Topological Degree via a Degree of Nondensifiability and Applications

Abstract

The goal of this work is to introduce the notion of topological degree via the principle of the degree of nondensifiability (DND for short). We establish some new fixed point theorems, concerning, Schaefer’s fixed point theorem and the nonlinear alternative of Leray–Schauder type. As applications, we study the existence of mild solution of functional semilinear integro-differential equations.

1. Introduction

Fixed point theory is a very active branch of mathematics, and plays a circular role in nonlinear analysis, since it is used for establishing the existence of solutions for many nonlinear problems arising in differential equations and inclusions in physics, economics, mechanics, and biology [1,2,3,4]. In fact, in many real problems, we seek solutions as fixed points of the original problem using hypotheses on the single and multivalued mappings involved in the problem or on the structure of the corresponding Banach space.

In 1930, Kuratowski [5] introduced the concept of measure of noncompactness (MNC), and this technique was used in functional analysis. After that, Darbo [6] developed a result on fixed point theory by using the concept of MNC and the generalized Banach principle of contraction [7]. The concept of MNC and its applications have been generalized in different directions, see [8,9,10,11] for example.

By using the notion of topological degree introduced by Brouwer [12], Leray–Schauder [13] defined this concept for compact perturbation of the identity map. Using the theory of measure of noncompactness, the different generalization of the Leray and Schauder degree was given by Nussbaum [14,15,16].

The theory of fixed point for multivalued applications is an important topic in set-valued analysis. For its developments and applications, one can see [17,18,19].

In 1997, Mora and Cherruault [20] introduced the concept of the $\alpha$-dense curve and densifiable set in metric spaces. This notion is a generalization of a space-filing curve (see [21]), and the class of densifiable sets is strictly comprised between the class of Peano continua and the class of connected and precompacts sets (for more information, see [22,23,24]).

Very recently, several authors have proven some fixed point theorems by using the concept of degree of nondensifiability based on $\alpha$-dense curves, which is an alternative method to $MNC$ to obtain fixed point results (see, e.g., [25,26] and the references therein).

In [27,28], Garcia gives a version Schauder fixed point theorem via DND. This paper contains a new approach to topological degree theory by introducing the concept of “degree of nondensifiability” (DND). Utilizing DND, we established some novel fixed point theorems, including a variant of the Leray–Schauder nonlinear alternative and a new version of Schaefer’s fixed point theorem.

The goal of this work is to introduce the topological degree by using the concept of degree of nondensifiability. This research paper is structured as follows: Section 2 provides the definitions, notations, basic propositions, and theorems from the literature that are used throughout this paper. In Section 3, by using the degree of nondensifiability, we introduced the topological degree of Leray–Schauder type. As an application, we prove Schaefer’s fixed points theorems and nonlinear alternative of Leray and Schauder. In Section 4, we apply our results to a functional semi linear integro-differential equations.

2. Preliminaries

In the first part of this work, we give several notations, definitions, and preliminary results facts that are used later.

Let X be a metric space (or normed space), and set ${\mathcal{P}}_{*}\left(X\right)=\{A\subset X:A\ne \varnothing \}$, ${\mathcal{P}}_{cl}\left(X\right)=\{A\in \mathcal{P}\left(X\right):A$ closed}, ${\mathcal{P}}_b\left(X\right)=\{A\in {\mathcal{P}}_{*}\left(X\right):A$ bounded}, ${\mathcal{P}}_{cv}\left(X\right)=\{A\in {\mathcal{P}}_{*}\left(X\right):A$ convex}, ${\mathcal{P}}_{cp}\left(X\right)=\{A\in {\mathcal{P}}_{*}\left(X\right):A$ compact}, and ${\mathcal{P}}_{arc}\left(X\right)=\{A\in {\mathcal{P}}_{*}\left(X\right):A$ path-connected}.

Let X and Y be two topological spaces, and $\mathcal{S}:X\to \mathcal{P}\ast \left(Y\right)$ be a multifunction. A single-valued function $h:X\to Y$ is called be a selection of $\mathcal{S}$, and we write $h\subset \mathcal{S}$ whenever

$$h\left(x\right)\in \mathcal{S}\left(x\right)\mathrm{for}\mathrm{each}x\in X.$$

$\mathcal{S}$ is considered lower semi-continuous (l.s.c.) if, for each $x_0\in X$, the set $\mathcal{S}\left(x_0\right)$ is a nonempty subset of Y, and if, for every open subset $\mathcal{O}$ of Y, such that $\mathcal{S}\left(x_0\right)\cap \mathcal{O}\ne \varnothing$, there exists an open U, such that $x_0\in U$,

$$\mathcal{S}\left(a\right)\cap \mathcal{O}\ne \varnothing \mathrm{for}\mathrm{every}a\in U.$$

Proposition 1.

Let $\mathcal{S}:X\to \mathcal{P}\left(Y\right)$ be a multivalued mapping. Then, the following statements are equivalent:

1. 

S is $l.s.c$.

2. 

For every open subset $\mathcal{O}$ in Y the sunset

$$S_{-}^{-1}\left(\mathcal{O}\right)=\{a\in X:\mathcal{S}\left(a\right)\cap \mathcal{O}\ne \varnothing \},$$

is an open subset of X.

3. 

For all closed sunset $\mathcal{C}$ in Y the set

$${\mathcal{S}}_{+}^{-1}\left(\mathcal{C}\right)=\{a\in X:\mathcal{S}\left(a\right)\subset \mathcal{C}\},$$

is any closed subset of X.

The concept of the measure of noncompactness permits us to characterize and compare the noncompactness of such sets; for more details in this direction, we refer the readers to [8,9,10,11].

Definition 1.

Consider a complete metric space $(\tilde{X},d)$. A function ${\phi }_{*}:{\mathcal{P}}_b\left(X\right)\to [0,+\infty )$ is assigned as a measure of noncompactness ($MNC$) defined on $\tilde{X}$ if:

(a) 

Regularity: for any $B_{*}\in {\mathcal{P}}_b\left(\tilde{X}\right)$, ${\phi }_{*}\left(B_{*}\right)=0⟺B_{*}$ is a relatively compact set.

(b) 

Invariant under closure: ${\phi }_{*}\left(B_{*}\right)={\phi }_{*}\left({\overline{B}}_{*}\right)$, for any $B_{*}\in {\mathcal{P}}_b\left(\tilde{X}\right)$.

(c) 

Semi-additivity: ${\phi }_{*}(B_1^{*}\cup B_2^{*})=max\{{\phi }_{*}\left(B_1^{*}\right),{\phi }_{*}\left(B_2^{*}\right)\}$, for any $B_1^{*},B_2^{*}\in {\mathcal{P}}_b\left(\tilde{X}\right)$.

The notion of $\alpha$-dense curve was introduced by G. Mora [20] in 1997, but the notion of $DND$ appeared in 2015 as an application of such a theory.

Definition 2.

Consider a metric space $(X,d)$, $\alpha \in {\mathbb{R}}_{+}$ and $B_{*}\in {\mathcal{P}}_b\left(X\right)$ a function $\widehat{\gamma }\in C([0,1],{\mathbb{R}}_{+})$ is called curve $\alpha \text{-}$dense in $B_{*}$ if

• $\widehat{\gamma }\left([0,1]\right)\subset B_{*}$.
• For all $x\in B_{*}$, there exists $\overline{t}\in [0,1]$, such that

  $$d(x,\widehat{\gamma }\left(\overline{t}\right))<\alpha .$$

The bounded subset $B_{*}$ of X is said to be densifiable, if for each $\alpha >0$ we can find an $\alpha \text{-}$dense curve in $B_{*}$.

