The antipodal mapping theorem and difference equations in Banach spaces

Size: px
Start display at page:

Download "The antipodal mapping theorem and difference equations in Banach spaces"

Transcription

1 Journal of Difference Equations and Applications Vol., No.,, 1 13 The antipodal mapping theorem and difference equations in Banach spaces Daniel Franco, Donal O Regan and Juan Peran * Departamento de Matemática Aplicada, Universidad Nacional de Educación a Distancia, Apartado 60149, Madrid 8080, Spain. Department of Mathematics. National University of Ireland. Galway. Ireland. () We employ the Borsuk-Krasnoselskii antipodal theorem to prove a new fixed point theorem in ordered Banach spaces. Then, the applicability of the result is shown by presenting sufficient conditions for the existence of solutions to initial value problems for first-order difference equations in Banach spaces. To prove that result we shall employ set valued analysis techniques. Keywords: difference equations; antipodal theorem; set valued analysis AMS Subject Classification (000): 39A05; 47H10 1 Introduction Let E be a Banach space and f a compact map from B = {x E : x 1} to E. Theorem 1.1 Borsuk-Krasnoselskii antipodal theorem. Suppose that f has no fixed points on B and that the antipodal condition f(x) x f(x) x f( x) + x f( x) + x, x B is satisfied. Then f has a fixed point in B. The above result has been taken from section 16.3 in [] and it was first This research has been supported in part by Ministerio de Ciencia y Tecnología (Spain), project MTM C03-03 * Corresponding author. jperan@ind.uned.es

2 Franco, O Regan and Peran proved by Borsuk in 1933 in the finite dimensional case [7] and this generalization to the infinite dimensional setting is due to Krasnoselskii [16]. There is a vast literature on applications of the Borsuk-Krasnoselskii antipodal theorem and its variants and generalizations to different fields of mathematics (we recommend the interested reader to the nice survey paper [1]). In particular, it has been employed extensively to prove existence and multiple results for nonlinear integral and differential equations (see [19, 0] and the references therein). In the first part of this paper, using Borsuk-Krasnoselskii s result we shall present a new fixed point theorem in Banach spaces with an order structure given by an normal order cone. More precisely, we shall introduce new abstract definitions of lower and upper solutions for the equation u = f(u) in the Banach space E. Then, we shall show the existence of maximal and minimal fixed points in a sector defined by the lower and the upper solution without assuming monotone conditions on the nonlinearity f. The second part of the paper deals with the application of the previous result to prove new sufficient conditions for the existence of solutions of initial and boundary value problems for first and higher order difference equations in Banach spaces. Evidently, if the dimension of the Banach space is finite we just have a difference equation or a system of difference equations (see [1, 15] for a general theory). Existence of solutions for difference equations, difference equations in Banach spaces or systems of difference equations have been considered by several authors using different techniques [, 8, 9, 11, 13, 17]. We note that in [9] the authors also introduce a cone in order to prove existence of single and multiple fixed sign solutions. In [9] the cone is needed so that a fixed point theorem of cone expansion and compression of norm type can be used. The approach in [9] is completely different from our approach. In [8,10,13,14] the lower and upper solution technique was employed. In Section 3 we prove an existence result for first order initial value problems in Banach spaces, assuming that there exist suitably defined upper an lower solutions. Then, we study higher order initial and boundary problems by reducing these problems to initial value ones. Finally, we note that our technique was inspired by methods employed to solve the first order scalar case in [10]. Existence principle Let E be an ordered Banach space whose order cone K E is normal and has non-empty interior. That is, K is a closed, convex, proper ({ x, x} K x = 0}), normal (there is a number r 0 > 0 such that 0 u v u r 0 v ) cone and int(k).

3 Antipodal theorem and difference equations 3 As usual, we will write u v for v u int(k) and [u, v] = (u + K) (v K) = {w E : u w v} (u, v) = int([u, v]) = (u + int(k)) (v int(k)) = {w E : u w v}. Notice that [u, v] is a bounded set, because for each w [u, v] one has w w u + u r 0 v u + u. We consider on E m, with m N, the norm x = x x m and the partial order relation correspondent to the cone K m (which is closed, convex, proper, normal and with non-empty interior), that is: x y if and only if x k y k for all k {1,..., m}. A map is said to be compact if it is continuous and maps bounded sets into relatively compact sets. Remark 1 A compact subset S E always has maximal and minimal elements. To see that, suppose J is a linearly ordered subset of S and consider the filter base on S formed by the sets F v = {u S : u v} with v J. Since S is a compact set, this filter base has a cluster point u 0 S, that is u 0 v J F v. Therefore, u 0 is an upper bound of J. By Zorn s Lemma, S has a maximal element. A slight change in the proof actually shows the minimal element existence. Theorem.1 If a b in E and f : [a, b] E is a compact map such that a f(a + v); b f(b v) for all v K with v b a, then f has a fixed point in (a, b) but none in [a, b]. Furthermore, the set of fixed points {u [a, b] : f(u) = u} is a non-empty compact subset of (a, b), having maximal and minimal elements. Proof First suppose that f has a fixed point u [a, b] such that u [a, b] = ((a + K) (b K)) ((a + K) (b K)). Then one has u a K with u a b a or b u K with b u b a. If u a K with u a b a, then a f(a + (u a)) = f(u) = u, so u a int(k), which is impossible. Similarly, if we assume b u K with b u b a, we have b u int(k), once more a contradiction. Now consider the Minkowski functional defined on E by p(w) = inf{r > 0 : w r [ a b, b a ]}.

4 4 Franco, O Regan and Peran The functional p is a norm on E, since [ a b, b a ] is a symmetric, closed, bounded, convex set with 0 ] a b, b a [. Furthermore, p and are equivalent norms. To see this notice 1 r 0 b a w p(w) 1 r 1 w, where r 0 comes from the normal cone condition and r 1 is the radius of a ball with centre 0 and contained in ( a b, b a ). Since ( a b, b a ) = {w E : p(w) < 1}, the result follows from the antipodal theorem if the antipodal condition holds for all u [ a b Recall that, b a f (u) u f (u) u f ( u) + u f ( u) + u ]; here f (u) = f ( u + b+a ) b+a. [ a b, b a ] = ( ( a b + K) ( b a K) ) ( ( a b + K) ( b a K) ). Suppose the antipodal condition does not hold. Then there is a number r > 0 and v K with v b a such that that is This gives f ( a b + v ) ( a b + v ) = r ( f ( b a v ) ( b a v )), f (a + v) (a + v) = r (f (b v) (b v)). v = a (a + v) f (a + v) (a + v) = r (f (b v) (b v)) r (b (b v)) = rv, and then (r + 1)v int(k), which is impossible because v K. Therefore S = {u [a, b] : f(u) = u} is a non-empty compact set. To see the compactness notice that f continuous implies S is closed, and also since S is bounded we have that S = f(s) is a relatively compact set. Finally, the minimal and maximal existence follows from Remark 1. In view of the last Theorem we make the following definition.

