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We consider the convex combinations cnα:=(1−α)an+αbn,n∈ℕ,α∈[0,1]$$ {c}_n^{\alpha}:= \left(1-\alpha \right){a}_n+\alpha {b}_n,n\in \mathrm{\mathbb{N}},\alpha \in \left[0,1\right] $$, of a pair of sequences of real numbers (an)n∈ℕ$$ {\left({a}_n\right)}_{n\in \mathrm{\mathbb{N}}} $$ and (bn)n∈ℕ$$ {\left({b}_n\right)}_{n\in \mathrm{\mathbb{N}}} $$ such that an≤bn,n∈ℕ$$ {a}_n\le {b}_n,n\in \mathrm{\mathbb{N}} $$, converging to ln2$$ \ln 2 $$, and study the location of the limit inside the intervals [an,bn]$$ \left[{a}_n,{b}_n\right] $$, for every n∈ℕ$$ n\in \mathrm{\mathbb{N}} $$ or for sufficiently large n$$ n $$. We also investigate the same problem for the case of two corresponding sequences converging to ln3$$ \ln 3 $$. Among other results, we prove some, a bit, unexpected ones. Namely, for each α∈[0,1]$$ \alpha \in \left[0,1\right] $$, we determine the exact index n0∈ℕ$$ {n}_0\in \mathrm{\mathbb{N}} $$ at which the sequence cnα$$ {c}_n^{\alpha } $$ changes the monotonicity, and we also determine the type of the monotonicity. A number of interesting remarks are also presented. [ABSTRACT FROM AUTHOR]

This manuscript is concerned about the study of the existence of solutions for the class of nonlinear neutral Caputo-Hadamard fractional differential equations including integral terms. In order to establish the necessary conditions of solvability for the proposed problem, we apply the semi-group property of Hadamard fractional integral operator. Also, under the appropriate conditions, we demonstrate that the solution set for the proposed problem is non-empty by using Arzelá-Ascoli theorem and the method of upper and lower solutions. In contrast to the fundamental results graphical example is also presented in order to validate the findings. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a new class of nonlinear fractional integrodifferential systems that includes the Ψ -Riemann–Liouville fractional integral term. Using the technique of upper and lower solutions, the solvability of the system is examined. We add two examples to demonstrate and validate the main result. The main results highlight crucial contributions to the general theory of fractional differential equations. [ABSTRACT FROM AUTHOR]

In this paper we consider double cosine series whose coefficients form a null sequence of bounded variation of order (p, 0), (0, p) and (p, p) with the weight (jk)p-1 for some p > 1. We study pointwise convergence, uniform convergence and convergence in Lr-norm of the series under consideration. In a certain sense our results extend the results of Young [7], Kolmogorov [3] and Móricz [4, 5]. [ABSTRACT FROM AUTHOR]

In this present work we concern with mathematical modelling of biological experiments. The fractional hybrid iterative differential equations are suitable for mathematical modelling of biology and also interesting equations since the structure are rich with particular properties. The solution technique is based on the Dhage fixed point theorem that describes the mixed solutions by monotone iterative technique in the nonlinear analysis. In this method we combine two solutions, namely, lower and upper solutions. It is shown an approximate result for the hybrid fractional differential equations in the closed assembly formed by the lower and upper solutions. [ABSTRACT FROM AUTHOR]

Given a set of points P⊆ℝ, a conflict-free coloring of P w.r.t. rectangle ranges is an assignment of colors to points of P, such that each nonempty axis-parallel rectangle T in the plane contains a point whose color is distinct from all other points in P∩ T. This notion has been the subject of recent interest and is motivated by frequency assignment in wireless cellular networks: one naturally would like to minimize the number of frequencies (colors) assigned to base stations (points) such that within any range (for instance, rectangle), there is no interference. We show that any set of n points in ℝ can be conflict-free colored with $O(n^{\beta^{*}+o(1)})$ colors in expected polynomial time, where $\beta^{*}=\frac{3-\sqrt{5}}{2} < 0.382$. [ABSTRACT FROM AUTHOR]

Abstract: We consider a nonlocal first order partial differential equation with time delay that models simultaneous cell replication and maturation processes. We establish a comparison principle and construct monotone sequences to show the existence and uniqueness of the solution to the equation. We then analyze the asymptotic behavior of the solution via upper–lower solution technique. [Copyright &y& Elsevier]

The problems of existence of solutions for Volterra and Fredholm integral equations are investigated in this paper. Using the monotone method, given are sufficient conditions when integral equations have their solutions in a corresponding sector. Some delay differential and integral inequalities are also discussed. [ABSTRACT FROM AUTHOR]

This paper concerns differential equations with boundary conditions. Given are sufficient conditions under which such problems with deviated arguments have a unique solution in a corresponding sector. To obtain existence results we apply a monotone iterative method. [Copyright &y& Elsevier]

Abstract: We investigate the existence of solutions for advanced differential equations with nonlinear boundary conditions. Sufficient conditions when the problem has extremal solutions or a unique solution are formulated. Linear advanced differential inequalities are also discussed. [Copyright &y& Elsevier]

Abstract: The quasilinearization method is used for nonlinear ordinary differential equations with nonlinear boundary conditions. Given are sufficient conditions when corresponding monotone sequences converge to the unique solution and this convergence is quadratic. [Copyright &y& Elsevier]

In this paper, we investigate the existence of extremal solutions of boundary value problems for second-order differential equations with delayed arguments. We show that a lower and upper solution method has been applied successfully to such problems. An one-sided Lipschitz condition (with corresponding constants) is important assumption in our research. [Copyright &y& Elsevier]

We study the existence and approximation of solution of boundary value problems for differential equations with delayed arguments. Sufficient conditions are established for the existence of a unique solution or extremal ones of the given problem. A monotone iterative technique is applied. [Copyright &y& Elsevier]

The method of quasilinearization is a procedure for obtaining approximate solutions of differential equations. In this paper, this technique is applied to a differential-algebraic problem. Under some natural assumptions, monotone sequences converge quadratically to a unique solution of our problem. [ABSTRACT FROM AUTHOR]

A numerical method for the calculation of equilibrium distributions of chemical species in aqueous solutions of electrolytes is presented. This method is constructed by transforming the problem of determining the set of unknown concentrations satisfying the mass balance and mass action equations into the equivalent problem of finding the limits of certain well-behaved mathematical sequences in a multivariable direct iteration scheme. The total (analytical) concentrations are taken as the starting estimates, and the recursive equations are constructed from the starting equations. It is shown that the sequences so constructed are both monotonic and bounded, hence convergent in an orderly fashion by a mathematical axiom. Each unknown free ion concentration (that is, each limit) is approached simultaneously from above and from below, being effectively 'sandwiched' in an ever tightening manner. Strict error bounds therefore are easily constructed. The method has been found to be exceedingly efficient in practice, with the error bound (maximum fractional error in taking the upper estimate as the final answer) decreasing in an approximately exponential fashion with respect to iteration number, commonly 0.5-1.5 orders of magnitude per iteration. Correction for nonideality is presented, and the possibility of this giving rise to more than one solution is discussed in connection with three examples of natural waters of low-, medium-, and high-ionic strengths. [ABSTRACT FROM AUTHOR]