Nonzero Solutions And Multiple Solutions Of Set-valued Mapping Equation

In this paper, under some suitable boundary conditions,we study the existence the solutions,nonzero solutions and multiple solutions of set-valued mapping equationby means of general fp - homotopy method.In Chapter 3 of this paper,we discuss the existence of the nonzero solutions of set-valued mapping (1.1) in Banach spaces .We obtained the following result:Theorem 3.2.1 Suppose that(1) T₀ : C → 2^X satisfies boundary condition (MS,(?)C, θ);(2) Exist l₀ ∈ (0,1) such that D (?) L₀C₀ and T₀ has not fixed points in D⁰;(3) T : C × I → 2^X is K mapping and satisfies the following conditions:Then either E_V{T₁, C⁰\D,θ,1+) is not bounded or there exists nonzero solutions of mapping (1.1) in C⁰\D.In Chapter 4,5 of this paper,by using general fp - condensing homotopy, we discuss the nonzero solutions and multiple solutions of set-valued mapping (1.1) in Hilbert spaces as the following condition is not satisfied:T₁(C∩K) (?) K,i.e.T₁(C∩K) (?) K.We had the following results:Theorem 4.3.1 Suppose that(1) g : BCk —* CK is a bounded continuous strictly set contractive mapping such that||5(x)|| < ||x|| (x € dCK)(2) To : Ck —* 2X is a closed convex valued mapping and satisfies boundary condition (MS,dCK,g);(3) To and T\ arc general fp - condensing homotopy in Ck and the corresponding homotopy mapping T(-, ?) satisfies the following condition:for any t e I, T(-,t) : Ck x / —> 2X is closed convex valued mapping with (KX) property.Then the equation (1.1) has solutions in CK. Theorem 5.4.1 Suppose that(1) 6 e K\{9};(2) g : OQk U 9Ck ^* CK is a bounded continuous strictly set contractive mapping such that\\g{x)\\ < \\x\\ (xedCKUdQK)(3) To : CK -> 2X satisfies boundary conditions (MS, dCK, g), (CS, dDk, 6) , (MS, dQK, 9) and ro(dDfc) C K;(4) T : / x Ck —* 2X is a K mapping with closed convex value, and satisfies the following conditions :(4.1) x i T(t,x) (x e dCK UdDk U dQK, t G /)Ti(x) = T(i,x) (x€CK,i = 0,1)(4.2) T] is a strictly set contractive mapping;(4.3) for any t € I,T(t,x) has (KX) property in CK\Q-Then equation (1.1) has at least three solutions in C^,and at least two solutions are nonzero.