5 Antipodal theorem and difference equations 5 Definition. We say that a b in E are a pair of lower and upper solution for equation u = f(u) iff a f(a + v); b f(b v) for all v K with v b a. 3 Applications to difference equations Before presenting our results in this section we need to recall some definitions and results of the theory of set valued analysis [4]. Let X be a non-empty set and let P(X) denote the set of all subsets of X. We define a multifunction F from X to an arbitrary set Y, written F : X Y, to be a map F : P(X) P(Y ) such that F (A) = a A F ({a}). Of course, given F ({x}) for each x X, the multifunction F is determined. We denote F ({x}) by F (x). The map F : P(Y ) P(X) defined by F (B) = {x X : F (x) B } is a multifunction, called the inverse multifunction of F. It is easily proved that the composition of two multifunctions F : X Y, G: Y Z is a multifunction G F : X Z such that G F (x) = y F (x) G(y). As usual, we identify each map f : X Y with the multifunction A X f(a). Obviously, the map usually denoted as f 1 : P(Y ) P(X) coincides with the multifunction f. Let X and Y be Hausdorff spaces. A multifunction F : X Y is called upper semicontinuous (u.s.c.) if F (C) is a closed subset of X for each closed subset C of Y. An u.s.c. multifunction with non-empty compact values is called an usco map. Lemma 3.1 Let D E be a closed bounded set, f : D E a compact map and h: D E defined by h(u) = u f(u). Then h is a closed map, that is, h 1 : E D is an u.s.c. multifunction. Proof Let C D be a closed set, (y n ) y 0 a convergent sequence with y n h(c) and x n C such that y n = h(x n ). Since cl(f(c)) is a compact set, there exists a convergent subsequence (f(x ni )) of (f(x n )) with limit w 0 cl(f(c)). Now, so w 0 + y 0 C and lim x ni = lim f(x ni ) + lim h(x ni ) = w 0 + y 0 f(w 0 + y 0 ) = f(lim x ni ) = lim f(x ni ) = w 0.

6 6 Franco, O Regan and Peran Finally, h(w 0 + y 0 ) = w 0 + y 0 f(w 0 + y 0 ) = w 0 + y 0 w 0 = y 0, so y 0 h(c) and h(c) = ( h 1) (C) is a closed subset of E. 3.1 First order difference equations. Consider the following difference equation: y k = f k (y k+1 ), k {1,..., m 1} (1) where y = (y 1,..., y m ) E m, y k = y k+1 y k and f k : E E is a compact map for each k {1,..., m 1}. Define h k : E E by h k (u) = u f k (u) for k {1,..., m 1}. Then rewrite (1) as y k = h k (y k+1 ), k {1,..., m 1}. () We define a continuous map H = (H 1,..., H m ): E E m as follows: H m (u) = u (3) H k (u) = h k (H k+1 (u)), k {1,..., m 1}. (4) It is easy to see that y = H(u) is a solution of (1) for each u E, and conversely, for each solution y of (1) there is a u E with y = H(u). Definition 3. We say that α β in E m are a pair of lower and upper solutions for equation (1) if for each k = 1,..., m 1 and v K such that v β k+1 α k+1 one has α k f k (α k+1 + v) and β k f k (β k+1 v). Remark 1 If α β are a pair of lower and upper solutions, we observe by considering v = 0 above that α is a subsolution and β is a supersolution for (1), in accordance with standard definitions (see for example [10]). But an arbitrary subsolution with an arbitrary supersolution, do not necessarily form a pair of lower and upper solutions in the sense of Definition 3.. Finally, notice that y E m is a solution of (1) if and only if it forms with itself a pair of lower and upper solutions. Remark Observe that α β are a pair of lower and upper solutions if and only if, for all k = 1,..., m 1 and v, v K such that v, v β k+1 α k+1, h k (α k+1 + v) v α k β k h k (β k+1 v ) + v (5)

7 Antipodal theorem and difference equations 7 Lemma 3.3 Let α β be a pair of lower and upper solutions for (1) and w k (α k, β k ) for each k {1,..., m 1}. Then, the set S k (w k ) = h 1 k (w k) [α k+1, β k+1 ] is a non-empty compact set contained in (α k+1, β k+1 ). Furthermore, the multifunction S k : (α k, β k ) (α k+1, β k+1 ) is an usco map. Proof For each v K with v β k+1 α k+1 the compact map f from [α k+1, β k+1 ] to E defined by f(u) = f k (u) + w k satisfies α k f(α k+1 + v) w k and β k f(β k+1 v) w k. This gives and α k+1 α k+1 + (w k α k ) f(α k+1 + v) β k+1 β k+1 + (w k β k ) f(β k+1 v). By Theorem.1, the set {u [α k+1, β k+1 ] : f(u) = u} is a non-empty compact subset of (α k+1, β k+1 ). But this is the desired conclusion, since from h k (u) = u f k (u) = u f(u) + w k we have S k (w k ) = {u [α k+1, β k+1 ] : f(u) = u}. It remains to be proved that S k is an u.s.c. multifunction. The set D = [α k+1, β k+1 ] is closed and bounded, the map f k : D E is compact and, by Lemma 3.1, h k : D E is a closed map. Suppose B int(d) = (α k+1, β k+1 ) is closed for the subspace topology induced on int(d), that is, there is a closed (in E) B D such that B = B int(d). Then S k (B) = S k (B ) = h k (B ) (α k, β k ) is a closed subset of (α k, β k ) for the induced topology on this set. Theorem 3.4 Let α β be pair of lower and upper solutions for (1). For each w (α 1, β 1 ), denote by S(w) the subset of [α, β] formed by the solutions y E m of (1) satisfying the initial value condition y 1 = w. Then S(w) is a non-empty compact subset of (α, β) having maximal and minimal elements. Furthermore, the multifunction S : (α 1, β 1 ) (α, β) is an usco map satisfying S = H S m 1 S m S 1. Proof The proof falls in four parts:

