Topological Degree via a Degree of Nondensifiability and Applications

: 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.


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 setvalued analysis.For its developments and applications, one can see [17][18][19].
In 1997, Mora and Cherruault [20] introduced the concept of the α-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 α-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.

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 Let X and Y be two topological spaces, and S : X → P * (Y) be a multifunction.A single-valued function h : X → Y is called be a selection of S, and we write h ⊂ S whenever h(x) ∈ S(x) for each x ∈ X.
S is considered lower semi-continuous (l.s.c.) if, for each x 0 ∈ X, the set S(x 0 ) is a nonempty subset of Y, and if, for every open subset O of Y, such that S(x 0 ) ∩ O ̸ = ∅, there exists an open U, such that x 0 ∈ U, S(a) ∩ O ̸ = ∅ for every a ∈ U. Proposition 1.Let S : X → P (Y) be a multivalued mapping.Then, the following statements are equivalent: S is l.s.c.

2.
For every open subset O in Y the sunset is an open subset of X.

3.
For all closed sunset C in Y the set 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].
The notion of α-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.
The bounded subset B * of X is said to be densifiable, if for each α > 0 we can find an α-dense curve in B * .
For any α ≥ 0 and B ∈ P b (X), we denote the sets α-dense curves by Γ α,B.
Definition 3. The function φ d : P b (X) → R + given by which defines the degree of nondensifiability (DND).

Remark 1.
• From Definition 3, we deduce where δ(B * ) = sup{d(x, y) : x, y ∈ B * }.This implies that φ d is well defined.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 φ d be DND, then (a) Regularity: for any B * ∈ P b,arc (X), Consider a Banach space X.Then, Example 1 ([30]).Consider a Banach space X and B(0, 1) ⊂ X to be a closed unit ball; then, 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 MNCs and the DND φ d .Proposition 3 ([22,31]).Let X be a metric space and B * ∈ P b,arc (X).Then, where χ is a Hausdorff measure defined as follows: where ϖ is a Kuratowskii MNC defined by where k ∈ (0, 1).The collection of these maps is shown by Proof.Let S : X → P ( Y) be a multivalued map defined by It is clear that, for all a ∈ X, S(a Therefore, S is l.s.c.and, by the Michael Selection Theorem, there is f * ∈ C( X, Y) such that f * (a) ∈ S(a) for all a ∈ X and f * ( X) ⊂ co f (C).

Topological Degree
Using the idea of the degree of nondensifiability, we define the topological degree in this section.
Proof.We set D 0 = O, and define by induction By Proposition 2, we obtain It is clear that ( D n ) n∈N is a decreasing sequence of bounded, convex, and nonempty subsets of X.By Proposition 2(h), D * = n∈N D n is nonempty, convex and compact.Also, We defined the following homotopy application Since f * and f both map O in D * , then For every x ∈ D * , we have By using the homotopy invariance of the Leray-Schauder degree, we can conclude that The topological degree via a degree of nondensifiability in normed space conserves the basic features of the Leray-Schauder degree.(1 and Some ramifications for this topological degree concept.x ̸ = λN(x) for every λ ∈ [0, 1), x ∈ ∂O.
Then, N possesses at least fixed points.
Proof.Let H : [0, 1] × O → X a homotopy given by By the Leray-Schauder condition, we have Then, According to Theorem 3, there is x ∈ O such that Next, as a result, we present the version Schaefer's fixed point type.
Theorem 5. Consider a Banach space X and a continuous map N : X → X and k 2 − φ d −contraction map.Then, one of the following statements holds: 1.
x = Nx possesses at least one solution.
Similar to how Theorem 4 is proven, Consequently, N possess a fixed point.
Then, there is x ∈ U 2 \U 1 with x = N(x).
Then, by Theorem 3, By (H 1 ), we can apply Theorem 6, Then, by homotopy proprieties of topological degree, we obtain

Semilinear Integro-Differential Equations with Finite Delay
In this section, we consider the following semi-linear functional differential equation problem: where A is the infinitesimal generator of a Here, U t (•) represents the history of the state from time t − r up to the present time t.
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 (R(t)) t∈R + ∈ L(E ) is called a resolvent operator associated with (2) if (a) R(0) is the identity map and ∥R(t)∥ ≤ Me βt for a certain a positive real constant M and β ∈ R. (b) For each U ∈ E , R(t)U is strongly continuous.
(c) For all U ∈ E , t → R(t)U s continuously differentiable, and In the following, we will need the following lemma.
In order to give the existence result of the problem (2), we shall need the following hypotheses: (NU 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.
The sequence ( f n ) n∈N , defined by t ), satisfies the conditions of the Lebesgue's theorem.Indeed, Utilizing that f is a Carathéodory function and by the separability of C([−r, 0], E) × E, we deduce that f (•, •) is measurable.So, for any n ∈ N, the function f n (•).Since the sequence (U (n) ) n∈N converges to U in C([0, b], E ) and f is a Carathéodory function, then there exists M * > 0 such that and from (H 3 ), By the Lebesgue theorem, we obtain Step Thus, Therefore, N(B r ) is bounded.
Step 3: Proving that N is φ d −contractive.

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.

Now, we areDefinition 5 . 2 (Proposition 4 .
in a position to give the definition topological degree based on degree of nondensifiability.Consider a Banach space X, O ⊂ X, nonempty bounded, open and f ∈ C(O, X) ∩ KC k O, X) and 0 ̸ ∈ (I − f )(∂O).We define the degree of I − f bydeg d (I − f , O, 0) = deg LS (I − f , O, 0),where deg LS is the Leray-Schauder degree and f * is defined in Lemma 1.•If b ̸ ∈ (I − f )(∂O) we define the topological degree bydeg d (I − f − b, O, b) = deg LS (I − f − b, O, b).The next proposition makes Definition 5 meaningful.The degree deg d is well defined.

Theorem 3 .
Let X be a Banach space, O ⊂ X be an open bounded subset, and f : O −→ X be a continuous k 2 − φ d −contractions map.If b ̸ ∈ (I − f )(∂O), then there exists an integer deg d (I − f , O, b) satisfying the following properties:

Theorem 4 .
Consider a Banach space X, where O is an open bounded subset of X with 0 ∈ O and N : O → X is a continuous k 2 − φ d −contraction map.Suppose the following Leray-Schauder condition:
is a given function, and ϕ * ∈ C([−r, 0], E ).For any t ∈ [0, b], S(t) is a closed linear operator on E , with domain D(A) ⊂ D(S(t)), which is independent of t.For any function U : [−r, b] → E and any t ∈ [0, b], we denote by U t the element of C([−r, 0], E ) defined by