Guiding potential method for differential inclusions with nonlocal conditions
aa r X i v : . [ m a t h . C A ] D ec GUIDING POTENTIAL METHOD FOR DIFFERENTIAL INCLUSIONS WITHNONLOCAL CONDITIONS
RADOSŁAW PIETKUNA bstract . The existence of solutions of some nonlocal initial value problems for di ff er-ential inclusions is established. The guiding potential method is used and the topologicaldegree theory for admissible multivalued vector fields is applied. Some conclusions con-cerning compactness of the solution set have been drawn.
1. I ntroduction
The aim of this paper is to formulate theorems about the existence of absolutely contin-uous solutions of the following nonlocal Cauchy problem ˙ x ( t ) ∈ F ( t , x ( t )) , a.e. t ∈ I = [0 , T ] , x (0) = g ( x ) , (1)where F : I × ’ N ⊸ ’ N is a multivalued map and g : C ( I , ’ N ) → ’ N . Proofs of thesetheorems rely on the use of C -guiding potential. This method, whose base was laid byM. A. Krasnosel’skii and A. I. Perov, generically employed to di ff erential equations andsubsequently expanded to inclusions, has demonstrated its e ff ectiveness to the study ofperiodic problems (see, e.g. [2, 6, 7] and especially [8], which contains a lot of referencesto subject literature). However, periodic condition constitutes just a single case of a muchwider class of so called nonlocal initial conditions. The consideration for nonlocal initialcondition, given by some g , is stimulated by the observation that this type of conditions ismore realistic than usual ones in treating physical problems.Let us mention here three important cases covered by our results: • g ( x ) = − x ( T ) (anti-periodicity condition); • g ( x ) = P ni = α i x ( t i ), where P ni = | α i | P ni = α i , < t < t < . . . < t n T (multi-point discrete mean condition); • g ( x ) = T R T h ( x ( t )) dt , with h : ’ N → ’ N such that | h ( x ) | | x | (mean valuecondition).In each of the above cases we are dealing with the situation when the function g pos-sesses a sublinear growth, and strictly speaking satisfies the condition: | g ( x ) | || x || for x ∈ C ( I , ’ N ). This means in particular that the existence of solutions to the problem (1)for such g’s can not be resolved by means of the fixed point theorem (see Theorem 2. be-low). Therefore arises a need for a di ff erent approach to boundary problems of this type.Such an approach is presented in Theorems 3., 4. and 5., which exploit the existence of asmooth guiding potential V for the multivalued right-hand side of the di ff erential inclusion Mathematics Subject Classification.
Key words and phrases. guiding potential, di ff erential inclusion, topological degree, Fredholm operator, non-local initial condition. ˙ x ( t ) ∈ F ( t , x ( t )). Next, we try to abstract properties of the map appearing in the nonlocalcondition by formulating Theorem 6., which generalizes the previous considerations.In Theorem 7. we give an example of the property of g , which also guarantees theexistence of solutions to the problem (1), but that excludes same time sublinear growth (6)with a constant c reliminaries Let X and Z be a Banach space. An open ball with center x ∈ X (resp. zero) and radius r > B ( x , r ) ( B ( r )). Symbol B ( r ) stands for a closed ball. If A ⊂ X , then A denotes the closure of A , ∂ A the boundary of A and co A the convex hull of A . The innerproduct in ’ N represents h· , ·i .For I ⊂ ’ , ( C ( I , ’ N ) , || · || ) is the Banach space of continuous maps I → ’ N equippedwith the maximum norm and AC ( I , ’ N ) is the subspace of absolutely continuous functions.By ( L ( I , ’ N ) , || · || ) we mean the Banach space of all Lebesgue integrable maps.A multivalued map F : X ⊸ Z assigns to any x ∈ X a nonempty subset F ( x ) ⊂ Z . Theset of all fixed points of the multivalued (or univalent) map F is denoted by Fix( F ). Recallthat a map F : X ⊸ Z with compact values is upper semicontinuous i ff given a sequence( x n , y n ) in the graph of F with x n → x in X , there is a subsequence y k n → y ∈ F ( x ). If theimage F ( X ) is relatively compact in Z , then we say that F is a compact multivalued map.A multimap F : I ⊸ ’ N is called measurable, if { t ∈ I : F ( t ) ⊂ A } belongs to the Lebesgue σ -field of I for every closed A ⊂ ’ N .A set-valued map F : X ⊸ Z is admissible (compare [5, Def.40.1]) if there is a metricspace Y and two continuous functions p : X → Y , q : Y → Z from which p is a Vietorismap such that F ( x ) = q ( p − ( x )) for every x ∈ X . It turns out that every acyclic multivaluedmap, i.e. an usc multimap with compact acyclic values, is admissible. In particular, everyusc multivalued map with compact convex values is admissible.Let M be the set of triples ( Id − F , Ω , y ) such that Ω ⊂ X is open bounded, Id isthe identity, F : Ω ⊸ X is a compact usc multimap with closed convex values, and y < ( Id − F )( ∂ Ω ). Then it is possible to define, using approximation methods for multivaluedmaps, a unique topological degree function deg : M → š (see [3, 5, 8] for details). Thisdegree inherits directly all the basic properties of the Leray-Schauder degree, among othersexistence, localization, normality, additivity, homotopy invariance and contractivity.Let L : dom L ⊂ X → Z be a linear Fredholm operator of zero index such that Im L ⊂ Z is a closed subspace. Consider continuous linear idempotent projections P : X → X and Q : Z → Z such that Im P = ker L , Im L = ker Q . By the symbol L P we denote therestriction of the operator L on dom L ∩ ker P and by K P , Q : Z → dom L ∩ ker P the operatorgiven by the relation K P , Q ( z ) = L − P ( Id − Q )( z ). Since dim ker L = dim coker L < ∞ wemay choose a linear homeomorphism Φ : Im Q → Im P .The proofs of main results of this paper will be based in particular on the followingcoincidence point theorem. In the wake of [4] we should specify it as the ContinuationTheorem. Theorem 1. [3, Lemma 13.1.]
