A Fractional-Order Sequential Hybrid System with an Application to a Biological System

With the help of Banach’s fixed-point approach and the Leray–Schauder alternative theorem, we produced existence results for a general class of fractional differential equations in this paper. The proposed problem is more comprehensive and applicable to real-life situations. As an example of how our problem might be used, we have created a fractional-order COVID-19 model whose solution is guaranteed by our results. We employed a numerical approach to solve the COVID-19 model, and the results were compared for different fractional orders. Our numerical results for fractional orders follow the same pattern as the classical example of order 1, indicating that our numerical scheme is accurate.

Global exponential stability of periodic solution of delayed discontinuous Cohen–Grossberg neural networks and its applications

This paper is concerned with the existence and global exponential stability of the periodic solution of delayed Cohen–Grossberg neural networks (CGNNs) with discontinuous activation functions. The activations considered herein are non-decreasing but not required to be Lipschitz or continuous. Based on differential inclusion theory, Lyapunov functional theory and Leary–Schauder alternative theorem, some sufficient criteria are derived to ensure the existence and global exponential stability of the periodic solution. In order to show the superiority of the obtained results, an application and some detailed comparisons between some existing related results and our results are presented. Finally, some numerical examples are also illustrated.

Existence results to a ψ- Hilfer neutral fractional evolution equation with infinite delay

In this paper, we prove the existence and uniqueness of a mild solution to the system of ψ- Hilfer neutral fractional evolution equations with infinite delay H 𝔻0 αβ;ψ [x(t) − h(t, xt )] = A x(t) + f (t, x(t), xt ), t ∈ [0, b], b > 0 and x(t) = ϕ(t), t ∈ (−∞, 0]. We first obtain the Volterra integral equivalent equation and propose the mild solution of the system. Then, we prove the existence and uniqueness of solution by using the Banach contraction mapping principle and the Leray-Schauder alternative theorem.

New results for a coupled system of ABR fractional differential equations with sub-strip boundary conditions

<abstract><p>In this article, we investigate sufficient conditions for the existence, uniqueness and Ulam-Hyers (UH) stability of solutions to a new system of nonlinear ABR fractional derivative of order $ 1 &lt; \varrho\leq 2 $ subjected to multi-point sub-strip boundary conditions. We discuss the existence and uniqueness of solutions with the assistance of Leray-Schauder alternative theorem and Banach's contraction principle. In addition, by using some mathematical techniques, we examine the stability results of Ulam-Hyers (UH). Finally, we provide one example in order to show the validity of our results.</p></abstract>

Technical Note Alternative Theorem: Equilibrium in Abstract Economies

We prove a new theorem for the existence of equilibrium in discontinuous games in which the players’ preferences are neither complete nor transitive. Our result is an alternative version of Shafer and Sonnenschein ([1975] J. Math. Econ. 2, 345–348), He and Yannelis ([2016] Econ. Theory 61, 497–513), Reny ([2016] Econ. Theory Bull.) and Carmona and Prodczeck ([2016] Econ. Theory 61, 457–478).

Optimality Conditions for a Nonsmooth Uncertain Multiobjective Programming Problem

This note is devoted to the investigation of optimality conditions for robust approximate quasi weak efficient solutions to a nonsmooth uncertain multiobjective programming problem (NUMP). Firstly, under the extended nonsmooth Mangasarian–Fromovitz constrained qualification assumption, the optimality necessary conditions of robust approximate quasi weak efficient solutions are given by using an alternative theorem. Secondly, a class of generalized convex functions is introduced to the problem (NUMP), which is called the pseudoquasi-type-I function, and its existence is illustrated by a concrete example. Finally, under the pseudopseudo-type-I generalized convexity hypothesis, the optimality sufficient conditions for robust approximate quasi weak efficient solutions to the problem (NUMP) are established.

Accurate Depth Inversion Method for Coastal Bathymetry: Introduction of Water Wave High-Order Dispersion Relation

This paper proposes a wave model for the depth inversion of sea bathymetry based on a new high-order dispersion relation which is more suitable for intermediate water depth. The core of this model, a high-order dispersion relation is derived in this paper. First of all, new formulations of wave over generally varying seabed topography are derived using Fredholm’s alternative theorem (FAT). In the new formulations, the governing equation is coupled with wave number and varying seabed effects. A new high-order dispersion relation can be obtained from the coupling equation. It is worth mentioning that both the slope square and curvature terms ( ( ∇ h ) 2 , ∇ 2 h , ( ∇ k ) 2 , ∇ 2 k , ∇ h ⋅ ∇ k ) of water wavenumber and seabed bottom are explicitly expressed in high-order dispersion relation. Therefore, the proposed method of coastal bathymetry reversion using the higher-order dispersion relation model is more accurate, efficient, and economic.

Relaxation of the game problem of guidance connected with alternative in guidance-evasion differential game

Differential game (DG) of guidance-evasion for a finite time interval is considered;as parameters, the target set (TS) and the set defining phase constraints (PC) are used.Player I interested in realization of guidance with TS under validity PC uses set-valuedquasistrategies (nonanticipating strategies) and Player II having opposite target uses strategieswith nonanticipating choice of correction instants and finite numbers of such instants.On informative level, the setting corresponds to alternative theorem of N. N. Krasovskii andA. I. Subbotin. For position not belonging to solvability set of Player I, determination ofthe least size of neighborhoods for set-parameters under that Player I guarantees guidance(under weakened conditions) is interested. In article, this scheme is supplemented by priorityelements in questions of TS attainment and PC validity; this is realized by special parameterdefining relation for sizes of corresponding neighborhoods. Under these conditions, a functionof the least size of TS neighborhood is defined by procedure used program iteration methodfor two variants. The above-mentioned function is fixed point for one of two used “program”operators. Special type of the quality functional for which values of the above-mentionedfunction coincide with values of the minimax-maximin games is established.

Robust stability of mixed Cohen–Grossberg neural networks with discontinuous activation functions

PurposeThe purpose of this paper is to develop a method for the existence, uniqueness and globally robust stability of the equilibrium point for Cohen–Grossberg neural networks with time-varying delays, continuous distributed delays and a kind of discontinuous activation functions.Design/methodology/approachBased on the Leray–Schauder alternative theorem and chain rule, by using a novel integral inequality dealing with monotone non-decreasing function, the authors obtain a delay-dependent sufficient condition with less conservativeness for robust stability of considered neural networks.FindingsIt turns out that the authors’ delay-dependent sufficient condition can be formed in terms of linear matrix inequalities conditions. Two examples show the effectiveness of the obtained results.Originality/valueThe novelty of the proposed approach lies in dealing with a new kind of discontinuous activation functions by using the Leray–Schauder alternative theorem, chain rule and a novel integral inequality on monotone non-decreasing function.