the authors obtained some sufficient conditions for the existence and multiplicity of symmetric positive solutions for a class of
-Laplacian fourth-order differential equations with integral boundary conditions.
For the case of
, problem (1.6) is related to fourth-order two-points boundary value problem of ODE. Under this case, problem (1.6) has received considerable attention (see, e.g., [40–42] and references cited therein). Aftabizadeh  showed the existence of a solution to problem (1.6) under the restriction that
is a bounded function. Bai and Wang  have applied the fixed point theorem and degree theory to establish existence, uniqueness, and multiplicity of positive solutions to problem (1.6). Ma and Wang  have proved that there exist at least two positive solutions by applying the existence of positive solutions under the fact that
is either superlinear or sublinear on
by employing the fixed point theorem of cone extension or compression.
Being directly inspired by [18, 34, 43], in the present paper, we consider some existence results for problem (1.6) in a specially constructed cone by using the fixed point theorem. The main features of this paper are as follows. Firstly, comparing with [39–43], we discuss the impulsive boundary value problem with integral boundary conditions, that is, problem (1.6) includes fourth-order two-, three-, multipoint, and nonlocal boundary value problems as special cases. Secondly, the conditions are weaker than those of [33, 34, 46], and we consider the case
. Finally, comparing with [33, 34, 39–43, 46], upper and lower bounds for these positive solutions also are given. Hence, we improve and generalize the results of [33, 34, 39–43, 46] to some degree, and so, it is interesting and important to study the existence of positive solutions of problem (1.6).
The organization of this paper is as follows. We shall introduce some lemmas in the rest of this section. In Section 2, we provide some necessary background. In particular, we state some properties of the Green's function associated with problem (1.6). In Section 3, the main results will be stated and proved. Finally, in Section 4, we offer some interesting discussion of the associated problem (1.6).
To obtain positive solutions of problem (1.6), the following fixed point theorem in cones is fundamental, which can be found in [47, page 94].
be two bounded open sets in Banach space
, such that
be a cone in
and let operator
be completely continuous. Suppose that one of the following two conditions is satisfied:
has at least one fixed point in