Solvable Theory of Nonlinear Differential Equations …math.bit.edu.cn/docs/2016-07/20160731020007724894.pdfSolvable Theory of Nonlinear Di erential Equations and Their Applications

Solvable Theory of Nonlinear DifferentialEquations and Their Applications

Meiqiang Feng(joint work with Prof. Xuemei Zhang)

Beijing Information Science & Technology University

July 20, 2016 at Beijing Institute of Technology

Solvable Theory of Nonlinear Differential Equations and Their Applications

Outline

Progress in Nonlinear Differential Equations

Main results

Summary

References


Outline


Main results

Summary

References


Outline


Main results

Summary

References


Outline


Main results

Summary

References



1. p-Laplace type equationsIn recent years many cases of the exactness, existence,

multiplicity and uniqueness of positive solution of differentialequations with p-Laplacian have attracted considerableattention.

Addou, Sanchez and Ubilla independently proved exactmultiplicity of positive solutions for a more general k-Laplacianproblem

−(φk(u′(x)))′ = λf (u), −1 < x < 1,u(−1) = u(1) = 0,

(1.1.1)

where f (u) = uq + up, k > 1, φk(y) = |y |k−2y and (φk(u′))′

is one-dimensional k-Laplacian. Using shooting method, for0 < q < k − 1 < p, they proved the existence of some λ∗ > 0such that (1.1.1) has exactly two positive solutions for0 < λ < λ∗, exactly one positive solution for λ = λ∗ and nopositive solution for λ > λ∗.



1. p-Laplace type equationsKajikiya et al . investigated the following one-dimensional

p-Laplacian problem(ϕp(u′))′ + λω(t)f (u) = 0, t ∈ (0, 1),u(0) = u(1) = 0,

(1.1.2)

and by virtue of the global bifurcation theory, they obtainedthe existence, nonexistence, uniqueness and multiplicity ofpositive solutions as well as sign-changing solutions undersuitable conditions imposed on the nonlinear term f .



1. p-Laplace type equationsDai et al . investigated the existence of one-sign solutions

for the following periodic p-Laplacian problem−(ϕp(u′))′ + q(t)ϕp(u) = λω(t)f (u), 0 < t < T ,u(0) = u(T ), u′(0) = u′(T ).

(1.1.3)The author also examined the uniqueness of the solution andits dependence on the parameter λ under condition(H) f (s)

ϕp(s)is strictly decreasing in (0,∞).



1. p-Laplace type equationsOn the other hand, we notice that there has been a

considerable attention on impulsive differential equations withone-dimensional p-Laplacian. For example, in [12], employingthe classical fixed point index theorem for compact maps,Zhang and Ge obtained some sufficient conditions for theexistence of multiple positive solutions of the problem

(ϕp(u′(t)))′ = −f (t, u(t)), 0 < t < 1, t 6= tk ,∆u|t=tk = −Ik(u(tk)),∆u′|t=tk = 0, k = 1, 2, . . . ,m,u(0) = Σm−2

i=1 aiu(ξi), u′(1) = 0.(1.1.4)



1. p-Laplace type equationsOf course, some questions are

Q 1.1.1 Whether or not the exactness of positive solutions forproblem (1.1.1) can be obtained when the nonlinearity f isgeneral form: f (u) = λg(u) + h(u)? Furthermore, can wegeneralize and improve the exactness results to nonlocalproblems?Q 1.2.2 Without a similar condition to that of (H), can weshow the dependence of positive periodic solution xλ(t) on theparameter λ for problem (1.1.3)?Q 1.2.3 If problem (1.1.4) with a parameter or multipleparameters, then can we investigate the existence, multiplicityand nonexistence of it?



2. Prescribed mean curvature equationsA typical model of prescribed mean curvature equation is −div

(∇u√

1+‖∇u‖2

)= λf (t, u), t ∈ R+, u ∈ Ω,

u = 0 on ∂Ω,(1.2.1)

where Ω is a bounded domain in RN and f : Ω× R+ → R+ iscontinuous.



2. Prescribed mean curvature equationsThe one-dimensional version of (1.2.1) is −

(u′√

1+u′2

)′= λf (t, u), a < t < b,

u(a) = u(b) = 0.(1.2.2)

There are some papers considering the exact number ofpositive solutions of (1.2.2) in special case of f . The studyderived from an open problem proposed by A. Ambrosetti, H.Brezis and G. Cerami, which concerned the exact number andthe detailed property of solutions of the semilinear equation

−u′′ = λ(up + uq), u > 0 in (0, 1),u(0) = u(1) = 0,

(1.2.3)

where 0 < p < 1 < q < +∞.



