JFSJournal of Function Spaces2314-88882314-8896Hindawi10.1155/2020/91681249168124Research ArticleThree-Order Multipoint Boundary Value Problems for p-Laplacian Operator on Time Scaleshttps://orcid.org/0000-0002-4578-8714SuHuaWangLiguangSchool of Mathematics and Quantitative EconomicsShandong University of Finance and EconomicsJinanShandong 250014Chinasdufe.edu.cn202014820202020260620202007202014820202020Copyright © 2020 Hua Su.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.

In this paper, the existence of positive solutions for the nonlinear four-point singular BVP for there-order with p-Laplacian operator on time scales will be studied. By using the fixed-point theory, the existence of positive solutions for nonlinear singular boundary value problem with p-Laplacian operator on time scales is obtained.

Shandong Province Higher Educational Science and Technology ProgramJ16LI01Project of National Social Science Fund of China18BTY015
1. Introduction

In recent years, the nonlinear boundary value problems have been extensively studied. Recently, for the existence of positive solutions of multipoint boundary value problems, some authors have obtained the existence results. The differential equations offer wonderful tools for describing various natural phenomena arising from natural sciences and engineering, many numerical and analytical results, for example . However, the multipoint boundary value problems treated in the above mentioned references do not discuss the problems with singularities and the there-order p-Laplacian operator. For the singular case of multipoint boundary value problems for higher-order p-Laplacian operator, with the author’s acknowledge, no one has studied the existence of positive solutions in this case.

In this paper, we study the following equation with p-Laplacian on time scale: (1)ϕpuΔΔ+gtfut,uΔt=0,0<t<T,with the following boundary value conditions: (2)u0=0,uΔ0M0uΔΔξ=0,uΔT+M1uΔΔη=0,where ϕps=sp2s,p>1, ϕq=ϕp1, 1/p+1/q=1. ξ,η0,T is prescribed and ξ<η, g:0,T0,, M0,M1 are both nondecreasing continuous odd functions defined on ,+.

A time scale T is a nonempty subset and closed subset of R. By an internal 0,T, we always mean the intersection of the real internal 0,T with the given time scale, that is 0,TT. The operators σ and ρ from T to T which defined by , (3)σt=infτTτ>tT,ρt=supτTτ<tT.are called the forward jump operator and the backward jump operator, respectively.

The point tT is left-dense, left-scattered, right-dense, right-scattered if ρt=t,ρt<t,σt=t,σt>t, respectively. If T has a right scattered minimum m, define Tk=Tm; otherwise set Tk=T. If T has a left scattered maximum M, define Tk=TM; otherwise set Tk=T.

In this paper, by constructing an integral equation which is equivalent to the problem (1), (2), we research the existence of positive solutions for nonlinear singular boundary value problem (1), (2) when g and f satisfy some suitable conditions.

2. Preliminaries and Lemmas

For convenience, in the rest of this article, T is a closed subset of R with 0Tk,TTk.

Letting (4)B=uCld0,T: u0=0.

Then, B is a Banach space with the norm u=maxt0,TuΔt. Suppose (5)K=uB:uΔt0,uΔtisconcavefunctionont0,T.

Obviously, K is a cone in B and 0uΔtu on 0,T. Set Kr=uK:ur.

In the rest of the paper, we make the following assumptions:

(H1) fC0,+2,0,+;

(H2) gCld0,T,0,+ and there exists t00,T which satisfy (6)gt0>0,0<0Tgtt<;

(H3) M0,M1,+,R are both increasing, continuous, odd functions, and at least one of them satisfies the condition that there exists one b>0 which satisfy (7)0<Mivbv,v0,i=0or1.

By direct account, From paper , we can easy to obtain the following results.

Lemma 1.

Suppose condition H2 holds. Then, there exists a constant L0,T/2 satisfies (8)0<LTLgtt<.

