A New Fixed Point Result and its Application to Existence Theorem for Nonconvex Hammerstein Type Integral Inclusions ∗

In this paper, a generalization of Nadler's fixed point theorem is presented for H + -type k-multi- valued weak contractive mappings. We consider a nonconvex Hammerstein type integral inclusion and prove an existence theorem by using an H + -type multi-valued weak contractive mapping.


Preliminaries and Definitions
Let (X, d) be a metric space.Let CB(X) and C(X) denote the collection of all nonempty closed and bounded subsets of X and the collection of all compact subsets of X, respectively.
For A, B ∈ CB(X), let H(A, B) = max ρ(A, B), ρ(B, A) , • multi-valued quasi-contraction mapping if there exists a fixed real number k, 0 < k < 1 such that H(T x, T y) ≤ k max{d(x, y), d(x, T x), d(y, T y), d(x, T y), d(y, T x)}, for all x, y ∈ X.
Notice that the two metrics H and H + are equivalent [11] since 1 2 H(A, B) ≤ H + (A, B) ≤ H(A, B).
In the light of this equivalence and referring to Kuratowski [11], we conclude that (CB(X), H + ) is complete whenever (X, d) is complete.Indeed, it is a simple consequence of the completeness of the Hausdorff metric H.Moreover, C(X) is a closed subspace of (CB(X), H + ).
Notice also that H + : CB(X) × CB(X) → R is a continuous function.To see this, we observe that the inequality holds for any A, B, C ∈ CB(X).Now pick any (A 0 , B 0 ) ∈ CB(X) × CB(X).Then for a given ǫ > 0, we can choose a positive number δ = ǫ 2 such that In [16], S. B. Nadler proved the following result, which he announced earlier.
Let (X, d) be a complete metric space and T : X → CB(X) a multi-valued contraction mapping.Then T has a fixed point.
In this paper, we intend to generalize this result by weakening the multi-valued contraction to an H + -type multi-valued weak contractive mapping.Our main result is summarized in Section 3. In Section 4, we consider a nonconvex Hammerstein type integral inclusion and prove an existence theorem by using an H + -type multi-valued weak contractive mapping.EJQTDE, 2012 No. 62, p. 2

Main results
We begin our discussion with the following definition.Definition 3.1.Let (X, d) be a metric space.A multi-valued mapping T : X → CB(X) is called H + -contraction if (1) there exists a fixed real number k, 0 < k < 1 such that H + (T x, T y) ≤ kd(x, y) for every x, y ∈ X, (2) for every x in X, y in T (x) and ǫ > 0, there exists z in T (y) such that In [18], Pathak and Shahzad proved the following result.Theorem 3.2.Every H + -type multi-valued contraction mapping T : X → CB(X) with Lipschitz constant 0 < k < 1 has a fixed point.
We now introduce the following definition.Definition 3.3.Let (X, d) be a metric space.A mapping T : X → CB(X) is called an H +type multi-valued weak contractive mapping if the condition (2) holds and there exists a fixed real number k, 0 < k < 1 such that for all x, y in X.
Now we state and prove our main result.
Theorem 3.4.Let (X, d) be a complete metric space and T : X → CB(X) an H + -type multivalued weak k-contractive mapping with 0 < k < 1.Then T has a fixed point.
Proof.Notice first that for each A, B ∈ CB(X), a ∈ A and α > 0 with for any x, y ∈ X, x = y.Now we choose a sequence {x n } recursively in X in the following way.Let x 0 ∈ X be arbitrary.Fix an element x 1 in T x 0 .From (2) it follows that we can choose x 2 ∈ T x 1 such that In general, if x n be chosen, then we choose x n+1 ∈ T x n such that Thus, we have for each n ∈ N. Thus, from (3.2) we have It follows that x n is a fixed point of T and we are finished.So, we may assume that d( where c = √ L. Repeating the same argument n-times as in (3.7), we obtain It is obvious that {x n } is bounded.Indeed, for any n ∈ N, we have Further, by virtue of (3.8), one may observe that {x n } is a Cauchy sequence.Since X is complete, there exists u ∈ X such that lim Since lim n→∞ d(x n+1 , u) = 0 exists, and a contradiction.This implies that d(u, T u) = 0, and, since T u is closed, it must be the case that u ∈ T u.
Notice that every multi-valued contraction mapping with respect to Pompeiu-Hausdorff metric H is an H + -type multi-valued weak contractive mapping but the converse implication need not be true.To see this, we have the following example: 2] and y ∈ [−2, −1), then we have • max{d(y, T y), d(x, T x)}.
It follows that for all x, y ∈ X and k ∈ [ 3 4 , 1).To check the condition (2), we consider the following cases: Case 1.If x ∈ [−2, −1), then for any y ∈ T x = {2}, there exists z ∈ T y = { 1  2 } such that for any ǫ > 0 Thus all the conditions of Theorem 3.4 are satisfied.Moreover, 0 ∈ T 0 = {0} is a fixed point of T .EJQTDE, 2012 No. 62, p. 5 Notice that the map T does not satisfy the assumptions of Theorem 2.2 and Theorem 3.2.Indeed, for x = −1 and y → −1 from the left we have for all k ∈ (0, 1).
We also notice that since [d(x, T y) + d(y, T x)]/2 ≤ max{d(x, T y), d(y, T x} for all x, y ∈ X, it follows that every weak contractive mapping is quasi-contraction.
Using the technique of the proof of Theorem 3.4, one can easily prove the following result.
Let (X, d) be a complete metric space.Let T : X → CB(X) be a H + -type k-multi-valued quasi-contraction mapping with 0 < k < 1 2 .Then, T has a fixed point.
Pathak and Shahzad [18] introduced the class of H + -type nonexpansive mappings Definition 3.7.Let (X, • ) be a Banach space.A multi-valued map T : ) for every x in X, y in T (x) and ǫ > 0, there exists z in T (y) such that Applying the main result of this section, we obtain the following result which plays a role in the next section.
Proposition 3.8.([18]).Let (X, d) be a complete metric space.Suppose that T i : X → CB(X), i = 1, 2, are two H + -type multi-valued contraction mappings with Lipschitz constant L < 1.Then if F ix(T 1 ) and F ix(T 2 ) denote the respective fixed point sets of T 1 and T 2 ,

