Multiple Positive Solutions for a Coupled System of p -Laplacian Fractional Order Two-Point Boundary Value Problems

Abstract

This paper establishes the existence of at least three positive solutions for a coupled system of p -Laplacian fractional order two-point boundary value problems, [superscript] D [superscript] 0 + [/superscript] [subscript] β 1 [/subscript] [/superscript] ( [subscript] [varphi] p [/subscript] ( [superscript] D [superscript] 0 + [/superscript] [subscript] α 1 [/subscript] [/superscript] u ( t ) ) ) = [subscript] f 1 [/subscript] ( t , u ( t ) , v ( t ) ) , t ∈ ( 0,1 ) , [superscript] D [superscript] 0 + [/superscript] [subscript] β 2 [/subscript] [/superscript] ( [subscript] [varphi] p [/subscript] ( [superscript] D [superscript] 0 + [/superscript] [subscript] α 2 [/subscript] [/superscript] v ( t ) ) ) = [subscript] f 2 [/subscript] ( t , u ( t ) , v ( t ) ) , t ∈ ( 0,1 ) , u ( 0 ) = [superscript] D [superscript] 0 + [/superscript] [subscript] q 1 [/subscript] [/superscript] u ( 0 ) = 0 , γ u ( 1 ) + δ [superscript] D [superscript] 0 + [/superscript] [subscript] q 2 [/subscript] [/superscript] u ( 1 ) = 0 , [superscript] D [superscript] 0 + [/superscript] [subscript] α 1 [/subscript] [/superscript] u ( 0 ) = [superscript] D [superscript] 0 + [/superscript] [subscript] α 1 [/subscript] [/superscript] u ( 1 ) = 0 , v ( 0 ) = [superscript] D [superscript] 0 + [/superscript] [subscript] q 1 [/subscript] [/superscript] v ( 0 ) = 0 , γ v ( 1 ) + δ [superscript] D [superscript] 0 + [/superscript] [subscript] q 2 [/subscript] [/superscript] v ( 1 ) = 0 , [superscript] D [superscript] 0 + [/superscript] [subscript] α 2 [/subscript] [/superscript] v ( 0 ) = [superscript] D [superscript] 0 + [/superscript] [subscript] α 2 [/subscript] [/superscript] v ( 1 ) = 0 , by applying five functionals fixed point theorem.

(ProQuest: ... denotes non-US-ASCII text omitted.)

K. R. Prasad 1 and B. M. B. Krushna 2

Academic Editor:Bashir Ahmad

1, Department of Applied Mathematics, Andhra University, Visakhapatnam 530 003, India
2, Department of Mathematics, MVGR College of Engineering, Vizianagaram 535 005, India

Received 24 February 2014; Accepted 21 April 2014; 7 May 2014

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

The theory of differential equations offers a broad mathematical basis to understand the problems of modern society which are complex and interdisciplinary by nature. Fractional order differential equations have gained importance due to their applications to almost all areas of science, engineering, and technology. Among all the theories, the most applicable operator is the classical p -Laplacian, given by _{[varphi] p} ( s ) = | s ^{| p - 2} s , p > 1 . These types of problems have a wide range of applications in physics and related sciences such as biophysics, plasma physics, and chemical reaction design.

The positive solutions of boundary value problems associated with ordinary differential equations were studied by many authors [1-3] and extended to p -Laplacian boundary value problems [4-6]. Later, these results are further extended to fractional order boundary value problems [7-15] by applying various fixed point theorems on cones. Recently, researchers are concentrating on the theory of fractional order boundary value problems associated with p -Laplacian operator.

In 2012, Chai [16] investigated the existence and multiplicity of positive solutions for a class of boundary value problem of fractional differential equation with p -Laplacian operator, [figure omitted; refer to PDF] by means of the fixed point theorem on cones.

