Abstract In this study, a multipoint boundary value problem for Volterra-Fredholm integro-differential equations is considered. The addition of a new function converts the system of Volterra-Fredholm integro-differential equations to a system of Fredholm integro-differential equations. In contrast to the original problem, the dimension of a Fredholm integro-differential equation is determined by the number of matrices in the degenerate kernel of the Volterra integral. A numerical algorithm of Dzhumabaev parameterization method for addressing a multipoint boundary value problem for Volterra-Fredholm integro-differential equations is proposed. The main advantage of the proposed method is splitting the problem into auxiliary Cauchy problems for ordinary differential equations and a system of algebraic equations with respect to the parameters. The conditions for the unique solvability of the multipoint boundary value problem for Fredholm integro-differential equations are established. Finally, various numerical examples are provided to demonstrate the efficiency and correctness of the suggested technique.

Abstract This article deals with the asymptotic behavior of fractional lattice systems with time-varying delays in weighted space. First, we establish some sufficient conditions for the existence and uniqueness of solutions. Subsequently, we demonstrate the existence of pullback attractors for the considered fractional lattice systems.

Abstract We investigate third-order strongly nonlinear differential equations of the type ( Φ ( k ( t ) u ″ ( t ) ) ) ′ = f ( t , u ( t ) , u ′ ( t ) , u ″ ( t ) ) , a.e. on [ 0 , T ] , \left(\Phi \left(k\left(t){u}^{^{\prime\prime} }\left(t)))^{\prime} =f\left(t,u\left(t),u^{\prime} \left(t),{u}^{^{\prime\prime} }\left(t)),\hspace{1em}\hspace{0.1em}\text{a.e. on}\hspace{0.1em}\hspace{0.33em}\left[0,T], where Φ \Phi is a strictly increasing homeomorphism, and the non-negative function k k may vanish on a set of measure zero. Using the upper and lower solution method, we prove existence results for some boundary value problems associated with the aforementioned equation. Moreover, we also consider second-order integro-differential equations like ( Φ ( k ( t ) v ′ ( t ) ) ) ′ = f t , ∫ 0 t v ( s ) d s , v ( t ) , v ′ ( t ) , a.e. on [ 0 , T ] , \left(\Phi \left(k\left(t)v^{\prime} \left(t)))^{\prime} =f\left(t,\underset{0}{\overset{t}{\int }}v\left(s){\rm{d}}s,v\left(t),v^{\prime} \left(t)\right),\hspace{1em}\hspace{0.1em}\text{a.e. on}\hspace{0.1em}\hspace{0.33em}\left[0,T], for which we provide existence results for various types of boundary conditions, including periodic, Sturm-Liouville, and Neumann-type conditions.

Abstract We find new upper bounds for the global cyclicity index, a variant of the Kirchhoff index, and discuss the wide family of graphs for which the bounds are attained.

Abstract For a given sequence a = ( a 1 , … , a n ) ∈ R n a=\left({a}_{1},\ldots ,{a}_{n})\in {{\mathbb{R}}}^{n} , our aim is to obtain an estimate of E n ≔ a 1 + a n 2 − 1 n ∑ i = 1 n a i {E}_{n}:= \left|\hspace{-0.33em},\frac{{a}_{1}+{a}_{n}}{2}-\frac{1}{n}{\sum }_{i=1}^{n}{a}_{i},\hspace{-0.33em}\right| . Several classes of sequences are studied. For each class, an estimate of E n {E}_{n} is obtained. We also introduce the class of convex matrices, which is a discrete version of the class of convex functions on the coordinates. For this set of matrices, new discrete Hermite-Hadamard-type inequalities are proved. Our obtained results are extensions of known results from the continuous case to the discrete case.

