若干非线性算子的理论及其应用研究
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摘要
本文的工作主要有两个方面,一方面讨论了两类非线性算子方程:一类为Banach空间中的非线性混合单调算子方程;另一类为有序的局部凸拓扑线性空间中集值(或多值)映象方程,所使用的方法主要为半序方法及单调迭代技巧等,另一方面是利用非线性算子理论讨论了微分方程解的存在性问题。
全文共分为三章。
在第一章中,我们对几类非线性算子(u_0-凹算子,混合单调算子,集值算子等)的研究现状进行了阐述,同时简明地介绍了我们在本文中将要做的主要工作。
第二章,我们给出了几类非线性算子的不动点定理。
在§2.1和§2.2中,我们给出了t-α(t)型凹凸及t-α(t,u,v)型凹凸的混合单调算子具有唯一不动点的新的存在性定理(参见定理2.1.1和定理2.2.1),本质性地改进了最近许多相关文献原有的条件和结论。在u_0-凹增算子的情形,与这些存在性定理相类似的结论实际上也是成立的,作为所得理论结果的一个应用,讨论了一类非线性积分方程解的存在性。
在§2.3中,我们引入了一类ω-凹凸型混合单调算子,给出了它存在唯一不动点的充分必要条件(参见定理2.3.1等),在§2.4中,讨论了混合单调算子和的不动点的存在性(参见定理2.4.3),所得结果改进并推广了相关文献中的部分结果。
在§2.5中,我们利用§2.1§2.2所得的理论结果并结合算子半群的性质,讨论了Banach空间非线性发展方程解的存在性(参见定理2.5.2和定理2.5.3),所得结果改进了相关文献中的工作。
在§2.6中,我们首先在局部凸拓扑线性空间中引入序,给出了集值(多值)凝聚映射的几个不动点定理(参见定理2.6.3和定理2.6.8等),推广了相关文献在Banach空间中所做的一些工作,然后利用所得结果,讨论了优化理论中的一个带约束条件的极小化问题:
x∈G(x),ω(x,x)=(?)ω(x,y)
其中ω为二元连续实函数,G为集值映象,给出了它存在正解的充分条件(参见定理2.6.9)。
There are mainly two works in this paper. On the one hand, we discuss two classes of nonlinear operator equations, including nonlinear mixed monotone operator equations in Bnanch space and the set- valued mapping equation in an ordered locally convex topological linear space. The methods employed are mainly partial ordering method and iterative techniques and so on. On the other hand, we will use more precise theory of nonlinear operator to discuss the existence problems of solutions for differential equations.This paper includes three chapters.In Chapter 1, we provide a research summary of several classes of nonlinear operators ( including u_0—concave operators, mixed monotone operators, set-valued (multivalued ) operators,etc). At the some time, we introduce briefly the main works in our papers.In Chapter 2, we present some fixed point theorems for several classes of nonlinear operators.In §2.1 and §2.2, we obtain new fixed points theorems of the existence and uniqueness for t — a(t) model concace convex or t — a(t,u,v) model concave convex mixed monotone operators (see Theorem 2.1.1 and Theorem 2.2.1). The resulting conclusion essentially improves many related conditions and results to date. In fact, similar results that associted to these theorems are also valid to u_0 concace increasing operators. Moreover,as an application of our results, we prove the existence and uniqueness of positive solution of a class of ninlinear integral equation.In §2.3, we introduced a class of w— concace convex mixed monotone operator, and give both necessary and sufficient conditions for the existence and uniqueness of fixed point of this kind of mixed monotone operator (see Theorem 2.3.1 etc );In §2.4, we discuss the existence of fixed point about the sum of mixed monotone operator (see Theorem 2.4.3 etc ). We improve and generalize some related results.In §2.5, by using the obtained results in §2.1 and §2.2, we discuss the existence of solutions of nonlinear evolution equations in Banach space (see Theorem 2.5.2 and Theorem 2.5.3 etc ), improve and generalize some related results in references.
