On Uniqueness of Fixed Points and Their Regularity

Abstract

In this paper, we study the problem of uniqueness of fixed points for operators acting from a Banach space X into a subspace Y with a stronger norm. Our main objective is to preserve the expected regularity of fixed points, as determined by the norm of Y, while analyzing their uniqueness without imposing the classical or generalized contraction condition on Y. The results presented here provide generalized uniqueness theorems that extend existing fixed-point theorems to a broader class of operators and function spaces. The results are used to study fractional initial value problems in generalized Hölder spaces.

1. Introduction

The study of fixed points of mappings constitutes a major branch of mathematics, with numerous applications to the solution of various types of equations (differential, integral, partial differential, operator) and optimization. In this paper, we focus on the uniqueness of fixed points in the case where their additional regularity is allowed. One motivation for studying this case is, among others, the integral operators used in the study of differential and integral equations, which in general have certain properties that improve the class of functions on which they operate. For an operator $T:X\to X$, by regularity, we mean that the fixed points belong to a subspace Y of X, which is generally equipped with a stronger norm and not induced from X. A prototype of such a situation is the case in which an operator is defined on $C[a,b]$ but its regularity improves when considered on $C^1[a,b]$ or Hölder spaces. The growing interest in the study of the uniqueness of fixed points is in turn motivated by the need for constructive methods that are iterative or numerical. Meanwhile, usually theorems guaranteeing the uniqueness, mainly provided by contractions or generalized contractions, lead either to a contraction in X and one loses the regularity, or a contraction in Y, which is a much stronger and more difficult condition to provide. It is worth noting that in certain function spaces—for example, $C^1[a,b]$ or Hölder spaces—the classical Banach fixed point theorem may fail to apply because of degeneracy phenomena (see [1,2]). However, if the operator acts continuously from the space Y into itself, it still satisfies the Darbo condition locally, meaning that the associated compactness assumption holds. In contrast, in $C[a,b]$ (cf. [1]), assuming that the operator is a contraction that remains natural, the additional conditions needed in these smoother spaces reduce to relatively mild compactness assumptions, which are simple to verify in practice, as noted in [1,3]. The results of this paper address the problem of preserving the regularity of fixed points without requiring that all conditions are imposed on the mappings in the space Y. To this end, we separate the necessary assumptions into two groups: those ensuring the construction of the iterative sequence in X and those ensuring that the limit of this sequence belongs to Y. Naturally, a central goal is to establish assumptions that are strictly weaker than those used in previous results.

The paper is organized as follows. In Section 2, we collect notation, definitions and auxiliary results which will be utilized in the remaining part of the paper. Section 3 reviews the key theorems on the uniqueness of fixed points in normed spaces and then presents some generalizations. In Section 4, we establish the conditions (UN), (BA) and (FPT) under which the uniqueness of fixed points and the convergence of iterates in the stronger norm ${\parallel ·\parallel }_Y$ can be ensured, summarize classical assumptions that guarantee conditions (UN) and (BA), and examine a series of assumption under which property (FPT) holds true in order to have the applicability of the main result. Finally, in Section 5, we illustrate our results considering fractional initial value problems involving Caputo derivatives and extending the result to the case of operators with a tempered continuity modulus.

2. Preliminaries

We denote by $(X,\parallel ·{\parallel }_X)$ a Banach space and by Y a linear subspace of X endowed with a stronger norm. We will study the special case of subspaces Y with the norm ${\parallel ·\parallel }_Y={\parallel ·\parallel }_X+S(·)$, where S is a seminorm on Y, with $S\left(y\right)<+\infty$ whenever $y\in Y$, and operators $T:X\to X$ for which the iterates $T^n\left(x_0\right)$ lie definitively in Y starting for any initial point $x_0\in X$. We will look for conditions mainly on the seminorm S, relatively to the operator $T:X\to X$, in such a way that T admits a unique fixed point $y\in Y$ which is the limit of iterates in the ${\parallel ·\parallel }_Y$-norm starting from any point in X.

We denote by $C[a,b]$ the space of continuous function on the interval $[a,b]$ with the usual supremum norm ${\parallel x\parallel }_{\infty }$, and by $C^1[a,b]$ the space of continuously differentiable functions on $[a,b]$ with the norm ${\parallel x\parallel }_{C^1}={\parallel x\parallel }_{\infty }+{\parallel x^{\prime }\parallel }_{\infty }$. Moreover, given $\gamma \in (0,1]$ we denote by ${\mathbb{H}}^{\gamma }[a,b]$ the Hölder space endowed with the norm ${\parallel x\parallel }_{\gamma }={\parallel x\parallel }_{\infty }+{\left[x\right]}_{\gamma }$, where ${\left[x\right]}_{\gamma }={sup}_{t\ne s}{\displaystyle \frac{|x(t)-x(s)|}{{|t-s|}^{\gamma }}}$$t,s\in [a,b]$. For $\gamma \in (0,1)$, ${\mathbb{H}}_0^{\gamma }[a,b]=\{x\in {\mathbb{H}}^{\gamma }[a,b]:x\left(a\right)=0\}$ (see [4]).

When choosing a certain class of operators $T:X\to X$, we often have a situation where the values of the operator are more regular than the elements of the domain, that is $T\left(x\right)\in Y$, and it is the preservation of these properties of regularity when studying the uniqueness of fixed points that is the aim of this paper. Of course, one can always assume conditions leading to the uniqueness of a fixed point immediately in Y, but due to the stronger norm, this is a rather restrictive method. A few simple examples where our approach is useful: for the Volterra operator T, we can consider, for example, $X=C[a,b]$ and Y the space $AC[a,b]$ of absolutely continuous functions on $[a,b]$ or $Y=C^1[a,b]$; for the Riemann–Liouville fractional operator $T=I^{\alpha }$, it could be Hölder spaces $X={\mathbb{H}}_0^{\alpha }[a,b]$ and $Y={\mathbb{H}}_0^{\zeta +\alpha }[a,b]$ provided $\alpha +\zeta <1$ (as $I^{\alpha }:{\mathcal{H}}_0^{\zeta }[a,b]\to {\mathcal{H}}_0^{\zeta +\alpha }[a,b]$) (see, for example, [5,6,7]). Now, we collect notations and auxiliary results which will be utilized in the sequel. We recall that a mapping $T:X\to Y$ is called Lipschitz if there exists a number $k\ge 0$ such that

$${\mathrm{for}\mathrm{all}x,y\in X,\parallel T\left(x\right)-T\left(y\right)\parallel }_Y\le k·{\parallel x-y\parallel }_X.$$

The constant k is called a Lipschitz constant for T, and the mapping T is said to be k-Lipschitz. If $k=1$ the mapping T is called nonexpansive. A k-Lipschitz mapping $T:X\to X$ with $0\le k<1$ is called a contraction. A mapping $T:X\to X$ satisfying the relation

$$\mathrm{for}\mathrm{all}x,y\in X,x\ne {y,\parallel T\left(x\right)-T\left(y\right)\parallel }_X<{\parallel x-y\parallel }_X,$$

is called condensing.

Definition 1.

A mapping ${T:(X,\parallel ·\parallel }_X{)\to (Y,\parallel ·\parallel }_Y)$, ${\parallel ·\parallel }_Y={\parallel ·\parallel }_X+S(·)$ is called S-0-continuous along iterates if for any sequence of iterates, $x_{n+1}=T\left(x_n\right)$, the following holds: $S(T\left(x_n\right)-y)\to 0$ as $n\to +\infty$ with $y\in Y$ implies $\parallel T\left(x_n\right)-\tilde{y}{\parallel }_X\to 0$, for some $\tilde{y}$ with $S(y-\tilde{y})=0$.

Of course, this is a much weaker assumption than the continuity ${T:(X,\parallel ·\parallel }_X{)\to (Y,\parallel ·\parallel }_Y)$. Indeed, the usual continuity of $T:X\to Y$ would require that for any sequence $x_n\to x$ in X, we have $T\left(x_n\right)\to T\left(x\right)$ in Y. Here, however, the convergence is considered only along iterates of T. Furthermore, we control the convergence of the seminorm part of the norm in Y to some point $\tilde{y}$, “S-close” to y.

Example 1.

Let $X:=c_{00}$ be the space of all real sequences with finite support and ${\parallel x\parallel }_X:={sup}_{k\ge 1}|x_k|$ the ${\ell }_{\infty }$-norm on $c_{00}$. Let $Y=X$, $S\left(x\right):={\sum }_{k=1}^{\infty }k|x_k|$ (finite on $c_{00}$; this is a seminorm) and ${\parallel x\parallel }_Y:={\parallel x\parallel }_X+S\left(x\right)$.

Note that the identity map ${\mathrm{Id}:(X,\parallel ·\parallel }_X{)\to (Y,\parallel ·\parallel }_Y)$ is not continuous (it would require ${\parallel x\parallel }_Y\le C{\parallel x\parallel }_X$ for all x, which fails on $e^{\left(N\right)}$). Take $T=\mathrm{Id}$. Then, the iterates are constant: for any initial point $x_0$,

$$x_{n+1}=T\left(x_n\right)=x_n\mathrm{i}mplies{x}_n\equiv x_0,T\left(x_n\right)\equiv x_0.$$

Check the S-0-continuity along iterates. Fix any $y\in Y$. Suppose that

$$S(T\left(x_n\right)-y)=S(x_0-y)\to 0.$$

Then, $S(x_0-y)=0$. However, here, $S\left(z\right)=0$ implies that $z=0$ (since all the weights are positive and S is a norm). Therefore, $y=x_0$. Choose $\tilde{y}:=x_0$ (which satisfies $S(y-\tilde{y})=0$). Then,

$$\parallel T\left(x_n\right)-\tilde{y}{\parallel }_X={\parallel x_0-x_0\parallel }_X=0.$$

Thus, T is S-0-continuous along iterates. We see that $T=\mathrm{Id}$ is not continuous as a map $X\to Y$ but is S-0-continuous along iterates. Therefore, the S-0-continuous along iterates condition is strictly (essentially) weaker than the usual continuity from X to Y: it only constrains very specific sequences (the iterates) and only via the S-component up to the S-null equivalence.

The S-0-continuous along iterates of T is a necessary condition for the existence of a unique fixed point of T.

Lemma 1.

If the operator $T:X\to X$ has a unique fixed point $x\in Y$, and for any initial point $x_0\in X$, the sequence of iterates $x_{n+1}=T\left(x_n\right)$ converges to x in the norm ${\parallel ·\parallel }_Y$, then T is S-0-continuous along iterates.

