We survey and announce some current results on the existence, the viability, and the topological structure of the viable solutions of differential equations and inclusion in Banach spaces under set constraints. Some new results concerning semilinear differential inclusions with state variables constrained to the so-called regular and strictly regular sets, together with their applications, are ...

In this paper we prove the existence of solutions of fractional impulsive semilinear evolution equations in Banach spaces. A nonlocal Cauchy problem is discussed for the evolution equations. The results are obtained using fractional calculus and fixed point theorems. An example is provided to illustrate the theory.

The existence of mild solutions of first order semilinear evolution inclusions on a compact real interval is investigated in Banach spaces with nonlocal conditions. A fixed point theorem for condensing maps due to Martelli is used. 2000 Mathematics Subject Classification: 34A60, 34G20, 35R10, 47H20

Sufficient conditions ensuring weak lower semi-continuity of the optimal value function are presented. To this end, refined inner semi-continuity properties of set-valued maps are introduced which meet the needs of the weak topology in Banach spaces. The results are applied to prove the existence of solutions in various worst-case robust optimal control problems governed by semilinear elliptic ...

Imperfect bifurcation phenomena are formulated in framework of analytical bifurcation theory on Banach spaces. In particular the perturbations of transcritical and pitchfork bifurcations at a simple eigenvalue are examined, and two-parameter unfoldings of singularities are rigorously established. Applications include semilinear elliptic equations, imperfect Euler buckling beam problem and pertu...

We prove convergence of the solutions Xn of semilinear stochastic evolution equations on a Banach space B, driven by a cylindrical Brownian motion in a Hilbert space H, dXn(t) = (AnX(t) + Fn(t,Xn(t))) dt+Gn(t,Xn(t)) dWH(t), Xn(0) = ξn, assuming that the operators An converge to A and the locally Lipschitz functions Fn and Gn converge to the locally Lipschitz functions F and G in an appropriate ...

We first obtain some properties of a fundamentally nonexpansive self-mapping on a nonempty subset of a Banach space and next show that if the Banach space is having the Opial condition, then the fixed points set of such a mapping with the convex range is nonempty. In particular, we establish that if the Banach space is uniformly convex, and the range of such a mapping is bounded, closed and con...

This paper present results of ω-order preserving partial contraction mapping generating a general class semilinear initial value problems. We consider the use fractional powers unbounded linear operators for its application by starting with some concerning such powers. assume A to be infinitesimal generator an analytic semigroup in Banach space X, 0 ∈ ρ(A) and defined < α ≤ 1. also show that...

In this paper we continue our investigations. begun in the previous paper, of describing the solution sets of constrained extremum problems inf Au). l&l-‘(p) (*) where fand f are twice continuously differentiable functionals on a reflexive Banach space V, and r-‘(p) denotes the level set of the functional f with value p E R. Considering p as a parameter in (*) we obtain results concerning the c...