On semilinear differential inclusions with lower semicontinuous nonlinearities

In questo lavoro si dà un teorema di esistenza di soluzioni per un inclusione differenziale semilineare in uno spazio di Banach. Le ipotesi sono deboli in quanto si suppone che l'operatore lineare A sia il generatore di un semigruppo fortemente continuo, senza ipotesi di compattezza, e la non linerità multivoca sia semicontinua inferiormente, non necessariamente a valori convessi. The work of the first two authors is partially supported by the Russian Foundation for basic Research Grant 99-01-00333. The work of the second author was carried out during his tenure at the University of Florence under a Fellowship from the Italian Ministry of Foreign Affairs through Landau Network—Centro Volta. The work of P. Zecca is supported by a National Grant 40% of MURST.     

Existence and controllability results for fractional semilinear differential inclusions

In this paper, we prove the existence and controllability results for fractional semilinear differential inclusions involving the Caputo derivative in Banach spaces. The results are obtained by using fractional calculation, operator semigroups and Bohnenblust–Karlin’s fixed point theorem. At last, an example is given to illustrate the theory.     

Existence and controllability results for nondensely defined impulsive semilinear functional differential inclusions

In this paper, we shall establish sufficient conditions for the existence of integral solutions and extremal integral solutions for some nondensely defined impulsive semilinear functional differential inclusions in separable Banach spaces. We shall rely on a fixed point theorem for the sum of completely continuous and contraction operators. The question of controllability of these equations and the topological structure of the solutions set are considered too.     

Multiplicity results for semilinear resonant elliptic problems with discontinuous nonlinearities

In the present paper, some generalized critical points theorems for locally Lipschitz functions are given; as applications, some multiplicity results for zero-boundary-value elliptic resonant problems are obtained. Two classical important theorems are improved.     

Controllability and strong controllability of differential inclusions

In this paper, we prove sufficient conditions for controllability and strong controllability in terms of the Mordukhovich subdifferential for two classes of differential inclusions. The first one is the class of sub-Lipschitz multivalued functions introduced by Loewen-Rockafellar (1994) [10]. The second one, introduced recently by Clarke (2005) [18], is the class of multivalued functions which are pseudo-Lipschitz and satisfy the so-called tempered growth condition. To do this, we establish an error bound result in terms of the Mordukhovich subdifferential outside Asplund spaces.     

Existence and controllability results for impulsive partial functional differential inclusions

In this paper we prove the existence and the controllability of mild and extremal mild solutions for first-order semilinear densely defined impulsive functional differential inclusions in separable Banach spaces with local and nonlocal conditions.     

Well posed cauchy problems for complex nonlinear equations must be semilinear

The first author was supported in part by CNPq     

Impulsive semilinear neutral functional differential inclusions with multivalued jumps

In this paper we establish sufficient conditions for the existence of mild solutions and extremal mild solutions for some densely defined impulsive semilinear neutral functional differential inclusions in separable Banach spaces. We rely on a fixed point theorem for the sum of completely continuous and contraction operators.     

Nonlocal problems for some partial differential equations

Abstract Nonlocal problems for polyharmonic functions and for a special third order system in a half plane are studied which have applications in elasticity. Some of them are solved explicitly on the basis of solutions to related classical problems, others are reduced to the RlEMANN problem for serveral holomorphic functions.     

Approximate controllability of Hilfer fractional differential inclusions with nonlocal conditions

In this paper, we investigate the approximate controllability of Hilfer fractional differential inclusions with nonlocal conditions. The main techniques rely on the fixed point theorem combined with the semigroup theory, fractional calculus, and multivalued analysis. An interesting example is provided to illustrate the obtained results. Copyright © 2016 John Wiley & Sons, Ltd.     

Approximate weak invariance for semilinear differential inclusions in Banach spaces

In this paper we give a criterion for a given set K in Banach space to be approximately weakly invariant with respect to the differential inclusion x′(t) ∈ Ax(t) + F(x(t)), where A generates a C ₀-semigroup and F is a given multi-function, using the concept of a tangent set to another set. As an application, we establish the relation between approximate solutions to the considered differential inclusion and solutions to the relaxed one, i.e., x′(t) ∈ Ax(t) + [`(co)]overline {co} F(x(t)), without any Lipschitz conditions on the multi-function F.     

Lower semi-continuity of the solution set for semilinear differential inclusions

In this paper we provide sufficient conditions for the solution set of a semilinear differential inclusion to be lower semi-continuous. We give an application to the propagation of continuity of the minimum time function associated with the given differential inclusion and the target zero.     

Existence results for fractional semilinear differential inclusions in Banach spaces

We investigate the existence of at least three positive solutions to a singular boundary value problem of fractional differential equation with first-order derivative. Our analysis relies on the Avery-Peterson fixed point theorem in a cone. An example is given to illustrate our results.     

Controllability for noninstantaneous impulsive semilinear functional differential inclusions without compactness

The first part of this paper considers the controllability for a functional semilinear differential inclusion governed by a family of operators $\{A\left(t\right):t\in [0,b]\}$ generating an evolution operator in a Banach space in the presence of noninstantaneous impulse effects. In the second part of this paper we study the controllability for a fractional noninstantaneous impulsive semilinear differential inclusion with delay, where the linear part is an infinitesimal generator of a $C_0?$semigroup. Using a weakly convergent criterion in the space of piecewise continuous functions and weak topology theory (for weak sequentially closed graph operators) we establish sufficient conditions to guarantee controllability results. Examples are given to illustrate the abstract results.     

Impulsive problems for semilinear differential equations with nonlocal conditions

This paper is concerned with the existence of impulsive semilinear differential equations with nonlocal conditions. Using the technique of fixed point theory, existence results are obtained, for mild solutions, when the nonlocal item is Lipschitz, is not Lipschitz and not compact, respectively.