This paper is concerned with the existence of positive solutions for a coupled system of p -Laplacian fractional order boundary value problems: [figure omitted; refer to PDF] where _{[varphi] p} ( s ) = | s ^{| p - 2} s , p > 1 , ^{[varphi] p - 1} = _{[varphi] q} , 1 / p + 1 / q = 1 , γ , δ are positive real numbers, 2 < _{α i} ...4; 3 , 1 < _{β i} , _{q i} ...4; 2 , _{f i} : [ 0,1 ] × ^{... 2} [arrow right] ^{... +} are continuous functions, and ^{D 0 + _{α i}} , ^{D 0 + _{β i}} , ^{D 0 + _{q i}} , for i = 1,2 are the standard Riemann-Liouville fractional order derivatives.

The rest of the paper is organized as follows. In Section 2, the Green functions for the homogeneous BVPs corresponding to (2), (4) are constructed and the bounds for the Green functions are estimated. In Section 3, sufficient conditions for the existence of at least three positive solutions for a coupled system of p -Laplacian fractional order BVP (2)-(5) are established, by using five functionals fixed point theorem. In Section 4, as an application, the results are demonstrated with an example.

2. Green Functions and Bounds

In this section, the Green functions for the homogeneous BVPs are constructed and the bounds for the Green functions are estimated, which are essential to establish the main results.

Let _{G 1} ( t , s ) be Green's function for the homogeneous BVP: [figure omitted; refer to PDF]

Lemma 1.

Let d = δ Γ ( _{α 1} ) + γ Γ ( _{α 1} - _{q 2} ) ...0; 0 . If h ∈ C [ 0,1 ] , then the fractional order BVP [figure omitted; refer to PDF] with (7) has a unique solution [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

Proof.

Let u ∈ ^{C [ _{α 1} ] + 1} [ 0,1 ] be the solution of fractional order BVP (8), (7). Then [figure omitted; refer to PDF] and hence [figure omitted; refer to PDF] Using the boundary conditions (7), _{c 1} , _{c 2} , and _{c 3} are determined as [figure omitted; refer to PDF] Hence, the unique solution of (8), (7) is [figure omitted; refer to PDF]

Lemma 2.

Let y ( t ) ∈ C [ 0,1 ] and 2 < _{α 1} ...4; 3 , 1 < _{β 1} ...4; 2 . Then the fractional order BVP [figure omitted; refer to PDF] with (4) has a unique solution [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

Proof.

An equivalent integral equation for (16) is given by [figure omitted; refer to PDF] Using the conditions ^{D 0 + _{α 1}} u ( 0 ) = 0 , ^{D 0 + _{α 1}} u ( 1 ) = 0 , _{c 1} , and _{c 2} are determined as _{c 1} = ( - 1 / Γ ( _{β 1} ) ) ^{∫ 0 1} ... ^{( 1 - τ ) _{β 1} - 1} y ( τ ) d τ and _{c 2} = 0 . Then, [figure omitted; refer to PDF] Therefore, [figure omitted; refer to PDF] Consequently, [figure omitted; refer to PDF] Hence, u ( t ) = ^{∫ 0 1} ... _{G 1} ( t , s ) _{[varphi] q} ( ^{∫ 0 1} ... _{H 1} ( s , τ ) y ( τ ) d τ ) d s is the solution of fractional order BVP (16) and (4).

Lemma 3.

Assume that δ ( _{q 2} - 1 ) > γ Γ ( _{α 1} - _{q 2} ) / Γ ( _{α 1} ) . Then Green's function _{G 1} ( t , s ) satisfies the following inequalities:

(i) _{G 1} ( t , s ) ...5; 0 , for all ( t , s ) ∈ [ 0,1 ] × [ 0,1 ] ,

(ii) _{G 1} ( t , s ) ...4; _{G 1} ( 1 , s ) , for all ( t , s ) ∈ [ 0,1 ] × [ 0,1 ] ,

(iii): _{G 1} ( t , s ) ...5; ( 1 / ^{4 _{α 1} - 1} ) _{G 1} ( 1 , s ) , for all ( t , s ) ∈ I × [ 0,1 ] ,

where I = [ 1 / 4,3 / 4 ] .

Proof.