8 8 Franco, O Regan and Peran a) We claim S k S k 1 S 1 (w) is a non-empty subset of (α k+1, β k+1 ) for k {1,..., m 1}. For k = 1 it follows directly from Lemma 3.3. Assuming the assertion holds for k 1, apply Lemma 3.3 for each w k S k 1 S 1 (w). b) We claim S(w) H S m 1 S m S 1 (w). If y S(w), then y 1 = w and y k = h k (y k+1 ) for k {1,..., m 1}. This gives y = H(y m ) and y m S m 1 (y m 1 ) S m 1 S m (y m )... S m 1 S m S 1 (w). Thus y H S m 1 S m S 1 (w) and so S(w) H S m 1 S m S 1 (w). c) We claim H S m 1 S m S 1 (w) S(w) (α, β). If y H S m 1 S m S 1 (w), then y = H(y m ), with y m S m 1 S m S 1 (w), which means that y is a solution of (1). Since y m S m 1 S m S 1 (w), there exist w, w 3..., w m 1 such that w S 1 (w) (α, β ),..., w k+1 S k (w k ) (α k+1, β k+1 ),... and y m S m 1 (w m 1 ) (α m, β m ). But, by the definition of S k, one has w m 1 = h m 1 (y m ) = y m 1, w m = h m (y m 1 ) = y m, w = h (y 3 ) = y, w = h 1 (y ) = y 1. Therefore, y S(w) (α, β). d) A continuous map, considered as a single-valued multifunction, is an usco map and finite compositions of usco maps are usco maps (see 6. of [6]). By Lemma 3.3 and the result above, S is an usco map. The maximal and minimal existence is a consequence of Remark 1. Remark 3 As we have mentioned in section 1, some authors have proposed various definitions of upper and lower solutions notions in the scalar case and have developed related existence results. Notice Theorem 3.4 is stated for the multidimensional case. The crucial, but simple, idea is that one can reduce the order of a problem by increasing its dimension (see Hartman s paper [1] for the linear case). Therefore, Definition 3. and Theorem 3.4 have a wide range of applications, as the following sections show. Observe that order reduction

9 Antipodal theorem and difference equations 9 and Definition 3. gives new and useful notions of upper and lower solutions for higher order problems. 3. Higher order difference equations. Let p > n be natural numbers and consider the following n-th order initial value problem on an ordered Banach space F whose order cone is normal and has non-empty interior: n z k = ϕ k (z k+1, z k+1, z k+1,..., n 1 z k+1 ), k {1,..., p n} (6) z 1 = w 1,..., z n = w n, (7) here n z k = ( n 1 z k ). As we have seen, this problem can be written as z k = ψ k (z k+1, z k+, z k+3,..., z k+n ), k {1,..., p n} (8) z 1 = w 1,..., z n = w n. (9) Assume p be a multiple of n, p = mn (if p wasn t a multiple of n, we would define ψ k (u 1,..., u n ) = u n for k {p n + 1,..., mn}, where mn is the smallest multiple of n greater than p). Consider the first order initial value problem in E = F n : y k = h k (y k+1 ), k {1,..., m 1} (10) y 1 = (w 1,..., w n ), (11) where the maps h k : E E are defined as follows: the n-th component of h k is defined by (h k ) n (u) = ψ nk (u). Assuming that (h k ) n, (h k ) n 1,..., (h k ) j+1 have been defined, let (h k ) j be (h k ) j (u) = ψ n(k 1)+j ( (h k ) j+1 (u), (h k ) j+ (u),..., (h k ) n (u), u 1, u,..., u j ). It is easily checked that, for each solution z 1, z,..., z p of (8)-(9), we obtain a solution of (10)-(11) by y k = (z n(k 1)+1, z n(k 1)+,..., z nk ) with k {1,,..., m} and vice versa. Now Section 3.1 can be applied to problem (10)-(11).

10 10 Franco, O Regan and Peran Example 3.5 The discrete analogous of the autonomous boundary value problem z + F (z) = 0; z(0) = z (0) = 0 can be formulated as z k = ψ(z k+1, z k+ ), k {1,..., m } (1) z 1 = z = 0 (13) where m N, ψ(u 1, u ) = g(u 1 ) u and g(x) = x F (x) 4m. Setting as above y k = (z k 1, z k ) for k {1,,..., m}, the equivalent bidimensional first order equation is y k = f(y k+1 ), k {1,..., m 1} (14) where f(u) = L(u) + γ(u), with γ(u 1, u ) = 1 4m (F (u 1) + F (g(u 1 ) u ), F (u 1 )) ( ) ( ) u1 and L(u 1, u ) =. u Consider the order relation given by K = {(x, x ) R : x x 1 0} and suppose F is such that there exist d m 0 with dm m w m γ(u) dm m w m, where w m = ( m 1/m, 3m 1/m 1 ). Remark gives us the condition for the upper and lower solutions to be inductively constructed from greater to smaller indexes. In such a way, we obtain α k = α k+1 L(α k+1 ) + d m m w m β k = α k + d mc k m w m, where c k = m k/m + m 1/m 1. Observe that c k+1 c k = m1/m 1 m c 1/m k+1 and that m1/m 1 m w 1/m m L(w m ). It is obvious that β k α k int(k). Furthermore,

11 Antipodal theorem and difference equations 11 for all v K, 0 v (β k+1 ) (α k+1 ), since L(v) (0, 0), one has α k = L(α k+1 ) d m m w m L(α k+1 + v) + γ(α k+1 + v) = f(α k+1 + v) β k = α k + d m(c k+1 c k ) m w m = L(α k+1 ) d m m w m + d m(c k+1 c k ) m w m = = L(α k+1 ) + d m m w m + d m(c k+1 c k ) m w m = = L(α k+1 ) + d m m w m + d mc k+1 (m 1/m 1) m m 1/m w m L(α k+1 ) L(v) + d m m w m + d mc k+1 m L(w m ) L(β k+1 v) + γ(β k+1 v) = f(β k+1 v). If we begin with α m = dm m ( m 1/m m + m + m 1/m 1, m 1/m m + m + 3m 1/m 1 ), we obtain α 1 = (I L) m 1 (α m ) + d m m ( m ) (I L) k (w m ) = k=0 = d ( ) ( m m 1 m + m 1/m m + m + m 1/m 1 m m m + 3 m 1/m m + m + 3m 1/m 1 + d ( ) ( ) m m 1 m m 1/m m m 3 m 3m 1/m 1 ) + (0, 0) α 1 + d mc 1 m w m = β 1 We conclude by Theorem 3.4 that there exists a solution satisfying y 1 = (0, 0), thus problem 1-13 has a solution located by α and β. 3.3 Other boundary conditions. Some boundary value problems for discrete equations can be changed into initial value problems with a greater order. For instance, consider the following first order problem with a periodic boundary condition: c k = φ k (w k+1 ), k {1,..., m 1} (15) w 1 = w m, (16)

