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Apr 6, 2016 - Academic Editor: Kanishka Perera. Copyright © 2016 Sabbavarapu Nageswara Rao. This is an open access article distributed under theΒ ...
Hindawi Publishing Corporation International Journal of Differential Equations Volume 2016, Article ID 6906049, 10 pages http://dx.doi.org/10.1155/2016/6906049

Research Article Multiplicity of Positive Solutions for Fractional Differential Equation with 𝑝-Laplacian Boundary Value Problems Sabbavarapu Nageswara Rao Department of Mathematics, Jazan University, P.O. Box 114, Jazan, Saudi Arabia Correspondence should be addressed to Sabbavarapu Nageswara Rao; sabbavarapu [email protected] Received 15 October 2015; Accepted 6 April 2016 Academic Editor: Kanishka Perera Copyright Β© 2016 Sabbavarapu Nageswara Rao. 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. We investigate the existence of multiple positive solutions of fractional differential equations with 𝑝-Laplacian operator 𝛽 π·π‘Ž+ (πœ™π‘ (π·π‘Žπ›Ό+ 𝑒(𝑑))) = 𝑓(𝑑, 𝑒(𝑑)), π‘Ž < 𝑑 < 𝑏, 𝑒(𝑗) (π‘Ž) = 0, 𝑗 = 0, 1, 2, . . . , 𝑛 βˆ’ 2, 𝑒(𝛼1 ) (𝑏) = πœ‰π‘’(𝛼1 ) (πœ‚), πœ™π‘ (π·π‘Žπ›Ό+ 𝑒(π‘Ž)) = 0 = 𝛽

π·π‘Ž+1 (πœ™π‘ (π·π‘Žπ›Ό+ 𝑒(𝑏))), where 𝛽 ∈ (1, 2], 𝛼 ∈ (𝑛 βˆ’ 1, 𝑛], 𝑛 β‰₯ 3, πœ‰ ∈ (0, ∞), πœ‚ ∈ (π‘Ž, 𝑏), 𝛽1 ∈ (0, 1], 𝛼1 ∈ {1, 2, . . . , 𝛼 βˆ’ 2} is a fixed integer, and πœ™π‘ (𝑠) = |𝑠|π‘βˆ’2 𝑠, 𝑝 > 1, πœ™π‘βˆ’1 = πœ™π‘ž , (1/𝑝) + (1/π‘ž) = 1, by applying Leggett–Williams fixed point theorems and fixed point index theory.

1. Introduction

with the boundary conditions

The goal of differential equations is to understand the phenomena of nature by developing mathematical models. Fractional calculus is the field of mathematical analysis, which deals with investigation and applications of derivatives and integrals of an arbitrary order. Among all, a class of differential equations governed by nonlinear differential operators appears frequently and generated a great deal of interest in studying such problems. In this theory, the most applicable operator is the classical 𝑝-Laplacian, given by πœ™π‘ (𝑒) = |𝑒|π‘βˆ’2 , 𝑝 > 1. The positive solutions of boundary value problems associated with ordinary differential equations were studied by many authors [1–4] and extended to 𝑝-Laplacian boundary value problems [5–8]. Later, these results are further extended to fractional order boundary value problems [9–12] 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 [13–19]. The above few papers motivated this work. In this paper, we are concerned with the existence of multiple positive solutions for the fractional differential equation with 𝑝-Laplacian operator 𝛽

π·π‘Ž+ (πœ™π‘ (π·π‘Žπ›Ό+ 𝑒 (𝑑))) = 𝑓 (𝑑, 𝑒 (𝑑)) , π‘Ž < 𝑑 < 𝑏,

(1)

𝑒(𝑗) (π‘Ž) = 0,

𝑗 = 0, 1, 2, . . . , 𝑛 βˆ’ 2,

𝑒(𝛼1 ) (𝑏) = πœ‰π‘’(𝛼1 ) (πœ‚) ,

(2)

𝛽

πœ™π‘ (π·π‘Žπ›Ό+ 𝑒 (π‘Ž)) = π·π‘Ž+1 (πœ™π‘ (π·π‘Žπ›Ό+ 𝑒 (𝑏))) = 0, where πœ™π‘ (𝑠) = |𝑠|π‘βˆ’2 𝑠, 𝑝 > 1, πœ™π‘βˆ’1 = πœ™π‘ž , (1/𝑝) + (1/π‘ž) = 1, 𝛽 ∈ (1, 2], 𝛼 ∈ (𝑛 βˆ’ 1, 𝑛], 𝑛 β‰₯ 3, 𝛽1 ∈ (0, 1], 𝛼1 ∈ [1, 𝛼 βˆ’ 2] is a fixed integer, and πœ‰ ∈ (0, ∞), πœ‚ ∈ (π‘Ž, 𝑏) are constants. The function 𝑓 : [π‘Ž, 𝑏] Γ— 𝑅+ β†’ 𝑅+ is continuous and π·π‘Žπ›Ό+ , 𝛽 𝛽 π·π‘Ž+ , π·π‘Ž+1 are the standard Riemann-Liouville fractional order derivatives. The rest of this paper is organized as follows. In Section 2, the Green functions for the homogeneous BVPs corresponding to (1)-(2) are constructed and the bounds for the Green functions are estimated. In Section 3, sufficient conditions for the existence of at least two or at least three positive solutions are established, by using fixed point index theory and LeggettWilliams fixed point theorems. In Section 4, as an application, an example is presented to illustrate our main result.

2

International Journal of Differential Equations

2. Green’s Function and Bounds In this section, we construct Green’s function for the homogeneous boundary value problem and estimate bounds for Green’s function that will be used to prove our main theorems. Let 𝐺(𝑑, 𝑠) be Green’s function for the homogeneous BVP

From 𝑒(𝑗) (π‘Ž) = 0, 0 ≀ 𝑗 ≀ 𝑛 βˆ’ 2, we have 𝑐𝑛 = π‘π‘›βˆ’1 = π‘π‘›βˆ’2 = β‹… β‹… β‹… = 𝑐2 = 0. Then 𝑒 (𝑑) =

βˆ’1 𝑑 ∫ (𝑑 βˆ’ 𝑠)π›Όβˆ’1 𝑦 (𝑠) 𝑑𝑠 + 𝑐1 (𝑑 βˆ’ π‘Ž)π›Όβˆ’1 , Ξ“ (𝛼) π‘Ž 𝛼1

𝑒(𝛼1 ) (𝑑) = 𝑐1 ∏ (𝛼 βˆ’ 𝑖) (𝑑 βˆ’ π‘Ž)π›Όβˆ’π›Ό1 βˆ’1

(9)

𝑖=1

βˆ’π·π‘Žπ›Ό+ 𝑒 (𝑑) = 0, π‘Ž < 𝑑 < 𝑏,

𝛼1

𝑒(𝑗) (π‘Ž) = 0, 0 ≀ 𝑗 ≀ 𝑛 βˆ’ 2,

(3)

π›Όβˆ’π›Ό1 βˆ’1

Lemma 1. Let 𝑑 = (𝑏 βˆ’ π‘Ž) βˆ’ πœ‰(πœ‚ βˆ’ π‘Ž) 𝐢[π‘Ž, 𝑏], then the fractional order BVP

> 0. If 𝑦 ∈

𝑒

𝛼1

𝛼1

𝑖=1

𝑖=1

𝑐1 ∏ (𝛼 βˆ’ 𝑖) (𝑏 βˆ’ π‘Ž)π›Όβˆ’π›Ό1 βˆ’1 βˆ’ ∏ (𝛼 βˆ’ 𝑖)

1 Ξ“ (𝛼)

𝑏

β‹… ∫ (𝑏 βˆ’ 𝑠)π›Όβˆ’π›Ό1 βˆ’1 𝑦 (𝑠) 𝑑𝑠

π·π‘Žπ›Ό+ 𝑒 (𝑑) + 𝑦 (𝑑) = 0, π‘Ž < 𝑑 < 𝑏, (𝑗)

𝑖=1

𝑑 1 ∫ (𝑑 βˆ’ 𝑠)π›Όβˆ’π›Ό1 βˆ’1 𝑦 (𝑠) 𝑑𝑠. Ξ“ (𝛼) π‘Ž

From 𝑒(𝛼1 ) (𝑏) = πœ‰π‘’(𝛼1 ) (πœ‚), we have

𝑒(𝛼1 ) (𝑏) = πœ‰π‘’(𝛼1 ) (πœ‚) . π›Όβˆ’π›Ό1 βˆ’1

βˆ’ ∏ (𝛼 βˆ’ 𝑖)

π‘Ž

(4)

(π‘Ž) = 0, 0 ≀ 𝑗 ≀ 𝑛 βˆ’ 2,

𝑒(𝛼1 ) (𝑏) = πœ‰π‘’(𝛼1 ) (πœ‚)

(10)

𝛼1

= πœ‰ [𝑐1 ∏ (𝛼 βˆ’ 𝑖) (πœ‚ βˆ’ π‘Ž) 𝑖=1

(5)

𝛼1

βˆ’ ∏ (𝛼 βˆ’ 𝑖) 𝑖=1

𝑏

has a unique solution, 𝑒(𝑑) = βˆ«π‘Ž 𝐺(𝑑, 𝑠)𝑦(𝑠)𝑑𝑠, where πœ‰ (𝑑 βˆ’ π‘Ž)π›Όβˆ’1 𝐺 (𝑑, 𝑠) = 𝐺1 (𝑑, 𝑠) + 𝐺2 (πœ‚, 𝑠) ; 𝑑

π›Όβˆ’π›Ό1 βˆ’1

πœ‚ 1 π›Όβˆ’π›Ό βˆ’1 ∫ (πœ‚ βˆ’ 𝑠) 1 𝑦 (𝑠) 𝑑𝑠] . Ξ“ (𝛼) π‘Ž

Therefore (6)

𝑐1 =

1 π›Όβˆ’π›Ό1 βˆ’1

Ξ“ (𝛼) [(𝑏 βˆ’ π‘Ž)π›Όβˆ’π›Ό1 βˆ’1 βˆ’ πœ‰ (πœ‚ βˆ’ π‘Ž)

]

𝑏

Γ— [∫ (𝑏 βˆ’ 𝑠)π›Όβˆ’π›Ό1 βˆ’1 𝑦 (𝑠) 𝑑𝑠

here

π‘Ž

πœ‚

π‘Ž

(𝑑 βˆ’ π‘Ž)π›Όβˆ’1 (𝑏 βˆ’ 𝑠)π›Όβˆ’π›Ό1 βˆ’1 { , π‘Ž ≀ 𝑑 ≀ 𝑠 ≀ 𝑏, { { { (𝑏 βˆ’ π‘Ž)π›Όβˆ’π›Ό1 βˆ’1 β‹…{ π›Όβˆ’π›Ό βˆ’1 π›Όβˆ’1 { { (𝑑 βˆ’ π‘Ž) (𝑏 βˆ’ 𝑠) 1 { βˆ’ (𝑑 βˆ’ 𝑠)π›Όβˆ’1 , π‘Ž ≀ 𝑠 ≀ 𝑑 ≀ 𝑏, { (𝑏 βˆ’ π‘Ž)π›Όβˆ’π›Ό1 βˆ’1 1 𝐺2 (πœ‚, 𝑠) = Ξ“ (𝛼) π›Όβˆ’π›Ό1 βˆ’1

