DDNS Discrete Dynamics in Nature and Society 1607-887X 1026-0226 Hindawi Publishing Corporation 10.1155/2015/418410 418410 Research Article Positive Solutions for a Class of Fourth-Order p -Laplacian Boundary Value Problem Involving Integral Conditions http://orcid.org/0000-0003-2105-5775 Sun Yan Diblík Josef Department of Mathematics Shanghai Normal University Shanghai 200234 China shnu.edu.cn 2015 2682015 2015 30 04 2015 19 07 2015 27 07 2015 2682015 2015 Copyright © 2015 Yan Sun. 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.

Under some conditions concerning the first eigenvalues corresponding to the relevant linear operator, we obtain sharp optimal criteria for the existence of positive solutions for p -Laplacian problems with integral boundary conditions. The main methods in the paper are constructing an available integral operator and combining fixed point index theory. The interesting point of the results is that the nonlinear term contains all lower-order derivatives explicitly. Finally, we give some examples to demonstrate the main results.

1. Introduction

In this paper, we discuss the existence of positive solutions for the following p -Laplacian integral boundary value problems:(1)ϕpyt=atft,yt,yt,yt,yt,0<t<1,y0=y1=01bsysds,y0=y1=ϕq01csϕpysds,where ϕ p ( t ) = | t | p - 2 t , p > 1 , ϕ q = ϕ p - 1 , 1 / p + 1 / q = 1 , a L 1 [ 0,1 ] is symmetric on the interval [ 0,1 ] (i.e., a ( 1 - t ) = a ( t ) and a ( t ) 0 for t [ 0,1 ] ), f : [ 0,1 ] × [ 0 , + ) × ( - , + ) × ( - , + ) × ( - , + ) [ 0 , + ) is continuous, and b , c L 1 [ 0,1 ] are nonnegative symmetric on [ 0,1 ] .

Boundary value problems of ordinary differential equations have become an important research field in recent years. Fourth-order p -Laplacian boundary value problems arise in applied mathematics, physics, gas diffusion through porous media, engineering, elastic mechanics, electromagnetic waves of gravity driven flows, and the various areas of adiabatic tubular reactor processes, as well as biological problems; see  and the references therein.

In , by using the fixed point theorem for strict set contraction operator, Zhang et al. considered the existence of positive solutions for the following n th-order impulsive boundary value problems with integral boundary conditions in Banach spaces:(2)znt+gt,zt,zt,,zn-2t=θ,tJ,ttk,k=1,,m,Δzn-1t=tk=-Ikzn-2tk,k=1,,m,zi0=θ,i=0,1,,n-3,zn-20=zn-21=01htzn-2tdt,where J = [ 0,1 ] , 0 < t 1 < t 2 < < t k < < t m < 1 , g C [ J × P n - 1 , P ] , I k C [ P , P ] , k = 1 , , m , h L 1 [ 0,1 ] , and Δ z ( n - 1 ) t = t k = z ( n - 1 ) ( t k + ) - z ( n - 1 ) ( t k - ) denotes the jump of z ( n - 1 ) ( t ) at t = t k . They also gave the upper and lower bounds for these positive solutions.

In , by means of the fixed point index theory, Sun and Zhang studied the following singular nonlinear Sturm-Liouville problems:(3)-Lϕx=axgϕx,0<x<1,R1ϕ=α1ϕ0+β1ϕ0=0,R2ϕ=α2ϕ1+β2ϕ1=0,-Lϕx=fx,ϕx,0<x<1,R1ϕ=α1ϕ0+β1pxϕ0=0,R2ϕ=α2ϕ1+β2pxϕ1=0,where ( L ϕ ) ( x ) = p x ϕ x + q ( x ) ϕ ( x ) and a ( x ) is allowed to be singular at x = 0 and x = 1 .

In , by making use of the theory of the spectrum, Sergejeva studied the regions of solvability for three-point boundary value problem:(4)-yt=μy+-λy-+ht,y,y,t0,1,y0=0,y1=cy12,where y + = max { y , 0 } , y - = max { - y , 0 } , and h is a bounded function. The solvability results are established for the problem with h 0 .