Let X and Z be real Banach spaces, L : dom L ⊂ X → Zbe Fredholm operator of index zero and with closed graph, Ω ⊂ X be open bounded andN : Ω ⊸ Z be such that QN and K P , Q N are compact usc multimaps with compact convexvalues. Assume also that
UIDING POTENTIAL METHOD FOR DIFFERENTIAL INCLUSIONS WITH NONLOCAL CONDITIONS 3 (a) Lx < λ N ( x ) for all λ ∈ (0 , and x ∈ dom L ∩ ∂ Ω , (b) 0 < QN ( x ) on ker L ∩ ∂ Ω and deg( Φ QN , ker L ∩ Ω , , .Then deg( Id − P − ( Φ Q + K P , Q ) N , Ω , , . In particular, Lx ∈ N ( x ) has a solution in Ω . We are able to place considered boundary value problem (1) in the context of Theorem1. and the above introduced general framework. Let X : = C ( I , ’ N ), Z : = L ( I , ’ N ) × ’ N ,dom L : = AC ( I , ’ N ); L : dom L ⊂ X → Z be such that Lx : = ( ˙ x , L = i ( ’ N ),where i : ’ N ֒ → C ( I , ’ N ) is defined by i ( x )( t ) = x , Im L = L ( I , ’ N ) × { } and coker L ≈ ’ N , i.e. L is a Fredholm mapping of index zero. Consider continuous linear operators P : X → X and Q : Z → Z such that P ( x )( t ) = x (0) and Q (( y , v )) = (0 , v ). It is clearthat ( P , Q ) is an exact pair of idempotent projections with respect to L . Define operator N F : X → Z by N F ( x ) : = n f ∈ L ( I , ’ N ) : f ( t ) ∈ F ( t , x ( t )) for a.a. t ∈ I o × { γ ( x ) } , where γ = ev − g (ev t : C ( I , ’ N ) → ’ N stands for the evaluation at point t ∈ I ). It is clearthat the nonlocal Cauchy problem (1) is equivalent to the operator inclusion Lx ∈ N F ( x ).For a given potential V ∈ C ( ’ N , ’ ) we define the induced vector field W V : ’ N → ’ N by the formula W V ( x ) = ∇ V ( x ) , if |∇ V ( x ) | , ∇ V ( x ) |∇ V ( x ) | , if |∇ V ( x ) | > . Then W V is continuous and bounded. Let N W V : X → Z be such that N W V ( x ) = ( W V ( x ( · )) , γ ( x )) . In this particular case we have K P , Q N W V ( x )( t ) = R t W V ( x ( s )) ds . Therefore N W V is L -compact (see [4] for more details).Throughout the rest of this paper Φ : Im Q → ker L will denote a fixed linear homeo-morphism, given by Φ ((0 , x )) = i ( x ).Regarding the guiding potential evoked in the title, we are obliged to introduce someauxiliary concepts. We say that potential V : ’ N → ’ is monotone , if (cid:2) | x | | y | ⇒ V ( x ) V ( y ) (cid:3) . Observe that, if potential V is monotone, then V satisfies [ | x | = | y | ⇒ V ( x ) = V ( y )]. Inparticular, V is also even. The function V : ’ N → ’ is coercive , if lim | x |→ + ∞ V ( x ) = + ∞ . Example 1. (i)
The classical Krasnosel’skii-Perov potential V ( x ) = | x | is monotone coerciveand continuously di ff erentiable. (ii) Suppose f ∈ C ( ’ , ’ ) is nondecreasing and there is R > such that f ( x ) > xfor x > R. Then the mapping V ( x ) = f (cid:16) | x | (cid:17) is a monotone coercive potential ofC -class. The method of guiding potential exploits also another generic notion:
Definition 1.
A continuously di ff erentiable function V : ’ N → ’ is called a nonsingularpotential, if ∃ R > ∀ | x | > R , ∇ V ( x ) , . (2)The symbol h A , B i − (or h A , B i + ) we use below stands for the lower inner product (upperinner product) of nonempty compact subsets of ’ n , i.e. h A , B i − = inf {h a , b i : a ∈ A , b ∈ B } , h A , B i + = sup {h a , b i : a ∈ A , b ∈ B } . RADOSŁAW PIETKUN
Let us specify the concept of a guiding potential for the multimap F : I × ’ N ⊸ ’ N (compare [2, 6]): Definition 2.
Suppose that V ∈ C ( ’ N , ’ ) is nonsingular. We will call the mapping Va weakly positively guiding potential for multimap F, if ∃ R > ∀ | x | > R ∀ t ∈ I ∃ y ∈ F ( t , x ) , h∇ V ( x ) , y i > , (3) a weakly negatively guiding potential for multimap F, if ∃ R > ∀ | x | > R ∀ t ∈ I ∃ y ∈ F ( t , x ) , h∇ V ( x ) , y i , (4) a strictly negatively guiding potential for multimap F, if ∃ R > ∀ | x | > R ∀ t ∈ I h∇ V ( x ) , F ( t , x ) i + < . (5)In what follows we shall permanently refer to a certain initial set of assumptions regard-ing the multimap F , which concretizes the following definition: Definition 3.
We will say that a nonempty convex compact valued map F : I × ’ N ⊸ ’ N is a Carathéodory map, if the following conditions are satisfied: ( F ) the multimap t F ( t , x ) is measurable for every fixed x ∈ ’ N , ( F ) the multimap x F ( t , x ) is upper semicontinuous for t ∈ I a.e. ( F ) sup {| y | : y ∈ F ( t , x ) } µ ( t )(1 + | x | ) for every ( t , x ) ∈ I × ’ N , where µ ∈ L ( I , ’ ) .
3. M ain R esults The first assertion illustrates why referring to guiding potential method is superfluous, ifthe function g : C ( I , ’ N ) → ’ N satisfies su ffi ciently strong assumption regarding sublineargrowth. Theorem 2.
Let F : I × ’ N ⊸ ’ N be a Carathéodory map. Let g : C ( I , ’ N ) → ’ N be acontinuous mapping with contractive sublinear growth, i.e. ∃ c ∈ (0 , ∃ d > ∀ x ∈ C ( I , ’ N ) | g ( x ) | c || x || + d . (6) Then the nonlocal initial value problem (1) possesses at least one solution.Proof.
