Resolvent Operators and Weak Solutions of Integrodifferential Equations
نویسنده
چکیده
Equations from heat conduction and viscoelasticity are written as x′(t) = A [ x(t) + ∫ t 0 F (t− s)x(s)ds ] + f(t), t ≥ 0, x(0) = x0, (1) in a Banach space X, where A is a closed and densely defined operator and F (t) is a bounded operator for t ≥ 0. We obtain the equivalence between the resolvent operator of Eq.(1) and solutions of d dt 〈x(t), v〉 = 〈x(t) + ∫ t 0 F (t− s)x(s)ds,A∗v〉+ 〈f(t), v〉, t ≥ 0, x(0) = x0. (2) where A∗ is the adjoint of A, v ∈ D(A∗), and 〈, 〉 denotes the pairing between X and its dual X∗. The result also enables us to unify many concepts about solutions of Eq.(1). AMS Subject Classification : 45D, 45J, 45N.
منابع مشابه
Zuomao Yan EXISTENCE OF SOLUTIONS FOR SOME NONLINEAR DELAY INTEGRODIFFERENTIAL EQUATIONS WITH NONLOCAL INITIAL CONDITIONS
The main purpose of this paper is the existence of mild solutions for a class of first-order nonlinear delay integrodifferential equations with nonlocal initial conditions in Banach spaces. We show that the solutions are given by the application of the theory of resolvent operators and the Sadovskii’s fixed point theorem. An example is presented in the end to show the applications of the obtain...
متن کاملNonlocal Problems for Delay Integrodifferential Equations in Banach Spaces
In this paper we study the existence of mild solutions for a class of first-order delay integrodifferential equations with nonlocal condition in a Banach space. The results are established by the application of the theory of resolvent operators, the contraction mapping principle and the Schaefer theorem. An example is presented in the end to show the applications of the obtained results. Mathem...
متن کاملImpulsive integrodifferential Equations and Measure of noncompactness
This paper is concerned with the existence of mild solutions for impulsive integro-differential equations with nonlocal conditions. We apply the technique measure of noncompactness in the space of piecewise continuous functions and by using Darbo-Sadovskii's fixed point theorem, we prove reasults about impulsive integro-differential equations for convex-power condensing operators.
متن کاملA Class of Nonlocal Integrodifferential Equations via Fractional Derivative and Its Mild Solutions
In this paper, we discuss a class of integrodifferential equations with nonlocal conditions via a fractional derivative of the type: D t x(t) = Ax(t) + t Z 0 B(t − s)x(s)ds + tf (t, x(t)) , t ∈ [0, T ], n ∈ Z, q ∈ (0, 1], x(0) = g(x) + x0. Some sufficient conditions for the existence of mild solutions for the above system are given. The main tools are the resolvent operators and fixed point the...
متن کاملAlmost Periodicity of Abstract Volterra Integro-differential Equations
The main purpose of this paper is to investigate almost periodic properties of various classes of (a, k)-regularized C-resolvent families in Banach spaces. We contemplate the work of many other authors working in this field, giving also some original contributions and applications. In general case, (a, k)regularized C-resolvent families under our considerations are degenerate and their subgener...
متن کامل