For any $\alpha \ge 0$ and $B\in {\mathcal{P}}_b\left(X\right)$, we denote the sets $\alpha$-dense curves by ${\mathsf{\Gamma }}_{\alpha ,B.}$

Definition 3.

The function ${\phi }_d:{\mathcal{P}}_b\left(X\right)\to {\mathbb{R}}_{+}$ given by

$${\phi }_d\left(B_{*}\right)=inf\{{\alpha }_{*}\ge 0:{\mathsf{\Gamma }}_{{\alpha }_{*},B_{*}}\ne \varnothing \},B_{*}\in {\mathcal{P}}_b\left(X\right),$$

which defines the degree of nondensifiability (DND).

Remark 1.

• From Definition 3, we deduce

  $${\phi }_d\left(B_{*}\right)\le \delta \left(B_{*}\right)\mathit{for}\mathit{every}{B}_{*}\in {\mathcal{P}}_b\left(X\right),$$

  where $\delta \left(B_{*}\right)=sup\{d(x,y):x,y\in B_{*}\}$. This implies that ${\phi }_d$ is well defined.
• From Hahn–Mazurkiewicz theorem (see, for example, [21,29]), we know that a subset $B_{*}\in {\mathcal{P}}_b\left(X\right)$ is a Peano Continuum, if and only if it is the continuous image of $[0,1]$. So, the DND ${\phi }_d$ quantifies the distance, in the given sense, between B and the class of Peano Continua that it contains.

We give some neutral properties of the DND in the following result proved in [22,26,27].

Proposition 2.

Consider a complete metric space $(X,d)$ and ${\phi }_d$ be $DND$, then

(a) 

Regularity: for any $B_{*}\in {\mathcal{P}}_{b,arc}\left(X\right)$, ${\phi }_d\left(B_{*}\right)=0$ if and only if ${\overline{B}}_{*}$ is a compact set.

(b) 

${\phi }_d\left(B\right)={\phi }_d\left({\overline{B}}_{*}\right)$, for any $B_{*}\in {\mathcal{P}}_{b,arc}\left(X\right)$.

Consider a Banach space X. Then,

(c) 

${\phi }_d(x_0+{\tilde{B}}_{*})={\phi }_d\left({\tilde{B}}_{*}\right)$, for any ${\tilde{B}}_{*}\in {\mathcal{P}}_b\left(X\right)$.

(d) 

${\phi }_d\left(\lambda {\tilde{B}}_{*}\right)=\left|\lambda \right|{\phi }_d\left({\tilde{B}}_{*}\right)$, for all ${\tilde{B}}_{*}\in {\mathcal{P}}_b\left(X\right)$ and $\lambda \in \mathbb{R}$.

(e) 

${\phi }_d\left(co\left({\tilde{B}}_{*}\right)\right)\le {\phi }_d\left({\tilde{B}}_{*}\right)$ for all ${\tilde{B}}_{*}\in {\mathcal{P}}_b\left(X\right)$, $co$ is convex hull.

(f) 

${\phi }_d\left(co({\tilde{B}}_{*}\cup C)\right)\le max({\phi }_d\left(co{\tilde{B}}_{*}\right),{\phi }_d\left(coC\right))$ for all ${\tilde{B}}_{*},C\in {\mathcal{P}}_b\left(X\right)$.

(g) 

${\phi }_d({\tilde{B}}_{*}+C)\le {\phi }_d\left({\tilde{B}}_{*}\right)+{\phi }_d\left(C\right)$, for all ${\tilde{B}}_{*},C\in {\mathcal{P}}_b\left(X\right)$.

(h) 

Generalized Cantor’s intersection theorem: if ${\left({\tilde{B}}_n\right)}_{n\in \mathbb{N}}$ such that for all $n\in \mathbb{N}$, ${\tilde{B}}_{n+1}\subseteq {\tilde{B}}_n$, ${\left({\tilde{B}}_n\right)}_{n\in \mathbb{N}}\in {\mathcal{P}}_{cl,cv,b}\left(X\right)$ and ${\displaystyle \underset{n\to \infty }{lim}{\phi }_d\left({\tilde{B}}_n\right)=0}$, then ${\displaystyle \bigcap _{n=1}^{\infty }{\tilde{B}}_n\in {\mathcal{P}}_{cp}\left(X\right)}$.

Example 1

([30]). Consider a Banach space X and $\overline{B}(0,1)\subset X$ to be a closed unit ball; then,

$${\phi }_d\left(\overline{B}(0,1)\right)=\left\{\begin{array}{ccc}1\hfill & \mathit{if}& \hfill dimX=+\infty ;\\ & & \\ 0\hfill & \mathit{if}& \hfill dimX<\infty .\end{array}\right.$$

Remark 2.

García and Mora ([26], Example 2.1), show that $DND$ is not $MNC$.

Now, we give some relationships between the Kuratowski and Hausdorff $MNC$s and the DND ${\phi }_d$.

Proposition 3

([22,31]). Let X be a metric space and ${\tilde{B}}_{*}\in {\mathcal{P}}_{b,arc}\left(X\right)$. Then,

$$\chi \left({\tilde{B}}_{*}\right)\le {\phi }_d\left({\tilde{B}}_{*}\right)\le 2\chi \left({\tilde{B}}_{*}\right)$$

where χ is a Hausdorff measure defined as follows:

$$\chi \left({\tilde{B}}_{*}\right)=inf\{ϵ>0:{\tilde{B}}_{*}\subset \bigcup _{i=1}^{n}B(x_i,ϵ)\};$$

and

$${\displaystyle \frac{1}{2}}\varpi \left({\tilde{B}}_{*}\right)\le {\phi }_d\left({\tilde{B}}_{*}\right)\le \varpi \left({\tilde{B}}_{*}\right).$$

where ϖ is a Kuratowskii $MNC$ defined by

$$\varpi \left({\tilde{B}}_{*}\right)=inf\{ϵ>0:{\tilde{B}}_{*}\subset \bigcup _{i=1}^n{\tilde{B}}_{*i};\delta \left({\tilde{B}}_{*i}\right)\le ϵ\}.$$

Remark 3.

Notice that for all ${\tilde{B}}_{*}\in {\mathcal{P}}_b\left(X\right)$,

$$\chi \left({\tilde{B}}_{*}\right)\le \varpi \left({\tilde{B}}_{*}\right)\le 2\chi \left({\tilde{B}}_{*}\right).$$

Definition 4.

Consider two metric spaces $\tilde{X},\tilde{Y}$, $C\in {\mathcal{P}}_{*}\left(\tilde{Y}\right)$, $N:\tilde{X}\to \tilde{Y}$, and ${\phi }_d$ is a degree of nondensifiability (DND). The operator N is $k-{\phi }_d-$contraction if

$${\phi }_d\left(N\left({\tilde{B}}_{*}\right)\right)\le k{\phi }_d\left({\tilde{B}}_{*}\right),{\tilde{B}}_{*}\in {\mathcal{P}}_b\left(C\right),$$

where $k\in (0,1)$. The collection of these maps is shown by ${\mathcal{KC}}_k(\tilde{X},\tilde{Y})$. For $\tilde{X}=\tilde{Y}$, ${\mathcal{KC}}_k(\tilde{X},\tilde{Y})={\mathcal{KC}}_k\left(\tilde{X}\right)$.

