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The existence of mild solutions and approximate controllability for Riemann–Liouville fractional neutral evolution systems with nonlocal conditions of a fractional order is investigated. The Laplace transform and semigroup theory are the tools used to prove the existence. In turn, approximate controllability is proved on the basis of a Nemytskii operator, a Mittag-Leffler function and certain hypotheses using fixed point theorems, as well as the construction of a Cauchy sequence. An example is provided to highlight the main results.

The continuous Hopfield network (CHN) is a common recurrent neural network. The CHN tool can be used to solve a number of ranking and optimization problems, where the equilibrium states of the ordinary differential equation (ODE) related to the CHN give the solution to any given problem. Because of the non-local characteristic of the "infinite memory" effect, fractional-order (FO) systems have been proved to describe more accurately the behavior of real dynamical systems, compared to the model's ODE. In this paper, a fractional-order variant of a Hopfield neural network is introduced to solve a Quadratic Knap Sac Problem (QKSP), namely the fractional CHN (FRAC-CHN). Firstly, the system is integrated with the quadratic method for fractional-order equations whose trajectories have shown erratic paths and jumps to other basin attractions. To avoid these drawbacks, a new algorithm for obtaining an equilibrium point for a CHN is introduced in this paper, namely the optimal fractional CHN (OPT-FRAC-CHN). This is a variable time-step method that converges to a good local minima in just a few iterations. Compared with the non-variable time-stepping CHN method, the optimal time-stepping CHN method (OPT-CHN) and the FRAC-CHN method, the OPT-FRAC-CHN method, produce the best local minima for random CHN instances and for the optimal feeding problem. [ABSTRACT FROM AUTHOR]

We prove existence of entropy solutions to general class of unilateral nonlinear parabolic equation in inhomogeneous Musielak-Orlicz spaces avoiding ceorcivity restrictions on the second lower order term. Namely, we consider

We are concerned with the study of the existence and multiplicity of solutions for Dirichlet boundary value problems, involving the $ (p(m), \, q(m))- $ equation and the nonlinearity is superlinear but does not fulfil the Ambrossetti-Rabinowitz condition in the framework of Sobolev spaces with variable exponents in a complete manifold. The main results are proved using the mountain pass theorem and Fountain theorem with Cerami sequences. Moreover, an example of a $ (p(m), \, q(m)) $ equation that highlights the applicability of our theoretical results is also provided. Note: The second old affiliation of fourth author has been removed. The Acknowledgments' section has been updated. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the existence of at least two nonnegative nontrivial solutions to a p z –Laplacian system involving critical nonlinearity in the context of Sobolev spaces with variable exponents on complete manifolds. We have established our main results by exploring both Nehari's method and doing a refined analysis on the associated fiber map and some variational techniques. [ABSTRACT FROM AUTHOR]

This paper is devoted to study the global existence of solutions of the hyperbolic Dirichlet equation u t ⁢ t = L ⁢ u + f ⁢ ( x , t ) in ⁢ Ω T = Ω × ( 0 , T ) , u_{tt}=Lu+f(x,t)\quad\text{in }\Omega_{T}=\Omega\times(0,T), where L is a nonlinear operator and ϕ ⁢ ( x , t , ⋅ ) {\phi(x,t,\cdot\,)} , f ⁢ ( x , t ) {f(x,t)} and the exponents of the nonlinearities p ⁢ ( x , t ) {p(x,t)} and μ ⁢ ( x , t ) {\mu(x,t)} are given functions. [ABSTRACT FROM AUTHOR]