Positive Solutions For Differential System In Banach Space

It is well known that singular boundary value problem (SBVP, for short) arises inthe fields of nuclear physics, gas dynamics, Newtonian ?uid mechanics, the theory ofboundary layer, nonlinear optics and so on. SBVP has been one of the important prob-lems that attract the attention of mathematicians and other technicians. The presentpaper mainly investigates positive solutions for di?erential system, including the singularboundary value problems. The existence and uniqueness of positive solutions for singulardi?erential system has been considered extensively in the last twenty years([9]-[28]). Thispaper discusses the problems of di?erential system more generally on the basis of abovereferences.The first chapter is considered with the existence of one positive solution of a classof two-point boundary value problems for system of second order nonlinear integro-di?erential equationswheret ∈ J = [0,1], a_i, b_i, c_i, d_i ∈ R~+, f, g ∈ C[J × P × P × P,P], Tu(t) =(?) K(t,s)u(s)ds, K ∈ C[D,R~+], D = {(t,s) : 0 ≤ s ≤ t ≤ 1}, R~+ = [0, +∞), i = 1, 2.The proper conditions (H1.1)-(H1.3) are given from page 7 to 8 in the article. The mainresult is given in follow:Theorem 1.2.1 Suppose (H1.1)-(H1.3) hold. Then (1.2.1) has at least one positivesolution (u,v) ∈ C2[J,E] × C2[J,E] satisfying u(t) ≥ u0, v(t) ≥ v0 as t ∈ I.The method used here is Mo¨nch fixed point theorem . And at last, an example ininfinite dimensional space is worked out to indicate our conditions are reasonable.In order to investigate the existence of multiple positive solutions of system (1.2.1)in the second chapter, we use fixed point index theorem and give suitable conditions(H2.1)-(H2.5) that can be seen from page 15 to 16. The main results are as follows:Theorem 2.2.1 Assume (H2.1)-(H2.4) hold . Then BVP (1.2.1) has at least two pos-itive solutions (u1,v1), (u2,v2) ∈ C2[J,E] × C2[J,E] satisfying u1(t) u0, v1(t) v0 ast ∈ I.Theorem 2.2.2 Assume (H2.1)-(H2.2) and (H2.5) hold. Then BVP (1.2.1) has at leastone positive solution (u,v) ∈ C2[J,E] × C2[J,E] satisfying u(t) ≥ u0, v(t) ≥ v0 as t ∈ I.And examples in infinite and finite dimensional space are given to show the application.The third chapter, by using cone expansion and compression theorem, we investigatesingular boundary value problems of coupled system of second and fourth order ordinarydi?erential equations???????????????u(4) = f(t,u,v);?v = g(t,u,v);u(0) = u(1) = u (0) = u (1) = 0;v(0) = v(1) = 0.(3.2.1)where t ∈ (0,1), f, g ∈ C[(0,1) × [0,∞) × [0,∞),[0,∞)], and the proper conditions(H3.1)-(H3.5) are given that can be seen from page 25 to 26. The main results are asfollows:Theorem 3.2.1 Suppose (H3.3)-(H3.5) hold. Then SBVP(1.1) has a C2[0,1] × C[0,1]positive solution (u,v), if and only if (H3.1) holds.Theorem 3.2.2 Suppose (H3.3)-(H3.5) hold. Then SBVP(1.1) has a C3[0,1]×C1[0,1]positive solution (u,v), if and only if (H3.2) holds.The last chapter deals with singular boundary value system with P-laplacian???????????????(?p(u )) (t) + λa(t)f(t,u(t),v(t)) = 0;(?q(v )) (t) + λb(t)g(t,u(t),v(t)) = 0;u(0) ? B0(u (0)) = u(1) + B0(u (1)) = 0;v(0) ? B1(v (0)) = v(1) + B1(v (1)) = 0.(4.1.1)where t ∈ (0,1), ?p(x) = |x|p?2x, ?q(x) = |x|q?2x, p, q > 1, λ > 0, a, b ∈ C[(0,1),(0,+∞)],f, g ∈ C[(0,1) × [0,∞) × [0,∞),(0,∞)], and we have produced the suitable conditions(H4.1)-(H4.5), which can be seen page 39 in the article. The main result is given in follow:Theorem 4.2.1 Suppose (H4.1)-(H4.5) hold. Then for any r > 0, there exists λˉ =λˉ(r) > 0 such that SBVP(4.1.1) has at least two positive solutions(u1,v1) and (u2,v2)satisfying 0 < (u1,v1) < r < (u2,v2) as λ ∈ (0, λˉ(r)).