12 1 Franco, O Regan and Peran with φ k : E E. Let the maps ψ k : E E, k {1,..., m 1}, be defined as follows: ψ 1 (u) = 0, ψ (u) = φ 1 (u ) u 1, ψ j 1 (u) = u, j {,..., m 1} ψ j (u) = φ j (u ), j {,..., m 1} ψ m 1 (u) = u 1. It can be easily shown that if z 1,..., z m, z m+1 is a solution for the initial value problem z k = ψ k (z k+1, z k+ ), k {1,..., m 1} (17) z 1 = z = 0, (18) then a solution for the problem (15)-(16) is obtained by setting w 1 = z m, c k = z k for k {,..., m}. Observe that (17)-(18) can be handle like (8)-(9). Notice that in order for ψ k to be a compact map, E must be a finite dimensional space. References [1] R.P. Agarwal, Difference equations and inequalities. Theory, methods, and applications, Marcel Dekker Inc., New York, 000. [] R.P. Agarwal and D. O Regan, Difference equations in abstract spaces, J. Austral. Math. Soc. (Series A) 64 (1998) [3] D.R. Anderson, A fourth-order nonlinear difference equation, J. Difference Equ. Appl. 9 (003), no. 1, [4] J.P. Aubin and H. Frankowska, Set-valued analysis, Birkhäuser, Boston, [5] R. I. Avery, Chuan Jen Chyan, J. Henderson, Twin solutions of boundary value problems for ordinary differential equations and finite difference equations, Comput. Math. Appl. 4 (001), no. 3-5, [6] G. Beer, Topologies on closed and closed convex sets, Kluwer Academic Publishers, Dordrecht, [7] K. Borsuk, Über stetige Abbildungen der euklidischen Räume, Fund. Math. 1 (1933) [8] A. Cabada and V. Otero-Espinar, Optimal existence results for nth order periodic boundary value difference equations, J. Math. Anal. Appl. 47 (000), no. 1, [9] C.J. Chyan, J. Henderson and H.C. Lo, Existence of triple solutions of discrete (n, p) boundary value problems, Appl. Math. Lett. 14 (001), no. 3,

13 Antipodal theorem and difference equations 13 [10] D. Franco, D. O Regan and J. Perán, Upper and lower solution theory for first and second order difference equations, Dynamic Systems and Applications, (to appear). [11] C. González and A. Jiménez-Melado, Asymptotic behavior of solutions of difference equations in Banach spaces, Proc. Amer. Math. Soc. 18 (000), no. 6, [1] P. Hartman, Difference equations: disconjugacy, principal solutions, Green s functions, complete monotonicity, Trans. Amer. Math. Soc. 46 (1978), [13] J. Henderson and H.B. Thompson, Existence of multiple solutions for secondorder discrete boundary value problems, Comput. Math. Appl. 43 (00), no , [14] J. Henderson and P.J.Y. Wong, On multiple solutions of a system of m discrete boundary value problems, Z. Angew. Math. Mech. 81 (001), no. 4, [15] W.G. Kelley and A.C. Peterson, Difference equations. An introduction with applications, Harcourt/Academic Press, San Diego CA, 001. [16] M.A. Krasnoselskii, On a fixed point principle for completely continous operators in functional spaces, Dokl. Akad. Nauk SSSR 73 (1950) (Russian) [17] C.V. Pao, Monotone methods for a finite difference system of reaction diffusion equation with time delay, Comput. Math. Appl. 36 (1998) [18] A. Peterson, Existence and uniqueness theorems for nonlinear difference equations, J. Math. Anal. Appl. 15 (1987), no. 1, [19] W.B. Song, On the solvability of a nonselfadjoint quasilinear elliptic boundary value problem, Taiwanese J. Math. 6 (00), no. 4, [0] S. Staněk, On some boundary value problems for second order functionaldifferential equations, Nonlinear Anal. 8 (1997), no. 3, [1] H. Steinlein, Borsuk s antipodal theorem and its generalizations and applications: a survey, Topological Methods in Nonlinear Analysis, , Sém. Math. Sup., 95, Presses Univ. Montréal, Montreal, [] E. Zeidler, Nonlinear functional analysis and its applications. Vol I. Fixed-point theorems, Springer-Verlag, New York, 1986.

ANTIPODAL FIXED POINT THEORY FOR VOLTERRA MAPS

ANTIPODAL FIXED POINT THEORY FOR VOLTERRA MAPS Dynamic Systems and Applications 17 (2008) 325-330 ANTIPODAL FIXED POINT THEORY FOR VOLTERRA MAPS YEOL JE CHO, DONAL O REGAN, AND SVATOSLAV STANEK Department of Mathematics Education and the RINS, College

More information

POSITIVE SOLUTIONS TO SINGULAR HIGHER ORDER BOUNDARY VALUE PROBLEMS ON PURELY DISCRETE TIME SCALES

POSITIVE SOLUTIONS TO SINGULAR HIGHER ORDER BOUNDARY VALUE PROBLEMS ON PURELY DISCRETE TIME SCALES Communications in Applied Analysis 19 2015, 553 564 POSITIVE SOLUTIONS TO SINGULAR HIGHER ORDER BOUNDARY VALUE PROBLEMS ON PURELY DISCRETE TIME SCALES CURTIS KUNKEL AND ASHLEY MARTIN 1 Department of Mathematics

More information

Epiconvergence and ε-subgradients of Convex Functions

Epiconvergence and ε-subgradients of Convex Functions Journal of Convex Analysis Volume 1 (1994), No.1, 87 100 Epiconvergence and ε-subgradients of Convex Functions Andrei Verona Department of Mathematics, California State University Los Angeles, CA 90032,

More information

TRIPLE POSITIVE SOLUTIONS FOR A CLASS OF TWO-POINT BOUNDARY-VALUE PROBLEMS

TRIPLE POSITIVE SOLUTIONS FOR A CLASS OF TWO-POINT BOUNDARY-VALUE PROBLEMS Electronic Journal of Differential Equations, Vol. 24(24), No. 6, pp. 8. ISSN: 72-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) TRIPLE POSITIVE

More information

On Monch type multi-valued maps and fixed points

On Monch type multi-valued maps and fixed points Applied Mathematics Letters 20 (2007) 622 628 www.elsevier.com/locate/aml On Monch type multi-valued maps and fixed points B.C. Dhage Kasubai, Gurukul Colony, Ahmedpur-413 515 Dist: Latur, Maharashtra,

More information

A NICE PROOF OF FARKAS LEMMA

A NICE PROOF OF FARKAS LEMMA A NICE PROOF OF FARKAS LEMMA DANIEL VICTOR TAUSK Abstract. The goal of this short note is to present a nice proof of Farkas Lemma which states that if C is the convex cone spanned by a finite set and if

