Generalizations of the Nonlinear Henry Inequality and the Leray–Schauder Type Fixed Point Theorem with Applications to Fractional Differential Inclusions

: The authors give some singular versions of the Gronwall–Bihari–Henry inequalities. They also establish a multivalued version of the Leray–Schauder alternative in strictly star-shaped sets. Based on these new fractional inequalities and ﬁxed point theorem, they study an initial value problem for fractional differential inclusions with delay.


Introduction
It is well-known that inequalities, such as the Gronwall-Bellman-Bihari-Henry inequality, play an important role in the study of existence, uniqueness, boundedness, stability, and other qualitative properties of solutions of differential equations, integral equations, and differential inclusions (see, for example, [1][2][3][4][5][6][7]). For nonlinear integral inequalities, perhaps one of the most important contributions was made by Bihari [8] for v(t) ≤ k + t 0 f (s)ψ(v(s))ds, k ∈ R + . (1) This integral inequality was generalized by many authors. For example, Agarwal et al. [9] replaced k, t, f , and ψ with the functions a(t), ( , respectively, and investigated the retarded Gronwall type inequality In recent years, ordinary and partial differential equations of fractional order have been investigated more in the literature due to their applicability to many problems in engineering and other scientific disciplines; see [10][11][12][13][14][15] and the references therein for recent work. The question of the existence of solutions and other mathematical aspects of fractional differential equations and inclusions have been extensively studied and have attracted much attention; many important contributions have been obtained so far (see the monographs [16][17][18] as well as papers listed in the references below).
Linear and nonlinear integral inequalities with singular kernels have received considerable attention in the literature since 1981 when Henry [3] established the following result: If w is a non-negative locally integrable function, a ≥ 0 is a constant, 0 < γ < 1, and v(t) ≤ w(t) + a t 0 v(s) (t − s) γ ds, then there exists a constant K = K(γ) such that Henry's result has been extended to more general linear integral inequalities by many authors such as in [3,[19][20][21][22][23]. All these results are proved by an iteration argument and the results expressed as integrals with singular kernels often defined by power series of very complicated forms that are sometimes not very convenient for applications.
In [24,25], Medved studied the nonlinear integral inequality of Henry type where α < 1 and ψ is a positive nondecreasing function. The aim of the present paper is to establish some new and useful nonlinear generalizations of the integral inequality given in [3]; we also generalize the singular inequality of Agarwal et al. [9]. In our proofs, we make use of the Young and Hölder inequalities combined with a classical Bihari type inequality to obtain our results.
It is also our goal to establish a new multivalued version of the Leray-Schauder fixed point theorem. In order to accomplish this, we first recall some notions from multivariate analysis in Section 3. In Section 4, we prove our Leray-Schauder type fixed point theorem for multivalued mappings. Then, in Section 5, we use the fractional inequalities derived in Section 2 and apply our fixed point theorem from Section 4 to an initial value problem for a fractional delay differential inclusion in star shaped sets.

A Nonlinear Integral Inequality
In this section, we wish to establish some nonlinear integral inequalities that can be used in the analysis of fractional differential equations and inclusions. The proofs are based on Young's and Hölder's inequalities. Theorem 1. Let 0 < α < 1, q > 1 α , and k 0 , k 1 , k 2 > 0. For i = 1, 2, . . . , n, let f i , λ i , γ i , and g be non-negative functions that are locally integrable on I = [0, T], and let ψ i : [0, ∞) → [0, ∞) be nondecreasing continuous functions. If u(t) is a non-negative continuous function on I satisfying , , ψ(y) = max{ψ q i (y) : i = 1, 2, . . . , n}, Ψ −1 is the inverse function of Ψ, Dom(Ψ −1 ) is the domain of Ψ −1 , and for every t ∈ [0, T], Proof. By Young's inequality, we see that Then, Since the ψ i are nondecreasing and where Integrating the above inequality from 0 to t, we obtain which proves the theorem.
In the particular case where g(t) ≡ 0, we have the following corollary to Theorem 1.