Motivated and inspired greatly by the above mentioned works, the aim of the paper is to improve, generalize, and supplement the previous results. We obtain the existence results on nonlinear fourth-order integral boundary value problems with one-dimensional p -Laplacian integral boundary value problem (1). The novelty of our results is in exploring some optimal criteria for the existence of positive solutions of problem (1). The methods used in our work will depend on an application of fixed point index theory together with the first eigenvalues corresponding to the relevant linear operator. It can also be seen that the nonlinear term involves all lower-order derivatives explicitly.

The paper is organized as follows. In Section 2, we state some lemmas and several preliminary results. The main results are formulated and proved in Section 3. Examples are given in Section 4.

2. Preliminaries

Throughout the paper, we make the following assumptions:

(H 1) f C ( [ 0,1 ] × [ 0 , + ) × ( - , + ) × ( - , + ) × ( - , + ) , [ 0 , + ) ) and h ( t , y ) f t , y , y , y , y g ( t , y ) , h , g : [ 0,1 ] × [ 0 , + ) [ 0 , + ) , are continuous and also h ( · , y ) and g ( · , y ) are symmetric on [ 0,1 ] for all y 0 (i.e., h ( 1 - t , y ) = h ( t , y ) , g ( 1 - t , y ) = g ( t , y ) ).

(H 2) a , b , c L 1 [ 0,1 ] are all nonnegative and symmetric on [ 0,1 ] , and a ( t ) 0 is continuous on ( 0,1 ) , and a ( t ) may be singular at t = 0 and/or t = 1 , and μ ( 0,1 ) and ν ( 0,1 ) , where(5)μ=01bsds,ν=01csds.

In what follows, we will consider the Banach space E = C [ 0,1 ] , C + [ 0,1 ] = { x C [ 0,1 ] x ( t ) 0 } , equipped with the maximum norm z = max 0 t 1 z t . Denote cone P by(6)P=zC+0,1zt  is  concave  for  t0,1;then, P is a positive cone in C [ 0,1 ] .

Let B r = z C + [ 0,1 ] z < r for r ( r > 0 ), and(7)Ht,s=Gt,s+11-μ01Gs,τbτdτ,H1t,s=Gt,s+11-ν01Gs,υcυdυ,Gt,s=s1-t,0st1,t1-s,0ts1.Here G ( t , s ) , H ( t , s ) , and H 1 ( t , s ) are Green’s functions of problem (1).

Lemma 1 (see [<xref ref-type="bibr" rid="B10">1</xref>]).

Suppose that (H 1) and (H 2) hold; then, for t , s [ 0,1 ] , we have

G ( t , s ) 0 , H ( t , s ) 0 , H 1 ( t , s ) 0 ;

e ( t ) e ( s ) G ( t , s ) e ( t ) 1 / 4 , where e ( t ) = t ( 1 - t ) ;

G ( 1 - t , 1 - s ) = G ( t , s ) , H ( 1 - t , 1 - s ) = H ( t , s ) , and H 1 ( 1 - t , 1 - s ) = H 1 ( t , s ) ;

ρ e ( s ) H ( t , s ) γ s ( 1 - s ) = γ e ( s ) 1 / 4 γ ; ρ 1 e ( s ) H 1 ( t , s ) γ 1 s ( 1 - s ) = γ 1 e ( s ) 1 / 4 γ 1 , where γ = 1 / 1 - μ , ρ = 0 1 e ( τ ) b ( τ ) d τ / 1 - μ , γ 1 = 1 / 1 - ν , ρ 1 = 0 1 e ( τ ) c ( τ ) d τ / 1 - ν .

Define an integral operator A : C + [ 0,1 ] C + [ 0,1 ] by (8) A y t = 0 1 H t , s ϕ q 0 1 H 1 s , τ a τ · f τ , y τ , y τ , y τ , y τ d τ d s .

It is well known that y C + [ 0,1 ] is a solution of problem (1) if and only if y is a fixed point of operator equation y = A y .

Lemma 2.

Suppose that (H 1) and (H 2) hold, if y E is a solution of the following integral equation: (9) y t = 0 1 H t , s ϕ q 0 1 H 1 s , τ a τ · f τ , y τ , y τ , y τ , y τ d τ d s ; then, y is a solution of problem (1).

Lemma 3.

Suppose that (H 1) and (H 2) hold; then, A : P P is a completely continuous operator.

Proof.

