On Convex Nonlinear Operators And Their Applications

This study has two main parts, the first part: discussing three categories of nonlinearoperators -α(> 1)- convex operators, concave or convex ordered contractive operators andφconvex -ψconcave mixed monotone operators; the second part: Using nonlinear operators theory, especially the conclusions that we obtained, we investigated the solutions of a class of singular boundary value problems for functional differentialequations and a class of initial value problems for impulsive integro-differential equations.Studying convex nonlinear operators in-depth is of great significance not only in theory but also in the application. For example, in the engineering problems, the nuclear issue and economic problems often relate to some convex nonlinear operators. It is regrettable that the operators have not been able to be studied in depth, and therefore solve the problems without powerful tool.Impulsive differential equation theory on the rise was in the late 1960s. Impulsive phenomena is widespread in the actual problems of modern technologies in various fields, and its mathematical models can often be attributed to impulsive differential systems. Impulsive differential systems can be more profound and more accurately reflect the change rules of things. In recent years, such systems have important applicationsto space technology, information science, control systems, communications, Life sciences, medical, economic and other fields. However, its theory and the applied research is still in the elementary stage. Discuss such issues is of great significance.The methods employed are mainly partial ordering method. Compared with the common literatures, the difference lies that the partial ordering method is employed for mappings' preimages which are set-valued instead of mappings' images which are single-valued.The major results are as follows.In§2.1, a class ofα(> 1)- convex operators were discussed. The existence and uniqueness of fixed points ofα(> 1)- convex operators is obtained under much weaker conditions. The concrete results are Theorem 2.1.1, Theorem 2.1.4, Theorem 2.1.6, Theorem 2.1.8, Theorem 2.1.11, Theorem 2.1.12 and Theorem 2.1.21. And applied to solving a class of partial differential equations, see Theorem 2.1.23. In§2.2, some concave or convex ordered contractive operators were discussed.Conditions for the existence and uniqueness of fixed points of some concave or convex ordered contractive operators are given. The concrete results are Theorem 2.2.15, Theorem 2.2.16, Theorem 2.2.18, Theorem 2.2.21, Theorem 2.2.24 - 2.2.27. And applied to discussing a class of decreasing operators, see Theorem 2.2.17 and Theorem 2.2.20.In§2.3, we introduceφconvexψconcave mixed monotone operator, for which,the questions discussed are convexity in essence. Conditions for the existence and uniqueness of positive fixed points ofφconvex -ψconcave mixed monotone operators are given under much weaker conditions. The concrete results are Theorem 2.3.2 -2.3.20. And applied to solving a class of integral equations, see Example 2.3.21.In§3.1, we discuss the singular boundary value problem for functional differentialequations in a Banach space:where y^t(s) = y(t + s),(?)s∈[-r, a].By using A very-Henderson's fixed point theorem, we establish sufficient conditions for the existence of at least two positive solutions to the singular boundary value problem(1), see Theorem 3.1.4. And by using the fixed point theorem ofα(∈(0,1))- concaveoperators, we give conditions for that the singular boundary value problem (1) has an unique positive solution, see Theorem 3.1.6.In§3.2, we discuss the initial value problems for first and second order impulsiveintegro-differential equations in a Banach space:where f∈C(J×P×P,P),I_k∈C(P,P)(k=1,2,…),x₀>θ,â–³x|_t=tk=x(t_k⁺)-x(t_k),x(t_k⁺)represents the right-hand limit of x(t) at t = t_k; Tx(t) =∫_t0^tK(t,s)x(s)ds((?)t∈J),K∈C(D,R⁺),,D={(t,s)∈J×J|t≥s},(R⁺=[0,+∞)); andwhere f∈C(J×P×P×P,P),I_k,I_k∈C(P,P)(k=1,2,…,),x₀>θ,x'₀≥θ,â–³x|_t=tk=x(t_k⁺)-x(t_k),â–³x'|_t=tk=x'(t_k⁺)-x'(t_k^-),x'(t_k^-)and x'(t_k⁺) represent theright and left-hand limit of x'(t) at t =t_k, respectively; Tx(t)=∫_t0^tK(t,s)x(s)ds((?)t∈J),K∈C(D,R⁺),D={(t,s)∈J×J|t≥s},(R⁺=[0,+∞));Sx(t)=∫_t0^+∞H(t,s)x(s)ds((?)t∈J),H∈C(J×J,R⁺).By using our results obtained in [58], we establish the existence and uniqueness of positive solutions for the initial value problems (2) and (3). Moreover, the iterative methods and error estimates of the solutions are given. See Theorem 3.2.5, Theorem 3.2.10, Theorem 3.2.14, Theorem 3.2.16, Theorem 3.2.18, Theorem 3.2.22, Theorem 3.2.26, Theorem 3.2.28 and Theorem 3.2.30.