More information

On Positive Solutions of Boundary Value Problems on the Half-Line

On Positive Solutions of Boundary Value Problems on the Half-Line Journal of Mathematical Analysis and Applications 259, 127 136 (21) doi:1.16/jmaa.2.7399, available online at http://www.idealibrary.com on On Positive Solutions of Boundary Value Problems on the Half-Line

More information

Scalar Asymptotic Contractivity and Fixed Points for Nonexpansive Mappings on Unbounded Sets

Scalar Asymptotic Contractivity and Fixed Points for Nonexpansive Mappings on Unbounded Sets Scalar Asymptotic Contractivity and Fixed Points for Nonexpansive Mappings on Unbounded Sets George Isac Department of Mathematics Royal Military College of Canada, STN Forces Kingston, Ontario, Canada

More information

Yuqing Chen, Yeol Je Cho, and Li Yang

Yuqing Chen, Yeol Je Cho, and Li Yang Bull. Korean Math. Soc. 39 (2002), No. 4, pp. 535 541 NOTE ON THE RESULTS WITH LOWER SEMI-CONTINUITY Yuqing Chen, Yeol Je Cho, and Li Yang Abstract. In this paper, we introduce the concept of lower semicontinuous

More information

A LEFSCHETZ FIXED POINT THEOREM FOR ADMISSIBLE MAPS IN FRÉCHET SPACES

A LEFSCHETZ FIXED POINT THEOREM FOR ADMISSIBLE MAPS IN FRÉCHET SPACES Dynamic Systems and Applications 16 (2007) 1-12 A LEFSCHETZ FIXED POINT THEOREM FOR ADMISSIBLE MAPS IN FRÉCHET SPACES RAVI P. AGARWAL AND DONAL O REGAN Department of Mathematical Sciences, Florida Institute

More information

FUNCTIONAL COMPRESSION-EXPANSION FIXED POINT THEOREM

FUNCTIONAL COMPRESSION-EXPANSION FIXED POINT THEOREM Electronic Journal of Differential Equations, Vol. 28(28), No. 22, pp. 1 12. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) FUNCTIONAL

More information

NONLINEAR EIGENVALUE PROBLEMS FOR HIGHER ORDER LIDSTONE BOUNDARY VALUE PROBLEMS

NONLINEAR EIGENVALUE PROBLEMS FOR HIGHER ORDER LIDSTONE BOUNDARY VALUE PROBLEMS NONLINEAR EIGENVALUE PROBLEMS FOR HIGHER ORDER LIDSTONE BOUNDARY VALUE PROBLEMS PAUL W. ELOE Abstract. In this paper, we consider the Lidstone boundary value problem y (2m) (t) = λa(t)f(y(t),..., y (2j)

More information

SCALARIZATION APPROACHES FOR GENERALIZED VECTOR VARIATIONAL INEQUALITIES

SCALARIZATION APPROACHES FOR GENERALIZED VECTOR VARIATIONAL INEQUALITIES Nonlinear Analysis Forum 12(1), pp. 119 124, 2007 SCALARIZATION APPROACHES FOR GENERALIZED VECTOR VARIATIONAL INEQUALITIES Zhi-bin Liu, Nan-jing Huang and Byung-Soo Lee Department of Applied Mathematics

More information

MATH 202B - Problem Set 5

MATH 202B - Problem Set 5 MATH 202B - Problem Set 5 Walid Krichene (23265217) March 6, 2013 (5.1) Show that there exists a continuous function F : [0, 1] R which is monotonic on no interval of positive length. proof We know there

More information

AW -Convergence and Well-Posedness of Non Convex Functions

AW -Convergence and Well-Posedness of Non Convex Functions Journal of Convex Analysis Volume 10 (2003), No. 2, 351 364 AW -Convergence Well-Posedness of Non Convex Functions Silvia Villa DIMA, Università di Genova, Via Dodecaneso 35, 16146 Genova, Italy villa@dima.unige.it

More information

Positive Periodic Solutions of Systems of Second Order Ordinary Differential Equations

Positive Periodic Solutions of Systems of Second Order Ordinary Differential Equations Positivity 1 (26), 285 298 26 Birkhäuser Verlag Basel/Switzerland 1385-1292/2285-14, published online April 26, 26 DOI 1.17/s11117-5-21-2 Positivity Positive Periodic Solutions of Systems of Second Order

More information

WEAK CONVERGENCE OF RESOLVENTS OF MAXIMAL MONOTONE OPERATORS AND MOSCO CONVERGENCE

WEAK CONVERGENCE OF RESOLVENTS OF MAXIMAL MONOTONE OPERATORS AND MOSCO CONVERGENCE Fixed Point Theory, Volume 6, No. 1, 2005, 59-69 http://www.math.ubbcluj.ro/ nodeacj/sfptcj.htm WEAK CONVERGENCE OF RESOLVENTS OF MAXIMAL MONOTONE OPERATORS AND MOSCO CONVERGENCE YASUNORI KIMURA Department

More information

A PROOF OF A CONVEX-VALUED SELECTION THEOREM WITH THE CODOMAIN OF A FRÉCHET SPACE. Myung-Hyun Cho and Jun-Hui Kim. 1. Introduction

A PROOF OF A CONVEX-VALUED SELECTION THEOREM WITH THE CODOMAIN OF A FRÉCHET SPACE. Myung-Hyun Cho and Jun-Hui Kim. 1. Introduction Comm. Korean Math. Soc. 16 (2001), No. 2, pp. 277 285 A PROOF OF A CONVEX-VALUED SELECTION THEOREM WITH THE CODOMAIN OF A FRÉCHET SPACE Myung-Hyun Cho and Jun-Hui Kim Abstract. The purpose of this paper

More information

Product metrics and boundedness

Product metrics and boundedness @ Applied General Topology c Universidad Politécnica de Valencia Volume 9, No. 1, 2008 pp. 133-142 Product metrics and boundedness Gerald Beer Abstract. This paper looks at some possible ways of equipping

More information

A fixed point theorem for multivalued mappings

A fixed point theorem for multivalued mappings Electronic Journal of Qualitative Theory of Differential Equations 24, No. 17, 1-1; http://www.math.u-szeged.hu/ejqtde/ A fixed point theorem for multivalued mappings Cezar AVRAMESCU Abstract A generalization

More information

Abdulmalik Al Twaty and Paul W. Eloe

Abdulmalik Al Twaty and Paul W. Eloe Opuscula Math. 33, no. 4 (23, 63 63 http://dx.doi.org/.7494/opmath.23.33.4.63 Opuscula Mathematica CONCAVITY OF SOLUTIONS OF A 2n-TH ORDER PROBLEM WITH SYMMETRY Abdulmalik Al Twaty and Paul W. Eloe Communicated