First, we prove that A : P P . For every y P , we have (10) A y t = - ϕ q 0 1 H 1 t , s a s · f s , y s , y s , y s , y s d s 0 , A y 0 = 0 1 H 0 , s ϕ q 0 1 H 1 s , τ a τ · f τ , y τ , y τ , y τ , y τ d τ d s 0 , A y 1 = 0 1 H 1 , s ϕ q 0 1 H 1 s , τ a τ · f τ , y τ , y τ , y τ , y τ d τ d s 0 . Thus, ( A y ) ( t ) 0 , for all t [ 0,1 ] . Hence, ( A y ) ( t ) is nonnegative concave functional on [ 0,1 ] .

By Lemma 1, for t [ 0,1 ] , we obtain (11) A y t = 0 1 H t , s ϕ q 0 1 H 1 s , τ a τ · f τ , y τ , y τ , y τ , y τ d τ d s 0 1 H t , s ϕ q 0 1 H 1 s , τ a τ · g τ , y τ d τ d s γ γ 1 q - 1 0 1 e s · ϕ q 0 1 e τ a τ g τ , y τ d τ d s < + . Hence, A y P , and A ( P ) P , which implies A : P P .

Next, we show that A : P P is completely continuous.

For any natural number n 2 , set (12) a n t = inf t s 1 / n a s , 0 t 1 n , a t , 1 n t n - 1 n , inf n - 1 / n s t a s , n - 1 n t 1 . Then, a n : [ 0,1 ] [ 0 , + ) is continuous, and a n ( t ) a ( t ) , t [ 0,1 ] . Let (13) A n y t = 0 1 H t , s ϕ q 0 1 H 1 s , τ a n τ · f τ , y τ , y τ , y τ , y τ d τ d s .

Obviously, A n : P P is a completely continuous operator. From Lemma 1, we obtain(14)limnAny-Aylimnmax0t101Ht,s·ϕq01H1s,τaτ-anτgτ,yτdτdsγγ1q-1limn01es·ϕq01eτaτgτ,yτdτds=γγ1q-1·limnenesϕq01eτaτgτ,yτdτds=0,where e ( n ) = [ 0 , 1 / n ] [ n - 1 / n , 1 ] . Therefore, A : P P is a completely continuous operator. The proof is completed.

Now we define linear integral operator as follows:(15)Tyt=01Ht,sϕq01H1s,τaτyτdτds,L=maxt0,101Ht,sϕq01H1s,τaτdτds.

It is easy to see that T : C + [ 0,1 ] C + [ 0,1 ] is a completely continuous linear operator, and T ( P ) P .

Lemma 4.

Assume that (H 1) and (H 2) hold. Then, the operator T : P P defined by (15) is a completely continuous linear operator and T ( P ) P , the spectral radius r ( T ) 0 , and T has a positive eigenfunction corresponding to its first eigenvalue λ 1 = r - 1 ( T ) .

Proof.

A simple modification of the argument in Lemma 3 yields that T : P P is a completely continuous linear operator and T ( P ) P . By (H 1) and (H 2), there is s 0 ( 0,1 ) such that H 1 ( s 0 , s 0 ) a ( s 0 ) > 0 . Choose constants u and v such that s 0 ( u , v ) [ u , v ] ( 0,1 ) and H 1 ( s , s ) a ( s ) > 0 , for all t , s [ u , v ] . Choose a nonnegative continuous functional g C [ 0,1 ] such that g ( t ) > 0 , for all t ( u , v ) . Then, for any t ( u , v ) , we have(16)Tgt=01Ht,sϕq01H1s,τaτgτdτdsuvHt,sϕquvH1s,τaτgτdτds0.

Thus, there exists a constant w > 0 such that w ( T g ) ( t ) g ( t ) , for all t [ 0,1 ] . By famous Krein-Rutman theorems, we know that the spectral radius r ( T ) 0 and T has a positive eigenfunction ψ corresponding to its first eigenvalue λ 1 = r - 1 ( T ) ; that is, ψ = λ 1 T ψ . This completes the proof.

Lemma 5 (see [<xref ref-type="bibr" rid="B3">8</xref>]).

Let P be a cone in Banach space E , and let Ω ( P ) be a bounded open set in P . Suppose that A : Ω ¯ ( P ) P is completely continuous. If there exists y 0 P θ , such that y - A y μ y 0 , for all y Ω ( P ) , μ 0 . Then, the fixed point index i ( A , Ω ( P ) , P ) = 0 .

