Let E be a uniformly convex and uniformly smooth real Banach space, and let E* be its dual. Let A : E → 2E* be a bounded maximal monotone map. Assume that A−1(0) ≠ Ø. A new iterative sequence is constructed which converges strongly to an element of A−1(0). The theorem proved complements results obtained on strong convergence of the proximal point algorithm for approximating an element of A−1(0) (assuming existence) and also resolves an important open question. Furthermore, this result is applied to convex optimization problems and to variational inequality problems. These results are achieved by combining a theorem of Reich on the strong convergence of the resolvent of maximal monotone mappings in a uniformly smooth real Banach space and new geometric properties of uniformly convex and uniformly smooth real Banach spaces introduced by Alber, with a technique of proof which is also of independent interest.

We consider relatively Meir–Keeler condensing operators to study the existence of best proximity points (pairs) by using the notion of measure of noncompactness, and extend a result of Aghajani et al. [‘Fixed point theorems for Meir–Keeler condensing operators via measure of noncompactness’, Acta Math. Sci. Ser. B35 (2015), 552–566]. As an application of our main result, we investigate the existence of an optimal solution for a system of integrodifferential equations.

We prove the existence of common fixed points for monotone contractive and monotone nonexpansive semigroups of nonlinear mappings acting in Banach spaces equipped with partial order. We also discuss some applications to differential equations and dynamical systems.

Extending recent results by Cascales et al. [‘Plasticity of the unit ball of a strictly convex Banach space’, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 110(2) (2016), 723–727], we demonstrate that for every Banach space $X$ and every collection $Z_{i},i\in I$ , of strictly convex Banach spaces, every nonexpansive bijection from the unit ball of $X$ to the unit ball of the sum of $Z_{i}$ by $\ell _{1}$ is an isometry.

Let $X$ and $Y$ be two normed spaces over fields $\mathbb{F}$ and $\mathbb{K}$, respectively. We prove new generalised hyperstability results for the general linear equation of the form $g(ax+by)=Ag(x)+Bg(y)$, where $g:X\rightarrow Y$ is a mapping and $a,b\in \mathbb{F}$, $A,B\in \mathbb{K}\backslash \{0\}$, using a modification of the method of Brzdęk [‘Stability of additivity and fixed point methods’, Fixed Point Theory Appl.2013 (2013), Art. ID 285, 9 pages]. The hyperstability results of Piszczek [‘Hyperstability of the general linear functional equation’, Bull. Korean Math. Soc.52 (2015), 1827–1838] can be derived from our main result.

We consider the split common null point problem in Hilbert space. We introduce and study a shrinking projection method for finding a solution using the resolvent of a maximal monotone operator and prove a strong convergence theorem for the algorithm.

We extend the results of Schu [‘Iterative construction of fixed points of asymptotically nonexpansive mappings’, J. Math. Anal. Appl.158 (1991), 407–413] to monotone asymptotically nonexpansive mappings by means of the Fibonacci–Mann iteration process

$$\begin{eqnarray}x_{n+1}=t_{n}T^{f(n)}(x_{n})+(1-t_{n})x_{n},\quad n\in \mathbb{N},\end{eqnarray}$$

$T$

$\{f(n)\}$

$L_{p}([0,1])$

$1<p<+\infty$

Through appropriate choices of elements in the underlying iterated function system, the methodology of fractal interpolation enables us to associate a family of continuous self-referential functions with a prescribed real-valued continuous function on a real compact interval. This procedure elicits what is referred to as an α-fractal operator on , the space of all real-valued continuous functions defined on a compact interval I. With an eye towards connecting fractal functions with other branches of mathematics, in this paper we continue to investigate the fractal operator in more general spaces such as the space of all bounded functions and the Lebesgue space , and in some standard spaces of smooth functions such as the space of k-times continuously differentiable functions, Hölder spaces and Sobolev spaces . Using properties of the α-fractal operator, the existence of Schauder bases consisting of self-referential functions for these function spaces is established.