Proof. 

Assume that $T:X\to X$ has a unique fixed point $x\in Y$ and that for any initial point $x_0\in X$, the sequence of iterates $x_{n+1}=T\left(x_n\right)$ converges to x in the norm ${\parallel ·\parallel }_Y$, i.e.,

$$\parallel x_n{-x\parallel }_Y\to 0\mathrm{as}n\to \infty .$$

Since ${\parallel ·\parallel }_Y={\parallel ·\parallel }_X+S(·)$, we have

$$\parallel x_n{-x\parallel }_X\to 0\mathrm{and}S(x_n-x)\to 0\mathrm{as}n\to \infty .$$

Now, suppose that for some $y\in Y$,

$$S(T\left(x_n\right)-y)\to 0.$$

However, since $T\left(x_n\right)=x_{n+1}$, by the triangle inequality for the seminorm S,

$$S(y-x)\le S(y-x_{n+1})+S(x_{n+1}-x)\to 0.$$

Thus, $S(y-x)=0$. Now, taking $\tilde{y}=x$ in the definition of S-0-continuity, the assertion follows. □

Definition 2.

A mapping ${T:(X,\parallel ·\parallel }_X{)\to (Y,\parallel ·\parallel }_Y)$, ${\parallel ·\parallel }_Y={\parallel ·\parallel }_X+S(·)$ is said to be S-asymptotically continuous along iterates if for any $x_0\in X$ and the sequence of iterates $x_{n+1}=T\left(x_n\right)$, we have the following: $\mathit{for} \mathit{all} \epsilon >0, \mathit{there} \mathit{exists} N\in \mathbb{N} \mathit{such} \mathit{that} \mathit{for} \mathit{all} m,n\ge N$

$$\parallel x_n-x_m{\parallel }_X<\epsilon \mathit{implies}S(T\left(x_n\right)-T\left(x_m\right))<\epsilon .$$

The latter is also a necessary condition for the existence of a unique fixed point of T when it is reached as the limit of a sequence of iterations converging in Y. This S-asymptotic continuity arises from the idea of checking only the iterates of T instead of checking all sequences (cf. [8]).

The following lemma provides some conditions that guarantee S-0-continuity.

Lemma 2.

A mapping $T:X\to Y$ is S-0-continuous along iterates if one of the following conditions holds:

1.

For every $y\in Y$, there exists $C\left(y\right)>0$ and $\delta \left(y\right)>0$ such that

$${\parallel T\left(x\right)-y\parallel }_X\le C\left(y\right)·S(T\left(x\right)-y)\mathit{for}\mathit{all}x\mathit{with}S(y-x)<\delta \left(y\right).$$

2.

There exists a function $\omega :[0,+\infty )\to [0,+\infty )$ with $\omega \left(0\right)=0$ such that

$$\parallel T^n{\left(x\right)-y\parallel }_X\le \omega \left(S(T^n\left(x\right)-y)\right)$$

for all $x,y\in X$ and $n\in \mathbb{N}$.

3.

We assume that $S(T^n\left(x\right)-T^n\left(y\right))\to 0$ for all $x,y\in X$, and there exists $C>0$ such that

$$\parallel T^n\left(x\right)-T^n{\left(y\right)\parallel }_X\le C·S(T^n\left(x\right)-T^n\left(y\right))\mathit{for}\mathit{all}x,y\in X,n\in \mathbb{N}.$$

Proof. 

The above conditions are quite classical. Only the first one needs to be justified. It is a generalized Fejér monotone condition for T with respect to ${\parallel ·\parallel }_X$ and S (i.e., generalized quasi-nonexpansive mappings). Let $\left(x_n\right)$ be a sequence of iterates, and suppose $S(T\left(x_n\right)-y)\to 0$. Since $x_{n+1}=T\left(x_n\right)$, we have $S(x_{n+1}-y)\to 0$. This implies that there exists $N\in \mathbb{N}$ such that for $n>N$, $S(y-x_n)<\delta \left(y\right)$. Thus, for $n>N$,

$$\parallel T\left(x_n\right){-y\parallel }_X={\parallel x_{n+1}-y\parallel }_X\le C\left(y\right)·S(T\left(x_n\right)-y)=C\left(y\right)·S(x_{n+1}-y).$$

Since $S(x_{n+1}-y)\to 0$, it implies that $\parallel x_{n+1}{-y\parallel }_X\to 0$. Clearly, $\tilde{y}=y$ in this case. □

3. Known Results and Some Generalization

In this section, we will first review some important theorems on the uniqueness of fixed points in normed spaces. Using some examples, we will show how they can be generalized in the case of interest, and in the next section, we will show how abstractly formulated assumptions can be checked in concrete spaces and how they are natural in many applications. The existence of fixed points can be deduced from a number of well-known results. Focusing on the problem of the uniqueness of the solution, the Banach fixed point theorem [9] (or: Banach-Caccioppoli, [10]) proves to be a very powerful tool.

Theorem 1

(Banach contraction theorem). Any contraction mapping $T:X\to X$ on a Banach space X has a unique fixed point. Moreover, for an arbitrary point $x_0\in X,$ the sequence of iterations $\left(x_n\right)$ defined by the recurrence relation

$$x_{n+1}=T\left(x_n\right),n\in \mathbb{N},$$

converges to the fixed point of the mapping T.

However, contractions are not the only source to have uniqueness of fixed points. It is known that we cannot simply replace a contraction condition by a condensing (or nonexpansive) condition and still keep the existence of fixed points. In such cases, to guarantee the uniqueness of a fixed point, we need additional conditions. These might be some compactness assumptions (on the space or by applying measures of noncompactness on operators), some generalized condensing conditions, or strict convexity or quasi-contraction properties (usually imposed on X). We will consider some of such results, with particular emphasis on those relevant to normed spaces, and in particular to the spaces Y under consideration here. A prototype theorem for this type of mappings is the following Edelstein theorem.

Theorem 2

(Edelstein [11,12]). Let $(X,\parallel ·{\parallel }_X)$ be a normed space and $T:X\to X$ a condensing mapping. If there exists $x_0\in X$ such that the sequence of iterates $\left(T^n\left(x_0\right)\right)$ has a limit point $x\in X,$ then x is the unique fixed point of T.

This theorem suggests the direction of our approach. We need some conditions allowing us to find a limit point for the sequence of iterates, and it should be in the space Y. We also recall the following corollary of Edelstein theorem for condensing mappings.

Corollary 1

([11]). If C is a compact subset of a normed space $(X,\parallel ·{\parallel }_X)$, then every condensing mapping $T:X\to C$ has a unique fixed point in X. Moreover, for any $x_0\in X$, the sequence of iterations defined by $\left(T^n\left(x_0\right)\right)$ converges to the fixed point of the mapping T.

This result indicates the role of compactness also when testing the uniqueness of fixed points for condensing mappings. For this purpose, we will use the target space Y and the operator T and some of its properties.

Proposition 1.

Let $(Y,\parallel ·{\parallel }_Y)$ be a subspace of $(X,\parallel ·{\parallel }_X)$ equipped with the norm ${\parallel x\parallel }_Y={\parallel x\parallel }_X+S\left(x\right)$, and let $C\subset Y$ be closed in $(X,\parallel ·{\parallel }_X)$. Let $T:X\to X$ be a contraction operator such that $T\left(C\right)\subseteq C$. Assume that for the sequence of iterations defined by $x_{n+1}=T\left(x_n\right)$, $n\in \mathbb{N}$ and any $y\in Y$, we have $S(T\left(x_n\right)-T\left(y\right))\to 0\mathit{whenever}{\parallel x_n-y\parallel }_X\to 0$ as $n\to +\infty$. Then, there exists a unique fixed point $x\in C$ of T, and for any $x_0\in X$, the sequence of iterations converges in ${\parallel ·\parallel }_Y$ to the fixed point of the mapping T.

Proof. 

First, it is obvious that the operator T satisfies all the assumptions of the Banach contraction theorem but when considered in the Banach space X. This is due to the contraction condition with respect to ${\parallel ·\parallel }_X$ of $T:X\to X$. Therefore, T has a unique fixed point in X and for an arbitrary point $x_0\in X,$ the sequence of iterations $\left(x_n\right)$ converges to the fixed point x of the mapping T. The next step is to show that this fixed point is indeed an element of Y. Consider an arbitrary sequence of iterations $\left(x_n\right)$ starting from a point $x_0$ in C so that it is a Cauchy sequence in ${\parallel ·\parallel }_X$. By our assumptions, $x_{n+1}=T\left(x_n\right)\in C$, so since C is closed in ${\parallel ·\parallel }_X$, we get $x\in C$.

Now, we want to prove that for any initial point $x_0\in X$, the sequence of iterates of T converges in ${\parallel ·\parallel }_Y$ to the fixed point. The sequence $\left(x_n\right)$ is convergent to x in X, so

$$\parallel x_n{-x\parallel }_X\to 0\mathrm{as}n\to +\infty .$$

By our assumptions, $S(T\left(x_n\right)-T\left(x\right))=S(x_{n+1}-T\left(x\right))\to 0$ as $n\to +\infty$. Thus, $S(x-T(x\left)\right)=0$. Finally,

$$\parallel x_n{-x\parallel }_Y={\parallel x_n-x\parallel }_X+S(x_n-x)\to 0\mathrm{as}n\to +\infty .$$

Hence, the sequence of iterates converges in ${\parallel ·\parallel }_Y$ to the fixed point $x\in C$ of the mapping T. □

The following simple case will help to illustrate a line of research based on the properties of the space Y while preserving the uniqueness shown in X. This will gradually reconcile the ease of checking fixed points in the weaker norm of the space X with the study of regularity in Y using the seminorm S to determine which elements belong to Y. A set $C\subset Y$ is called S-complete if every S-Cauchy sequence is convergent.

Proposition 2.

Let $(Y,\parallel ·{\parallel }_Y)$ be a subspace of a Banach space $(X,\parallel ·{\parallel }_X)$ equipped with the norm ${\parallel x\parallel }_Y={\parallel x\parallel }_X+S\left(x\right)$, where S is a given seminorm on X, and let $C\subset Y$ be a closed subset of $(X,\parallel ·{\parallel }_X)$. Let $T:X\to Y$ be a contraction operator such that $T\left(C\right)\subseteq C$. Moreover, suppose that T is S-asymptotically continuous along iterates and S-0-continuous along iterates and C is S-complete. Then, T has a unique fixed point in Y. Moreover, for any $x_0\in X$, the sequence of iterates of T converges to the fixed point of the mapping T in ${\parallel ·\parallel }_Y$.