Lemma 6 (see [<xref ref-type="bibr" rid="B3">8</xref>]).

Let P be a cone in Banach space E , and let Ω ( P ) be a bounded open set in P with θ Ω ( P ) . Suppose that A : Ω ¯ ( P ) P is a completely continuous operator. If A y μ y , for all y Ω ( P ) , μ 1 . Then, the fixed point index i ( A , Ω ( P ) , P ) = 1 .

Lemma 7 (see [<xref ref-type="bibr" rid="B2">9</xref>]).

Let P and P 1 be cones in Banach space E , and let Ω ( P ) be a bounded open set in P . Suppose that A : Ω ¯ ( P ) P is a completely continuous operator, and A has no fixed point on Ω ( P ) . If there exist linear operators L 1 , L 2 , L 3 : E E which satisfy L 1 ( P ) P , L 2 ( P ) P 1 , L 3 ( P 1 ) P 1 , and y 0 P θ , such that

L 2 L 1 n y 0 L 2 y 0 , for certain natural number n ,

L 2 L 1 y = L 3 L 2 y , for all y P ,

L 2 A y L 2 L 1 y , for all y Ω P ,

then the fixed point index i ( A , Ω ( P ) , P ) = 0 .

3. Main Results

In this section, we establish sharp optimal criteria for the existence of positive solutions to problem (1) under superlinear cases and sublinear cases, respectively.

Theorem 8.

Suppose that (H 1) and (H 2) hold. In addition, assume that(17)liminfy+ht,yty>λ1,uniformlyont0,1,(18)limsupy0+gt,yty<λ1,uniformlyont0,1,where λ 1 is the first eigenvalue of T defined by (15); then, problem (1) has at least one positive solution.

To prove our main result, we need some preliminary results.

Lemma 9.

Suppose that (H 1) and (H 2) hold. Define(19)P1=yP01y~tytdtλ1-1δy,where y ~ ( t ) = y ( t ) a ( t ) , y P , is the positive eigenfunction corresponding to its first eigenvalue λ 1 = r - 1 ( T ) ; that is, y = λ 1 T y , and(20)δ01ytatytdt01Ht,sϕq01H1s,τaτyτdτds;then, P 1 is a cone in C + [ 0,1 ] , and T ( P ) P 1 .

Proof.

It is obvious that P 1 is a cone in C + [ 0,1 ] . For any y P , we obtain(21)01y~tTytdt=01y~tdt·01Ht,sϕq01H1s,τaτyτdτds=λ1-101y~tytdt=λ1-101ytatytdtλ1-1δTy.Thus, T ( P ) P 1 . The proof is completed.

Proof of Theorem <xref ref-type="statement" rid="thm3.1">8</xref>.

It follows from (17) that there exists ɛ > 0 , such that h ( t , y ( t ) ) ( λ 1 + ɛ ) y , for y is sufficiently large. In view of (H 1), we see that there exists b 0 such that(22)ht,ytλ1+ɛy-b,0y<+.Let R > L b λ 1 2 ( ɛ δ ) - 1 0 1 y ~ ( t ) d t , where y ~ and δ are defined by Lemma 9.

In the following, we prove that(23)y-Ayμy,yBRP,μ0,where y P is a positive eigenfunction corresponding to its first eigenvalue λ 1 = r - 1 ( T ) .

Otherwise, there exist y 1 B R P and μ 0 0 , such that(24)y1-Ay1=μ0y.Since A ( P ) P 1 and T ( P ) P 1 , we know from (24) that y 1 P 1 . Therefore, it follows from (19) and (22) that(25)01y~tAy1tdt-01y~ty1tdt01dt·01y~tHt,s·ϕq01H1s,τaτhτ,y1τdτds-01y~ty1tdtλ1+ɛ01dt01y~t·Ht,sϕq01H1s,τaτy1τdτds-b01dt01y~tHt,s·ϕq01H1s,τaτdτds-01y~t·y1tdt=λ1+ɛλ1-101y~ty1tdt-bL01y~tdt-01y~ty1tdt=ɛλ1-101y~ty1tdt-bL01y~tdtɛλ1-1λ1-1δy1-bL01y~tdt=ɛλ1-2δR-bL01y~tdt>0.On the other hand, it follows from (24) that(26)01y~ty1tdt-01y~tAy1tdt=μ001y~tytdt0,which is a contradiction; thus, by Lemma 5, we see(27)iA,BRP,P=0.