π›Όβˆ’π›Ό1 βˆ’1

βˆ’ πœ‰ ∫ (πœ‚ βˆ’ 𝑠)

1 𝐺1 (𝑑, 𝑠) = Ξ“ (𝛼)

𝑦 (𝑠) 𝑑𝑠] =

1 Ξ“ (𝛼) [(𝑏 βˆ’ π‘Ž)π›Όβˆ’π›Ό1 βˆ’1 ]

𝑏

β‹… ∫ (𝑏 βˆ’ 𝑠)π›Όβˆ’π›Ό1 βˆ’1 𝑦 (𝑠) 𝑑𝑠 π‘Ž

π›Όβˆ’π›Ό1 βˆ’1

(7)

+

πœ‰ (πœ‚ βˆ’ π‘Ž)

Ξ“ (𝛼) (𝑏 βˆ’ π‘Ž)π›Όβˆ’π›Ό1 βˆ’1 [(𝑏 βˆ’ π‘Ž)π›Όβˆ’π›Ό1 βˆ’1 βˆ’ πœ‰ (πœ‚ βˆ’ π‘Ž) 𝑏

(πœ‚ βˆ’ π‘Ž) (𝑏 βˆ’ 𝑠) { { , πœ‚ ≀ 𝑠 ≀ 𝑏, { { π›Όβˆ’π›Ό1 βˆ’1 { (𝑏 βˆ’ π‘Ž) β‹…{ π›Όβˆ’π›Ό βˆ’1 { { (πœ‚ βˆ’ π‘Ž) 1 (𝑏 βˆ’ 𝑠)π›Όβˆ’π›Ό1 βˆ’1 π›Όβˆ’π›Ό βˆ’1 { { βˆ’ (πœ‚ βˆ’ 𝑠) 1 , π‘Ž ≀ 𝑠 ≀ πœ‚. (𝑏 βˆ’ π‘Ž)π›Όβˆ’π›Ό1 βˆ’1 {

π‘Ž

βˆ’

πœ‚

Proof. Assume that 𝑒 ∈ 𝐢[𝛼]+1 [π‘Ž, 𝑏] is a solution of fractional order BVP (4)-(5) and is uniquely expressed as πΌπ‘Žπ›Ό+ π·π‘Žπ›Ό+ 𝑒(𝑑) = βˆ’πΌπ‘Žπ›Ό+ 𝑦(𝑑), so that βˆ’1 𝑑 𝑒 (𝑑) = ∫ (𝑑 βˆ’ 𝑠)π›Όβˆ’1 𝑦 (𝑠) 𝑑𝑠 + 𝑐1 (𝑑 βˆ’ π‘Ž)π›Όβˆ’1 Ξ“ (𝛼) π‘Ž + 𝑐2 (𝑑 βˆ’ π‘Ž)

π›Όβˆ’2

+ β‹… β‹… β‹… + 𝑐𝑛 (𝑑 βˆ’ π‘Ž)

π›Όβˆ’π‘›

.

πœ‰ Ξ“ (𝛼) [(𝑏 βˆ’ π‘Ž)π›Όβˆ’π›Ό1 βˆ’1 βˆ’ πœ‰ (πœ‚ βˆ’ π‘Ž) π›Όβˆ’π›Ό1 βˆ’1

β‹… ∫ (πœ‚ βˆ’ 𝑠) π‘Ž

(8)

π›Όβˆ’π›Ό1 βˆ’1

𝑏

π‘Ž

πœ‰ [(𝑏 βˆ’ π‘Ž) 𝑏

π›Όβˆ’π›Ό1 βˆ’1

π‘Ž

π›Όβˆ’π›Ό1 βˆ’1

βˆ’ πœ‰ (πœ‚ βˆ’ π‘Ž)

β‹… ∫ 𝐺2 (πœ‚, 𝑠) 𝑦 (𝑠) 𝑑𝑠.

]

1 Ξ“ (𝛼) (𝑏 βˆ’ π‘Ž)π›Όβˆ’π›Ό1 βˆ’1

𝑦 (𝑠) 𝑑𝑠 =

β‹… ∫ (𝑏 βˆ’ 𝑠)π›Όβˆ’π›Ό1 βˆ’1 𝑦 (𝑠) 𝑑𝑠 +

]

(11)

β‹… ∫ (𝑏 βˆ’ 𝑠)π›Όβˆ’π›Ό1 βˆ’1 𝑦 (𝑠) 𝑑𝑠

π›Όβˆ’π›Ό1 βˆ’1

π›Όβˆ’π›Ό1 βˆ’1

]

International Journal of Differential Equations

3 Here 𝐻(𝑑, 𝑠) is Green’s function for

Thus, the unique solution of (4)-(5) is

𝛽

βˆ’π·π‘Ž+ (πœ™π‘ (π‘₯ (𝑑))) = 0, π‘Ž < 𝑑 < 𝑏,

βˆ’1 𝑑 𝑒 (𝑑) = ∫ (𝑑 βˆ’ 𝑠)π›Όβˆ’1 𝑦 (𝑠) 𝑑𝑠 Ξ“ (𝛼) π‘Ž +

+

πœ™π‘ (π‘₯ (π‘Ž)) = 0, 𝛽

𝑏 (𝑑 βˆ’ π‘Ž)π›Όβˆ’1 ∫ (𝑏 βˆ’ 𝑠)π›Όβˆ’π›Ό1 βˆ’1 𝑦 (𝑠) 𝑑𝑠 π›Όβˆ’π›Ό1 βˆ’1 π‘Ž Ξ“ (𝛼) (𝑏 βˆ’ π‘Ž)

π·π‘Ž+1 (πœ™π‘ (π‘₯ (𝑏))) = 0. Proof. An equivalent integral equation for (13) is given by

πœ‰ (𝑑 βˆ’ π‘Ž)π›Όβˆ’1 [(𝑏 βˆ’ π‘Ž)π›Όβˆ’π›Ό1 βˆ’1 βˆ’ πœ‰ (πœ‚ βˆ’ π‘Ž)

π›Όβˆ’π›Ό1 βˆ’1

] (12)

𝑏

𝑏

π‘Ž

π‘Ž

πœ™π‘ (π·π‘Žπ›Ό+ 𝑒 (𝑑)) =

𝑑 1 ∫ (𝑑 βˆ’ 𝜏)π›½βˆ’1 β„Ž (𝜏) π‘‘πœ Ξ“ (𝛽) π‘Ž

+ 𝑐1 (𝑑 βˆ’ π‘Ž)

β‹… ∫ 𝐺2 (πœ‚, 𝑠) 𝑦 (𝑠) 𝑑𝑠 = ∫ 𝐺1 (𝑑, 𝑠) 𝑦 (𝑠) 𝑑𝑠 +

[(𝑏 βˆ’ π‘Ž)

π›½βˆ’1

+ 𝑐2 (𝑑 βˆ’ π‘Ž)

π›½βˆ’2

(18) .

By (14), one can determine 𝑐2 = 0 and 𝑐1 = (βˆ’1/(𝑏 βˆ’

πœ‰ (𝑑 βˆ’ π‘Ž)π›Όβˆ’1 π›Όβˆ’π›Ό1 βˆ’1

(17)

𝑏

βˆ’ πœ‰ (πœ‚ βˆ’ π‘Ž)

𝑏

𝑏

π‘Ž

π‘Ž

π›Όβˆ’π›Ό1 βˆ’1

π‘Ž)π›½βˆ’π›½1 βˆ’1 )(1/Ξ“(𝛽)) βˆ«π‘Ž (𝑏 βˆ’ 𝜏)π›½βˆ’π›½1 βˆ’1 β„Ž(𝜏)π‘‘πœ. Thus, the unique solution of (13), (2) is

]

πœ™π‘ (π·π‘Žπ›Ό+ 𝑒 (𝑑))

β‹… ∫ 𝐺2 (πœ‚, 𝑠) 𝑦 (𝑠) 𝑑𝑠 = ∫ 𝐺 (𝑑, 𝑠) 𝑦 (𝑠) 𝑑𝑠,

= where 𝐺(𝑑, 𝑠) is given in (6).

𝑑 1 ∫ (𝑑 βˆ’ 𝜏)π›½βˆ’1 β„Ž (𝜏) π‘‘πœ Ξ“ (𝛽) π‘Ž

βˆ’

Lemma 2. If β„Ž ∈ 𝐢[π‘Ž, 𝑏], then the fractional order differential equation

(𝑑 βˆ’ π‘Ž)π›½βˆ’1

𝑏 1 ∫ (𝑏 βˆ’ 𝜏)π›½βˆ’π›½1 βˆ’1 β„Ž (𝜏) π‘‘πœ π›½βˆ’π›½1 βˆ’1 Ξ“ (𝛽) π‘Ž (𝑏 βˆ’ π‘Ž)

(19)

𝑏

= βˆ’ ∫ 𝐻 (𝑑, 𝜏) β„Ž (𝜏) π‘‘πœ.

𝛽

π·π‘Ž+ (πœ™π‘ (π·π‘Žπ›Ό+ 𝑒 (𝑑))) = β„Ž (𝑑) ,

π‘Ž < 𝑑 < 𝑏,

π‘Ž

(13)

𝑏

Therefore, πœ™π‘βˆ’1 (πœ™π‘ (π·π‘Žπ›Ό+ 𝑒(𝑑))) = βˆ’πœ™π‘βˆ’1 (βˆ«π‘Ž 𝐻(𝑑, 𝜏)β„Ž(𝜏)π‘‘πœ). Con𝑏

sequently, π·π‘Žπ›Ό+ 𝑒(𝑑) + πœ™π‘ž (βˆ«π‘Ž 𝐻(𝑑, 𝜏)β„Ž(𝜏)π‘‘πœ) = 0. Hence,

satisfying (5) and πœ™π‘ (π·π‘Žπ›Ό+ 𝑒 (π‘Ž)) = 0, (14)

𝛽

π·π‘Ž+1 (πœ™π‘ (π·π‘Žπ›Ό+ 𝑒 (𝑏))) = 0

𝑏

𝑒 (𝑑) = ∫ 𝐺 (𝑑, 𝑠) πœ™π‘ž (∫ 𝐻 (𝑠, 𝜏) β„Ž (𝜏) π‘‘πœ) 𝑑𝑠, π‘Ž

π‘Ž

(15)

where 𝐻 (𝑑, 𝑠) =

𝑏

π‘Ž

π‘Ž

(20)

Lemma 3. Green’s function 𝐺(𝑑, 𝑠) satisfies the following inequalities:

has a unique solution, 𝑏

𝑏

𝑒 (𝑑) = ∫ 𝐺 (𝑑, 𝑠) πœ™π‘ž (∫ 𝐻 (𝑠, 𝜏) β„Ž (𝜏) π‘‘πœ) 𝑑𝑠.

