Sufficient conditions for the existence of at least one solution of a nonlinear integral equation with a general kernel are established. The existence result is proved in $C([0,T],E)$, where $E$ denotes an arbitrary Banach space. We use the Darbo-Sadovskii fixed point theorem and techniques of measure of noncompactness. We extend and generalize results obtained by other authors in the context of fractional differential equations. One example illustrates the theoretical results.

The present work deals with some spectral properties of the problem

\medskip $(\mathcal{P} )$ $\Bigg \{$\vbox {$D^{\alpha }_{b,-}(p(x)D^{\alpha }_{a,+}y)(x)+\lambda q(x)\,y(x)=0$,\quad $a\lt x\lt b$,

\vspace {-2pt} \qquad \quad $\displaystyle \lim _{\stackrel {x\rightarrow a}{>}}(x-a)^{1-\alpha }y(x)=0=y(b)$,} \smallskip

\noindent where $p,q \in C([a,b])$, $p(x)>0$, $q(x)>0$, for all $x \in [a,b]$ and ${1}/{2} \lt \alpha \lt 1$. $D^{\alpha }_{b,-}$ and $D^{\alpha }_{a,+}$ are the right- and left-sided Riemann-Liouville fractional derivatives of order $\alpha \in (0,1)$, respectively. $\lambda $ is a scalar parameter.

First, we prove, using the spectral theory of linear compact operators, that this problem has an infinite sequence of real eigenvalues and the corresponding eigenfunctions form a complete orthonormal system in the Hilbert space $L^{2}_q[a,b]$. Then, we investigate some asymptotic properties of the spectrum as $\alpha \underset {\lt }{\rightarrow } 1$. We give, in particular, the asymptotic expansion of the first eigenvalue.

The existence and approximation of a solution to a delay equation with a random forcing term and non local conditions is studied by using a stochastic version of the well-known Banach fixed point theorem and semigroup theory. Moreover, the convergence of Faedo-Galerkin approximations of the solution is shown. An example is given which illustrates the results.

We demonstrate the existence of at least one positive solution to the perturbed Hammerstein integral equation \[ y(t)=\gamma _1(t)H_1\big (\varphi _1(y)\big )+\gamma _2(t)H_2\big (\varphi _2(y)\big )\] \[\qquad \qquad \qquad \quad +\lambda \int _0^1G(t,s)f\big (s,y(s)\big )\, ds,\] where certain asymptotic growth properties are imposed on the functions $f$, $H_1$ and $H_2$. Moreover, the functionals $\varphi _1$ and $\varphi _2$ are realizable as Stieltjes integrals with signed measures, which means that the nonlocal elements in the Hammerstein equation are possibly of a very general, sign-changing form. We focus here on the case where the kernel $(t,s)\mapsto G(t,s)$ is allowed to change sign and demonstrate the existence of at least one positive solution to the integral equation. As applications, we demonstrate that, by choosing $\gamma _1$ and $\gamma _2$ in particular ways, we obtain positive solutions to boundary value problems, both in the ODEs and elliptic PDEs setting, even when the Green's function is sign-changing, and, moreover, we are able to localize the range of admissible values of the parameter~$\lambda $. Finally, we also provide a result that for each $\lambda >0$ yields the existence of at least one positive solution.

In this paper, we are concerned with the controllability of linear and nonlinear Caputo impulsive fractional order damped systems in Banach spaces. Our main purpose is to establish some necessary and sufficient conditions for controllability for this kind of impulsive control system by using Mittag-Leffler matrix functions and the Schauder fixed point theorem.

In this paper, we obtain some estimates of essential norm of the Volterra-type integral operator $T_g$, where \[ T_gf(z)=\int ^z_0f(\zeta )g'(\zeta )\,d\zeta , \] from Hardy spaces to the BMOA space, Besov spaces, Berg\-man spaces and Bloch-type spaces.