Theorem 1.

Consider two Banach spaces $\tilde{X}$, $\tilde{Y}$, $C\in {\mathcal{P}}_{b,cl}\left(\tilde{X}\right)$ and $f\in \mathcal{K}(C,\tilde{Y})$; then, there is ${\tilde{f}}_{*}\in \mathcal{K}(\tilde{X},\tilde{Y})$ such that

$${\tilde{f}}_{{*|}_C}=f\mathit{and}{\tilde{f}}_{*}\left(X\right)\subset cof\left(C\right).$$

Proof. 

Let $\mathcal{S}:\tilde{X}\to \mathcal{P}\left(\tilde{Y}\right)$ be a multivalued map defined by

$$\mathcal{S}\left(a\right)=\left\{\begin{array}{ccc}\left\{f\right(a\left)\right\}\hfill & & a\in C\hfill \\ & & \\ \overline{co}f\left(C\right)\hfill & & a\in X\setminus C.\hfill \end{array}\right.$$

It is clear that, for all $a\in \tilde{X}$, $\tilde{S}\left(a\right)\in {\mathcal{P}}_{cl,cv}\left(\tilde{X}\right)$. Now, demonstrate that $\mathcal{S}$ is $l.s.c$. Indeed, let $\mathcal{O}\in {\mathcal{P}}_{cl}\left(\tilde{Y}\right)$, then

$${\mathcal{S}}_{+}^{-1}\left(\mathcal{O}\right)=\{a\in \tilde{X}:\tilde{S}\left(a\right)\subset \mathcal{O}\}.$$

If $\overline{co}f\left(C\right)\subset \mathcal{O}$, hence

$${\mathcal{S}}_{+}^{-1}\left(\mathcal{O}\right)=\tilde{Y}\in {\mathcal{P}}_{cl}\left(\tilde{X}\right).$$

If $\overline{co}f\left(C\right)\neg \subset \tilde{O}$, thus

$${\mathcal{S}}_{+}^{-1}\left(\mathcal{O}\right)=f^{-1}\left(\mathcal{O}\right)\cap C\in {\mathcal{P}}_{cl}\left(\tilde{X}\right).$$

Therefore, $\mathcal{S}$ is $l.s.c$. and, by the Michael Selection Theorem, there is ${\tilde{f}}_{*}\in C(\tilde{X},\tilde{Y})$ such that

$${\tilde{f}}_{*}\left(a\right)\in \mathcal{S}\left(a\right)\mathrm{for}\mathrm{all}a\in \tilde{X}\mathrm{and}{\tilde{f}}_{*}\left(\tilde{X}\right)\subset \overline{co}f\left(C\right).$$

Since f is compact, then, by the Mazur theorem,

$$\overline{co}f\left(C\right)\in {\mathcal{P}}_{cp}\left(\tilde{Y}\right).$$

We conclude that

$${\tilde{f}}_{*}\in \mathcal{K}(\tilde{X},\tilde{Y}),{\tilde{f}}_{{*|}_C}=f\mathrm{and}{\tilde{f}}_{*}\left(\tilde{X}\right)\subset cof\left(C\right).$$

□

Theorem 2

([26]). Consider a Banach space $(X,\Vert ·\Vert )$ and ${\tilde{B}}_{*}\in P_{cv,b}\left(\tilde{X}\right)$. Presume that $N:{\tilde{B}}_{*}⟶{\tilde{B}}_{*}$ is a continuous and k−${\phi }_d$−contraction of measure DND. Then, $FixN\ne \varnothing$.

3. Topological Degree

Using the idea of the degree of nondensifiability, we define the topological degree in this section.

Lemma 1.

Consider a Banach space, where $\mathcal{O}$ an open bounded subset of $\tilde{X}$, $f_{*}\in C(\overline{\mathcal{O}},\tilde{X})\cap {\mathcal{KC}}_{{\displaystyle \frac{k}{2}}}(\overline{\mathcal{O}},\tilde{X})$, and $0\notin (I-f)(\partial \mathcal{O})$. Then, there is ${\tilde{D}}_{*}\in \mathcal{K}\left(\tilde{X}\right)$ and ${\tilde{f}}_{*}\in C(\overline{\mathcal{O}},{\tilde{D}}_{*})$ such that

$${\tilde{f}}_{{*|}_{{\tilde{D}}_{*}\cap \overline{\mathcal{O}}}}=f\mathit{and}{\tilde{f}}_{*}\left(\overline{\mathcal{O}}\right)\subset \overline{co}f({\tilde{D}}_{*}\cap \overline{\mathcal{O}}).$$

Proof. 

We set ${\tilde{D}}_0=\overline{\mathcal{O}}$, and define by induction

$${\tilde{D}}_n=\overline{co}f({\tilde{D}}_{n-1}\cap \overline{\mathcal{O}})\mathrm{for}\mathrm{all}n\in \mathbb{N}.$$

By Proposition 2, we obtain

$$\begin{array}{ccc}\hfill {\phi }_d\left({\tilde{D}}_n\right)& =& {\phi }_d\left(\overline{co}f({\tilde{D}}_{n-1}\cap \overline{\mathcal{O}})\right)\hfill \\ & =& {\phi }_d\left(cof({\tilde{D}}_{n-1}\cap \overline{\mathcal{O}})\right)\hfill \\ & \le & {\phi }_d\left(f({\tilde{D}}_{n-1}\cap \overline{\mathcal{O}})\right)\le \chi \left(f({\tilde{D}}_{n-1}\cap \overline{\mathcal{O}})\right)\hfill \\ & \le & 2\chi \left(f\left({\tilde{D}}_{n-1}\right)\right)\le 2{\phi }_d\left(f\left({\tilde{D}}_{n-1}\right)\right)\hfill \\ & \le & k{\phi }_d\left({\tilde{D}}_{n-1}\right).\hfill \end{array}$$

Therefore,

$${\phi }_d\left({\tilde{D}}_n\right)\le k^n{\varphi }_d\left({\tilde{D}}_1\right)\to 0\mathrm{as}n\to \infty .$$

It is clear that ${\left({\tilde{D}}_n\right)}_{n\in \mathbb{N}}$ is a decreasing sequence of bounded, convex, and nonempty subsets of $\tilde{X}$. By Proposition 2(h), ${\displaystyle {\tilde{D}}_{*}=\bigcap _{n\in \mathbb{N}}{\tilde{D}}_n}$ is nonempty, convex and compact. Also, $f\in \mathcal{K}({\tilde{D}}_{*}\cap \overline{\mathcal{O}},\tilde{X})$. Using Theorem 1, we can find ${\tilde{f}}_{*}\in \mathcal{K}(\overline{\mathcal{O}},{\tilde{D}}_{*})$, such that

$${\tilde{f}}_{{*|}_{\tilde{D}\cap \overline{\mathcal{O}}}}=f\mathrm{and}{\tilde{f}}_{*}\left(\overline{\mathcal{O}}\right)\subset cof(\tilde{D}\cap \overline{\mathcal{O}}).$$

□

Now, we are in a position to give the definition topological degree based on degree of nondensifiability.

