In this paper, four types of new nonlinear Henry–Gronwall type integral inequalities have been established. As for the first type, by employing inequality techniques, we overcome the limitation of traditional methods in which dividing the range of parameter β ∈ ( 0 , 1 ) into two parts is needed. For the second type, we derive a bound that is more precise than previous studies by comparative analysis. Regarding the third type and the fourth type, they are new models that are studied in our work. Specifically, the third type extends our proposed inequality to case when β ≥ 1 , and the fourth type constitutes a new variant of the nonlinear Bihari-type inequality with time-varying delay that offers greater generality. As applications of the derived results, the existence of solutions to the fractional differential equations has been discussed by fixed point theorems and two examples are provided to illustrate the validity of the theorems.

In this paper, we study the multiplicity of solutions to a class of Kirchhoff-type equation with critical growth − ( a + b ∫ R 3 | ∇ u | 2 d x ) Δ u + V ( x ) u = λ h ( x ) f ( u ) + g ( x ) u 5 in R 3 , where a , b > 0 , λ is a positive parameter and f is a continuous nonlinearity with subcritical growth. Under suitable conditions on the potentials V ( x ) , h ( x ) and g ( x ) , we prove the multiplicity results and investigate the relation between the number of solutions with the topology of the set where g ( x ) attains its maximum value for small values of the parameter λ . The proofs are based on Nehari manifold and Lusternik–Schnirelmann theory.

We correct and update a result of R. G. D. Richardson [Amer. J. Math. 40(1918), 283–316] dealing with the separation of zeros of the real and imaginary parts of non-real eigenfunctions of non-definite Sturm–Liouville eigenvalue problems. We then extend it to the case where the weight function is allowed to be identically zero on a subinterval that excludes the end-points and study the behavior of the zeros of the real and imaginary parts when the end-points are included. Examples are given illustrating the sharpness of the results along with open questions.

In this paper, under the condition that there exists an ordered interval composed of two internal ordered intervals which have the location similar to that of Amann's three-solution theorem, we add some simple conditions and then we obtain some results about the existence of multiple critical points inside and outside the ordered interval. The main results of this paper can be regarded as an extension of the classical Amann three-solution theorem and the mountain pass lemma on the ordered interval of Shujie Li and Zhiqiang Wang. To show our main results, we extend the method of invariant sets of descending flow that proposed by Jingxian Sun for smooth functionals to the locally Lipschitz functionals. Our main results can be applied to the study of differential inclusion problems with concave-convex nonlinearity. In this way, we partially extend some relevant results concerning the differential equation boundary value problems with a concave-convex nonlinearity that was first studied by A. Ambrosetti, H. Brezis and G. Cerami.

This paper is concerned with the existence of solutions for parameters dependent Schrödinger–Kirchhoff system driven by nonlocal integro-differential operators with singular Trudinger–Moser nonlinearity in the whole Euclidean space R N . These parameters have a major impact on the produced analysis. It is noted that, we also study the asymptotic behaviour of solutions depending upon these parameters. The proofs of the existence results to the aforementioned system rely on the mountain pass theorem, the Ekeland variational principle, the classical deformation lemma, and the Krasnoselskii genus theory. The salient feature and novelty of this paper is that it also covers the so-called degenerate case of the Kirchhoff function, that is, it could vanish at zero.

In this paper, we characterize a family of planar polynomial differential systems of degree greater or equal than n + 1 , by presenting polynomial curves of degree n , which generally contain closed components. These systems admit precisely the bounded components of the curve as hyperbolic limit cycles.

We establish that uniformly exponentially stable random dynamical systems on the half line have equivalent dynamics through a C m -conjugacy. This result was obtained for random differential equations as well as for random dynamical systems with a uniformly exponentially stable linear part.

In this paper, we investigate the existence and regularity of positive solutions for certain singular problems that involve an anisotropic ( p , q ) -Laplacian-type operator and a singular term with a variable exponent, under zero Dirichlet boundary conditions on ∂ Ω . The main equation we analyze is − ∑ i = 1 N ∂ i ( | ∂ i u ( x ) | p i − 2 ∂ i u ( x ) ) − ∑ i = 1 N ∂ i ( | ∂ i u ( x ) | q i − 2 ∂ i u ( x ) ) = f ( x ) u ( x ) γ ( x ) in Ω , where Ω is a bounded, regular domain in R N , f is a positive function belonging to a specific Lebesgue space, and γ ( x ) is a positive continuous function on Ω ¯ . In our study, we do not make comparisons between p i and q i , and as a result, we show that the solution belongs to either W 0 1 , p → ( Ω ) ∩ W 0 1 , q → ( Ω ) or W l o c 1 , p → ( Ω ) ∩ W l o c 1 , q → ( Ω ) depending on the summability of f ( x ) and the values of γ ( x ) . The results are achieved using approximation techniques that include truncation, comparison, and variational methods.

In our work, we prove the existence and uniqueness of positive 1 -periodic solutions of a first order differential equation that contains a piecewise constant term. Such equations admit periodic solutions and hence they can be used for numerical modelling of temperature changes in a space surrounded by environment with different temperature. For some types of functional equations, it is possible to use the comparison principle method of differential equations to obtain existence, uniqueness, and asymptotic stability results.

We present a detailed study of a scalar differential equation with threshold state-dependent delayed feedback. This equation arises as a simplification of a gene regulatory model. There are two monotone nonlinearities in the model: one describes the dependence of delay on state, and the other is the feedback nonlinearity. Both increasing and decreasing nonlinearities are considered. Our analysis is exhaustive both analytically and numerically as we examine the bifurcations of the system for various combinations of increasing and decreasing nonlinearities. We identify rich bifurcation patterns including Bautin, Bogdanov–Takens, cusp, fold, homoclinic, and Hopf bifurcations whose existence depend on the derivative signs of nonlinearities. Our analysis confirms many of these patterns in the limit where the nonlinearities are switch-like and change their value abruptly at a threshold. Perhaps one of the most surprising findings is the existence of a Hopf bifurcation to a periodic solution when the nonlinearity is monotone increasing and the time delay is a decreasing function of the state variable.