THE ROLE OF DIFFERENTIAL INEQUALITIESIN A WIDE RANGE OF NONLINEAR PROBLEMS

V. Lakshmikantham

Abstract. The author’s scientific work in Nonlinear Sciences is sketchedbriefly. It starts in 1956 without guidance from anyone and with self study,and stretches up to the present time. It deals with a wide variety of nonlinearproblems in finite and infinite dimensions and with the classical and moderntechniques. It is collected in several research monographs and survey papers,in addition to many research papers.

1. Introduction

The author started in 1956 to embark on research work in differential equations,without any guidance. It was a common practice in the English speaking world toutilize integral inequalities to obtain the basic and qualitative results in differentialequations such as uniqueness, convergence of successive approximations, bounded-ness and asymptotic behavior of solutions. This approach demands an assumptionof monotonic increasing property in the functions involved, whereas employing dif-ferential inequalities dispenses with the monotone requirement. The author’s initialwork uses this observation and has been a forerunner for a variety of results in sev-eral types of nonlinear problems in which the theory of differential inequalities playsa crucial role. Let us briefly trace the developments of these ideas.

2. Differential Inequalities

Consider the differential system

(2.1)dx

dt= f(t, x), x(t0) = x0,

where f ∈ C[R+ ×Rn, Rn] satisfies an estimate

(2.2) ‖f(t, x)‖ 6 g(t, ‖x‖)

Key words and phrases: Nonlinear problems, differential inequalities, classical and modernmethods.

6

V. LAKSHMIKANTHAM 7

with g ∈ C[R+ × R+, R+]. Converting (2.1) to an integral equation using (2.2),one obtains the integral inequality

(2.3) m(t) 6 m(t0) +∫ t

t0

g(s,m(s))ds, t > t0,

where m(t) = ‖x(t)‖ and x(t) is any solution of (2.1). If g(t, u) is monotonenondecreasing, using the theory of integral inequalities, we get

(2.4) m(t) 6 r(t; t0, u0), t > t0,

provided m(t0) 6 u0, where r(t; t0, u0) is the maximal solution of the scalar differ-ential equation

(2.5)du

dt= g(t, u), u(t0) = u0

existing on [t0,∞). Once estimate (2.4) is obtained, certain qualitative propertiesof solutions of (2.1) can be deduced from the corresponding properties of solutions(2.5). Note that g has to be nonnegative and this is an unfavorable restriction.

On the other hand, instead of (2.3) one can use for sufficiently small h > 0, theinequality

|m(t + h)−m(t)| 6∫ t+h

t

g(s,m(s))ds.

from which it follows that

(2.6)∣∣∣dm(t)

dt

∣∣∣ 6 g(t,m(t)) a.e.

Using the theory of differential inequality, one can get estimate (2.4) from (2.6)without demanding the nondecreasing nature of g. In order to obtain (2.4), it iseven enough to have

(2.7) Dm(t) 6 g(t,m(t)),

where Dm(T ) is any one of the Dini derivatives. This observation implies that allwe need is an assumption of the type

(2.8) ‖x + hf(t, x)‖ 6 ‖x‖+ hg(t, ‖x‖) + o(h),

in place of (2.2) to derive the desired inequality (2.7). The advantage of assumption(2.8) is that g need not be nonnegative either, which yields a better estimate for‖x(t)‖. Observing (2.7) and (2.8), it is clear that one could use a nonnegativefunction with some properties of norm and this leads to

V (t + h, x + hf(t, x)) 6 V (t, x) + h(t, V (t, x)) + o(h),

where V ∈ C[R+ × Rn, R+] and V (t, x) is locally Lipschitz in x. Setting m(t) =V (t, x(t)), we arrive at the differential inequality (2.7) from which the estimate(2.4) is obtained.

We see this systematic development of the theory of differential and integralinequalities employing the norm and Lyapunov-like functions as candidates to mea-sure. This has led to several results such as uniqueness and nonuniqueness, con-vergence of successive approximation, existence of periodic solutions, stability and

8 V. LAKSHMIKANTHAM

boundedness. The comparison method in terms of Lyapunov-like functions is thecentral theme in all generalizations, extensions and refinements of Lyapunov’s sec-ond method. Extensions of these ideas to differential equations with finite andinfinite delay, differential equations in a Banach space, parabolic differential equa-tions, complex differential equations, integrodifferential equations and control sys-tems were carried out successfully. AT the same time, several new concepts instability theory such as asymptotically self-invariant sets, conditionally invariantsets, shift invariant sets and strict stability, have been introduced. It was recog-nized that the notions of Lyapunov stability and boundedness of these differentialinvariant sets lead to weaker concepts of stability and boundedness. Moreover, theadvantage of employing vector Lyapunov functions, introduced by Bellman andMatrosov, was recognized and utilized to investigate and to refine the Lyapunovtheory of stability.

The intense research activity resulted initially in a two volume monograph[15, 16] entitled, Differential and Integral Inequalities, which, although similar toSzarski’s and Walter’s book with the same title, is much more comprehensive. Vol-ume 1 is, in fact, a book on the theory of differential equations and Volterra integralequations. It contains for the first time a chapter on the method of vector Lyapunovfunctions which inspired a lot of work on large-scale dynamic systems. Volume IIcontains the theory of differential equations with delay, partial differential equa-tions, differential equations in a Banach space and complex differential equations.Moreover, in Volume II, the basic theory of differential equations with finite delayappears for the first time in the English language together with the extension ofLyapunov stability theory in terms of Lyapunov functions and functionals. Thetheory of Razumikhin’s method in terms of differential inequalities was also devel-oped. These two volumes stimulated further work and have an enormous impacton research works in this general area. Furthermore, these two volumes are stillpopular after thirty years.

3. Differential Equations in Abstract Spaces

The initial interest in differential equations in Banach spaces started with theextension of the theory of differential inequalities and Lyapunov’s method to suchequations in which the conditions are imposed on the resolvent operator in termsof a Lyapunov function to obtain a variety of results such as uniqueness, boundson norms of solutions and approximate solutions, global existence, and stabilityand boundedness of perturbed systems, generalizing the results of Mlak and Kato.The work continued in this direction and fundamental results such as uniqueness,continuous dependence and differentiability of solutions with respect to initial data,nonlinear variation of constants formula, and asymptotic equilibrium were estab-lished. The significant results concerning lower bounds and uniqueness of solutionsof evolution inequalities in a Hilbert space generalize the works of Agmon andNirenberg and Ogawa and provide a wide range of applicability to partial differen-tial equations.

V. LAKSHMIKANTHAM 9

By this time, the theory of linear evolution equations was developed satisfac-torily by Sobolevskii and Tanabe. Foreseeing the importance of this fascinatingarea of research, a monograph [12] was written which is a self-contained text thatappeals to the beginner as well as the specialist. As a natural consequence, themonograph [12] generated a lot of interest and provided a great impetus for fur-ther development of the theory of nonlinear evolution equations resulting in a largenumber of works.