Definition 5.

Consider a Banach space $\tilde{X}$, $\tilde{O}\subset \tilde{X}$, nonempty bounded, open and $f\in C(\overline{\mathcal{O}},\tilde{X})\cap {\mathcal{KC}}_{{\displaystyle \frac{k}{2}}}(\overline{\tilde{O}},\tilde{X})$ and $0\notin (I-f)(\partial \mathcal{O})$. We define the degree of $I-f$ by

$$de{g}_d(I-f,\mathcal{O},0)=de{g}_{LS}(I-\tilde{f},\mathcal{O},0),$$

where $de{g}_{LS}$ is the Leray–Schauder degree and ${\tilde{f}}_{*}$ is defined in Lemma 1.

• If $b\notin (I-f)(\partial \mathcal{O})$ we define the topological degree by

  $$de{g}_d(I-f-b,\mathcal{O},b)=de{g}_{LS}(I-\tilde{f}-b,\mathcal{O},b).$$

The next proposition makes Definition 5 meaningful.

Proposition 4.

The degree $de{g}_d$ is well defined.

Proof. 

Let ${\tilde{f}}_{*},\widehat{f}\in K(\overline{\mathcal{O}},\tilde{D})$ such that

$${\tilde{f}}_{{*|}_{\tilde{D}\cap \overline{\mathcal{O}}}}=f={\widehat{f}}_{{|}_{\tilde{D}\cap \overline{\mathcal{O}}}}.$$

We defined the following homotopy application $\mathcal{H}:[0,1]\times \overline{\mathcal{O}}\to \tilde{X}$ given by

$$\mathcal{H}(t,x)=t\tilde{f}\left(x\right)+(1-t)\widehat{f}\left(x\right),x\in \overline{\mathcal{O}}.$$

Let $(t,x)\in [0,1]\times \partial \mathcal{O}$ such that

$$\mathcal{H}(t,x)=x,x\in \partial \mathcal{O}.$$

Since ${\tilde{f}}_{*}$ and $\widehat{f}$ both map $\overline{\mathcal{O}}$ in ${\tilde{D}}_{*}$, then

$$x=t{\tilde{f}}_{*}\left(x\right)+(1-t)\widehat{f}\left(x\right)\in {\tilde{D}}_{*}.$$

For every $x\in {\tilde{D}}_{*}$, we have

$$\tilde{f}\left(x\right)=f\left(x\right)=\widehat{f}\left(x\right),$$

so, $0=x-f\left(x\right)$, which is in contradiction with $0\notin (I-f)(\partial \mathcal{O})$. Therefore,

$$x\ne \mathcal{H}(t,x)\mathrm{for}\mathrm{all}(t,x)\in [0,1]\times \partial \mathcal{O}.$$

By using the homotopy invariance of the Leray–Schauder degree, we can conclude that

$$deg(I-{\tilde{f}}_{*},\mathcal{O},0)=deg(I-\widehat{f},\mathcal{O},0).$$

□

The topological degree via a degree of nondensifiability in normed space conserves the basic features of the Leray–Schauder degree.

Theorem 3.

Let $\tilde{X}$ be a Banach space, $\mathcal{O}\subset X$ be an open bounded subset, and $f:\overline{\mathcal{O}}⟶\tilde{X}$ be a continuous ${\displaystyle \frac{k}{2}}-{\phi }_d-$contractions map. If $b\notin (I-f)(\partial \mathcal{O})$, then there exists an integer ${\mathit{deg}}_d(I-f,\mathcal{O},b)$ satisfying the following properties:

$\left(1\right)$

(Solvability) ${\mathit{deg}}_d(f,\mathcal{O},b)\ne 0$ then $x-f\left(x\right)=b$ has a solution in $\mathcal{O}$;

$\left(2\right)$

(Additivity). Suppose that ${\mathcal{O}}_1,{\mathcal{O}}_2$ are two disjoint open subsets of $\mathcal{O}$ and $b\notin f(\overline{\mathcal{O}}\setminus {\mathcal{O}}_1\cup {\mathcal{O}}_2)$. Then,

$${\mathit{deg}}_d(I-f,\mathcal{O},b)={\mathit{deg}}_d(I-f,{\mathcal{O}}_1,b)+{\mathit{deg}}_d(I-f,{\mathcal{O}}_2,b);$$

$\left(3\right)$

(Homotopy invariance) $If\mathcal{H}:\left[01\right]\times \overline{\mathcal{O}}⟶\tilde{X}$ is continuous, ${\displaystyle b\notin \bigcup _{t\in [0,1]}(I-\mathcal{H}(t,\partial \mathcal{O}))}$ and

$${\phi }_d\left(\mathcal{H}(t,B)\right)\le {\displaystyle \frac{k}{2}}{\phi }_d\left(B\right)\mathit{for}\mathit{all}B\in {\mathcal{P}}_b\left(\overline{\mathcal{O}}\right),$$

then ${\mathit{deg}}_d(I-\mathcal{H}(t,·),\mathcal{O},b)$ does not depend on $t\in [0,1]$;

$\left(4\right)$

${\mathit{deg}}_d(I-f,\mathcal{O},b)$ is a constant on each connected component of $X\setminus f(\partial \mathcal{O})$;

$\left(5\right)$

${\mathit{deg}}_d(I-f+y_0,\mathcal{O},b+y_0)={\mathit{deg}}_d(I-f,\mathcal{O},b)$ for all $y_0\in X$;

$\left(6\right)$

Let $g\in C(\overline{\mathcal{O}},\tilde{X})$ be a compact application. If $f_{{|}_{\partial \mathcal{O}}}=g_{{|}_{\partial \mathcal{O}}}$, then

$${\mathit{deg}}_d(I-f,\mathcal{O},b)={\mathit{deg}}_d(I-g,\mathcal{O},b);$$

$\left(7\right)$

(Excision). Let $K\subset \overline{\mathcal{O}}$ is closed and $b\notin f\left(K\right)$, then

$${\mathit{deg}}_d(I-f,\mathcal{O},b)={\mathit{deg}}_d(I-f,\mathcal{O}\setminus K,b).$$

Some ramifications for this topological degree concept.

Theorem 4.

Consider a Banach space $\tilde{X}$, where $\mathcal{O}$ is an open bounded subset of $\tilde{X}$ with $0\in \mathcal{O}$ and $N:\overline{\mathcal{O}}\to \tilde{X}$ is a continuous ${\displaystyle \frac{k}{2}}-{\phi }_d-$contraction map. Suppose the following Leray–Schauder condition:

$$x\ne \lambda N\left(x\right)\mathrm{for}\mathrm{every}\lambda \in [0,1),x\in \partial \mathcal{O}.$$

Then, N possesses at least fixed points.

Proof. 

Let $\mathcal{H}:[0,1]\times \overline{\mathcal{O}}\to \tilde{X}$ a homotopy given by

$$\mathcal{H}(t,x)=x-tN\left(x\right),(t,x)\in [0,1]\times \overline{\mathcal{O}}.$$

By the Leray–Schauder condition, we have

$$x\ne \lambda \mathcal{H}(t,x),\mathrm{for}\mathrm{all}t\in [0,1),x\in \partial \mathcal{O}.$$

Then,

$$de{g}_d(x-N\left(x\right),\mathcal{O},0)=de{g}_{d}d(I,\mathcal{O},0)=1.$$

According to Theorem 3, there is $x\in \mathcal{O}$ such that

$$N\left(x\right)=x.$$

□

Next, as a result, we present the version Schaefer’s fixed point type.