Furthermore, the function (9)At=Ltϕqstgs1s1Δs+tTLϕqtsgs1s1Δs,tL,TL,is positive ld-continuous function on L,TL, therefore, At has minimum on L,TL. Hence, we suppose that there exists constant B>0 which satisfy AB on tL,TL.

Lemma 2.

Suppose that conditions H1,H2,H3 hold, utKC20,T is a solution of boundary value problems (1), (2) if and only if utB is a solution of the following integral equation (10)ut=0twsΔs,where (11)wt=M0ϕqξδgsfus,uΔsΔs+0tϕqsδgrfur,uΔrΔrs,0tδ,M1ϕqδηgsfus,uΔsΔs+tTϕqδsgrfur,uΔrΔrs,δtT.

Here, δ is a unique solution of the equation g1t=g2t, where (12)g1t=M0ϕqξδgsfus,uΔsΔs+0tϕqsδgrfur,uΔrΔrs,g2t=M1ϕqδηgsfus,uΔsΔs+tTϕqδsgrfur,uΔrΔrs.

Equation g1t=g2t has a unique solution in 0,T. Because g1t is strictly monotone increasing on 0,T, and g10=0,g2t is strictly monotone decreasing on 0,T, and g2T=0.

Proof.

Necessity. By the equation of the boundary condition and H3, we have (13)uΔΔξ0,uΔΔη0.

Then, there exist a constant δξ,η0,T which satisfy uΔΔδ=0. Firstly, by integrating the equation of the problems (1) on δ,t, we have (14)ϕpuΔΔt=ϕpuΔΔδδtgsfus,uΔsΔs,then (15)uΔΔt=ϕqδtgsfus,uΔsΔs,(16)uΔt=uΔδδtϕqδsgrfur,uΔrΔrs.

By uΔΔδ=0 and condition (2), let t=η on (15), we have (17)uΔΔ1=M1uΔΔη=M1ϕqδηgsfus,uΔsΔs.

Then, we have (18)uΔδ=M1ϕqδηgsfus,uΔsΔs+δTϕqδsgrfur,uΔrΔrs.

Then (19)uΔΔt=M1ϕqδηgsfus,uΔsΔs+tTϕqδsgrfur,uΔrΔrs.

Therefore, by integrating the above equation (19) on 0,t, we can east to have (20)ut=0tM1ϕqδηgsfus,uΔsΔst+0ts1Tϕqδsgrfur,uΔrΔrss1.

Similarly, for t0,δ, by integrating the equation of problems (1) on 0,δ, we have (21)ut=0tM0ϕqξδgsfus,uΔsΔst+0ts1Tϕqsδgrfur,uΔrΔrss1.

Therefore, for any t0,T, ut can be expressed as equation ut=0twsΔs, where wt is expressed as Lemma 3.

Sufficiency. Suppose ut=0twsΔs. Then we have (22)uΔΔt=ϕqtδgsfus,uΔsΔs0,0t<δ,ϕqδtgsfus,uΔsΔs0,δ<tT,

So, ϕpuΔΔ+gtfut,uΔt=0,0<t<T,tδ. These imply that the equation (1) holds. Furthermore, we can easily obtain the boundary value equations of (2). This completes the proof of Lemma 3.

Now, we define an operator T:KC20,T given by (23)Tut=0twsΔs,where wt is given by (15). And we can easily obtain the following Lemma.

Lemma 3.

Let uK and L in Lemma 1. Then (24)uΔtLu,tL,TL.

Lemma 4.

Suppose that conditions H1,H2 hold, then for L0,T/2 in Lemma 1, we have (25)ut1LuΔt,tL,TL.

Proof.

If ut is the solution of problem (1), (2), then uΔt is a concave function, and ut0,uΔt0,t0,T.

Thus for tL,TL, we have uΔtLuΔ. Then by ut=0tuΔsΔsuΔ, we have (26)ut1LuΔt,tL,1L.

The proof is complete.

Remark.

: Obviously, we can obtain the following results, (27)w0M0wξ=0,w1+M1wη=0.

Furthermore, by Arzela-Ascoli Theorem, it is easy to obtain the following Lemma.