Define so-called solution set map S F : ’ N ⊸ C ( I , ’ N ), associated with the Cauchyproblem ˙ x ( t ) ∈ F ( t , x ( t )) , a.e. t ∈ I , x (0) = x , (7)by the formula S F ( x ) : = n x ∈ C ( I , ’ N ) : x is a solution of (7) o . As it is well known the multimap S F is admissible (in terms of [5, Def. 40.1]). In fact, it isan usc multimap with compact R δ values. Now we are able to introduce the Poincaré-likeoperator P : ’ N ⊸ ’ N related to problem (1), given by P = g ◦ S F .It is clear that S F ( x ) ⊂ B ( | x | + || µ || ) for every x ∈ ’ N . Thus S F ( B ( r )) ⊂ B ( r + || µ || ).On the other hand we have g ( B ( M )) ⊂ B ( cM + d ), due to (6). Therefore P ( B ( r )) ⊂ B ( c ( r + || µ || ) + d ). By the fact that c <
1, we can take any r such that r > c || µ || + d − c > P ( B ( r )) ⊂ B ( r ). Of course, the multivalued operator P : B ( r ) ⊸ B ( r ) iscompact admissible. As such, this operator possesses a fixed point x ∈ B ( r ) in view of the UIDING POTENTIAL METHOD FOR DIFFERENTIAL INCLUSIONS WITH NONLOCAL CONDITIONS 5 generalized Schauder fixed point theorem ([6, Th.41.13]). This point corresponds to thesolution x of the nonlocal Cauchy problem (1) in such a way that x (0) = x = g ( x ). (cid:3) Corollary 1.
Suppose all the assumptions of Theorem 2. are satisfied. Then the set S F ( g ) of solutions to the problem (1) is nonempty and compact as a subset of C ( I , ’ N ) .Proof. Retaining the notation of the proof of Theorem 2., we claim that the fixed point setFix( P ) is compact. Indeed, take x ∈ Fix( P ) and observe that | x | = | g ( x ) | c || x || + d forsome x ∈ S F ( x ). Thus, | x | c ( | x | + || µ || ) + d and as a result | x | c || µ || + d − c . Fix( P ) is obviously closed, hence it must be compact.Define a multivalued map Ψ : Fix( P ) ⊸ C ( I , ’ N ), by the formula Ψ ( x ) = { x ∈ S F ( x ) : g ( x ) = x } . It easy to see that Ψ is a compact valued usc multimap and that S F ( g ) coincides with Ψ (Fix( P )). Therefore, the solution set S F ( g ) is compact. (cid:3) Remark 1.
The following exemplary classes of mappings have sublinear growth: compactmaps, linear continuous maps, Lipschitzian maps, uniformly continuous maps. A su ffi cientcondition for a map to have a contractive sublinear growth is, for instance: compactness,to be linear continuous with norm less then one, contractivity. The next theorem addresses the issue of the existence of solutions to the problem (1),when mapping g = − ev T . Theorem 3.
If F : I × ’ N ⊸ ’ N is a Carathéodory map and there exists an even coerciveweakly positively guiding potential V : ’ N → ’ for the map F, then the antiperiodicproblem ˙ x ( t ) ∈ F ( t , x ( t )) , a.e. t ∈ I , x (0) = − x ( T ) (8) possesses at least one solution.Proof. Choose R > r > max { V ( x ) : | x | R } and put G : = V − (( −∞ , r ))). Then Ω : = C ( I , G ) is open in X . Since V is coercive, Ω is alsobounded.Suppose that x ∈ Ω = C ( I , G ) is a solution to the following antiperiodic problem ˙ x ( t ) = λ W V ( x ( t )) , t ∈ I , x (0) = − x ( T )for some λ ∈ (0 , t ∈ (0 , T ) such that x ( t ) ∈ ∂ G , then t is a critical point of V ◦ x in view of Fermat lemma, i.e. ( V ◦ x ) ′ ( t ) =
0. On the other hand however ( V ◦ x ) ′ ( t ) = h∇ V ( x ( t )) , λ W V ( x ( t )) i >
0, since | x ( t ) | > R . Thus x ( t ) ∈ ∂ G is contradicted. Assumingthat x (0) ∈ ∂ G we see that 1 h ( V ( x ( h )) − V ( x (0))) h >
0. It means that the right-hand derivative ( V ◦ x ) ′ (0)
0. However, atthe same time h∇ V ( x (0)) , λ W V ( x (0)) i >
0. Once again a contradiction. This implies that x (0) < ∂ G . Because V is even the boundary ∂ G of the set G is symmetric with respect to RADOSŁAW PIETKUN the origin. Thus it follows also that x ( T ) < ∂ G . Summing up, we see that x ( I ) ∩ ∂ G = ∅ ,i.e. x < ∂ Ω . In fact, we have shown that Lx , λ N W V ( x ) for every x ∈ dom L ∩ ∂ Ω and λ ∈ (0 , < QN W V (ker L ∩ ∂ Ω ) is equivalent to x , − x for every x ∈ ∂ G . The latter is obviously true, since ∂ G ⊂ ’ N \ B ( R ) ⊂ ’ N \ { } .Let us notice that i : i − ( Ω ) → ker L ∩ Ω is a di ff eomorphism of C -class. Using standardproperty of the Brouwer degree we getdeg( Φ QN W V , ker L ∩ Ω , = deg( i − Φ QN W V i , i − ( Ω ) , = deg( γ ◦ i , G , = deg((ev + ev T ) i , G , . Observe that (ev + ev T ) i ( x ) = x and define U ∈ C ( ’ N , ’ ), by U ( x ) = | x | . Since U is acoercive potential we know, by [7, Th.12.9.], that Ind( U ) =