2. Prescribed mean curvature equationsRecently, P.Habets and P.Omari [18] considered the

existence, non-existence and multiplicity of positive solutionsof the Dirichlet problem for the one-dimensional prescribedcurvature equation −

(u′√

1+u′2

)′= f (u), u > 0 in (0, 1),

u(0) = u(1) = 0,(1.2.4)

in connection with the changes of concavity of the function f ,such as f (u) = maxλup, µuq and f (u) = minλup, µuq,where 0 < p < 1 < q. In particular, they obtained thefollowing results for f (u) = λup, where 0 < p < 1, p = 1 orp > 1.



2. Prescribed mean curvature equationsVery recently, W.Li and Z.Liu [19] examined problem

(1.2.4) when f (u) = up + uq, and obtained the followingtheorems.Theorem 1.2.2 (See W.Li and Z.Liu [19].) Assume that1 < p < q < +∞. The following conclusions hold:(i) for any λ > 0, (1.2.4) has at most one solution;(ii) there exist 0 < λ1 < λ2 < +∞ such that (1.2.4) has nosolution for 0 < λ < λ1 and has exactly one solution forλ > λ2;(iii) if, in addition

q ≤ p − 2 +√p2 + 20p + 20

2,

then there exists 0 < λ∗ < +∞ such that (1.2.4) has nosolution for λ ≤ λ∗ and has exactly one solution for λ > λ∗.



2. Prescribed mean curvature equationsTheorem 1.2.3 (See W.Li and Z.Liu [19].) Assume that

0 < p < q < 1. The following conclusions hold:(i) for any λ > 0, (1.2.4) has at most two solutions;(ii) there exist 0 < λ1 < λ2 < +∞ such that (1.2.4) hasexactly one solution for 0 < λ < λ1 and has no solution forλ > λ2;(iii) if, in addition

p ≥(

π + 2√2 ln(√

2 + 1) + 2

)1/2

− 1,

then there exist λ∗ and λ∗ with 0 < λ∗ < λ∗ such that (1.2.4)has exactly one solution for λ ∈ (0, λ∗] ∪ λ∗, exactly twosolutions for λ ∈ (λ∗, λ

∗), and no solution for (λ∗,+∞).



2. Prescribed mean curvature equationsOn the other hand, W.Li and Z.Liu [19] raised an open

problem: It would be very interesting to study exact number ofsolution of (1.2.4) if p, q satisfy 0 < p < 1 < q < +∞ (see,(d) of Remark 1.4). The purpose here is to solve this problem.

Lienard type equations have a wide range of applicationsin applied science, such as physics, biology, mechanics, and theengineering technique fields. However, to the best of ourknowledge, the corresponding theory for prescribed meancurvature Lienard equation with a deviating argument is notinvestigated till now.



3. Impulsive differential equationsImpulsive differential equations, which provide a natural

description of observed evolution processes, are regarded asimportant mathematical tools for the better understanding ofseveral real world problems in applied sciences, such aspopulation dynamics, ecology, biological systems,biotechnology, industrial robotic, pharmacokinetics, optimalcontrol, etc. Therefore, the study of this class of impulsivedifferential equations has gained prominence and it is a rapidlygrowing field.



3. Impulsive differential equationsSome classical tools such as bifurcation theory, fixed point

theorems in cones, the method of lower and upper solutionsand the theory of critical point theory and variational methodshave been widely used to study impulsive differential equations.

At the same time, we notice that a new technique viaappropriate transformation is proved to be very effective instudying the solvability of impulsive differential equations.Such techniques have attracted the attention of many authors.X.Zhang, J.Yan and A.Zhao considered the following equation

x ′(t) = −a(t)x(t) + p(t)f (t, x(t − τ0(t)),x(t − τ1(t)), · · · , x(t − τn(t))),

a.e. t > 0, t 6= tk ,x(t+

k )− x(tk) = bkx(tk), k = 1, 2, · · ·

(1.3.1)



3. Impulsive differential equationsHowever it is quite difficult to apply this approach to

second impulsive differential equation, especially for secondimpulsive differential equation with deviating arguments.

Another question is that there are only a few articles whichdealt with some impulsive differential equations withquasilinear operator.



