| In this thesis, we mainly study properties of several class of nonlinear operatorswith applications.The thesis consists of 4 chapters.In Chapterâ… , we mainly study the nonlinear operators with the form C=A+B,where B is a constant operator, a linear operator or anα-concave operator (0＜α＜1).Some problems, such as three-point boundary value problems, singular boundary valueproblems and impulse problems, are often transformed into this class of operators. Re-searching deeply the class of operators is helpful for discussing the above problems.We introduce the notion of locally u0-concave operators. This is a class of operatorscontaining u0-concave oprators with wider range than u0-concave operators. We provethat C is a locally u0-concave operator when A satisfies some certain conditions, andobtain some existence and uniqueness theorems of fixed points for the class of oper-ators. In these theorems, we do not suppose that the operator C has both an uppersolution and a lower solution, we also do not suppose that the operator C has anycompactness and continuity. Main results are as follows.Let E be a real Banach space, P a normal cone in E, h＞θ, f∈Ph and M＞0.Assume that A: P→P is an increasingα-homogeneous operator (α＞1). Theoperator C is given by Cu=Au+Mf, u∈P. If there exists v0∈Ph such that(â…°) Cv0≤v0;(â…±) Av0≤mf, where m∈(0,M/(α-1),then(â…°) C has a unique fixed point x* in [θ,v0], and x*∈Ph; there exists v′0∈Ph withv′0＞v0 such that C has not any fixed points in [θ, v′0]\[θ, v0];(â…±) for any x0∈[θ, v0], writing xn+1=Cxn, n=0,1, 2,…, we have (?) xn=x*.Moreover, there exist (?),γ∈(0,1)such that‖xn-x*‖≤2N(1-(?)γn)‖vo‖, n=1,2,…,where N is the normal constant of P.Next we apply these theorems to discuss the three-point boundary value problems andobtain some results about the existence and uniqueness of solutions for them. It needto be pointed out that for nonlinear three-point boundary value problems, existencetheory does not usually give uniqueness. At last, we apply the obtained results todiscuss the equations including the nonlinear Volterra integral equationSince its parameterα＞1, very little is known of the properties of its solutions. Wegive some existence and uniqueness theorems of solutions for it. Main results are asfollows.Let E=C[0, 1], P={x|x∈E,x(t)≥0,t∈[0,1]},κ(t,s) is a nonnegativecontinuous function which is not identically zero on D={(t, s)|0≤s≤t≤1}.h(t)=integral from n=0 to tκ(t,s)ds, t∈[0,1]. g∈C([0,+∞), [0,+∞)) is incresing and satisfies thatg(lu)≥lαg(u)(α＞1) for any u＞0, l∈(0,1). Assume that v0, f∈Ph satisfies(â…°) v0(t)≥α/(α-1)f(t), t∈[0,1];(â…±) there existsε∈(0,1/(α-1)) such thatthen the above equation has a unique solution x* such that 0≤x*(t)≤v0(t), t∈[0,1],and x*∈Ph; and there exists (?)0∈Ph with (?)0＞v0 such that the above equation has nosolutions in [θ, (?)0]\[θ, v0]. Moreover, for any x0∈[θ, v0], making an iterated sequencewe have the sequence {xn(t)} converges uniformly to x*(t) on [0, 1], and there existl,γ∈(0,1) such that (?)|xn(t)-x*(t)|≤(1-lγn)(?)|v0(t)|,n=1,2,….In Chapterâ…¡, we mainly study mixed monotone operators. Mixed monotoneoperators were introduced by Dajun Guo and V. Lakshmikantham in 1987. Theyhave great singnificance for studying nonlinear analysis, variational calculus, nonlineardifferential equations and integral equations. They have already been widely appliedin many fields such as engineering technology, nuclear physics and biological chemistry technology—for example, an infectious disease model. In applications, people usuallyneed to study existence and uniqueness of fixed points for relevant mixed monotoneoperators. We study the mixed monotone operators satisfying the conditionorand obtain several existence and uniqueness theorems for their fixed points. In thesetheorems, we do not suppose that the operators have an upper solution or a lowersolution, and we also do not suppose that the operators have any continuity andcompactness. Our results imply many relevant conclusions to date. Main results areas follows.Let E be a real Banach space, P a normal cone in E and h＞θ. Suppose thattwo positive-valued continuous functions f(t),g(t) and a real-valued function w(t) on(a, b) satisfy that(â…°) (?){f(t)}=1=(?){g(t)};(â…±) (?)f(t)=(?)1/g(t)=0 or (?)f(t)=(?)1/g(t)=0(â…²) (f(t1)-f(t2))(g(t1)-g(t2))≤0 for any t1, t2∈(a, b);(â…³) w(t)＞f(t) and w(t)g(t)＞1 for any t∈(a, b), where a, b are real numbers with a＜b. Assume that A: Ph×Ph→Ph is a mixedmonotone operator satisfying thatfor any t∈(a, b) and u, v∈Ph.Then A has a unique fixed point x* in Ph. Moreover, for any x0, y0∈Ph, makingiterated sequenceswe have (?) xn=x* and (?) yn=x*. In addition, if ((?),(?))∈Ph×Ph is a couplefixed point of A, then (?)=(?)=x*.At last, we apply them to discuss existence and uniqueness of solutions for the integral equation x(t)=integral from n=Gκ(t,s){[F(x(s))]α+[G(x(s))]-β}ds.In Chapterâ…¢, we discuss the problems about existence of positive eigenvaluesfor some nonlinear operators. The notion of eigenvalues is very important in nonlinearfunctional analysis, whose existence is a fundamental problem in nonlinear functionalanalysis. We apply Banach contractive mapping principle to obtain a series of resultsabout the existence of positive eigenvalues and positive eigenvectors for Lipschitz map-pings on cones by functional. We also apply the fixed point theorems for condensingmappings to obtain some results about positive eigenvalues ofκ-α-contractive map-pings, which improve the conclusions in the recent relevant references. Main resultsare as follows.Let (E,‖·‖) be a reflexive real Banach space, P a cone in E, (?)=P\{θ},α,β＞0are constants. Assume that the mapping G: E×E→R satisfies the followingconditions:(g1) G(λx, y)≤λαG(x,y), x, y∈E,λ＞0;(g2)‖x‖β≤G(x, x), x∈E.Let f: P→(?) be a Lipechtz mapping whose parameter isÏ＞0,ÏI-f a demi-closedmapping. IfthenÏis an eigenvalue of f associated to an eigenvector in (?).In Chapterâ…£, we study convex functionals. They are a most important class offunctionals in convex analysis. Convex set seperation theorem is an important toolstudying convex functionals. We first provide a ball seperation property of boundedconvex sets in Hilbert spaces. On the basis of it, we present two representation formsof convex closures and two results about convex functional.
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