Some of the important areas of investigation that were taken up during thisperiod are:

(i) existence of solutions under compactness-type conditions using Kura-towski measure of noncompactness and Lyapunov-like functions gener-alizing many existing results;

(ii) existence of extremal solutions and comparison theorems using propertiesof abstract cones and measure of noncompactness;

(iii) existence of solutions of equations of retarded type in Banach space;(iv) existence of solutions on closed subsets of a Banach space employing

dissipative-type conditions and certain boundary condition;(v) existence of weak solutions in nonreflexive Banach spaces;(vi) nonlinear contraction principle and fixed-point theorems in abstract cones;(vii) fixed-point theorems for operators whose domain and range are in different

Banach spaces;(viii) measure of nonconvexity introduced to study the convexity of solutions;(ix) extension of the method of quasilinearization;(x) boundary-value problems;(xi) quasisolutions and Muller-type results;(xii) Volterra-integral equations; and(xiii) reaction-diffusion equations.

In addition to these works, we have some survey articles which unify the resultsand provide a state-of-the-art for the abstract Cauchy problem, comparison results,abstract boundary value problems, and lecture notes on stability and asymptoticbehavior of solutions [5].

The lecture notes [5] generated further interest in the study of qualitative be-havior of solutions of nonlinear differential equations. This study, supported by thetheory of cones, the notion of measure of noncompactness and existence results inclosed sets, developed to a satisfactory level very soon. Several basic results weredeveloped in the theory of nonlinear evolution equations with continuous right-handsides. On the other hand, the study of evolution equations with set-valued, dis-continuous right-hand sides resulted in fundamental works of Crandall and Liggett,Kobayashi, and Benilan. This important state of the art in nonlinear evolutionequations formed a monograph [17] which together with the books of Barbu, Mar-tin, and Deimling gives up-to-date information on the theory of evolution equations.

10 V. LAKSHMIKANTHAM

4. Boundary Value Problems

The theory of nonlinear boundary value problems is an extremely importantand interesting area of research in differential equations. Due to the entirely dif-ferent nature of the underlying physical processes, its study is substantially moredifficult than that of initial-value problems. In the early seventies, attention isfocused on this area of activity. The important contributions are:

(i) use of vector Lyapunov-like functions and the theory of differential in-equalities to study boundary value problems on finite or infinite intervals;

(ii) showing that whatever is achieved by the application of Leray-Schauder’salternative can also be realized by the modified function technique, underthe same set of assumptions, thus shedding more light on the apparentlydiverse approaches to study boundary value problems;

(iii) use of vector Lyapunov functions to study boundary value problems offunctional differential equations;

(iv) using the angular function technique together with Lyapunov-like func-tions and differential inequalities to obtain existence of solutions for non-linear boundary value problems with generalized boundary conditions.

The substantial results that were obtained in these works paved the way fortheir compilation as a monograph [8], which contains much updated informationand a detailed discussion of various techniques that are employed in boundary valueproblems and covers the contribution of the Russian school of which the English-speaking world was not well aware.

5. Stochastic Differential Equations

Differential systems can involve random behavior in many ways, and the studyof stochastic differential equations is an important and challenging area of research.Strongly motivated and influenced by the idea that the theory of differential in-equalities and Lyapunov functions together provide a universally effective tool tostudy various problems of differential equations, the theory of random differentialinequalities and the concept of random Lyapunov functions are successfully devel-oped in order to study random differential systems by various probabilistic modes ofapproach. The monograph [13] that resulted from these investigations has severalimportant features such as:

(i) the study of random differential equations through suitable calculus;(ii) the unified treatment of stability theory through random Lyapunov func-

tions and random differential inequalities;(iii) the emphasis on the role of the method of variation of parameters; and(iv) the development of the theory of random differential equations through

various modes of probabilistic analysis.

V. LAKSHMIKANTHAM 11

6. Monotone Iterative Techniques

The method of upper and lower solutions, coupled with the monotone iterativetechnique, manifests itself as an effective and flexible mechanism that offers theo-retical as well as constructive, existence results of nonlinear problems in a closetset, namely the sector. The upper and lower solutions that generate the sectorserve as upper and lower bounds for solutions which are improved by monotone it-erative procedure. Moreover, iteration schemes are also useful for the investigationof qualitative properties of solutions.

The development of corresponding theory for discontinuous nonlinear problemsis no longer applicable. To investigate only theoretical existence results, one canconvert the given function which is discontinuous in the dependent variable intoa set-valued map and then consider the set-valued differential equations. Owingto the intrinsic difficulties of relating the partial ordering with inclusion so as toreveal the monotone character of set-valued maps, no real progress is made in thisdirection. A natural question is therefore whether one can develop some generalizedmonotone iterative technique with discontinuous problems to yield constructiveresults. The answer is positive. One can develop generalized monotone iterativetechnique in the context of partially ordered sets based on set theory and using well-ordered chains of iterations. One can then apply it to derive suitable fixed pointtheorems in ordered abstract spaces which are important tools in the systematicinvestigation of generalized monotone iterative method in terms of upper and lowersolutions of a variety of discontinuous nonlinear problems.

If we utilize the method of lower and upper solutions together with the methodof quasilinearization discussed by Bellman and Kalaba, and employ the idea ofNewton-Fourier, it is possible to construct concurrently lower and upper boundingmonotone sequences which converge quadratically. This unification also providesa mechanism to enlarge the class of nonlinear problems considerably to which themethod is applicable. For example, it is not necessary to impose convexity assump-tion as in quasilinearization. Moreover, several possibilities can be investigatedwith this unified methodology which is now known as method of generalized quasi-linearization.

The different ideas described above have played an important role in unifyinga variety of nonlinear problems. These works have been incorporated successivelyin the three monographs [11, 14, 34] which are of immense value.

7. Stability Analysis

The interesting and fruitful technique of Lyapunov’s second method gainedincreasing significance and has given decisive impetus for modern development ofstability theory. It is recognized that the concept of Lyapunov-like functions serveas a vehicle to transform a given complicated dynamic system into a relativelysimpler system and then it is enough to investigate the properties of this simplersystem. It is also realized that the same versatile tools are adaptable to discussentirely different nonlinear problems providing effective methods to investigate.

12 V. LAKSHMIKANTHAM

The concept of Lyapunov stability has given rise to many new notions that areimportant in applications. For example, eventual stability, partial stability, relativestability, perfect stability, conditional stability, and total stability, to name a few.Also, to unify and include, the variety of stability concepts, the notion of stabilityin terms of two different measures was introduced. The interplay of two differentmeasures offers new insights into the stability problem and provides an opportunityfor new applications.

From the practical point of view, a concrete system will be considered stable ifthe deviations of the motions from the equilibrium remain within certain boundsdetermined by the physical situation. The desired state of the system may beunstable in the sense of Lyapunov and yet the system may oscillate sufficientlynear this state such that its performance is acceptable. Many aircraft and missilesbehave in this manner. The notion of practical stability is introduced to cover thissituation which is neither weaker nor stronger than Lyapunov stability.