(ii) 𝐺(𝑑, 𝑠) β‰₯ ((πœ‚ βˆ’ π‘Ž)/(𝑏 βˆ’ π‘Ž))π›Όβˆ’1 𝐺(𝑏, 𝑠), π‘“π‘œπ‘Ÿ π‘Žπ‘™π‘™ (𝑑, 𝑠) ∈ [πœ‚, 𝑏] Γ— [π‘Ž, 𝑏]. Proof. Consider Green’s function given by (6). Let π‘Ž ≀ 𝑑 ≀ 𝑠 ≀ 𝑏. Then, we have

1 Ξ“ (𝛽)

(𝑑 βˆ’ π‘Ž)π›½βˆ’1 (𝑏 βˆ’ 𝑠)π›½βˆ’π›½1 βˆ’1 { { , { { { (𝑏 βˆ’ π‘Ž)π›½βˆ’π›½1 βˆ’1 β‹…{ π›½βˆ’π›½ βˆ’1 { { (𝑑 βˆ’ π‘Ž)π›½βˆ’1 (𝑏 βˆ’ 𝑠) 1 { { βˆ’ (𝑑 βˆ’ 𝑠)π›½βˆ’1 , π›½βˆ’π›½1 βˆ’1 βˆ’ π‘Ž) (𝑏 {

(i) 𝐺(𝑑, 𝑠) ≀ 𝐺(𝑏, 𝑠), π‘“π‘œπ‘Ÿ π‘Žπ‘™π‘™ (𝑑, 𝑠) ∈ [π‘Ž, 𝑏] Γ— [π‘Ž, 𝑏],

1 πœ• (𝑑 βˆ’ π‘Ž)π›Όβˆ’2 (𝑏 βˆ’ 𝑠)π›Όβˆ’π›Ό1 βˆ’1 [ ] 𝐺1 (𝑑, 𝑠) = πœ•π‘‘ Ξ“ (𝛼 βˆ’ 1) (𝑏 βˆ’ π‘Ž)π›Όβˆ’π›Ό1 βˆ’1

π‘Ž ≀ 𝑑 ≀ 𝑠 ≀ 𝑏, (16) π‘Ž ≀ 𝑠 ≀ 𝑑 ≀ 𝑏.

β‰₯

(1 βˆ’ π‘Ž) π‘‘π›Όβˆ’2 [ Ξ“ (𝛼 βˆ’ 1)

β‹… (1 βˆ’ 𝑠)π›Όβˆ’2 β‰₯ 0.

π›Όβˆ’2 π›Όβˆ’π›Ό1 +1

𝑏

(1 βˆ’ (1 βˆ’ 𝛼1 ) 𝑠 + 𝑂 (𝑠2 ))

(𝑏 βˆ’ π‘Ž)π›Όβˆ’π›Ό1 βˆ’1

]

(21)

4

International Journal of Differential Equations

Let π‘Ž ≀ 𝑠 ≀ 𝑑 ≀ 𝑏. Then, we have πœ• 1 (𝑑 βˆ’ π‘Ž)π›Όβˆ’2 (𝑏 βˆ’ 𝑠)π›Όβˆ’π›Ό1 βˆ’1 βˆ’ (𝑑 βˆ’ 𝑠)π›Όβˆ’2 ] 𝐺1 (𝑑, 𝑠) = [ πœ•π‘‘ Ξ“ (𝛼 βˆ’ 1) (𝑏 βˆ’ π‘Ž)π›Όβˆ’π›Ό1 βˆ’1 β‰₯

π›Όβˆ’2

𝑑 [ Ξ“ (𝛼 βˆ’ 1)

(1 βˆ’ π‘Ž)π›Όβˆ’2 π‘π›Όβˆ’π›Ό1 +1 (1 βˆ’ (1 βˆ’ 𝛼1 ) 𝑠 + 𝑂 (𝑠2 )) βˆ’ (𝑏 βˆ’ π‘Ž)π›Όβˆ’π›Ό1 βˆ’1 (𝑏 βˆ’ π‘Ž)π›Όβˆ’π›Ό1 βˆ’1

Therefore 𝐺1 (𝑑, 𝑠) is increasing in 𝑑, which implies 𝐺1 (𝑑, 𝑠) ≀ 𝐺1 (𝑏, 𝑠). Now we prove 𝐺2 (πœ‚, 𝑠) β‰₯ 0, 𝑠 ∈ [π‘Ž, 𝑏] .

β‰₯

(πœ‚ βˆ’ π‘Ž) 1 (𝑏 βˆ’ 𝑠) 1 [ Ξ“ (𝛼) (𝑏 βˆ’ π‘Ž)π›Όβˆ’π›Ό1 βˆ’1 (πœ‚ βˆ’ π‘Ž) 1 [ Ξ“ (𝛼)

] (1 βˆ’ 𝑠)π›Όβˆ’2 β‰₯ 0.

In fact, if 𝑠 β‰₯ πœ‚, obviously, (23) holds. If 𝑠 ≀ πœ‚, one has

(23)

π›Όβˆ’π›Ό βˆ’1

𝐺2 (πœ‚, 𝑠) =

(22)

π›Όβˆ’π›Ό1 βˆ’1

(𝑏 βˆ’ 𝑠)

π›Όβˆ’π›Ό1 βˆ’1

π›Όβˆ’π›Ό1 βˆ’1

βˆ’ (πœ‚ βˆ’ 𝑠)

π›Όβˆ’π›Ό1 βˆ’1

βˆ’ (πœ‚ βˆ’ πœ‚π‘ )

]

π›Όβˆ’π›Ό1 βˆ’1

(24) π›Όβˆ’π›Ό1 βˆ’1

(𝑏 βˆ’ π‘Ž)

π›Όβˆ’π›Ό1 βˆ’1

(𝑏 βˆ’ π‘Ž)

That implies that (23) is also true. Therefore, by (6), (21), and (23), we find

If π‘Ž ≀ 𝑑 ≀ 𝑠 ≀ 𝑏, we have 𝐺1 (𝑑, 𝑠) =

πœ• πœ• (𝛼 βˆ’ 1) πœ‰ (𝑑 βˆ’ π‘Ž)π›Όβˆ’2 𝐺 (𝑑, 𝑠) = 𝐺1 (𝑑, 𝑠) + 𝐺2 (πœ‚, 𝑠) πœ•π‘‘ πœ•π‘‘ 𝑑 (25) β‰₯ 0. Therefore 𝐺(𝑑, 𝑠) is increasing with respect to 𝑑 ∈ [π‘Ž, 𝑏]. Hence the inequality (i) is proved. Now, we establish the inequality (ii). On the other hand, if π‘Ž ≀ 𝑠 ≀ 𝑑 ≀ 𝑏, we have

] β‰₯ 0.

=

1 (𝑑 βˆ’ π‘Ž)π›Όβˆ’1 (𝑏 βˆ’ 𝑠)π›Όβˆ’π›Ό1 βˆ’1 [ ] Ξ“ (𝛼) (𝑏 βˆ’ π‘Ž)π›Όβˆ’π›Ό1 βˆ’1

𝑑 βˆ’ π‘Ž π›Όβˆ’1 (𝑏 βˆ’ π‘Ž)π›Όβˆ’1 (𝑏 βˆ’ 𝑠)π›Όβˆ’π›Ό1 βˆ’1 1 ( ] ) [ Ξ“ (𝛼) 𝑏 βˆ’ π‘Ž (𝑏 βˆ’ π‘Ž)π›Όβˆ’π›Ό1 βˆ’1

β‰₯(

(27)

πœ‚ βˆ’ π‘Ž π›Όβˆ’1 ) 𝐺1 (𝑏, 𝑠) . π‘βˆ’π‘Ž

Therefore 𝐺1 (𝑑, 𝑠) β‰₯ (

πœ‚ βˆ’ π‘Ž π›Όβˆ’1 ) 𝐺1 (𝑏, 𝑠) . π‘βˆ’π‘Ž

(28)

From (6) and (28) we have 𝐺1 (𝑑, 𝑠)

𝐺 (𝑑, 𝑠) = 𝐺1 (𝑑, 𝑠) +

=

1 (𝑑 βˆ’ π‘Ž)π›Όβˆ’1 (𝑏 βˆ’ 𝑠)π›Όβˆ’π›Ό1 βˆ’1 [ βˆ’ (𝑑 βˆ’ 𝑠)π›Όβˆ’1 ] Ξ“ (𝛼) (𝑏 βˆ’ π‘Ž)π›Όβˆ’π›Ό1 βˆ’1

β‰₯

1 𝑑 βˆ’ π‘Ž π›Όβˆ’1 ( ) Ξ“ (𝛼) 𝑏 βˆ’ π‘Ž

(𝑏 βˆ’ π‘Ž)π›Όβˆ’1 (𝑏 βˆ’ 𝑠)π›Όβˆ’π›Ό1 βˆ’1 β‹…[ βˆ’ (𝑏 βˆ’ 𝑠)π›Όβˆ’1 ] (𝑏 βˆ’ π‘Ž)π›Όβˆ’π›Ό1 βˆ’1 β‰₯(

πœ‚βˆ’π‘Ž ) π‘βˆ’π‘Ž

π›Όβˆ’1

𝐺1 (𝑏, 𝑠) .

(26)

πœ‰ (𝑑 βˆ’ π‘Ž)π›Όβˆ’1 𝐺2 (πœ‚, 𝑠) 𝑑

β‰₯(

πœ‚ βˆ’ π‘Ž π›Όβˆ’1 πœ‰ (𝑑 βˆ’ π‘Ž)π›Όβˆ’1 ) 𝐺1 (𝑑, 𝑠) + 𝐺2 (πœ‚, 𝑠) (29) π‘βˆ’π‘Ž 𝑑

β‰₯(

πœ‚ βˆ’ π‘Ž π›Όβˆ’1 ) 𝐺 (𝑏, 𝑠) . π‘βˆ’π‘Ž

Therefore 𝐺 (𝑑, 𝑠) β‰₯ (

πœ‚ βˆ’ π‘Ž π›Όβˆ’1 ) 𝐺 (𝑏, 𝑠) π‘βˆ’π‘Ž

(30) βˆ€ (𝑑, 𝑠) ∈ [πœ‚, 𝑏] Γ— [π‘Ž, 𝑏] .

Hence the inequality (ii) is proved.