In §2.6, firstly, we discuss several fixed point theorems about set-valued mapping in an ordered locally convex topological linear space (see Theorem 2.6.3 and Theorem 2.6.8 etc ), generalize some related results in reference. Secondly, By using the obtained results, we discuss the minimum problem with constraint condition in superior theoreyxeG(x), w(x,x)= min w(x, y),GG()where ui is continuous function, and 6* is set-valued mapping. We give its sufficient conditione for existence of solution (see Theorem 2.6.9 ).In §2.7, by using fixed point theorems of set -valued mappings and Schauder , we discuss the differential equations x'(t) + g(t, x(t)) = 0. The existence and properties of solutions of this equation is obtained (see Theorem 2.7.2 ). To the knowledge of the author, this discussion method are very few in reference.In Chapter 3, we study the existence,nonexistence,multiplicity of periodic positive solutions for functional differential equation and the existence multiplicity of positive solutions for a kind of mixed boundary value problem by employing the fixed-point theorem of Krasnoselskii's cone, the fixed point index in cones and cone expansion or compression type fixed point theorem, etc.In §3.1, we will use more precise theory of the fixed point index in cones to discuss the existence of positive periodic solutions for the functional differential equations with parametery'(t) = -a{t)f(y(t))y(t) + Xg(t,y(t - r(t))),where A > 0 is a parameter.In §3.2, we study the existence,multiplicity and nonexistence of positive u— periodic solutions for a kind of second order functional differential equation with parameterU"(t) + a(t)u(t) = Xf(t, U(t - T0(t)), U(t - Tl(t)), ■ ■;U{t - Tn{t)))by employing the fixed point theorem of cone expansion or compression , where A > 0 is a parameter.As far as I know, in §3.1, §3.2, our conclusions (see Theorem 3.1.5- 3.1.7 and Theorem 3.2.1- 3.2.3 )are new and original.
In §3.3, by using the fixed point theorem of cone expansion and compression and fixed point index theory, we establish the existence and multiplicity of positive solutions for a class of mixed boundary value problems as follows:(?(?'))' + f{t,u) = 0, 0 < i < 1,a
全文共分为三章。
在第一章中,我们对几类非线性算子(u_0-凹算子,混合单调算子,集值算子等)的研究现状进行了阐述,同时简明地介绍了我们在本文中将要做的主要工作。
第二章,我们给出了几类非线性算子的不动点定理。
在§2.1和§2.2中,我们给出了t-α(t)型凹凸及t-α(t,u,v)型凹凸的混合单调算子具有唯一不动点的新的存在性定理(参见定理2.1.1和定理2.2.1),本质性地改进了最近许多相关文献原有的条件和结论。在u_0-凹增算子的情形,与这些存在性定理相类似的结论实际上也是成立的,作为所得理论结果的一个应用,讨论了一类非线性积分方程解的存在性。
在§2.3中,我们引入了一类ω-凹凸型混合单调算子,给出了它存在唯一不动点的充分必要条件(参见定理2.3.1等),在§2.4中,讨论了混合单调算子和的不动点的存在性(参见定理2.4.3),所得结果改进并推广了相关文献中的部分结果。
在§2.5中,我们利用§2.1§2.2所得的理论结果并结合算子半群的性质,讨论了Banach空间非线性发展方程解的存在性(参见定理2.5.2和定理2.5.3),所得结果改进了相关文献中的工作。
在§2.6中,我们首先在局部凸拓扑线性空间中引入序,给出了集值(多值)凝聚映射的几个不动点定理(参见定理2.6.3和定理2.6.8等),推广了相关文献在Banach空间中所做的一些工作,然后利用所得结果,讨论了优化理论中的一个带约束条件的极小化问题:
x∈G(x),ω(x,x)=(?)ω(x,y)
其中ω为二元连续实函数,G为集值映象,给出了它存在正解的充分条件(参见定理2.6.9)。
There are mainly two works in this paper. On the one hand, we discuss two classes of nonlinear operator equations, including nonlinear mixed monotone operator equations in Bnanch space and the set- valued mapping equation in an ordered locally convex topological linear space. The methods employed are mainly partial ordering method and iterative techniques and so on. On the other hand, we will use more precise theory of nonlinear operator to discuss the existence problems of solutions for differential equations.This paper includes three chapters.In Chapter 1, we provide a research summary of several classes of nonlinear operators ( including u_0—concave operators, mixed monotone operators, set-valued (multivalued ) operators,etc). At