Proof. 

As in the first part of the proof of Proposition 1, we have that T has a unique fixed point $x\in C$ and $\parallel x_n{-x\parallel }_X\to 0$ as $n\to +\infty$, for any starting point $x_0\in X$, with $x_{n+1}=T\left(x_n\right)$, $n\in \mathbb{N}$.

The sequence $\left(x_n\right)$ is ${\parallel ·\parallel }_X$-Cauchy, by the S-asymptotic continuity of T, we obtain that $\left(T\left(x_n\right)\right)$ is S-Cauchy. Since C is S-complete, the entire sequence $\left(T\left(x_n\right)\right)$ is S-convergent to some $y\in C$,

$$S(x_n-y)\to 0asn\to +\infty .$$

(1)

On the one hand, from (1) using the S-0-continuity along iterates, we have

$$\parallel x_n-\tilde{y}{\parallel }_X\to 0asn\to +\infty ,$$

with $S(y-\tilde{y})=0$. But $\parallel x_n{-x\parallel }_X\to 0$ as $n\to +\infty$, hence $\tilde{y}=x$, so we find $S(x-y)=0$. Now, from $S(x_n-x)\le S(x_n-y)+S(x-y)$, we obtain $S(x_n-x)\to 0$ as $n\to +\infty$. Therefore, $\parallel x_n{-x\parallel }_Y={\parallel x_n-x\parallel }_X+S(x_n-x)\to 0$ as $n\to +\infty$, which completes the proof. □

It is important to have in mind that the operator T need not be a contraction in the sense of the norm ${\parallel ·\parallel }_Y$. Recall that if this were the case, the Banach fixed point theorem would be directly applicable in Y. But to exclude this case definitively, it is enough to recall that there are situations in which the Banach contraction theorem, in certain spaces, can be applied to some problem involving superposition operators only if this problem is linear. We refer, in particular, to the classical situation where the space Y is the Hölder–Lipschitz space [4] embedded in $C[a,b]$. For example, if the operator T under consideration is a composition with a nonlinear superposition operator, then such operators in this space (globally) satisfy the Lipschitz condition with respect to ${\parallel ·\parallel }_Y$ if and only if their generating function is affine $f(x,y)=a\left(x\right)y+b\left(x\right)$, for some $a,b\in Y$ [2,3,13].

The following is an example of operators taking X into Y with a unique fixed point but not necessarily norm contraction.

Example 2.

Consider $X=(C[a,b],\parallel ·{\parallel }_{\infty })$ and $Y=(C^1[a,b],\parallel ·{\parallel }_Y)$ with the norm ${\parallel x\parallel }_Y={\parallel x\parallel }_{\infty }+{\parallel x^{\prime }\parallel }_{\infty }$. Let us define the operator $T:X\to Y$ as

$$T\left(x\right)\left(s\right)={\int }_a^s{\displaystyle \frac{1}{2}}x\left(t\right)dt.$$

Let $x,y\in X$. Then,

$$\begin{array}{ccc}\hfill {\parallel T\left(x\right)-T\left(y\right)\parallel }_{\infty }& =& \underset{s\in [a,b]}{sup}\left|{\int }_a^s{\displaystyle \frac{1}{2}}(x\left(t\right)-x\left(t\right))dt\right|\hfill \\ & \le & \underset{s\in [a,b]}{sup}{\int }_a^s{\displaystyle \frac{1}{2}}|x\left(t\right)-y\left(t\right)|dt\le {\displaystyle \frac{1}{2}}(b-a){\parallel x-y\parallel }_{\infty }.\hfill \end{array}$$

If $b-a<2$, the operator T is a contraction from X to X with contraction constant ${\displaystyle \frac{1}{2}}(b-a)<1$. For an operator T to be a contraction in $C^1[a,b]$, there must exist a constant $c<1$ such that

$${\parallel T\left(x\right)-T\left(y\right)\parallel }_Y\le c{\parallel x-y\parallel }_Y.$$

Let $x\in X$. Therefore, $T\left(x\right)\in Y$ and the derivative ${\left(T\left(x\right)\right)}^{\prime }\left(s\right)={\displaystyle \frac{1}{2}}x\left(s\right)$, $s\in (a,b)$. Then,

$${\parallel \left(T\left(x\right)\right)}^{\prime }-{\left(T\left(y\right)\right)}^{\prime }{\parallel }_{\infty }=\underset{s\in [a,b]}{sup}|{\displaystyle \frac{1}{2}}x\left(s\right)-{\displaystyle \frac{1}{2}}y\left(s\right)|={\displaystyle \frac{1}{2}}{\parallel x-y\parallel }_{\infty }.$$

If $x\left(s\right)=s$ and $y\left(s\right)=0$, then ${\parallel x-y\parallel }_{\infty }={sup}_{s\in [a,b]}|s|=max\left(|a|,|b|\right)$, while $\parallel x^{\prime }-y^{\prime }{\parallel }_{\infty }=1$. In this case,

$$\begin{array}{ccc}\hfill {\parallel T\left(x\right)-T\left(y\right)\parallel }_Y& =& {\parallel T\left(x\right)-T\left(y\right)\parallel }_{\infty }+{\parallel {\left(T\left(x\right)\right)}^{\prime }-{\left(T\left(y\right)\right)}^{\prime }\parallel }_{\infty }\hfill \\ & =& {\displaystyle \frac{1}{4}}{(max\left(|a|,|b|\right))}^2+{\displaystyle \frac{1}{2}}max\left(|a|,|b|\right)\hfill \end{array}$$

and

$${\parallel x-y\parallel }_Y={\parallel x-y\parallel }_{\infty }+||x^{\prime }-y^{\prime }{||}_{\infty }=max\left(|a|,|b|\right)+1.$$

Hence, there must exist $0\le c<1$ such that

$${\displaystyle \frac{1}{4}}{(max\left(|a|,|b|\right))}^2+{\displaystyle \frac{1}{2}}max\left(|a|,|b|\right)\le c(max\left(|a|,|b|\right)+1).$$

However, for sufficiently large values of $max\left(|a|,|b|\right)$, this inequality will not hold for any $c<1$. Thus, the operator T is not necessarily a contraction in $Y=C^1[a,b]$. Nevertheless, this operator has a unique fixed point $T\left(0\right)=0$. It is important to realize that in order to obtain the fixed point, we can apply the Banach fixed point theorem to this operator in $C[a,b]$ but not in $C^1[a,b]$ (since it is not a contraction operator). However, in this case, the smoothness of the fixed point, $C^1$, is lost (unless it is determined constructively).

The example given is also an illustration of the usefulness of the results obtained here. It can be verified that Proposition 2 applies. In this case, the set $C=\{x\in C^1[a,b]:\parallel f{\parallel }_{\infty }\le 2M,\parallel x^{\prime }{\parallel }_{\infty }\le M\}$ is S-compact, but it is not compact in Y. Moreover, T is S-asymptotically continuous and S-0-continuous along iterates but not condensing with respect to S or in the norm in Y.

There are also many other extensions of the Banach fixed point theorem based on some generalization of the contraction condition (see, for example, [14]). We will mention two important classical results related to the idea of controlling iterates. However, it is important to note that all the results mentioned below apply to the original space X and do not allow us to study other fixed point properties. It is also important to note that a significant development in the study of fixed point existence and uniqueness was the shift in focus from the behavior of the operator T on arbitrary bounded sequences to controlling only the sequences of its iterations. This approach significantly expands the classes of operators that can be studied. In this paper, we study only the behavior of strings of iterations. Let us recall the basic results obtained in this area of study that motivate us.

Theorem 3

([12,14,15] Boyd–Wong–Matkowski fixed point theorem). Let X be a Banach space and C be a bounded and closed subset of X. If $T:C\to X$ satisfies the following:

$${\parallel T\left(x\right)-T\left(y\right)\parallel }_X\le {\phi (\parallel x-y\parallel }_X),\mathit{for}\mathit{all}x,y\in X,$$

where $\phi :[0,+\infty )\to [0,+\infty )$ is a nondecreasing function with $\phi \left(t\right)<t$ and ${\phi }^n\left(t\right)\to 0$ as $n\to +\infty$ for all $t>0$, then T has a unique fixed point x, and for any $x_0\in C$, $x={lim}_{n\to +\infty }{T}^n\left(x_0\right)$.

It is also important to note the following:

Theorem 4

([15,16,17] Caristi fixed point theorem). Let $(X,\parallel ·{\parallel }_X)$ be a Banach space, and let $\phi :X\to \mathbb{R}$ be a lower semicontinuous real-valued function satisfying the following:

$${\parallel x-T\left(x\right)\parallel }_X\le \phi \left(x\right)-\phi \left(T\left(x\right)\right).$$

Then, T has a fixed point. If $\phi \left(x\right)$ is strictly decreasing along iterates, then a fixed point is unique.

In our approach, we split the assumptions on the operator into two parts: the first part is about studying uniqueness with assumptions in the space X; the second part is about the regularity of fixed points with assumptions on the mapping and its properties related to the seminorm S.

Recall also a result dealing with the case when T is not necessarily a contraction but can be decomposed into a contraction and a compact operator. Also recall the result regarding the case when T is not necessarily contractive but can be decomposed into a contractive operator and a compact operator (the Krasnoselskii fixed point theorem [12,15]). To obtain the uniqueness and regularity of fixed points, these assumptions must occur in Y. This directs us toward the next stage of research. The Darbo theorem and measures of noncompactness generalize these types of results. Unfortunately, they do not generally guarantee the uniqueness of the fixed point. In the paper, we will demonstrate how to unify these results with our decomposition of assumptions regarding the uniqueness of the fixed point in X and its regularity as well as the behavior of T in Y. Therefore, we will devote the final section of the paper to presenting results in this direction.

Remark 1.

The next idea is to consider the local compactness and asymptotic compactness of iterates. As claimed in the Matkowski fixed point theorem, to achieve the goals of the paper, we also want to make sure that the sequence of iterates of the operator T is convergent in Y. We are most interested in the case where Y is not a complete subset of X, and we concentrate on S-complete subsets of Y. In this case, we need to ensure that the entire sequence of iterates converges by another method. In Proposition 2, this is simply the assumption that the values of the operator are in the S-complete set C, which is achieved by relating the properties of the seminorm and the operator T on its iteration sequence. In Proposition 1, we replace this by considering an appropriate type of continuity of the operator T. In the rest of the paper, we will point out the general properties of the operator T and the seminorm S that ensure the convergence of sequences of iterates in the norm ${\parallel ·\parallel }_Y$.

