We study the existence of multiple nonnegative solutions for the doubly singular three-point boundary value problem with derivative dependent data function . Here, with on and is allowed to be discontinuous at . The fixed point theory in a cone is applied to achieve new and more general results for existence of multiple nonnegative solutions of the problem. The results are illustrated through examples.
In this paper, we consider the following three-point boundary value problem of Sturm-Liouville type:
with boundary conditions
where and .
Throughout this paper, we assume the following conditions on the functions , , and : with on and ;, is not identically zero on and ; and is not identically zero.Note that condition (E2) allows be discontinuous at , and if , then the differential equation (1.1) is called doubly singular .
Nonlocal boundary value problem have variety of applications in the area of applied mathematics and physical sciences. The design of a large size bridge with multipoint supports can be considered as an application of these types of boundary value problem . Some more applications can be found in [3–5] and the references therein. Recently, motivated by the wide application of boundary value problems in physical and applied mathematics, the study of multipoint boundary value problems has received increasing interest (see [2, 6–12] and the references therein).
Nonsingular multipoint boundary value problems have been extensively studied in literature, see [13–16] for derivative dependent data function and [8, 10, 12] for derivative independent data function .
Some attention has been devoted to singular multipoint boundary value problems (see [17, 18] and the references therein). When and may have singularity at , and , differential equation (1.1) with boundary conditions is considered by Chen et al.  and Agarwal et al. . Chen et al. proved the existence of at least one positive solution while Agarwal et al. established that this problem may have at least two positive solutions and also may have no positive solutions under some conditions on and .
Bai and Ge  have generalized the Leggett-Williams fixed point theory and applied to
to achieve at least three positive solutions of the two-point boundary value problem.
In this work, we consider the problem (1.1)-(1.2) with unbounded coefficient of along with singularity in the data function .
Existence of nonnegative solution(s) of the problem (1.1)-(1.2) may be established either directly or by reducing the problem to
and applying the existing results. But direct consideration of the problem provides better results, especially as the order of singularity increases. This may be demonstrated by the following simple linear three-point boundary value problem:
The problem (1.5) can be reduced to the following boundary value problem:
by change of variable .
Now we apply the result (Theorem 4.2) of this work to the problem (1.5) and conclude that the problem has at least one nonnegative solution with
Further, for , Theorems 4.2 and 4.3 may be regarded as extension of Theorem 3.1 in  for three-point singular boundary value problem. Now applying Theorem 4.2 with to the reduced problem (1.6), we get that the problem (1.5) has at least one nonnegative solution with
Now as approaches to one, that is, the order of singularity increases, the upper bound for in (1.8) approaches to while in (1.7) approaches to , which can be seen from Figure 1. As smaller upper bound for will enable to find nonnegative solution(s) faster and hence will be helpful in constructing efficient numerical algorithms to find multiple nonnegative solutions, thus it is justified to consider the singular problem directly. A detailed working is given in Example 5.1.
Figure 1: Variation of bounds for in both cases.
In this work, we are concerned with existence of multiple nonnegative solutions of the three-point doubly singular boundary value problem (1.1)-(1.2). To achieve this, we use generalized Leggett-Williams fixed point theorem established by Bai and Ge .
For this purpose, we first establish certain properties of Green’s function of the corresponding homogeneous boundary value problem. Then fixed point theorem of functional type (generalized Leggett-Williams fixed point theorem) is applied to yield multiple nonnegative solutions for the boundary value problem (1.1)-(1.2).
We organize this work as follows. In Section 2, we present some definitions and basic results required for this work. Section 3 deals with nonnegativity of Green’s function and some basic properties. Section 4 is devoted to existence of at least one and three or odd number of nonnegative solutions. In Section 5, we demonstrate the results through examples.
2. Background and Definitions
The proof of main results is based on fixed point theorem of functional type in a cone given by Bai and Ge , which deals with three fixed points of completely continuous nonlinear operators defined in a cone of an ordered Banach space. In this section, we provide some background material from the theory of cone in Banach spaces to make the paper self-contained.
Definition 2.1. A subset of Banach space is said to be retract of if a continuous map such that for every .
