This work is intended for a generalization of Darbo’s fixed-point theorem by using multivalued condensing operators, and a measure of weak noncompactness which does not necessarily possess the maximum property. Moreover, we prove some fixed-point theorems in Banach algebras which satisfy an appointed weak sequential condition. As an application, we establish the existence of solutions for a functional integral inclusion.

This paper focuses on the approximate controllability of semilinear neutral stochastic integrodifferential systems with fractional Brownian motion in a Hilbert space. The theory of analytic resolvent operators of linear neutral integrodifferential equations and the Banach fixed point theorem are employed to obtain the main results. In order to improve the regularity of solutions, the theory of fractional powers and the $\alpha$-norm are also utilized to discuss the problem. An example is presented in the end to illustrate the applications of the obtained results.

This paper is concerned with the numerical treatment of second kind Volterra integral equations whose integrands present diagonal and/or endpoint algebraic singularities. A projection method based on an optimal interpolating operator is developed in the spaces of weighted continuous functions endowed with the supremum norm. In such spaces, the uniqueness of the solution is discussed and suitable conditions are determined to assure the stability and the convergence of the method. Several numerical tests are presented to show the efficiency of the method and the agreement with the theoretical estimates.

We study the existence of a strong solution for a class of second-order nonlinear retarded differential equations involving a lower-order fractional term. We prove the existence result by using Rothe’s time-discretization method. We also provide some examples to examine the existence of solutions for a class of second-order fractional differential equations.

We consider an integrodifferential Dirac system with an integral delay on a finite interval. We obtain the asymptotical formula for the nodal points of the first components of the eigenfunctions, formulate a uniqueness theorem and prove that the kernel of the Dirac operator can be uniquely determined from a dense subset of the nodal set. We also present examples for reconstructing the kernel by using the nodal points.

We consider a new approach to investigating the mixed boundary value problem (BVP) for the Helmholtz equation in the case of a three-dimensional unbounded domain ${\mathrm{\Omega }}^{-}\subset {ℝ}^3$ with compact boundary surface $S=\partial {\mathrm{\Omega }}^{-}$, which is divided into two disjoint parts, $S_D$ and $S_N$, where the Dirichlet and Neumann type boundary conditions are prescribed respectively. Our approach is based on the classical potential method. We look for a solution to the mixed boundary value problem in the form of a linear combination of the single layer and double layer potentials with densities supported on the Dirichlet and Neumann parts of the boundary, respectively. This approach reduces the mixed BVP to a system of integral equations containing neither extensions of the Dirichlet or Neumann data nor a Steklov–Poincaré type operator involving the inverse of the single layer boundary integral operator, which is not available explicitly for arbitrary boundary surfaces. The right-hand sides of the resulting boundary integral equations system are functions coinciding with the given Dirichlet and Neumann data of the problem under consideration. We show that the corresponding pseudodifferential matrix operator is bounded and coercive in the appropriate $L_2$-based Bessel potential spaces. Consequently, the operator is invertible, which implies the unconditional unique solvability of the mixed BVP in the class of functions belonging to the Sobolev space $W_{2,\mathrm{\text{ loc}}}^1({\mathrm{\Omega }}^{-})$ and satisfying the Sommerfeld radiation conditions. We also show that the pseudodifferential matrix operator thus obtained is invertible in the $L_p$-based Besov spaces and prove that under appropriate boundary data a solution to the mixed BVP possesses the $C^{\alpha }$-Hölder continuity property in the closed domain ${\overline{\mathrm{\Omega }}}^{-}$ with $\alpha =\frac{1}{2}-𝜀$, where $𝜀>0$ is an arbitrarily small number.

This study probes into the existence of a unique solution and the numerical approximation of a nonlinear functional Volterra–Fredholm integral equations of the mixed type and second kind. Based on the Lipschitz constants of the functional and kernel, a Bielecki’s norm is defined and used to modify a distance inequality on a constructed self-map. The map is shown to be contractive, thereby establishing solvability. The problem is then approximated by collocating at discrete points and use of a composite multidimensional numerical quadrature approximation. A new Grönwall-type inequality is proposed, and used, to prove the second order of convergence of the numerical scheme. Numerical experiments are provided to verify the theoretical results.

We discuss the ellipticity of the single layer boundary integral operator for the wave equation in one space dimension. This result not only generalizes the well-known ellipticity of the energetic boundary integral formulation in $L^2$, but it also turns out to be a particular case of a recent result on the inf-sup stability of boundary integral operators for the wave equation. Instead of the time derivative in the energetic formulation, we use a modified Hilbert transformation, which allows us to stay in Sobolev spaces of the same order. This results in the applicability of standard boundary element error estimates, which are confirmed by numerical results.