The method of vector Lyapunov functions which has proved to be an efficienttool in theory and applications for large-scale interconnected dynamic systems oc-curring in diverse areas of real world problems suffers from one drawback, namely,it requires the comparison system to be quasimonotone nondecreasing. In the caseof linear comparison systems, this means requiring nondiagonal elements of thecomparison matrix to be nonnegative. Since quasimonotonity of the matrix is nota necessary condition for the matrix to be stable, the disadvantage is clear. Itwas observed that this difficulty can be remedied by choosing an appropriate coneother than Rn

+ in which the partial ordering is considered. This idea led to thedevelopment of the theory of cone-valued Lyapunov functions which is very usefulin removing the disadvantage indicated above.

Another approach to deal with the lack of quasimonotonicity in the method ofvector Lyapunov functions is using the concept of quasisolutions of the comparisonsystem. The use of coupled quasisolutions together with the theory of cone-valuedLyapunov functions can provide greater flexibility in studying large-scale systems.

Another idea introduced in the study of nonuniform properties of solutionsand the preservation of those properties under constantly acting perturbations isperturbing the Lyapunov functions. Since the challenge involved in constructingsuitable Lyapunov functions for a given problem is well known, if the functionsfound do not satisfy the desired assumptions, then perturbing them is very useful.Moreover, employing the solutions of unperturbed systems, in developing the vari-ational Lyapunov method is another idea one can use in perturbation theory. Thisapproach combines the method of variation of parameters and of Lyapunov secondmethod in a fruitful way

The ideas and concepts described above have been incorporated in the mono-graphs [20, 21, 22, 23, 25, 26], which have contributed enormously in the de-velopment of stability analysis. Several ideas need further investigation to fullyexplore their applicability.

V. LAKSHMIKANTHAM 13

8. Systems with Finite and Infinite Delay

Systems with delay include differential equations and integrodifferential equa-tions where finite and infinite or unbounded delays play an important role. Thedevelopment of the theory of differential equations with finite delay did progressdramatically and there are now several books dealing with this branch. The theoryof differential equations with unbounded delay lagged behind, till it became neces-sary to create an adequate theory for materials with memory surfaced. Then theinvestigation proceeded quickly and the survey paper [2] provided further impetusfor the development.

In extending the method of Lyapunov to equations with delay, one has thechoice of employing Lyapunov functionals or functions. Using Lyapunov function-als demands prior knowledge of solutions of the equations under consideration.On the other hand, utilizing Lyapunov functions depends crucially on choosingappropriate minimal sets of a suitable space, along which the derivative of Lya-punov functions admits a convenient estimate. Initially, research work progressedalong these two directions. If we examine, however, the Lyapunov functionals con-structed in examples, we find that one always employs a combination of a Lyapunovfunction and a functional in such a way that the corresponding derivative can beestimated suitably without demanding prior knowledge of solutions or the minimalsets. This observation to the development of the method of Lyapunov functions onproduct spaces, which serves equally well for all types of equations with boundedor unbounded delays.

Most results in the oscillation theory of delay equations started with extendingthe corresponding results for equations without delay. The work on oscillationresults that are caused by the equations with delay, which do not exhibit oscillationbehavior for corresponding equations without delay, generated a lot of interest andhelped to create much significant work.

The three monographs [30, 34, 35] related to the ideas described above haveclearly influenced further work in the area of delay equations.

9. Nonlinear Problems in Abstract Cones

In many problems that arise from models of chemical reactors, neutron trans-port, population dynamics, infectious diseases, economics, and other systems, oneneeds to consider the existence of nonnegative solutions with desired qualitativeproperties. What one normally understands by nonnegativity can be developedthrough arbitrary cones, which define the partial order in the space to be discussed.We already know the advantage of the theory of cone-valued Lyapunov functionsin stability analysis. It is therefore clear that investigating nonlinear problemsthrough abstract cones is an important branch of nonlinear analysis. Thus employ-ing the theory of cones coupled with the fixed point index, one can discuss positivefixed points of nonlinear operators, which can be utilized to investigate positivesolutions of nonlinear differential and integral equations of Fredholm, Volterra andHammerstein types. The two monographs [10, 24] deal with the investigation of

14 V. LAKSHMIKANTHAM

nonlinear problems in abstract cones and provide several interesting applicationsto real world problems.

10. Dynamic Systems on Time Scales

The mathematical modeling of many important dynamic processes has beenvia difference equations or differential equations. Difference equations also appearin the study of discretization methods for differential equations. In recent years, thetheory of difference equations has assumed a greater importance as a well deserveddiscipline. In spite of the apparent independence, there is a striking similarityor even duality between the theories of continuous and discrete dynamic systems.More results in the theory of difference equations have been obtained as more or lessnatural analogues of corresponding results of differential equations. Nonetheless,the theory of difference equations is a lot richer than the corresponding theory ofdifferential equations. For example, a simple difference equation resulting from afirst order differential equation exhibits the chaotic behavior which can only happenfor higher order differential equations. To understand the interplay between the twodifferent theories, it is more realistic to model a phenomenon by a dynamic systemthat incorporates both continuous and discrete times. This can be achieved takingtime as an arbitrary closed set of reals called time scale or measure chain. Recently,the work in this direction has progressed and the theory of dynamic systems ontime scales or measure chains has been systematically developed. The timelyrecognition of the importance of the theory of difference equations did result in amonograph [32] which has been the forerunner for a lot of work in this direction.The recent monograph [31] demonstrates the interplay of the two different theories,namely, the theories of continuous and discrete dynamic systems when imbeddedin one unified framework.

11. Impulsive Differential Equations

Many evolution processes are characterized by the fact that at certain mo-ments of time they experience a change of state abruptly. This is due to shortterm perturbations whose duration is negligible in comparison with the duration ofthe process. It is natural, therefore, to assume that such perturbations act instan-taneously, that is, in the form of impulses. Thus impulsive differential equations,namely, differential equations involving impulse effects, appear as natural descrip-tion of observed evolution phenomenon of several real world problems. Moreover,the theory of impulsive differential equations is much richer than the correspond-ing theory of differential equations without impulse effects. Realizing the futureimportance of impulsive dynamic systems, a systematic development of the basictheory of impulsive differential equations was investigated where impulses occur atfixed as well as variable times. The monograph [7] generated much interest andis responsible to make this branch of knowledge popular. The theory of impulsivedifferential equations relative to impulses occurring at variable times has not yetprogressed well because of inherent difficulties which require new ideas.

V. LAKSHMIKANTHAM 15

12. Uniqueness and Nonuniqueness

The question of existence and uniqueness of solutions of differential equations,including the convergence of successive approximations, is an age-old problem ofgreat importance. Research has continued to discover, refine, extend and generalizeuniqueness conditions, including nonuniqueness conditions. This has resulted in anenormous amount of literature offering various sufficient conditions for uniquenessproblems, with implied conditions for the existence of solutions. Moreover, it is alsoknown that existing uniqueness criteria cannot offer any answer to the pathologicalexample x′ = xα + tβ , where 0 < α < 1 and β > 0 and therefore there are stilluniqueness problems of merit which need new ideas. A recent monograph [6] is anattempt to codify the existing results with a view to enable the readers to compareand contrast existing results and explore to discover new ideas that can solve openproblems.

