The Solutions Of Bvps For Nonlinear Diffenential Equation And Systems

Nonlinear functional analysis has been one of the most important branches of learning in modern mathematics at present. It provides an effect theoretical tool for studying many nonlinear problems. It plays an important role in dealing with nonlinear equations and differential equations arising in applied mathematics.The boundary value problems of third order ordinary differential equations have been studied in much literature([l]-[16]). Compared with boundary value problems of ordinary differential equations, a few authors have been studied the boundary value problems for systems of ordinary differential equations, for the demand of practical problems ([17][19]), the research on systems of ordinary differential equations has certain value.Some authors studied four-order boundary value problems, such as [33]-[39]. In [33]-[36], these authors using upper and lower solutions method study the existence problem as followsf exclude u". Recently, in [37]-[39] these authors study equation as followsf exclude u',u'". Compared with four-order boundary value problems, a few authors have been studied the boundary value problems which f include u' and u'". It is very necessary to study the problem.In the second chapter, We consider the following system of third-order boundary value problemsu':'(t) = ft(t,ui{t),u2(t),---,un{t))1 te[0,l],WJ(O) = <(0) = Ui{\) = 0, i = 1, 2,3, ? ? ? , n.Using a nonlinear alternative of Leray-Schauder type and Krasnosel'skii's fixed point theorem, we establish the Multiple constant-sign solutions of BVPs for third order differential systems.For clarity, we list some conditions used later as follows. In these conditions, the number 6t ￡ {1,-1} (1 < i < n) is fixed and the sets K,K are defined byK = {u￡ B\ 9iUi(t) >0,t￡ [0,1], 1 < i < n},K = {u e K\ djUj(t) > 0 for somej ￡ {1,2, ? ? ? n} and some t e [0,1]} = A"\{0}(Ci) For each i (1 < i < n), /, is continuous on [0,1] x k, with &ifi(t, u) > 0, (t, u) G [0, l]x K and ej^t, u) > 0, (t, u) € [0,1] x K.(C2) For each i (1 < i < n),6ifi(t,u) < gi(<)wii(M)u>i2(M) ? ■ -iwin(|?n|), (t,u) ￡ [0,1] x K. where (&, Wij, 1 < j < n are continuous, ?/?? : [0, oo) —> [0, oo) are nondecreasing, and q{ : [0,1] —>■ [0, oo).(C3) There exists a > 0 such that for each i (1 < i < n),a > diWni^w^ia) ■ ■?win(a),f1 where di = sup / Gi(t,s)qi(s)ds, 1 < i < n.te[o,i] Jote[o,i] (C4) For each j (1 < j < n) and some i e {1,2, ? ? ■ n} ( z depends on j ),> TijWWijilUjW (t,U) E fj, ^1 X K,fl 3]where t^ : -, - —>? (0, 00) is continuous. 4 4(C5) There exists /3 > 0 such that for each j (1 < j < n), the following-holds for some i (E {1, 2, ? ? -n}, (i depends on j and is same as in (C4)):& ) /where a^ G [0,1] is defined by/ Gi(alj,s)Tij(s)ds = sup / Gi(t,s)Tij(s)ds. J\ je[o,i]JiMain Results:Theorem 2.2.1 Let f\ : [0,1] x Rn -> R, 1 < i < n be continuous. Suppose there exists a constant p, independent of A, such that ||u|| ^ p for any solution u G (C[0,1])" of the systemUi{t) = \ [ Gi{t,s)fi(s,u(s))ds, te[0,l], l -a for some j G {1, 2, ? ? -n}, if a < /3;?e(|,|] 4(b) (3 < \\u*\\ < a and min 0jU*(t) > -(3 for some j G {1, 2, ? ? ? n}, if a > 0.Theorem 2.2.4 Let (Ci) - (C5) hold with a < /3. Then (1) has at least two constant-sign solutions ul,u2 G (C[0, l])n such that0<\\u1\\ -a for some j G {1, 2, ? ? -n}. *e[i,f] 4Theorem 2.2.5 Let (Ci)-(C5) and (C5)\^ hold, where 0 < /3 < a < j3. Then (1) has (at least) two constant-sign solutions u1, u2 G (C[0,1])? such that0 < P < \\v}\\ < a < \\u2\\ < P, min dku\{t) > -J3, min 9jU2(t) > -afor some j, k G {1, 2, ? ? ? , n}.Theorem 2.2.6 Assume (Ci), (C2) and (C4) hold. Let (C3) be satisfied for a = en, I = 1, 2, 3,..., k, and (C5) be satisfied for P = Pt, I — 1, 2, 3,..., m.(a) if m = k + 1 and 0 < Pi < ai < ■ ■ ■ < Pk < uk < Pk+i, then (1) has (at least) 2k constant-sign solutions it1, ■ ■ ? , u2k G (C[0, l])n such that0 < A < II^H < on < \\u2\\ ■ R is continuous and there exists two nonnegative constants Ai, A2, 0 < X\ + A2 < 2, such thatf(t,Ui,Vi)-f(t,U2,V2)>-\l{ui-U2)-\2{v1-V2). (3)for t G [0,1], Ui > u2, vx > v2Main Results:Theorem 3.2.1 If (2) has a lower solution x and an upper solution y such that x"{t) < y"{t) for t G [0,1], then (2) has a solution u G C4[0,1] which satisfiesx"(t) < u"(t) < y"(t), te[0,l].Corollary 3.2.2(1) If min /(￡, 0, 0) > 0, and there exists c > 0, such thatmax{f(t,u,v) : (t,u,v) G [0,1] x [0,c] x [0,c]} < 12c,then (2) has a nonnegative solution u*.(2) If max f(t, 0, 0) < 0, and there exists c > 0, such thatmin {f(t, u, v) : (t, u, v) G [0,1] x [-c, 0] x [-c, 0]} > -12c,then (2) has negative u*.In the four chapter, We consider the following four-order boundary value problem with one-sided Nagumo condition :',u",u'"), 0 < i < 1,(4) u'(Q) = 0, u'"(0) = 0, u"'(l) = 0.Theorem 4.2.1 Assume that there exist a, f3 G C4[0,1] lower and upper solutions of problem (4), respectively, such that?"(i) < P"(t), Vte[0,l].Define the setE, = {(t,x,y,z) e [0,1] x R* : a'(t) R be a continuous function that satisfies the one-sided Nagumo condition in E* and such thatf(t, a'(t), y, z) > f(t, x, y, z) > f(t, P'(t), y, z)for (t, y, z) e [0,1] x R2 and a'(t) 0, such that max /[0, c]< 6c, and inf f(t, 0) > 0. Then (5) has one non-negative solution u such that"" o 0, such that max /[—c, 0] > —6c, and max f[t, 0] < 0, Then (5) has one negative solution u such that ||a|| 0, such that max f[—c, c]< 6c, and min /[—c, c] > —6c. Then (5) has one solution u such that llull < c. ~~ o -6c and min |/(t,0)| > 0 hold , then u 4 0.1 J ~ o 0 andl->oo0oo0 0>oo0