Theorem 5.

Consider a Banach space $\tilde{X}$ and a continuous map $N:\tilde{X}\to \tilde{X}$ and ${\displaystyle \frac{k}{2}}-{\phi }_d-$contraction map. Then, one of the following statements holds:

1. 

$\mathcal{E}\left(N\right)=\{x\in \tilde{X}:\lambda Nx=x;\lambda \in (0,1)\}$ unbounded.

2. 

$x=Nx$ possesses at least one solution.

Proof. 

Assume that $\mathcal{E}\left(N\right)$ is bounded, then there exists $r>0$, $\mathcal{E}\left(N\right)\subset B(0,r)$. If $N\left(x\right)=x$ for some $x\in B(0,r)$, then $\left(2\right)$ holds. Suppose that

$$N\left(x\right)\ne x\mathrm{for}\mathrm{all}x\in \partial B(0,r).$$

Similar to how Theorem 4 is proven,

$$de{g}_d(I-N,B(0,r),0)=1.$$

Consequently, N possess a fixed point. □

Theorem 6.

Consider a Banach space $(\tilde{X},\Vert ·\Vert )$, $\mathcal{O}\subset \tilde{X}$ bounded nonempty subset with $0\in \mathcal{O}$, where $N\in C(\overline{\mathcal{O}}\to \tilde{X})$ is a ${\displaystyle \frac{k}{2}}-\phi -$contraction application and $L:\partial \mathcal{O}\to X$ compact continuous mapping. Suppose

$$\lambda N\left(x\right)+(1-{\lambda }_{*})L\left(x\right)\ne x\mathit{and}\Vert x\Vert \le \Vert L\left(x\right)\Vert \mathit{for}\mathit{each}x\in \partial \mathcal{O},{\lambda }_{*}\in [0,1].$$

(1)

Then,

$${\mathit{deg}}_d(I-N,\mathcal{O},0)=0.$$

Proof. 

By Theorem 1, there exists ${\overline{S}}_{*}:\overline{\mathcal{O}}\to C$ a compact, continuous mapping such that ${\overline{S}}_{{*|}_{\partial \mathcal{O}}}={\overline{S}}_{*}$. Let $\mathcal{H}:[0,1]\times \overline{\mathcal{O}}\to \tilde{X}$ be a mapping of homotopy provided by

$$\mathcal{H}(t,x)=\lambda N\left(x\right)+(1-\lambda ){\overline{S}}_{*}\left(x\right),(t,x)\in [0,1]\times \overline{\mathcal{O}}.$$

From Condition (1), we have

$$0\notin (I-L)(\partial \mathcal{O})\mathrm{and}0\notin (I-\mathcal{H}(t,\partial \mathcal{O}\left)\right)\mathrm{for}\mathrm{all}x\in \partial \mathcal{O}.$$

Thus,

$${\mathrm{deg}}_d(I-N,\mathcal{O},0)={\mathrm{deg}}_d(I-{\overline{S}}_{*},\mathcal{O},0)={\mathrm{deg}}_{LS}(I-{\overline{S}}_{*},\mathcal{O},0).$$

By [32], (Lemma 2.2.11), we obtain

$${\mathrm{deg}}_{LS}(I-{\overline{S}}_{*},\mathcal{O},0)=0.$$

□

Theorem 7.

Let X be a infinite dimensional Banach space, and $U_i\subset X,i=1,2$ be two open bounded nonempty subset of X, $0\in U_1\subset U_2,x_0\in X$, such that

$$\underset{x\in \partial U_2}{sup}\Vert x\Vert \le \Vert x_0\Vert .$$

Let $N:{\overline{U}}_2\to X$ be a continuous ${\displaystyle \frac{k}{2}}-{\phi }_d-$contraction map satisfying

$\left({\mathcal{H}}_1\right)$

$\Vert x\Vert \le \Vert N\left(x\right)\Vert$ and $\lambda N\left(x\right)\ne x+(1-\lambda )x_0$ for all $x\in \partial U_2$.

$\left({\mathcal{H}}_2\right)$

$\Vert N\left(x\right)\Vert \le \Vert x\Vert$ for all $x\in \partial U_1$.

Then, there is $x\in {\overline{U}}_2\setminus U_1$ with $x=N\left(x\right)$.

Proof. 

It is clear that $U_2\setminus {\overline{U}}_1$ is a open set and $U_2=U_2\setminus {\overline{U}}_1\cup U_1$. Then, by Theorem 3,

$$de{g}_d(I-N,U_2\setminus {\overline{U}}_1,0)=de{g}_d(I-N,U_2,0)-de{g}_d(I-N,U_1,0).$$

By $\left({\mathcal{H}}_1\right)$, we can apply Theorem 6,

$$de{g}_d(I-N,U_2,0)=de{g}_{LS}(I-\tilde{N},U_2,0)=0.$$

By $\left({\mathcal{H}}_2\right)$, we have

$$x-\lambda N\left(x\right)\ne x,\lambda \in [0,1]andx\in \partial U_1.$$

Then, by homotopy proprieties of topological degree, we obtain

$$de{g}_d(I-N,U_1,0)=de{g}_d(I,U_2,0)=1,$$

which implies $de{g}_d(I-N,U_2\setminus {\overline{U}}_1,0)=-1$; hence, there is $x\in U_2\setminus {\overline{U}}_1$ satisfying

$$x=N\left(x\right).$$

□

4. Semilinear Integro-Differential Equations with Finite Delay

In this section, we consider the following semi-linear functional differential equation problem:

$$\left\{\begin{array}{c}{\displaystyle {\mathcal{U}}^{\prime }\left(t\right)=\mathcal{A}\mathcal{U}\left(t\right)+{\int }_0^{t}S(t-s)\mathcal{U}\left(s\right)ds+f(t,{\mathcal{U}}_t),\mathrm{a}.\mathrm{e}.t\in [0,b],}\hfill \\ \mathcal{U}\left(t\right)={\varphi }_{*}\left(t\right),t\in [-r,0],\hfill \end{array}\right.$$

(2)

where $\mathcal{A}$ is the infinitesimal generator of a $C_0-\mathrm{semigroup}$${\left\{T\left(t\right)\right\}}_{t\ge 0}$ on separable Banach space $(\mathcal{E},\Vert ·\Vert )$ with domain $\mathcal{D}\left(\mathcal{A}\right)$, $f:[0,b]\times \mathcal{C}\left(\right[-r,0],\mathcal{E})\times \mathcal{E}\to \mathcal{E}$, is a given function, and ${\varphi }_{*}\in \mathcal{C}([-r,0],\mathcal{E})$. For any $t\in [0,b]$, $S\left(t\right)$ is a closed linear operator on $\mathcal{E}$, with domain $\mathcal{D}\left(\mathcal{A}\right)\subset \mathcal{D}\left(S\right(t\left)\right)$, which is independent of t. For any function $\mathcal{U}:[-r,b]\to \mathcal{E}$ and any $t\in [0,b]$, we denote by ${\mathcal{U}}_t$ the element of $\mathcal{C}\left(\right[-r,0],\mathcal{E})$ defined by

$${\mathcal{U}}_t\left(\theta \right)=U(t+\theta ),\theta \in [-r,0].$$

Here, ${\mathcal{U}}_t(·)$ represents the history of the state from time $t-r$ up to the present time t.