Lemma 5.

T:KK is completely continuous.

For convenience, we set (28)R=2B1,R=b+1ϕq0TgrΔr1.where B and L are given as Lemma 1.

3. The Existence of Single and Many Positive Solution

In this section, we present the following five main results.

Theorem 6.

Suppose that condition (H1), (H2), (H3) hold. Assume that f also satisfy.

(A1) For Lru2r,0u11/Lu2, we have fu1,u2mrp1;

(A2) For 0u2R,0u11/Lu2, we have fu1,u2MRp1,where mR,,M0,R. Then, the boundary value problem (1), (2) has at last one solution u such that u lies between r and R.

The proof of Theorem 6. From Condition H3, for v0, we can suppose that 0<M0vbv, and r<R. By Lemma 3, for any uK, we can obtain that (29)uΔtLu,tL,TL.

We define the following two open subset Ω1 and Ω2 of E: (30)Ω1=uK:u<r,Ω2=uK:u<R.

For any uΩ1, by (29), we have (31)r=uuΔtLu=Lr,tL,TL.

For tL,TL and uΩ1, we shall discuss it from three perspectives.

If δL,TL, thus for uΩ1, by (A1) and Lemma 3, we have

(32)2TuΔδ0δϕqsδgrfur,uΔrrΔs+δTϕqδsgrfur,uΔrrΔsLδϕqsδgrfur,uΔrrΔs+δTLϕqδsgrfur,uΔrrΔsmrAδmrB>2r=2u.

Then, with the case of δL,TL and uΩ1, we have Tuu.

If δTL,T, thus for uΩ1, by (A1) and Lemma 3, we have

(33)TuΔδM0ϕqξδgrfur,uΔrr+0δϕqsδgrfur,uΔrrΔsLTLϕqsTLgrfur,uΔrrΔsmrA1LmrB>2r>r=u.

Then, with the case of δTL,T and uΩ1, we have Tuu.

If δ0,L, thus for uΩ1, by (A1) and Lemma 3, we have

(34)TuΔδM1ϕqδηgrfur,uΔrr+δ1ϕqδsgrfur,uΔrrΔsLTLϕqLsgrfur,uΔrrΔsmrALmrB>2r>r=u.

Then, with the case of δ0,L and uΩ1, we have Tuu.

Therefore, for any case of δ0,TL, we all easy to obtain that (35)Tu>u,uΩ1.

Then, by fixed point theorem of cone expansion-compression type in [23, 24], we have (36)iT,Ω1,K=0.

Secondly, for uΩ2, using uΔtu=R, from (A2), we can easily know that (37)TuΔδM0ϕq0Tgrfur,uΔrΔr+0Tϕqsδgrfur,uΔrrΔsbMRϕq0TgrΔr+MRϕq0TgrΔr=b+1MRϕq0TgrΔrR=u.

Thus, we have (38)Tu<u,uΩ2.

Then, by fixed point theorem of cone expansion-compression type in [23, 24], we have (39)iT,Ω2,K=1.

Therefore, by (36), (39), r<R we have (40)iT,Ω2Ω¯1,K=1.

Then, operator T has at last one fixed point uΩ2\Ω¯1, and ruR. This completes the proof of Theorem 6.

Theorem 7.

Suppose that condition (H1), (H2), (H3) hold. Assume that f also satisfy.

(A3) f0=limu20max0u11/Lu2fu1,u2/u2p1=φ0,R/4p1;

(A4) f=limu2min0u11/Lu2fu1,u2/u2p1=ψ2R/Lp1,.

Then, the boundary value problem (1), (2) has at last one solution u.

The proof of Theorem 7.

First, by 0limu20max0u11/Lu2fu1,u2/u2p1=φ<R/4p1, letting ε=R/4p1φ>0, we know that there exists an appropriately small positive number ρ which satisfy as 0u2ρ,u20, and we have (41)fu1,u2φ+εu2p1R/4p1ρp1=R/4ρp1.