Corollary 2.2. Every close convex set of a Banach space is a retract of Banach space.
Definition 2.3. Let be a Banach space, is nonempty convex, closed set, is said to be cone provided that(1) for all , and(2) implies .
Note. From Corollary 2.2, a cone of Banach Space is retract of .
Definition 2.4. A subset of Banach space is called relatively compact if (closure of ) is compact.
Definition 2.5. Consider two Banach spaces and , a subset of , and a map . Then is said to be completely continuous operator if(1) is continuous, and(2) maps bounded subset of into relatively compact sets.
Definition 2.6. The map is said to be a nonnegative continuous convex functional on provided that is continuous and
for all and . Similarly, the map is said to be a nonnegative continuous concave functional on provided that is continuous and
for all , and .
Definition 2.7. Suppose are two continuous convex functionals satisfying
where is positive constant, and
From (2.3) and (2.4), is a bounded nonempty open subset of .
Definition 2.8. Let , be given constants, two nonnegative continuous convex functionals satisfying (2.3) and (2.4), and a nonnegative continuous concave functional on the cone . Then bounded convex sets are defined as
Theorem 2.9 (see ). Let be retract of real Banach space . Then for every bounded relatively open subset of and every completely continuous operator which has no fixed point on (relative to ), there exists an integer such that if , then has at least one fixed point in . Moreover, is uniquely defined.
Theorem 2.10 (see ). Let be Banach space, retract of , a bounded convex retract of , and nonempty open subset, such that . If is completely continuous, , such that there is no fixed point of in , then .
Theorem 2.11 (see ) (fixed point theorem of functional type). Let be Banach space, a cone, and given constants. Assume that are nonnegative continuous convex functionals on such that (2.3) and (2.4) are satisfied. is a nonnegative continuous concave functional on such that for all and let be a completely continuous operator. Suppose that(1) and for ,(2), for all ,(3) for all with .Then has at least three fixed points such that
3. Some Preliminary Results
In this section, we construct the Green’s function and establish some properties, required to establish the main results in Section 4.
Lemma 3.1. The Green’s function for the following boundary value problem:
is given by
Proof. Consider the following linear differential equation:
where . Integrating the above differential equation twice first from to 1 and then from 0 to , changing the order of integration, and applying the boundary conditions, we get
For , can be written as
Similarly, for , can be written as
From (3.7) and (3.9), we may write
where is given in the lemma. It is easy to see that satisfies all the properties of Green’s function. Hence is the Green’s function for the boundary value problem (3.1).
Lemma 3.2. The Green’s function satisfies the following properties:(i),(ii) for all ,(iii)there exist a constant in (0,1) such that , where
Since is independent of , therefore . (ii) For , and
it is easy to show that , for . Next we show , and as follows:
Thus for all . (iii) We prove the inequality for the following cases: (a) and (b) , (a) for , we further divide this case in two parts as follows. (1) When , (2) when . Case 1 (For ). It is easy to see that
Thus for ,
Case 2 (For ). It is easy to see that
Thus for ,
Combining (3.18) and (3.21), we may write for ,
(b) For . For this case, and are considered. From (3.2), it can be easily seen that for ,
Thus from (3.22) and (3.23), we get
It can be easily seen that . This completes the proof.
4. Existence of Multiple Nonnegative Solutions
Let be endowed with ordering if for all and , where
Let be bounded subset of . is Banach Space.
Now define a cone as
The boundary value problem (1.1)-(1.2) has a solution if and only if solves the following operator equation:
where the operator is given by
Here is the Green’s function of the problem (3.1) defined in Lemma 3.1.
Lemma 4.1. Let (E1)–(E3) hold, then the operator is well defined and is completely continuous.
Proof. First we show that the operator is well defined. For this, we take . From (E2), (E3), and , it follows that . Now applying Lemma 3.2, we get
It is easy to show that . Thus is well defined. We now show that is completely continuous. Let be a sequence in and with . Then, there exists a constant such that for all . Thus as implies as . So and as . Since is continuous on , so
From (4.6) and (4.7),
Hence is a continuous operator. Next we prove that maps every bounded subset of into relatively compact set. Let be any bounded subset of . For ,
Therefore is uniformly bounded. Further, equicontinuity of follows from
Thus from Arzela-Ascoli Theorem, is relatively compact subset of and also is completely continuous.
