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Existence of periodic and non-periodic solutions to systems of boundary value problems for first-order differential inclusions with super-linear growth. (English) Zbl 1165.34004

The paper deals with the first-order differential inclusion subject to the periodic conditions

x ' (t)b(t)x(t)+F(t,x(t)),t(0,1),x(0)=x(1)(1)

and also with the first-order differential inclusion subject to the non-periodic conditions

x ' (t)F(t,x(t)),t(0,1),Ax(0)+Bx(1)=0·(2)

Here, I=[0,1], F:I× n 2 n is a Carathéodory multifunction, b:I is continuous and does not vanish on the whole interval I. Further, A, B are n×n matrices with real elements such that det(A+B)0, and either det(A)0, A -1 B<1, or det(B)0, B -1 A<1. The authors provide new sufficient conditions under which solutions of problem (1) or problem (2) exist. The results apply to differential inclusions that may have a right-hand side with a super-linear growth in its second variable and also apply to systems of first-order differential inclusions. The proofs are based on novel differential inequalities and the Leray-Schauder nonlinear alternative. Some new results for ordinary differential equations with Carathéodory single-valued right-hand sides are also obtained.

34A60Differential inclusions
34B15Nonlinear boundary value problems for ODE
47N20Applications of operator theory to differential and integral equations