Periodic-integral Boundary Value Problems For Higher Order Differential Equations

The theory of boundary value problems with integral conditions for or-dinary diferential equations has widely application in areas of applied math-ematics and physics such as heat conduction, plasma physics. The existenceof solutions for boundary value problems with integral conditions has beenwidely studied. The main content of this paper is to study the existence anduniqueness of solutions for several types of periodic-integral boundary valueproblems for high diferential equations by using a combination of the nonlin-ear functional analysis theory and method such as degree theory, homotopycontinuation method.In the first part, we research the existence of solutions of periodic-integralboundary value problems for high diferential equations. we extend the re-sults of periodic integral boundary value problem of the second order Dufngequation. We obtain some sufcient conditions for the existence and unique-ness of solutions of periodic-integral boundary value problems for even-orderdiferential equations. Furthermore we consider the case of x∈Rn. In this part,we first consider the2nth-order differential equation: where t∈[0,2Ï€],x∈R,ki,i=0,1,…,n-1are some constants,and f:[0,2Ï€]×Râ†'R.Theorem2.1.1The function f is continnous and differentiable,and there exist constants k,M1,M2,such that where is monotone increasing on Z+Then periodic integral boundary value problem(0.0.6)has a unique solution.Furthermore,we also consider the vector equations where,x(t)=(x1(t),x2(t),…,xn(t)),t∈[0,2Ï€],αj,j=0,1,…,k-1are some constants.We need the following hypotheses:(H1) f∈C1([0,2Ï€]×Rn),and Jacobian matrix fx is a symmetric n×n matrix.(H2)There exist two constant symmetric n×n matrices A and B such that and, if λ1≤λ2≤,…,λn and μ1≤μ2≤,…,μn are the eigenvalues of A and B, respectively, then there exist integers Ni, i=1,2,…, n, satisfying the condition where is monotone increasing on Z+.Theorem2.4.1Assumed (H1) and (H2) are satisfied, the problem (0.0.7) has a unique solution.In the second part, we study the periodic-integral boundary value problem of nonlinear differential equation By the transform of the periodic-integral boundary value problem of differential equation (0.0.8) is equivalent to where p(t)∈C1([0,2Ï€], R) and f∈C1([0,2Ï€]×R,R).By applying the homotopy method we obtain the existence and uniqueness of solutions to this kind boundary value problems.We need the following hypotheses:(Al) there exist two constants M1>0and M2>0such that p(t) satisfies for al t∈[0,2Ï€]; (A2) there exist constants a and b, such that for all (t,x)∈[0,T]×R;(A3) there exists∈E Z+, such thatTheorem3.3.1Assume that (Al),(A2) and (A3) are satisfied. Then problem (0.0.9) has a unique solution.In the third part, we consider periodic integral boundary value problem under the linear increasing conditions. We obtain the existence of solutions for periodic integral boundary value problem.In this part, we main consider the following second order Duffing equation: where t E [0,2Ï€],x∈R.Assume that(A1)f∈C([0,2Ï€]×R,R);(A2) there exist N E Z+and ε>0, such that for all (t,x)∈[0,2Ï€]×((-∞,-m]∪[m,∞)). we have the following theorem:Theorem4.2.7 Assume that (A1) and (A2) are satisfied. Then periodic integral boundaryvalue problem (0.0.10) has at least one solution.