Then, letting R=ρ,0<M=R/4<R, thus by the above equation, we can have (42)fu1,u2MRp1,0u2R,0u11Lu2.

So condition (A2) in Theorem 6 holds.

Next, by 2R/Lp1<limu2min0u11/Lu2fu1,u2/u2p1=ψ<, letting ε=ψ2R/Lp1, we know that there exists an adequately big positive number rR which satisfy as u2Lr,0u11/Lu2, and we have (43)fu1,u2ψεu2p12RLp1Lrp1=2Rrp1,

Letting m=2R>R, thus by the above equation, we have that (A1) in Theorem 6 holds. Therefore, by Theorem 6, we can easily obtain the results of Theorem 7 holds. The proof of Theorem 3.2 is complete.

Corollary 8.

Suppose that condition (H1), (H2), (H3) hold. Assume that f also satisfy.

(A5) f=limu2max0u11/Lu2fu1,u2/u2p1=λ0,R/4p1;

(A6) f0=limu20min0u11/Lu2fu1,u2/u2p1=φ2R/Lp1,.

Then, the boundary value problem (1), (2) has at last one solution u.

The proof of Corollary 8. Similar to the proof of Theorem 7, we can obtain Corollary 8.

Theorem 9.

Suppose that conditions (H1), (H2), (H3), and A2 in Theorem 6 hold. Assume that f also satisfy.

(A7) f0=limu20min0u11/Lu2fu1,u2/u2p1=+;

(A8) f=limu2min0u11/Lu2fu1,u2/u2p1=+.

Then, the boundary value problem (1), (2) have at least two solutions u1,u2 which satisfy (44)0<u1<R<u2.

The proof of Theorem 9.

Firstly, by limu20min0u11/Lu2fu1,u2/u2p1=+, for any M>2/BL, there exists a constant ρ0,R which satisfy (45)fu1,u2Mu2p1,0<u2ρ,0u11Lu2.

Set Ωρ=uK:u<ρ, similar to the previous proof of Theorem 6, for any uΩρ, from the above discussion and Lemma 2, we can have from three perspectives (46)Tuu,uΩρ.

Then, by fixed point theorem of cone expansion-compression type, we can have (47)iT,Ωρ,K=0.

Secondly, for any M¯>2/BL, by limu2min0u11/Lu2fu1,u2/u2p1=+, there exists a constant ρ0>0 which satisfy (48)fu1,u2M¯u2p1,u2>ρ0,0u11Lu2.

Therefore, we choose a constant ρ>maxR,ρ0/L, obviously ρ<R<ρ. Set Ωρ=uK:u<ρ. For any uΩρ, by Lemma 2, we can easily obtain (49)uΔtLu=Lρ>ρ0,tL,TL.

Then, by the above discussion and also similar to the previous proof of Theorem 6, we can also have from three perspectives (50)Tuu,uΩρ.

Then, by fixed point theorem of cone expansion-compression type, we have (51)iT,Ωρ,K=0.

Finally, imitating the latter proof of Theorem 6, for any uΩR, by A2, setting ΩR=uK:u<R, we can also easy to have (52)Tuu,uΩR.

Then, by fixed point theorem of cone expansion-compression type, we have (53)iT,ΩR,K=1.

Therefore, by (47), (51), (53), ρ<R<ρ we have (54)iT,ΩR\Ω¯ρ,K=1,iT,Ωρ\Ω¯R,K=1.

Then, T have fixed point u1ΩR\Ω¯ρ, and fixed point u2Ωρ\Ω¯R.

Obviously, u1,u2 are all positive solutions of problem (1), (2) and 0<u1<R<u2. The proof of Theorem 9 is complete.

Theorem 10.

Suppose that conditions (H1), (H2), (H3) and A1 in Theorem 6 hold. Assume that f also satisfy.

(A9) f0=limu20max0u11/Lu2fu1,u2/u2p1=0;

(A10) f=limu2max0u11/Lu2fu1,u2/u2p1=0.