4. Functional differential equationsFunctional differential equations with periodic delays

appear in some ecological models. An important functionaldifferential equation is Rayleigh equation with two deviatingarguments in the form of

x ′′(t)+f (x ′(t))+g1(t, x(t−τ1(t)))+g2(t, x(t−τ2(t))) = e(t),(1.4.1)

which has been studied by authors under the assumption off (0) = 0 or f (t, 0) = 0.



4. Functional differential equationsIt is not difficult to see that if g1(t, 0) + g2(t, 0) 6≡ e(t),

then the periodic solution obtained in [48-50] must benontrivial. But if g1(t, 0) + g2(t, 0) ≡ e(t), then the periodicsolution obtained in [48-50] may be trivial under theassumption of f (0) = 0 or f (t, 0) = 0. And if the periodicsolution is unique, then it must be trivial.



4. Functional differential equationsThus, it is worth discussing the existence of the nontrivial

periodic of Rayleigh equations with two deviating arguments inthis case.

At the same time, we notice that there exist only fewresults for the existence of anti-periodic solutions for Rayleighequation and Rayleigh type equations with and withoutdeviating arguments in the literature. The main difficulty liesin the middle term f (x ′(t)) of Eq.(1.4.1), the existence ofwhich obstructs the usual method of finding a priori boundsfor delay Duffing or Lienard equations from working. Thus, itis worthwhile to continue to investigate the anti-periodicsolutions of Rayleigh equation in this case.



4. Functional differential equationsRecently, differential equations with deviating arguments

have received much attention. For example, in [51], C.Yang,C.Zhai and J.Yan studied the existence and multiplicity ofpositive solutions to a three-point boundary value problemwith an advanced argument

x′′(t) + a(t)f (x(α(t))) = 0, t ∈ (0, 1),

x(0) = 0, bx(η) = x(1),(1.4.2)

where 0 < η < 1, b > 0 and 1− bη > 0. The main tool is thefixed point index theory. It is clear that the solution of [51] isconcave when a(t) ≥ 0 on [0, 1] and f (x) ≥ 0 on [0,∞).



4. Functional differential equationsHowever, few papers have been reported on the same

problems when the solution without concavity.Very recently, a class of p-Laplacian differential equations

with deviating arguments both of an advanced or delayed typehave received much attention. For example, in [52], T.Jankowski considered the following third order p-Laplaciandifferential equation (ϕp(u′′(t)))′ + h(t)f (t, u(t), u(α(t))) = 0 t ∈ J ′,

u′(t+k ) = u′(t−k ) + Qk(u(tk)), k = 1, 2, · · · ,m,

βu(0)− γu′(0) = 0, δu(1) + ηu′(1) = 0, u′′(0) = 0,(1.4.3)

where α(t) 6≡ t on J . The author obtained the existence of atleast three positive solutions.



4. Functional differential equationsOf course, a natural question is

Q 1.4.1 Whether or not the existence of positive solutions fora second order p-Laplacian differential equation with deviatingarguments both of an advanced or delayed type can be proved?Remark 1.4.1 In [52], by means of the properties of Green’sfunction, T. Jankowski obtain the inequality

mint∈[ξ,1]

u(t) ≥ ρ‖u‖,

where ξ ∈ (0, 1) and ρ ∈ (0, 1).In fact, the calculation of ρ is very difficult when t ∈ [ξ, 1].

This is probably the main reason that there is almost nopaper to study the existence of positive solutions for class ofsecond order p-Laplacian impulsive differential equations withtwo parameters and deviating arguments both of an advancedor delayed type.



4. Functional differential equationsT. Jankowski obtained a constant number ρ by means of

the properties of Green’s function. However, it is well knownthat there is not any Green’s function in one-dimensionalp-Laplacian boundary value problems of second orderdifferential equations. This implies the following questionQ 1.4.2 Whether or not a similar inequality can be obtained ifthere is no Green’s function when t ∈ [ξ, 1]?

This needs to open a new technique to deal with secondorder p-Laplacian equations with deviating arguments,especially for second order p-Laplacian equations withimpulsive effects.



5. Fractional differential equationsFractional derivatives provide an excellent tool for the

description of memory and hereditary properties of variousmaterials and processes. This is the main advantage offractional differential equations in comparison with classicalinteger-order models. Therefore, there are many authors havestudied fractional differential equations in recent years.

However, the theory of boundary value problems fornonlinear fractional differential equations is still in the initialstages and many aspects of this theory need to be explored.