More information

Fixed Point Theorems for Condensing Maps

Fixed Point Theorems for Condensing Maps Int. Journal of Math. Analysis, Vol. 2, 2008, no. 21, 1031-1044 Fixed Point Theorems for Condensing Maps in S-KKM Class Young-Ye Huang Center for General Education Southern Taiwan University 1 Nan-Tai

More information

On restricted weak upper semicontinuous set valued mappings and reflexivity

On restricted weak upper semicontinuous set valued mappings and reflexivity On restricted weak upper semicontinuous set valued mappings and reflexivity Julio Benítez Vicente Montesinos Dedicated to Professor Manuel Valdivia. Abstract It is known that if a Banach space is quasi

More information

A fixed point theorem for weakly Zamfirescu mappings

A fixed point theorem for weakly Zamfirescu mappings A fixed point theorem for weakly Zamfirescu mappings David Ariza-Ruiz Dept. Análisis Matemático, Fac. Matemáticas, Universidad de Sevilla, Apdo. 1160, 41080-Sevilla, Spain Antonio Jiménez-Melado Dept.

More information

Journal of Mathematical Analysis and Applications. Riemann integrability and Lebesgue measurability of the composite function

Journal of Mathematical Analysis and Applications. Riemann integrability and Lebesgue measurability of the composite function J. Math. Anal. Appl. 354 (2009) 229 233 Contents lists available at ScienceDirect Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa Riemann integrability and Lebesgue measurability

More information

MULTIPLICITY OF CONCAVE AND MONOTONE POSITIVE SOLUTIONS FOR NONLINEAR FOURTH-ORDER ORDINARY DIFFERENTIAL EQUATIONS

MULTIPLICITY OF CONCAVE AND MONOTONE POSITIVE SOLUTIONS FOR NONLINEAR FOURTH-ORDER ORDINARY DIFFERENTIAL EQUATIONS Fixed Point Theory, 4(23), No. 2, 345-358 http://www.math.ubbcluj.ro/ nodeacj/sfptcj.html MULTIPLICITY OF CONCAVE AND MONOTONE POSITIVE SOLUTIONS FOR NONLINEAR FOURTH-ORDER ORDINARY DIFFERENTIAL EQUATIONS

More information

APPROXIMATE ISOMETRIES ON FINITE-DIMENSIONAL NORMED SPACES

APPROXIMATE ISOMETRIES ON FINITE-DIMENSIONAL NORMED SPACES APPROXIMATE ISOMETRIES ON FINITE-DIMENSIONAL NORMED SPACES S. J. DILWORTH Abstract. Every ε-isometry u between real normed spaces of the same finite dimension which maps the origin to the origin may by

More information

arxiv: v1 [math.na] 9 Feb 2013

arxiv: v1 [math.na] 9 Feb 2013 STRENGTHENED CAUCHY-SCHWARZ AND HÖLDER INEQUALITIES arxiv:1302.2254v1 [math.na] 9 Feb 2013 J. M. ALDAZ Abstract. We present some identities related to the Cauchy-Schwarz inequality in complex inner product

More information

Czechoslovak Mathematical Journal

Czechoslovak Mathematical Journal Czechoslovak Mathematical Journal Oktay Duman; Cihan Orhan µ-statistically convergent function sequences Czechoslovak Mathematical Journal, Vol. 54 (2004), No. 2, 413 422 Persistent URL: http://dml.cz/dmlcz/127899

More information

Solution existence of variational inequalities with pseudomonotone operators in the sense of Brézis

Solution existence of variational inequalities with pseudomonotone operators in the sense of Brézis Solution existence of variational inequalities with pseudomonotone operators in the sense of Brézis B. T. Kien, M.-M. Wong, N. C. Wong and J. C. Yao Communicated by F. Giannessi This research was partially

More information

Initial value problems for singular and nonsmooth second order differential inclusions

Initial value problems for singular and nonsmooth second order differential inclusions Initial value problems for singular and nonsmooth second order differential inclusions Daniel C. Biles, J. Ángel Cid, and Rodrigo López Pouso Department of Mathematics, Western Kentucky University, Bowling

More information

CONVERGENCE OF HYBRID FIXED POINT FOR A PAIR OF NONLINEAR MAPPINGS IN BANACH SPACES

CONVERGENCE OF HYBRID FIXED POINT FOR A PAIR OF NONLINEAR MAPPINGS IN BANACH SPACES International Journal of Analysis and Applications ISSN 2291-8639 Volume 8, Number 1 2015), 69-78 http://www.etamaths.com CONVERGENCE OF HYBRID FIXED POINT FOR A PAIR OF NONLINEAR MAPPINGS IN BANACH SPACES

More information

NONLINEAR FREDHOLM ALTERNATIVE FOR THE p-laplacian UNDER NONHOMOGENEOUS NEUMANN BOUNDARY CONDITION

NONLINEAR FREDHOLM ALTERNATIVE FOR THE p-laplacian UNDER NONHOMOGENEOUS NEUMANN BOUNDARY CONDITION Electronic Journal of Differential Equations, Vol. 2016 (2016), No. 210, pp. 1 7. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu NONLINEAR FREDHOLM ALTERNATIVE FOR THE p-laplacian

More information

POSITIVE SOLUTIONS FOR A SECOND-ORDER DIFFERENCE EQUATION WITH SUMMATION BOUNDARY CONDITIONS

POSITIVE SOLUTIONS FOR A SECOND-ORDER DIFFERENCE EQUATION WITH SUMMATION BOUNDARY CONDITIONS Kragujevac Journal of Mathematics Volume 41(2) (2017), Pages 167 178. POSITIVE SOLUTIONS FOR A SECOND-ORDER DIFFERENCE EQUATION WITH SUMMATION BOUNDARY CONDITIONS F. BOUCHELAGHEM 1, A. ARDJOUNI 2, AND

More information

On the Midpoint Method for Solving Generalized Equations

On the Midpoint Method for Solving Generalized Equations Punjab University Journal of Mathematics (ISSN 1016-56) Vol. 40 (008) pp. 63-70 On the Midpoint Method for Solving Generalized Equations Ioannis K. Argyros Cameron University Department of Mathematics

