Existence Of Solutions For Differential Equations (System)

It is well known that singular boundary value promble has been one of the important probelms that attract the attention of mathematics and other technicians. It arises in the fields of nuclear physics, gas dynamics, Newtonian fluid mechanics, the theory of boundary layer, nonliear optics and so on. This paper mainly investigates positive solutions for differential equations(system).In the first chapter, the existance of the multiple positive solutions for second order-three point singular boundary value problems in Banach spaceis considered, where f(t,x) is singular at t = 0, t = 1,x =θ. by constructing a special cone, and using the fixed point index theory on cone,we get the existence of at least two positive solutions. The main result is given bellow:Theorem 1.2.2 suppose (H₁)-(H₅) hold , then (1.2.1) has at least two positive solutions at C[J,E)∩C²[(0,1),E).An example is worked out to indicate our conditions are reasonable.In the second chapter, we consider the existance of positive solutions for singular impulsive differential equation on half-line in Banach space.x"(t) + f{t,x(t)) = e,t^tk,te (o, +00),Az|i=tjfc = h(x(tk)), {k = 1,2, â–  ? ? m)(2.2.1)where 9 denotes the zero element of E, f(t,x) maybe singular at t = 0, that is limt-> 0+||/(V)II = 00. / € C[J0 x P,P],Ik e C[P,P],k = 1,2,- --m, where Jo = (0,+oo).by constructing a special cone,and using cone compression and expansion fixed point theorem, we get the existence of at least one positive solutions. The main result is given bellow:Theorem 2.2.1 suppose (Hi)-(H5) hold, then (2.2.1) has at least one positive solution.An example is worked out to indicate our conditions are reasonable.In the third chapter, we investigate the existence of solutions for singular impulsive differential equations on unbounded domains in a Banach spacex"(t) + f(t,x(t),x(t)) = 0, t # tk,t 6 (0, +00),Ax\t=tk = Ik(x(tk)), (k = 1,2, ? ? ? m) Ax'\t=tk=9, x{0) =xo> 9,x'(+oo) = yoo>9;where 9 denotes the zeno element of E, f(t,x,x) maybe singular at t = 0, that is limt-> 0+||/(t, ?, -)]| = 00. / : Jo x P x P -> P, satistics Caratheodory s conditions, (i.e., for each (x, y) € P x P, the function /(-,x, y) is measureable on Jo;for a.e. t e Jo, the function /(ï¿¡, ?, ?) is continuous on P x P).Ik € C[P, P], A;= 1,2, ? ? ? m, where Jo = (0, +00).By using Monch fixed point theorem, we get the existence of at least one positive solutions.The main result is given bellow:Theorem 3.2.1 suppose (Hi) and (H2) hold, then (3.2.1) has at least one positive solution satisfing x(t) € DCl[J,P] and x(t) > x0.An example is worked out to indicate our conditions are reasonable.In the last chapter, we investigate the existance of at least three positive solutions for the second order quasilinear differential equation system(Vp(x'))'{t) + a(t)f{t,x(t),y{t)) = 0,te (0,1),( 1,/,g are nonnegative continuous functions.Using the five functionals fixed point theorem, we get the existence of three positive solutions. The main result is given bellow:Theorem 4.2.1 suppose (Hi) - (#3) hold, and exist 0 < a < b < lf-b < c such that /(ï¿¡, x, y),g{t, x, y) satisfying the following conditions:(H4) te[o,i],\\(*,v)\\<%f(t,x,y) > t J2a{u)dub0 mea methen (4.1.1) has at least three positive solutions (zi,yi), (^2,2/2), (^3)?/3) such thatmaxmg ||(ar3(*),Sft(<))|| > ^mir^ \\(x3(t),y3(t))\\ < 6}(g|x ||(*<(*),yi(t))|| < c, (? = 1,2,3).Theorem 4.2.2 suppose (Hi) — (H3) hold, and exist 0 -i ,, ., t>,(.—r) / 6(u)dti 1 ithen (4.1.1)has at least three positive solutions (xi,yi), (x2,?/2), (^3,2/3) such that max ||(xi(<),yi(<))ll < ^^g, \\Mt)ty2(t))\\ < b,vMx\\(x3(t),y3(t))\\ > d,trmnj\(x3(t),y3(t))\\ < 6, m? ||(ar*(t),?,(t))|| < c, (t = 1,2,3).