13. Method of Variation of Parameters

The method of variation of parameters (MVP) has been a well known anduseful tool in the investigation of the qualitative properties of solutions of variousdynamic systems. For a long time, it was believed that the MVP is applicable onlyto nonlinear problems with linear parts giving rise to linearization techniques. Thisbelief was shaken and now there are methods for nonlinear variation of parameterswhich can be utilized for the study of nonlinear systems that have no linear partsbut have smooth nonlinear parts. These developments have given rise to a

flood of activity in refining, extending and generalizing MVP to a variety ofdynamic systems, continuous as well as discrete. Because of the fact that themethods of variation of parameters and the comparison principle via Lyapunov-likefunctions are both extremely useful and effective techniques in the investigation ofnonlinear problems, it is natural to combine these two approaches in order to utilizethe benefits of these two important methods. Such a unification has recently beenachieved and awaits further development. Moreover, the monograph [9] providesa unified development of the theory of MVP to a variety of nonlinear problems aswell as describes the blending of the two important techniques, namely, MVP andmethod of Lyapunov-like functions.

14. Monotone Flows and Rapid Convergence

We have so far described briefly the areas of investigation which are sufficientlyknown and have influenced further work. We shall next indicate the more recentdevelopments that are not yet popular.

Let us first introduce the combined methodology employed to partial differentialequations by the classical and variational methods in one unified way so that itincludes several known results and covers some new results as well [4]. We shallonly indicate monotone iterative technique and generalized quasilinearization forelliptic problems by variational method.

16 V. LAKSHMIKANTHAM

We shall consider the BVP

(14.1)[

Lu = F (x, u) in Ω,u = 0 on ∂Ω (in the sense of trace);

where L denotes the second order partial differential operator in the divergenceform

(14.2) Lu = −n∑

i,j

(aij(x)uxi)xj

+ c(x)u

and Ω is an open, bounded subset of Rn. We assume that aij , c ∈ L∞(Ω), i, j =1, 2, . . . , n, aij = aji,

n∑

i,j=j1

aij(x)ξiξj > θ|ξ|2 for x ∈ Ω, a.e. ξ ∈ Rn

with θ > 0, (uniform elliptic condition), and c(x) > 0. We shall always mean thatthe boundary condition is in the sense of trace and hence we shall not repeat it toavoid monotony. Also, F : Ω×R → R, F (x, u) is a Caratheodory function that isF (·, u) measurable for all u ∈ R and F (x, ·) is continuous a.e. x ∈ Ω. The bilinearform B[ , ] associated with the operator L is

(14.3) B[u, v] =∫

Ω

[ n∑

i,j=1

aij(x)uxivxj + c(x)uv

]dx,

for u, v ∈ H10 (Ω).

Definition 14.1. The function u ∈ H10 (Ω) is said to be a weak solution of

(14.1) if F (x, u) ∈ L1(Ω), F (x, u)u ∈ L1(Ω), and

B[u, v] = (F, v)

for all v ∈ H10 (Ω) where ( , ) denotes inner product in L2(Ω).

Definition 14.2. The function α0 ∈ H1(Ω) is said to be a weak lower solutionof (14.1) if, α0 6 0 on ∂Ω and

∫

Ω

[ n∑

i,j=1

aij(x)(α0)xivxj + c(x)α0v

]dx 6

∫

Ω

F (x, α0)vdx,

for each v ∈ H10 (Ω), v > 0. If the inequalities are reversed, then α0 is said to be a

weak upper solution of (14.1).

In this setup, one can prove the following result on monotone iterative tech-nique.

Theorem 14.1. Assume that

(i) α0, β0 ∈ H1(Ω) are weak lower and upper solutions of (14.1) such thatα0 6 β0 in Ω a.e.;

V. LAKSHMIKANTHAM 17

(ii) F : Ω×R → R is a Caratheodory function satisfying

F (x, u)− F (x, v) > −M(x)(u− v), u > v, a.e. in x ∈ Ω,

where M(x) > 0, M ∈ L∞(Ω);(iii) 0 < N 6 c(x) + M(x) a.e., x ∈ Ω and for any η ∈ H1

0 (Ω),

F (x, η) + M(x)η ∈ L2(Ω) where α0 6 η 6 β0 in Ω, a.e.

Then there exist monotone sequences αn(x), βn(x) ∈ H10 (Ω) such that

αn → ρ, βn → r weakly in H10 (Ω) with ρ, r ∈ H1

0 (Ω), (ρ, r) are minimal andmaximal weak solutions of (14.1) respectively verifying α0 6 ρ 6 r 6 β0 in Ω, a.e.

The special case F (x, u) is monotone nondecreasing in u is included in Theorem14.1 with M(x) ≡ 0. We shall state below a more general result, when F (x, u)admits a splitting of a difference of two monotone functions, which includes severalcases of interest.

An extension of the method of generalized quasilinearization is also valid in thepresent framework. We shall merely state a result.

Theorem 14.2. Assume that(i) condition (i) of Theorem 14.1 holds; suppose further that(ii) F : Ω × R → R is a Caratheodory function and Fu(x, u), Fuu(x, u) exist

and are Caratheodory functions such that Fuu(x, u) > 0 a.e., x ∈ Ω;(iii) 0 < N 6 c(x) − Fu(x, β0) a.e., x ∈ Ω and for any η ∈ H1

0 (Ω) withα0 6 η 6 β0, F (x, η)− Fu(x, η)η ∈ L2(Ω).

Then there exist monotone sequences αn(x), βn(x) ∈ H10 (Ω) such that

αn → ρ, βn → r weakly in H10 (Ω) with ρ, r ∈ H1

0 (Ω), ρ(x) = r(x) = u(x) isthe unique weak solution of (14.1) satisfying α0 6 u 6 β0 in Ω, a.e. and theconvergence is quadratic.

A dual result when F (x, u) is concave in u is also true. We shall show later thatwhen F (x, u) admits a splitting of a difference of two convex or concave functions,we can construct monotone sequences that converge to the unique weak solutionand the convergence is quadratic. This would include the foregoing results as wellas several interesting cases.

Let us next state the basic comparison result in the variational setup.

Theorem 14.3. Let α0, β0 be weak lower and upper solutions of (14.1). Sup-pose further that F satisfies

F (x, u1)− F (x, u2) 6 K(u1 − u2)

whenever u1 > u2 a.e., x ∈ Ω and K > 0. Then, if 0 < c−K ∈ L1(Ω), we have

α0(x) 6 β0(x) in Ω, a.e.

We shall now describe monotone iterative technique in a unified way proving amore general theorem which contains as special cases, several important results ofinterest.

18 V. LAKSHMIKANTHAM

Let us consider the following semilinear elliptic boundary value problem in thedivergence form

(14.4)[

Lu = f(x, u) + g(x, u) in Ω,u = 0 on ∂Ω, (in the sense of trace),

where L denotes the same second order partial differential operator in the di-vergence form (14.2) with the bilinear form B[u, v] described in (14.3). Heref, g : Ω×R → R are Caratheodory functions, other assumptions being as before.

