Upper and lower solutions method for first-order impulsive differential inclusions with nonlinear boundary conditions
The authors study a first-order impulsive differential inclusion with nonlinear boundary conditions of the type \[ y'(t)\in F(t,y(t)), \quad t\in J=[0,T], \quad t\neq t_k, \quad k=1,\dots,m,quad y(t_k^+)=I_k(y(t_k^-)), \quad k=1,\dots,m, \] \[ L(y(0),y(T))=0, \] where \(F:J\times \mathbb R \to 2^{\mathbb R}\) is a multifunction with compact and convex values, \(L: \mathbb R^2 \to \mathbb R\) is a single-valued map, \(0=t_0<t_1<\cdots<t_m<t_{m+1}=T, I_k\in C(\mathbb R, \mathbb R) (k=1,\dots,m)\) are bounded and \(y(t_k^-)\) and \(y(t_k^+)\) represent the left and the right limits of \(y(t)\) at \(t=t_k\), respectively. Combining a fixed-point theorem for condensing multivalued maps with the concept of upper and lower solutions, they prove the existence of solutions for the above problem.
scientific article; zbMATH DE number 2142368
scientific article