On Lipschitz Perturbations of a Self-Adjoint Strongly Positive Operator

In this paper we study semilinear equations of the form Au + λF(u) = f, where A is a linear self-adjoint operator, satisfying a strong positivity condition, and F is a nonlinear Lipschitz operator. As applications we develop Krasnoselskii and Ky Fan type approximation results for certain pair of maps and to illustrate the usability of the obtained results, the existence of solution of an integral equation is provided.


Introduction and Preliminaries
The study of abstract operator equations involving linear or nonlinear operators has generated over time useful instruments in the approach of some concrete equations.Therefore, we consider as interesting to present some aspects regarding the semilinear abstract operator equations in Hilbert spaces.
Let  be a real Hilbert space endowed with the inner product ⟨⋅, ⋅⟩ and the norm ‖ ⋅ ‖.
In [1,2] semilinear equations of the form  − () = 0 were studied, where  : () ⊂  →  is a self-adjoint linear operator with the resolvent set () and  :  →  is a Gateaux differentiable gradient operator.If there exist real numbers  <  such that [, ] ⊂ () and for all , V ∈ ,  ̸ = V (i.e.,  interacts suitably with the spectrum of ), then it is proved in [2] that the equation  − () = 0 has exactly one solution.
The author in [3] presented an existence and uniqueness result for the semilinear equation  + () = , where  : () ⊂  →  is a linear maximal monotone operator, satisfying a strong positivity condition, and the nonlinearity  :  →  is a Lipschitz operator.
Let  be a real Banach space, ordered by a cone .A cone  is a closed convex subset of  with  ⊆  ( ≥ 0), and  ∩ (−) = {0}.As usual  ≤  ⇔  −  ∈ .Definition 1.Let  be a nonempty subset of an ordered Banach space  with order ≤.Two mappings ,  :  →  are said to be weakly isotone increasing if  ≤  and  ≤  hold for all  ∈ .Similarly, we say that  and  are weakly isotone decreasing if  ≥  and  ≥  hold for all  ∈ .The mappings  and  are said to be weakly isotone if they are either weakly isotone increasing or weakly isotone decreasing.
In our considerations, the following definition will play an important role.Let B() denote the collection of all nonempty bounded subsets of  and W() the subset of B() consisting of all weakly compact subsets of .Also, let   denote the closed ball centered at 0 with radius .Definition 2 (see [4]).A function  : B() → R + is said to be a measure of weak noncompactness if it satisfies the following conditions.
(5) If (  ) ≥1 is a sequence of nonempty weakly closed subsets of  with  1 bounded and The family ker  described in ( 1) is said to be the kernel of the measure of weak noncompactness .Note that the intersection set  ∞ from (5) belongs to ker  since ( ∞ ) ≤ (  ) for every , and lim  → ∞ (  ) = 0. Also, it can be easily verified that the measure  satisfies where   is the weak closure of .
A measure of weak noncompactness  is said to be regular if and set additive (or has the maximum property) if The first important example of a measure of weak noncompactness has been defined by de Blasi [5] as follows: for each  ∈ B().
By a measure of noncompactness on a Banach space , we mean a map  : B() → R + which satisfies conditions (1)-( 5) in Definition 2 relative to the strong topology instead of the weak topology.Definition 3. Let  be a Banach space and  a measure of (weak) noncompactness on .Let  : () ⊆  →  be a mapping.If (()) is bounded and for every nonempty bounded subset  of () with () > 0, we have (()) < (); then  is called -condensing.If there exists , 0 ≤  ≤ 1, such that (()) is bounded and for each nonempty bounded subset  of (), we have (()) ≤ (); then  is called --contractive.Definition 4 (see [6]).A map  : () →  is said to be ws-compact if it is continuous, and for any weakly convergent sequence (  ) ∈N in () the sequence (  ) ∈N has a strongly convergent subsequence in .Definition 5. A map  : () →  is said to be wwcompact if it is continuous, and for any weakly convergent sequence (  ) ∈N in () the sequence (  ) ∈N has a weakly convergent subsequence in .Definition 6.Let  be a Banach space.A mapping  : () ⊆  →  is called a nonlinear contraction if there exists a continuous and nondecreasing function  : for all ,  ∈ (), where () <  for  > 0.
In this paper we consider the semilinear equation where  :  →  is a linear self-adjoint operator, satisfying a strong positivity condition,  :  →  is a nonlinear Lipschitz operator, and  is a positive parameter.Using the Banach fixed point theorem, we prove an existence and uniqueness result about the considered equation.Thus, we obtain here the same type of result as in [2], replacing the maximal monotonicity of linear part  of the semilinear equation with the hypothesis that  is self-adjoint.So, the principal result of this paper can be applied in the study of nonlinear Lipschitz perturbations of a linear integral operator with symmetric kernel.Further a result of continuous dependence on the free term and a fixed point theorem are presented.As applications we present some common fixed point theorems and approximation results for a pair of nonlinear mappings.Finally, the existence of solution of an integral equation is provided to illustrate the usability of the obtained results.