International Journal of Differential Equations Lemma 4. Green’s function 𝐻(𝑑, 𝑠) satisfies the following inequalities: (i) 𝐻(𝑑, 𝑠) ≀ 𝐻(𝑠, 𝑠), π‘“π‘œπ‘Ÿ π‘Žπ‘™π‘™ (𝑑, 𝑠) ∈ [π‘Ž, 𝑏] Γ— [π‘Ž, 𝑏], (ii) 𝐻(𝑑, 𝑠) β‰₯ 𝛾𝐻(𝑠, 𝑠), π‘“π‘œπ‘Ÿ π‘Žπ‘™π‘™ (𝑑, 𝑠) ∈ 𝐼 Γ— [π‘Ž, 𝑏],

5 is a compact map such that π‘₯ =ΜΈ 𝑇π‘₯ for π‘₯ ∈ πœ•Ξ©π‘ž . Thus, one has the following conclusions: (𝐷1) If β€–π‘₯β€– < ‖𝑇π‘₯β€– for π‘₯ ∈ πœ•Ξ©π‘ž , then 𝑖(𝑇, Ξ©π‘ž , 𝑃) = 0. (𝐷2) If β€–π‘₯β€– β‰₯ ‖𝑇π‘₯β€– for π‘₯ ∈ πœ•Ξ©π‘ž , then 𝑖(𝑇, Ξ©π‘ž , 𝑃) = 1.

where 𝐼 = [(3π‘Ž + 𝑏)/4, (π‘Ž + 3𝑏)/4] and 𝛾 = (1/4)π›½βˆ’1 .

3. Main Results

The method of proof is similar to that [20], and we omit it here.

In this section, the existence of at least two or at least three positive solutions for fractional differential equation with 𝑝-Laplacian operator BVP (1)-(2) is established by using fixed point index theory and Leggett-Williams fixed point theorems. Let 𝐸 = {𝑒 : 𝑒 ∈ 𝐢[π‘Ž, 𝑏]} be the real Banach space equipped with the norm ‖𝑒‖ = maxπ‘‘βˆˆ[π‘Ž,𝑏] |𝑒(𝑑)|. Define the cone 𝑃 βŠ‚ 𝐸 by

Theorem 5 (Leggett-Williams [3]). Let 𝑇 : 𝑃𝑐 β†’ 𝑃𝑐 be completely continuous and let πœ™ be a nonnegative continuous concave functional on 𝑃 such that πœ™(𝑦) ≀ ‖𝑦‖ for all 𝑦 ∈ 𝑃𝑐 . Suppose that there exist π‘Ž, 𝑏, 𝑐, and 𝑑 with 0 < π‘Ž < 𝑏 < 𝑑 ≀ 𝑐 such that (𝐴1) {𝑦 ∈ 𝑃(πœ™, 𝑏, 𝑑) : πœ™(𝑦) > 𝑏} =ΜΈ 0 and πœ™(𝑇𝑦) > 𝑏 for 𝑦 ∈ 𝑃(πœ™, 𝑏, 𝑑),

𝑃 = {𝑒 ∈ 𝐸 : 𝑒 (𝑑) β‰₯ 0, βˆ€π‘‘ ∈ [π‘Ž, 𝑏] , min 𝑒 (𝑑) π‘‘βˆˆ[πœ‚,𝑏]

(𝐴3) πœ™(𝑇𝑦) > 𝑏 for 𝑦 ∈ 𝑃(πœ™, 𝑏, 𝑐) with ‖𝑇𝑦‖ > 𝑑. Then 𝑇 has at least three fixed points 𝑦1 , 𝑦2 , and 𝑦3 in 𝑃𝑐 satisfying ‖𝑦1 β€– < π‘Ž, 𝑏 < πœ™(𝑦2 ), ‖𝑦3 β€– > π‘Ž, and πœ™(𝑦3 ) < 𝑏. Theorem 6 (see [3]). Let 𝑇 : 𝑃𝑐 β†’ 𝑃 be a completely continuous operator and let πœ™ be a nonnegative continuous concave functional on 𝑃 such that πœ™(𝑦) ≀ ‖𝑦‖ for all 𝑦 ∈ 𝑃𝑐 . Suppose that there exist π‘Ž, 𝑏, and 𝑐 with 0 < π‘Ž < 𝑏 < 𝑐 such that (𝐡1) {𝑦 ∈ 𝑃(πœ™, 𝑏, 𝑐) : πœ™(𝑦) > 𝑏} =ΜΈ 0 and πœ™(𝑇𝑦) > 𝑏 for 𝑦 ∈ 𝑃(πœ™, 𝑏, 𝑐), (𝐡2) ‖𝑇𝑦‖ < π‘Ž for ‖𝑦‖ ≀ π‘Ž, (𝐡3) πœ™(𝑇𝑦) > (𝑏/𝑐)‖𝑇𝑦‖ for 𝑦 ∈ 𝑃𝑐 with ‖𝑇𝑦‖ > 𝑐. Then 𝑇 has at least two fixed points 𝑦1 and 𝑦2 in 𝑃𝑐 satisfying ‖𝑦1 β€– < π‘Ž, ‖𝑦2 β€– > π‘Ž and πœ™(𝑦2 ) < 𝑏. Theorem 7 (see [21]). Let 𝑃 be a closed convex set in a Banach space 𝐸 and let Ξ© be a bounded open set such that Ω𝑝 fl Ξ©βˆ©π‘ƒ =ΜΈ 0. Let 𝑇 : Ω𝑝 β†’ 𝑃 be a compact map. Suppose that π‘₯ =ΜΈ 𝑇π‘₯ for all π‘₯ ∈ πœ•π‘ :

(31)

πœ‚ βˆ’ π‘Ž π›Όβˆ’1 β‰₯( ) ‖𝑒‖} . π‘βˆ’π‘Ž

(𝐴2) ‖𝑇𝑦‖ < π‘Ž for ‖𝑦‖ ≀ π‘Ž,

Let 𝑇 : 𝑃 β†’ 𝐸 be the operator defined by 𝑇𝑒 (𝑑) 𝑏

𝑏

π‘Ž

π‘Ž

= ∫ 𝐺 (𝑑, 𝑠) πœ™π‘ž (∫ 𝐻 (𝑠, 𝜏) 𝑓 (𝜏, 𝑒 (𝜏)) π‘‘πœ) 𝑑𝑠.

If 𝑒 ∈ 𝑃 is a fixed point of 𝑇, then 𝑒 satisfies (32) and hence 𝑒 is a positive solution of 𝑝-Laplacian fractional order BVP (1)-(2). Lemma 9. The operator 𝑇 defined by (32) is a self-map on 𝑃. Proof. Let 𝑒 ∈ 𝑃. Clearly, 𝑇𝑒(𝑑) β‰₯ 0, for all 𝑑 ∈ [π‘Ž, 𝑏] and 𝑏

𝑏

𝑇𝑒 (𝑑) = ∫ 𝐺 (𝑑, 𝑠) πœ™π‘ž (∫ 𝐻 (𝑠, 𝜏) 𝑓 (𝜏, 𝑒 (𝜏)) π‘‘πœ) 𝑑𝑠 π‘Ž

π‘Ž

𝑏

𝑏

π‘Ž

π‘Ž

(33)

≀ ∫ 𝐺 (𝑏, 𝑠) πœ™π‘ž (∫ 𝐻 (𝑠, 𝜏) 𝑓 (𝜏, 𝑒 (𝜏)) π‘‘πœ) 𝑑𝑠 so that 𝑏

𝑏

π‘Ž

π‘Ž

‖𝑇𝑒‖ ≀ ∫ 𝐺 (𝑏, 𝑠) πœ™π‘ž (∫ 𝐻 (𝑠, 𝜏) 𝑓 (𝜏, 𝑒 (𝜏)) π‘‘πœ) 𝑑𝑠. (34)

(𝐢1) Existence: if 𝑖(𝑇, Ω𝑝 , 𝑃) =ΜΈ 0, then 𝑇 has a fixed point in Ω𝑝 .

On the other hand, by Lemma 3, we have

(𝐢2) Normalization: if 𝑒 ∈ Ω𝑝 , then 𝑖(Μ‚ 𝑒, Ω𝑝 , 𝑃) = 1, where Μ‚ (π‘₯) = 𝑒 for π‘₯ ∈ Ω𝑝 . 𝑒

min 𝑇𝑒 (𝑑) = min ∫ 𝐺 (𝑑, 𝑠)

(𝐢3) Homotopy: let V : [0, 1] Γ— Ω𝑝 β†’ 𝑃 be a compact map such that π‘₯ =ΜΈ V(𝑑, π‘₯) for π‘₯ ∈ πœ•Ξ©π‘ and 𝑑 ∈ [0, 1]. Then 𝑖(V(0, β‹…), Ω𝑝 , 𝑃) = 𝑖(V(1, β‹…), Ω𝑝 , 𝑃). (𝐢4) Additivity: if π‘ˆ1, π‘ˆ2 are disjoint relatively open subsets of Ω𝑝 such that π‘₯ =ΜΈ 𝑇π‘₯ for π‘₯ ∈ Ω𝑝 \ (π‘ˆ1 βˆͺ π‘ˆ2), then 𝑖(𝑇, Ω𝑝 , 𝑃) = 𝑖(𝑇, π‘ˆ1, 𝑃) + 𝑖(𝑇, π‘ˆ2, 𝑃), where 𝑖(𝑇, π‘ˆπ‘—, 𝑃) = 𝑖(𝑇 | 𝑒𝑗, π‘ˆπ‘—, 𝑃) (𝑗 = 1, 2). Theorem 8 (see [22]). Let 𝑃 be a cone in a Banach space 𝐸. For π‘ž > 0, define Ξ©π‘ž = {π‘₯ ∈ 𝑃 : β€–π‘₯β€– < π‘ž}. Assume that 𝑇 : Ξ©π‘ž β†’ 𝑃

(32)

𝑏

π‘‘βˆˆ[πœ‚,𝑏]

π‘‘βˆˆ[πœ‚,𝑏] π‘Ž

𝑏

β‹… πœ™π‘ž (∫ 𝐻 (𝑠, 𝜏) 𝑓 (𝜏, 𝑒 (𝜏)) π‘‘πœ) 𝑑𝑠 π‘Ž

β‰₯(

πœ‚ βˆ’ π‘Ž π›Όβˆ’1 𝑏 ) ∫ 𝐺 (𝑏, 𝑠) π‘βˆ’π‘Ž π‘Ž 𝑏

β‹… πœ™π‘ž (∫ 𝐻 (𝑠, 𝜏) 𝑓 (𝜏, 𝑒 (𝜏)) π‘‘πœ) 𝑑𝑠 π‘Ž

β‰₯(

πœ‚ βˆ’ π‘Ž π›Όβˆ’1 ) ‖𝑇𝑒‖ . π‘βˆ’π‘Ž

(35)

6

International Journal of Differential Equations

Hence 𝑇𝑒 ∈ 𝑃 and so 𝑇 : 𝑃 β†’ 𝑃. Standard argument involving the Arzela-Ascoli theorem shows that 𝑇 is completely continuous.