1. Thereforedeg( Φ QN W V , ker L ∩ Ω , = deg( ∇ U , G , = Ind( U ) , . We are in position to apply the Continuation Theorem (Theorem 1.) to deduce thatdeg( Id − P − ( Φ Q + K P , Q ) N W V , Ω , = deg( Id − P − Φ QN W V , Ω , = deg( − Φ QN W V , ker L ∩ Ω , , . (9)Of course, the integer deg( Id − P − ( Φ Q + K P , Q ) N W V , Ω ,
0) is nothing more than the coin-cidence degree deg(( L , N W V ) , Ω ).Following proof of Theorem 4.4. in [6] we bring in a useful auxiliary multivalued map F V : I × ’ N ⊸ ’ N defined by: F V ( t , x ) = F ( t , x ) ∩ { y ∈ ’ N : h α ( x ) ∇ V ( x ) , y i > } , where α is the Urysohn function α ( x ) = , | x | R , , | x | > R . So defined map F V has the property that F V ( t , x ) ⊂ F ( t , x ), while the function V is a guidingpotential for F V in the strict sense, i.e. h∇ V ( x ) , F V ( t , x ) i − > t , x ) ∈ I × ’ N , with | x | > R . It is routine to check that F V is a Carathéodory map.Now we define another map G : I × ’ N × [0 , ⊸ ’ N by G ( t , x , λ ) : = λ W V ( x ) + (1 − λ ) F V ( t , x ) . It is easy to see that G is also a Carathéodory map. In particular, the map G ( t , · , · ) is uppersemicontinuous for a.a t ∈ I and sup {| y | : y ∈ G ( t , x , λ ) } max { µ ( t ) , } (1 + | x | ).Define N : Ω × [0 , ⊸ Z by the formula N ( x , λ ) : = { f ∈ L ( I , ’ N ) : f ( t ) ∈ G ( t , x ( t ) , λ ) for a.a. t ∈ I } × { γ ( x ) } , where γ = ev + ev T . Now we may introduce a homotopy H : Ω × [0 , ⊸ X in thefollowing way: H ( x , λ ) : = Px + ( Φ Q + K P , Q ) N ( x , λ ) . (11)Observe that QN ( x , λ ) = (0 , γ ( x )) and K P , Q N ( x , λ )( t ) = R t G ( s , x ( s ) , λ ) ds . It is clear that QN is completely continuous. Standard and plain arguments justify that K P , Q N is an uscmultimap with compact convex values and the range K P , Q N ( Ω × [0 , P is linear continuous and has a finite dimensional range it is UIDING POTENTIAL METHOD FOR DIFFERENTIAL INCLUSIONS WITH NONLOCAL CONDITIONS 7 clear that P ( Ω ) is also relatively compact. All these observations allow to conclude that thehomotopy H is a compact usc multimap with convex compact values.We claim that x < H ( x , λ ) on ∂ Ω × [0 , x ∈ Ω = C ( I , G ) be a solution to theproblem ˙ x ( t ) ∈ G ( t , x ( t ) , λ ) , a.e. t ∈ I , x (0) = − x ( T )for some λ ∈ (0 , t ∈ [0 , T ) such that x ( t ) ∈ ∂ G . This means that | x ( t ) | > R and as aresult there is δ ∈ (0 , T − t ] such that | x ( t ) | > R for t ∈ [ t , t + δ ]. It implies that h∇ V ( x ( t )) , ˙ x ( t ) i > h∇ V ( x ( t )) , λ W V ( x ( t )) + (1 − λ ) F V ( t , x ( t )) i − > λ h∇ V ( x ( t )) , W V ( x ( t )) i + (1 − λ ) h∇ V ( x ( t )) , F V ( t , x ( t )) i − > λ h∇ V ( x ( t )) , W V ( x ( t )) i (by (10)) > t ∈ [ t , t + δ ]. Therefore we get: V ( x ( t + δ )) − V ( x ( t )) = Z t + δ t h∇ V ( x ( t )) , ˙ x ( t ) i dt > V ( x ( t )) < V ( x ( t + δ )) r . We arrive at contradiction with x ( t ) ∈ ∂ G . Oncemore we use the fact that V is even to infer that x ( T ) < ∂ G . Summing up, x < ∂ Ω . Infact, the above reasoning proves that Lx < N ( x , λ ) for all x ∈ dom L ∩ ∂ Ω and λ ∈ (0 , Lx < N ( x ,
0) for every x ∈ ∂ Ω (otherwise (8) has a solutionand there is nothing to prove). Taking into account that Lx ∈ N ( x , λ ) ⇔ x ∈ Px + ( Φ Q + K P , Q ) N ( x , λ ) we conclude that x < H ( x , λ ) on ∂ Ω × [0 ,
1] is verified.Relying on the homotopy invariance of the topological degree for the class M (or if theReader considers it more appriopriate, of coincidence degree theory) we obtain the equalitydeg( Id − P − ( Φ Q + K P , Q ) N ( · , , Ω , = deg( Id − H ( · , , Ω , = deg( Id − H ( · , , Ω , = deg( Id − P − ( Φ Q + K P , Q ) N W V , Ω , . In view of (9) we see that deg( Id − P − ( Φ Q + K P , Q ) N ( · , , Ω , ,
0. This means that thereis a coincidence point x ∈ Ω ∩ dom L of the inclusion Lx ∈ N ( x , (cid:3) Corollary 2.
Assume that F : I × ’ N ⊸ ’ N is a Carathéodory map and there exists aneven coercive weakly negatively guiding potential V : ’ N → ’ for the map F. Then theantiperiodic problem (8) has at least one solution.Proof. Define a map W : ’ N → ’ , by W ( x ) = − V ( x ). Then W is an even coercive weaklypositively guiding potential for the map F . Thus, the thesis of Theorem 3. applies. (cid:3) Remark 2.
Let us notice that the coercivity condition in Theorem 3. could not be dropped.In fact, V ( x ) −→ | x |→ + ∞ + ∞ ⇐⇒ ∀ r > V − (( −∞ , r )) is bounded . The following theorem refers to the case when the function g takes the form of a linearcombination of evaluations at fixed points of division of segment I . Note that the anti-periodicity condition is a special case of the multi-point discrete mean condition. RADOSŁAW PIETKUN
Theorem 4.
