Using the recently defined generalized derivative, we present a generalized formulation of variation of calculus, which includes the classical and conformable formulation as particular cases. In the first part of the article, through the properties of this generalized derivative, we discuss the generalized versions of the Bois–Reymond lemma, a Tonelli-type existence theorem, Euler–Lagrange equation, d’Alembert principle, du Bois–Reymond optimality condition and Noether’s theorem. In the second part, we discuss the Picard–Lindelöf theorem, Grönwall’s inequality, Pontryagin’s maximum principle and Noether’s principle for optimal control. We end with an application involving the time fractional Schrödinger equation.

Our purpose is to extend the work of Lois-Prados and Precup (2020) and use the Krasnoselskii-type homotopy fixed-point theorem to prove the existence of positive periodic solutions of nonautonomous Lotka–Volterra dynamic systems with a general attack rate on time scales.

We study an inverse source problem for a multiterm time-fractional diffusion equation from a noisy final data in a general bounded domain. This problem is ill-posed. Uniqueness and a conditional stability for the inverse problem are derived based on an expression of the solution and some properties of the multinomial Mittag-Leffler function. Further we introduce the modified quasiboundary regularization method and the Landweber iterative regularization method to solve the inverse source problem. Convergence estimates between the regularization solution and the exact solution are given under the a priori regularization parameter choice rule and the a posteriori regularization parameter choice rule, respectively. Finally, we use the finite difference method to solve the direct problem and the inverse source problem in the one-dimensional case, and apply the finite element method to solve them in the two-dimensional case. Numerical examples are provided to show the effectiveness of the proposed method in the one- and two-dimensional cases.

We aim to study clopen linear subspaces and connectedness properties of the space $C(X)$ of all real-valued continuous functions defined on a metric space $(X,d)$ equipped with various topologies. In particular, we consider the topologies of strong Whitney and strong uniform convergence on bornology. We also examine when these topologies on $C(X)$ are locally convex. While studying clopen subspaces, we give new characterizations for the notion of a shield and for a bornology to be shielded from closed sets.

We show some existence results for the system of nonlocal Neumann problems with the Minkowski-curvature operator

$${\left(r^{N-1}\frac{u^{\prime }}{\sqrt{1-{u^{\prime }}^2}}\right)}^{\prime }=r^{N-1}f(r,u,u^{\prime }),r\in (0,1),u^{\prime }(0)=0,u^{\prime }(1)={\displaystyle {\int }_0^1{u}^{\prime }(s)dg(s)},$$

where $N\ge 1$ is an integer, $f:[0,1]\times {ℝ}^k\times I^k\to {ℝ}^k$ is continuous and bounded, $I≔(-1,1)$, and $g:[0,1]\to {ℝ}^k$ is a function of bounded variation. The proof is based on topological-degree arguments and extends to a larger class of nonlinearities.

We study the bifurcation structure to Serrin’s overdetermined problem such that

$$-\mathrm{\Delta }u=1\text{ in }\mathrm{\Omega },u=0,{\partial }_{\nu }u=\text{ const on }\partial \mathrm{\Omega }.$$

We prove that the bifurcation from the straight cylinder $B_{{\mathrm{\lambda }}_1}\times ℝ$ with is critical at the bifurcation point. Moreover, we obtain the global structure of bifurcation branches. To study the global structure of bifurcation branches, we establish a global bifurcation theorem in finite dimensional space.

We consider a Caputo-type hybrid functional fractional differential equation of order with nonlocal boundary conditions. By using the fixed-point theorem in Banach algebra due to Dhage (1988), we study the existence of the solutions.

We deal with entropy solutions to degenerate elliptic equations of the form

where the Carathéodory function $\mathcal{𝒜}:\mathrm{\Omega }\times ℝ\times {ℝ}^n\to {ℝ}^n$ satisfies degenerate coercivity condition

$$\mathcal{𝒜}(x,s,\xi )\cdot \xi \ge \alpha \frac{|\xi {|}^p}{{(1+|s|)}^{\tau }}$$

and controllable growth condition

$$|\mathcal{𝒜}(x,s,\xi )|\le \beta |\xi {|}^{p-1}$$

for almost all $x\in \mathrm{\Omega }$ and all $(s,\xi )\in ℝ\times {ℝ}^n$. We let , , , we let $f$ and $E$ belong to some Marcinkiewicz spaces, and we give some regularity properties for entropy solutions. We derive a generalized version of Stampacchia’s lemma in order to prove the main theorem.

We study the existence of homoclinic orbits of the second order quasilinear Schrödinger equations

$$\ddot{u}(t)-V(t)u(t)+2[\ddot{u}(t)u^2(t)+{\dot{u}}^2(t)u(t)]+g(t,u(t+\tau ),u(t),u(t-\tau ))=h(t).$$

containing both advance and retardation terms. By using critical point theory and variational approaches, we establish two different existence results. The first is based on $g$ which does not satisfy the Ambrosetti–Rabinowitz growth condition. The second is based on $g$ satisfying the Ambrosetti–Rabinowitz growth condition.

We give the generalization of $\alpha$-Baskakov Durrmeyer operators depending on a real parameter . We present the approximation results in Korovkin and weighted Korovkin spaces. We also prove the order of approximation and rate of approximation for these operators. In the end, we verify our results with the help of numerical examples by using Mathematica.