5. Fractional differential equationsZhang used a fixed point theorem for the mixed monotone

operator to show the existence of positive solutions to thefollowing singular fractional differential equation

Dα0+u(t) +q(t)f (t, x(t), x

′(t), · · · , x (n−2)(t)) = 0, 0 < t < 1,

subject to the boundary conditions

u(0) = u′(0) = u

′′(0) = · · · = u(n−2)(0) = u(n−1)(1) = 0,

where Dα0+ is the standard Rimann-Liouville fractional

derivative of order n − 1 < α ≤ n, n ≥ 2, the nonlinearity fmay be singular at u = 0, u

′= 0, · · · , u(n−2) = 0, and

function q(t) may be singular at t = 0.



5. Fractional differential equationsShe derived the corresponding Green’s function-named by

fractional Green’s function and obtained some properties asfollows.Proposition 1.5.1 The Green’s function G (t, s) satisfies thefollowing conditions:(i)G (t, s) ≥ 0, G (t, s) ≤ tα−n+2/Γ(α−n+2), G (t, s) ≤ G (s, s)for all 0 ≤ t, s ≤ 1;(ii) There exists a positive function ρ ∈ C (0, 1) such that

minγ≤t≤δ

G (t, s) ≥ ρ(s)G (s, s), s ∈ (0, 1), (1.5.1)

where 0 < γ < δ < 1 and

ρ(s) =

[δ(1−s)]α−n+1−(δ−s)α−n+1

(s(1−s))α−n+1 , s ∈ (0, r ],(γs

)α−n+1

, s ∈ [r , 1),

here γ < r < δ.Solvable Theory of Nonlinear Differential Equations and Their Applications


5. Fractional differential equationsIt is well known that the cone theoretic techniques play a

very important role in applying the Green’s function in thestudy of solutions to boundary value problems. In [61], theauthor can not acquire a positive constant taking instead ofthe role of positive function ρ(s) with n − 1 < α ≤ n, n ≥ 2in (1.5.1). At the same time, we notice that many authorsobtained the similar properties to (1.5.1). Naturally, onewishes to find whether there exists a positive constant ρ suchthat

minγ≤t≤δ

G (t, s) ≥ ρG (s, s), s ∈ [0, 1],

for the fractional order cases?



5. Fractional differential equationsAt the same time, it is worth mentioning that it is an

important method to express the solution of differentialequations by Green’s function. According to the previous work,we find that the solution of impulsive differential equationswith integer order can be expressed by the Green’s function ofthe case without impulse. For example, the Green’s functionof the following boundary value problem

−x ′′(t) = σ(t), t ∈ (0, 1),x(0) = x(1) = 0

(1.5.2)

can be expressed by

G (t, s) =

s(1− t), 0 ≤ s ≤ t ≤ 1,t(1− s), 0 ≤ t ≤ s ≤ 1,

where σ ∈ C [0, 1].Solvable Theory of Nonlinear Differential Equations and Their Applications


5. Fractional differential equationsThe solution of problem (1.5.2) can be expressed by

x(t) =

∫ 1

0

G (t, s)σ(s)ds.

If we consider impulsive differential equations−x ′′(t) = σ(t), t ∈ (0, 1), t 6= tk ,∆x |t=tk = Ik(x(tk)), k = 1, 2, . . . , n,∆x ′|t=tk = −Lk(x(tk)), k = 1, 2, . . . , n,x(0) = x(1) = 0,

(1.5.3)

then the solution of problem (1.5.3) can be expressed by

x(t) =

∫ 1

0

G (t, s)σ(s)ds+n∑

k=1

G (t, tk)Lk(xtk )+n∑

k=1

G ′s(t, tk)Ik(xtk ),

where Ik , Lk ∈ C [0, 1], k = 1, 2, . . . , n.Solvable Theory of Nonlinear Differential Equations and Their Applications


5. Fractional differential equationsNaturally, one wishes to know whether or not the same

result holds for the fractional order case? We first study thefractional order differential equations with Caputo derivatives

−cDq0+x(t) = σ(t), t ∈ (0, 1), 1 < q ≤ 2,

x(0) = x(1) = 0,(1.5.4)

where cDq0+ is the Caputo’s fractional derivative. Then

x(t) =

∫ 1

0

G (t, s)σ(s)ds,

where

G (t, s) =1

Γ(q)

t(1− s)q−1 − (t − s)q−1, 0 ≤ s ≤ t ≤ 1,t(1− s)q−1, 0 ≤ t ≤ s ≤ 1.