Results
Theorem 7. Let  :  →  be a linear self-adjoint operator and  :  →  nonlinear, satisfying the following conditions: for all ,  ∈ ; (ii)  is a strongly positive operator, that is, there is a constant  > 0 such that for all  ∈ .
Proof.Let us choose  in the spectrum of .We have and we obtain that every real number  ∈ (−∞; ) is in the resolvent set of the operator .Consequently, we have for all  < , where  is the identity of .
Let  < 0. We write (9) in the equivalent form where for all ,  ∈ .Also From ( 16) we obtain Consequently, there exists  −1  :  →  which is linear and continuous, that is  −1  ∈ L(), the Banach space of all linear and bounded operators from  to .Moreover, we have Now ( 14) can be equivalently written as We consider the operator  :  →  defined by Therefore ( 19) becomes and so, the problem of the solvability of ( 9) is reduced to the study of fixed points of the operator .We have It results that  is a strict contraction from  to  because  < .According to the Banach fixed point theorem,  has a unique fixed point, and thus the proof of Theorem 7 is complete.
Let us consider now the dependence of solution of (9) on the data .Theorem 8.Under the assumptions from the hypothesis of Theorem 7, let  ∈ {1, 2}, and let   be the unique solution of the equation Then Proof.According to the equivalent form (19) of ( 9), we have It results that and thus our assertion is proved.
In fact, Theorem 8 establishes the continuous dependence of the solution of ( 9) on the free term and signifies the stability of the solution.
Theorem 9. Let  :  →  be a linear self-adjoint operator and  :  →  nonlinear, satisfying the following conditions: for all ,  ∈ ; (ii)  is a strongly positive operator, that is, there is a constant  > 0 such that for all  ∈ .
The way used in obtaining Theorem 9 can be applied in the study of the following problem: extracting operators which have the unique fixed point property from a family of operators {  / ∈ R}.
Let  be a Banach space, and F = {  :  → / ∈ R},   satisfying some constraints for any  ∈ R. Our intention is to extract a subfamily G ⊂ F, so that   can have a unique fixed point for all   ∈ G.It is easy to observe, using the same method as in obtaining Theorem 9, that the following result holds.

Theorem 10. If
(i) there exists  ⊂ R,  ̸ = , so that   is invertible and for all  ∈ ; (ii) there exists  ⊂ ,  ̸ = , so that () < 1 for all  ∈ , then   has a unique fixed point for all  ∈ .
As an application of the observations established above, we develop here the Krasnoselskii and Ky Fan type approximation results for certain pair of maps.
Theorem 11.Let , , ,  and  be as in Theorem 9,  = (+ ) −1 , and let  be an ordered Hilbert space.Let  be a nonempty closed convex subset of  and  a set additive measure of noncompactness on .Let  :  →  be a mapping satisfying the following: Thus  is a shrinking mapping.Now all of the conditions of Corollary 1.18 [7] are satisfied so there exists an  * ∈  such that  * =  * =  * which implies that ( + ) * =  * , and hence  * +  * =  * .Theorem 12. Let , , , , and  be as in Theorem 9,  = ( + ) −1 , and let  be an ordered Hilbert space.Let  be a nonempty closed convex subset of  and  a set additive measure of noncompactness on .Let  :  →  be a mapping satisfying the following: (i) () ⊆  and () ⊆ , (ii)  is a nonlinear contraction, (iii)  and  are weakly isotone.
As an application of Corollaries 1.24 or 1.25 and 1.30 or 1.31 [7], we obtain the following results, respectively Theorem 13.Let , , , , and  be as in Theorem 9,  = ( + ) −1 , and let  be an ordered Hilbert space.Let  be a nonempty closed convex subset of  and  a set additive measure of weak noncompactness on .Assume that ,  :  →  are sequentially weakly continuous mappings satisfying the following: (i) () ⊆  and () ⊆ , (ii)  is -condensing or  is a nonlinear contraction, (iii)  and  are weakly isotone.
Then there exists a unique point  * ∈  such that  * +  * =  * .Theorem 14.Let , , , , and  be as in Theorem 9,  = ( + ) −1 , and let  be an ordered Hilbert space.Let  be a nonempty closed convex subset of  and  a set additive measure of weak noncompactness on .Assume that ,  satisfy the following: (i)  is a ww-compact mapping, (ii)  is continuous ws-compact and -condensing or  is continuous ws-compact and nonlinear contraction, (iii)  and  are weakly isotone, (iv) () ⊆  and () ⊆ .
Following the proof of Corollary 4.5 [8] and using Theorem 16, we obtain the following common fixed point theorem.

An Application
Fixed point theorems for certain operators have found various applications in differential and integral equations (see [7][8][9][10] and references therein).In this section, we present an application of our Theorem 7 to establish a solution of a nonlinear integral equation.