It follows from (18) that there exists 0 r R , such that(28)gt,yt<λ1y,0yr.Suppose that there exist y 2 B r P , μ 1 1 , such that A y 2 = μ 1 y 2 . Without loss of generality, we may suppose that μ 1 > 1 (otherwise, the proof is completed). Then,(29)μ1y2t01Ht,s·ϕq01H1s,τaτgτ,y2τdτds<λ101Ht,s·ϕq01H1s,τaτy2τdτds.Multiplying y ~ and then integrating by (29), we have(30)μ101y~ty2tdt<λ101dt01y~tHt,s·ϕq01H1s,τaτy2τdτds=λ101dt·01Ht,sy~t·ϕq01H1s,τaτy2τdτds=01y~t·y2tdt.By maximum principle, y ( t ) > 0 and y 2 t > 0 , t 0,1 ; then 0 1 y ~ ( t ) y 2 ( t ) d t > 0 . Hence, (30) implies μ 1 < 1 which is a contradiction. So by Lemma 6, we have(31)iA,BrP,P=1.It follows from (27) and (31) that(32)iA,BRPB¯rP,P=iA,BRP,P-iA,BrP,P=0-1=-1.Then, A has at least one fixed point on ( B R P ) ( B ¯ r P ) . This means problem (1) has at least one positive solution. The proof is completed.

Corollary 10.

Suppose that (H 1) and (H 2) hold. In addition, assume that(33)0limsupy0+gt,yty=g0<h=liminfy+ht,yty+.Then, for any(34)ξλ1h,λ1g0,where λ 1 is the first eigenvalue of T defined by (15), the following p -Laplacian integral boundary value problem(35)ϕpyt=ξatft,yt,yt,yt,yt,0<t<1,y0=y1=01bsysds,ϕpy0=ϕpy1=01csϕpysdshas at least one positive solution.

Proof.

By (34), we obtain(36)liminfy+ξht,yty>λ1,limsupy0+ξgt,yty<λ1.Therefore, it follows from Theorem 8 that Corollary 10 holds.

Theorem 11.

Suppose that (H 1) and (H 2) hold. In addition, assume that(37)liminfy0+ht,yty>λ1,uniformlyont0,1;(38)limsupy+gt,yty<λ1,uniformlyont0,1,where λ 1 is the first eigenvalue of T defined by (15); then, problem (1) has at least one positive solution.

Proof.

It follows from (37) that there exists r 1 > 0 such that(39)gt,ytλ1y,0yr1.Let y P be the positive eigenfunction corresponding to its first eigenvalue λ 1 = r - 1 ( T ) . Then, y = λ 1 T y ; that is,(40)yt=λ1Tyt=λ101Ht,sϕq01H1s,τaτyτdτds,yt=λ1Tyt=-λ1ϕq01H1t,sasysds0,t0,1,y0=λ101H0,sϕq01H1s,τaτyτdτds0,y1=λ101H1,sϕq01H1s,τaτyτdτds0.Hence, y ( t ) 0 , for all t [ 0,1 ] . Then, y ( t ) is nonnegative concave on [ 0,1 ] . Thus, y P θ .

Let ( T 2 y ) ( t ) = λ 1 ( T y ) ( t ) , y C + [ 0,1 ] . Then, T 2 : C + [ 0,1 ] C + [ 0,1 ] is a completely continuous linear operator, and T 2 ( P 1 ) P , T 2 y = λ 1 T y = y .

For any y B r 1 P , it follows from (39) that(41)Ayt01Ht,sϕq01H1s,τaτhτ,yτdτdsλ101Ht,sϕq01H1s,τaτyτdτds=T2yt,t0,1.

Setting Ω ( P ) = B r 1 P , L 1 = L 3 = T 2 , L 2 = I denotes the identical operator, and n = 1 , in Lemma 7. We have(42)iA,Br1P,P=0.

It follows from (38) that there exist r 2 > r 1 and 0 < σ < 1 , such that(43)gt,ytσλ1y,yr2.