4. Main Results

In this section, we provide an algorithm in which we propose conditions to guarantee the existence, uniqueness and regularity of fixed points. We will use the classical iteration sequence; although some modifications of it [18,19] could be also considered, our restriction is only to increase the readability of the idea of the paper. Then, the subsequent theorem shows that these conditions suffice to guarantee a unique fixed point in the subspace Y. Moreover, we provide a list of illustrative cases in which the conditions of the algorithm are fulfilled. Note that asymptotic fixed point theory involves assumptions about the iterates of the mapping in question. In fact, the notion of “asymptotic contractions” was introduced by one of the earliest versions of the Banach fixed point theorem. We will prefer here to keep the assumption of the S-asymptotic continuity of T (cf. Definition 2) being a weaker and more flexible concept then classical continuity notions in the context of iterative sequences. The assumptions about the regularity of fixed points will be explicitly specified; precisely in terms of the norm in Y or the seminorm S, we concentrate on the sets of iterates (and usually not on the set $T\left(X\right)\subset Y$). The following scheme concerns certain assumptions that are considered sufficient to prove separately the existence and uniqueness of fixed points as well as their regularity. It is based on the steps of our algorithm.

• Part 1: Properties of T on the space X.
  • For some $x_0\in X$ (or in $C\subset X$), the sequence of iterations is a Cauchy sequence.
  • This sequence is convergent in X to some $x\in X$.
  • The point x is a fixed point of T.
  • This point is unique in X.
  • T maps Cauchy sequences in X to Cauchy sequences in Y.
• Part 2: Properties of Y or T acting on this space.

  6.

  The Cauchy sequence of iterates of T in Y has an S-convergent subsequence.

  7.

  The accumulation point y is an element of Y.

  8.

  Prove that $x=y$.

  9.

  Conclude that $y\in Y$ is the unique fixed point of T.

The following algorithm produces a fixed point test scheme with a grouping of assumptions corresponding to tests for classes of operators, and therefore it is natural in the case of the seminorm S and the space Y. The division into groups is intended to highlight the possibilities of weakening assumptions with respect to classical fixed point theorems. Indeed, to preserve the regularity of fixed points, the space X should be replaced in existing theorems by the space Y, and due to the stronger norm, these assumptions would be more restrictive. We will show how to extract the assumptions that are nevertheless preserved in the space X and those that allow us to transfer the required properties to the space Y (see condition (FPT) in the Algorithm).

Algorithm 

Let $T:X\to Y$. The algorithm is based on the following conditions:

• (UN) T satisfies some condition guaranteeing that the sequence of iterates $x_{n+1}=T\left(x_n\right)$ is a Cauchy sequence in X for any choice of $x_0\in X$.
  (BA) The Cauchy sequence of iterates $\left(x_n\right)$ has a unique limit point $x\in X$, which is common to all choices of $x_0\in C\subseteq X$.
  (FPT) Suppose that all the iterates $\left(x_n\right)$ of T belong to Y. There exists a subsequence $\left(x_{n_k}\right)$ which is S-convergent to some $y\in Y$, i.e., such that $S(x_{n_k}-y)\to 0$ as $k\to +\infty$.

The next result is an example of a theorem that implements the use of this algorithm.

Theorem 5.

Let $(Y,\parallel ·{\parallel }_Y)$ be a subspace of a Banach space $(X,\parallel ·{\parallel }_X)$ equipped with the norm ${\parallel x\parallel }_Y={\parallel x\parallel }_X+S\left(x\right)$. Let ${T:(X,\parallel ·\parallel }_X{)\to (Y,\parallel ·\parallel }_Y)$ and suppose that T is S-0-continuous and S-asymptotically continuous along iterates. Under the assumptions (UN), (BA) and (FTP), there exists a unique fixed point of the operator T. The fixed point is an element of Y, and it can be obtained as a limit in ${\parallel ·\parallel }_Y$ of iterates of T independent of the choice of the first element.

Proof. 

We will divide the proof into its natural parts.

Step 1. Properties of T from X to X. Take any point $x_0\in X$. Since $T\left(X\right)\subset Y$, by (UN), the sequence of iterates $x_{n+1}=T\left(x_n\right)=T^n\left(x_0\right)$ is a Cauchy sequence in X. Since X is complete, $\left(x_n\right)$ is convergent in X. Condition (BA) implies that the limit is independent of the choice of the initial point $x_0$. Therefore, we obtain a limit point $x\in X$, and $\parallel x_n{-x\parallel }_X\to 0$ as $n\to +\infty$.

Step 2. We will now take a look at the properties that connect the operator $T:X\to X$ and the seminorm S. By Assumption (FTP), we can find a subsequence $\left(x_{n_k}\right)$ which is S-convergent to some $y\in Y$, i.e., $S(x_{n_k}-y)\to 0$ as $k\to +\infty$. We prove the same property for the entire sequence $\left(x_n\right)$. Fix $\epsilon >0$. From the S-asymptotic continuity of T, we obtain the existence of N such that for all $n,m\ge N$,

$$\parallel x_n-x_m{\parallel }_X<\epsilon \mathrm{implies}S(T\left(x_n\right)-T\left(x_m\right))<\epsilon /2.$$

Since $S(T\left(x_{n_k}\right)-y)\to 0$, there exists K such that for $k\ge K$,

$$S(T\left(x_{n_k}\right)-y)<\epsilon /2.$$

Now, for $n\ge N$ and $n_k\ge N$ with $k\ge K$,

$$S(T\left(x_n\right)-y)\le S(T\left(x_n\right)-T\left(x_{n_k}\right))+S(T\left(x_{n_k}\right)-y)<\epsilon /2+\epsilon /2=\epsilon .$$

Therefore, the entire sequence $\left(x_n\right)$ is S-convergent to y. Since $S(x_n-y)\to 0$ as $n\to +\infty$, by the S-0-continuity of T, we obtain $\parallel x_n-\tilde{y}{\parallel }_X\to 0$, for some $\tilde{y}$ with $S(y-\tilde{y})=0$. Now, $S(x_n-\tilde{y})\le S(x_n-\tilde{y})+S(y-\tilde{y})$, hence $S(x_n-\tilde{y})\to 0$. By (BA), the limit in X is uniquely determined, so $\tilde{y}=x$. Therefore, we can assert

$$\parallel x_n{-x\parallel }_Y={\parallel x_n-x\parallel }_X+S(x_n-x)\to 0(n\to +\infty ),$$

which completes this part of the proof. Thus, we have found the unique fixed point $x\in Y$, which is obtained as the limit in ${\parallel ·\parallel }_Y$ of a sequence of iterations of the operator T.

Step 3. To ensure that for sequences of iterates starting from arbitrary points there is a common limit point in the norm ${\parallel ·\parallel }_Y$, since we just proved the convergence of iterates in the seminorm S, it is sufficient to apply the condition (BA). □

Note also that an essential part of proving the convergence of the sequence of iterates in Y is checking that $S(x-y)=0$, since the topology generated by S need not be Hausdorff, and we have $S(x_n-x)\le S(x_n-y)+S(y-x)$, so the use of the (FPT) condition depends on checking this.

The remaining part of the section is devoted to supplementing the algorithm with a list of concrete cases in which (UN), (BA), and (FPT) occur in a way that allows us to derive generalizations of existing results. We will briefly summarize only the most important of the classical theorems which guarantee (UN) and (BA). The second part concerning (FPT) is an important novelty, so we will discuss it in detail. This serves as the basis for both the indication of the role of known results and the derivation of new ones, and a special case will be considered in the last section. We remark that if the truncation of the operator T on the space Y is a contraction in Y or it has a compact set of fixed points, then (FPT) is satisfied. The asymptotic continuity of T does not imply any continuity of T and is useful in many natural cases, such as Sobolev seminorms S. This algorithm is then used to deal with concrete operators and spaces. We will now show how useful this form of theorem is when we use different compactness results for the spaces under consideration. We will also justify how this theorem can be used in practice and how it allows us to generalize the results used so far.

Condition (UN). First, there are some special cases of assumptions implying the condition (UN), so the properties of T from X to X are considered. We do not discuss the interesting question of the conditions and their relations for the existence of fixed points in detail, but we will collect a selection of them below and refer the interested reader to the papers [20,21].

Proposition 3.

Let ${T:(X,\parallel ·\parallel }_X{)\to (X,\parallel ·\parallel }_X)$ be a given mapping. In each of the following cases, condition (UN) is applicable.

1.

[Banach] T is a contraction mapping on X;

2.

[Edelstein] $T:C\to C$ is nonexpansive, where C is a compact subset of X;

3.

[Matkowski] T is weakly contracting, i.e., ${\parallel T\left(x\right)-T\left(y\right)\parallel }_X\le {\phi (\parallel x-y\parallel }_X)$ for $x,y\in X$$\phi :[0,+\infty )\to [0,+\infty )$ is a nonincreasing function with $\phi \left(t\right)\to 0$ as $t\to 0^{+}$;

4.

[Kannan] T is a Kannan contraction, i.e., there exists $0\le k<{\displaystyle \frac{1}{2}}$ such that ${\parallel T\left(x\right)-T\left(y\right)\parallel }_X\le {k(\parallel x-T\left(x\right)\parallel }_X{+\parallel y-T\left(y\right)\parallel }_X)$, for $x,y\in X$;

5.

[Chatterjea] T is a Chatterjea contraction, i.e., there exists $0\le k<{\displaystyle \frac{1}{2}}$ such that ${\parallel T\left(x\right)-T\left(y\right)\parallel }_X\le {k(\parallel x-T\left(y\right)\parallel }_X{+\parallel y-T\left(x\right)\parallel }_X)$, for all $x,y\in X$;

6.

[Ćirić] T is a Ćirić contraction, i.e., there exist constants $0\le \alpha ,\beta ,\gamma <1$ such that ${\parallel T\left(x\right)-T\left(y\right)\parallel }_X\le {\alpha \parallel x-y\parallel }_X{+\beta \parallel x-T\left(x\right)\parallel }_X+\gamma {\parallel y-T\left(y\right)\parallel }_X$, for $x,y\in X$;

7.