Green's function _{G 1} ( t , s ) is given in (10). For 0 ...4; t ...4; s ...4; 1 , [figure omitted; refer to PDF] For 0 ...4; s ...4; t ...4; 1 , [figure omitted; refer to PDF] Hence, the inequality (i) is proved. For 0 ...4; t ...4; s ...4; 1 , [figure omitted; refer to PDF] Therefore, _{G 11} ( t , s ) is increasing with respect to t , which implies that _{G 11} ( t , s ) ...4; _{G 11} ( 1 , s ) . Now, for 0 ...4; s ...4; t ...4; 1 , [figure omitted; refer to PDF] Therefore, _{G 12} ( t , s ) is increasing with respect to t , which implies that _{G 12} ( t , s ) ...4; _{G 12} ( 1 , s ) . Hence, the inequality (ii) is proved. Now, the inequality (iii) can be established.

Let 0 ...4; t ...4; s ...4; 1 and t ∈ I . Then [figure omitted; refer to PDF] Let 0 ...4; s ...4; t ...4; 1 and t ∈ I . Then [figure omitted; refer to PDF]

Hence the inequality (iii) is proved.

Lemma 4.

For t , s ∈ [ 0,1 ] , Green's function _{H 1} ( t , s ) satisfies the following inequalities:

(i) _{H 1} ( t , s ) ...5; 0 ,

(ii) _{H 1} ( t , s ) ...4; _{H 1} ( s , s ) .

Proof.

Green's function _{H 1} ( t , s ) is given in (18). Clearly, it is observed that, for 0 ...4; t ...4; s ...4; 1 , _{H 1} ( t , s ) ...5; 0 .

For 0 ...4; s ...4; t ...4; 1 , [figure omitted; refer to PDF] Hence, the inequality (i) is proved. Now we establish the inequality (ii), for 0 ...4; t ...4; s ...4; 1 , [figure omitted; refer to PDF] Therefore, _{H 1} ( t , s ) is increasing with respect to t , for s ∈ [ 0,1 ) , which implies that _{H 1} ( t , s ) ...4; _{H 1} ( s , s ) . Similarly, it can be proved that _{H 1} ( t , s ) ...4; _{H 1} ( s , s ) for 0 ...4; s ...4; t ...4; 1 . Hence the inequality (ii) is proved.

Lemma 5.

Green's function _{H 1} ( t , s ) satisfies the following inequality: ( A ) there exists a positive function ^{γ 1 *} ( s ) ∈ C ( 0,1 ) such that [figure omitted; refer to PDF]

Proof.

Since _{H 1} ( t , s ) is monotonic function, for all t , s ∈ [ 0,1 ] , we have [figure omitted; refer to PDF] From (i) of Lemma 4, _{H 1} ( t , s ) ...5; 0 , for t , s ∈ [ 0,1 ] . For s ∈ ( 0,1 / 4 ) , _{H 1} ( t , s ) is increasing with respect to t for t ∈ ( 0 , s / ( 1 - ^{( 1 - s ) ( _{β 1} - 1 ) / ( _{β 1} - 2 )} ) ) and decreasing with respect to t for t ∈ ( s / ( 1 - ^{( 1 - s ) ( _{β 1} - 1 ) / ( _{β 1} - 2 )} ) , 1 / 4 ) . For s ∈ ( 1 / 4,1 ) , _{H 1} ( t , s ) is decreasing with respect to t for s ...4; t and increasing with respect to t for s ...5; t . If we define [figure omitted; refer to PDF] Then, [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and ξ ∈ ( 1 / 4,3 / 4 ) satisfy the equation [ ( 3 / 4 ) ( 1 - ξ ) ^{] _{β 1} - 1} - ^{( ( 3 / 4 ) - ξ ) _{β 1} - 1} = ^{( ( 1 - ξ ) / 4 ) _{β 1} - 1} . In particular, ξ = 0.5 if _{β 1} = 2 ; ξ [arrow right] 0.5 as _{β 1} [arrow right] 2 ; and ξ [arrow right] 0.76 as _{β 1} [arrow right] 1 . Hence the inequality in (31) holds.

