The authors deal with the problem of recovering the scalar unknown kernel \(h\) in the nonlinear integro-differential equation \[ u'(t) = G(u(t)) + \int_0^t h(t-s)F(u(t),u(s)) ds + f(t),\qquad t\in [0,T], \tag{1} \] related to the Banach space \(X\) and subject to the initial condition \[ u(0) = u_0,\tag{2} \] and to the additional condition \[ \Phi[u(t)] = g(t), \quad t\in [0,T], \tag{3} \] \(\Phi\) being a continuous linear functional on \(X\).
The authors assume that the nonlinear operators \(G\) and \(F\) are of class \(C^2\) with Lipschitz-continuous second-order derivatives, that the Frechét derivative \(G'(u_0):D\subset X\to X\) is a generator of an analytic semigroup of linear bounded operators as well as suitable regularity and consistency conditions involving the data.
Under the solvability condition \(\Phi[F(u_0,u_0)]\neq 0\) the identification problem (1)–(3) admits, for some \(\tau \in (0,T]\), a unique solution \((u,h)\in [W^{2+\beta,p}(0,\tau;X)\cap W^{1+\beta,p}(0,\tau;D)]\times W^{\beta,p}(0,\tau)\), \(p\in (1,+\infty]\), \(\beta \in (0,1)\backslash \{1/p\}\), where \(W^{2+\beta,\infty}(0,\tau;Z) =C^{2+\beta}(0,\tau;Z)\). Furthermore a global in time uniqueness result is proved.
Finally, the abstract results are applied to identification problems related to population dynamics, combustion of a material with memory and to the fully nonlinear problem (1)–(3), where \[ G(u_1)=d(A(x,D_x)u_1),\qquad F(u_1,u_2)=e(A(x,D_x)u_2) \] \(d\) and \(e\) being smooth scalar functions and \(A(x,D_x)\) being an elliptic operator of order \(2m\) endowed with general homogeneous boundary conditions \(B_j(x,D_x)u=0\), \(j=1,\ldots,m\).