In this paper, we introduce a new class of nonsmooth pseudo-invex and nonsmooth quasi-invex functions to non-smooth variational problems. By using these concepts, numbers of necessary and sufficient conditions are established for a nonsmooth variational problem wherein Clarke’s generalized gradient is used. Also, weak, strong and converse duality are established. Keywords—Variational problem; N...

The aim of this paper is to extend to variational inequalities a result of Amann and Zehnder’s, refined by Chang (see [1, 6, 14]), concerning the existence of nontrivial solutions for a semilinear elliptic boundary value problem, when the nonlinearity is asymptotically linear, is zero at zero and its derivative has a suitable jump between zero and infinity. In the constrained problem studied he...

This paper considers the state observation problem for autonomous nonlinear systems. An observation mapping is introduced, which is defined by applying a linear integral operator (rather than a differential operator) to the output of the system. It is shown that this observation mapping is well suited to capture the observability nature of smooth as well as nonsmooth systems, and to construct o...

Abstract We present a new proof of the turnpike property for nonlinear optimal control problems, when running target is steady control-state pair underlying system. Our strategy combines construction quasi-turnpike controls via controllability, and bootstrap argument, does not rely on analyzing optimality system or linearization techniques. This in turn allows us to address several problems fin...

A nonsmooth modeling paradigm for dynamic simulation and optimization of process operations is advocated. Nonsmooth differential-algebraic equations (DAEs) naturally model a wide range of physical systems encountered in chemical engineering conventionally viewed as exhibiting hybrid continuous/discrete behavior. Due to recent advancements in nonsmooth analysis, nonsmooth DAEs now have a suitabl...

Implicit Runge-Kutta(IRK) methods for solving the nonsmooth ordinary differential equation (ODE) involve a system of nonsmooth equations. We show superlinear convergence of the slanting Newton method for solving the system of nonsmooth equations. We prove the slanting differentiability and give a slanting function for the involved function. We develop a new code based on the slanting Newton met...

Abstract. We consider a semilinear elliptic equation with a nonsmooth, locally Lipschitz potential function (hemivariational inequality). Our hypotheses permit double resonance at infinity and at zero (double-double resonance situation). Our approach is based on the nonsmooth critical point theory for locally Lipschitz functionals and uses an abstract multiplicity result under local linking and...

We consider minimizing a nonsmooth objective subject to nonsmooth constraints. The nonsmooth functions are approximated by a bundle of subgradients. The novel idea of a filter is used to promote global convergence.

This paper deals with a general system of equations and conditions arising from mathematical model prostate cancer growth chemotherapy antiangiogenic therapy that has been recently introduced analyzed; see P. Colli et al. \[Math. Models Methods Appl. Sci. 30 (2020), 1253–1295]. The related includes two evolutionary operator involving fractional powers selfadjoint, nonnegative, unbounded linear ...

Given $$n \in \{ 3,4,5 \}$$ and $$k > 1$$ (resp. $$ \frac{4}{3}> k ) if 3,4 $$n=5$$ ), we prove scattering of the radial $${\tilde{H}}^{k} := {\dot{H}}^{k} ({\mathbb {R}}^{n}) \cap {\dot{H}}^{1} {R}}^{n})-$$ solutions focusing log energy-supercritical Schrödinger equation i \partial _{t} u + \triangle = -|u|^{\frac{4}{n-2}} \log ^{\gamma } ( 2 |u|^{2})$$ for a range positive $$\gamma \, s$$ dep...