In a similar manner, the results of the Green functions _{G 2} ( t , s ) and _{H 2} ( t , s ) for the homogeneous BVPs corresponding to the fractional order BVP (3) and (5) are obtained.

Remark 6.

Consider the following.

_{G 1} ( t , s ) ...5; η _{G 1} ( 1 , s ) and _{G 2} ( t , s ) ...5; η _{G 2} ( 1 , s ) , for all ( t , s ) ∈ I × [ 0,1 ] , where η = min ... ... { 1 / ^{4 _{α 1} - 1} , 1 / ^{4 _{α 2} - 1} } .

Remark 7.

Consider the following.

_{H 1} ( t , s ) ...5; ^{γ *} ( s ) _{H 1} ( s , s ) and _{H 2} ( t , s ) ...5; ^{γ *} ( s ) _{H 2} ( s , s ) , for all ( t , s ) ∈ I × [ 0,1 ] , where ^{γ *} ( s ) = min ... { ^{γ 1 *} ( s ) , ^{γ 2 *} ( s ) } .

3. Existence of Multiple Positive Solutions

In this section, the existence of at least three positive solutions for a coupled system of p -Laplacian fractional order BVP (2)-(5) is established by using five functionals fixed point theorem.

Let γ , β , θ be nonnegative continuous convex functionals on P and let α , ψ be nonnegative continuous concave functionals on P ; then for nonnegative numbers ^{h [variant prime]} , ^{a [variant prime]} , ^{b [variant prime]} , ^{d [variant prime]} , and ^{c [variant prime]} , convex sets are defined: [figure omitted; refer to PDF]

In obtaining multiple positive solutions of the p -Laplacian fractional order BVP (2)-(5), the following so-called five functionals fixed point theorem is fundamental.

Theorem 8 (see [17]).

Let P be a cone in the real Banach space B . Suppose that α and ψ are nonnegative continuous concave functionals on P and γ , β , θ are nonnegative continuous convex functionals on P , such that, for some positive numbers ^{c [variant prime]} and ^{e [variant prime]} , α ( y ) ...4; β ( y ) and || y || ...4; ^{e [variant prime]} γ ( y ) , for all y ∈ P ( γ , ^{c [variant prime]} ) ¯ . Suppose further that T : P ( γ , ^{c [variant prime]} ) ¯ [arrow right] P ( γ , ^{c [variant prime]} ) ¯ is completely continuous and there exist constants ^{h [variant prime]} , ^{d [variant prime]} , ^{a [variant prime]} , and ^{b [variant prime]} ...5; 0 with 0 < ^{d [variant prime]} < ^{a [variant prime]} such that each of the following is satisfied:

(B1) { y ∈ P ( γ , θ , α , ^{a [variant prime]} , ^{b [variant prime]} , ^{c [variant prime]} ) : α ( y ) > ^{a [variant prime]} } ...0; ∅ and α ( T y ) > ^{a [variant prime]} for y ∈ P ( γ , θ , α , ^{a [variant prime]} , ^{b [variant prime]} , ^{c [variant prime]} ) ,

(B2) { y ∈ Q ( γ , β , ψ , ^{h [variant prime]} , ^{d [variant prime]} , ^{c [variant prime]} ) : β ( y ) > ^{d [variant prime]} } ...0; ∅ and β ( T y ) > ^{d [variant prime]} for y ∈ Q ( γ , β , ψ , ^{h [variant prime]} , ^{d [variant prime]} , ^{c [variant prime]} ) ,

(B3) α ( T y ) > ^{a [variant prime]} provided that y ∈ P ( γ , α , ^{a [variant prime]} , ^{c [variant prime]} ) with θ ( T y ) > ^{b [variant prime]} ,

(B4) β ( T y ) < ^{d [variant prime]} provided that y ∈ Q ( γ , β , ψ , ^{h [variant prime]} , ^{d [variant prime]} , ^{c [variant prime]} ) with ψ ( T y ) < ^{h [variant prime]} .