Let < t < t < . . . < t n T be arbitrary, but fixed, P ni = | α i | and P ni = α i , . If F : I × ’ N ⊸ ’ N is a Carathéodory map and there exists a monotonecoercive weakly negatively guiding potential V : ’ N → ’ for the map F, then the followingnonlocal initial value problem with multi-point discrete mean condition ˙ x ( t ) ∈ F ( t , x ( t )) , a.e. t ∈ I , x (0) = n P i = α i x ( t i ) (13) possesses at least one solution.Proof. We keep the notations introduced in the proof of Theorem 3. Let x ∈ Ω be suchthat ˙ x ( t ) = − λ W V ( x ( t )) , t ∈ I , x (0) = n P i = α i x ( t i ) , for some λ ∈ (0 , h∇ V ( x ( t )) , − λ W V ( x ( t )) i < t ∈ I , with | x ( t ) | > R , we infer in a strictly analogous manner to the proof of Theorem 3 that x ( t ) < ∂ G for t ∈ (0 , T ). If x ( T ) ∈ ∂ G , then the left-hand derivative ( V ◦ x ) ′ ( T ) >
0, which is incontradiction with h∇ V ( x ( T )) , − λ W V ( x ( T )) i <
0. Notice that | x (0) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X i = α i x ( t i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X i = | α i || x ( t i ) | max i n | x ( t i ) | = | x ( t i ) | , where t i ∈ (0 , T ]. Keeping in mind that V is nondecreasing we get V ( x (0)) V ( x ( t i )) < r .Thus x (0) < ∂ G and x < ∂ Ω . These considerations show that Lx , λ N ( − W V ) ( x ) for every x ∈ dom L ∩ ∂ Ω and λ ∈ (0 , < QN ( − W V ) (ker L ∩ ∂ Ω ) is equivalent to γ ( x ) , x ∈ ker L ∩ ∂ Ω . In the cur-rent case γ = ev − P ni = α i ev t i . Therefore, we demand that ev ( i ( x )) , P ni = α i ev t i ( i ( x )),i.e. x , P ni = α i x for every x ∈ ∂ G . The latter is true, since P ni = α i , W ( x ) = (1 − P ni = α i ) | x | . Applying again Continuation Theorem we getdeg( Id − P − ( Φ Q + K P , Q ) N ( − W V ) , Ω , = deg( Id − P − Φ QN ( − W V ) , Ω , = deg( − Φ QN ( − W V ) , ker L ∩ Ω , = deg( − γ ◦ i , G , = deg − ev − n X i = α i ev t i i , G , = deg( −∇ W , G , = ( − N + deg( ∇ W , B ( R ) , = ( − N + Ind( W ) , . (14)Let us modify definition of the multimap F V : I × ’ N ⊸ ’ N in the following way: F V ( t , x ) = F ( t , x ) ∩ { y ∈ ’ N : h α ( x ) ∇ V ( x ) , y i } . (15)Then (10) takes the form h∇ V ( x ) , F V ( t , x ) i + t , x ) ∈ I × ’ N , with | x | > R . We change respectively definition of multimap G : I × ’ N × [0 , ⊸ ’ N , namely G ( t , x , λ ) : = λ ( − W V )( x ) + (1 − λ ) F V ( t , x ) . Our aim is to show that x < H ( x , λ ) on ∂ Ω × [0 , H is a homotopy defined by (11).Let x ∈ Ω be a solution to the nonlocal Cauchy problem ˙ x ( t ) ∈ G ( t , x ( t ) , λ ) , a.e. t ∈ I , x (0) = P ni = α i x ( t i ) UIDING POTENTIAL METHOD FOR DIFFERENTIAL INCLUSIONS WITH NONLOCAL CONDITIONS 9 for some λ ∈ (0 , t ∈ (0 , T ] such that x ( t ) ∈ ∂ G , then there is δ ∈ (0 , t ] suchthat | x ( t ) | > R for t ∈ [ t − δ, t ]. Now (12) assumes the form h∇ V ( x ( t )) , ˙ x ( t ) i h∇ V ( x ( t )) , λ ( − W V ( x ( t ))) + (1 − λ ) F V ( t , x ( t )) i + λ h∇ V ( x ( t )) , − W V ( x ( t )) i + (1 − λ ) h∇ V ( x ( t )) , F V ( t , x ( t )) i + λ h∇ V ( x ( t )) , − W V ( x ( t )) i (by (16)) < t ∈ [ t − δ, t ]. Consequently V ( x ( t )) − V ( x ( t − δ )) = Z t t − δ h∇ V ( x ( t )) , ˙ x ( t ) i dt < . Thus V ( x ( t )) < V ( x ( t − δ )) r and x ( t ) ∈ ∂ G is contradicted. Recall that the boundarycondition implies | x (0) | | x ( t i ) | for some t i ∈ (0 , T ]. Thus V ( x (0)) < r , since V ismonotone. Consequently x < ∂ Ω . Analogous reasoning like in the proof of Theorem 3.justifies x < H ( x , λ ) on ∂ Ω × [0 , Id − P − ( Φ Q + K P , Q ) N ( · , , Ω , = deg( Id − H ( · , , Ω , = deg( Id − H ( · , , Ω , = deg( Id − P − ( Φ Q + K P , Q ) N ( − W V ) , Ω , , . This indicates that there is a coincidence point x ∈ Ω ∩ dom L of the operator inclusion Lx ∈ N ( x , (cid:3) Substitution of the notion of a weakly negatively guiding potential in the hypotheses ofTheorem 4. leads to the following conclusion:
Corollary 3.
Let t < t < . . . < t n < T be arbitrary, but fixed, P ni = | α i | and P ni = α i , . If F : I × ’ N ⊸ ’ N is a Carathéodory map and there exists a monotonecoercive weakly positively guiding potential V : ’ N → ’ for the map F, then the followingnonlocal boundary value problem with multi-point discrete mean condition ˙ x ( t ) ∈ F ( t , x ( t )) , a.e. t ∈ I , x ( T ) = n P i = α i x ( t i ) possesses at least one solution. Subsequent result concerns the situation where the mapping g, determining the bound-ary condition, has the form of mean value of the composition of its argument and somesubsidiary function h . Unfortunately, the argumentation contained in the proof of thistheorem, based on the evaluation of the Brouwer degree of Id − h , excludes the case g ( x ) = T R T x ( t ) dt . Theorem 5.
Let h : ’ N → ’ N be a continuous mapping such that | h ( x ) | | x | for everyx ∈ ’ N . Assume further that the fixed point set of h is compact. If F : I × ’ N ⊸ ’ N is a Carathéodory map and there exists a monotone coercive weakly negatively guidingpotential V : ’ N → ’ for the map F, then the following nonlocal Cauchy problem withmean value condition ˙ x ( t ) ∈ F ( t , x ( t )) , a.e. t ∈ I , x (0) = T R T h ( x ( t )) dt (17) possesses at least one solution. Proof.
