This paper discusses the existence of mild solutions for a class of semilinear fractional evolution equations with nonlocal initial conditions in an arbitrary Banach space. We assume that the linear part generates an equicontinuous semigroup, and the nonlinear part satisfies noncompactness measure conditions and appropriate growth conditions. An example to illustrate the applications of the abstract result is also given.

This paper is concerned with the existence of mild solutions for impulsive semilinear differential equations with nonlocal conditions. Using the technique of measures of noncompactness in Banach and Fréchet spaces of piecewise continuous functions, existence results are obtained both on bounded and unbounded intervals, when the impulsive functions and the nonlocal item are not compact in the space of piecewise continuous functions but they are continuous and Lipschitzian with respect to some measure...

In this paper we investigate the existence of mild solutions defined on a semiinfinite interval for initial value problems for a differential equation with a nonlocal condition. The results is based on the Schauder-Tychonoff fixed point theorem and rely on a priori bounds on solutions.

The paper deals with the existence of multiple positive solutions for the boundary value problem $$\left\{\begin{array}{c}{\left(\varphi \left(p\left(t\right)u^{(n-1)}\right)\left(t\right)\right)}^{\text{'}}+a\left(t\right)f(t,u\left(t\right),u^{\text{'}}\left(t\right),...,u^{(n-2)}\left(t\right))=0,0<t<1,\hfill \\ u^{\left(i\right)}\left(0\right)=0,i=0,1,...,n-3,\hfill \\ u^{(n-2)}\left(0\right)=\sum _{i=1}^{m-2}{\alpha }_i{u}^{(n-2)}\left({\xi }_i\right),u^{(n-1)}\left(1\right)=0,\hfill \end{array}\right.$$ where $\varphi :ℝ\to ℝ$ is an increasing homeomorphism and a positive homomorphism with $\varphi \left(0\right)=0$. Using a fixed-point theorem for operators on a cone, we provide sufficient conditions for the existence of multiple positive solutions to the above boundary value problem.

We study the existence of principal eigenvalues for differential operators of second order which are not necessarily in divergence form. We obtain results concerning multiplicity of principal eigenvalues in both the variational and the general case. Our approach uses systematically the Krein-Rutman theorem and fixed point arguments for the spectral radius of some associated problems. We also use a variational characterization for both the self-adjoint and the general case.

The fixed point theorem of Krasnoselskii and the concept of large contractions are employed to show the existence of a periodic solution of a nonlinear integro-differential equation with variable delay $$x^{\text{'}}\left(t\right)=-{\int }_{t-\tau \left(t\right)}^{t}a(t,s)g\left(x\left(s\right)\right)ds+\frac{d}{dt}Q(t,x(t-\tau \left(t\right)))+G(t,x\left(t\right),x(t-\tau \left(t\right))).$$ We transform this equation and then invert it to obtain a sum of two mappings one of which is completely continuous and the other is a large contraction. We choose suitable conditions for $\tau$, $g$, $a$, $Q$ and $G$ to show that this sum of mappings fits into the framework of a modification of Krasnoselskii’s...

The existence of nonzero nonnegative solutions are established for semilinear equations at resonance with the zero solution and possessing at most linear growth. Applications are given to nonlinear boundary value problems of ordinary differential equations.