Let denote the Banach space of all continuous functions from into ℝ endowed with the usual supremum norm.
In view of Lemma 2.3, we define an operator
Observe that the problem (1.1) has solutions if and only if the operator equation has fixed points.
In the sequel, we use the following notation:
where , with () given by (2.3).
Our first result is based on the Leray-Schauder nonlinear alternative .
Lemma 3.1 (Nonlinear alternative for single valued maps)
Let E be a Banach space
, C a closed
, convex subset of E
, U an open subset of C
. Suppose that is a continuous
, is a relatively compact subset of C
. Then either
F has a fixed point in , or
there is a (the boundary of U in C) and with .
Theorem 3.2 Let be a jointly continuous function. Assume that:
() there exist a function and a nondecreasing function such that , ;
) there exists a constant such that
Then the boundary value problem (1.1) has at least one solution on .
Consider the operator
defined by (3.1). We show that F maps bounded sets into bounded sets in
. For a positive number r
be a bounded set in
Next, we show that F maps bounded sets into equicontinuous sets of
is a bounded set of
. Then we obtain
Obviously, the right-hand side of the above inequality tends to zero independently of as . As ℱ satisfies the above assumptions, therefore, it follows by the Arzelá-Ascoli theorem that is completely continuous.
be a solution. Then for
, using the computations in proving that ℱ
is bounded, we have
Consequently, we have
In view of (
), there exists M
. Let us set
Note that the operator is continuous and completely continuous. From the choice of U, there is no such that for some . Consequently, by the nonlinear alternative of Leray-Schauder type (Lemma 3.1), we deduce that ℱ has a fixed point which is a solution of the problem (1.1). This completes the proof. □
In the special case when and (κ and N are suitable constants) in the statement of Theorem 3.2, we have the following corollary.
Corollary 3.3 Let be a continuous function. Assume that there exist constants , where ω is given by (3.2) and such that for all , . Then the boundary value problem (1.1) has at least one solution.
Next, we prove an existence and uniqueness result by means of Banach’s contraction mapping principle.
Theorem 3.4 Suppose that is a continuous function and satisfies the following assumption:
() , , , .
Then the boundary value problem
(1.1) has a unique solution provided
where ω is given by (3.2).
, we define
is given by (3.2). Then we show that
, we have
, the above expression yields
where we used (3.2). Now, for
and for each
, we obtain
Note that ω depends only on the parameters involved in the problem. As , therefore, ℱ is a contraction. Hence, by Banach’s contraction mapping principle, the problem (1.1) has a unique solution on . □
Now, we prove the existence of solutions of (1.1) by applying Krasnoselskii’s fixed point theorem .
Theorem 3.5 (Krasnoselskii’s fixed point theorem)
Let M be a closed, bounded, convex, and nonempty subset of a Banach space X. Let A, B be the operators such that (i) whenever ; (ii) A is compact and continuous; (iii) B is a contraction mapping. Then there exists such that .
Theorem 3.6 Let be a jointly continuous function satisfying the assumption (). In addition we assume that:
() , , and .
Then the problem
(1.1) has at least one solution on if
, we choose a real number
satisfying the inequality
. We define the operators
, we find that
. It follows from the assumption (
) together with (3.4) that
is a contraction mapping. Continuity of f
implies that the operator
is continuous. Also,
is uniformly bounded on
Now, we prove the compactness of the operator .
In view of (
), we define
, and consequently, for
, we have
which is independent of x. Thus, is equicontinuous. Hence, by the Arzelá-Ascoli theorem, is compact on . Thus, all the assumptions of Theorem 3.5 are satisfied. So, the conclusion of Theorem 3.5 implies that the boundary value problem (1.1) has at least one solution on . □