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Regularity and existence of solutions to parabolic equations with nonstandard p(x,t),q(x,t)-growth conditions

El Bahja Hamid

Opuscula Mathematica

2023

Vol. 43, no. 6

759--788

We study the Cauchy–Dirichlet problem for a class of nonlinear parabolic equations driven by nonstandard p(x, t), q(x, t)-growth condition. We prove theorems of existence and uniqueness of weak solutions in suitable Orlicz-Sobolev spaces, derive global and local in time L∞ bounds for the weak solutions.

Existence and stability of impulsive coupled system of fractional integrodifferential equations

Zada Akbar, Waheed Hira, Alzabut Jehad, Wang Xiaoming

Demonstratio Mathematica

2019

Vol. 52, nr 1

296--335

In this manuscript, we deal with a class and coupled system of implicit fractional differential equations, having some initial and impulsive conditions. Existence and uniqueness results are obtained by means of Banach’s contraction mapping principle and Krasnoselskii’s fixed point theorem. Hyers–Ulam stability is investigated by using classical technique of nonlinear functional analysis. Finally, we provide illustrative examples to support our obtained results.

Discrete approximation of nonconvex hyperbolic optimal control problems with state constraints

Chryssoverghi I., Bacopoulos A., Coletsos J., Kokkinis B.

Control and Cybernetics

1998

Vol. 27, no 1

29-50

We consider an opitmal control problem for systems defined by nonlinear hyperbolic partial differential equations with state constraints. Since no convexity assumptions are made on the data, we also consider the control problem in relaxed form. We discretize both the classical and the relaxed problenms by using a finite element method in space and a finite difference scheme in time, the controls being approximated by piecevise constant ones. We develop the existence theory and the necessary conditions for optimality, for the continous and the discrete problems. Finally, we study the behaviour in the limit of discrete optimality, admissibility and extremality properties.