[Branciari] T is a generalized integral-type condition of the following form:

$${\parallel T\left(x\right)-T\left(y\right)\parallel }_X\le {\mathsf{\Psi }(\parallel x-y\parallel }_X{,\parallel x-T\left(x\right)\parallel }_X),$$

where $\mathsf{\Psi }:[0,+\infty )\times [0,+\infty )\to [0,+\infty )$ is continuous in both arguments, $\mathsf{\Psi }(r,s)\le r$ for all $r,s\ge 0$, and $\mathsf{\Psi }(r,s)=r$ only if $r=0$ or $s=0$.

Example 3.

In order to demonstrate to the reader that the condition (UN) is not sufficient for the purposes at hand, an example of a mapping that satisfies (UN) and has a fixed point in X will be provided. However, it will be demonstrated that this mapping has no fixed point in Y. Consider the Hammerstein operator $T:C[0,1])\to C[0,1]$:

$$T\left(x\right)\left(s\right)={\displaystyle \frac{1}{2}}{\int }_0^{1}K(s,t)x\left(t\right)dt,$$

with a suitable choice of the kernel $K(s,t)$. If ${sup}_{s\in [0,1]}{\int }_0^1|K(s,t)|dt\le {\displaystyle \frac{1}{2}}$, then for any $x\in C[0,1]$

$${\parallel T\left(x\right)\parallel }_{\infty }\le {\displaystyle \frac{1}{2}}{\parallel x\parallel }_{\infty },$$

so being linear means that T is a contraction in $X=C[0,1]$ with constant $L={\displaystyle \frac{1}{2}}$ and satisfies the condition (UN). Intriguingly, for such an operator type, it is possible to provide kernel examples (e.g., $K(s,t)={\displaystyle \frac{1}{2}}\left(1+{\displaystyle \frac{sin(1/t)}{1+s}}\right)$) where the operator T is a contraction in the supremum norm, it maps $C[0,1]$ such that the iterates $T^n\left(x\right)$ belong to the Hölder space ${\mathbb{H}}^{\alpha }[0,1]$ with some exponent $\alpha \in (0,1]$, and it has a unique fixed point x which, however, does not belong to this Hölder space.

Condition (BA). The case of condition (BA) is similar to that in (UN), and it still concerns the properties of T in X.

Proposition 4.

Let $(Y,\parallel ·{\parallel }_Y)$ be a subspace of a Banach space $(X,\parallel ·{\parallel }_X)$ equipped with the norm ${\parallel x\parallel }_Y={\parallel x\parallel }_X+S\left(x\right)$, where S is lower semicontinuous with respect to ${\parallel ·\parallel }_X$ seminorm on X. Let $T:X\to Y$.

In each of the following cases, condition (BA) holds true.

1.

T is a contraction on X.

2.

The mapping $T:X\to X$ is nonexpansive, and $T\left(X\right)$ is compact and invariant under T.

3.

The mapping $T:X\to X$ is a weak contraction (Matkowski).

4.

The mapping $T:X\to X$ is asymptotic regular and continuous:

$$\underset{n\to +\infty }{lim}{\parallel x_{n+1}-x_n\parallel }_X=0.$$

5.

$T:X\to Y\subset X$ satisfies a Kannan-type condition, i.e., there exists $k\in [0,1/2)$ such that

$${\parallel T\left(x\right)-T\left(y\right)\parallel }_X\le k\left({\parallel x-T\left(x\right)\parallel }_X+{\parallel y-T\left(y\right)\parallel }_X\right)\mathit{for}\mathit{all}x,y\in X.$$

6.

$T:X\to Y\subset X$ is a Meir–Keeler contraction, i.e., for every $ϵ>0$, there exists $\delta >0$ such that

$$ϵ\le {\parallel x-y\parallel }_X<ϵ+\delta \mathit{implies}{\parallel T\left(x\right)-T\left(y\right)\parallel }_X<ϵ.$$

The conditions (UN) and (BA) refer to the properties of the operator T in X and its norm ${\parallel ·\parallel }_X$, which is weaker than the norm in Y. We need to ‘transfer’ some properties of sequences in X to the stronger topology of Y. We concentrate on the properties depending on the seminorm S and iterates of T (cf. Proposition 2).

Definition 3.

A mapping ${T:(X,\parallel ·\parallel }_X{)\to (X,\parallel ·\parallel }_X)$ is said to be asymptotically S-compact with respect to its iterates $x_{n+1}=T\left(x_n\right)$ if every sequence of iterates $\left(x_n\right)$ for which $\left(S\left(x_n\right)\right)$ is a convergent sequence contains a subsequence $\left(x_{n_k}\right)$ convergent to y in the seminorm S, i.e., $S(x_{n_k}-y)\to 0$ as $k\to +\infty$, and $y\in Y$.

Definition 4.

A set $C\subset Y$ is called S-compact (S-sequentially compact) if every bounded sequence $\left(x_n\right)$ in C has an S-convergent subsequence.

Definition 5.

The mapping ${T:(X,\parallel ·\parallel }_X{)\to (Y,\parallel ·\parallel }_Y)$ is said to be S-compact if it maps bounded sets in X into relatively S-compact sets in Y.

In the case we are considering, all of the above notions are more general than those with respect to the norm ${\parallel ·\parallel }_Y$, because we do not need to control the properties with respect to the norm ${\parallel ·\parallel }_X$. The following example illustrates the difference between conditions expressed in terms of the seminorm S and the norm ${\parallel ·\parallel }_Y$.

Example 4.

To construct an example of a subset $C\subset Y$ that is S-complete and satisfies the inclusions $T\left(C\right)\subset T\left(X\right)\subset C\subset Y$, let us choose $X=C[0,1]$ and define a seminorm S on X as the Lipschitz seminorm, i.e.,

$$S\left(x\right)=\underset{s\ne t}{sup}{\displaystyle \frac{|x(s)-x(t)|}{|s-t|}}.$$

Let Y be the subspace of X consisting of functions with finite Lipschitz seminorm: $Y=\{x\in X:S(x)<+\infty \}$ and then it is Hölder–Lipschitz space $Y={\mathbb{H}}^1[0,1]$. Since ${\parallel x\parallel }_Y={\parallel x\parallel }_X+S\left(x\right)$, the identity operator $T:X\to X$ satisfies $T\left(X\right)=X$. The expected S-complete set C can be defined as follows:

$$C=\{x\in Y:S(x)\le 1\}.$$

A sequence $\left(x_n\right)\subset C$ that is Cauchy in S will converge to a function x with $S\left(x\right)\le 1$, ensuring C is complete in the seminorm S. Clearly, it is not complete in X. All the expected inclusions hold.

Condition (FPT). Now, we include a discussion of the (FPT) condition. Instead of the norm contraction on Y (or other (UN)-type conditions), we can use some weaker conditions to describe how the properties of T in X should be transferred to Y in a way that preserves the (unique) fixed point in Y. This is where our approach differs from previous results. The following result naturally omits the trivial case where the operator T is compact from X to Y. The assumptions are related to the properties of the operator T, the seminorm S and the properties of the space Y in their various interdependencies.

Theorem 6.

Let $(Y,\parallel ·{\parallel }_Y)$ be a subspace of a Banach space $(X,\parallel ·{\parallel }_X)$ equipped with the norm ${\parallel x\parallel }_Y={\parallel x\parallel }_X+S\left(x\right)$.

Assume that the conditions (UN) and (BA) hold true for $T:X\to X$. In each of the following cases, the condition (FPT) is satisfied.

1.

$T\left(X\right)$ is S-compact or there exists an S-compact set C with $T\left(X\right)\subset C$;

2.

For any sequence $\left(x_n\right)$ of iterates of T, there exist $C>0$ and $N\in \mathbb{N}$ such that for all $n,m\ge N$, we have $S(x_n-x_m)\le C·{\parallel x_n-x_m\parallel }_X$ and $S\left(x_n\right)<+\infty$, $T\left(X\right)$ is S-complete;

3.

The mapping ${T:(X,\parallel ·\parallel }_X{)\to (Y,\parallel ·\parallel }_Y)$ is S-asymptotically continuous with respect to its iterates, and $T\left(X\right)\subset Y$ is S-complete;

4.

T is ${\parallel ·\parallel }_X$-S-continuous along its iterates;

5.

T is asymptotically S-compact with respect to its iterates, and is S-nonexpansive along iterates, i.e., $S\left(T\left(x_n\right)\right)\le S\left(x_n\right)$ for all n;

6.

S satisfies a contraction-type property along the iterates of T, i.e., there exists a constant $0\le \lambda <1$ and a subsequence $\left(x_{n_k}\right)$ of the sequence of iterates is S-convergent to some z such that for sufficiently large k, $S(x_{n_{k+1}}-z)\le \lambda ·S(x_{n_k}-z)$;

7.

For any sequence $\left(x_n\right)$ of iterates of T and any $z\in Y$ with $S(x-z)=0$, we have

$$\underset{n\to +\infty }{lim\; inf}{\displaystyle \frac{S(x_n-z)}{\parallel x_n{-z\parallel }_X}}<+\infty ,$$

8.

S is frequently bounded along iterates, i.e., by taking a sequence $v_n={\displaystyle \frac{x_n-z}{\parallel x_n{-z\parallel }_X}}$ in a unit sphere in X, there exists $C>0$ such that for infinitely many n, we have $S\left(v_n\right)\le C$ for any sequence $\left(x_n\right)$ of iterates of T and any $z\in Y$ with $S(x-z)=0$;

9.

T is bounded and S-compact from X into Y;

10.

There exists $n\in \mathbb{N}$ such that $T^n$ is an S-contraction; i.e., there exists $\lambda \in (0,1)$ such that for all $x,y\in X,S(T^n\left(x\right)-T^n\left(y\right))\le \lambda S(x-y)$, and $T\left(X\right)$ is S-complete.

Proof. 

[1.]

This is a very general but useful condition, as we will show later in this paper. Every sequence bounded in $T\left(X\right)$ contained in an S-compact set contains an S-convergent subsequence. In particular, for any sequence $\left(x_n\right)$ of iterates of the operator T, we obtain the existence of a subsequence $\left(x_{n_k}\right)$ that S-converges to some element y of $T\left(X\right)$.

[2.]

