Solutions Of Nonlinear Operator Equations And Applications

Nonlinear operator theory is one of important parts of nonlinear functional analysis. It does not only provide powerful tools for studying nonlinear differen-tial and integral equations, but also brings them into unified structure. Therefore, it has wide applications in the applied sciences such as physical sciences, engi-neering, biological chemistry technology as well as in mathematics. The problems on numbers and types of solutions for nonlinear operator equations have received a lot of attention. In this paper, firstly, we study fixed point theorems for some mixed monotone operators with perturbation. Secondly, we prove the existence of multiple fixed points for two classes of nonlinear operators. Thirdly, we discuss multiple solutions and sign-changing solutions of asymptotically linear operator equations.Chapter 1 introduces some preliminaries, which will be used in the remaining chapters. Section 1.1 introduces the concepts of partial ordering and cone. In Section 1.2, we give some basic results on calculations of time scales. In Section 1.3, we introduce some concepts of topological degree theory and fixed point index, and some related lemmas are also given.Tn Chapter 2, we employ the partial ordering theory and monotone iterative technique to study the existence and uniqueness of solutions of the following operator equation where A is a mixed monotone operator, B is sublinear, and E is a real ordered Banach space. The operator A has the following convexity and concavity condi- tions: whereτ:(a,b)(?)(0,1) is a surjection,φ(t,x,y)>τ(t),(?)(a,b),x,y(?)P, and P is a normal cone in E. We should point out that we do not require the operator A to have coupled upper-lower solutions,compactness and continuity conditions.As applications, the existence and uniqueness of a kind of integral equations is discussed. Moreover, a class of second-order boundary value problem on time scales is considered, we do not only obtain the existence and uniqueness of positive solutions of this problem, but also establish the iterative schemes for approximating the solution.In Chapter 3,firstly,by using the fixed point index theory,we study multiple fixed points of nonlinear operator A under the following condition of two pairs of paralleled lower and upper solutions We assume that E is a real Banach space,P,Q are both normal cones in E, Q(?)P,Q≠{0},whereθdenotes the zero dement of E. Let A:P→P be a condense increasing operator. A(P)(?)Q. Suppose that the following conditions are satisfied:(ⅰ)there exist h∈P\{θ} and a functional f:Q→R~+ with f(x)→+∞(||x||→+∞),such that Ax≥f(x)h.(?)x∈Q;(ⅱ)A|Q is e-continuous, and c∈Q\{θ};(ⅲ) there existλ1,μ1,γ1,λ2,μ2,γ2>0 and positive integers m1,n1,m2,n2 such that Then A has at least six fixed points in P. Furthermore, we apply the obtained abstract result to superlinear Hammerstein integral equation,and the existence theorem of at least six solutions for this problem is established. Secondly, we combineτ-φ-concave operators withτ-φ-convex operators, and the existence of two positive fixed points for some nonlinear operators is considered.Suppose that the following conditions are satisfied:(a)P is a normal cone of real Banach space E, N is the normal constant of P, A:P→P is a strict set contraction, which satisfies that(b)there exist operators Ai:P→P such that(c)A1 is aτ1-φ1-concave operator, and where If there exists a positive number c such that A2 is aτ2-φ2-convex operator,and Then A has at least two fixed points x1*,x2* in P\{θ},such that ||x1*||<1<||x2*||. Our tools are based on the properties of normal cones and the fixed point theorem of cone expansion and compression. As corollaries, we also obtain some fixed point theorems for the sum of aφ1-concave operator and aφ2-convex operator. Finally, the main fixed point theorem is applied to a class of multi-point boundary value problems for second-order differential equations.Tn Chapter 4,firstly, under the assumption that: asymptotically linear oper-ator A has the following two pairs of lower and upper solutions(ⅰ) there exist u1∈(-P)\{θ} and v1∈P\{θ} such that u1≤Au1 and Av1≤v1;(ⅱ) there exist u2∈(-P)\{θ}, v2∈P\{θ},andδ>0 such that u10,μ2>0. For some concrete problems, it is easy to check the condition (ⅱ)'.Secondly,under the condition of a known result that topological degree is 1, employing the theorems of index calculation of Frechet differentiable operators and asymptotically linear operators, we obtain that this class of nonlinear oper-ator equations have at least two sign-changing solutions, two positive solutions and two negative solutions. Thirdly,sign-changing fixed point and multiple fixed points of unilaterally asymptotically linear operators under the lattice structure are also discussed. Finally, we apply the abstract results to nonlinear Hammer-stein integral equations and a class of boundary value problems for elliptic partial differential equations. And the multiple sign-changing solutions of a class of dis-crete boundary value problems are also considered. In this chapter, we do not only abstract some general conditions from those of some concrete differential equations, but also improve some known conditions. Some known results are equipped with the wider meanings.