Corollary 1.
Let k 0 , k 1 > 0, q > 1 α , and the functions f i , λ i , ψ i , and u be as in Theorem 1. If Ψ −1 is the inverse function of Ψ, and for every t ∈ [0, T], Proof. Proceeding exactly as in the proof of Theorem 1, we again arrive at (3). Define the function ψ by Integrating the above inequality from 0 to t gives In view of the fact that u(t) ≤ v(t), this proves the corollary.

Corollary 2.
Let k 0 , k 1 > 0, and the functions λ i , ψ i , and u be as in Theorem 1. If the constants k i are given by and T 1 ∈ I is the largest number such that For n = 1, g ≡ 0, and f 1 ≡ 1 ≡ λ 1 in Theorem 1, we obtain the following fractional Bihari type inequality on bounded intervals. Corollary 3. Let k 0 , k 1 > 0, q > 1 α , and the functions ψ and u be as in Theorem 1 such that Then, for t ∈ [0, T], we have , If n = 1 and λ 1 (t) ≡ 1 in Corollary 1, then by using Hölder's inequality, we have the following result. where Next, for convenience, we introduce a class of functions to be used to obtain new fractional Bihari type inequalities. For additional details on this class of functions, see [26]. Theorem 2. Let 0 < α < 1, q > 1 α , and assume that for i = 1, 2, . . . , n, the functions f , λ i , γ i , h, and g are non-negative and locally integrable, f and g are increasing, and ( f g)/h is decreasing on I = [0, T]. Additionally, let ψ i ∈ H, i = 1, 2, . . . , n, with corresponding multiplier functions φ i . If u(t) is a non-negative continuous function on I satisfying , , and ψ(y) = max{ψ q i (y), i = 1, . . . , n}.
Proof. From (6), we have With z(t) = u(t) h(t) , we see that this last inequality is equivalent to If we replace f i by in Theorem 1, we obtain Therefore, which completes the proof.
For n = 1, g ≡ 0, and f (t) ≡ k, we next have a another fractional Bihari type inequality on a bounded interval. then Proof. From (7) and the fact that h is nondecreasing, Applying Young's inequality gives and so which completes the proof.
As another result in this same spirit, we have the following theorem.
Theorem 3. Let 0 < α < 1, q > 1 α , and assume that u, λ i , and h are non-negative functions, i = 1, 2, . . . , n, that are locally integrable on I = [0, T], and let ψ i ∈ H with corresponding multiplier functions φ i . In addition, assume that the function h is nondecreasing and satisfies ( H) for each t ∈ I there exists a continuous function χ such that h(z(t)) ≤ χ(t) z(t) . If , , and ψ(y) = max{ψ Proof. From (8) and the facts that Ψ satisfies (H 2 ) and h is increasing, we obtain With z(t) = u(t) h(u(t)) , this becomes By Young's inequality, we define the functions m and ψ by Since the ψ i are nondecreasing, and integrating from 0 to t, we obtain With v(0) = k, we have and ( H) holds, we finally obtain Thus, and so This completes the proof of the theorem.