By assumption (UN), the sequence of iterates $\left(x_n\right)$ is a Cauchy sequence in X (and converges to some $x\in X$). If S is dominated along Cauchy sequences by the norm ${\parallel ·\parallel }_X$, then there exists a constant $C>0$ and $N\in \mathbb{N}$ such that $S(x_n-x_m)\le C·{\parallel x_n-x_m\parallel }_X$ for $m,n\ge N$. It means that since $\left(x_n\right)$ is a Cauchy sequence in X, it is also an S-Cauchy sequence. By the property S-completeness of $T\left(X\right)$, it is S-convergent (so we obtain an S-convergent subsequence too) to some y: $S(x_n-y)\to 0$ as $n\to +\infty$. But $S\left(y\right)\le S(x_n-y)+S\left(x_n\right)$, and both terms on the right-hand side are finite, so $S\left(y\right)<+\infty$ and then $y\in Y$.

[3.]

In this case, we obtain an S-Cauchy condition for $\left(x_n\right)$ by applying the S-asymptotic continuity of T with respect to its iterates. The rest of the conclusion is the same as in the previous point.

[4.]

This assumption only concerns T. Since T maps convergent sequences in X to S-convergent sequences in Y, we obtain the expected thesis for the entire sequence $\left(x_n\right)$ and for $y=x$. Since for a sequence of iterates $\left(x_n\right)$ of T we have both $\parallel x_n{-x\parallel }_X\to 0$ and $S(T\left(x_n\right)-T\left(x\right))=S(x_{n+1}-x)\to 0$ as $n\to +\infty$, we finally obtain $\parallel x_n{-x\parallel }_Y\to 0$ as $n\to +\infty$. Then, $x\in Y$.

[5.]

First, note that this assumption only concerns the behavior of iterates of T. Since the operator T is nonexpansive along iterates, we have $S\left(x_{n+1}\right)=S\left(T\left(x_n\right)\right)\le S\left(x_n\right)$ for all n. Being nonincreasing and bounded below by $0=S\left(0\right)$, the sequence $\left(S\left(x_n\right)\right)$ is convergent.

But the operator T is asymptotically S-compact with respect to its iterates, so $\left(x_n\right)$ contains a subsequence $\left(x_{n_k}\right)$ with $S(x_{n_k}-y)\to 0$ as $k\to +\infty$ and $y\in Y$.

[6.]

Here, we assume that the sequence of iterates is S-Fejér-monotone (more precisely: contractive) with respect to a set of points being S-equivalent to a fixed point of T. By our assumption, there exists a subsequence $\left(x_{n_k}\right)$ and $\lambda \in [0,1)$ such that

$$S(x_{n_{k+1}}-x)\le \lambda S(x_{n_k}-x).$$

Let us restrict our attention to this subsequence. Since we know that $\parallel x_n{-x\parallel }_X\to 0$ as $n\to +\infty$, for any of its subsequences $\left(x_{n_k}\right)$, we can apply our assumption to $z=x$ for any choice of a subsequence. Let $a_k=S(x_{n_k}-x)$. Then, for $k\ge K$, we have $a_{k+1}=S(x_{n_{k+1}}-x)\le \lambda ·a_k$.

Repeat this for $k\ge K$ and $m>0$

$$a_{k+m}\le \lambda ·a_{k+m-1}\le {\lambda }^2·a_{k+m-2}\le \cdots \le {\lambda }^m·a_k.$$

So, for $k\ge K$, $0\le S(x_{n_k}-x)\le {\lambda }^{k-K}S(x_{n_K}-x)$. Since $0\le \lambda <1$, we know that ${lim}_{k\to +\infty }{\lambda }^{k-K}=0$. Therefore, for $k\ge K$,

$$S(x_{n_k}-x)\le {\lambda }^{k-K}S(x_{n_K}-x),$$

and then

$$0\le \underset{k\to +\infty }{lim}S(x_{n_k}-x)\le \underset{k\to +\infty }{lim}\left({\lambda }^{k-K}S(x_{n_K}-x)\right)=0·S(x_{n_K}-x)=0.$$

Thus, ${lim}_{k\to +\infty }S(x_{n_k}-x)=0$. Moreover, each subsequence of $\left(x_n\right)$ is convergent to x, i.e., $\parallel x_{n_k}{-x\parallel }_X\to 0$, so $\parallel x_{n_k}{-x\parallel }_Y\to 0$ as $k\to +\infty$, and then $x\in Y$.

[7.]

The asymptotic S-boundedness of T implies that there exists $C>0$ and $\epsilon >0$ such that there exists a subsequence $(x_{n_k}-z)$ of $(x_n-z)$, and we have $S(x_{n_k}-z)\le (C+\epsilon )·{\parallel x_{n_k}-z\parallel }_X$ for sufficiently large k, i.e., there is a natural number $N\in \mathbb{N}$ such that $n_k>N$. In particular, we can consider $z=x$ (which is the limit of the sequence of iterates), and we obtain $\parallel x_{n_k}{-x\parallel }_X\to 0$ as $k\to +\infty$, so $S(x_{n_k}-x)\to 0$ as $k\to +\infty$. Finally, $\parallel x_{n_k}{-x\parallel }_Y\to 0$ as $k\to +\infty$, so $x\in Y$.

[8.]

The difference between this case and the previous one is that $\parallel v_n{\parallel }_X=1$ for all n, since $v_n$ is the normalization of $x_n-z$. Put $z=x$. Thus, the seminorm preserves only the directional information about the convergence of $\left(x_n\right)$ to z and ignores the magnitude, which is not important in our case. It is a property of S that guarantees that it is not infinite along infinitely many directions $v_n$. Obviously,

$${\displaystyle \frac{S(x_n-x)}{\parallel x_n{-x\parallel }_X}}=S\left(v_n\right).$$

Let $\mathcal{N}$ be the set of all natural numbers for which $S\left(v_n\right)\le C$. So, take a subsequence $\left(x_{n_k}\right)$ of $\left(x_n\right)$ such that $n_k\in \mathcal{N}$. Next, for each k, we have the following:

$$S\left(v_{n_k}\right)\le M.$$

Multiplying both sides of this inequality by $\parallel x_{n_k}{-z\parallel }_X$, we obtain the following:

$$S(x_{n_k}-x)=S\left(v_{n_k}\right)·\parallel x_{n_k}{-x\parallel }_X\le M{\parallel x_{n_k}-x\parallel }_X$$

and we have the expected S-convergent subsequence of $\left(x_n\right)$.

[9.]

This assumption is not limited to iterations. So it is more classical but less general. However, we do not need to ensure the existence of the S-complete image set C beforehand, since we can construct it here and use it as in the previous argument.

Since T is S-compact, the image of any bounded set under T is relatively S-compact. The sequence of iterates $\left(x_n\right)$, being convergent, is bounded in X, and by the boundedness of T, is also S-bounded in Y, i.e., there exists $M>0$ such that $S\left(x_n\right)\le M<+\infty$. Thus, by the S-compactness of T, there exists a subsequence $\left(x_{n_k}\right)$ such that $S(x_{n_k}-y)\to 0$ for some y. But $S\left(y\right)\le S(x_{n_k}-y)+S\left(x_{n_k}\right)$. The first term is convergent to zero (so bounded) and the second is bounded by M, so $S\left(y\right)<+\infty$ and $y\in Y$.

[10.]

It is a well-known case for metric spaces that under this assumption, T need be neither S-contraction nor S-continuous. However, here we consider a seminorm, which is still sufficient for our purposes. Define the subsequence: $y_k:=T^{\left(n_k\right)}{x}_0.$ So, $y_{k+1}=T^n\left(y_k\right)$.

Then, for all $k\in \mathbb{N}$, by our assumption

$$S(y_{k+1}-y_k)=S(T^n\left(y_k\right)-T^n\left(y_{k-1}\right))\le \lambda S(y_k-y_{k-1}),$$

and then

$$S(y_{k+1}-y_k)\le {\lambda }^{k}S(y_1-y_0).$$

Therefore, for $p\in \mathbb{N}$, we can estimate

$$S(y_{k+p}-y_k)\le \sum _{j=0}^{p-1}S(y_{k+j+1}-y_{k+j})\le \sum _{j=0}^{p-1}{\lambda }^{k+j}S(y_1-y_0).$$

This sum is bounded above,

$$S(y_{k+p}-y_k)\le {\lambda }^{k}S(y_1-y_0)\sum _{j=0}^{+\infty }{\lambda }^j={\lambda }^k·{\displaystyle \frac{1}{1-\lambda }}·S(y_1-y_0).$$

Thus,

$$\underset{p\in \mathbb{N}}{sup}S(y_{k+p}-y_k)\to 0\mathrm{as}k\to +\infty .$$

Therefore, $\left(y_k\right)$ is a Cauchy sequence in the seminorm S. As $y_k=T^n\left(y_{k-1}\right)\in T\left(X\right)$ and $T\left(X\right)$ is S-complete, it converges to some y. Since $S\left(y\right)\le S(y-y_k)+S\left(y_k\right)<+\infty$, we have $y\in T\left(X\right)\subset Y$.

□

5. Applications

The study of differential and integral problems is not the purpose of this paper, but it allows us to illustrate the obtained results (see [22], for instance). Therefore, we will not discuss the problems but only their equivalent operator forms. We are mainly concerned with nonlinear differential equations of a fractional order. Many results give the existence and uniqueness of continuous solutions despite the fact that the space of continuous functions is not very suitable for such kinds of problems. As an example, consider the problem $D^{\tau }x\left(t\right)=f(t,x\left(t\right))$, $x\left(a\right)=0,x\left(b\right)=B$, $t\in [a,b]$, where $B\in \mathbb{R},\tau >0$, and $D^{\tau }$ denotes the fractional Riemann–Liouville derivative of order $\tau$. The solving operator $T:C[a,b]\to C[a,b]$ will be of the form

$$T\left(x\right)\left(t\right)=B{\displaystyle \frac{{(t-a)}^{\alpha -1}}{{(b-a)}^{\alpha -1}}}+{\int }_a^{b}G(t,s)f(s,x\left(s\right))ds,$$