In order to develop monotone iterative technique for the BVP (14.4), we needto utilize appropriate coupled lower and upper solutions and it is sufficient to define,in the present setup, only the following.

Definition 14.3. Relative to the BVP (14.4), the functions α0, β0 ∈ H1(Ω)are said to be

(i) weakly coupled lower and upper solutions of type I if

B[α0, v] 6 (f(x, α0) + g(x, β0), v),

B[β0, v] > (f(x, β0) + g(x, α0), v),for each v ∈ H1

0 (Ω), v > 0 a.e. in Ω;(ii) weakly coupled lower and upper solutions of type II if

B[α0, v] 6 (f(x, β0) + g(x, α0), v),

B[β0, v] > (f(x, α0) + g(x, β0), v),for each v ∈ H1

0 (Ω), v > 0, a.e. in Ω.

We are now in a position to state the following result.

Theorem 14.4. Assume that(A1) α0, β0 ∈ H1(Ω) are the weak coupled lower and upper solutions of type I

with α0(x) 6 β0(x) a.e. in Ω;(A2) f, g : Ω × R → R are Caratheodory functions such that f(x, u) is nonde-

creasing in u, g(x, u) is nonincreasing in u for x ∈ Ω, a.e.;(A3) c(x) > N > 0 in Ω, a.e. and for any η, µ ∈ H1(Ω) with α0 6 η, µ 6 β0,

the function h(x) = f(x, η) +g(x, µ) ∈ L2(Ω).Then there exist monotone sequences αn(x), βn(x) ∈ H1

0 (Ω) such thatαn → ρ, βn → r weakly in H1

0 (Ω) as n → ∞ and (ρ, r) are weak coupled minimaland maximal solutions of (14.4) respectively, that is,

Lρ = f(x, ρ) + g(x, r) in Ω, ρ = 0 on ∂Ω,

Lr = f(x, r) + g(x, ρ) in Ω, r = 0 on ∂Ω.

Concerning the method of generalized quasilinearization in a unified way, wehave the following result.

Theorem 14.5. Assume that(B1) α0, β0 ∈ H1(Ω) are lower and upper solutions of (14.4) such that α0 6 β0

in Ω, a.e.;

V. LAKSHMIKANTHAM 19

(B2) f, g : Ω×R → R are Caratheodory functions, fu(x, u), gu(x, u), fuu(x, u),guu(x, u) exist and are Caratheodory functions, and fuu(x, u) > 0, guu(x, u)6 0 for α0 6 u 6 β0 a.e. in Ω;

(B3) 0 < N 6 c(x)− fu(x, β0)− gu(x, α0) a.e. in Ω, and for any µ, η ∈ H1(Ω)satisfying α0 6 µ 6 η 6 β0, the function h ∈ L2(Ω) where

h(x) = f(x, η) + g(x, η)− fu(x, µ)η − gu(x, η)η.

Then there exist monotone sequences αk, βk ∈ H10 (Ω) such that αk → ρ,

βk → r weakly in H10 (Ω) as k →∞, with ρ = r = u is the unique weak solution of

(14.4) satisfying α0 6 u 6 β0, a.e. in Ω and the convergence is quadratic.

For further details, see [4].

15. Set Differential Equations

Let K(Rn)(Kc(Rn)) denote the collection of all nonempty, compact (compact,convex) subsets of Rn. Define the Hausdorff metric

(15.1) D[A,B] = max[

supx∈B

d(x, A), supy∈A

d(y, B)]

where d[x, A] = inf[d(x, y) : y ∈ A], A,B are bounded sets in Rn. We note thatK(Rn), (Kc(Rn)), with the metric is a complete metric space. It is known that ifthe space Kc(Rn) is equipped with the natural algebraic operations of addition andnonnegative scalar multiplication, then Kc(Rn) becomes a semilinear metric spacewhich can be embedded as a complete cone into a corresponding Banach space [37].

The Hausdorff metric (15.1) satisfies the following properties.

D[A + C,B + C] = D[A,B] and D[A, B] = D[B,A],(15.2)D[λA, λB] = λD[A,B],(15.3)D[A,B] 6 D[A,C] + D[C,B],(15.4)

for all A, B,C ∈ Kc(Rn) and λ ∈ R+.Let A,B ∈ Kc(Rn). The set C ∈ Kc(Rn) satisfying A = B + C is known as

the geometric difference of the sets A and B and is denoted by the symbol A−B.We say that the mapping F : I → Kc(Rn) has a Hukuhara derivative DHF (t0) ata point t0 ∈ I, if there exists an element DHF (t0) ∈ Kc(Rn) such that the limits

limh→0+

F (t0 + h)− F (t0)h

, and limh→0+

F (t0)− F (t0 − h)h

exist in the topology of Kc(Rn) and are equal to DHF (t0). Here I is any intervalin R.

By embedding Kc(Rn) as a complete cone in a corresponding Banach spaceand taking into account the result on differentiation of Bochner integral, we findthat if

F (t) = X0 +∫ t

0

Φ(s) ds, X0 ∈ Kc(Rn),

20 V. LAKSHMIKANTHAM

where Φ : I → Kc(Rn) is integrable in the sense of Bochner, then DHF (t) existsand the equality

DHF (t) = Φ(t), a.e on I,

holds. Also, the Hukuhara integral∫

I

F (s) ds =[∫

I

f(s) ds : f is a continuous selector of F

],

for any compact set I ⊂ R+. With these preliminaries, we consider the set differ-ential equation

(15.5) DHU = F (t, U), U(t0) = U0 ∈ Kc(Rn), t0 > 0,

where F ∈ C[R+ ×Kc(Rn),Kc(Rn)].The mapping U ∈ C1[J,Kc(Rn)], J = [t0, t0 + a] is said to be a solution of

(15.5) on J if it satisfies (15.5) on J . Since U(t) is continuously differentiable, wehave

U(t) = U0 +∫ t

t0

DHU(s) ds, t ∈ J.

Thus we associate with the initial value problem (IVP) (15.5) the following

(15.6) U(t) = U0 +∫ t

t0

F (s, U(s))ds, t ∈ J,

where the integral is the Hukuhara integral. Observe also that U(t) is a solutionof (15.5) iff it satisfies (15.6) on J .

The investigation of set differential equation (15.5) as an independent subjecthas some advantages. For example, when U(t) is a singlevalued mapping, it is easyto see that Hukuhara derivative and the integral reduce to the ordinary vector de-rivative and the integral, and therefore, the results obtained in this new frameworkof (15.5) become the corresponding results of ordinary differential systems. Also,we have only semilinear complete metric space to work with, in the present setup,compared to the complete normed linear space one employs in the study of ordinarydifferential systems. Furthermore, set differential equations, that are generated bymultivalued differential inclusions, when the multivalued functions involved do notpossess convex values, can be used as a tool for studying multivalued differentialinclusions. See [37]. Moreover one can utilize set differential equations of the type(15.5) to investigate profitably fuzzy differential equations, since the original for-mulation of which suffers from grave disadvantages and does not reflect the richbehavior of corresponding differential equations without fuzziness which we shalldescribe in the next section [18].