Multivalued Analysis
In order to apply the inequalities obtained above to fractional differential inclusions, we need to recall some basic notions from multivalued analysis (see, for example, [27,28]).
For any set X, we employ the following notation.
. A multifunction is lower semi-continuous (l.s.c.) provided that is lower semi-continuous at every point x ∈ X.
Definition 4. ( [27], Definition 2.1.1) A mapping G : X → Y is closed if the graph Gr(G) is a closed subset of X × Y, i.e., for sequences (x n ) n∈N ⊂ X and (y n ) n∈N ⊂ Y, if x n → x * and y n → y * as n → ∞ with y n ∈ G(x n ), then y * ∈ G(x * ).
A mapping G is said to be completely continuous if it is u.s.c. and for every bounded set A ⊆ X, G(A) is relatively compact, i.e., there exists a relatively compact set Definition 5. ( [27], Definition 2.6.2) Let X be a metric space and E be a Banach space. A multivalued map F : x * , y is upper semicontinuous.
Next, we recall the following results.
Conversely, if G is locally compact, has nonempty compact values, and a closed graph, then it is u.s.c.
Theorem 4. ( [31]) Let X be a reflexive Banach space and F : X → P cl,cv (X) be a upperhemicontinuous multivalued map. Let J be a finite interval of R and the sequences (x n : J → X) n∈N and (y n : J → X) n∈N satisfy the following conditions: 1) (x n ) n∈N converges to a function x : J → X; 2) (y n ) n∈N converges weakly to y ∈ L p (J, E), 1 ≤ p < ∞; 3) y n (t) ∈ coB(F(B(x n (t), n ), n )) for a.e. all t ∈ J, where n → 0 as n → ∞. Then y(t) ∈ F(x(t)) for a.e. t ∈ J.
is theČech-homology functor with compact carriers and coefficients in the field of rationals Q.

Remark 1.
If R δ − is a compact connected space that is acyclic with respect to theČech-homology functor, then it has the same homology as a one-point space. (ii) L is surjective; (iii) the set L −1 (y) is acyclic for every y ∈ Y.
It is clear that any multivalued operator F : X → P (Y) admits the standard factorization through the graph Gr(F), i.e., there exists a diagram where p F : Gr(F) → X and q F : Gr(F) → Y are the projections such that This suggests the following definition and properties. Given a Banach space (X, · ), for a multivalued map F : J × X → P (X), set A multivalued map G : J −→ P cp (X) is said to be measurable if for each x ∈ R the function Y : J −→ R defined by A multifunction F : [a, b] → P cp (X) is strongly measurable if there exists a sequence {F n : n ∈ N} of step multifunctions such that where µ denotes a Lebesgue measure on [a, b] and H d is the Hausdorff metric on P cp (X).
In what follows, L p ([a, b], X) denotes the Banach space of functions y : J −→ X, that are Bochner integrable with norm For each y ∈ C(J, X), the set is known as the set of selection functions for F.

Lemma 4.
( [32], Theorem 19.7) Let X be a separable metric space and G be a multivalued map with nonempty closed values. Then G has a measurable selection.

Definition 11. ([28], Definition 2.80) A multivalued map F is a Carathéodory function if
(a) the function t → F(t, x) is strong measurable for each x ∈ X; (b) for a.e. t ∈ J, the map x → F(t, x) is upper semi-continuous.
Furthermore, F is L 1 −Carathéodory if it is locally integrably bounded, i.e., for each positive r, there exists h r ∈ L 1 (J, R + ) such that F(t, x) P ≤ h r (t), for a.e. t ∈ J and all x ≤ r.

Measures of Noncompactness (MNC)
For more details on measure of noncompactness than given below, we refer the reader to [33][34][35] and the references therein.
for every a ∈ X, Ω ∈ P (X).  It should be mentioned that these MNC satisfy all above-mentioned properties except regularity.

Definition 14.
( [28]) Let M be a closed subset of a Banach space X and β : P (X) → (A, ≥) be a MNC on E. A multivalued map F : M → P cp (X) is said to be β−condensing if for every Ω ⊂ M, the relation β(Ω) ≤ β(F (Ω)) implies the relative compactness of Ω.
Some important results on fixed point theory with MNCs are recalled next (see, for example, [34] for proofs and additional details). The first one is a compactness criterion.
(S 2 ) N is weakly-strongly sequentially continuous on compact subsets: for any compact K ⊂ X and