Existence Theorem for Nonconvex Hammerstein Type Integral Inclusions
Let 0 < T < ∞, I := [0, T ] and L(I) denote the σ-algebra of all Lebesgue measurable subsets of I. Let E be a real separable Banach space with the norm • .Let P(E) denote the family of all nonempty subsets of E and B(E) the family of all Borel subsets of E.
In what follows, as usual, we denote by C(I, E) the Banach space of all continuous functions x(•) : I → E endowed with the norm x(•) C = sup t∈I x(t) .Consider the following integral equation ) is a positive real single-valued function, while g : ), and let r ∈ [1, ∞) be the conjugate exponent of q, that is 1/q + 1/r = 1.Let • p denote the p-norm of the space L p (I, E) and is defined by u p = ( T 0 u(s) p ds) 1/p for all u ∈ L p (I, E).Consider the Nemitsky operator associated to g, p, q and G : L p (I, E) → L q (I, E) given by G(u) = g(t, s, u(s)) a.e. on I.
Consider the linear integral operator of kernel k, S : L q (I, E) → L p (I, E) given by where V : C(I, E) → C(I, E) is a given mapping.In the sequel, we also use the following: For any x ∈ E, λ ∈ C(I, E), σ ∈ L p (I, E), we define the set-valued maps M λ,σ (t In order to study problem (4.1)-( 4.2) we introduce the following assumption.
for all x, y in E, and for any x, y ∈ X, w ∈ F (t, x) and any ǫ > 0, there exists z ∈ F (t, y) such that and T λ (•) satisfies the condition: For any σ ∈ L p (I, E), σ 1 ∈ T λ (σ) and any given ǫ > 0, there exists σ 2 ∈ T λ (σ 1 ) such that Assume that U is an open bounded subset of R n (or Y , a subset of E homeomorphic to R n ) and U T = (0, T ]×U for some fixed T > 0. We say that the partial differential operator ∂ ∂t +L is parabolic if there exists a constant θ > 0 such that n i,j=1 a ij (t, x)ξ i ξ j ≥ θ|ξ| 2 for all (t, x) ∈ U T , ξ ∈ R n .The letter L denotes for each time t a second order partial differential operator, having either the divergence form Lu = − n i,j=1 (a ij (t, x)u x 0 , where {G(t)} t≥0 is a C 0 -semigroup with an infinitesimal generator A. Then a solution of system (4.1)-(4.2) represents a mild solution of In particular, this problem includes control systems governed by parabolic partial differential equations as a special case.When A = 0, the relation (4.3) reduces to Then the integral inclusion system (4.1)-(4.2) reduces to the form which may be written in more "compact" form as a.e.(I).Now we recall the following: E) and satisfy relation (S).
For our further discussion, we denote by S(λ) the solution set of (4.1) − (4.2).
For given α ∈ R we denote by L p (I, E) the Banach space of all Bochner integrable functions u(•) : I → E endowed with the norm where p(•) ∈ L p (I, R + ) and y(t) = µ(t) + Φ(v)(t), ∀t ∈ I.
It is well known that the set-valued map M λ,σ (•) is measurable.For example the map t → M λ,σ (t) can be approximated by step functions and so we can apply Theorem III.40 in [1].As the values of F are closed, with the measurable selection theorem we infer that M λ,σ (•) is nonempty.
Further, we note that the set T λ (σ) is bounded and closed.Indeed, if ψ n ∈ T λ (•) and ψ n −ψ p → 0, then there exists a subsequence ψ n k such that ψ n k (t) → ψ(t) for a.e.t ∈ I and we find that ψ ∈ T λ (σ).
Next, we prove the following estimate: Then one has the following inequality: Combining the last inequality with (4.12) we obtain This completes the proof.
s)u(s)ds a.e. on I. Thus the Hammerstein type integral equation (4.1) is transformed into the form x = SG(u), u ∈ L p (I, E) a.e. on I (4.1 ′ )
It is worth mentioning that the system (4.1)-(4.2) includes a large variety of differential inclusions and control systems.