Here we shall state some recent results [28] in this direction. Let us start witha comparison result.

Theorem 15.1. Assume that F ∈ C[R+ × Kc(Rn),Kc(Rn)] and for t ∈R+, A, B ∈ Kc(Rn)

D[F (t, A), F (t, B)] 6 g(t, D[A,B]),

V. LAKSHMIKANTHAM 21

where g ∈ C[R2+, R+]. Suppose further that the maximal solution r(t, t0, w0) of the

scalar differential equation

w′ = g(t, w), w(t0) = w0 > 0,

exists for t > t0. Then, if U(t) = U(t, t0, U0), V (t) = V (t, t0, V0) are any twosolution of equation (15.5) such that U(t0) = U0, V (t0) = V0, U0, V0 ∈ Kc(Rn)existing for t > t0, we have:

D[U(t), V (t)] 6 r(t, t0, w0), t > t0,

provided D[U0, V0] 6 w0.

The next result is an existence and uniqueness Theorem more general than Lip-schitz type condition the proof of which exhibits the idea of comparison principle.

Theorem 15.2. Assume that(a) F ∈ C[R0,Kc(Rn)], where R0 = J × B(U0, b), J = [t0, t0 + a], a > 0,

B(U0, b) = [U ∈ Kc(Rn) : D[U,U0] 6 b] and D[F (t, U), θ] 6 M0 on R0,where θ is the zero element of Rn regarded as a one point set;

(b) g ∈ C [J × [0, 2b], R+], g(t, w) 6 M1 on J × [0, 2b], g(t, 0) ≡ o, g(t, w) isnondecreasing in w for each t ∈ J and w(t) ≡ 0 is the only solution of

w′ = g(t, w), w(t0) = 0;

(c) D[F (t, U), F (t, V )] 6 g(t,D[U, V ]) on R0.Then the successive approximations defined by

Un+1(t) = U0 +∫ t

t0

F (s, Un(s))ds, n = 0, 1, 2, . . . ,

exist on J0 = [t0, t + α], α = min(a, b/M ], M = max(M0,M1), as continuousfunctions and converge uniformly to the unique solution U(t) = U(t, t0, U0) of IVP(15.5) on J0.

The following global existence result is a special case of Theorem 5.2 in [28]which serves our purpose.

Theorem 15.3. Assume that(1) F ∈ C[R+ ×Kc(Rn),Kc(Rn)] and for (t, A) ∈ R+ ×Kc(Rn),

D[F (t, A), θ] 6 q(t, D[A, θ]),

where q ∈ C[R2+, R+], q(t, w) is nondecreasing in w for each t ∈ R+ and

the maximal solution r(t, t0, w0) of

w′ = q(t, w), w(t0) = w0,

exists for t > t0 and for every w0 > 0;(2) there exists a local solution U(t) = U(t, t0, U0) of (15.5) for every (t0, U0) ∈

R+ ×Kc(Rn).Then for every U0 ∈ Kc(Rn) such that D[U0, θ] 6 w0, the IVP (15.5) possesses

a solution U(t) = U(t, t0, U0) defined for t > t0, satisfying

D[U(t), θ] 6 r(t, t0, w0), t > t0.

22 V. LAKSHMIKANTHAM

A result that relates the solution of set differential equation to the attainableset of multivalued differential inclusion is the following result. See [37]

Theorem 15.4. Assume that F ∈ C[R+ ×Rn,Kc(Rn)];

D[F (t, x), F (t, y)] 6 g(t, ‖x− y‖), (t, x, y) ∈ R+ ×Rn ×Rn,

and D[F (t, x), θ] 6 q(t, ‖x‖), (t, x) ∈ R+ × Rn, where g and q satisfy the assump-tions listed in Theorem 15.2 and Theorem 15.3 respectively, except that conditionsrelative to g hold for R+ ×R+.

Then there exist a unique solution U(t) = U(t, t0, U0) on [t0,∞) of IVP (15.5)and the attainable set A(U0, t) of differential inclusion

x′ ∈ F (t, x), x(t0) ∈ U0,

satisfying A(U0, t) ⊂ U(t), t0 6 t < ∞.

Finally, we need the following representation result [18].

Theorem 15.5. Let Yβ ⊂ Rn, 0 6 β 6 1 be family of compact subsets satisfying(a) Yβ ∈ K(Rn) for all 0 6 β 6 1;(b) Yβ ⊆ Yα whenever α 6 β, α, β ∈ [0, 1];

(c) Yβ =∞∏

i=1

Yβi , for any nondecreasing sequence βi → β in [0, 1].

Then there is a fuzzy set u ∈ Dn, such that [u]β = Yβ. If Yβ are also convex, thenu ∈ En. (Here Dn denotes the set of usc normal fuzzy set with compact supportand thus En ⊂ Dn). Conversely, the level sets [u]β, of any u ∈ En, are convex andsatisfy these conditions.

16. Fuzzy Differential Equations

In the mathematical modeling of real world phenomena, we encounter two in-conveniences. The first is caused by the excessive complexity of the model. As thecomplexity of the system being modeled increases, our ability to make precise andyet relevant statements about its behavior diminishes until a threshold is reachedbeyond which precision and significance become almost mutually exclusive charac-teristics. As a result, we are either not able to formulate the mathematical modelor the model is too complicated to be useful in practice.

The second inconvenience relates to the indeterminacy caused by our inabilityto differentiate events in a real situation exactly, and therefore to define instru-mental notions in precise form. This indeterminacy is not an obstacle, when weuse natural language, because its main property is the vagueness of its semanticsand therefore capable of working with vague notions. Classical mathematics, onthe other hand, cannot cope with such vague notions. It is therefore necessary tohave some mathematical apparatus to describe vague and uncertain notions andthereby help to overcome the foregoing obstacles in the mathematical modeling ofimprecise real world systems.

In 1965, Zadeh initiated the development of the modified set theory knownas fuzzy set theory, which is a tool that makes possible the description of vague

V. LAKSHMIKANTHAM 23

notions and manipulations with them. The basic idea of fuzzy set theory is simpleand natural. Let us describe the present state of it.

Let us denote by En = [u : Rn → [0, 1] such that u satisfies (i) to (iv) givenbelow].

(i) u is normal, that is, there exists an x0 ∈ Rn such that u(x0) = 1;(ii) u is fuzzy convex, that is, for x, y ∈ Rn and 0 6 λ 6 1,

u(λx + (1− λ)y) > min(u(x), u(y));

(iii) u is upper semi-continuous(usc);(iv) [u]0 = cl[u ∈ En : u(x) > 0] is compact.For 0 < α 6 1, we denote [u]α = [x ∈ Rn : u(x) > α]. Then from (i) to (iv), it

follows that the α - level set [u]α ∈ Kc(Rn) for 0 6 α 6 1. We set

D0[u, v] = sup06α61

D [[u]α, [v]α]

which defines a metric in En and (En, D0) is also a semilinear complete metricspace [18]. Also, D0[u, v] satisfies similar properties as D[A, B] listed in (15.2) to(15.4). One can also define Hukuhara derivative DHF (t) ∈ En for a fuzzy mapF : J → En, where J = [t0, t0 + a], a > 0, similarly as before, taking limits in themetric space (En, D0). Moreover, if F : J → En is continuous, it is integrable,the integral G(t) =

∫ t

t0F (s) ds is differentiable and DHG(t) = F (t). Furthermore

F (t) = F (t0) +∫ t

t0DHF (s) ds, t ∈ J .