𝑏

β‰₯ 𝐴𝑏󸀠 ∫ 𝐺 (πœ‚, 𝑠) πœ™π‘ž (∫ πœ‚

= 2𝑏󸀠 (

For convenience of the reader, we denote

𝜏∈𝐼

𝛾𝐻 (𝜏, 𝜏) π‘‘πœ) 𝑑𝑠

𝑏 βˆ’ π‘Ž π›Όβˆ’1 𝑏 βˆ’ π‘Ž π›Όβˆ’1 ) ) . > 𝑏󸀠 ( πœ‚βˆ’π‘Ž πœ‚βˆ’π‘Ž (38)

π›Όβˆ’1

𝐴=

𝐡=

2 ((𝑏 βˆ’ π‘Ž) / (πœ‚ βˆ’ π‘Ž)) 𝑏

βˆ«πœ‚ 𝐺 (πœ‚, 𝑠) πœ™π‘ž (∫𝜏∈𝐼 𝛾𝐻 (𝜏, 𝜏) π‘‘πœ) 𝑑𝑠 1 𝑏

𝑏

βˆ«π‘Ž 𝐺 (𝑏, 𝑠) πœ™π‘ž (βˆ«π‘Ž 𝐻 (𝜏, 𝜏) π‘‘πœ) 𝑑𝑠

𝑓0 = lim+ min

𝑒→0 π‘‘βˆˆ[π‘Ž,𝑏]

,

Consequently, min 𝑇𝑒 (𝑑) β‰₯ (

π‘‘βˆˆ[πœ‚,𝑏]

,

πœƒ (𝑇𝑒) > 𝑏󸀠 ,

Theorem 10. Let 𝑓(𝑑, 𝑒) be nonnegative continuous on [π‘Ž, 𝑏] Γ— [0, ∞). Assume that there exist constants π‘ŽσΈ€  , 𝑏󸀠 with 𝑏󸀠 > π‘ŽσΈ€  > 0 such that the following conditions are satisfied: σΈ€ 

σΈ€ 

σΈ€ 

(𝐻1) 𝑓(𝑑, 𝑒(𝑑)) β‰₯ πœ™π‘ (𝐴𝑏 ) for all (𝑑, 𝑒) ∈ [πœ‚, 𝑏] Γ— [𝑏 , 𝑏 ((𝑏 βˆ’ π‘Ž)/(πœ‚ βˆ’ π‘Ž))2(π›Όβˆ’1) ]. (𝐻2) 𝑓(𝑑, 𝑒(𝑑)) < πœ™π‘ (π΅π‘ŽσΈ€  ) for all (𝑑, 𝑒) ∈ [π‘Ž, 𝑏] Γ— [0, π‘ŽσΈ€  ].

Proof. Let πœƒ : 𝑃 β†’ [0, ∞) be the nonnegative continuous concave functional defined by πœƒ(𝑒) = minπ‘‘βˆˆ[πœ‚,𝑏] 𝑒(𝑑), 𝑒 ∈ 𝑃. Evidently, for each 𝑒 ∈ 𝑃, we have πœƒ(𝑒) ≀ ‖𝑒‖. It is easy to see that 𝑇 : 𝑃𝑏󸀠 ((π‘βˆ’π‘Ž)/(πœ‚βˆ’π‘Ž))2(π›Όβˆ’1) β†’ 𝑃 is completely continuous and 𝑏󸀠 ((𝑏 βˆ’ π‘Ž)/(πœ‚ βˆ’ π‘Ž))2(π›Όβˆ’1) > 𝑏󸀠 > π‘ŽσΈ€  > 0. We choose 𝑒(𝑑) = 𝑏󸀠 ((𝑏 βˆ’ π‘Ž)/(πœ‚ βˆ’ π‘Ž))2(π›Όβˆ’1) ; then

π‘βˆ’π‘Ž πœƒ (𝑒) = 𝑏 ( ) πœ‚βˆ’π‘Ž σΈ€ 

𝑏

π‘‘βˆˆ[π‘Ž,𝑏]

π‘‘βˆˆ[π‘Ž,𝑏]

𝑏

𝑏

π‘Ž

π‘Ž

π‘Ž

𝑏

𝑏

π‘Ž

π‘Ž

< π΅π‘ŽσΈ€  ∫ 𝐺 (𝑏, 𝑠) πœ™π‘ž (∫ 𝐻 (𝜏, 𝜏) π‘‘πœ) 𝑑𝑠 = π‘ŽσΈ€  , which shows that 𝑇 : π‘ƒπ‘ŽσΈ€  β†’ π‘ƒπ‘ŽσΈ€  , that is, ‖𝑇𝑒‖ < π‘ŽσΈ€  for 𝑒 ∈ π‘ƒπ‘ŽσΈ€  . This shows that condition (𝐡2) of Theorem 6 is satisfied. Finally, we show that (𝐡3) of Theorem 6 also holds. Assume that 𝑒 ∈ 𝑃𝑏󸀠 ((π‘βˆ’π‘Ž)/(πœ‚βˆ’π‘Ž))2(π›Όβˆ’1) with ‖𝑇𝑒‖ > 𝑏󸀠 ((π‘βˆ’π‘Ž)/(πœ‚βˆ’ π‘Ž))2(π›Όβˆ’1) ; then by the definition of cone 𝑃, we have

>( =

) | πœƒ(𝑒) > 𝑏 } =ΜΈ 0. So {𝑒 ∈ 𝑃(πœƒ, 𝑏 , 𝑏 ((𝑏 βˆ’ π‘Ž)/(πœ‚ βˆ’ π‘Ž)) 2(π›Όβˆ’1) σΈ€  σΈ€  Hence, if 𝑒 ∈ 𝑃(πœƒ, 𝑏 , 𝑏 ((π‘βˆ’π‘Ž)/(πœ‚βˆ’π‘Ž)) ), then 𝑏󸀠 ≀ 𝑒(𝑑) ≀ 2(π›Όβˆ’1) σΈ€  𝑏 ((𝑏 βˆ’ π‘Ž)/(πœ‚ βˆ’ π‘Ž)) for 𝑑 ∈ [πœ‚, 𝑏]. Thus for 𝑑 ∈ [πœ‚, 𝑏], from assumption (𝐻1), we have

(41)

β‹… πœ™π‘ž (∫ 𝐻 (𝜏, 𝜏) 𝑓 (𝜏, 𝑒 (𝜏)) π‘‘πœ) 𝑑𝑠

σΈ€ 

σΈ€ 

π‘Ž

β‹… (∫ 𝐻 (𝑠, 𝜏) 𝑓 (𝜏, 𝑒 (𝜏)) π‘‘πœ) 𝑑𝑠] ≀ ∫ 𝐺 (𝑏, 𝑠)

π‘‘βˆˆ[πœ‚,𝑏]

(37)

(40)

‖𝑇𝑒‖ = max |𝑇𝑒 (𝑑)| = max [∫ 𝐺 (𝑑, 𝑠) πœ™π‘ž

>𝑏.

2(π›Όβˆ’1)

σΈ€ 

𝑏 βˆ’ π‘Ž 2(π›Όβˆ’1) ). ) πœ‚βˆ’π‘Ž

Therefore, condition (𝐡1) of Theorem 6 is satisfied. Now if 𝑒 ∈ π‘ƒπ‘ŽσΈ€  , then ‖𝑒‖ ≀ π‘ŽσΈ€  . By assumption (𝐻2), we have

πœƒ (𝑇𝑒) = min 𝑇𝑒 (𝑑) β‰₯ (

𝑏 βˆ’ π‘Ž 2(π›Όβˆ’1) ), ) πœ‚βˆ’π‘Ž

2(π›Όβˆ’1)

βˆ€π‘’ ∈ 𝑃 (πœƒ, 𝑏󸀠 , 𝑏󸀠 (

𝑏

Then fractional order BVP (1)-(2) has at least two positive solutions 𝑒1 and 𝑒2 satisfying ‖𝑒1 β€– < π‘ŽσΈ€  , minπ‘‘βˆˆ[πœ‚,𝑏] 𝑒2 (𝑑) < 𝑏󸀠 , and ‖𝑒2 β€– > π‘ŽσΈ€  .

σΈ€ 

(39)

for πœ‚ ≀ 𝑑 ≀ 𝑏, 𝑏󸀠 ≀ 𝑒(𝑑) ≀ 𝑏󸀠 ((𝑏 βˆ’ π‘Ž)/(πœ‚ βˆ’ π‘Ž))2(π›Όβˆ’1) . That is,

𝑓 (𝑑, 𝑒) π‘“βˆž = lim max . π‘’β†’βˆž π‘‘βˆˆ[π‘Ž,𝑏] πœ™ (𝑒) 𝑝

𝑒 ∈ 𝑃 (πœƒ, 𝑏󸀠 , 𝑏󸀠 (

πœ‚ βˆ’ π‘Ž π›Όβˆ’1 𝑏 βˆ’ π‘Ž π›Όβˆ’1 = 𝑏󸀠 >( ) Γ— 𝑏󸀠 ( ) π‘βˆ’π‘Ž πœ‚βˆ’π‘Ž

(36)

𝑓 (𝑑, 𝑒) , πœ™π‘ (𝑒)

πœ‚ βˆ’ π‘Ž π›Όβˆ’1 ) ‖𝑇𝑒‖ π‘βˆ’π‘Ž

πœ‚ βˆ’ π‘Ž (π›Όβˆ’1) ) ‖𝑇𝑒‖ π‘βˆ’π‘Ž

πœ‚ βˆ’ π‘Ž 2(π›Όβˆ’1) ) ‖𝑇𝑒‖ π‘βˆ’π‘Ž

(42)

𝑏󸀠 2(π›Όβˆ’1)

𝑏󸀠 ((𝑏 βˆ’ π‘Ž) / (πœ‚ βˆ’ π‘Ž))

‖𝑇𝑒‖ .

So condition (𝐡3) of Theorem 6 is satisfied. Thus using Theorem 6, 𝑇 has at least two fixed points. Consequently, boundary value problem (1)-(2) has at least two positive solutions 𝑒1 and 𝑒2 in 𝑃𝑏󸀠 ((π‘βˆ’π‘Ž)/(πœ‚βˆ’π‘Ž))2(π›Όβˆ’1) satisfying σ΅„©σ΅„© σ΅„©σ΅„© σΈ€  󡄩󡄩𝑒1 σ΅„©σ΅„© < π‘Ž ,

𝑇𝑒 (πœ‚) 𝑏

𝑏

π‘Ž

π‘Ž

𝑏

𝑏

πœ‚

π‘Ž

= ∫ 𝐺 (πœ‚, 𝑠) πœ™π‘ž (∫ 𝐻 (𝑠, 𝜏) 𝑓 (𝜏, 𝑒 (𝜏)) π‘‘πœ) 𝑑𝑠 β‰₯ ∫ 𝐺 (πœ‚, 𝑠) πœ™π‘ž (∫ 𝐻 (𝑠, 𝜏) 𝑓 (𝜏, 𝑒 (𝜏)) π‘‘πœ) 𝑑𝑠

min 𝑒2 (𝑑) < 𝑏󸀠 ,

π‘‘βˆˆ[πœ‚,𝑏]

σ΅„©σ΅„© σ΅„©σ΅„© σΈ€  󡄩󡄩𝑒2 σ΅„©σ΅„© > π‘Ž .