Fixed Point Theory
We begin with some basics of fixed point theory. The next result is concerned with the nonlinear alternative for β−condensing u.s.c. multi-valued maps. Lemma 7. Let V ⊂ X be a bounded open neighborhood of zero and N : V → P (X) be a βadmissible multi-valued map, where β is a nonsingular measure of noncompactness defined on subsets of X, that satisfies the Leray-Schauder boundary condition for all x ∈ ∂V and 0 < λ < 1. Then FixN is nonempty and compact.
Proof. Let C be the set defined by It is clear that C is nonempty set since 0 ∈ C. To show that C is closed, let (x n ) n∈N ⊂ C be a sequence converging to x; then there exists (λ n ) n∈N ⊂ [0, 1] such that x n ∈ λ n N(x n ) for n ∈ N.
If λ ∈ (0, 1], then using the fact that N is admissible, we see that N has a closed graph, so x ∈ λN(x).
If λ = 0, it is clear that N {x n : n ∈ N} is a compact set, so there exists M > 0 such that y ≤ M for any y ∈ N {x n : n ∈ N} .
This implies x n ≤ λ n M, n ∈ N, that is, x = 0 ∈ 0N(0), and we conclude that C is closed set in X.
Since C ∩ X\V = ∅, Urysohn's lemma guarantees the existence of a continuous function µ : X → [0, 1] such that µ(x) = 1 for x ∈ C and µ(x) = 0 for x ∈ X\V. Let r : X → V be a retraction of the space X onto V.
We introduce the multivalued operator N : X → P (X) defined by Observe that Let C * = co N(V) ∪ {0} ; it is easy to show that It follows from the definition of N that it is an admissible multivalued map. Next, we show that N is β−condensing. Let D ∈ P b (C * ) be such that If D ∩ V = ∅, then from the definition of N, we have From the definition of N, From Definition 13, for every x ∈ X, we have Since N is β−condensing, we obtain that D ∩ V is relatively compact. Moreover, since N is u.s.c., β(D) = 0, so D is relatively compact. Therefore, N is β−admissible since it is admissible and β-condensing.
We now use Theorem 5 to show that N has at least one fixed point x ∈ C * such that To see that Fix(N) is compact, first note that (Fix(N)).
An additional result on the set of fixed points of F is contained in the following proposition. Proposition 2. Let X be a Banach space and N : X → P (X) be a β−admissible multivalued map, where β is a nonsingular measure of noncompactness defined on subsets of X. If the set M = {x ∈ X : x ∈ λN(x), for some λ ∈ (0, 1)} is bounded, then FixN = ∅ and is compact.
Proof. Since M is a bounded set, there exists M > 0 such that We can easily prove that N : B(0, M) → P (X) is a β−admissible operator and Consequently, from Lemma 7, the set Fix(N) = ∅ and is compact.
Similarly, we have the following result. In what follows, we wish to replace the Leray-Schauder boundary condition with the following "star-shaped" condition.
This function satisfies the properties:

Remark 3.
For a strictly star set, the Minkowski function can equivalently be defined as: µ(0) = 0 and, for

Proposition 3. ([37]) Let X be a Banach space and V be an open bounded neighborhood of zero.
If V is strictly star-shaped, then the Minkowski function µ V is continuous and the mapping R V : X → V given by is a continuous retract of X into closure of V.
In 2005, Jiménez-Melado and Morales [36] introduced the so-called interior condition for single valued maps. In the following definition, we introduce a multivalued version of this condition.