5. Fractional differential equationsThen whether the solution of fractional order impulsive

differential equations with Caputo derivatives−cDq

0+x(t) = σ(t), t ∈ (0, 1), 1 < q ≤ 2,∆x |t=tk = Ik(x(tk)), k = 1, 2, . . . , n,∆x ′|t=tk = −Lk(x(tk)), k = 1, 2, . . . , n,x(0) = x(1) = 0

(1.5.5)

can be expressed by

x(t) =

∫ 1

0

G (t, s)σ(s)ds+n∑

k=1

G (t, tk)Lk(xtk )+n∑

k=1

G ′s(t, tk)Ik(xtk )?

Thus, it is an interesting problem, and so it is worthwhile tostudy.



6. Elastic beam equationsOwing to its importance in modeling the stationary states

of the deflection of an elastic beam, fourth order differentialequations with various boundary conditions have attractedmuch attention from many authors.

Yang considered the following fourth order two-pointboundary value problem

x (4)(t) = g(t)f (x(t)), 0 ≤ t ≤ 1,x(0) = x

′(0) = x

′′(1) = x

′′′(1) = 0.

(1.6.3)



6. Elastic beam equationsAnderson and Avery considered the following fourth-order

four-point right focal boundary value problem−x (4)(t) = f (x(t)), t ∈ [0, 1],x(0) = x

′(q) = x

′′(r) = x

′′′(1) = 0.

(1.6.4)



6. Elastic beam equationsRecently, Zhang, Feng and Ge [64] studied the existence

of positive solutions of the following fourth-order boundaryvalue problem with integral boundary conditions

x (4)(t)− λf (t, x(t)) = θ, 0 < t < 1,

x(0) = x(1) =∫ 1

0g(s)x(s)ds,

x′′(0) = x

′′(1) =

∫ 1

0h(s)x(s)ds,

(1.6.5)

where θ is the zero element of E .However, to the best of our knowledge, no paper has

considered the existence of multiple positive solutions offourth-order impulsive differential equations with integralboundary conditions and one-dimensional p-Laplacian till now.

On the other hand, if the nonlinear term f involves thebending term, then we require a special technique to tacklethe problem.


Main results

1. p-Laplace type equationsIn this chapter, we discuss the exactness and bifurcation

diagrams of positive solutions and parameter dependence ofpositive solutions for p-Laplacian type equations, respectively.

We first deal with the exact number and bifurcationdiagrams of positive solutions for the one-dimensionalp-Laplacian in a class of two-point boundary value problems.The interesting point is that the nonlinearity f is general form:f (u) = λg(u) + h(u).

Meanwhile, some properties of the solutions are given indetails. The arguments are based upon a time-map analysis.


Main results

1. p-Laplace type equations

−(φp(u′(x)))′ = λg(u(x)) + h(u(x)), a < x < b,u(x) > 0, a < x < b,u(a) = u(b) = 0.

(3.1.1)


Main results

1. p-Laplace type equationsSecondly, using detail time-map analysis, we establish the

exact number of pseudo-symmetric positive solutions for aclass of three-point boundary value problems withone-dimensional p-Laplacian. The interesting point is that thesolution is pseudo-symmetric and the boundary condition isnonlocal.

−(φp(u′(x)))′ = λg(u(x)) + h(u(x)), 0 < x < 1,u(0) = 0, u(η) = u(1).

(3.2.1)


Main results

1. p-Laplace type equationsFinally, we consider the one-dimensional singular

p-Laplacian problems with a parameter. Applying fixed pointtechniques combining with partially ordered structure ofBanach space, some new and more general existence,multiplicity and nonexistence results are established. Inparticular, the dependence of positive solution uλ(t) on theparameter λ is also studied, i.e.,

limλ→+∞

‖uλ‖ = +∞ or limλ→+∞

‖uλ‖ = 0.

λ(ϕp(u′))′ + ω(t)f (t, u) = 0, t ∈ (0, 1),

au(0)− bu′(0) =∫ 1

0g(t)u(t)dt, u′(1) = 0,

(3.3.1)

where λ is a positive parameter


Main results

2. Prescribed Mean Curvature Operator EquationsConsidering a conjecture raised by Li and Liu. We first

consider various properties of time-map for theone-dimensional prescribed mean curvature equation withconcave-convex nonlinearities. Next, we study the bifurcationdiagrams and exact multiplicity of positive solutions for theone-dimensional prescribed mean curvature equation withconcave-convex nonlinearities.

Then we discuss the one-dimensional prescribed meancurvature equation with concave-convex nonlinearities in the

form of −(

u′√1+u′2

)′= λ(up + uq), u(x) > 0, 0 < x <

1, u(0) = u(1) = 0, where λ > 0 is a parameter and p, qsatisfy −1 < p < q < +∞, and we obtain new exact results ofpositive solutions. Our methods are based on a detailedanalysis of time maps.