Abstract Let ℜ \Re be a commutative ring with unity and ℑ \Im be a left ℜ \Re -module. We define the secondary-like spectrum of ℑ \Im to be the set of all secondary submodules K K of ℑ \Im such that the annihilator of the socle of K K is the radical of the annihilator of K K , and we denote it by Spec L ( ℑ ) {{\rm{Spec}}}^{L}\left(\Im ) . In this study, we introduce a topology on Spec L ( ℑ ) {{\rm{Spec}}}^{L}\left(\Im ) having the Zariski topology on the second spectrum Spec s ( ℑ ) {{\rm{Spec}}}^{s}\left(\Im ) as a subspace topology and study several topological structures of this topology.

Abstract It is shown that the expression as A 2 = α A + β P {A}^{2}=\alpha A+\beta P for generalized quadratic matrices is not unique by numerical examples. Then it is proven that the uniqueness of expression for generalized quadratic matrices is concerned not only with the properties of A A but also with the rank of P P . Furthermore, the sufficient and necessary conditions for the uniqueness of generalized quadratic matrices’expression are obtained. Finally, some related discussions about generalized quadratic matrices are also given.

Abstract We establish weighted Hermite-Hadamard-type inequalities for some classes of differentiable functions without assuming any symmetry property on the weight function. Next, we apply our obtained results to the approximation of some classes of weighted integrals.

Abstract The atom-bond sum-connectivity (ABS) index of a graph G G with edges e 1 , … , e m {e}_{1},\ldots ,{e}_{m} is the sum of the numbers 1 − 2 ( d e i + 2 ) − 1 \sqrt{1-2{\left({d}_{{e}_{i}}+2)}^{-1}} over 1 ≤ i ≤ m 1\le i\le m , where d e i {d}_{{e}_{i}} is the number of edges adjacent to e i {e}_{i} . In this article, we study the maximum values of the ABS index over graphs with given parameters. More specifically, we determine the maximum ABS index of connected graphs of a given order with a fixed (i) minimum degree, (ii) maximum degree, (iii) chromatic number, (iv) independence number, or (v) number of pendent vertices. We also characterize the graphs attaining the maximum ABS values in all of these classes.

Abstract In this article, we prove the following result. Let n ≥ 3 n\ge 3 be some fixed integer and let R R be a prime ring with char ( R ) ≠ ( n + 1 ) ! 2 n − 2 {\rm{char}}\left(R)\ne \left(n+1)\!{2}^{n-2} . Suppose there exists an additive mapping D : R → R D:R\to R satisfying the relation 2 n − 2 D ( x n ) = ∑ i = 0 n − 2 n − 2 i x i D ( x 2 ) x n − 2 − i + ( 2 n − 2 − 1 ) ( D ( x ) x n − 1 + x n − 1 D ( x ) ) + ∑ i = 1 n − 2 ∑ k = 2 i ( 2 k − 1 − 1 ) n − k − 2 i − k + ∑ k = 2 n − 1 − i ( 2 k − 1 − 1 ) n − k − 2 n − i − k − 1 x i D ( x ) x n − 1 − i \begin{array}{rcl}{2}^{n-2}D\left({x}^{n})& =& \left(\mathop{\displaystyle \sum }\limits_{i=0}^{n-2}\left(\genfrac{}{}{0.0pt}{}{n-2}{i}\right){x}^{i}D\left({x}^{2}){x}^{n-2-i}\right)+\left({2}^{n-2}-1)\left(D\left(x){x}^{n-1}+{x}^{n-1}D\left(x))\\ & & +\mathop{\displaystyle \sum }\limits_{i=1}^{n-2}\left(\mathop{\displaystyle \sum }\limits_{k=2}^{i}\left({2}^{k-1}-1)\left(\genfrac{}{}{0.0pt}{}{n-k-2}{i-k}\right)+\mathop{\displaystyle \sum }\limits_{k=2}^{n-1-i}\left({2}^{k-1}-1)\left(\genfrac{}{}{0.0pt}{}{n-k-2}{n-i-k-1}\right)\right){x}^{i}D\left(x){x}^{n-1-i}\end{array} for all x ∈ R . x\in R. In this case, D D is a derivation. This result is related to a classical result of Herstein, which states that any Jordan derivation on a prime ring with char ( R ) ≠ 2 {\rm{char}}\left(R)\ne 2 is a derivation.