Abstract
In the paper, we consider the existence criteria for positive solutions of the nonlinear p-Laplacian fractional differential equation whose nonlinearity contains the first-order derivative explicitly
where is the p-Laplacian operator, i.e., , , and , . is the standard Caputo derivative and satisfies the Carathéodory type condition. The nonlinear alternative of Leray-Schauder type and the fixed-point theorems in Banach space are used to investigate the existence of at least single, twin, triple, n or positive solutions for p-Laplacian fractional order differential equations. As an application, two examples are given to illustrate our theoretical results.
MSC:34A08, 34B18, 34K37.
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1 Introduction
Fractional calculus is a generalization of ordinary differentiation and integration to arbitrary noninteger order. The increasing interest of fractional equations is motivated by their applications in various fields of science such as physics, fluid mechanics, heat conduction in materials with memory, chemistry and engineering. Fractional derivatives and integrals are proved to be more useful for the formulation of certain electrochemical problems than the classical models [1–8]. In consequence, the subject of fractional differential equations is gaining diverse and continuous attention. For more details of some recent theoretical results on fractional differential equations and their applications, we refer the reader to [9–16] and the references therein.
Turbulent flow in a porous medium is a fundamental mechanics problem. For studying this type of problem, Leibenson [17] introduced the following p-Laplacian differential equation:
where , , and , . The study of differential equation (1) is of significance theoretically and practically [7], then many important results relative to differential equation (1) with different boundary value conditions have been obtained. It is quite natural that to investigate arbitrary noninteger order differential equation relative to equation (1).
Motivated by the references [17–19], in this paper, we consider the following p-Laplacian fractional differential equations with Caputo fractional derivative:
where is the p-Laplacian operator, i.e., , , and , . is the standard Caputo derivative and satisfies the Carathéodory type condition. is real and denotes the integer part of the real number α, , , () are constants satisfying and . The existence criteria of at least one or two positive solutions are established by using the nonlinear alternative of Leray-Schauder type and the Krasnosel’skii’s fixed-point theorem, and the existence of at least n or distinct positive solutions are obtained by using of the Leggett-Williams fixed-point theorem, the generalized Avery-Henderson fixed-point theorem as well as the Avery-Peterson fixed-point theorem.
The rest of the paper is organized as follows. In Section 2, we present some basic definitions and several fixed-point theorems. In Section 3, we give and discuss the completely continuous operator of p-Laplacian fractional differential equation (2). In Section 4, by using the nonlinear alternative of Leray-Schauder type and the Krasnosel’skii’s fixed-point theorem, some new sufficient conditions of the existence of at least one or two positive solutions of p-Laplacian fractional differential equation (2) are obtained. In Section 5, the existence criteria for at least three or arbitrary n or positive solutions of p-Laplacian fractional differential equation (2) are established. In Section 6, we present two examples.
In this study, we assume that satisfies the following conditions of Carathéodory type:
(S1) is Lebesgue measurable with respect to t on ;
(S2) for a.e. , is continuous on .
2 Preliminaries
In this section, we list some basic definitions and the several fixed-point theorems, which help us to better understand our proofs presented in next a few sections.
Definition 1 [1]
Let , the fractional integral of order α of function is defined by
provided the integral exists.
Definition 2 [1]
The Caputo derivative of a function is given by
provided that the right side is pointwise defined on , where and n is a integer.
The Gamma function is given by
and the Beta function is given by
In addition,
The following are two fixed point theorems. The former one is the so-called nonlinear alternative of Leray-Schauder type and the latter one is the Krasnosel’skii’s fixed-point theorem [20, 21].
Lemma 1 Let X be a Banach space with being closed and convex. Assume that U is a relatively open subset of C with and is a completely continuous operator, then either
-
(i)
A has a fixed point in , or
-
(ii)
there exists and with .
Lemma 2 Let P be a cone in a Banach space E. Assume and are open subsets of E with and . If is a completely continuous operator such that either
-
(i)
, and , , or
-
(ii)
, and , .
Then A has a fixed point in .