Main results

2. Prescribed Mean Curvature Operator EquationsFinally, we consider a prescribed mean curvature Lienard

equation with a deviating argument. Some new criteria for theexistence of periodic solutions are established by using thecoincidence degree theory.

(

x ′√1+x ′2

)′+ f (x(t))x ′(t) + g(t, x(t − τ(t))) = e(t), t ∈ J ,

x(0) = x(T ), x′(0) = x ′(T ),

(4.3.1)where J = [0,T ], T > 0, g ∈ C (R2,R1), f , e, τ ∈C (R1,R1), g(t + T , x) = g(t, x), ande(t + T ) = e(t), τ(t + T ) = τ(t).


Main results

3. Green’s Function and Fractional DifferentialEquations

This chapter is devoted to nonlinear fractional orderdifferential equations with impulses and integral boundaryconditions.

We consider a nonlinear fractional differential equation withintegral boundary conditions. After establishing the expressionand properties of Green’s function of the above equation, weobtain some results on the existence of positive solutions byusing fixed point theorem in cones. Then, we investigate theexistence and multiplicity of positive solutions for a class ofhigher-order nonlinear fractional differential equations withintegral boundary conditions.


Main results


Consider the following nonlinear boundary value problemof fractional impulsive differential equations:

cDq0+x(t) = ω(t)f (t, x(t), x ′(t)), 1 < q ≤ 2, t ∈ J1 = J \ t1, t2, · · · , tn,

∆x |t=tk = Ik(x(tk)), ∆x ′|t=tk = Jk(x(tk)), tk ∈ (0, 1), k = 1, 2, . . . , n,α1x(0)− β1x

′(0) = 0, α2x(1) + β2x′(1) = 0,

where cDq0+ is the Caputo’s fractional derivative

We first discuss the expression and properties of Green’sfunction for boundary value problems of nonlinearSturm-Liouville-type fractional order impulsive differentialequations. And then, we study the expression of the solutionof fractional impulsive differential equations by using Green’sfunction.


Main results


For any σ(t) ∈ C [0, 1], the unique solution of boundaryvalue problem

cDq0+x(t) = σ(t), 1 < q ≤ 2, t ∈ J1 = J \ t1, t2, · · · , tn,

∆x |t=tk = Ik(x(tk)), ∆x ′|t=tk = Jk(x(tk)), tk ∈ (0, 1), k = 1, 2, . . . , n,α1x(0)− β1x

′(0) = 0, α2x(1) + β2x′(1) = 0,

(5.3.2)is given by

x(t) =∫ t

tk

(t−s)q−1

Γ(q)σ(s)ds +

n+1∑i=1

G ′1s(t, ti)∫ titi−1

(ti−s)q−1

Γ(q)σ(s)ds

−n∑

i=1

G1(t, ti)∫ titi−1

(ti−s)q−2

Γ(q−1)σ(s)ds +

n∑i=1

G ′1s(t, ti)Ii(x(ti))

−n∑

i=1

G1(t, ti)Ji(x(ti)),

t ∈ (tk , tk+1], k = 0, 1, 2, . . . , n, t0 = 0, tn+1 = 1,Solvable Theory of Nonlinear Differential Equations and Their Applications

Main results


where

G1(t, s) = −1

η

(β1 + α1t)(α2 + β2 − α2s), t ≤ s,(β1 + α1s)(α2 + β2 − α2t), t ≥ s,

andη = α1α2 + α1β2 + α2β1.


Main results

4. Elasticity Beam EquationsWe discuss p-Laplacian elasticity problems with the

bending term. We first transform p-Laplacian elasticityproblems with the bending term into a differential systemswithout bending term.

(φp(y

′′(t)))

′′= F (t, y(t), y

′′(t)), 0 < t < 1,

ay(0)− by′(0) =

∫ 1

0g(s)y(s)ds,

ay(1) + by′(1) =

∫ 1

0g(s)y(s)ds,

φp(y′′(0)) = φp(y

′′(1)) =

∫ 1

0h(s)φp(y

′′(s))ds,

(6.2.1)

where a, b > 0, φp(t) = |t|p−2t, p > 1, φq = φ−1p , 1

p+ 1

q=

1, F : [0, 1]× R × R → R is continuous.