Then, T has at least three fixed points _{y 1} , _{y 2} , _{y 3} ∈ P ( γ , ^{c [variant prime]} ) ¯ such that β ( _{y 1} ) < ^{d [variant prime]} , ^{a [variant prime]} < α ( _{y 2} ) and ^{d [variant prime]} < β ( _{y 3} ) with α ( _{y 3} ) < ^{a [variant prime]} .

Consider the Banach space B = E × E , where E = { u : u ∈ C [ 0,1 ] } equipped with the norm || ( u , v ) || = _{|| u || 0} + _{|| v || 0} , for ( u , v ) ∈ B and the norm, is defined as [figure omitted; refer to PDF] Define a cone P ⊂ B by [figure omitted; refer to PDF] Define the nonnegative continuous concave functionals α , ψ and the nonnegative continuous convex functionals β , θ , γ on P by [figure omitted; refer to PDF] where _{I 1} = [ 1 / 3,2 / 3 ] . For any ( u , v ) ∈ P , [figure omitted; refer to PDF] Let [figure omitted; refer to PDF]

Theorem 9.

Suppose that there exist 0 < ^{a [variant prime]} < ^{b [variant prime]} < ^{b [variant prime]} / η < ^{c [variant prime]} such that _{f i} , for i = 1,2 satisfies the following conditions:

(A1) _{f i} ( t , u ( t ) , v ( t ) ) < _{[varphi] p} ( ^{a [variant prime]} L / 2 ) , t ∈ [ 0,1 ] and u , v ∈ [ η ^{a [variant prime]} , ^{a [variant prime]} ] ,

(A2) _{f i} ( t , u ( t ) , v ( t ) ) > _{[varphi] p} ( ^{b [variant prime]} M / 2 ) , t ∈ I and u , v ∈ [ ^{b [variant prime]} , ^{b [variant prime]} / η ] ,

(A3) _{f i} ( t , u ( t ) , v ( t ) ) < _{[varphi] p} ( ^{c [variant prime]} L / 2 ) , t ∈ [ 0,1 ] and u , v ∈ [ 0 , ^{c [variant prime]} ] .

Then, the fractional order BVP (2)-(5) has at least three positive solutions, ( _{x 1} , _{x 2} ) , ( _{y 1} , _{y 2} ) , and ( _{z 1} , _{z 2} ) such that β ( _{x 1} , _{x 2} ) < ^{a [variant prime]} , ^{b [variant prime]} < α ( _{y 1} , _{y 2} ) and ^{a [variant prime]} < β ( _{z 1} , _{z 2} ) with α ( _{z 1} , _{z 2} ) < ^{b [variant prime]} .

Proof.