for $t\in [a,b]$ and some kernel G. In [23] (see also ([3], Theorem 1.1)), it has been proved, under the assumptions of continuity and a Lipschitz condition in the second variable, for the function f and of a suitable bound for $b-a$, that the problem has a unique continuous solution. This was proven using the Banach fixed point theorem in $C[a,b]$. In [24], the fractional differential equation $D_c^{\tau }x\left(t\right)=f(t,x\left(t\right))+D_c^{\tau -1}g(t,x\left(t\right))$$x\left(a\right)={\theta }_1,x^{\prime }\left(a\right)={\theta }_2$, $t\in [a,b]$ has been taken into consideration, where $\tau >1$, $f,g:[a,b]\times \mathbb{R}\to \mathbb{R}$ are given continuous functions, ${\theta }_1,{\theta }_2\in \mathbb{R}$, and $D_c^{\tau }$ denotes the Caputo fractional derivative. Recall that the Caputo fractional derivative of order $\tau \ge 0$ of a function $x:[a,b]\to \mathbb{R}$ is defined by $D_c^{\tau }x\left(t\right)={\displaystyle \frac{1}{\mathrm{\Gamma }(n-\tau )}}{\int }_a^t{(t-s)}^{n-\tau -1}{x}^{\left(n\right)}\left(s\right)ds$, for $\tau >0$, and by $D_c^{0}x\left(t\right)=x\left(t\right)$, for any $t\in [a,b]$, where $n=\left[\tau \right]+1$, $\left[\tau \right]$ denotes the integer part of $\tau$ and $\mathrm{\Gamma }$ denotes the Euler Gamma function, provided that the right side of this formula is pointwise defined. Let us recall that the Riemann–Liouville fractional integral of order $\tau >0$ of a function $x\in L^1[a,b]$ is defined for a.e. t by $I^{\tau }x\left(t\right):={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\tau \right)}}{\int }_a^t{(t-s)}^{\tau -1}x\left(s\right)ds$. Then, the solving operator is of the form

$$T\left(x\right)\left(t\right)={\theta }_1+({\theta }_2-g(a,{\theta }_1))(t-a)+(I^{\alpha }\circ F)\left(x\right)\left(t\right)+(V\circ G)\left(x\right)\left(t\right),$$

where $F,G$ represent the superposition operators generated by F and G, respectively. In the existence and uniqueness proof for solutions, it is not required that f and g satisfy the Lipschitz condition, but the more general condition Matkowski condition $|f(t,u)-f(t,v)|\le \varphi (|u-v|)$, and similarly for g, where $\varphi$ is a comparison function, which we discussed earlier. Note that again the solution is continuous. However, imposing either global Lipschitz or Matkowski conditions on the nonlinear operators involved leads to degeneracy phenomena.

In order to apply our approach to fixed point theorems, it is first necessary to recall the following definition. Let ${\mathcal{M}}_X$ denote the family of all nonempty and bounded subsets of X. In the following, an axiomatic approach to the notion of a measure of noncompactness is used.

Definition 6

([4]). A mapping $\mu :{\mathcal{M}}_X\to [0,+\infty )$ is said to be a measure of noncompactness in X if it satisfies the following conditions: (i) $\mu \left(A\right)=0\Rightarrow A$ is relatively compact; (ii) $A\subset B\Rightarrow \mu \left(A\right)\le \mu \left(B\right)$; (iii) $\mu \left(\overline{A}\right)=\mu \left(convA\right)=\mu \left(A\right)$; (iv) $\mu \left(\lambda A\right)=|\lambda |\mu \left(A\right)$ for $\lambda \in \mathbb{R}$; (v) $\mu (A+B)\le \mu \left(A\right)+\mu \left(B\right)$; (vi) $\mu (A\cup B)=max\left\{\mu \right(A),\mu (B\left)\right\}$; (vii) If $A_n$ is a sequence of nonempty, bounded, closed subsets of X such that $A_{n+1}\subset A_n,$$n=1,2,3,\cdots$, and ${lim}_{n\to +\infty }\mu \left(A_n\right)=0$, then the set $A_{\infty }={\cap }_{n=1}^{+\infty }{A}_n$ is nonempty.

The Darbo fixed point theorem is a common generalization of the Schauder and Banach fixed point theorems with its application to the measure of noncompactness settings. However, in contrast to the guarantees provided by the Banach contraction theorem, the Darbo theorem does not generally guarantee the uniqueness of the fixed point (cf. [25]). The present approach allows the use of this result to guarantee the condition (FPT), and by using our results, uniqueness is obtained (cf. [3]). We refer the reader to [26] for a discussion on Darbo’s theorem and related topics. We restrict ourselves to the following lemma, which allows to cover classical compactness-type assumptions of the Darbo type, i.e., when the operators are contractions with respect to some measure of noncompactness in Y (see, for example, [3]). It guarantees the existence of a compact invariant set (see [27]).

Lemma 3.

Let $(Y,\parallel ·{\parallel }_Y)$ be a Banach space and μ be a measure of noncompactness in Y. Assume that $M\subset Y$ is nonempty, bounded, closed, and convex, and $T:M\to M$ is continuous and satisfies

$$\mu \left(T\right(B\left)\right)\le k·\mu \left(B\right),B\subseteq M,$$

where $0\le k<1$. Then, there exists a compact set K in M such that $T:K\to K$ and T has a fixed point in K.

Now, a reference paper to illustrate our results is the work by Appell et al. [3], where it is worth noticing that for the study of a general fractional initial value problem, both the space Y and the seminorm S (which is actually a norm) were constructed according to the natural action of the operator T under investigation. The space Y is defined as a Hölder-type space involving moduli of continuity in integral form. Due to the goals of our paper, we will slightly modify the results of the mentioned paper by considering the scenario of operators tempered, for example, by the function $\vartheta \left(t\right)=t^{\alpha }·e^{-\nu t}$) with $\nu \ge 0$. We will use the results from [28] to show that modifying the seminorm from the case of $\nu =0$ to $\nu >0$ allows us to preserve the proof steps from the original paper [3] (without repeating the technical details) while obtaining new results with a wide range of applications. We will only complete those parts of the proof that differentiate these cases and those that relate to the assumptions of Theorem 5.

From our point of view, the new result in [3] is relevant to the Darbo fixed point theorem when applied in the new “smoother” space Y (the direct use of the Schauder fixed point theorem is rather complicated and not natural). This preserves the regularity, but we lose the uniqueness property of the solution. We are ready to recall some ideas from the paper [3] related to fixed points of fractional order operators and which we will use here for the application of our algorithm.

For our purposes, we denote by $\vartheta :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ a continuous increasing function with $\vartheta \left(0\right)=0$ and ${lim}_{t\to 0+}{\displaystyle \frac{\vartheta \left(t\right)}{t}}=+\infty$, and we define the space of (generalized) Hölder functions ${\mathbb{H}}^{\vartheta }[0,1]$ as the family of all $x\in C[0,1]$, such that $|x\left(t\right)-x\left(s\right)|\le L\vartheta (|t-s|),L>0$. The role of the $\vartheta$ function is to control the growth rate of the continuity modulus, which is “tempered” by $\vartheta$, so we set

$${\omega }_{\vartheta }(x,\sigma )=\underset{t\ne s}{sup}\left\{{\displaystyle \frac{|x\left(t\right)-x\left(s\right)|}{\vartheta (|t-s|)}}:|t-s|<\sigma \right\}.$$

Then, as in the classical case of spaces tempered by a modulus of continuity, we can prove that (see, [3,28,29]) the functional ${\omega }_{\vartheta }:C[0,1]\to {\mathbb{R}}^{+}$

$${\omega }_{\vartheta }\left(x\right)=\underset{\sigma \to 0^{+}}{lim}{\omega }_{\vartheta }(x,\sigma )$$

is a seminorm on $C[0,1]$. Obviously, this seminorm vanishes for any constant function. The special choices $\vartheta \left(t\right)=t^{\alpha }$, $\alpha \in (0,1)$ lead naturally to the classical Hölder spaces ${\mathbb{H}}^{\alpha }[0,1]$ [30]. Next, for $0<\alpha \le 1,\alpha <\beta <+\infty$ and $0<s\le 1$, we set

$$j_{\alpha ,\beta }^{\vartheta }(x,[0,s])={\int }_0^s{\sigma }^{-(\beta +1)}{\omega }_{\vartheta }{(x,\sigma )}^{{\displaystyle \frac{\beta }{\alpha }}}d\sigma ,$$

and we consider the space $J_{\alpha ,\beta }^{\vartheta }[0,1]$ consisting of all of functions x with finite values of the functional

$${|||x|||}_{\alpha ,\beta }^{\vartheta }={\parallel x\parallel }_{\infty }+\left(j_{\alpha ,\beta }^{\vartheta }\right(x,[0,1]){)}^{{\displaystyle \frac{\alpha }{\beta }}}.$$

Then, ${|||x|||}_{\alpha ,\beta }^{\vartheta }$ is a norm on that space and $(J_{\alpha ,\beta }^{\vartheta }[0,1],|||x||{|}_{\alpha ,\beta }^{\vartheta })$ is a Banach space ([31], Proposition 4). Therefore, we have $X=C[0,1]$ with the supremum norm, $Y=(J_{\alpha ,\beta }^{\vartheta }[0,1],|||x||{|}_{\alpha ,\beta }^{\vartheta })$, so that the seminorm $S:Y\to [0,+\infty ]$ is defined by $S\left(x\right)=\left(j_{\alpha ,\beta }^{\vartheta }\right(x,[0,1]){)}^{{\displaystyle \frac{\alpha }{\beta }}}$. Note that

$$\left(j_{\alpha ,\beta }^{\vartheta }\right(x,[0,1]){)}^{{\displaystyle \frac{\alpha }{\beta }}}\le C·{\parallel x\parallel }_{\infty },$$

(2)

with the constant $C={\displaystyle \frac{2^{\beta /\alpha }}{\beta }}$. It was proved in ([32], Proposition 6.3) that the Hausdorff measure of noncompactness ${\beta }_Y$ in the space $Y=J_{\alpha ,\beta }^{\vartheta }$ is equivalent to the set function

$$\eta \left(M\right):=\underset{s\to 0}{lim\; sup}\underset{x\in M}{sup}{j}_{\alpha ,\beta }^{\vartheta }(x,[0,s]).$$

In particular, a subset $M\subset J_{\alpha ,\beta }^{\vartheta }[0,1]$ is compact if and only if it is bounded, closed, and satisfies $\eta \left(M\right)=0$. Note the measure $\eta$ is obviously related to the S-completeness and S-compactness of sets in Y. The purpose of this section is to illustrate how natural the assumptions on the operator are. To this end, the assumptions are presented here, but the constants (only their bounds are relevant) are not focused on. Those interested in the proofs can find them in [3] for fractional Riemann–Liouville operators or [28] for tempered fractional operators.