Assume that and , where the map q is a nonnegative continuous concave functional on P. The following two theorems are the Leggett and Williams fixed-point theorems [22] and the generalized Avery-Henderson fixed-point theorem [23], respectively.
Lemma 3 Suppose that is completely continuous and there exists a concave positive functional q on P such that for . Suppose that there exist constants such that:
-
(i)
and if ;
-
(ii)
if ;
-
(iii)
for with .
Then A has at least three fixed points , and such that
For each , let , where γ is a nonnegative continuous functional on a cone P of a real Banach space E.
Lemma 4 Let P be a cone in a real Banach space E. Let α, β and γ be increasing, nonnegative continuous functionals on P such that for some and , and for all . Suppose that there exist positive numbers a and b with , and is a completely continuous operator such that:
-
(i)
for all ;
-
(ii)
for all ;
-
(iii)
and for .
Then A has at least three fixed points , and belonging to such that
Let β and ϕ be nonnegative continuous convex functionals on P, λ be a nonnegative continuous concave functional on P and φ be a nonnegative continuous functional on P. We define the following convex sets:
and
We are ready to recall the Avery-Peterson fixed-point theorem [24].
Lemma 5 Let P be a cone in a real Banach space E, and β, ϕ, λ and φ be defined as the above. Moreover, φ satisfies for such that for some positive numbers h and d,
holds for all . Suppose that is completely continuous and there exist positive real numbers a, b, c, with such that:
-
(i)
and for ;
-
(ii)
for with ;
-
(iii)
and for all with .
Then A has at least three fixed points such that
3 Completely continuous operator
In this section, we firstly present some lemmas, which will be used in our discussions. Then we establish the completely continuous operator for our p-Laplacian fractional differential equation, and obtain that solving the solutions of p-Laplacian fractional differential equation (2) are equivalent to finding the fixed points of the associated completely continuous operator.
Lemma 6 Let , assume that , then the following fractional differential equation:
has the unique solution
Proof It follows from Definition 2 that the result is true. □
Lemma 7 Assume that , then the p-Laplacian fractional differential equation
has a unique solution
Proof It follows from (5) that
The fractional integral of order α of function u denotes by , then
the latter inequality holds since , . In addition,
hence
The boundary conditions of (5) reduce to
and
Now, plugging (8) and (9) into (7), then (6) is satisfied. □
Suppose that , then is a Banach space endowed with norm
where
and
The cone is defined by
The operator is defined by
Then the solutions of fractional differential equation (2) are the corresponding fixed points of the operator A.
Lemma 8 Suppose that conditions (S1) and (S2) are satisfied. For any and all , we assume that there exist two nonnegative real-value functions such that
or
Then the operator is completely continuous.
Proof of Lemma 8 Firstly, we show that is continuous.
Let , it is obvious that for arbitrary . By using the property of the fractional integral and derivative, we obtain that
and
Since Ax is nonnegative, increasing and convex on , we have
and
Moreover,
and
Therefore,
which implies that .
Suppose that , , and uniformly converges to u on , that is
So, we have
which implies that
Then (S1) and the continuous of imply that
then, we obtain
and
By virtue of (14) and (15), we have
which means that A is continuous.
Secondly, we show that A maps bounded sets into bounded sets in . It suffices to show that for any , there is a positive constant such that for each , we have .
Let
According to (11) and (13), we have
Using (12) and (13) yields
In addition, using (11) and (12), we also have
Hence, we have .
Thirdly, we consider that A maps bounded sets into equicontinuous sets of . Since and t are uniformly continuous on , then for any , there exists , whenever , we have
we also obtain
For convenience, we assume . For any , according to (11) and (12), we get
and
Consequently, we obtain
which implies that the family of functions is equicontinuous. It follows from the virtue of the Arzela-Ascoli theorem that the operator is completely continuous. □
Remark 1 If is continuous, we can obtain that is completely continuous by using a similar argument as the above.
4 Existence of one or two solutions
In this section, we discuss the existence of single or twin positive solutions to problem (2).
Theorem 1 Assume that all assumptions of Lemma 8 and
hold, then the fractional differential equation (2) has at least one positive solution.