The existence of mild solutions for integro-differential equations in infinite dimensional space has been investigated (see, e.g., [33,34,35,36,37,38] and the references therein).

Here, we investigate the existence of the mild solutions for the above partial integro-differential evolution equations with finite delay where the semi group is not necessarily compact.

Existence Result

We recall some knowledge on resolvent operators in Banach space.

Definition 6

([39]). A family of bounded linear operators ${\left(R\left(t\right)\right)}_{t\in {\mathbb{R}}_{+}}\in \mathcal{L}\left(\mathcal{E}\right)$ is called a resolvent operator associated with (2) if

(a) 

$R\left(0\right)$ is the identity map and $\Vert R\left(t\right)\Vert \le M{e}^{\beta t}$ for a certain a positive real constant M and $\beta \in \mathbb{R}$.

(b) 

For each $\mathcal{U}\in \mathcal{E}$, $R\left(t\right)\mathcal{U}$ is strongly continuous.

(c) 

For all $\mathcal{U}\in \mathcal{E}$, $t\to R\left(t\right)U$ s continuously differentiable, and

$$\begin{array}{cc}{R}^{\prime }\left(t\right)\mathcal{U}& {\displaystyle =AR\left(t\right)u+{\int }_0^{t}S(t-s)R\left(s\right)\mathcal{U}ds}\hfill \\ & {\displaystyle =R\left(t\right)\mathcal{A}\mathcal{U}+{\int }_0^{t}R(t-s)S\left(s\right)\mathcal{U}ds.}\end{array}$$

Theorem 8

([39]). Let ${\varphi }_{*}\left(0\right)\in D\left(A\right)$. Then, $\left(2\right)$ possesses a resolvent operator. Also, if $\mathcal{U}$ is a solution of $\left(2\right)$, then

$$\mathcal{U}\left(t\right)=\left\{\begin{array}{cc}{\displaystyle R\left(t\right){\varphi }_{*}\left(0\right)+{\int }_0^{t}R(t-s)f(s,{\mathcal{U}}_s)ds,}\hfill & t\in [0,b],\hfill \\ & \\ {\varphi }_{*}\left(t\right),\hfill & t\in [-r,0].\hfill \end{array}\right.$$

Definition 7.

A function $\mathcal{U}\in \mathcal{C}\left(\right[-r,b],\mathcal{E})$ is called to be a mild solution of Problem (2) if

$$U\left(t\right)=\left\{\begin{array}{cc}{\displaystyle R\left(t\right){\varphi }_{*}\left(0\right)+{\int }_0^{t}R(t-s)f(s,{\mathcal{U}}_s)ds,}\hfill & t\in [0,b],\hfill \\ \hfill & \\ {\varphi }_{*}\left(t\right),\hfill & t\in [-r,0].\hfill \end{array}\right.$$

(3)

In the following, we will need the following lemma.

Lemma 2

([27]). Let $B_{*}\in {\mathcal{P}}_b\left(\mathcal{C}([0,c],\mathcal{E})\right)$, then

$$sup\{{\phi }_d\left(\{x_{*}\left(t\right):x_{*}\in B_{*}\}\right):t\in [0,c]\}\le {\phi }_d\left(B_{*}\right).$$

In order to give the existence result of the problem (2), we shall need the following hypotheses:

$\left({\overline{\mathcal{H}}}_1\right)$

f is a is Carathéodory from $[0,b]\times \mathcal{C}\left(\right[-r,0],\mathcal{E})\times \mathcal{E}$ to $\mathcal{E}$.

$\left({\overline{\mathcal{H}}}_2\right)$

${\left\{T\left(t\right)\right\}}_{t\ge 0}$ is operator-norm continuous for $t>0$.

$\left({\overline{\mathcal{H}}}_3\right)$

There exists $h\in L^1([0,b],{\mathbb{R}}_{+})$ such that

$$\Vert f(t,\xi )\Vert \le h\left(t\right)(\Vert \xi {\Vert }_{\infty }+1),\mathrm{for}\mathrm{every}\xi \in \mathcal{C}([-r,0],\mathcal{E}),\mathrm{a}.\mathrm{e}.t\in [0,b].$$

$\left({\overline{\mathcal{H}}}_4\right)$

There exists $g\in L^1([0,b],{\mathbb{R}}_{+})$ such that for all $D_{*}\in {\mathcal{P}}_b\left(\mathcal{C}([-r,0],\mathcal{E})\right)$, we have

$${\phi }_d\left(f(t,D_{*})\right)\le g\left(t\right)\left(\underset{\theta \in [-r,0]}{sup}{\phi }_d\left(D\left(\theta \right)\right)\right),\mathrm{for}\mathrm{a}.\mathrm{e}.t\in [0,b]$$

where

$$D\left(\theta \right)=\left\{\varphi \right(\theta ):\varphi \in D\},\theta \in [-r,0].$$

Theorem 9.

If $\left({\overline{\mathcal{H}}}_1\right)-\left({\overline{\mathcal{H}}}_4\right)$ hold, then Problem (2) admits at least one mild solution.

Proof. 

Let us consider the operator $N:\mathcal{C}\left(\right[-r,b],E)\to \mathcal{C}\left(\right[-r,b],E)$ as follows:

$$\left(N\mathcal{U}\right)\left(t\right)=\left\{\begin{array}{cc}{\displaystyle R\left(t\right){\varphi }_{*}\left(0\right)+{\int }_0^{t}R(t-s)f(s,{\mathcal{U}}_s)ds,}\hfill & t\in [0,b],\hfill \\ \hfill & \hfill \\ {\varphi }_{*}\left(t\right),\hfill & t\in [-r,0].\hfill \end{array}\right.$$

This is evidence that the fixed points of N are solutions of Problem (2). Utilizing Banach’s fixed point theorem, we prove that N possesses a fixed point.

• Step 1: Demonstrating the continuity of N.

Indeed, let ${\left({\mathcal{U}}^{\left(n\right)}\right)}_{n\in \mathbb{N}}\subset \mathcal{C}([-r,b],\mathcal{E})$ such that ${\mathcal{U}}^{\left(n\right)}\to \mathcal{U}$ in $\mathcal{C}\left(\right[-r,b],\mathcal{E})$. Then,

$$\begin{array}{cc}\Vert \left(N{\mathcal{U}}^{\left(n\right)}\right)\left(t\right)-\left(N\mathcal{U}\right)\left(t\right)\Vert \hfill & =‖{\displaystyle {\int }_0^{t}R(t-s)f(s,{\mathcal{U}}_s^{\left(n\right)})ds-{\int }_0^{t}R(t-s)f(s,{\mathcal{U}}_s)ds}‖\hfill \\ \hfill & \\ \hfill & {\displaystyle \le {\int }_0^t\Vert R(t-s)\Vert ‖f(s,{\mathcal{U}}_s^{\left(n\right)})-f(s,{\mathcal{U}}_s)‖ds.}\hfill \end{array}$$