In this setup, the IVP for fuzzy differential equations originally proposed is ofthe type

(16.1) DHu = f(t, u), u(t0) = u0 ∈ En,

where f ∈ C[R+ × En, En], for which basic results are discussed. See [18]. Thisapproach is based on the fuzzification of the differential operator, whose values arein En and therefore suffers from the disadvantage, since the solution u(t) of (16.1)has the property that diam[u(t)] is nondecreasing as t increases. Consequently,this formulation can not fully reflect the rich behavior of solutions of correspondingordinary differential equations.

Recently, Hullermeier has suggested an alternative formulation of fuzzy IVPsby replacing the R.H.S of a system of ordinary differential equation by a fuzzyfunction

f : R+ ×Rn → En,

and with the initial fuzzy set x0 ∈ En, so that one can consider the fuzzy multival-ued differential inclusion

(16.2) x′ ∈ f(t, x), x(t0) = x0 ∈ En,

on J , where now f is defined from R+×Rn → En rather than R+×En → En as in(16.1). However, instead of (16.2), a sequence of multivalued differential inclusions

(16.3) x′β ∈ F (t, xβ ;β), xβ(t0) ∈ [x0]β , 0 6 β 6 1,

24 V. LAKSHMIKANTHAM

is investigated on J , where F (t, x; β) ≡ [f(t, x)]β and F (t, x, 0) = supp(f(t, x)).The idea is that the set of all solutions Sβ(x0, T ), t0 6 t 6 T , would be β-level ofa fuzzy set S(x0, T ), in the sense that all attainable sets Aβ(x0, t), t0 < t 6 T , arelevels of a fuzzy set on Rn. Considering S(x0, T ) to be the solution of (16.1) thuscaptures both uncertainty and the rich behavior of differential inclusion in one andthe same technique.

For this purpose, the standard results of multivalued differential inclusions,under the usual conditions on F in (16.3) yield that the attainable set Aβ(x0, t)is compact subset of Rn. If F is assumed to be quasiconcave in addition, one canconclude, under reasonable assumptions, utilizing the representation theorem, theexistence of a fuzzy set u(t) such that [u(t)]β = Aβ(x0, t) with a similar relation forthe solution set Sβ(x0, T ). See for details [18].

Let us consider the often quoted simple example to show the advantage gainedby the alternative approach when compared to the original one.

Consider the crisp initial value problem with unknown initial value x0, that is,

(16.4) x′ = −x, x(0) = x0 ∈ [−1, 1].

The solution of (16.4) when restricted to the interval [−1, 1] is x(t) = [−e−t, e−t],t > 0.

The fuzzy differential equation corresponding to (16.4) in E1 is

(16.5) DHx = −x, x(0) = x0 = [−1, 1], x0 ∈ E1.

Suppose that [x]β = [xβ1 , xβ

2 ], [DHx]β = [dxβ1

dt ,dxβ

2dt ], are the β - level sets for 0 6

β 6 1. By extension principle, (16.5) becomes

(16.6)dxβ

1

dt= −xβ

2 ,dxβ

2

dt= −xβ

1 , 0 6 β 6 1.

The solution of (16.6) is given by xβ1 (t) = −et, xβ

2 (t) = et and therefore thefuzzy function x(t) solving (16.5) is x(t) = [−et, et], t > 0, which shows thatthe diam(x(t)) →∞ as t →∞.

In the framework of Hullermeier, on the other hand, fuzzy differential equation(16.5) is replaced by the family of inclusions

(16.7) x′β ∈ F (t, xβ ; β) = −[xβ2 , xβ

1 ], x(0) = [−1, 1];

which has a fuzzy solution set S([−1, 1], T ), 0 6 t 6 T and fuzzy attainable setAβ([−1, 1], t) respectively are defined by β-level sets

Sβ([−1, 1], T ) = [x(·) : x(t) ∈ [−e−t, e−t]] 0 6 t 6 T,

(16.8) Aβ [[−1, 1], t) = [−e−t, e−t],

which matches the kind of behavior a fuzzyfications of the crisp differential equation(16.4) should have.

We now propose another formulation of fuzzy differential equation (16.1) by aset differential equation which is generated by β-level set of the R.H.S. of (16.1),

V. LAKSHMIKANTHAM 25

where f : R+ × Rn → En, as before. For this purpose, consider the level set foreach β, 0 6 β 6 1, and write

F (t, x;β) = [f(t, x)]β ∈ Kc(Rn).

Next generate the mapping H : R+×Kc(Rn)× I → Kc(Rn), I = [0, 1] by defining

H(t, A; β) = F (t, A; β)

for each A ∈ Kc(Rn). Then consider the sequence of set differential equations givenby

(16.9) DHUβ = H(t, Uβ ; β), Uβ(t0) = U0β ∈ Kc(Rn),

on [t0, T ], T > t0, where DHUβ is the Hukuhara derivative for each β.Let us list the following conditions.(1) F (t, x; β) is quasi-concave, that is,

(a) for (t, x) ∈ R+×Rn, α, β ∈ I, F (t, x; α) ⊇ F (t, x, β) whenever α 6 β(b) if βn is nondecreasing sequence in I, converging to β, then for (t, x) ∈

R+ ×Rn,∞⋂

n=1F (t, x;βn) = F (t, x; β).

(2) D[H(t, A;β), H(t, B; β)] 6 g(t,D[A,B])for t ∈ R+, A,B ∈ Kc(Rn), β ∈ I;

(3) g ∈ C[R2+, R+], g(t, 0) ≡ 0, g(t, w) is nondecreasing in w for each t ∈ R+

and w(t) ≡ 0 is the only solution of

w′ = g(t, w), w(t0) = 0,

for t > t0;(4) D[H(t, A;α), H(t, A;β)] 6 L|α − β|, α, β ∈ I, (t, A) ∈ R+ × Kc(Rn),

L > 0.We are now in a position to state the following result.

Theorem 16.1. Suppose that the assumptions (1) to (4) are satisfied. Thenthere exists a unique solution Uβ(t) = Uβ(t, t0, U0β) ∈ Kc(Rn), β ∈ I of (16.9) andUβ(t) is quasiconcave in β for t > t0. Moreover, there exists a fuzzy set u(t) ∈ En

such that[u(t)]β = Uβ(t), t > t0.

To find the connection between the solution Uβ(t), of (16.9) and the attainabil-ity set Aβ(U0, t) of (16.3), we have the following result.

Theorem 16.2. Let F ∈ C[R+ × Rn × I, Kc(Rn)] satisfy the assumptions ofTheorem 15.4 for each β ∈ I = [0, 1] and assume that it is also quasiconcave in βas well. Then there exist a unique solution Uβ(t) = Uβ(t, t0, U0) of (16.9) for t > t0and the attainable set Aβ(U0, t) of the inclusion (16.3) such that

(16.10) Aβ(U0, t) ⊂ Uβ(t, t0, U0), t > t0.