Proof of Theorem 1
Let
where
Assume that there exist and such that . Then we find
and
Thus, we have
which means that and .
By virtue of Lemma 1, we conclude that the fractional differential equation (2) has at least one positive solution. □
Remark 2 ‘All assumptions of Lemma 8 hold’ can be replaced by ‘ is continuous’ in Theorem 1.
Theorem 2 Assume that all assumptions of Lemma 8 and the following conditions hold:
-
(i)
there exists a constant such that for , where ;
-
(ii)
there exists a constant such that for , where , and .
Then the fractional differential equation (2) has at least one positive solution u such that lies between m and e.
Proof of Theorem 2 Without loss of generality, we assume that .
Let
For any , there is
It follows from condition (i) that
which implies that
We define
for arbitrary , and find
On the other hand, it follows from condition (ii) that
which implies that
By using (18) and (20), it follows from Lemma 2 that the fractional differential equation (2) has a positive solution u in . □
Let
and
Now, we have the following two results.
Theorem 3 Assume that all assumptions of Lemma 8 hold. In addition, when and hold too, then the fractional differential equation (2) has at least one positive solution.
Proof of Theorem 3
It is easy to obtain that
then, according to the assumption , there exists a sufficiently small such that
we can obtain (17) is true. That is, if we let
then
It follows from that there exists an such that
where and .
Set
then we see that .
For any , we have . Equation (21) gives
Consequently, it follows from the virtue of Lemma 2 that the fractional differential equation (2) has a positive solution u in . □
Theorem 4 Assume that all assumptions of Lemma 8 hold. In addition, and are satisfied. Then the fractional differential equation (2) has at least one positive solution.
Proof of Theorem 4 It follows from that there exists a sufficiently small such that
When , we get
we can obtain (19). Take
Then
Let (>0). Since , there exists a () such that
where .
Note that
So there exists a such that
Equations (22) and (23) reduce to
Let
and
If , one has and
This implies our desired result. □
Next, we deal with the existence of at least two distinct positive solutions to the fractional differential equation (2).
Theorem 5 Assume that all assumptions of Lemma 8 hold. Moreover, suppose that and , and the condition (i) in Theorem 2 is satisfied. Then the fractional differential equation (2) has at least two distinct positive solutions .
Proof of Theorem 5 In view of , there exists an such that and
where h is given by
Take
If with , it means that
and
It follows from (24) and (25) that
which implies that
Let
Then we obtain that (18) holds by using the condition (i) of Theorem 2. According to Lemma 2, the fractional differential equation (2) has a positive solution in .
It follows from that there exists an such that
where and . Moreover, k satisfies that
Let
then we see that .
For any , we have . According to (26), we deduce that
Thus, it follows from (i) of Lemma 2 that the fractional differential equation (2) has at least a single positive solution in with
It is easily seen that and are distinct. □
By a closely similar way, we can obtain the following result.
Theorem 6 Assume that all assumptions of Lemma 8 hold. Moreover, suppose that and , and the condition (ii) in Theorem 2 is satisfied, then the fractional differential equation (2) has at least two distinct positive solutions .
5 Existence of triple or multiple solutions
In this section, we will further discuss the existence of at least 3, n or positive solutions to p-Laplacian fractional differential equation (2) by using different fixed point theorems in cone.
For the notational convenience, we define
and
5.1 Existence of three solutions
In this subsection, we investigate the existence of at least three distinct positive solutions of equation (2).
Theorem 7 Let a, b and c be constants such that and . In addition, if all assumptions of Lemma 8 hold and satisfies the following conditions:
-
(i)
for ;
-
(ii)
for ;
-
(iii)
for .
Then the fractional differential equation (2)has at least triple positive solutions such that
Proof of Theorem 7 By the virtue of the completely continuous operator A and Lemma 3, we need to show that all conditions of Lemma 3 with respect to A are satisfied.
Let
then is a nonnegative continuous concave function and satisfies
Since
it follows from condition (iii) that
which implies .
When , it implies that and , from this, we can easily obtain that the conditions (ii) of Lemma 3 is true.