Definition 16.
( [36], Page 501) Let X be a real Banach space and V be open subset of X with 0 ∈ V. We say that the multivalued map N : V → P (X) satisfies the interior condition if there exists δ > 0 such that The following result is taken from González, Jiménez-Melado, and Llorens-Fuster ([37], Proposition 2). Proposition 4. Let X be a Banach space and V be a strictly star-shaped open bounded neighborhood of the origin. Let 0 ≤ k ≤ K, where k = d(0, ∂V) and K = sup{ x : x ∈ ∂V}, and choose δ ∈ (0, k]. Define L δ V : X → X by x, x ∈ X\V, x, x ∈ V.
The following fixed point theorem is for multivalued maps satisfying the interior condition.
Theorem 6. Let X be a real Banach space and let V be a bounded open and strictly star-shaped subset of X with 0 ∈ V. If the multivalued map N : V → P (X) is β−condensing, admissible, and satisfies the interior condition, then N has at least one fixed point.
Proof. Let R V : X → V be a retraction of X into V. Define the multivalued operator N * : V → P (X) by We can write N * as are continuous functions. Then, by Proposition 1, N * is an admissible operator, and we see that ; then from the definition of N * , β(N * (D)) = β (N(D)).
Using the fact that N * is β-condensing, we see that D is relatively compact. Next, we show that N * satisfies the Leray-Schauder condition. Since N satisfies the interior condition, there exists δ ∈ I := (0, K) such that As in the proof of ([36], Theorem 1), for any t ∈ (1 − δ K , 1), the set V t = {tx : x ∈ V} is an open subset of V, tV = tV, and tV ⊂ V.
Additionally, we can easily show that Now suppose there exist λ t > 1 and x t ∈ ∂V t such that λ t x t ∈ N * (x t ). Then by the definition of N * , λ t x t ∈ N(x t ), λ t > 1, and x t ∈ ∂V t .
Since V is strictly star-shaped and x t ∈ ∂V t ⊂ V, there exists a unique z t ∈ ∂V such that x t = tz t (see Remark 3). Hence, we obtain and consequently, by the interior condition, we have λ t x t ∈ V. This implies that λ t x t ∈ [x t , z t ] since that is a strictly star-shaped set. Thus, we conclude that Therefore, the set D * = {x t : x t satisfies the relation (10), t ∈ I} is bounded and D * ⊂ co{N(D * ) ∪ {0}}. Since N is β−condensing, D * is relatively compact, and hence there exists a sequence (x n ) n∈N converging to x in V and a sequence (λ n ) n∈N in R such that λ n x n ∈ N(x n ), λ n → 1, t n → 1, and x n → x as n → ∞.
It is easy to see that (11), implies lim n→∞ x n = lim n→∞ z n = x and there exists n ∈ N such that x n ∈ V. We then have that The interior condition of N implies that λ n x n ∈ N(x n ) and N(x n ) ∩ X\V = ∅, which contradicts (10).
Therefore, N * satisfies the Leray-Schauder condition. Since V is a retract of X, N * is an admissible map and satisfies Mönch's theorem. Then, by ([39], Theorem 5), there exists x ∈ V such that x ∈ N * (x). From the definition of N * , it follows that x ∈ N(x), and this completes the proof of the theorem.