Main results

4. Elasticity Beam EquationsLet φp(y

′′(t)) = −x(t). Then (6.2.1) can be changed into

x′′(t) = −F (t,

∫ 1

0H1(t, s)φq(x(s))ds,−φq(x(t))), 0 < t < 1,

x(0) = x(1) =∫ 1

0h(s)x(s)ds.

(6.2.2)


Main results

4. Elasticity Beam EquationsAssume that (H1)− (H3) hold. Then we have the

following conclusions.(i) The problem (6.2.2) has (at least) one positive solutionx ∈ K such that

δr ≤ x(t) ≤ 1

δR , t ∈ J . (6.2.3)

(ii) The problem (6.2.1) has (at least) one positive solution ysuch that y(t) =

∫ 1

0H1(t, s)φq(x(s))ds, t ∈ J ;

‖y‖ ≤ ρ2φq(‖x‖);y(t) ≥ σφq(‖x‖), t ∈ J .

(6.2.4)

We further have

σφq(r) ≤ y(t) ≤ ρ2φq(1

δR), t ∈ J . (6.2.5)


Main results

5. Functional Differential EquationsSection 7.1 deals with the existence of positive periodic

solution for a functional differential equation with a parameter

λ

(x′(t)− a(t)g(x(t))x(t)

)= −b(t)f (x(t − τ(t))). (7.1.1)

Using a well-known fixed point theory, we establish severalcriteria for the optimal intervals of the parameter λ so as toensure existence of positive periodic solutions.

The dependence of positive periodic solution xλ(t) on theparameter λ is also studied, i.e.,

limλ→+∞

‖xλ‖ = +∞ or limλ→+∞

‖xλ‖ = 0.


Main results

5. Functional Differential EquationsSection 7.2 deals with first order impulsive functional

differential equationsu′(t) = −a(t)g(u(h1(t)))u(t)

+λb(t)f

(t, u(h2(t)),

∫ 0

−ζ e(s)u(t + s)ds

),

t ∈ R, t 6= tk ,u(t+

k )− u(tk) = µIk(tk , u(tk)), k ∈ Z,

(7.2.1)

where λ > 0 and µ > 0 are parameters,e ∈ C ([−ζ, 0],R),

∫ 0

−ζ e(s)ds = 1 and tk , k ∈ Z is anincreasing sequence of real number with lim

k→±∞tk = ±∞.


Main results

5. Functional Differential EquationsWe first study the existence and multiplicity results of

positive periodic solutions under the case g is bounded. Andthen, we establish the dependence results of the positiveperiodic solution on the parameter. On the other hand, westudy the existence of positive periodic solutions under thecase g is not necessarily bounded.

we study problem (7.2.1) under three cases:

0 < l ≤ g(u) ≤ L <∞, for all u > 0. (7.2.2)

0 < l ≤ g(u) <∞, for all u > 0. (7.2.3)

0 < g(u) ≤ L, for all u > 0, (7.2.4)

where l and L are two positive constants.


Main results

5. Functional Differential EquationsSection 7.3 and Section 7.4 study the existence of periodic

solutions and anti-periodic solutions for the Rayleigh typeequation with two deviating arguments, respectively

x ′′(t)+f (x ′(t))+g1(t, x(t−τ1(t)))+g2(t, x(t−τ2(t))) = e(t),(7.3.1)

where

f ∈ C (R ,R), gi ∈ C (R×R ,R), i = 1, 2, e, τi ∈ C (R ,R), i = 1, 2,

gi(t+T , x) = gi(t, x), τi(t+T ) = τi(t), i = 1, 2, e(t+T ) = e(t).

f ∈ C (R ,R), gi ∈ C (R×R ,R), i = 1, 2, e, τi ∈ C (R ,R), i = 1, 2,

gi(t+T , x) = gi(t, x), gi(t+T

2,−x) = −gi(t, x), τi(t+T ) = τi(t), τi(t+

T

2) = −τi(t), i = 1, 2,

and

e(t + T ) = e(t), e(t +T

2) = −e(t).


Main results

5. Functional Differential EquationsSection 7.5 considers positive solutions for a second-order

singular differential equations with a delayed argumentLx = ω(t)f (t, x(α(t))), t ∈ (0, 1),

x ′(0) = 0, x(1)−∫ 1

0h(t)x(t)dt = 0,

(7.5.1)

where L denotes the linear operator Lx := −x ′′ − ax′

+ bx .Generally, when y(t) ≥ 0 on J , the solution x is not

concave for the linear equation Lx − y(t) = 0. This meansthat the method depending on concavity is no longer valid,and need to introduce a new method to study this kind ofproblems.