Let $0<\tau <1$, $0<\alpha <\tau$, $\beta >\alpha$ and $\vartheta \left(t\right)=t^{\tau }·e^{-\nu t}$. We consider the problem

$$\left\{\begin{array}{cc}{D}_c^{\tau ,\nu }x\left(t\right)=f(t,x\left(t\right))\hfill & (0<t<1),\hfill \\ x\left(0\right)={\theta }_1,\hfill & x^{\prime }\left(0\right)={\theta }_2,\hfill \end{array}\right.$$

where $f:[0,1]\times \mathbb{R}\to \mathbb{R}$, and $D_c^{\tau ,\nu }$ is the tempered Caputo fractional derivative of order $\tau$. A continuous function $x:[0,1]\to \mathbb{R}$ solves this Cauchy problem if and only if x is a fixed point of the operator

$$T\left(x\right)\left(t\right)={\theta }_1+{\theta }_{2}t+(I^{\tau ,\nu }\circ F)\left(x\right)\left(t\right),$$

(3)

where $I^{\tau ,\nu }$ is the generalized fractional Riemann–Liouville operator of order $\tau >0$ and parameter $\nu \in {\mathbb{R}}^{+}$, which is defined by $I^{\tau ,\nu }x\left(t\right)={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\tau \right)}}{\int }_a^t{\left(t-s\right)}^{\tau -1}{e}^{-\nu \left(t-s\right)}x\left(s\right)ds$ and F is a superposition operator generated by f. This operator is a generalization of the one studied in ([3], Section 6) in the case $\nu =0$.

If we use the Darbo fixed point theorem instead of the Banach fixed point theorem, we will not be able to prove the uniqueness of the fixed point. This means that the result obtained is only valid for its existence. At the same time, however, the example from the paper ([3], Example 7.2) is even more interesting, because the uniqueness of the fixed point is obtained in the case of $f(t,x)=\lambda sin(t+x)$, for $t\in [0,1]$.

Now, let us give directly a version of the fixed point theorem that we can apply here. We can choose conditions (UN), (BA) and (FPT) which are related to the operator under consideration. We can choose suitable conditions directly from our propositions. The following corollary is a particular version of our Theorem 5.

Corollary 2.

Let $0<\alpha \le 1,\alpha <\beta <+\infty$, $\tau \in (0,1)$, and $\vartheta \left(t\right)=t^{\tau }·e^{-\nu t}$, $\nu \ge 0$.

Let $X=(C[0,1],\parallel ·{\parallel }_{\infty })$ and $Y=(J_{\alpha ,\beta }^{\vartheta }{,\parallel ·\parallel }_{\infty }+\left(j_{\alpha ,\beta }^{\vartheta }\right(x,[0,1]){)}^{{\displaystyle \frac{\alpha }{\beta }}})$. Let M be a bounded and convex set in X. Also, let $T:M\to M$ be a contraction with respect to the norm ${\parallel ·\parallel }_{\infty }$ and the measure of noncompactness η. Then, there exists a fixed point in Y that is unique.

The conditions (UN) and (BA) are guaranteed because the operator T defined in (3) is a contraction in M. Since T is contractive with respect to the measure of noncompactness $\eta$, according to Lemma 3, the condition (FPT) holds by virtue of Theorem 6 (item [1.]). It remains to prove that ${T:(X,\parallel ·\parallel }_X{)\to (Y,\parallel ·\parallel }_Y)$ is S-asymptotically continuous and S-0-continuous along iterates, using the property of the seminorm $j_{\alpha ,\beta }^{\vartheta }$.

Note that the sequence of the iterates $\left(x_n\right)$, for any starting point, is a Cauchy sequence in ${\parallel ·\parallel }_{\infty }$. Since $I^{\tau ,\nu }$ maps $C[0,1]$ into $J_{\alpha ,\beta }^{\vartheta }[0,1]$, we obtain $T\left(x_n\right)\in J_{\alpha ,\beta }^{\vartheta }[0,1]$.

Given $\epsilon >0$, choose $N\in \mathbb{N}$ such that for all $m,n\ge N$, we have $\parallel x_n-x_m{\parallel }_{\infty }<\delta$ for some small $\delta >0$. Let $\tilde{C}>0$; then for $\delta <{(\epsilon /\tilde{C})}^{1/\alpha }$, we obtain the estimate (cf. ([3], Proposition 6.3), ([28], Proposition 3.1))

$$S(T\left(x_n\right)-T\left(x_m\right))={\left(j_{\alpha ,\beta }^{\vartheta }(T\left(x_n\right)-T\left(x_m\right),[0,1])\right)}^{\alpha /\beta }\le \tilde{C}{\delta }^{\alpha /\beta }<\epsilon .$$

Thus, T is S-asymptotically continuous along iterates.

Next, from $S(T\left(x_n\right)-y)=j_{\alpha ,\beta }^{\vartheta }{((T\left(x_n\right)-y),[0,1])}^{\alpha /\beta }\to 0$, as $n\to +\infty$,

$$\mathrm{for}\mathrm{all}\epsilon >0,\mathrm{there}\mathrm{exists}N\mathrm{such}\mathrm{that}n\ge N\mathrm{implies}{j}_{\alpha ,\beta }^{\vartheta }((T\left(x_n\right)-y),[0,1])<{\epsilon }^{\beta /\alpha }.$$

By definition of $j_{\alpha ,\beta }^{\vartheta }$, for all $t,s\in [0,1]$, for any $\epsilon >0$ there exists N such that for $n\ge N$

$$\underset{t,s}{sup}{\displaystyle \frac{|(T\left(x_n\right)-y)\left(t\right)-(T\left(x_n\right)-y)\left(s\right)|}{{|t-s|}^{\alpha }{(t^{\tau }+s^{\tau })}^{\beta }}}<{\epsilon }^{\beta /\alpha }.$$

For fixed $t\in [0,1]$, the seminorm bound implies $|(T\left(x_n\right)-y)\left(t\right)-(T\left(x_n\right)-y)\left(s\right)|\le {\epsilon }^{\beta /\alpha }{|t-s|}^{\alpha }{\left(2t^{\tau }\right)}^{\beta }$ for $s\approx t$, proving local Hölder continuity.

Define $\tilde{y}\left(t\right):={lim}_{n\to +\infty }T\left(x_n\right)\left(t\right)$ (the bound exists pointwise because $\left(T\left(x_n\right)\left(t\right)\right)$ is a Cauchy sequence). For fixed t, the Hölder bound gives $|T\left(x_n\right)\left(t\right)-T\left(x_m\right)\left(t\right)|\le 2{\epsilon }^{\beta /\alpha }{t}^{\tau \beta }$. Thus, $\left(T\left(x_n\right)\left(t\right)\right)$ is a Cauchy sequence in $\mathbb{R}$, so the limit $\tilde{y}\left(t\right)$ exists. Since $t^{\tau \beta }\le 1$ for $t\in [0,1]$, we have ${sup}_t|T\left(x_n\right)\left(t\right)-\tilde{y}\left(t\right)|\le {\epsilon }^{\beta /\alpha }$. From $j_{\alpha ,\beta }^{\vartheta }((T\left(x_n\right)-y),[0,1])\to 0$ and $\parallel T\left(x_n\right)-\tilde{y}{\parallel }_{\infty }\to 0$, the triangle inequality implies

$$j_{\alpha ,\beta }^{\vartheta }((y-\tilde{y}),[0,1])\le \underset{n\to +\infty }{lim}(j_{\alpha ,\beta }^{\vartheta }((T\left(x_n\right)-y),[0,1])+j_{\alpha ,\beta }^{\vartheta }\left((T\left(x_n\right)-\tilde{y})\right),[0,1])=0,$$

where $j_{\alpha ,\beta }^{\vartheta }((T\left(x_n\right)-\tilde{y}),[0,1])\to 0$ by uniform convergence. Thus, $S(y-\tilde{y})=0$. Since $I^{\tau ,\nu }$ maps X into Y, the uniform convergence preserves continuity of each $T\left(x_n\right)$. Moreover,

$$j_{\alpha ,\beta }^{\vartheta }(\tilde{y},[0,1]))\le \underset{n\to +\infty }{lim\; inf}{j}_{\alpha ,\beta }^{\vartheta }(T\left(x_n\right),[0,1]))<+\infty$$

so $\tilde{y}\in Y$. So we have checked that the operator T is S-0-continuous along iterates.

From our results, we can now conclude that this problem has a unique solution in $J_{\alpha ,\beta }^{\vartheta }[0,1]$. We should note that all the steps, to ensure that the operator T defined in (3) is invariant on a ball, continuous, bounded, and $\eta$-contractive on Y, should follow the line of those in ([3], Section 7). Therefore, we outline the necessary steps and corresponding results for tempered fractional operators.

(1)

We need to prove that T is a contraction mapping on $X=C[0,1]$. Due to properties of superposition operators acting on this space [13], this is easily verifiable. Thus, conditions (UN) and (BA) are verified (see ([28], Lemma 3.4), for the case of tempered operators).

(2)

The operator T maps $B_r\left(J_{\alpha ,\beta }^{\vartheta }\right)$ to $J_{\alpha ,\beta }^{\vartheta }$ and is bounded on this ball. Necessary conditions for constants and bounds for a given f should be checked (see ([28], Theorem 3.2)).

(3)

The operator T is continuous on $B_r\left(J_{\alpha ,\beta }^{\vartheta }\right)$. The continuity of $I^{\tau ,\nu }$ is equivalent to its boundedness (cf. [28], Theorem 3.2), while the continuity of F should be proved in the constructed space Y (cf. [3], Theorem 4.5).

(4)

The operator T is a contraction with respect to the measure of noncompactness $\eta$ (related to S-compactness, [Proposition 6.3 [3]) on $B_r\left(J_{\alpha ,\beta }^{\vartheta }\right)$ (cf. ([3], Theorem 6.4) and appropriate changes in the tempered case as in ([31], proof of Theorem 3)).

(5)

The operator T maps $B_r\left(J_{\alpha ,\beta }^{\vartheta }\right)$ into itself (see ([3], Theorem 3.5, Theorem 4.1) and appropriate changes in ([28], Proposition 3.1)). As claimed in Lemma 3, it implies the existence of an invariant compact set in Y. Finally, we obtain the (FTP) condition, and we are finished.

Note that the proof of S-asymptotic continuity and S-0-continuity along iterates of T presented above, as well as points (1) and (5), are new and necessary to prove the uniqueness of a fixed point in Y. Points (2)–(4) are kept in the original proof in [3] and are also necessary to apply the Darbo fixed point theorem to prove the existence of a fixed point in Y.