We show the global structure of positive solutions for second order periodic boundary value problem

$$\{\begin{array}{l}-{\mathrm{\Delta }}^{2}u(t-1)=\lambda a(t)g(u(t)),t\in {\mathbb{N}}_1^T,\hfill \\ u(0)=u(T),u(1)=u(T+1),\hfill \end{array}$$

where ${\mathbb{N}}_1^T=\{1,2,\dots ,T\},T\ge 3$ is an integer, is a parameter, $g:[0,\infty )\to [0,\infty )$ is a continuous function with $g(0)=0$ and $a:{\mathbb{N}}_1^T\to ℝ$ is sign-changing. Depending on the behavior of $g$ near $0$ and $\infty$, we obtain that there exist such that above problem has at least two positive solutions for and no solution for $\lambda \in (0,{\lambda }_0)$. The proof of our main results is based upon bifurcation technique.

We discuss a class of $p$-Laplacian-type fractional four-point boundary-value problems with a parameter. We use the Green’s function, the Schauder fixed-point theorem and the Guo–Krasnoselskii fixed-point theorem on cones. Some examples are presented to show the validity of the conditions of our main theorem.

A class of commutative Banach algebras which satisfy a Bochner–Schoenberg–Eberlein-type inequality was introduced by Takahasi and Hatori. We generalize this property for the commutative Fréchet algebra ${(\mathcal{𝒜},p_{\ell })}_{\ell \in \mathbb{N}}$. Moreover, we verify and generalize some of the main results in the class of Banach algebras, for the Fréchet case. We prove that all Fréchet ${\mathrm{C}}^{*}$-algebras and also uniform Fréchet algebras are BSE algebras. Also, we show that $C^{\infty }[0,1]$ is not a Fréchet BSE algebra.

A graph is split if its vertex set can be partitioned into a clique and an independent set. A split graph is $(x,y)$-bidegreed if each of its vertex degrees is equal to either $x$ or $y$. Each connected split graph is of diameter at most 3. In 2017, Nikiforov proposed the $A_{\alpha }$-matrix, which is the convex combination of the adjacency matrix and the diagonal matrix of vertex degrees of the graph $G$. It is well-known that a connected graph of diameter $l$ contains at least $l+1$ distinct $A_{\alpha }$-eigenvalues. A graph is said to be $l_{\alpha }$-extremal with respect to its $A_{\alpha }$-matrix if the graph is of diameter $l$ having exactly $l+1$ distinct $A_{\alpha }$-eigenvalues. In this paper, using the association of split graphs with combinatorial designs, the connected $2_{\alpha }$-extremal (resp. $3_{\alpha }$-extremal) bidegreed split graphs are classified. Furthermore, all connected bidegreed split graphs of diameter 2 having just 4 distinct $A_{\alpha }$-eigenvalues are identified.

We study focal surfaces of (wave) fronts associated to unbounded principal curvatures near nondegenerate singular points of initial fronts. We give characterizations of singularities of those focal surfaces in terms of types of singularities and geometrical properties of initial fronts. Moreover, we investigate behavior of the Gaussian curvature of the focal surface.

We introduce and study the regularity properties of the commutator of the bilinear maximal operator and the bilinear maximal commutator. We establish some new bounds for the weak gradient of the above commutators. We also obtain the boundedness and continuity of the above commutators on the Triebel–Lizorkin spaces and Besov spaces.

Lutwak and Zhang introduced $L_p$-centroid bodies and established the $L_p$-centro-affine inequality. Thereafter, Lutwak, Yang and Zhang established the $L_p$-Busemann–Petty centroid inequality. In this paper, we give the inequalities of dual quermassintegrals, dual affine quermassintegrals and dual harmonic quermassintegrals for polar $L_p$-centroid bodies.

We obtain the quantitative weighted strong-type and weak-type estimates for variation operators associated with heat semigroups in the Schrödinger setting. In particular, we first established the quantitative endpoint bound for such operators in the Schrödinger setting, which is the main novelty of our results.

Let $I\subset R=K[x_1,\dots ,x_n]$ be a monomial ideal, $\mathfrak{𝔪}=(x_1,\dots ,x_n)$, $t$ a positive integer, and $y_1,\dots ,y_s$ be distinct variables in $R$ such that, for each $i=1,\dots ,s$, $\mathfrak{𝔪}\setminus y_i\notin \mathrm{Ass}(R∕{(I\setminus y_i)}^t)$, where $I\setminus y_i$ denotes the deletion of $I$ at $y_i$. It is shown in Theorem 3.4 of the article in question that $\mathfrak{𝔪}\in \mathrm{Ass}(R∕I^t)$ if and only if $\mathfrak{𝔪}\in \mathrm{Ass}\left(R∕\left(I^t:{\prod }_{i=1}^s{y}_i\right)\right)$. As an application of Theorem 3.4, it is argued in Theorem 3.6 that under certain conditions, every unmixed König ideal is normally torsion-free. In addition, Theorem 3.7 states that under certain conditions a square-free monomial ideal is normally torsion-free. It turns out that these conditions are not enough to obtain the desired statements in Theorems 3.6 and 3.7. We update these conditions to validate the conclusions of Theorems 3.6 and 3.7. For this purpose, it is enough for us to replace the expression “$\mathfrak{𝔪}\setminus x_i\notin \mathrm{Ass}(R∕{(I\setminus x_i)}^t)$” with the new expression “$I\setminus x_i$ is normally torsion-free”. It should be noted that the previous proofs are still correct.