In the following we denote by the space of -Lebesgue measurable functions from to with the norm .
Theorem 3.1 Let be continuous functions satisfying the following conditions:
() f satisfies the Lipschitz condition
where , , , ;
() there exist a function and a nondecreasing function such that
Then the boundary value problem (1) has at least one solution, provided
Proof Let be a closed bounded and convex subset of , where r will be fixed later. Using (5), we define a map as
Observe that the problem (1) is equivalent to a fixed point problem .
Let us decompose Φ as , where
Step 1. .
For that, we select , where
Using for , , we get
In a similar manner, we have that
which implies that .
Step 2. is continuous and γ-contractive.
To show the continuity of for , let us consider a sequence converging to x. Then, by the assumption (), we have
Next, we show that is γ-contractive. For , we get
By the given assumption,
it follows that is γ-contractive.
Step 3. is compact.
In Step 1, it has been shown that is uniformly bounded. Now we show that maps bounded sets into equicontinuous sets of . Let with and . Then we obtain
Obviously, the right-hand side of the above inequality tends to zero independently of as . Therefore it follows by the Arzelá-Ascoli theorem that is completely continuous. Thus is compact on .
Step 4. Φ is condensing.
Since is continuous, γ-contraction and is compact, therefore, by Lemma 2.2, with is a condensing map on .
Consequently, by Theorem 2.1, the map Φ has a fixed point which implies that the problem (1) has a solution. □
In the special case when , L a constant, we have the following.
Corollary 3.1 Let be continuous functions. Assume that g satisfies () and f satisfies the following condition:
()′ for all , is a constant.
then the boundary value problem (1) has at least one solution.
Example 3.1 Consider a boundary value problem of integro-differential equations of fractional order with nonlocal fractional boundary conditions given by
where , , , , .
Then we have
Also, for all and each , we have
and with () and . Selecting and using the given data, we find that
As , therefore, by the conclusion of Theorem 3.1, the problem (6) has a solution.