More information

COMPLETENESS THEOREM FOR THE DISSIPATIVE STURM-LIOUVILLE OPERATOR ON BOUNDED TIME SCALES. Hüseyin Tuna

COMPLETENESS THEOREM FOR THE DISSIPATIVE STURM-LIOUVILLE OPERATOR ON BOUNDED TIME SCALES. Hüseyin Tuna Indian J. Pure Appl. Math., 47(3): 535-544, September 2016 c Indian National Science Academy DOI: 10.1007/s13226-016-0196-1 COMPLETENESS THEOREM FOR THE DISSIPATIVE STURM-LIOUVILLE OPERATOR ON BOUNDED

More information

Solvability of Discrete Neumann Boundary Value Problems

Solvability of Discrete Neumann Boundary Value Problems Solvability of Discrete Neumann Boundary Value Problems D. R. Anderson, I. Rachůnková and C. C. Tisdell September 6, 2006 1 Department of Mathematics and Computer Science Concordia College, 901 8th Street,

More information

ASYMPTOTICALLY NONEXPANSIVE MAPPINGS IN MODULAR FUNCTION SPACES ABSTRACT

ASYMPTOTICALLY NONEXPANSIVE MAPPINGS IN MODULAR FUNCTION SPACES ABSTRACT ASYMPTOTICALLY NONEXPANSIVE MAPPINGS IN MODULAR FUNCTION SPACES T. DOMINGUEZ-BENAVIDES, M.A. KHAMSI AND S. SAMADI ABSTRACT In this paper, we prove that if ρ is a convex, σ-finite modular function satisfying

More information

Generalized Monotonicities and Its Applications to the System of General Variational Inequalities

Generalized Monotonicities and Its Applications to the System of General Variational Inequalities Generalized Monotonicities and Its Applications to the System of General Variational Inequalities Khushbu 1, Zubair Khan 2 Research Scholar, Department of Mathematics, Integral University, Lucknow, Uttar

More information

Filters in Analysis and Topology

Filters in Analysis and Topology Filters in Analysis and Topology David MacIver July 1, 2004 Abstract The study of filters is a very natural way to talk about convergence in an arbitrary topological space, and carries over nicely into

More information

Stability of efficient solutions for semi-infinite vector optimization problems

Stability of efficient solutions for semi-infinite vector optimization problems Stability of efficient solutions for semi-infinite vector optimization problems Z. Y. Peng, J. T. Zhou February 6, 2016 Abstract This paper is devoted to the study of the stability of efficient solutions

More information

B. Appendix B. Topological vector spaces

B. Appendix B. Topological vector spaces B.1 B. Appendix B. Topological vector spaces B.1. Fréchet spaces. In this appendix we go through the definition of Fréchet spaces and their inductive limits, such as they are used for definitions of function

More information

BOUNDARY VALUE PROBLEMS OF A HIGHER ORDER NONLINEAR DIFFERENCE EQUATION

BOUNDARY VALUE PROBLEMS OF A HIGHER ORDER NONLINEAR DIFFERENCE EQUATION U.P.B. Sci. Bull., Series A, Vol. 79, Iss. 4, 2017 ISSN 1223-7027 BOUNDARY VALUE PROBLEMS OF A HIGHER ORDER NONLINEAR DIFFERENCE EQUATION Lianwu Yang 1 We study a higher order nonlinear difference equation.

More information

Topologies, ring norms and algebra norms on some algebras of continuous functions.

Topologies, ring norms and algebra norms on some algebras of continuous functions. Topologies, ring norms and algebra norms on some algebras of continuous functions. Javier Gómez-Pérez Javier Gómez-Pérez, Departamento de Matemáticas, Universidad de León, 24071 León, Spain. Corresponding

More information

Abstract. The connectivity of the efficient point set and of some proper efficient point sets in locally convex spaces is investigated.

Abstract. The connectivity of the efficient point set and of some proper efficient point sets in locally convex spaces is investigated. APPLICATIONES MATHEMATICAE 25,1 (1998), pp. 121 127 W. SONG (Harbin and Warszawa) ON THE CONNECTIVITY OF EFFICIENT POINT SETS Abstract. The connectivity of the efficient point set and of some proper efficient

More information

Existence and multiple solutions for a second-order difference boundary value problem via critical point theory

Existence and multiple solutions for a second-order difference boundary value problem via critical point theory J. Math. Anal. Appl. 36 (7) 511 5 www.elsevier.com/locate/jmaa Existence and multiple solutions for a second-order difference boundary value problem via critical point theory Haihua Liang a,b,, Peixuan

More information

On Semicontinuity of Convex-valued Multifunctions and Cesari s Property (Q)

On Semicontinuity of Convex-valued Multifunctions and Cesari s Property (Q) On Semicontinuity of Convex-valued Multifunctions and Cesari s Property (Q) Andreas Löhne May 2, 2005 (last update: November 22, 2005) Abstract We investigate two types of semicontinuity for set-valued

More information

On Ideal Convergent Sequences in 2 Normed Spaces

On Ideal Convergent Sequences in 2 Normed Spaces Thai Journal of Mathematics Volume 4 006 Number 1 : 85 91 On Ideal Convergent Sequences in Normed Spaces M Gürdal Abstract : In this paper, we investigate the relation between I-cluster points and ordinary

More information

MULTIPLE POSITIVE SOLUTIONS FOR FOURTH-ORDER THREE-POINT p-laplacian BOUNDARY-VALUE PROBLEMS

MULTIPLE POSITIVE SOLUTIONS FOR FOURTH-ORDER THREE-POINT p-laplacian BOUNDARY-VALUE PROBLEMS Electronic Journal of Differential Equations, Vol. 27(27, No. 23, pp. 1 1. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp MULTIPLE POSITIVE

More information

arxiv:math/ v1 [math.fa] 21 Mar 2000

arxiv:math/ v1 [math.fa] 21 Mar 2000 SURJECTIVE FACTORIZATION OF HOLOMORPHIC MAPPINGS arxiv:math/000324v [math.fa] 2 Mar 2000 MANUEL GONZÁLEZ AND JOAQUÍN M. GUTIÉRREZ Abstract. We characterize the holomorphic mappings f between complex Banach

More information

The small ball property in Banach spaces (quantitative results)

The small ball property in Banach spaces (quantitative results) The small ball property in Banach spaces (quantitative results) Ehrhard Behrends Abstract A metric space (M, d) is said to have the small ball property (sbp) if for every ε 0 > 0 there exists a sequence

More information

Shih-sen Chang, Yeol Je Cho, and Haiyun Zhou

Shih-sen Chang, Yeol Je Cho, and Haiyun Zhou J. Korean Math. Soc. 38 (2001), No. 6, pp. 1245 1260 DEMI-CLOSED PRINCIPLE AND WEAK CONVERGENCE PROBLEMS FOR ASYMPTOTICALLY NONEXPANSIVE MAPPINGS Shih-sen Chang, Yeol Je Cho, and Haiyun Zhou Abstract.

