Volume 25, Issue 1

In this paper, we develop two-step collocation (2-SC) methods to solve two-dimensional nonlinear Volterra integral equations (2D-NVIEs) of the second kind. Here we convert a 2D-NVIE of the second kind to a one-dimensional case, and then we solve the resulting equation numerically by two-step collocation methods. We also study the convergence and stability analysis of the method. At the end, the accuracy and efficiency of the method is verified by solving two test equations which are stiff. In examples, we use the well-known differential transform method to obtain starting values.

This paper studies a linear hyperbolic system with boundary conditions that was first studied under some weaker conditions in [8, 11]. Problems on the expansion of a semigroup and a criterion for being a Riesz basis are discussed in the present paper. It is shown that the associated linear system is the infinitesimal generator of a C0{C_{0}}-semigroup; its spectrum consists of zeros of a sine-type function, and its exponential system {eλn⁢t}n≥1{\{e^{\lambda_{n}t}\}_{n\geq 1}} constitutes a Riesz basis in L2⁢[0,T]{L^{2}[0,T]}. Furthermore, by the spectral analysis method, it is also shown that the linear system has a sequence of eigenvectors, which form a Riesz basis in Hilbert space, and hence the spectrum-determined growth condition is deduced.

A weak selection on ℝ{\mathbb{R}} is a function f:[ℝ]2→ℝ{f\colon[\mathbb{R}]^{2}\to\mathbb{R}} such that f⁢({x,y})∈{x,y}{f(\{x,y\})\in\{x,y\}} for each {x,y}∈[ℝ]2{\{x,y\}\in[\mathbb{R}]^{2}}. In this article, we continue with the study (which was initiated in [1]) of the outer measures λf{\lambda_{f}} on the real line ℝ{\mathbb{R}} defined by weak selections f . One of the main results is to show that CH is equivalent to the existence of a weak selection f for which λf⁢(A)=0{\lambda_{f}(A)=0} whenever |A|≤ω{\lvert A\rvert\leq\omega} and λf⁢(A)=∞{\lambda_{f}(A)=\infty} otherwise. Some conditions are given for a σ-ideal of ℝ{\mathbb{R}} in order to be exactly the family 𝒩f{\mathcal{N}_{f}} of λf{\lambda_{f}}-null subsets for some weak selection f . It is shown that there are 2𝔠{2^{\mathfrak{c}}} pairwise distinct ideals on ℝ{\mathbb{R}} of the form 𝒩f{\mathcal{N}_{f}}, where f is a weak selection. Also, we prove that the Martin axiom implies the existence of a weak selection f such that 𝒩f{\mathcal{N}_{f}} is exactly the σ-ideal of meager subsets of ℝ{\mathbb{R}}. Finally, we shall study pairs of weak selections which are “almost equal” but they have different families of λf{\lambda_{f}}-measurable sets.

In the paper, using the generalized Marichev–Saigo–Maeda fractional operator, the authors establish some fractional differential equations associated with generalized hypergeometric functions and, by employing integral transforms, present some image formulas of the resulting equations.

A new method is given for proving the global existence of the solution to nonlinear Volterra integral equations. A bound on the solution is derived. The results are based on a nonlinear inequality proved by the author earlier.

Some developments of the second-order characterizations of convex functions are investigated by using the coderivative of the subdifferential mapping. Furthermore, some applications of the second-order subdifferentials in optimization problems are studied.

In the present paper, a new class of generalized beta (r,g){(r,g)}-preinvex functions is introduced and some new integral inequalities for the left-hand side of Gauss–Jacobi type quadrature formula involving generalized beta (r,g){(r,g)}-preinvex functions are given. Moreover, some generalizations of Hermite–Hadamard type inequalities for generalized beta (r,g){(r,g)}-preinvex functions that are twice differentiable via k -fractional integrals are established. These general inequalities give us some new estimates for Hermite–Hadamard type k -fractional integral inequalities and also extend some results appeared in the literature; see [A. Kashuri and R. Liko, Ostrowski type fractional integral inequalities for generalized (s,m,φ)(s,m,\varphi)-preinvex functions, Aust. J. Math. Anal. Appl. 13 2016, 1, Article ID 16]. At the end, some applications to special means are given.

This paper is motivated by the recent progress on the Hermite–Hadamard inequality for convex functions defined on the bounded closed interval, obtained by Z. Pavić [Z. Pavić, Improvements of the Hermite–Hadamard inequality, J. Inequal. Appl. 2015 2015, Article ID 222]. As a generalization, we obtained a new refinement of the Hermite–Hadamard inequality for co-ordinated convex functions defined on the rectangle.

In this paper, we establish fractional Ostrowski’s inequalities for functions whose certain power of modulus of the first derivatives are pre-quasi-invex via power mean inequality.

Let K be a closed convex cone in a real Banach space, H:K→cc⁡(K){H\colon K\to\operatorname{cc}(K)} a continuous sublinear correspondence with nonempty, convex and compact values in K , and let f:ℝ→ℝ{f\colon\mathbb{R}\to\mathbb{R}} be defined by f⁢(t)=∑n=0∞an⁢tn{f(t)=\sum_{n=0}^{\infty}a_{n}t^{n}}, where t∈ℝ{t\in\mathbb{R}}, an≥0{a_{n}\geq 0}, n∈ℕ{n\in\mathbb{N}}. We show that the correspondence Ft(x):=∑n=0∞antnHn(x),(x∈K){F^{t}(x)\mathrel{\mathop{:}}=\sum_{n=0}^{\infty}a_{n}t^{n}H^{n}(x),(x\in K)} is continuous and sublinear for every t≥0{t\geq 0} and Ft∘Fs⁢(x)⊆∑n=0∞cn⁢Hn⁢(x){F^{t}\circ F^{s}(x)\subseteq\sum_{n=0}^{\infty}c_{n}H^{n}(x)}, x∈K{x\in K}, where cn=∑k=0nak⁢an-k⁢tk⁢sn-k{c_{n}=\sum_{k=0}^{n}a_{k}a_{n-k}t^{k}s^{n-k}}, t,s≥0{t,s\geq 0}.

In this paper, we consider a nonlinear quasilinear system of two coupled viscoelastic equations and investigate the asymptotic behavior of this system. We establish an explicit and general formula for the energy decay rates. Our result allows a wider class of relaxation functions, which improves earlier results existing in the literature.

In this article, we study a coupled Allen–Cahn–Navier–Stokes model in a two-dimensional domain. The model consists of the Navier–Stokes equations for the velocity, coupled with an Allen–Cahn model for the order (phase) parameter. We present an equivalent weak formulation for the model, and we prove a new regularity result for the weak solutions.