Next, define functionals such that
Clearly, are nonnegative continuous convex functionals such that satisfying (2.3) and (2.4), and is nonnegative concave functional with .
Now we state the main results of this work.
Theorem 4.2. Suppose that (E1)–(E3) are satisfied and satisfies the following condition.(H1) if there exist real constants and such that for ,then boundary value problem (1.1)-(1.2) has at least one nonnegative solution such that with .
Theorem 4.3. Suppose that (E1)–(E3) are satisfied. There exist real constants with , such that and satisfies following conditions.(H1) for ,(H2) for ,(H3) for .Then boundary value problem (1.1)-(1.2) has at least three nonnegative solutions , , and in such that
Proof of Theorem 4.2. Let be open subset of . We now show that . For ,
implies that . Consider that
implies that . Thus . Next, we show that has no fixed point on . On contrary, suppose there exists a fixed point on such that . Then from (4.14) and (4.15), and , which are not possible. So the operator has no fixed point on and from Theorem 2.10. Thus the operator has at least one fixed point in and also the boundary value problem (1.1)-(1.2) has at least one nonnegative solution such that with .
Proof of Theorem 4.3. It is easy to see that for each . We now show that is well defined. For ,
From (4.16) and (4.17),
Thus, is well defined, and by Lemma 4.1, it is completely continuous. Now Condition (2) of Theorem 2.11 can be proved by similar manner. Choose , , then , . Thus, . Further if , then for . Then by definition of and assumption , we have
Thus, Condition (1) of Theorem 2.11 is satisfied. We finally show that condition (3) of Theorem 2.11 holds, too. Suppose with . Then by definition of and , we have
So, Condition (3) of Theorem 2.11 is also satisfied. Therefore, Theorem 2.11 yields that boundary value problem (1.1)-(1.2) has at least three nonnegative solutions , , and in such that
Corollary 4.4. Suppose that (E1)–(E3) are satisfied. If there exist constants , with such that satisfies the following conditions:(M1),
then boundary value problem (1.1)-(1.2) has at least nonnegative solutions.
Proof. When , the result follows from Theorem 4.2. When , it is clear that all the conditions of Theorem 4.3 hold (with ). Thus the boundary value problem (1.1)-(1.2) has at least three positive solutions , , and . Following this way, we complete the proof by induction method.
Finally, we demonstrate these results through examples.
In Example 5.1, we demonstrate the detailed working of the boundary value problem (1.5) mentioned in the introduction. Example 5.2 verifies our results.
Example 5.1. Consider the boundary value problem (1.5). Here,
Following the notations of this work, it is easy to see that
Now for ,. Then from Theorem 4.2, the problem has at least one nonnegative solution with
Next we reduce the problem and then apply Theorem 4.2 for . Using the transformation , the boundary value problem (1.5) can be reduced to regular boundary value problem as
Now following the notation of this work for ,
Now for ,. So the problem has at least one nonnegative solution with
Hence the boundary value problem (1.5) has at least one nonnegative solution with
Now in this case it is easy to show that if approaches one, that is, the order of singularity increases, upper bound for approaches while in case of direct solving, upper bound for approaches 4.125. As smaller upper bound for will enable to find nonnegative solution(s) faster and hence will be helpful in constructing efficient numerical algorithms to find multiple nonnegative solutions.
Example 5.2. Consider the following boundary value problem:
then the boundary value problem (5.9) has at least one nonnegative solution.(ii) Further, if
then the boundary value problem (5.9) has at least three nonnegative solutions.
Proof. Here, and . After simple calculation, we get , , , and . (i) At least one nonnegative solution: we choose and . Here, ;
Thus, condition is satisfied. Now from Theorem 4.2 the problem has at least one nonnegative solution such that with . (ii) At least three nonnegative solutions: we choose constants , , , and