Mawhin’s Continuation Technique for a Nonlinear BVP of Variable Order at Resonance via Piecewise Constant Functions

In this paper, we establish the existence of solutions to a nonlinear boundary value problem (BVP) of variable order at resonance. The main theorem in this study is proved with the help of generalized intervals and piecewise constant functions, in which we convert the mentioned Caputo BVP of fractional variable order to an equivalent standard Caputo BVP at resonance of constant order. In fact, to use the Mawhin’s continuation technique, we have to transform the variable order BVP into a constant order BVP. We prove the existence of solutions based on the existing notions in the coincidence degree theory and Mawhin’s continuation theorem (MCTH). Finally, an example is provided according to the given variable order BVP to show the correctness of results.

Periodic Solution of A Delayed Intraguild Predation Impulsive System with Strong Allee Effect

With periodic coefficients and strong Allee effects, we establish a delayed intraguild predation impulsive model. We obtain a set of sufficient conditions for the existence of positive periodic solution of the model using Mawhin’s continuation theorem and analysis techniques. Finally, we identify the effectiveness of the theoretical results through some numerical simulations.

Periodic solution for inertial neural networks with variable parameters

<abstract><p>We discuss periodic solution problems and asymptotic stability for inertial neural networks with $ D- $operator and variable parameters. Based on Mawhin's continuation theorem and Lyapunov functional method, some new sufficient conditions on the existence and asymptotic stability of periodic solutions are established. Finally, a numerical example verifies the effectiveness of the obtained results.</p></abstract>

On the existence of almost periodic solutions of impulsive non-autonomous Lotka-Volterra predator-prey system with harvesting terms

<abstract><p>In this paper, by using the Mawhin's continuation theorem, some easily verifiable sufficient conditions are obtained to guarantee the existence of almost periodic solutions of impulsive non-autonomous Lotka-Volterra predator-prey system with harvesting terms. Our result corrects the result obtained in ^{[<xref ref-type="bibr" rid="b13">13</xref>]}. An example and some remarks are given to illustrate the advantage of this paper.</p></abstract>

Solutions to Neumann boundary value problems with a generalized 𝑝-Laplacian

The purpose of this work is to investigate the existence of solutions for various Neumann boundary value problems associated to the Laplacian-type operators. The main results are obtained using the extension of Mawhin’s continuation theorem.

Positive periodic solution for inertial neural networks with time-varying delays

In this paper the problems of the existence and stability of positive periodic solutions of inertial neural networks with time-varying delays are discussed by the use of Mawhin’s continuation theorem and Lyapunov functional method. Some sufficient conditions are obtained for guaranteeing the existence and stability of positive periodic solutions of the considered system. Finally, a numerical example is given to illustrate the effectiveness of the obtained results.

Periodic solution for neutral-type inertial neural networks with time-varying delays

In this paper the problems of the existence and stability of periodic solutions of neutral-type inertial neural networks with time-varying delays are discussed by applying Mawhin’s continuation theorem and Lyapunov functional method. Finally, two numerical examples are given to illustrate our theoretical results.

Existence of positive periodic solutions for super-linear neutral Liénard equation with a singularity of attractive type

In this paper, the existence of positive periodic solutions is studied for super-linear neutral Liénard equation with a singularity of attractive type $$ \bigl(x(t)-cx(t-\sigma)\bigr)''+f\bigl(x(t) \bigr)x'(t)-\varphi(t)x^{\mu}(t)+ \frac{\alpha(t)}{x^{\gamma}(t)}=e(t), $$ ( x ( t ) − c x ( t − σ ) ) ″ + f ( x ( t ) ) x ′ ( t ) − φ ( t ) x μ ( t ) + α ( t ) x γ ( t ) = e ( t ) , where $f:(0,+\infty)\rightarrow R$ f : ( 0 , + ∞ ) → R , $\varphi(t)>0$ φ ( t ) > 0 and $\alpha(t)>0$ α ( t ) > 0 are continuous functions with T-periodicity in the t variable, c, γ are constants with $|c|<1$ | c | < 1 , $\gamma\geq1$ γ ≥ 1 . Many authors obtained the existence of periodic solutions under the condition $0<\mu\leq1$ 0 < μ ≤ 1 , and we extend the result to $\mu>1$ μ > 1 by using Mawhin’s continuation theorem as well as the techniques of a priori estimates. At last, an example is given to show applications of the theorem.

Periodic Property and Asymptotic Behavior for a Discrete Ratio-Dependent Food-Chain System with Delays

In this paper, a discrete ratio-dependent food-chain system with delay is investigated. By using Gaines and Mawhin’s continuation theorem of coincidence degree theory and the method of Lyapunov function, a set of sufficient conditions for the existence of positive periodic solutions and global asymptotic stability of the model are established.

Dynamics of two-species delayed competitive stage-structured model described by differential-difference equations

Overf the last few years, by utilizing Mawhin’s continuation theorem of coincidence degree theory and Lyapunov functional, many scholars have been concerned with the global asymptotical stability of positive periodic solutions for the non-linear ecosystems. In the real world, almost periodicity is usually more realistic and more general than periodicity, but there are scarcely any papers concerning the issue of the global asymptotical stability of positive almost periodic solutions of non-linear ecosystems. In this paper we consider a kind of delayed two-species competitive model with stage structure. By means of Mawhin’s continuation theorem of coincidence degree theory, some sufficient conditions are obtained for the existence of at least one positive almost periodic solutions for the above model with nonnegative coefficients. Furthermore, the global asymptotical stability of positive almost periodic solution of the model is also studied. The work of this paper extends and improves some results in recent years. An example and simulations are employed to illustrate the main results of this paper.