In this section, our aim is to discuss the existence and uniqueness of solutions to the problem (1.1).

Let
be a Banach space of all continuous functions from
with the norm
.

Theorem 3.1.

Assume that

(H1) there exists a real-valued function

for some

such that

where
,
,
,
, then problem (1.1) has a unique solution.

Proof.

Define an operator

by

Let

and choose

where
is such that
.

Now we show that

, where

. For

, by Hölder inequality, we have

Take notice of Beta functions:

Therefore,
.

For

and for each

, based on Hölder inequality, we obtain

Since
, consequently
is a contraction. As a consequence of Banach fixed point theorem, we deduce that
has a fixed point which is a solution of problem (1.1).

Corollary 3.2.

Assume that

(H1)′ There exists a constant

such that

then problem (1.1) has a unique solution.

Theorem 3.3.

Suppose that (H1) and the following condition hold:

(H2) There exists a constant

and a real-valued function

such that

Then the problem (1.1) has at least one solution on

if

Proof.

here,

; consider

, then

is a closed, bounded, and convex subset of Banach space

. We define the operators

and

on

as

For

, based on Hölder inequality, we find that

Thus,
, so
.

For

and for each

, by the analogous argument to the proof of Theorem 3.1, we obtain

it follows that
is a contraction mapping.

The continuity of

implies that the operator

is continuous. Also,

is uniformly bounded on

as

On the other hand, let

, for all

, setting

For each

, we will prove that if

and

, then

In the following, the proof is divided into two cases.

Case 1.

For

, we have

Case 2.

for

,

, we have.

Therefore,
is equicontinuous and the Arzela-Ascoli theorem implies that
is compact on
, so the operator
is completely continuous.

Thus, all the assumptions of Lemma 2.7 are satisfied and the conclusion of Lemma 2.7 implies that the boundary value problem (1.1) has at least one solution on
.

Corollary 3.4.

Suppose that the condition (H1)′ hold and, assume that

Further assume that

(H2)′ there exists a constant

such that

then problem (1.1) has at least one solution on
.