Let d be a fixed constant such that , then we have and . This means that
For any , we get
which implies that
Hence, the condition (ii) gives that
which means that the condition (i) of Lemma 3 holds.
For any with , it gives and . By using the same argument as the above, we see that . This implies that the condition (iii) of Lemma 3 is fulfilled.
Consequently, all conditions of Lemma 3 are verified. That is, the fractional differential equation (2) has at least three distinct solutions distributed as (27). □
Corollary 1 Assume that all assumptions of Lemma 8 hold. If the condition (iii) in Theorem 7 is replaced by
then (27) in Theorem 7 also holds.
Proof of Corollary 1 We only need to prove that the condition (iii′) implies the condition (iii) in Theorem 7. That is, assume that (iii′) holds, then there exists a number such that
Conversely, we suppose that for any , there exists
such that
Take
Then there exists
such that
and
Since the condition (iii′) holds, there is a such that
Thus, we have
and
Otherwise, if
it follows from (31) that
which contradicts inequality (29).
Let
so we have
This apparently contradicts formula (30). Consequently, we complete the proof. □
5.2 Existence of arbitrary n solutions
In this subsection, the existence criteria for at least three or arbitrary n positive solutions to p-Laplacian fractional differential equation (2) are obtained.
We define the nonnegative, increasing, continuous functionals , and by
and
so
Since
and
Hence, we have
where .
Theorem 8 Assume that there exist real numbers , , with such that . In addition, if all assumptions of Lemma 8 hold and satisfies the following conditions:
-
(i)
for ;
-
(ii)
for ;
-
(iii)
for .
Then the fractional differential equation (2) has at least three distinct positive solutions such that
Proof of Theorem 8 We only need to check all conditions of Lemma 4 are fulfilled with respect to the operator A. By using a similar way as to the proof of (28), we can obtain that
For arbitrary , one has
and
This implies that
and
According to the condition (i), it gives
We see that for .
For any , it gives
and
This implies that
and
Making use of the condition (ii), we get
So we have for .
We now show that and for arbitrary . Since , for , we have
which gives
It follows from the assumption (iii) that
All conditions in Lemma 4 are satisfied. From (S1) and (S2), we know that solutions of equation (2) do not vanish identically on any closed subinterval of . Consequently, equation (2) has at least three distinct positive solutions , , and belonging to distributed as (32). □
The following result is regarded as a corollary of Theorem 8.
Corollary 2 Assume that all assumptions of Lemma 8 hold and f satisfies the following conditions:
-
(i)
and ;
-
(ii)
there exists a such that
Then the fractional differential equation (2) has at least three distinct positive solutions.
Proof of Corollary 2 Let . It follows from the condition (ii) that
which implies that the condition (ii) of Theorem 8 holds.
We choose a sufficiently small such that
In view of , there exists a sufficiently small such that
Without loss of generality, let . Because of , we have . Thus, it follows from (33) and (34) that
which implies that the condition (iii) of Theorem 8 holds.
Choose sufficiently small such that
By using the continuity of f, there exists a constant such that
Since , there exists a sufficiently large such that
Without loss of generality, let and , so we see if
then
Moreover, in view of (35), one has
From (36) and (37), we see that the condition (i) of Theorem 8 is fulfilled. Hence, equation (2) has at least three distinct positive solutions according to Theorem 8. □
According to Theorem 8, we can prove that the existence for multiple positive solutions to the equation (2) when conditions (i), (ii) and (iii) are modified appropriately on f.
Theorem 9 If there exist constant numbers , and such that together with
where . In addition, if all assumptions of Lemma 8 hold and the function f satisfies:
-
(i)
for ;
-
(ii)
for ;
-
(iii)
for .
Then the fractional differential equation (2) has at least n distinct positive solutions.
Proof of Theorem 9 By using almost same technique as to the proof of Theorem 9 in [15]. □
By virtue of Lemma 4, we can obtain the following results by using the similar way as to those of Theorem 8.
Theorem 10 Assume that there exist positive numbers , , with such that . In addition, if all assumptions of Lemma 8 hold and satisfies the following conditions:
-
(i)
for ;
-
(ii)
for ;
-
(iii)
for .