The fixed point set Fix( h ) is compact i ff there is M > h ) ⊂ B ( M ).Choose R > M according to (2) and (4). Following the scheme presented in the proof ofTheorem 4., take x ∈ Ω such that ˙ x ( t ) = − λ W V ( x ( t )) , t ∈ I , x (0) = T R T h ( x ( t )) dt , for some λ ∈ (0 , x ( t ) < ∂ G for t ∈ (0 , T ]. Suppose | x (0) | > | x ( t ) | for t ∈ (0 , T ]. Then | x (0) | = T Z T | x (0) | dt > T Z T | x ( t ) | dt > T Z T | h ( x ( t )) | dt > (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T Z T h ( x ( t )) dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = | x (0) | . Therefore there exists t ∈ (0 , T ] such that | x (0) | | x ( t ) | . Since V is monotone we have V ( x (0)) V ( x ( t )) < r , i.e. x (0) < ∂ G . This proves that x < ∂ Ω resulting in: Lx , λ N ( − W V ) ( x ) for every x ∈ dom L ∩ ∂ Ω and λ ∈ (0 , < QN ( − W V ) (ker L ∩ ∂ Ω ) is equivalent to x , T R T h ( i ( x )( t )) dt , i.e. x , h ( x ) for x ∈ ∂ G . The latter requirement is fulfilled in our case, because x ∈ ∂ G ⇒ | x | > R ⇒ x < B ( M ) ⇒ x < Fix( h ) ⇒ x , h ( x ) . Take x ∈ ∂ G . Then | x − h ( x )) | > | h ( x ) | | x | . Thus | x − h ( x )) − x | = | h ( x ) | < | x − h ( x ) | + | x | . The last inequality is an equivalent formulation of the Poincaré-Bohl theorem ([7, Th.2.1.]), which means that λ ( Id − h )( x ) + (1 − λ ) Id ( x ) , x , λ ) ∈ ∂ G × [0 , Id − h and Id are joined by the linear homotopy,which has no zeros on the boundary ∂ G . Apply Continuation Theorem to see thatdeg( Id − P − ( Φ Q + K P , Q ) N ( − W V ) , Ω , = deg( Id − P − Φ QN ( − W V ) , Ω , = deg( − Φ QN ( − W V ) , ker L ∩ Ω , = deg( − ( Id − h ) , G , = ( − N + deg( Id , G , = ( − N + , , (18)as 0 ∈ B ( R ) ⊂ G .Following proof of Theorem 4. we consider a solution x ∈ Ω of the succeeding bound-ary value problem ˙ x ( t ) ∈ G ( t , x ( t ) , λ ) , a.e. t ∈ I , x (0) = T R T h ( x ( t )) dt , for some λ ∈ (0 , x ( t ) < ∂ G for t ∈ (0 , T ]. The boundary condition as before implies that x (0) < ∂ G . Therefore, x < ∂ Ω ,which means that the homotopy H has no fixed points on ∂ Ω × [0 , Id − P − ( Φ Q + K P , Q ) N ( · , , Ω , = deg( Id − H ( · , , Ω , = deg( Id − H ( · , , Ω , = deg( Id − P − ( Φ Q + K P , Q ) N ( − W V ) , Ω , , . Summarizing, there exists a coincidence point x ∈ Ω ∩ dom L of the operator inclusion Lx ∈ N ( x , (cid:3) Corollary 4.
Let h : ’ N → ’ N be a continuous mapping such that | h ( x ) | | x | for everyx ∈ ’ N . Assume further that the fixed point set of h is compact. If F : I × ’ N ⊸ ’ N is a Carathéodory map and there exists a monotone coercive weakly positively guiding UIDING POTENTIAL METHOD FOR DIFFERENTIAL INCLUSIONS WITH NONLOCAL CONDITIONS 11 potential V : ’ N → ’ for the map F, then the following nonlocal boundary value problemwith mean value condition ˙ x ( t ) ∈ F ( t , x ( t )) , a.e. t ∈ I , x ( T ) = T R T h ( x ( t )) dtpossesses at least one solution. A cursory look at the proof of Theorem 4. and 5. leads to the following generalization.
Theorem 6.
Assume that F : I × ’ N ⊸ ’ N is a Carathéodory map and there exists a mono-tone coercive weakly negatively guiding potential V : ’ N → ’ for the map F. Let R > bechosen with accordance to (2) and (4) and r > max { V ( x ) : | x | R } . Let g : C ( I , ’ N ) → ’ N be a continuous function, which maps bounded sets into bounded sets and satisfies addi-tionally the following conditions: (i) ∀ x ∈ dom L ∩ C ( I , V − (( −∞ , r ])) [ x (0) = g ( x )] ⇒ ∃ t ∈ (0 , T ] | g ( x ) | | x ( t ) | , (ii) | g ( i ( x )) | | x | for all x ∈ V − ( { r } ) , (iii) Fix( g ◦ i ) ∩ V − ( { r } ) = ∅ .Then the nonlocal Cauchy problem (1) possesses at least one solution.Proof. Our reasoning will reproduce exactly the proceedings of the proof of Theorem 4.Let x ∈ Ω be such that ˙ x ( t ) = − λ W V ( x ( t )) , t ∈ I , x (0) = g ( x ) , for some λ ∈ (0 , x ( t ) < ∂ G for t ∈ (0 , T ]. Notice that G = V − (( −∞ , r ]), due to (2). Thus, x ∈ dom L ∩ C ( I , V − (( −∞ , r ])). It follows from condition (i) that there exists t ∈ (0 , T ] suchthat | x (0) | | x ( t ) | . Since V is monotone we get V ( x (0)) V ( x ( t )) < r . Therefore x (0) < ∂ G and x < ∂ Ω . Consequently, Lx , λ N ( − W V ) ( x ) for every x ∈ dom L ∩ ∂ Ω and λ ∈ (0 , < QN ( − W V ) (ker L ∩ ∂ Ω ) is equivalent to ev ( x ) , g ( x ) for x ∈ ker L ∩ ∂ Ω .In other words: x , g ( i ( x )) for every x ∈ ∂ G = V − ( { r } ). Of course, condition (iii) isformulated so that the latter is true.Take x ∈ ∂ G . Then | x − g ( i ( x )) | >
0, by condition (iii) and | g ( i ( x )) | | x | , bycondition (ii). Whence | x − g ( i ( x )) − x | < | x − g ( i ( x )) | + | x | . The last inequality isa simple reformulation of the Poincaré-Bohl condition ([7, Th.2.1.]), which means that λ ( Id − g ◦ i )( x ) + (1 − λ ) Id ( x ) , x , λ ) ∈ ∂ G × [0 , Id − g ◦ i and Id are joined by the homotopy, nonsingular on the boundary ∂ G . Now we can applyContinuation Theorem:deg( Id − P − ( Φ Q + K P , Q ) N ( − W V ) , Ω , = deg( Id − P − Φ QN ( − W V ) , Ω , = deg( − Φ QN ( − W V ) , ker L ∩ Ω , = deg( − ( Id − g ◦ i ) , G , = ( − N + deg( Id , G , = ( − N + , , (19)as 0 ∈ B ( R ) ⊂ G .The rest of the proof is strictly analogous to the remaining part of the proof of Theorem4. In particular, if x ∈ Ω is a solution to the problem ˙ x ( t ) ∈ G ( t , x ( t ) , λ ) , a.e. t ∈ I , x (0) = g ( x )for some λ ∈ (0 , x (0) < ∂ G . In this way we prove that x < ∂ Ω , which means that x < H ( x , λ ) on ∂ Ω × [0 , ff ering to the property (19) we conclude thatdeg( Id − P − ( Φ Q + K P , Q ) N ( · , , Ω , = deg( Id − H ( · , , Ω , = deg( Id − H ( · , , Ω , = deg( Id − P − ( Φ Q + K P , Q ) N ( − W V ) , Ω , , . The coincidence point x ∈ Ω ∩ dom L of the operator inclusion Lx ∈ N ( x ,
0) determinesthe solution of the nonlocal Cauchy problem (1). (cid:3)
Corollary 5.