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International Journal of Differential Equations

7

Theorem 11. Let 𝑓(𝑑, 𝑒) be nonnegative continuous on [π‘Ž, 𝑏] Γ— [0, ∞). Assume that there exist constants π‘ŽσΈ€  , 𝑏󸀠 , 𝑐󸀠 with ((πœ‚ βˆ’ π‘Ž)/(𝑏 βˆ’ π‘Ž))2(π›Όβˆ’1) 𝑐󸀠 > 𝑏󸀠 > π‘ŽσΈ€  > 0 such that σΈ€ 

σΈ€ 

(𝐻3) 𝑓(𝑑, 𝑒(𝑑)) < πœ™π‘ (π΅π‘Ž ) for all (𝑑, 𝑒) ∈ [π‘Ž, 𝑏] Γ— [0, π‘Ž ], (𝐻4) 𝑓(𝑑, 𝑒(𝑑)) β‰₯ πœ™π‘ (𝐴𝑏󸀠 ) for all (𝑑, 𝑒) ∈ [πœ‚, 𝑏] Γ— [𝑏󸀠 , 𝑏󸀠 ((𝑏 βˆ’ π‘Ž)/(πœ‚ βˆ’ π‘Ž))2(π›Όβˆ’1) ], (𝐻5) 𝑓(𝑑, 𝑒(𝑑)) ≀ πœ™π‘ (𝐡𝑐󸀠 ) for all (𝑑, 𝑒) ∈ [π‘Ž, 𝑏] Γ— [0, 𝑐󸀠 ]. Then fractional order BVP (1)-(2) has at least three positive solutions 𝑒1 , 𝑒2 , and 𝑒3 with ‖𝑒1 β€– < π‘ŽσΈ€  , minπ‘‘βˆˆ[πœ‚,𝑏] 𝑒2 (𝑑) > 𝑏󸀠 , ‖𝑒3 β€– > π‘ŽσΈ€  , and minπ‘‘βˆˆ[πœ‚,𝑏] 𝑒3 (𝑑) < 𝑏󸀠 . σΈ€ 

Proof. If 𝑒 ∈ 𝑃𝑐󸀠 , then ‖𝑒‖ ≀ 𝑐 . By assumption (𝐻5), we have

π‘‘βˆˆ[π‘Ž,𝑏]

(𝐻6) 𝑓0 = π‘“βˆž = ∞; (𝐻7) there exists a constant πœ‡1 > 0 such that 𝑓(𝑑, 𝑒) < πœ™π‘ (π΅πœ‡1 ), for (𝑑, 𝑒) ∈ [π‘Ž, 𝑏] Γ— [0, πœ‡1 ], then boundary value problem (1)-(2) has at least two positive solutions 𝑒1 and 𝑒2 such that 0 < ‖𝑒1 β€– < πœ‡1 < ‖𝑒2 β€–. Proof. From Lemma 1, we obtain 𝑇 : 𝑃 β†’ 𝑃 being completely continuous. In view of 𝑓0 = ∞, there exists 𝜎1 ∈ (0, πœ‡1 ) such that 𝑓(𝑑, 𝑒) β‰₯ πœ™π‘ (πœ‚1 𝑒), for π‘Ž ≀ 𝑑 ≀ 𝑏, 0 < 𝑒 ≀ 𝜎1 , where πœ‚1 ∈ (𝐴/2, ∞). Let Ω𝜎1 = {𝑒 ∈ 𝑃 | ‖𝑒‖ < 𝜎1 }. Then, for any 𝑒 ∈ πœ•Ξ©πœŽ1 , we have 𝑏

𝑏

π‘Ž

π‘Ž

𝑇𝑒 (πœ‚) = ∫ 𝐺 (πœ‚, 𝑠) πœ™π‘ž (∫ 𝐻 (𝑠, 𝜏) 𝑏

𝑏

β‹… 𝑓 (𝜏, 𝑒 (𝜏)) π‘‘πœ) 𝑑𝑠 β‰₯ ∫ 𝐺 (πœ‚, 𝑠)

‖𝑇𝑒‖ = max |𝑇𝑒 (𝑑)| = max [∫ 𝐺 (𝑑, 𝑠) πœ™π‘ž π‘‘βˆˆ[π‘Ž,𝑏]

Theorem 12. Let 𝑓(𝑑, 𝑒) be nonnegative continuous on [π‘Ž, 𝑏] Γ— [0, ∞). If the following assumptions are satisfied:

πœ‚

π‘Ž

𝑏

𝑏

π‘Ž

π‘Ž

𝑏

𝑏

π‘Ž

πœ‚

β‹… πœ™π‘ž (∫ 𝐻 (𝑠, 𝜏) πœ™π‘ (πœ‚1 𝑒) π‘‘πœ) 𝑑𝑠 β‰₯ ∫ 𝐺 (πœ‚,

β‹… (∫ 𝐻 (𝑠, 𝜏) 𝑓 (𝜏, 𝑒 (𝜏)) π‘‘πœ) 𝑑𝑠] ≀ ∫ 𝐺 (𝑏, 𝑠) (44)

𝑏

𝑠) πœ™π‘ž (∫

β‹… πœ™π‘ž (∫ 𝐻 (𝜏, 𝜏) 𝑓 (𝜏, 𝑒 (𝜏)) π‘‘πœ) 𝑑𝑠

𝜏∈𝐼

π‘Ž

σΈ€ 

𝑏

𝑏

π‘Ž

π‘Ž

β‹… πœ™π‘ (πœ‚1 (

σΈ€ 

≀ 𝐡𝑐 ∫ 𝐺 (𝑏, 𝑠) πœ™π‘ž (∫ 𝐻 (𝜏, 𝜏) π‘‘πœ) 𝑑𝑠 = 𝑐 . This shows that 𝑇 : 𝑃𝑐󸀠 β†’ 𝑃𝑐󸀠 . Using the same arguments as in the proof of Theorem 10, we can show that 𝑇 : 𝑃𝑐󸀠 β†’ 𝑃𝑐󸀠 is a completely continuous operator. It follows from conditions (𝐻3) and (𝐻4) in Theorem 11 that 𝑐󸀠 > 𝑏󸀠 ((𝑏 βˆ’ π‘Ž)/(πœ‚ βˆ’ π‘Ž))2(π›Όβˆ’1) > 𝑏󸀠 > π‘ŽσΈ€  . Similarly to the proof of Theorem 10, we have 𝑇 : π‘ƒπ‘ŽσΈ€  β†’ π‘ƒπ‘ŽσΈ€  and 𝑏 βˆ’ π‘Ž 2(π›Όβˆ’1) ) | πœƒ (𝑒) > 𝑏󸀠 } =ΜΈ 0, {𝑒 ∈ 𝑃 (πœƒ, 𝑏 , 𝑏 ( ) πœ‚βˆ’π‘Ž σΈ€ 

(45)

πœƒ (𝑇𝑒) > 𝑏󸀠 , for all 𝑒 ∈ 𝑃(πœƒ, 𝑏󸀠 , 𝑏󸀠 ((𝑏 βˆ’ π‘Ž)/(πœ‚ βˆ’ π‘Ž))2(π›Όβˆ’1) ). Moreover, for 𝑒 ∈ 𝑃(πœƒ, 𝑏󸀠 , 𝑐󸀠 ) and ‖𝑇𝑒‖ > 𝑏󸀠 ((𝑏 βˆ’ π‘Ž)/(πœ‚ βˆ’ π‘Ž))2(π›Όβˆ’1) , we have

𝑏 βˆ’ π‘Ž π›Όβˆ’1 > 𝑏󸀠 . >𝑏 ( ) πœ‚βˆ’π‘Ž

πœ‚ βˆ’ π‘Ž π›Όβˆ’1 ) ‖𝑒‖) π‘‘πœ) 𝑑𝑠 π‘βˆ’π‘Ž

𝑏 πœ‚ βˆ’ π‘Ž π›Όβˆ’1 ) ‖𝑒‖ ∫ 𝐺 (πœ‚, 𝑠) π‘βˆ’π‘Ž πœ‚

β‹… πœ™π‘ž (∫

𝜏∈𝐼

𝛾𝐻 (𝜏, 𝜏) π‘‘πœ) 𝑑𝑠 =

2πœ‚1 ‖𝑒‖ > ‖𝑒‖ , 𝐴

which implies ‖𝑇𝑒‖ > ‖𝑒‖ for 𝑒 ∈ πœ•Ξ©πœŽ1 . Hence, Theorem 8 implies 𝑖 (𝑇, Ω𝜎1 , 𝑃) = 0.

σΈ€ 

πœ‚ βˆ’ π‘Ž π›Όβˆ’1 πœƒ (𝑇𝑒) = min 𝑇𝑒 (𝑑) β‰₯ ( ) ‖𝑇𝑒‖ π‘‘βˆˆ[πœ‚,𝑏] π‘βˆ’π‘Ž

= πœ‚1 (

(47)

𝛾𝐻 (𝜏, 𝜏)

On the other hand, since π‘“βˆž = ∞, there exists 𝜎3 > πœ‡1 such that 𝑓(𝑑, 𝑒) β‰₯ πœ™π‘ (πœ‚2 𝑒), for 𝑒 β‰₯ 𝜎3 , where πœ‚2 ∈ (𝐴/2, ∞). Let 𝜎2 > max{𝜎3 ((𝑏 βˆ’ π‘Ž)/(πœ‚ βˆ’ π‘Ž))π›Όβˆ’1 , πœ‡1 } and Ω𝜎2 = {𝑒 ∈ 𝑃 | ‖𝑒‖ < 𝜎2 }. Then minπ‘‘βˆˆ[πœ‚,𝑏] 𝑒(𝑑) β‰₯ ((𝑏 βˆ’ π‘Ž)/(πœ‚ βˆ’ π‘Ž))π›Όβˆ’1 ‖𝑒‖ > 𝜎3 , for any 𝑒 ∈ πœ•Ξ©πœŽ2 . By using the method to get (48), we obtain 𝑇𝑒(πœ‚) > (2πœ‚1 /𝐴)‖𝑒‖ > ‖𝑒‖, which implies ‖𝑇𝑒‖ > ‖𝑒‖ for 𝑒 ∈ πœ•Ξ©2 . Thus, from Theorem 8, we have 𝑖 (𝑇, Ω𝜎2 , 𝑃) = 0.