Then the fractional differential equation (2) has at least three distinct positive solutions such that
It also follows from Theorem 10 that we can obtain the following corollary and theorem, respectively.
Corollary 3 Assume that all assumptions of Lemma 8 hold and f satisfies conditions:
-
(i)
, ;
-
(ii)
there exists such that for .
Then the fractional differential equation (2) has at least three distinct positive solutions.
Theorem 11 Assume that all assumptions of Lemma 8 hold and there are positive numbers , , such that together with
where . In addition, satisfies the following conditions:
-
(i)
for ;
-
(ii)
for ;
-
(iii)
for .
Then the fractional differential equation (2) has at least n distinct positive solutions.
5.3 Existence of arbitrary solutions
In this subsection, the existence of at least three or arbitrary odd positive solutions to p-Laplacian differential equation (2) are established.
Define the nonnegative continuous convex functionals ϕ and β, concave functional λ and functional φ on by
and
Theorem 12 Assume that all assumptions of Lemma 8 hold and there exist constants , , such that . In addition, satisfies the following conditions:
-
(i)
for all ;
-
(ii)
for all ;
-
(iii)
for all .
Then the fractional differential equation (2) has at least three distinct positive solutions , , such that
Proof of Theorem 12 It suffices to show that all conditions of Lemma 5 hold with respect to the completely continuous operator A.
For arbitrary , we have and . This implies that the inequality (3) of Lemma 5 is satisfied.
In the following, we show that .
For any , from and the assumption (i), we have
It remains to show that assumptions (i)-(iii) of Lemma 5 are fulfilled with respect to the operator A.
Let , where . It is obvious that , and . We see that that . So we have
For any , we get and for all . It follows from the assumption (ii) that
which implies that assumption (i) of Lemma 5 is satisfied.
For any with , we have and for . So we have
This implies that assumption (ii) of Lemma 5 is fulfilled.
Since , we have . If
it reduces to
Hence, we have
All assumptions of Lemma 5 are satisfied. Consequently, we complete the proof. □
Corollary 4 Assume that all assumptions of Lemma 8 hold and the condition (i) in Theorem 12 is replaced by (i′), then the conclusion of Theorem 12 also holds.
Similar to the proof of Theorem 9 by mathematical induction, we have the following.
Theorem 13 Assume that all assumptions of Lemma 8 hold and there exist constants , , such that
where . In addition, f satisfies the following conditions:
-
(i)
for all ;
-
(ii)
for all ;
-
(iii)
for all .
Then the fractional differential equation (2) has at least positive solutions.
6 Examples
In this section, we present two simple examples to illustrate our theoretical results. In Example 1, it shows the difference between two cases in Section 4.
Example 1
Consider
Case 1: when takes the form as
It is easy to see that
then and . Moreover, we see that
It follows from Theorem 1 that the fractional differential equation (39) has at least one positive solution. However, it is difficult for us to obtain the existence of at least one positive solution to the fractional differential equation (39) by using theorems of the super-linearity and sub-linearity in our paper.
Case 2: when takes the form as
By using the continuity of f, we obtain that the operator A is completely continuous. It is easy to check that and . According to Theorem 3, we see that fractional differential equation (39) has at least one positive solution. But it is difficult for us to know the existence of positive solution to the fractional differential equation (39) if we use Theorem 1.
Example 2
Consider
where
Since , , , , , , , , a straightforward calculation gives
and
Taking , , and , we see that and
By means of Theorem 7, we obtain that the fractional differential equation (40) has at least three distinct positive solutions , , such that
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This work is supported by Natural Science Foundation of Qinghai Province (No. 2012-Z-910) and Jiangsu Undergraduate Scientific and Technological Innovation Project.
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Su, Y., Li, Q. & Liu, XL. Existence criteria for positive solutions of p-Laplacian fractional differential equations with derivative terms. Adv Differ Equ 2013, 119 (2013). https://doi.org/10.1186/1687-1847-2013-119
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DOI: https://doi.org/10.1186/1687-1847-2013-119