Assume that F : I × ’ N ⊸ ’ N is a Carathéodory map and there existsa monotone coercive weakly positively guiding potential V : ’ N → ’ for the map F.Let R > be chosen with accordance to (2) and (3) and r > max { V ( x ) : | x | R } . Letg : C ( I , ’ N ) → ’ N be a continuous function, which maps bounded sets into bounded setsand satisfies conditions (ii)-(iii) of Theorem 6. along with: (iv) ∀ x ∈ dom L ∩ C ( I , V − (( −∞ , r ])) [ x ( T ) = g ( x )] ⇒ ∃ t ∈ (0 , T ] | g ( x ) | | x ( t ) | .Then the nonlocal boundary value problem ˙ x ( t ) ∈ F ( t , x ( t )) , a.e. t ∈ I , x ( T ) = g ( x ) possesses at least one solution. Example 2.
The mappings, which were used to define nonlocal initial conditions in Theo-rems 3.,4. and 5. satisfy conditions (i)-(iii) . Example 3.
The following requirement ∀ , x ∈ dom L ∩ C ( I , V − (( −∞ , r ])) ∃ t ∈ (0 , T ] | g ( x ) | < | x ( t ) | . implies conditions (i)-(iii) . Example 4.
If g : C ( I , ’ N ) → ’ N is such that | g ( x ) | < || x || for every x , , then inparticular conditions (i)-(iii) are satisfied. All previous theorems apply to the cases where the function g satisfies the estimation | g ( x ) | || x || for every x ∈ C ( I , ’ N ). The next result is based on such a property of g , whichprevents the function g from fulfillment of this estimation. Theorem 7.
Let F : I × ’ N ⊸ ’ N be a Carathéodory map. Assume that g : C ( I , ’ N ) → ’ N is continuous, maps bounded sets into bounded sets and satisfies the following condi-tions: (a) lim inf || x ||→ + ∞ | g ( x ) ||| x || > , (b) the counter image ( g ◦ i ) − ( { } ) is compact and deg( g ◦ i , B ( R ) , , , where ( g ◦ i ) − ( { } ) ⊂ B ( R ) .Then the solution set S F ( g ) of nonlocal initial value problem (1) is nonempty and compactas a subset of C ( I , ’ N ) .Proof. Let R > g ◦ i ) − ( { } ) ⊂ B ( R ). Property (a) of the mapping g meansthat there is M > | g ( x ) | > || x || for all || x || > M . Assume that M > R and put Ω : = C ( I , B ( M )). Define N F : Ω ⊸ Z , by N F ( x ) : = n f ∈ L ( I , ’ N ) : f ( t ) ∈ F ( t , x ( t )) for a.a. t ∈ I o × { γ ( x ) } , where γ = ev − g . Taking into account that QN F ( x ) = (0 , γ ( x )) and K P , Q N F ( x )( t ) = R t F ( s , x ( s )) ds we infer that QN F and K P , Q N F are compact usc multimaps with compactconvex values. UIDING POTENTIAL METHOD FOR DIFFERENTIAL INCLUSIONS WITH NONLOCAL CONDITIONS 13
Suppose x ∈ Ω is a solution to the following nonlocal Cauchy problem ˙ x ( t ) ∈ λ F ( t , ( x ( t )) , a.e. t ∈ I , x (0) = g ( x )for some λ ∈ (0 , || x || < M . Otherwise x (0) , g ( x ). Thus x < ∂ Ω and inclusion Lx ∈ λ N F ( x ) has no solution on ∂ Ω for every λ ∈ (0 , < QN F (ker L ∩ ∂ Ω ) is equivalent to x , g ( i ( x )) for every x ∈ ∂ B ( M ).However x ∈ ∂ B ( M ) ⇒ | x | = || i ( x ) || > M ⇒ | g ( i ( x )) | > | x | ⇒ x , g ( i ( x )) . Take x ∈ ∂ B ( M ). Then | x | < | g ( i ( x )) | . In other words | x − g ( i ( x )) − ( − g ( i ( x ))) | < | − g ( i ( x )) | . It follows from a corollary to the theorem of Poincaré-Bohl ([7, Th.2.3.]) thatvector fields Id − g ◦ i and − g ◦ i are mutually homotopic. Using standard properties of theBrouwer degree we see thatdeg( id − g ◦ i , B ( M ) , = deg( − g ◦ i , B ( M ) , = ( − N + deg( g ◦ i , B ( M ) , = ( − N + deg( g ◦ i , B ( R ) , , , by (b). Applying the same reasoning as in the proof of Theorem 4. we getdeg( Φ QN F , ker L ∩ Ω , = deg( γ ◦ i , B ( M ) , = deg( Id − g ◦ i , B ( M ) , , . In view of Theorem 1. there is a solution of the inclusion Lx ∈ N F ( x ) in Ω . This is ofcourse also a solution of the problem (1).It is easy to see that compactness of S F ( g ) is equivalent to the existence of a prioribounds on the solutions to (1). Indeed, suppose S F ( g ) is bounded. Then this set mustalso be equicontinuous in view of ( F ). Therefore it is relatively compact. The closed-ness of S F ( g ) is a straightforward consequence of Compactness Theorem ([1, Th.0.3.4.]),Convergence Theorem ([1, Th.1.4.1.]) and continuity of g .Suppose that for every r > x ∈ S F ( g ) such that || x || > r . Choose r : = M . Then | g ( x ) | > || x || > | x (0) | for some x ∈ S F ( g ), which yields a contradiction with x (0) = g ( x ). Therefore the solution set S F ( g ) must be bounded, completing the proof. (cid:3) Lemma 1.