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σΈ€ 

So all the conditions of Theorem 5 are satisfied. Thus using Theorem 5, 𝑇 has at least three fixed points. So, th boundary value problem (1)-(2) has at least three positive solutions 𝑒1 , 𝑒2 , and 𝑒3 with ‖𝑒1 β€– < π‘ŽσΈ€  , minπ‘‘βˆˆ[πœ‚,𝑏] 𝑒2 (𝑑) > 𝑏󸀠 , ‖𝑒3 β€–(𝑑) > π‘ŽσΈ€  , and minπ‘‘βˆˆ[πœ‚,𝑏] ‖𝑒3 β€–(𝑑) < 𝑏󸀠 .

(48)

(49)

Finally, let Ξ©πœ‡1 = {𝑒 ∈ 𝑃 | ‖𝑒‖ < πœ‡1 }. Then, for any 𝑒 ∈ πœ•Ξ©πœ‡1 , by (𝐻7), we then get 𝑏

𝑏

π‘Ž

π‘Ž

𝑇𝑒 (𝑑) = ∫ 𝐺 (𝑑, 𝑠) πœ™π‘ž (∫ 𝐻 (𝑠, 𝜏) 𝑓 (𝜏, 𝑒 (𝜏)) π‘‘πœ) 𝑑𝑠 𝑏

𝑏

π‘Ž

π‘Ž

𝑏

𝑏

π‘Ž

π‘Ž

≀ ∫ 𝐺 (𝑏, 𝑠) πœ™π‘ž (∫ 𝐻 (𝜏, 𝜏) 𝑓 (𝜏, 𝑒 (𝜏)) π‘‘πœ) 𝑑𝑠 < ∫ 𝐺 (𝑏, 𝑠) πœ™π‘ž (∫ 𝐻 (𝜏, 𝜏) πœ™π‘ (π΅πœ‡1 ) π‘‘πœ) 𝑑𝑠

8

International Journal of Differential Equations 𝑏

𝑏

π‘Ž

π‘Ž

Next, since π‘“βˆž = 0, there exists 𝛿3 > πœ‡2 such that 𝑓(𝑑, 𝑒) ≀ πœ™π‘ (πœ‚3 𝑒), for 𝑒 β‰₯ 𝛿3 , where πœ‚3 ∈ (0, 𝐡). We consider two cases.

= π΅πœ‡1 ∫ 𝐺 (𝑏, 𝑠) πœ™π‘ž (∫ 𝐻 (𝜏, 𝜏) π‘‘πœ) 𝑑𝑠 = πœ‡1 = ‖𝑒‖ , (50) which implies ‖𝑇𝑒‖ < ‖𝑒‖ for 𝑒 ∈ πœ•Ξ©πœ‡1 . Using Theorem 8 again, we get 𝑖 (𝑇, Ξ©πœ‡1 , 𝑃) = 1.

(51)

Note that 𝜎1 < πœ‡1 < 𝜎2 , by the additivity of fixed point index and (48)–(51); we obtain 𝑖 (𝑇, Ξ©πœ‡1 \ Ω𝜎1 , 𝑃) = 𝑖 (𝑇, Ξ©πœ‡1 , 𝑃) βˆ’ 𝑖 (𝑇, Ω𝜎1 , 𝑃) = 1, 𝑖 (𝑇, Ω𝜎2 \ Ξ©πœ‡1 , 𝑃) = 𝑖 (𝑇, Ω𝜎2 , 𝑃) βˆ’ 𝑖 (𝑇, Ξ©πœ‡1 , 𝑃)

(52)

= βˆ’1.

𝑇𝑒 (𝑑) 𝑏

𝑏

π‘Ž

π‘Ž

𝑏

𝑏

π‘Ž

π‘Ž

≀ ∫ 𝐺 (𝑏, 𝑠) πœ™π‘ž (∫ 𝐻 (𝜏, 𝜏) 𝑓 (𝜏, 𝑒 (𝜏)) π‘‘πœ) 𝑑𝑠 ≀ ∫ 𝐺 (𝑏, 𝑠) πœ™π‘ž (∫ 𝐻 (𝜏, 𝜏) πœ™π‘ (𝑁) π‘‘πœ) 𝑑𝑠 𝑏

𝑏

π‘Ž

π‘Ž

= 𝑁 ∫ 𝐺 (𝑏, 𝑠) πœ™π‘ž (∫ 𝐻 (𝜏, 𝜏) π‘‘πœ) 𝑑𝑠 =

(55)

𝑁 < 𝛿4 𝐡

= ‖𝑒‖ .

Hence, 𝑇 has a fixed point 𝑒1 in Ξ©πœ‡1 \ Ω𝜎1 , and it has a fixed point 𝑒2 in Ω𝜎2 \ Ξ©πœ‡1 . Clearly, 𝑒1 and 𝑒2 are positive solutions of boundary value problem (1)-(2) and 0 < ‖𝑒1 β€– < πœ‡1 < ‖𝑒2 β€–. Theorem 13. Let 𝑓(𝑑, 𝑒) be nonnegative continuous on [π‘Ž, 𝑏] Γ— [0, ∞). If the following assumptions are satisfied: (𝐻8) 𝑓0 = π‘“βˆž = 0; (𝐻9) there exists a constant πœ‡2 > 0 such that 𝑓(𝑑, 𝑒) > πœ™π‘ (π΄πœ‡2 ), for (𝑑, 𝑒) ∈ [πœ‚, 𝑏]Γ—[((πœ‚βˆ’π‘Ž)/(π‘βˆ’π‘Ž))π›Όβˆ’1 πœ‡2 , πœ‡2 ], then boundary value problem (1)-(2) has at least two positive solutions 𝑒1 and 𝑒2 such that 0 < ‖𝑒1 β€– < πœ‡2 < ‖𝑒2 β€–. Proof. From Lemma 9, we obtain 𝑇 : 𝑃 β†’ 𝑃 being completely continuous. In view of 𝑓0 = 0, there exists 𝛿1 ∈ (0, πœ‡2 ) such that 𝑓(𝑑, 𝑒) ≀ πœ™π‘ (πœ‚2 𝑒), for π‘Ž ≀ 𝑑 ≀ 𝑏, 0 < 𝑒 ≀ 𝛿1 , where πœ‚2 ∈ (0, 𝐡). Let Ω𝛿1 = {𝑒 ∈ 𝑃 | ‖𝑒‖ < 𝛿1 }. Then, for any 𝑒 ∈ πœ•Ξ©π›Ώ1 , we have

Case (ii). Suppose that 𝑓 is unbounded. In view of 𝑓 : [π‘Ž, 𝑏] Γ— [0, ∞) β†’ [0, ∞) being continuous, there exist 𝑑⋆ ∈ [π‘Ž, 𝑏] and 𝛿5 > max{((𝑏 βˆ’ π‘Ž)/(πœ‚ βˆ’ π‘Ž))π›Όβˆ’1 𝛿3 , πœ‡2 } such that 𝑓(𝑑, 𝑒) ≀ 𝑓(𝑑⋆ , 𝛿5 ), for π‘Ž ≀ 𝑑 ≀ 𝑏, 0 ≀ 𝑒 ≀ 𝛿5 . Then, for 𝑒 ∈ 𝑃 with ‖𝑒‖ = 𝛿5 , we obtain 𝑇𝑒 (𝑑) 𝑏

𝑏

π‘Ž

π‘Ž

𝑏

𝑏

π‘Ž

π‘Ž

𝑏

𝑏

π‘Ž

π‘Ž

≀ ∫ 𝐺 (𝑏, 𝑠) πœ™π‘ž (∫ 𝐻 (𝜏, 𝜏) 𝑓 (𝜏, 𝑒 (𝜏)) π‘‘πœ) 𝑑𝑠 ≀ ∫ 𝐺 (𝑏, 𝑠) πœ™π‘ž (∫ 𝐻 (𝜏, 𝜏) 𝑓 (𝑑⋆ , 𝛿5 ) π‘‘πœ) 𝑑𝑠 (56)

≀ ∫ 𝐺 (𝑏, 𝑠) πœ™π‘ž (∫ 𝐻 (𝜏, 𝜏) πœ™π‘ (πœ‚3 ‖𝑒‖) π‘‘πœ) 𝑑𝑠 𝑏

𝑏

π‘Ž

π‘Ž

≀ πœ‚3 𝛿5 ∫ 𝐺 (𝑏, 𝑠) πœ™π‘ž (∫ 𝐻 (𝜏, 𝜏) π‘‘πœ) 𝑑𝑠 =

πœ‚3 𝛿5 𝐡

< 𝛿5 = ‖𝑒‖ .

𝑇𝑒 (𝑑) 𝑏

𝑏

π‘Ž

π‘Ž

𝑏

𝑏

π‘Ž

π‘Ž

So, in either case, if we always choose Ω𝛿2 = {𝑒 ∈ 𝐸 | ‖𝑒‖ < 𝛿2 = max{𝛿4 , 𝛿5 }}, then we have ‖𝑇𝑒‖ < ‖𝑒‖, for 𝑒 ∈ πœ•Ξ©2 . Thus, from Theorem 8, we have

≀ ∫ 𝐺 (𝑏, 𝑠) πœ™π‘ž (∫ 𝐻 (𝜏, 𝜏) 𝑓 (𝜏, 𝑒 (𝜏)) π‘‘πœ) 𝑑𝑠 ≀ ∫ 𝐺 (𝑏, 𝑠) πœ™π‘ž (∫ 𝐻 (𝜏, 𝜏) πœ™π‘ (πœ‚2 𝑒) π‘‘πœ) 𝑑𝑠 𝑏

𝑏

π‘Ž

π‘Ž

(57)

Finally, let Ξ©πœ‡2 = {𝑒 ∈ 𝑃 | ‖𝑒‖ < πœ‡2 }. Then, for any 𝑒 ∈ πœ•Ξ©πœ‡2 , minπ‘‘βˆˆ[πœ‚,𝑏] 𝑒(𝑑) β‰₯ ((πœ‚βˆ’π‘Ž)/(π‘βˆ’π‘Ž))π›Όβˆ’1 ‖𝑒‖ = ((πœ‚βˆ’π‘Ž)/(π‘βˆ’π‘Ž))π›Όβˆ’1 πœ‡2 , by (𝐻9), and we then obtain

πœ‚2 ‖𝑒‖ < ‖𝑒‖ , 𝐡

𝑇𝑒 (πœ‚)

which implies ‖𝑇𝑒‖ < ‖𝑒‖ for 𝑒 ∈ πœ•Ξ©π›Ώ1 . Hence, Theorem 8 implies 𝑖 (𝑇, Ω𝛿1 , 𝑃) = 0.

𝑖 (𝑇, Ω𝛿2 , 𝑃) = 1.