Set ( T 1 y ) ( t ) = σ λ 1 ( T y ) ( t ) . Then, T 1 : C + [ 0,1 ] C + [ 0,1 ] is a completely continuous linear operator, and T 1 ( P 1 ) P . Let r 4 > r 2 > r 1 .

Now we prove that i ( A , B r 4 P , P ) = 1 .

Suppose that there exist y 2 B r 4 P , μ 0 1 , such that A y 2 = μ 0 y 2 . Let e y 2 = t 0,1 y 2 t > r 2 ; then, from (43) we have(44)Ay2t01Ht,s·ϕq01H1s,τaτgτ,y2τdτds=ey2Ht,s·ϕq01H1s,τaτgτ,y2τdτds+0,1ey2Ht,s·ϕq01H1s,τaτgτ,y2τdτdsσλ101Ht,s·ϕq01H1s,τaτy2τdτds+01Ht,s·ϕq01H1s,τaτgτ,y2τdτdsT1y2t+M,where M = sup y B r 2 P 0 1 H ( t , s ) ϕ q [ 0 1 H 1 ( s , τ ) a ( τ ) g ( τ , y 2 τ ) d τ ] d s . Thus, 0 μ 0 y 2 t = A y 2 t T 1 y 2 t + M , t [ 0,1 ] . Since T 1 ( C + [ 0,1 ] ) C + [ 0,1 ] , we have 0 ( T 1 j ( A y 2 ) ) ( t ) ( T 1 j ( T 1 y 2 + M ) ) ( t ) , t [ 0,1 ] , for j = 1 , , N - 1 . It means that T 1 j ( A y 2 ) T 1 j ( T 1 y 2 + M ) , j = 1 , , N - 1 . Let(45)ɛ=121-rT1.For any y C + [ 0,1 ] , denote(46)y=i=1NrT1+ɛN-iT1i-1y.It is easy to see that(47)μ0y2<rT1+3ɛ2y2.Since μ 0 1 , we have from (47) that 1 r ( T 1 ) + 3 ɛ / 2 , which contradicts (45). Thus, for any y B r 4 P , μ 1 , we have that A y μ y . Thus, from Lemma 6, we know that(48)iA,Br4P,P=1.

From (42) and (48), we see that(49)iA,Br4PB¯r1P,P=iA,Br4P,P-iA,Br1P,P=1-0=1.Then, A has at least one fixed point on ( B r 4 P ) ( B ¯ r 1 P ) . It means problem (1) has at least one positive solution. The proof is completed.

Corollary 12.

Suppose that (H 1) and (H 2) hold. In addition, assume that(50)0g=limsupy+gt,ytyliminfy0+gt,yty=g0+.Then, for any(51)ξλ1h0,λ1g,where λ 1 is the first eigenvalue of T defined by (15), problem (35) has at least one positive solution.

Proof.

By (51), we obtain(52)liminfy0+ξht,yty>λ1,limsupy+ξgt,yty<λ1.It follows from Theorem 11 that Corollary 12 holds. The proof is completed.

4. Examples Example 1.

Consider the existence of positive solutions for the following p -Laplacian integral boundary value problems:(53)ϕpyt=1+tπ-1y2+52016ln1+y,0<t<1,y0=y1=01bsysds,y0=y1=ϕq01csϕpysds,where f t , y t , y t , y t , y t = ( 1 + t ) ( π - 1 ) y 2 + 5 / 2016 ln ( 1 + y ) . ϕ p ( t ) = | t | p - 2 t , p > 1 , ϕ q = ϕ p - 1 , 1 / p + 1 / q = 1 , a L 1 [ 0,1 ] is symmetric on the interval [ 0,1 ] , a ( t ) = 1 for t [ 0,1 ] is continuous, and b , c L 1 [ 0,1 ] are nonnegative symmetric on [ 0,1 ] .

Let h ( t , y ) = 5 / 2016 ln ( 1 + y ) + ( π - 1 ) y 2 and let g ( t , y ) = ( 1 + t ) ( π - 1 ) y 2 + 5 / 2016 ln ( 1 + y ) . Obviously, h ( t , y ) f t , y t , y t , y t , y t g ( t , y ) , and(54)liminfy+ht,yty=+,uniformly  on  t0,1,limsupy0+gt,yty=52016,uniformly  on  t0,1.