Let _{T 1} , _{T 2} : P [arrow right] E and T : P [arrow right] B be the operators defined by [figure omitted; refer to PDF] It is obvious that a fixed point of T is the solution of the fractional order BVP (2)-(5). Three fixed points of T are sought. First, it is shown that T : P [arrow right] P . Let ( u , v ) ∈ P . Clearly, _{T 1} ( u , v ) ( t ) ...5; 0 and _{T 2} ( u , v ) ( t ) ...5; 0 , for t ∈ [ 0,1 ] . Also, for ( u , v ) ∈ P , [figure omitted; refer to PDF] Similarly, _{min ... ... t ∈ I T 2} ( u , v ) ( t ) ...5; η _{|| T 2 ( u , v ) || 0} . Therefore, [figure omitted; refer to PDF] Hence, T ( u , v ) ∈ P and so T : P [arrow right] P . Moreover, T is completely continuous operator. From (40), for each ( u , v ) ∈ P , α ( u , v ) ...4; β ( u , v ) , and || ( u , v ) || ...4; ( 1 / η ) γ ( u , v ) . It is shown that T : P ( γ , ^{c [variant prime]} ) ¯ [arrow right] P ( γ , ^{c [variant prime]} ) ¯ . Let ( u , v ) ∈ P ( γ , ^{c [variant prime]} ) ¯ . Then 0 ...4; | u | + | v | ...4; ^{c [variant prime]} . Condition (A3) is used to obtain [figure omitted; refer to PDF] Therefore T : P ( γ , ^{c [variant prime]} ) ¯ [arrow right] P ( γ , ^{c [variant prime]} ) ¯ . Now conditions (B1) and (B2) of Theorem 8 are to be verified. It is obvious that [figure omitted; refer to PDF] Next, let ( u , v ) ∈ P ( γ , θ , α , ^{b [variant prime]} , ^{b [variant prime]} / η , ^{c [variant prime]} ) or ( u , v ) ∈ Q ( γ , β , ψ , η ^{a [variant prime]} , ^{a [variant prime]} , ^{c [variant prime]} ) . Then, ^{b [variant prime]} ...4; | u ( t ) | + | v ( t ) | ...4; ^{b [variant prime]} / η and η ^{a [variant prime]} ...4; | u ( t ) | + | v ( t ) | ...4; ^{a [variant prime]} . Now, condition (A2) is applied to get [figure omitted; refer to PDF] Clearly, condition (A1) leads to [figure omitted; refer to PDF] To see that (B3) is satisfied, let ( u , v ) ∈ P ( γ , α , ^{b [variant prime]} , ^{c [variant prime]} ) with θ ( T ( u , v ) ( t ) ) > ^{b [variant prime]} / η . Then [figure omitted; refer to PDF] Finally, it is shown that (B4) holds. Let ( u , v ) ∈ Q ( γ , β , ^{a [variant prime]} , ^{c [variant prime]} ) with ψ ( T ( u , v ) ) < η ^{a [variant prime]} . Then, we have [figure omitted; refer to PDF] It has been proved that all the conditions of Theorem 8 are satisfied. Therefore, the fractional order BVP (2)-(5) has at least three positive solutions, ( _{x 1} , _{x 2} ) , ( _{y 1} , _{y 2} ) , and ( _{z 1} , _{z 2} ) such that β ( _{x 1} , _{x 2} ) < ^{a [variant prime]} , ^{b [variant prime]} < α ( _{y 1} , _{y 2} ) , and ^{a [variant prime]} < β ( _{z 1} , _{z 2} ) with α ( _{z 1} , _{z 2} ) < ^{b [variant prime]} . This completes the proof of the theorem.

4. Example

In this section, as an application, the result is demonstrated with an example.

Consider a coupled system of p -Laplacian fractional order BVP: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Then the Green functions _{G i} ( t , s ) and _{H i} ( t , s ) , for i = 1,2 , are given by [figure omitted; refer to PDF] Clearly, the Green functions _{G i} ( t , s ) and _{H i} ( t , s ) , for i = 1,2 , are positive and _{f 1} , _{f 2} are continuous and increasing on [ 0 , ∞ ) . By direct calculations, η = 0.08 , p = 2 , L = 33.16 , and M = 1677.73 . Choosing ^{a [variant prime]} = 1 , ^{b [variant prime]} = 10 and ^{c [variant prime]} = 900 and then 0 < ^{a [variant prime]} < ^{b [variant prime]} < ^{b [variant prime]} / η ...4; ^{c [variant prime]} and _{f i} , for i = 1,2 satisfies

(i) _{f i} ( t , u , v ) < 16.5845 = _{[varphi] p} ( ^{a [variant prime]} L / 2 ) , t ∈ [ 0,1 ] and u , v ∈ [ 0.08,1 ] ,

(ii) _{f i} ( t , u , v ) > 8388.65 = _{[varphi] p} ( ^{b [variant prime]} M / 2 ) , t ∈ [ 1 / 4,3 / 4 ] and u , v ∈ [ 10,125 ] ,

(iii): _{f i} ( t , u , v ) < 14926.09 = _{[varphi] p} ( ^{c [variant prime]} L / 2 ) , t ∈ [ 0,1 ] and u , v ∈ [ 0,900 ] .

Then, all the conditions of Theorem 9 are satisfied. Therefore, it follows from Theorem 8 that the p -Laplacian fractional order BVP (51) has at least three positive solutions.

Acknowledgment

The authors thank the referees for their valuable comments and suggestions.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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