The nonlinear alternative of the Leray Schauder type and the Banach contraction principle are used to investigate the existence of solutions for second-order differential equations with integral boundary conditions. The compactness of solutions set is also investigated.
1. Introduction
This paper is concerned with the existence of solutions for the second-order boundary value problem
(11)where 
is a given function and 
is an integrable function.
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. For boundary value problems with integral boundary conditions and comments on their importance, we refer the reader to the papers [1–9] and the references therein. Moreover, boundary value problems with integral boundary conditions have been studied by a number of authors, for example [10–14]. The goal of this paper is to give existence and uniqueness results for the problem (1.1). Our approach here is based on the Banach contraction principle and the Leray-Schauder alternative [15].
2. Preliminaries
In this section, we introduce notations, definitions, and preliminary facts that will
be used in the remainder of this paper. Let 
be the space of differentiable functions 
whose first derivative, 
, is absolutely continuous.
We take 
to be the Banach space of all continuous functions from 
into 
with the norm
(21)and we let 
denote the Banach space of functions 
that are Lebesgue integrable with norm
(22)Definition 2.1.
A map 
is said to be 
-Carathéodory if
(i)
is measurable for each 
(ii)
is continuous for almost each 
(iii)for every 
there exists 
such that
(23)3. Existence and Uniqueness Results
Definition 3.1.
A function 
is said to be a solution of (1.1) if 
satisfies (1.1).
In what follows one assumes that 
One needs the following auxiliary result.
Lemma 3.2.
. Let 
. Then the function defined by
(31)is the unique solution of the boundary value problem
(32)where
(33)Proof.
Let 
be a solution of the problem (3.2). Then integratingly, we obtain
(34)Hence
(35)
(36)where
(37)Now, multiply (3.6) by 
and integrate over 
, to get
(38)Thus,
(39)Substituting in (3.6) we have
(310)Therefore
(311)Set 
Note that
(312)Our first result reads
Theorem 3.3.
Assume that 
is an 
-Carathéodory function and the following hypothesis
(A1) There exists 
such that
(313)holds. If
(314)then the BVP (1.1) has a unique solution.
Proof.
Transform problem (1.1) into a fixed-point problem. Consider the operator 
defined by
(315)We will show that 
is a contraction. Indeed, consider 
Then we have for each 
(316)Therefore
(317)showing that, 
is a contraction and hence it has a unique fixed point which is a solution to (1.1).
The proof is completed.
We now present an existence result for problem (1.1).
Theorem 3.4.
Suppose that hypotheses
(H1) The function 
is an 
-Carathéodory,
(H2) There exist functions 
and 
such that
(318)are satisfied. Then the BVP (1.1) has at least one solution. Moreover the solution set
(319)is compact.
Proof.
Transform the BVP (1.1) into a fixed-point problem. Consider the operator 
as defined in Theorem 3.3. We will show that 
satisfies the assumptions of the nonlinear alternative of Leray-Schauder type. The
proof will be given in several steps.
Step 1 (
is continuous).
Let 
be a sequence such that 
in 
Then
(320)Since 
is 
-Carathéodory and 
then
(321)Hence
(322)Step 2 (
maps bounded sets into bounded sets in 
).
Indeed, it is enough to show that there exists a positive constant 
such that for each 
one has 
.
Let 
. Then for each 
, we have
(323)By (H2) we have for each 
(324)Then for each 
we have
(325)Step 3 (
maps bounded set into equicontinuous sets of 
).
Let 
,
and 
be a bounded set of 
as in Step 2. Let 
and 
we have
(326)As 
the right-hand side of the above inequality tends to zero. Then 
is equicontinuous. As a consequence of Steps 1 to 3 together with the Arzela-Ascoli
theorem we can conclude that 
is completely continuous.
Step 4 (A priori bounds on solutions).
Let 
for some 
. This implies by 
that for each 
we have
(327)Then
(328)If 
we have
(329)Thus
(330)Hence
(331)Set
(332)and consider the operator 
From the choice of 
, there is no 
such that 
for some 
As a consequence of the nonlinear alternative of Leray-Schauder type [15], we deduce that 
has a fixed point 
in 
which is a solution of the problem (1.1).
Now, prove that 
is compact. Let 
be a sequence in 
, then
(333)As in Steps 3 and 4 we can easily prove that there exists 
such that
(334)and the set 
is equicontinuous in 
hence by Arzela-Ascoli theorem we can conclude that there exists a subsequence of
converging to 
in 
Using that fast that 
is an 
-Carathédory we can prove that
(335)Thus 
is compact.
4. Examples
We present some examples to illustrate the applicability of our results.
Example 4.1.
Consider the following BVP
(41)Set
(42)We can easily show that conditions (A1), (3.14) are satisfied with
(43)Hence, by Theorem 3.3, the BVP (4.1) has a unique solution on 
.
Example 4.2.
Consider the following BVP
(44)Set
(45)We can easily show that conditions (H1), (H2) are satisfied with
(46)Hence, by Theorem 3.4, the BVP (4.4) has at least one solution on 
. Moreover, its solutions set is compact.
Acknowledgment
The authors are grateful to the referees for their remarks.
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