Recalling (16.7) in the example considered earlier, let us generate the set dif-ferential equation from F in (16.7), that is,

H(t, U ;β) = [F (t, x;β) : x ∈ U ∈ Kc(Rn)]

26 V. LAKSHMIKANTHAM

and thusDHUβ = −Uβ , Uβ(0) = U0β ∈ Kc(Rn)

so that Uβ(t) = U0βe−t, t > 0. If we choose U0β = [−1, 1] only, then

Uβ(t) = [−e−t, e−t] = Aβ([−1, 1], t),

which matches with (16.8). If the β-level set of x0 in (16.5) is [x0]β = [xβ1 , xβ

2 ], thenUβ(0) = U0β = [[xβ

1 , xβ2 ] : x0 ∈ U0β ], x0 ∈ R, then we have (16.10) satisfied.

We have only provided below the references of research monographs and thesurvey of collected works, which contain many references to the literature.

References

[1] Agarwal, R. and Leela, S., A brief biography and survey of collected works of V. Lakshmikan-tham, J. of Nonl. Anal. 40 (2000), 1-19.

[2] Corduneanu, C. and Lakshmikantham, V., Equations with unbounded delay: A survey, J.Nonl. Anal. 4 (1980), 831-877.

[3] Diamond, P. and Kloeden, P., Metric Spaces of Fuzzy Sets, World Scientific, Singapore 1992.[4] Koksal, S. and Lakshmikantham, V., Monotone Flows and Rapid Convergence for Nonlinear

Partial Differential Equations, Taylor and Francis, Ltd., Abington, UK (to appear).[5] Lakshmikantham, V., Stability and Asymptotic Behavior of Solutions of Differential Equa-

tions in a Banach Space, Lecture Notes, C.I.M.E., Italy 1974.[6] Lakshmikantham, V. and Agarwal, R.P., Uniqueness and Nonuniqueness Criteria for Ordi-

nary Differential Equations, World Scientific, Singapore 1993.[7] Lakshmikantham, V. and Bernfeld, S.R., An Introduction to Nonlinear Boundary Value

Problems, Academic Press, New York, 1974.[8] Lakshmikantham, V. and Deo, S.G., Method of Variation of Parameters for Nonlinear Dy-

namics Systems, Gordon and Breach, England 1997.[9] Lakshmikantham, V. and Guo, D., Nonlinear Problems in Abstract Cones, Academic Press,

New York 1988.[10] Lakshmikantham, V. and Heikkila, S., Monotone Iterative Techniques for Discontinuous

Differential Equations, Marcel Dekker, New York 1994.[11] Lakshmikantham, V. and Ladas, G., Differential Equations in Abstract Spaces, Academic

Press, New York 1972.[12] Lakshmikantham, V. and Ladde, G.S., Random Differential Inequalities, Academic Press

1980.[13] Lakshmikantham, V. and Leela, S., Differential and Integral Inequalities: Theory and Ap-

plications, Vol. I, Academic Press, New York 1969.[14] Lakshmikantham, V. and Leela, S., Differential and Integral Inequalities: Theory and Ap-

plications, Vol II, Academic Press, New York 1969.[15] Lakshmikantham, V. and Leela, S., An Introduction to Nonlinear Differential Equations in

Abstract Spaces, Pergamon Press, 1981.[16] Lakshmikantham, V. and Leela, S., The Origin of Mathematics, 2000.[17] Lakshmikantham, V. and Liu, X., Stability Analysis in Terms of Two Measures, World

Scientific, Singapore, 1993.[18] Lakshmikantham, V. and Mohapatra, R., Theory of Fuzzy Differential Equations and Inclu-

sions, Taylor and Francis, Abington, UK (to appear).[19] Lakshmikantham, V. and Rao, M.R.M., Theory of Integro-Differential Equations, Gordon

and Breach, 1994.[20] Lakshmikantham, V. and Trigiante, D., Theory of Difference Equations and Numerical Anal-

ysis, Academic Press, New York, 1988.[21] Lakshmikantham, V. and Vatsala, A.S., Generalized Quasilinearization for Nonlinear Prob-

lems, Kluwer, 1998.

V. LAKSHMIKANTHAM 27

[22] Lakshmikantham, V., Bainov, D. and Simeonov, P., Theory of Impulsive Differential Equa-tions, World Scientific, Singapore 1989.

[23] Lakshmikantham, V., Deo, S.G., and Raghavendra, V., Ordinary Differential Equations,Tata McGraw-Hill, India, 1997.

[24] Lakshmikantham, V., Ladde, G.S. and Vatsala, A.S., Monotone Iterative Techniques in Non-linear Differential Equations, Pitman, 1985.

[25] Lakshmikantham, V., Leela, S. and Martynyuk, A., Stability of Motion: The Method ofIntegral Inequalities, Nauka Dumka, Kiev, USSR 1989. (Russian)

[26] Lakshmikantham, V., Leela, S. and Martynyuk, A.A., Practical Stability of Nonlinear Sys-tems, World Scientific Publishers, Singapore 1990.

[27] Lakshmikantham, V., Leela, S. and Martynyuk, A.A., Stability of Motion: ComparisonMethod, Nauka Dumka, Kiev, USSR 1991. (Russian)

[28] Lakshmikantham, V., Leela, S. and Vatsala, A.S., Set-valued differential equations and sta-bility in terms of two measures, J. Hybrid Sys. 2 (2002), 169–187.

[29] Lakshmikantham, V., Leela, S. and Vatsala, A.S., Interconnection between set and fuzzydifferential equations, J. Nonl. Anal. (to appear).

[30] Lakshmikantham, V., Liu, X. and Guo, D., Nonlinear Integral Equations in Abstract Spaces,Kluwer, 1996.

[31] Lakshmikantham, V., Matrosov, V.M. and Sivadundaram, S., Vector Lyapunov Functionsand Stability Analysis of Nonlinear Systems, Kluwer, New York 1991.

[32] Lakshmikantham, V., Martynyuk, A. and Leela, S., Stability Analysis of Nonlinear Problems,Marcel Dekker, New York 1989.

[33] Lakshmikantham, V., Sivasundaram, S. and Kaymacalan, B., Dynamic Systems on MeasureChains, Kluwer, 1996.

[34] Lakshmikantham, V., Wen, L. and Zhang, B.G., Theory of Differential Equations with Un-bounded Delay, Kluwer, 1994.

[35] Lakshmikantham, V., Zhang, B. and Ladde, G.S., Theory of Oscillation for DifferentialEquations with Deviating Arguments, Marcel Dekker, New York, 1987.

[36] Leela, S., Mitropol’ski, Yu.A., and Martynyuk, A.A., A survey of collected works of Laksh-mikantham, Differential’nye Uravneniya, Vol. 22 (1986), 555-572.

[37] Tolstonogov, A., Differential Inclusions in a Banach Space, Kluwer, Dordrecht, 2000.