The sequence ${\left(f_n\right)}_{n\in \mathbb{N}}$, defined by $f_n:t\in [0,b]\to f_n\left(t\right)=f(t,{\mathcal{U}}_t^{\left(n\right)})$, satisfies the conditions of the Lebesgue’s theorem. Indeed,

• Utilizing that f is a Carathéodory function and by the separability of $\mathcal{C}\left(\right[-r,0],\mathcal{E})\times \mathcal{E}$, we deduce that $f(·,·)$ is measurable. So, for any $n\in \mathbb{N}$, the function $f_n(·)$.
• Since the sequence ${\left({\mathcal{U}}^{\left(n\right)}\right)}_{n\in \mathbb{N}}$ converges to $\mathcal{U}$ in $\mathcal{C}\left(\right[0,b],\mathcal{E})$ and f is a Carathéodory function, then there exists $M_{*}>0$ such that

  $$\Vert {\mathcal{U}}_n{\Vert }_{\infty }\le M_{*},f_n\left(t\right)⟶f(t,{\mathcal{U}}_t)n\to \infty ,\mathrm{for}\mathrm{all}t\in [0,b],$$

  and from $\left({\overline{\mathcal{H}}}_3\right)$,

  $$\Vert f_n\left(t\right)\Vert \le (M_{*}+1)h\left(t\right),\mathrm{for}\mathrm{all}n\in \mathbb{N},t\in [0,b].$$

By the Lebesgue theorem, we obtain

$$\Vert N{\mathcal{U}}_n{-N\mathcal{U}\Vert }_{\infty }\le M{e}^{\left|\beta \right|b}{\int }_0^b\Vert f_n(s,{\mathcal{U}}_s^n)-f(s,{\mathcal{U}}_s)\Vert ds\to 0n\to \infty .$$

• Step 2: N transforms bounded sets of $\mathcal{C}\left(\right[-r,b],\mathcal{E})$ into bounded subsets of $\mathcal{C}\left(\right[-r,b],\mathcal{E})$.

In fact, by letting $r>0$ and $B_r={\{u\in \mathcal{C}([-r,b],\mathcal{E})/\Vert u\Vert }_{\infty }\le r\}$, we show that $N\left(B_r\right)$ is bounded. By Definition (6) and using the hypothesis $\left({\overline{\mathcal{H}}}_4\right)$ and $\left({\overline{\mathcal{H}}}_2\right)$, we obtain, for any $t\in [0,b]$,

$$\begin{array}{cc}\Vert \left(N\mathcal{U}\right)\left(t\right)\Vert \hfill & {\displaystyle \le M{e}^{t\beta }\Vert \varphi \left(0\right)\Vert +M{e}^{t\beta }{\int }_0^t{e}^{-s\beta }\Vert f(s,{\mathcal{U}}_s)\Vert ds.}\hfill \end{array}$$

Thus,

$${\Vert N\left(\mathcal{U}\right)\Vert }_{\infty }\le M{e}^{\left|\beta \right|b}\left({\Vert \varphi \Vert }_{\infty }+(r+1){\int }_0^b{e}^{-s\beta }h\left(s\right)ds\right)=l_1.$$

Therefore, $N\left(B_r\right)$ is bounded.

• Step 3: Proving that N is ${\phi }_d-$contractive.

Now, let $B\in {\mathcal{P}}_b\left(\mathcal{C}([-r,b],\mathcal{E})\right)$ and, for each $s\in [0,b]$, we obtain ${\displaystyle {\beta }_s=\underset{\theta \in [-r,0]}{sup}{\phi }_d\left(\{{\mathcal{U}}_s\left(\theta \right):\mathcal{U}\in B\}\right)}$; then, by $\left(C_2\right)$, we have

$${\phi }_d\left(\{f(s,{\mathcal{U}}_s):\mathcal{U}\in B\}\right)\le g\left(s\right){\beta }_s.$$

Thus, for $s\in [0,b]$ and $ϵ>0$ there exists ${\tilde{\gamma }}_s\in C([0,1],\mathcal{E})$ such that

$${\tilde{\gamma }}_s\left([0,1]\right)\subset \{f(s,{\mathcal{U}}_s):\mathcal{U}\in B\}$$

and for every $\mathcal{U}\in B$, there exists $\tau \in [0,1]$ with

$$\Vert f(s,{\mathcal{U}}_s)-{\tilde{\gamma }}_s\left(\tau \right)\Vert \le g\left(s\right){\beta }_s+ϵ.$$

Thus, for ${\tau }_{*}>0$, we obtain

$$e^{-\left|\beta \right|s-{\tau }_{*}{\int }_0^{s}g\left(t\right)dt}\Vert f(s,{\mathcal{U}}_s)-{\tilde{\gamma }}_s\left(\tau \right)\Vert \le g\left(s\right){\beta }_s+ϵ.$$

We define the following mapping ${\gamma }_{*}:[0,1]\to \mathcal{C}([-r,b],\mathcal{E})$ as follows:

$$\left({\gamma }_{*}\left(\tau \right)\right)\left(t\right)=\left\{\begin{array}{cc}{\displaystyle R\left(t\right){\varphi }_{*}\left(0\right)+{\int }_0^{t}R(t-s){\tilde{\gamma }}_s\left(\tau \right)ds,}\hfill & t\in [0,b],\hfill \\ \hfill & \hfill \\ {\varphi }_{*}\left(t\right),\hfill & t\in [-r,0].\hfill \end{array}\right.$$

It clear that ${\gamma }_{*}$ is a continuous mapping and ${\gamma }_{*}\left([0,1]\right)\subset N\left(B\right)$. Furthermore, for ally $\mathcal{U}\in B$, there exists $\tau \in [0,1]$ such that

$$\begin{array}{ccc}\hfill \Vert \left(N\mathcal{U}\right)\left(t\right)-\left({\gamma }_{*}\left(\tau \right)\right)\left(t\right)\Vert & \le & M{\int }_0^t{e}^{\beta (-s+t)}{e}^{{\tau }_{*}{\int }_0^{s}g\left(\xi \right)d\xi }{e}^{-{\tau }_{*}{\int }_0^{s}g\left(\xi \right)d\xi }\Vert f(s,{\mathcal{U}}_s)-{\tilde{\gamma }}_s\left(\tau \right)\Vert ds\hfill \\ & \le & M{\int }_0^t{e}^{\left|\beta \right|(-s+t)}{e}^{{\tau }_{*}{\int }_0^{s}g\left(\xi \right)d\xi }{e}^{-{\tau }_{*}{\int }_0^{s}g\left(\xi \right)d\xi }\Vert f(s,{\mathcal{U}}_s)-{\tilde{\gamma }}_s\left(\tau \right)\Vert ds\hfill \\ & \le & M{\int }_0^t{e}^{\left|\beta \right|(-s+t)}{e}^{{\tau }_{*}{\int }_0^{s}g\left(\xi \right)d\xi }{e}^{-{\tau }_{*}{\int }_0^{s}g\left(\xi \right)d\xi }\Vert f(s,{\mathcal{U}}_s)-{\tilde{\gamma }}_s\left(\tau \right)\Vert ds\hfill \\ & \le & e^{\left|\beta \right|t}M{\int }_0^{t}g\left(s\right)e^{{\tau }_{*}{\int }_0^{s}g\left(\xi \right)d\xi }ds{\phi }_d\left(B\right)+ϵe^{\left|\beta \right|t}M{\int }_0^t{e}^{{\tau }_{*}{\int }_0^{s}g\left(\xi \right)d\xi }ds.\hfill \end{array}$$