Main results

5. Functional Differential EquationsSection 7.6 considers the one-dimensional p-Laplacian

differential equation with deviating arguments

−(φp(u′(t)))

′= ω(t)f (t, u(α(t))), t 6= tk , t ∈ (0, 1),

(7.6.1)subject to one of the following impulsive and boundaryconditions:

−4u|t=tk = Ik(u(tk)), k = 1, 2 · · · , n,u′(0) = 0, u(1) =

∫ 1

0g(t)u(t)dt,

(7.6.2)

or 4u|t=tk = Ik(u(tk)), k = 1, 2 · · · , n,u(0) =

∫ 1

0h(t)u(t)dt, u′(1) = 0.

(7.6.3)


Main results

5. Functional Differential EquationsIt is interesting to point that we open a new technique to

deal with second order p-Laplacian equations with deviatingarguments, especially for second order p-Laplacian equationswith impulsive effects.


Main results

6. Impulsive Differential EquationsThis chapter is devoted to impulsive differential equations

with parameters. Another contribution of this chapter is tointroduce a new method for dealing with impulse term ofsecond impulsive differential equations. −y

′′(t) + My(t) = λω(t)f (t, y(t)), t ∈ J , t 6= tk ,

−∆y′ |t=tk = λIk(tk , y(tk)), k = 1, 2, . . . ,m,

y ′(0) = y ′(1) = 0,(8.1.1)

−(ϕp(u′))′ = λω(t)f (t, u), 0 < t < 1, t 6= tk ,∆u|t=tk = µIk(tk , u(tk)),∆u′|t=tk = 0, k = 1, 2, . . . , n,

au(0)− bu′(0) =∫ 1

0g(t)u(t)dt, u′(1) = 0,

(8.2.1)


Main results

6. Impulsive Differential Equations

(φm(y′′(t)))

′′= λω(t)f (t, y(α(t))), t ∈ J , t 6= tk , k = 1, 2, . . . , n,

∆y′ |t=tk = −µIk(tk , y(tk)), k = 1, 2, . . . , n,

ay(0)− by′(0) =

∫ 1

0g(s)y(s)ds,

ay(1) + by′(1) =

∫ 1

0g(s)y(s)ds,

φp(y′′(0)) = φp(y

′′(1)) =

∫ 1

0h(t)φp(y

′′(t))dt,

(8.3.1)where λ > 0 and µ > 0 are two parameters,a, b > 0, J = [0, 1], φm(s) is m-Laplace operator, i.e.,φm(s) = |s|m−2s,m > 1, (φm)−1 = φm∗ ,

1m

+ 1m∗

= 1, tk(k =1, 2, . . . , n) (where n is fixed positive integer) are fixed pointswith 0 = t0 < t1 < t2 < · · · < tk < · · · < tn < tn+1 = 1,∆y

′∣∣t=tk

= y′(t+

k )− x′(t−k ), where y

′(t+

k ) and y′(t−k )

represent the right-hand limit and left-hand limit of y′(t) at

t = tk , respectively.Solvable Theory of Nonlinear Differential Equations and Their Applications

Main results

6. Impulsive Differential EquationsWe study the existence of positive solutions of the following

boundary value problem with impulse effects:λx ′′(t) + f (t, x(t)) = 0, t ∈ J , t 6= tk ,x(t+

k )− x(tk) = ckx(tk), k = 1, 2, · · · , n,ax(0)− bx ′(0) = ax(1)− bx ′(1) =

∫ 1

0h(s)x(t)dt.

(8.4.1)We shall reduce problem (8.4.1) to a system without impulse.To this goal, firstly by means of the transformation

x(t) = c(t)y(t),

we convert problem (8.4.1) into−λy ′′(t) = c−1(t)f (t, c(t)y(t)), t ∈ J ,

ay(0)− by ′(0) =∫ 1

0h(s)c(s)y(s)ds,

ac(1)y(1)− bc(1)y ′(1) =∫ 1

0h(s)c(s)y(s)ds.

(8.4.2)


Summary

This book gives a systematic study to the exactness,existence and multiplicity of solutions of p-Laplace typeequations, prescribed mean curvature equations, fractionaldifferential equations, impulsive differential equations,elasticity beam equations and functional differential equations.


References

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B.S. Lalli, B. Zhang, On a periodic delay population model,Quart. Appl. Math. 52 (1994) 35-42.

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References

T. Jankowski, Solvability of three point boundary valueproblems for second order differential equations with deviatingarguments, J. Math. Anal. Appl. 312 (2005) 620-636.

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References

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