Suppose F : I × ’ N ⊸ ’ N has sublinear growth, i.e. ( F ) is satisfied. Letx ∈ AC ( I , ’ N ) be a solution to the following Cauchy problem ˙ x ( t ) ∈ F ( t , x ( t )) , a.e. on I , x (0) = x . Then | x ( t ) | > r > for every t ∈ I, provided | x | > e || µ || ( r + − .Proof. Taking into account that x ∈ AC ( I , ’ N ) and | ˙ x ( t ) | µ ( t )(1 + | x ( t ) | ) a.e. on I , in wievof ( F ) we have | x ( t ) − x ( s ) | Z ts µ ( τ )(1 + | x ( τ ) | ) d τ for 0 s t T . Thus | x ( t ) | > | x ( s ) | − Z ts µ ( τ )(1 + | x ( τ ) | ) d τ for 0 s t T . Define f : [0 , t ] → ’ + by f ( s ) = | x ( t ) | + Z ts µ ( τ )(1 + | x ( τ ) | ) d τ + . Then f ( s ) > | x ( s ) | + > f ( t ) = | x ( t ) | + . (20)Obviously ln f ( t ) − ln f ( s ) = Z ts f ′ ( τ ) f ( τ ) d τ. Same time f ′ ( s ) = − µ ( s )(1 + | x ( s ) | )a.e. on [0 , t ]. Thereforeln f ( t ) f ( s ) = ln f ( t ) − ln f ( s ) = − Z ts µ ( τ )(1 + | x ( τ ) | ) f ( τ ) d τ (20) > − Z ts µ ( τ )(1 + | x ( τ ) | ) | x ( τ ) | + d τ = − Z ts µ ( τ ) d τ. From here f ( t ) > f ( s ) exp − Z ts µ ( τ ) d τ ! which implies | x ( t ) | + > ( | x ( s ) | +
1) exp − Z ts µ ( τ ) d τ ! for 0 s t T , by (20). Substituting s = | x ( t ) | > ( | x | +
1) exp − Z t µ ( τ ) d τ ! − > ( | x | + e −|| µ || − . Now the assertion of the lemma is already visible. (cid:3)
The purpose of the last theorem is to provide the conditions that ensure the compactnessof the set of solutions to nonlocal Cauchy problems that were the subject of interest inTheorems 3., 4. and 5.
Theorem 8.
Assume that F : I × ’ N ⊸ ’ N is a Carathéodory map. Suppose there existsa monotone coercive strictly negatively guiding potential V : ’ N → ’ for the map F.Let R > be chosen with accordance to (2) and (5) and r > max { V ( x ) : | x | R } . Letg : C ( I , ’ N ) → ’ N be a continuous function, which maps bounded sets into bounded setsand satisfies additionally the following conditions: (i) ∀ x ∈ AC ( I , ’ N ) [ x (0) = g ( x )] ⇒ ∃ t ∈ (0 , T ] | g ( x ) | | x ( t ) | , (ii) | g ( i ( x )) | | x | for all x ∈ V − ( { r } ) , (iii) Fix( g ◦ i ) ∩ V − ( { r } ) = ∅ .Then the solution set S F ( g ) of nonlocal initial value problem (1) is nonempty and compactas a subset of C ( I , ’ N ) .Proof. That the set S F ( g ) is not empty results from Theorem 6. Just as in the proof ofTheorem 7. it su ffi ces to show the boundedness of the solution set S F ( g ) to be certain of itscompactness.Take x ∈ S F ( g ) and suppose that | x (0) | > e || µ || ( R + −
1. From Lemma 1. it followsthat | x ( t ) | > R for every t ∈ I . Assume that | x (0) | | x ( t ) | for some t ∈ (0 , T ]. Then V ( x ( t )) − V ( x (0)) >
0, due to the monotonicity of V . On the other hand V ( x ( t )) − V ( x (0)) = R t h∇ V ( x ( s )) , ˙ x ( s ) i ds <
0, because h∇ V ( x ( s )) , ˙ x ( s ) i h∇ V ( x ( s )) , F ( s , x ( s )) i + < s ∈ [0 , t ] . UIDING POTENTIAL METHOD FOR DIFFERENTIAL INCLUSIONS WITH NONLOCAL CONDITIONS 15
We arrive at contradiction. If we assume the opposite, i.e. | x (0) | > | x ( t ) | for every t ∈ (0 , T ],then condition (i) is contradicted. Therefore it is impossible that the supposition | x (0) | > e || µ || ( R + − F ) along with the Gronwall inequality implies | x ( t ) | h || µ || + ( R + e || µ || − i e R t µ ( s ) ds for t ∈ I . Thus S F ( g ) ⊂ B ( M ), where M : = h || µ || + ( R + e || µ || − i e || µ || . (cid:3) R eferences [1] J. Aubin, A. Cellina, Di ff erential Inclusions , Springer, Berlin, 1984.[2] F. S. de Blasi, L. Górniewicz, G. Pianigiani, Topological degree and periodic solutions of di ff erential inclu-sions , Nonlinear Analysis 37 (1999), 217-245.[3] K. Deimling, Multivalued di ff erential equations , Walter de Gruyter, Berlin-New York, 1992.[4] R. Gaines, J. Mawhin, Coincidence Degree and Nonlinear Di ff erential Equations , Lecture Notes in Mathe-matics, 568, Springer-Verlag, Berlin, 1977.[5] L. Górniewicz, Topological fixed point theory of multivalued mappings , Second ed., Springer, Dordrecht,2006.[6] L. Górniewicz, S. Plaskacz,
Periodic solutions of di ff erential inclusions in ’ n , Boll. Un. Mat. Ital. 7(7-A)(1993), 409-420.[7] M. A. Krasnosel’skii, P. P. Zabreiko, Geometrical Methods of Nonlinear Analysis , Springer-Verlag, Berlin,1984.[8] V. Obukhovskii, P. Zecca, N. van Loi, S. Kornev,
Method of Guiding Functions in Problems of NonlinearAnalysis , Lecture Notes in Mathematics, 2076, Springer-Verlag, Berlin, 2013.T oru ´ n , P oland E-mail address ::