(53)

≀ πœ‚2 ‖𝑒‖ ∫ 𝐺 (𝑏, 𝑠) πœ™π‘ž (∫ 𝐻 (𝜏, 𝜏) π‘‘πœ) 𝑑𝑠 =

Case (i). Suppose that 𝑓 is bounded, which implies that there exists 𝑁 > 0 such that 𝑓(𝑑, 𝑒) ≀ πœ™π‘ (𝑁) for all 𝑑 ∈ [π‘Ž, 𝑏] and 𝑒 ∈ [0, ∞). Take 𝛿4 > max{𝑁/𝐡, 𝛿3 }. Then, for 𝑒 ∈ 𝑃 with ‖𝑒‖ = 𝛿4 , we get

(54)

𝑏

𝑏

π‘Ž

π‘Ž

𝑏

𝑏

πœ‚

π‘Ž

= ∫ 𝐺 (πœ‚, 𝑠) πœ™π‘ž (∫ 𝐻 (𝑠, 𝜏) 𝑓 (𝜏, 𝑒 (𝜏)) π‘‘πœ) 𝑑𝑠 β‰₯ ∫ 𝐺 (πœ‚, 𝑠) πœ™π‘ž (∫ 𝐻 (𝑠, 𝜏) πœ™π‘ (π΄πœ‡2 ) π‘‘πœ) 𝑑𝑠

International Journal of Differential Equations 𝑏

β‰₯ π΄πœ‡2 ∫ 𝐺 (πœ‚, 𝑠) πœ™π‘ž (∫ πœ‚

𝜏∈𝐼

9 (ii) 𝑓(𝑑, 𝑒(𝑑)) β‰₯ 68.364 = πœ™π‘ (𝐴𝑏󸀠 ) for all (𝑑, 𝑒) ∈ [0.5, 2] Γ— [2, 16.0049],

𝛾𝐻 (𝜏, 𝜏) π‘‘πœ) 𝑑𝑠

(iii) 𝑓(𝑑, 𝑒(𝑑)) ≀ 104.52 = πœ™π‘ (𝐡𝑐󸀠 ) for all (𝑑, 𝑒) ∈ [0, 1] Γ— [0, 100].

𝑏 βˆ’ π‘Ž π›Όβˆ’1 ) πœ‡2 > πœ‡2 = ‖𝑒‖ , = 2( πœ‚βˆ’π‘Ž (58) which implies ‖𝑇𝑒‖ > ‖𝑒‖ for 𝑒 ∈ πœ•Ξ©πœ‡2 . An application of Theorem 8 again shows that 𝑖 (𝑇, Ξ©πœ‡2 , 𝑃) = 0.

Consequently, all of the conditions of Theorem 11 are satisfied. With the use of Theorem 11, boundary value problem (61) has at least three positive solutions 𝑒1 , 𝑒2 , and 𝑒3 with σ΅„©σ΅„© σ΅„©σ΅„© 󡄩󡄩𝑒1 σ΅„©σ΅„© < 1,

(59)

min 𝑒2 (𝑑) > 2,

π‘‘βˆˆ[1/2,1]

Note that 𝛿1 < πœ‡2 < 𝛿2 by the additivity of fixed point index and (54)–(59); we obtain

σ΅„©σ΅„© σ΅„©σ΅„© 󡄩󡄩𝑒3 σ΅„©σ΅„© > 1,

𝑖 (𝑇, Ξ©πœ‡2 \ Ω𝛿1 , 𝑃) = 𝑖 (𝑇, Ξ©πœ‡2 , 𝑃) βˆ’ 𝑖 (𝑇, Ω𝛿1 , 𝑃) = βˆ’1,

min 𝑒3 (𝑑) < 2.

π‘‘βˆˆ[1/2,1]

(60)

Competing Interests

𝑖 (𝑇, Ω𝛿2 \ Ξ©πœ‡2 , 𝑃) = 𝑖 (𝑇, Ω𝛿2 , 𝑃) βˆ’ 𝑖 (𝑇, Ξ©πœ‡2 , 𝑃) = 1. Hence, 𝑇 has a fixed point 𝑒1 in Ξ©πœ‡2 \ Ω𝛿1 , and it has a fixed point 𝑒2 in Ω𝛿2 \ Ξ©πœ‡2 . Consequently, 𝑒1 and 𝑒2 are positive solutions of boundary value problem (1)-(2) and 0 < ‖𝑒1 β€– < πœ‡2 < ‖𝑒2 β€–.

4. Example In this section, we consider boundary value problem of the fractional differential equation 2.5 𝐷01.5 + (πœ™π‘ (𝐷0+ 𝑒 (𝑑))) = 𝑓 (𝑑, 𝑒 (𝑑)) ,

σΈ€ 

𝑒 (0) = 0, 1 1 𝑒󸀠 (1) = 𝑒󸀠 ( ) , 3 2

(61)

The author expresses his gratitude to his guide Professor K. Rajendra Prsasd.

References

[2] D. R. Anderson and J. M. Davis, β€œMultiple solutions and eigenvalues for third-order right focal boundary value problems,” Journal of Mathematical Analysis and Applications, vol. 267, no. 1, pp. 135–157, 2002.

[4] Y. Sun, Q. Sun, and X. Zhang, β€œExistence and nonexistence of positive solutions for a higher-order three-point boundary value problem,” Abstract and Applied Analysis, vol. 2014, Article ID 513051, 7 pages, 2014.

= 0,

where 𝑑 { 0 ≀ 𝑒 ≀ 2, + 12𝑒5 , { { 100 𝑓 (𝑑, 𝑒) = { { { 𝑑 + 𝑒 + 58, 𝑒 β‰₯ 2. { 100

Acknowledgments

[3] R. W. Leggett and L. R. Williams, β€œMultiple positive fixed points of nonlinear operators on ordered Banach spaces,” Indiana University Mathematics Journal, vol. 28, no. 4, pp. 673–688, 1979.

πœ™π‘ (𝐷02.5 + 𝑒 (0)) = 0, (πœ™π‘ (𝐷02.5 + 𝑒 (1)))

The author declares that he has no competing interests regarding the publication of this paper.

[1] R. P. Agarwal, D. O’Regan, and P. J. Wong, Positive Solutions of Differential, Difference and Integral Equations, Kluwer Academic, Dordrecht, The Netherlands, 1999.

0 < 𝑑 < 1,

𝑒 (0) = 0,

𝐷00.5 +

(63)

(62)

Let 𝑝 = 1/2. We note that π‘Ž = 0, 𝑏 = 1, πœ‚ = 1/2, πœ‰ = 1/3, 𝛽 = 3/2, 𝛽1 = 1/2, 𝑛 = 3, 𝛼 = 5/2, and 𝛼1 = 1. By a simple calculation, we obtain ((πœ‚ βˆ’ π‘Ž)/(𝑏 βˆ’ π‘Ž))π›Όβˆ’1 = 0.3535, 𝛾 = 0.5, 𝐡 = 1.0452, and 𝐴 = 34.1482. Choosing π‘ŽσΈ€  = 1, 𝑏󸀠 = 2, and 𝑐󸀠 = 100, then 0 < π‘ŽσΈ€  < 𝑏󸀠 < ((πœ‚ βˆ’ π‘Ž)/(𝑏 βˆ’ π‘Ž))2(π›Όβˆ’1) 𝑐󸀠 and 𝑓 satisfies (i) 𝑓(𝑑, 𝑒(𝑑)) < 1.0452 = πœ™π‘ (π΅π‘ŽσΈ€  ) for all (𝑑, 𝑒) ∈ [0, 1] Γ— [0, 1],

[5] S. Nageswararao, β€œSolvability of second order delta-nabla pLaplacian m-point eigenvalue problem on time scales,” TWMS Journal of Applied and Engineering Mathematics, vol. 5, no. 1, pp. 98–109, 2015. [6] M. Ramaswamy and R. Shivaji, β€œMultiple positive solutions for classes of p-Laplacian equations,” Differential and Integral Equations, vol. 17, no. 11-12, pp. 1255–1261, 2004. [7] C. Yang and J. Yan, β€œPositive solutions for third-order SturmLiouville boundary value problems with p-Laplacian,” Computers & Mathematics with Applications, vol. 59, no. 6, pp. 2059– 2066, 2010. [8] C. Zhai and C. Guo, β€œPositive solutions for third-order SturmLiouville boundary-value problems with p-Laplacian,” Electronic Journal of Differential Equations, vol. 2009, no. 154, pp. 1–9, 2009.

10 [9] S. Liang and J. Zhang, β€œPositive solutions for boundary value problems of nonlinear fractional differential equation,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 11, pp. 5545–5550, 2009. [10] S. Nageswararao, β€œMultiple positive solutions for a system of Riemann-Liouville fractional order two-point boundary value problems,” Panamerican Mathematical Journal, vol. 25, no. 1, pp. 66–81, 2015. [11] S. Nageswararao, β€œExistence and multiplicity for a system of fractional higher-order two-pointboundary value problem,” Journal of Applied Mathematics and Computing, vol. 53, pp. 93– 107, 2016. [12] Y. Zhao, S. Sun, Z. Han, and M. Zhang, β€œPositive solutions for boundary value problems of nonlinear fractional differential equations,” Applied Mathematics and Computation, vol. 217, no. 16, pp. 6950–6958, 2011. [13] G. Chai, β€œPositive solutions for boundary value problem of fractional Differential equation with p-Laplacian operator,” Boundary Value Problems, vol. 2012, article 18, 2012. [14] Z. Han, H. Lu, S. Sun, and D. Yang, β€œPositive solutions to boundary-value problems of p-Laplacian fractional differential equations with a parameter in the boundary conditions,” Electronic Journal of Differential Equations, vol. 2012, no. 213, pp. 1– 14, 2012. [15] H. Lu, Z. Han, and S. Sun, β€œMultiplicity of positive solutions for Sturm-Liouville boundary value problems of fractional differential equations with p-Laplacian,” Boundary Value Problems, vol. 2014, article 26, 2014. [16] C. T. Ledesma, β€œBoundary value problem with fractional pLaplacian operator,” Advances in Nonlinear Analysis, 2015. [17] Y. Li and A. Qi, β€œPositive solutions for multi-point boundary value problems of fractional differential equations with pLaplacian,” Mathematical Methods in the Applied Sciences, vol. 39, no. 6, pp. 1425–1434, 2016. [18] K. R. Prasad and B. M. B. Krushna, β€œExistence of multiple positive solutions for p-Laplacian fractional order boundary value problems,” International Journal of Analysis and Applications, vol. 6, no. 1, pp. 63–81, 2014. [19] B. Zhang, β€œSolvability of fractional boundary value problems with p-Laplacian operator,” Advances in Difference Equations, vol. 2015, article 352, 2015. [20] A. Kameswararao and S. Nageswararao, β€œEigenvalue problems for systems of nonlinear higher order boundary value problems,” Journal of Applied Mathematics and Informatics, vol. 28, no. 3-4, pp. 711–721, 2010. [21] D. J. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, Orlando, Fla, USA, 1988. [22] K. Deimling, Nonlinear Functional Analysis, Springer, New York, NY, USA, 1985.

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