All conditions of Theorem 8 are satisfied. Consequently, following from Theorem 8, we know that the boundary value problem (53) has at least one positive solution.

Example 2.

Consider the existence of positive solutions for the following p -Laplacian integral boundary value problems:(55)ϕpyt=y3+expyt-yt-yt-yt-1-6siny-sin6yt+yt+yt+yt-sinπt1-tπt1-t,0<t<1,y0=y1=01bsysds,y0=y1=ϕq01csϕpysds,where ϕ p ( t ) = | t | p - 2 t , p > 1 , ϕ q = ϕ p - 1 , 1 / p + 1 / q = 1 , a L 1 [ 0,1 ] is symmetric on the interval [ 0,1 ] , a ( t ) = 1 for t [ 0,1 ] is continuous, and b , c L 1 [ 0,1 ] are nonnegative symmetric on [ 0,1 ] . For t , y , y , y , y ( 0,1 ) × [ 0 , + ) × ( - , + ) × ( - , + ) × ( - , + ) ,(56)ft,yt,yt,yt,yt=y3+expyt-yt-yt-yt-1-6siny-sin6yt+yt+yt+yt-sinπt1-tπt1-t,0<t<1.Let(57)gt,y=y3+ey-1-6siny,ht,y=y3-216-3sinπt1-tπt1-t,0<t<1.

Obviously, h ( t , y ) f t , y t , y t , y t , y t g ( t , y ) , and(58)liminfy+ht,yty=+,uniformly  on  t0,1,limsupy0+gt,yty=-5λ1,λ1  is  the  first  eigenvalue  of  T,  uniformly  on  t0,1.

All conditions of Theorem 8 are satisfied. Consequently, following from Theorem 8, we know that the boundary value problem (55) has at least one positive solution.

Conflict of Interests

The author declares that she has no competing interests.

Acknowledgments

The author is very grateful to the editor and the anonymous referee for his/her careful reading of the first draft of the paper and making very valuable comments and helpful suggestions. She would like to express her gratitude to Professor Lishan Liu and Professor R. P. Agarwal for their many valuable comments. The author was supported financially by the Foundation of Shanghai Natural Science (13ZR1430100) and the Foundation of Shanghai Municipal Education Commission (DYL201105) and the NNSF of China (no. 11371095 and no. 2013M541455).

Zhang X. Feng M. Ge W. Symmetric positive solutions for p -laplacian fourth-order differential equations with integral boundary conditions Journal of Computational and Applied Mathematics 2008 222 2 561 573 10.1016/j.cam.2007.12.002 MR2474648 2-s2.0-53449100562 Sergejeva N. The regions of solvability for some three point problem Mathematical Modelling and Analysis 2013 18 2 191 203 10.3846/13926292.2013.780189 MR3046083 ZBL1275.34024 2-s2.0-84876317527 Sapagovas M. P. Stikonas A. D. On the structure of the spectrum of a differential operator with a nonlocal condition Differential Equations 2005 41 7 1010 1018 10.1007/s10625-005-0242-y MR2201989 2-s2.0-26044479412 Sergejeva N. On some problems with nonlocal integral condition Mathematical Modelling and Analysis 2010 15 1 113 126 10.3846/1392-6292.2010.15.113-126 MR2641930 2-s2.0-77951694241 Webb J. R. L. Infante G. Non-local boundary value problems of arbitrary order Journal of the London Mathematical Society 2009 79 1 238 258 10.1112/jlms/jdn066 MR2472143 2-s2.0-58449108384 Zhang X. Yang X. Ge W. Positive solutions of nth-order impulsive boundary value problems with integral boundary conditions in Banach spaces Nonlinear Analysis. Theory, Methods & Applications 2009 71 12 5930 5945 10.1016/j.na.2009.05.016 MR2566498 2-s2.0-72149101252 Sun J. X. Zhang G. W. Positive solutions of singular nonlinear Sturm-Liouville problems Acta Mathematica Sinica 2005 48 6 1095 1104 MR2205050 Guo D. J. Lakshmikantham V. Nonlinear Problems in Abstract Cones 1988 San Diego, Calif, USA Academic Press MR959889 Guo D. J. Sun J. X. Nonlinear Integral Equations Shandong Science and Technology Press 1987 (Chinese)