Let $X$ be a nonempty subset of a normed space such that $0\notin X$ and $X$ is symmetric with respect to $0$ and let $Y$ be a Banach space. We study the generalised hyperstability of the Drygas functional equation

$$\begin{eqnarray}f(x+y)+f(x-y)=2f(x)+f(y)+f(-y),\end{eqnarray}$$

$f$

$X$

$Y$

$x,y\in X$

$x+y,x-y\in X$

The Krasnosel’skiĭ–Mann (KM) iteration is a widely used method to solve fixed point problems. This paper investigates the convergence rate for the KM iteration. We first establish a new convergence rate for the KM iteration which improves the known big-$O$ rate to little-$o$ without any other restrictions. The proof relies on the connection between the KM iteration and a useful technique on the convergence rate of summable sequences. Then we apply the result to give new results on convergence rates for the proximal point algorithm and the Douglas–Rachford method.

We investigate a system of singular–degenerate parabolic equations with non-local terms, which can be regarded as a spatially heterogeneous competition model of Lotka–Volterra type. Applying the Leray–Schauder fixed-point theorem, we establish the existence of coexistence periodic solutions to the problem, which, together with the existing literature, gives a complete picture for such a system for all parameters.

In this paper we present a simple (fixed point) method that yields various results concerning approximate solutions of some difference equations. The results are motivated by the notion of Ulam stability.

In this paper we prove Caristi’s fixed point theorem using only purely metric techniques.

We consider the boundary-value problem

where H: [0,+∞) → ℝ and f : [0, 1] × ℝ → ℝ are continuous and λ > 0 is a parameter. We show that if H satisfies a boundedness condition on a specified compact set, then this, together with an assumption that H is either affine or superlinear at +∞, implies existence of at least one positive solution to the problem, even in the case where we impose no growth conditions on f. Finally, since it can hold that f(t, y) < 0 for all (t, y) ∈ [0, 1]×ℝ, the semipositone problem is included as a special case of the existence result.

In this paper we first give a negative answer to a question of Amini-Harandi [‘Best proximity point theorems for cyclic strongly quasi-contraction mappings’, J. Global Optim.56 (2013), 1667–1674] on a best proximity point theorem for cyclic quasi-contraction maps. Then we prove some new results on best proximity point theorems that show that results of Amini-Harandi for cyclic strongly quasi-contractions are true under weaker assumptions.

In this paper, we generalize monotone operators, their resolvents and the proximal point algorithm to complete CAT(0) spaces. We study some properties of monotone operators and their resolvents. We show that the sequence generated by the inexact proximal point algorithm $\unicode[STIX]{x1D6E5}$-converges to a zero of the monotone operator in complete CAT(0) spaces. A strong convergence (convergence in metric) result is also presented. Finally, we consider two important special cases of monotone operators and we prove that they satisfy the range condition (see Section 4 for the definition), which guarantees the existence of the sequence generated by the proximal point algorithm.

We study the existence of a weak solution of a nonlocal problem

$$\begin{eqnarray}\displaystyle & \displaystyle -{\mathcal{L}}_{K}u-\unicode[STIX]{x1D707}ug_{1}+h(u)g_{2}=f\quad \text{in }\unicode[STIX]{x1D6FA}, & \displaystyle \nonumber\\ \displaystyle & \displaystyle u=0\quad \text{in }\mathbb{R}^{n}\setminus \unicode[STIX]{x1D6FA}, & \displaystyle \nonumber\end{eqnarray}$$

${\mathcal{L}}_{k}$

$\unicode[STIX]{x1D707}$

$\unicode[STIX]{x1D6FA}$

$\mathbb{R}^{n}$

$n>2s$

$s\in (0,1)$

$\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}$

$f,g_{1},g_{2}:\unicode[STIX]{x1D6FA}\rightarrow \mathbb{R}$

$h:\mathbb{R}\rightarrow \mathbb{R}$

The aim of this paper is to introduce a new measure of noncompactness on the Sobolev space $W^{n,p}[0,T]$. As an application, we investigate the existence of solutions for some classes of functional integro-differential equations in this space using Darbo’s fixed point theorem.

Metric regularity theory lies in the very heart of variational analysis, a relatively new discipline whose appearance was, to a large extent, determined by the needs of modern optimization theory in which such phenomena as nondifferentiability and set-valued mappings naturally appear. The roots of the theory go back to such fundamental results of the classical analysis as the implicit function theorem, Sard theorem and some others. The paper offers a survey of the state of the art of some principal parts of the theory along with a variety of its applications in analysis and optimization.

We study the periodic and Neumann boundary-value problems associated with the second-order nonlinear differential equation

where is a sublinear function at infinity having superlinear growth at zero. We prove the existence of two positive solutions when

and λ > 0 is sufficiently large. Our approach is based on Mawhin's coincidence degree theory and index computations.