Fractional Differential Inclusions with Delay
In this section we wish to apply some of the inequalities obtained in Section 2 and the new Leray-Schauder type fixed point theorem obtained in Section 4 to proving the existence of solutions to the Cauchy problem for the fractional delay differential inclusion Here x t (·) represents the history of the state from time t − r up to the present time t. The study of differential inclusions has emerged as an important area of research due to their applicability to problems in optimal control theory and other areas; see, for example, the monographs [40,41] and the references contained therein. Additionally, differential inclusions can incorporate differential equations with discontinuities in the right hand side (or even for the case where the right hand side is inaccurately known) [34,[42][43][44]. Additional background on differential inclusions and multivalued analysis can be found in [27][28][29]32,[45][46][47][48]. For recent results on the existence of solutions to fractional differential equations and inclusions with various types of delays, we refer the reader to [49][50][51][52][53][54][55].
We begin by recalling the definitions of a fractional integral and the Caputo fractional derivative. Here, Γ denotes the well-known gamma function.
Definition 17. ([49], Definition 1.4) The fractional integral of order β > 0 with lower limit 0 for a function f is defined as The Caputo fractional derivative of order β with 0 ≤ n − 1 < β < n and lower limit 0 for a function f is given by We will need to make use of the following assumptions in our results in this section.
(H 1 )The mulivalued map F(t, ·) has a strong measurable selection for u ∈ C(J 0 , E).
for every t ∈ J and u ∈ C(J 0 , E). (H 4 )There exists g ∈ L q (J, R + ) such that for all bounded D ∈ C(J 0 , E), we have In the following proposition we establish some properties of the selection function operator.
Hence, v n ∈ S q F (x n ) for each n ∈ N, and (v n ) is bounded sequence in L q (J, E). Since E is a reflexive space, by the duality theorem ( [56], Theorem V.1.1), the space L q (J, E) is also reflexive. By the Eberlein-Smulian Theorem (see Brezis [57]), there exists a subsequence, still denoted by (v n ), that converges weakly to v ∈ L q (J, E). Additionally, we have Since F(t, ·) is upper hemicontinuous, By (14)- (16), and applying the Lebesgue dominated convergence theorem, we obtain Applying a similar argument, we can prove that the Nemytskii operator is weakly upper semicontinuous.
where v ∈ S q F (x). Notice that N can be written in the form where L : Next, we establish an important property of the operator N(x) defined in (17).
Proof. We will divide the proof into several steps. We begin by showing that N is u.s.c. and has nonempty convex values.
Step 1: N(·) ∈ P cv (E). Since L is single valued, K is a bounded linear operator, and S q F has nonempty convex values, for each x ∈ C([−r, T], E) we have N(x) ∈ P cv (E).
Step 2: N is u.s.c. We first show that N maps bounded sets into bounded subsets of C([−r, T], E). Let B ρ := {x ∈ C([−r, T], E) : x ∞ ≤ ρ}. For x ∈ B ρ and h ∈ N(x), there exists v ∈ S F,x such that Then, by (H 3 ) and Hölder's inequality Hence, Next, we wish to show that N maps bounded sets into equicontinuous subsets of C([−r, T], E). Let t 1 , t 2 ∈ J with t 1 < t 2 and let B ρ be a bounded subset of C([−r, T], E). If x ∈ B ρ , then for each h ∈ N(x), we have From the Hölder and Biernacki inequalities, it follows that The left-hand side tends to zero as t 2 − t 1 → 0, so N(B ρ ) is equicontinuous in C([−r, T], E).
Step 3: N is a condensing operator for a suitable MNC γ. Given a bounded subset To show that B is relatively compact, let {x n : n ∈ N} ⊂ B. From (18), each h n in N(x n ) can be represented as h n = L(x n ) + K(v n ), with v n ∈ S F (x n ). (21) Moreover, (20) yields γ({h n : n ∈ N}) ≥ γ({x n : n ∈ N}).
Hence, (21) implies that χ({h n } ∞ n=1 ) = 0. By an argument similar to the one we used to prove that N maps bounded sets into equicontinuous sets, we can prove that the set {h n } is equicontinuous, and so mod C (B) = 0. It follows that γ({h n } ∞ n=1 ) = 0, which by (22) implies that γ({x n } ∞ n=1 ) = 0. Thus, B is relatively compact. By the Arzelà-Ascoli theorem, N is completely continuous, from which we have that N is γ-condensing.
Step 4: N has a closed graph. Let {x n } n∈N ∈ C([−r, T], E) be a sequence such that {x n } n∈N converges to x, h n ∈ N(x n ), and {h n } n∈N converges to h. We need to show that h ∈ N( x). So for each n ∈ N there exist v n ∈ S q F (x n ) such that h n (t) = L(x n (t)) + K(v n (t)), t ∈ J 0 ∪ J.
Similar to the proof of Proposition 5, we can conclude that there exists a subsequence of v n converging weakly to v in L q and satisfying v(t) ∈ F(t, x t ), a.e. t ∈ J.
Since K is a continuous linear operator, K(v n (t)) → K(v(t)) as n → ∞.
On the other hand, the continuity of L implies that L(x n (t)) → L( x(t)) as n → ∞.
Hence, the multivalued operator N has a closed graph. In view of Steps 1-3, the proof of the proposition is complete.
We are now ready to give our main existence result for problem (12).

Theorem 7.
Assume that (H 1 )-(H 4 ) hold. Then the problem (12) has at least one solution on [−r, T] and the set FixN is compact.