Solutions Of Some Nolinear Equations In Abstract Spaces And Application

This paper is divided into throe sections.In section l.we study the existence of positive solutions of the following sigular equation for second order boundary value problem.Firstly,we list some conditions for conveniece.The definition of 0 is given from the next part. H(3') one of the following conditions holdsTheoreme 1.1 Assume that (H1) - (H3) hold ,then the BVP (1.3) has at least one positive solution.Corallaryl.l Suppose f(t,u) = p(t)q(t) satisfing (H1)(H2)(H'3), then the BVP(1.3)has at least one positive solution,Remark 1.1 We have the same conclusion as in Theoreme 3.1 provided that where pi(t),q(x) satisfy (H1) - (H3).Remark 1.2 From corallary and remark 1.1 we know that the result obtained here contains that in [11]. The method we use to proof the theoreme is different from that in [11] in essence.Remark 1.3 The limitations in [11] are replaced by the lower and supper limitation. The limitation in Thoreme can be an arbitary constants in [0,),however,the limitation in [11] can only be 0 or .So the result here improves and generalizes that in [11].In section 2, by establishing a new comparison result and using a lemma in [14], we obtain the existence of extremal solutions of the following initial value problem:The result obtained here weakens the coefficents in [15]. As an application of the main result,we consider the IVP of infinite system for nonlinear impulsive intgro-differential equations in Banach spaces.The main theorernes are as follows:Lemma(Comparison result) Assume that sat isfiesM(t),N(t) are bounded integrable nonnagtive functions, are constants andThen p(t) , for t J.Let us list some conditions.whore constants are lower and upper solutions of IVP(1.1) respectively.(H2) There exist bounded integrable nonnegative functions M(t) and N(t) such that(H3) For any t J,and bounded,equicontinuous on each ,mo not one sequence there exist nonncgative constants such thatP1, P2, P3 satisfy one of the following conditionswhereTheoreme 2.1 Let cone P be normal and conditions (H1) - (H3) be satisfied,suppose that and(2.5)hold. Then there exist monotone sequences which converge uniformly and monotonically on J to the minimal and maximal solutions respectively. Moreover sequences converge uniformly and monotonically on respectively, thenIn section 3,firstly,we consider the relationship between the cone P of E and the cone P of LC[R+, E].The sufficient condition and necessary condition are given for relative compactness of a class of abstract continuous fuction groups at infinite intervals. As the application of the theorem.the existence of solutions of terminal value problems for nonlinear first order differential equation and solutions for Fredholm integro-eqution in Banach spaces are discussed.The main theoremes are as follows:Theoreme 3.1 P is normal if and only if P is normal.Theoreme 3.2 Let P be regular there exists such thatTheoreme 3.3 Assume that P is regular then P is reguar. Theoreme 3.4 Let P be a regular cone in E and A :be an increasing operator such thatThen A has a maximal fixed point u* and a minimal fixed point it, in [u0, u0]i moreoverTheoreme 3.5 H C LC[R+,E] is relatively compact if and only if 10 The function in H is equicontinuous in [0, 6] for any b > 0, 20 H(t) =is relatively compact in E for any t R+, 30 when is uniform for any x € HTheoreme 3.7 H C FC'[R+, E] is relatively compact is and only if 10 The function in H is equicontinuous in [a. b] for any b > a. 20 H(t] = is relatively compact in E for any t R+. 30 when is uniform for any .r e H.Theoreme 3.8 Assume that(H1) is uniformly continuous in t on . where where, whereThen Eq.has at least one solution under previous conditions.