Then, the boundary value problem (1), (2) have at least two solutions u1,u2 which satisfy 0<u1<r<u2.

The proof of Theorem 10.

Firstly, by limu2max0u11/Lu2fu1,u2/u2p1=0, for η10,R, there exists a constant ρ0,r which satisfy (55)fu1,u2η1u2p1,0<u2ρ,0u11Lu2.

Set Ωρ=uK:u<ρ, for any uΩρ, by (23), we have (56)Tu=Tun2δM0ϕq0Tgrfur,uΔrr+0Tϕqsδgrfur,uΔrrΔsM0ϕq0Tgrfur,uΔrr+ϕq0Tgrfur,uΔrrb+1η1ρϕq0Tgrdrρ=u.

Then, by fixed point theorem of cone expansion-compression type, we have (57)iT,Ωρ,K=1.

Secondly, letting fx=max0un1x,0u11/Lu2fu1,u2, we can easy to know that fx is monotone increasing with respect to x0.

Therefore by limu2max0u11/Lu2fu1,u2/u2p1=0, we can easy to have limxfx/xp1=0.

Therefore, for any η20,R, there exists a constant ρ>r which satisfy (58)fxη2xp1,xρ.

Set Ωρ=uK:u<ρ, for any uΩρ, by (4.8), we have (59)Tu=TuΔδM0ϕq0Tgrfur,uΔrr+0Tϕqsδgrfur,uΔrrΔsM0ϕq0Tgrfur,uΔrr+ϕq0Tgrfur,uΔrrb+1ϕq0Tgrfρdrb+1η2ρϕq0Tgrdrρ=u.

Then, by fixed point theorem of cone expansion-compression type, we have (60)iT,Ωρ,K=1.

Finally, imitating the previous proof of Theorem 6, for any uΩr, by A1, setting Ωr=uK:u<r, For any uΩr, we can also easy to have (61)Tuu,uΩr.

Then, by fixed point theorem of cone expansion-compression type, we have (62)iT,Ωr,K=0.

Therefore, by (57), (60), (62), ρ<r<ρ, we have (63)iT,Ωr\Ω¯ρ,K=1,iT,Ωρ\Ω¯r,K=1.

Then, T have fixed point u1Ωr\Ω¯ρ, and fixed point u2Ωρ\Ω¯r.

Obviously, u1,u2 are all positive solutions of problem (1),(2) and 0<u1<r<u2. The proof of Theorem 10 is complete.

4. ApplicationExample.

Consider the following three-order BVP with p-Laplacian (64)ϕpu+164π4t121tlnu5+eu2=0,0<t<1,u0=0,u0u0.25=0,u1+5u0.3=0,where p=4,ξ=0.25,η=0.3,B=1/4, (65)gt=164π4t121t,fu1,u2=lnu5+eu2.

Then obviously, q=4/3,01gtdt=1/64π3,f=+,f0=+, (66)M0v=v<2v=bv,M1v=5v,v0,so conditions (H1), H2, H3, (A7), A8 hold.

Next, ϕq01gtdt=1/4π,R=4π/3, we choose R=3,M=2 and for B=1/4, because of the monotone increasing of fu1,u2 on 0,×0,, then (67)fu1,u2f12,3=286,0u23,0u14u2.

Therefore, using 0<M<R, we have MRp1=328, we know (68)fu1,u2MRp1,0u23,0u14u2,so conditions A2 holds. Then, by Theorem 9, the Example has at least two positive solutions v1,v2 and 0<v1<3<v2.

Data Availability

No data were used to support this study.

Conflicts of Interest

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

Authors’ Contributions

The study was carried out in collaboration among all authors. All authors read and approved the final manuscript.

Acknowledgments

The authors really appreciate the anonymous reviewers for their pertinent comments and suggestions, which were helpful to improve the earlier manuscript. The author was supported by the Project of National Social Science Fund of China (NSSF) (18BTY015) and Shandong Province Higher Educational Science and Technology Program (J16LI01).

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