Thus,

$$\begin{array}{ccc}\hfill e^{-\left|\beta \right|t-{\tau }_{*}{\int }_0^{s}g\left(\xi \right)d\xi }\Vert \left(N\mathcal{U}\right)\left(t\right)-\left({\gamma }_{*}\left(\tau \right)\right)\left(t\right)\Vert & \le & {\displaystyle \frac{M}{{\tau }_{*}}}{\phi }_d\left(B\right)+ϵMb.\hfill \end{array}$$

Therefore,

$$\begin{array}{ccc}\hfill \Vert \left(N\mathcal{U}\right)-\left({\gamma }_{*}\left(\tau \right)\right){\Vert }_{*}& \le & {\displaystyle \frac{M}{{\tau }_{*}}}{\phi }_d\left(B\right)+ϵMb,\hfill \end{array}$$

where

$${\Vert \mathcal{U}\Vert }_{*}=\underset{t\in [-r,b]}{sup}{e}^{-\left|\beta \right|t-{\tau }_{*}{\int }_0^{s}g\left(\xi \right)d\xi }\Vert \mathcal{U}\left(t\right)\Vert .$$

Letting $ϵ\to 0$, we obtain

$$\begin{array}{ccc}\hfill \Vert N\left(\mathcal{U}\right)-{\gamma }_{*}{\left(\tau \right)\Vert }_{*}& \le & {\displaystyle \frac{M}{{\tau }_{*}}}\alpha .\hfill \end{array}$$

Since ${\tau }_{*}$ is arbitrary, we choose ${\tau }_{*}$ such that $0<{\displaystyle \frac{2M}{{\tau }_{*}}}<1$, then

$${\phi }_d\left(B\right)\le {\displaystyle \frac{k}{2}}{\phi }_d\left(B\right),k={\displaystyle \frac{2M}{{\tau }_{*}}}B\in {\mathcal{P}}_b\left(\mathcal{C}([-r,b],\mathcal{E})\right).$$

Consequently, N is a ${\displaystyle \frac{k}{2}}-{\phi }_d-$contraction. It is clear that $(\mathcal{C}([-r,b],\mathcal{E}),\Vert ·{\Vert }_{*})$ is a Banach space, and ${N:(\mathcal{C}([-r,b],\mathcal{E}),\Vert ·\Vert }_{*}{)\to (\mathcal{C}([-r,b],\mathcal{E}),\Vert ·\Vert }_{*})$ are continuous operators.

• Step 4: A priori estimation.

Let $\mathcal{U}\in \mathcal{C}\left(\right[-r,b],\mathcal{E})$ such that $\mathcal{U}=\lambda N\left(\mathcal{U}\right)$, and $0<\lambda <1$. Then, for all $t\in [0,b]$, we obtain

$$\begin{array}{ccc}\hfill \mathcal{U}\left(t\right)& =& \lambda \left(R\left(t\right){\varphi }_{*}\left(0\right){\int }_0^{t}R(t-s)f(s,{\mathcal{U}}_s)ds\right),u\left(t\right)=\varphi \left(t\right),t\in [-r,0].\hfill \end{array}$$

Thus,

$$\begin{array}{ccc}\hfill \Vert \mathcal{U}\left(t\right)\Vert & \le & M{e}^{\beta t}{\Vert \varphi \Vert }_{\infty }+{\int }_0^{t}M{e}^{\beta (-s+t)}\Vert f(s,{\mathcal{U}}_s)\Vert ds\hfill \\ & \le & M{e}^{\beta t}{\Vert \varphi \Vert }_{\infty }+{\int }_0^{t}M{e}^{\beta (-s+t)}h\left(s\right)(\Vert {\mathcal{U}}_s{\Vert }_{\infty }+1)ds\hfill \\ & \le & M{e}^{\left|\beta \right|t}{\Vert \varphi \Vert }_{\infty }+M{e}^{\left|\beta \right|b}{\Vert h\Vert }_{L^1}+{\int }_0^{t}M{e}^{\beta (-s+t)}h\left(s\right){\Vert {\mathcal{U}}_s\Vert }_{\infty }ds\hfill \\ & \le & M{e}^{\left|\beta \right|b}{\Vert \varphi \Vert }_{\infty }+M{e}^{\left|\beta \right|b}{\Vert h\Vert }_{L^1}+{\int }_0^{t}M{e}^{\beta (-s+t)}h\left(s\right)\underset{\theta \in [-r,0]}{sup}\Vert \mathcal{U}(\theta +s)\Vert ds.\hfill \end{array}$$

Therefore,

$$\Vert \mathcal{U}\left(t\right)\Vert \le M{e}^{\left|\beta \right|b}{\Vert \varphi \Vert }_{\infty }+M{e}^{\left|\beta \right|b}{\Vert h\Vert }_{L^1}+{\int }_0^{t}M{e}^{\beta (-s+t)}h\left(s\right)\underset{\theta \in [-r,0]}{sup}\Vert \mathcal{U}(\theta +s)\Vert ds.$$

(4)

Set ${\displaystyle \mathcal{V}\left(t\right)=\underset{s\in [0,t]}{sup}\Vert \mathcal{U}\left(s\right)\Vert }$. Hence, the inequality (4) implies

$$\mathcal{V}\left(t\right)\le B_{*}+M{e}^{\left|\beta \right|b}{\int }_0^{t}h\left(s\right)\mathcal{V}\left(s\right)ds,$$

where

$$B_{*}=M{e}^{\left|\beta \right|b}{\Vert \varphi \Vert }_{\infty }+M{e}^{\left|\beta \right|b}{\Vert \varphi \Vert }_{\infty }{\Vert h\Vert }_{L^1}.$$

Gronwall’s inequality,

$$\mathcal{V}\left(t\right)\le B_{*}exp\left(M{e}^{\left|\beta \right|b}{\int }_0^{t}h\left(s\right)ds\right),t\in [-r,b].$$

Therefore,

$${\Vert \mathcal{U}\Vert }_{\infty }\le B_{*}exp\left(M{e}^{\left|\beta \right|b}{\int }_0^{b}h\left(s\right)ds\right):={\overline{M}}_{*}.$$

Set

$$\mathfrak{F}=\{\mathcal{U}\in \mathcal{C}([-r,b],\mathcal{E}):\Vert \mathcal{U}{\Vert }_{*}<M_{*}+1\}.$$

So, $0\in \mathfrak{F}$ is a bounded open. Then, from above steps, we can conclude that N satisfies all the conditions of Theorem 4. As a result, N has at least a fixed point $\mathcal{U}\in \mathfrak{F}$, representing the solution to Problem (2). □

5. Conclusions

This paper contains a new approach to topological degree theory by introducing the concept of “degree of nondensifiability” (DND). Utilizing DND, we established some novel fixed point theorems, including a variant of the Leray–Schauder nonlinear alternative and a new version of Schauder’s fixed point theorem. In the end, this work shows that, without the compactness of the Nemytskii operator, some class of semi linear integro differential with delay hast at least one solution under some sufficient conditions. I hope that these results extend some previous ones in the literature.