the some time, we introduce briefly the main works in our papers.In Chapter 2, we present some fixed point theorems for several classes of nonlinear operators.In §2.1 and §2.2, we obtain new fixed points theorems of the existence and uniqueness for t — a(t) model concace convex or t — a(t,u,v) model concave convex mixed monotone operators (see Theorem 2.1.1 and Theorem 2.2.1). The resulting conclusion essentially improves many related conditions and results to date. In fact, similar results that associted to these theorems are also valid to u_0 concace increasing operators. Moreover,as an application of our results, we prove the existence and uniqueness of positive solution of a class of ninlinear integral equation.In §2.3, we introduced a class of w— concace convex mixed monotone operator, and give both necessary and sufficient conditions for the existence and uniqueness of fixed point of this kind of mixed monotone operator (see Theorem 2.3.1 etc );In §2.4, we discuss the existence of fixed point about the sum of mixed monotone operator (see Theorem 2.4.3 etc ). We improve and generalize some related results.In §2.5, by using the obtained results in §2.1 and §2.2, we discuss the existence of solutions of nonlinear evolution equations in Banach space (see Theorem 2.5.2 and Theorem 2.5.3 etc ), improve and generalize some related results in references.
In §2.6, firstly, we discuss several fixed point theorems about set-valued mapping in an ordered locally convex topological linear space (see Theorem 2.6.3 and Theorem 2.6.8 etc ), generalize some related results in reference. Secondly, By using the obtained results, we discuss the minimum problem with constraint condition in superior theoreyxeG(x), w(x,x)= min w(x, y),GG()where ui is continuous function, and 6* is set-valued mapping. We give its sufficient conditione for existence of solution (see Theorem 2.6.9 ).In §2.7, by using fixed point theorems of set -valued mappings and Schauder , we discuss the differential equations x'(t) + g(t, x(t)) = 0. The existence and properties of solutions of this equation is obtained (see Theorem 2.7.2 ). To the knowledge of the author, this discussion method are very few in reference.In Chapter 3, we study the existence,nonexistence,multiplicity of periodic positive solutions for functional differential equation and the existence multiplicity of positive solutions for a kind of mixed boundary value problem by employing the fixed-point theorem of Krasnoselskii's cone, the fixed point index in cones and cone expansion or compression type fixed point theorem, etc.In §3.1, we will use more precise theory of the fixed point index in cones to discuss the existence of positive periodic solutions for the functional differential equations with parametery'(t) = -a{t)f(y(t))y(t) + Xg(t,y(t - r(t))),where A > 0 is a parameter.In §3.2, we study the existence,multiplicity and nonexistence of positive u— periodic solutions for a kind of second order functional differential equation with parameterU"(t) + a(t)u(t) = Xf(t, U(t - T0(t)), U(t - Tl(t)), ■ ■;U{t - Tn{t)))by employing the fixed point theorem of cone expansion or compression , where A > 0 is a parameter.As far as I know, in §3.1, §3.2, our conclusions (see Theorem 3.1.5- 3.1.7 and Theorem 3.2.1- 3.2.3 )are new and original.
In §3.3, by using the fixed point theorem of cone expansion and compression and fixed point index theory, we establish the existence and multiplicity of positive solutions for a class of mixed boundary value problems as follows:(?(?'))' + f{t,u) = 0, 0 < i < 1,a
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