Continuous dependence on data for bilocal difference equations

Abstract

The continuous dependence on data is studied for a class of second order difference equations governed by a maximal monotone operator A in a Hilbert space.A nonhomogeneous term f appears in the equation and some bilocal boundary conditions a, b are added. One shows that the function which associates to {a, b, A, f} the solution of this boundary value problem is continuous in a specific sense. One uses the convergence of a sequence of operators in the sense of the resolvent. The problem studied here is the discrete variant of a problem from the continuous case.Keywords: maximal monotone operator; the resolvent of an operator; Yosida approximation; convergence in the sense of resolvent

A note on the continuous dependence on data for second-order difference inclusions

“…By k·k, we denote the norm in H. The problem we study in the present paper is the discrete variant of the results obtained in [2]. This result complements the main theorem from [1], where a similar study was done on finite sets of integers i. Other properties of second-order evolution equations associated to maximal monotone operators were studied in [5,7,8], while their discrete variants are treated in [6,8,9].…”

Second order difference inclusions of monotone type

“…Existence and asymptotic behavior results for equation (1.1) for i 1 and various boundary conditions have been obtained in [7]. For finite sets of i (1 i N ), in [6] the authors analyzed the continuous dependence of the solution on the operator A, the sequence f i and the boundary conditions u 0 = a, u N +1 = b.…”

A Convergence Result for Second Order Difference Equations of Monotone Type

“…For second order difference inclusions with different boundary conditions, we mention the papers [2] . But unlike the paper [2] , where the boundary conditions are bilocal, the second boundary condition from the present paper is defined with the aid of an one-to-one maximal monotone operator β. The domain of β is bounded, while A is supposed to be β−dissipative.…”

“…Moreover, the proof of the main result is here simplified. Instead of the four auxiliary problems considered in [2] , in the present paper we employ only two approximating problems, which are simpler than those from [2] . They are defined with the aid of the Yosida approximations of A.…”