More information

Relationships between upper exhausters and the basic subdifferential in variational analysis

Relationships between upper exhausters and the basic subdifferential in variational analysis J. Math. Anal. Appl. 334 (2007) 261 272 www.elsevier.com/locate/jmaa Relationships between upper exhausters and the basic subdifferential in variational analysis Vera Roshchina City University of Hong

More information

7 Complete metric spaces and function spaces

7 Complete metric spaces and function spaces 7 Complete metric spaces and function spaces 7.1 Completeness Let (X, d) be a metric space. Definition 7.1. A sequence (x n ) n N in X is a Cauchy sequence if for any ɛ > 0, there is N N such that n, m

More information

Subdifferential representation of convex functions: refinements and applications

Subdifferential representation of convex functions: refinements and applications Subdifferential representation of convex functions: refinements and applications Joël Benoist & Aris Daniilidis Abstract Every lower semicontinuous convex function can be represented through its subdifferential

More information

HeadMedia Interaction in Magnetic Recording

HeadMedia Interaction in Magnetic Recording Journal of Differential Equations 171, 443461 (2001) doi:10.1006jdeq.2000.3844, available online at http:www.idealibrary.com on HeadMedia Interaction in Magnetic Recording Avner Friedman Departament of

More information

Smarandachely Precontinuous maps. and Preopen Sets in Topological Vector Spaces

Smarandachely Precontinuous maps. and Preopen Sets in Topological Vector Spaces International J.Math. Combin. Vol.2 (2009), 21-26 Smarandachely Precontinuous maps and Preopen Sets in Topological Vector Spaces Sayed Elagan Department of Mathematics and Statistics Faculty of Science,

More information

Remark on a Couple Coincidence Point in Cone Normed Spaces

Remark on a Couple Coincidence Point in Cone Normed Spaces International Journal of Mathematical Analysis Vol. 8, 2014, no. 50, 2461-2468 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijma.2014.49293 Remark on a Couple Coincidence Point in Cone Normed

More information

Trigonometric Recurrence Relations and Tridiagonal Trigonometric Matrices

Trigonometric Recurrence Relations and Tridiagonal Trigonometric Matrices International Journal of Difference Equations. ISSN 0973-6069 Volume 1 Number 1 2006 pp. 19 29 c Research India Publications http://www.ripublication.com/ijde.htm Trigonometric Recurrence Relations and

More information

ON THE HYPERBANACH SPACES. P. Raja. S.M. Vaezpour. 1. Introduction

ON THE HYPERBANACH SPACES. P. Raja. S.M. Vaezpour. 1. Introduction italian journal of pure and applied mathematics n. 28 2011 (261 272) 261 ON THE HYPERBANACH SPACES P. Raja Department O Mathematics Shahid Beheshti University P.O. Box 1983963113, Tehran Iran e-mail: pandoora

More information

Journal of Inequalities in Pure and Applied Mathematics

Journal of Inequalities in Pure and Applied Mathematics Journal of Inequalities in Pure and Applied Mathematics http://jipam.vu.edu.au/ Volume 4, Issue 4, Article 67, 2003 ON GENERALIZED MONOTONE MULTIFUNCTIONS WITH APPLICATIONS TO OPTIMALITY CONDITIONS IN

More information

On intermediate value theorem in ordered Banach spaces for noncompact and discontinuous mappings

On intermediate value theorem in ordered Banach spaces for noncompact and discontinuous mappings Int. J. Nonlinear Anal. Appl. 7 (2016) No. 1, 295-300 ISSN: 2008-6822 (electronic) http://dx.doi.org/10.22075/ijnaa.2015.341 On intermediate value theorem in ordered Banach spaces for noncompact and discontinuous

More information

FRACTIONAL BOUNDARY VALUE PROBLEMS ON THE HALF LINE. Assia Frioui, Assia Guezane-Lakoud, and Rabah Khaldi

FRACTIONAL BOUNDARY VALUE PROBLEMS ON THE HALF LINE. Assia Frioui, Assia Guezane-Lakoud, and Rabah Khaldi Opuscula Math. 37, no. 2 27), 265 28 http://dx.doi.org/.7494/opmath.27.37.2.265 Opuscula Mathematica FRACTIONAL BOUNDARY VALUE PROBLEMS ON THE HALF LINE Assia Frioui, Assia Guezane-Lakoud, and Rabah Khaldi

More information

ON ω-independence AND THE KUNEN-SHELAH PROPERTY. 1. Introduction.

ON ω-independence AND THE KUNEN-SHELAH PROPERTY. 1. Introduction. ON ω-independence AND THE KUNEN-SHELAH PROPERTY A. S. GRANERO, M. JIMÉNEZ-SEVILLA AND J. P. MORENO Abstract. We prove that spaces with an uncountable ω-independent family fail the Kunen-Shelah property.

More information

On the Local Convergence of Regula-falsi-type Method for Generalized Equations

On the Local Convergence of Regula-falsi-type Method for Generalized Equations Journal of Advances in Applied Mathematics, Vol., No. 3, July 017 https://dx.doi.org/10.606/jaam.017.300 115 On the Local Convergence of Regula-falsi-type Method for Generalized Equations Farhana Alam

More information

LECTURE 6. CONTINUOUS FUNCTIONS AND BASIC TOPOLOGICAL NOTIONS

LECTURE 6. CONTINUOUS FUNCTIONS AND BASIC TOPOLOGICAL NOTIONS ANALYSIS FOR HIGH SCHOOL TEACHERS LECTURE 6. CONTINUOUS FUNCTIONS AND BASIC TOPOLOGICAL NOTIONS ROTHSCHILD CAESARIA COURSE, 2011/2 1. The idea of approximation revisited When discussing the notion of the

More information

Positive solutions of BVPs for some second-order four-point difference systems

Positive solutions of BVPs for some second-order four-point difference systems Positive solutions of BVPs for some second-order four-point difference systems Yitao Yang Tianjin University of Technology Department of Applied Mathematics Hongqi Nanlu Extension, Tianjin China yitaoyangqf@63.com

More information

MONOTONE POSITIVE SOLUTION OF NONLINEAR THIRD-ORDER TWO-POINT BOUNDARY VALUE PROBLEM

MONOTONE POSITIVE SOLUTION OF NONLINEAR THIRD-ORDER TWO-POINT BOUNDARY VALUE PROBLEM Miskolc Mathematical Notes HU e-issn 177-2413 Vol. 15 (214), No. 2, pp. 743 752 MONOTONE POSITIVE SOLUTION OF NONLINEAR THIRD-ORDER TWO-POINT BOUNDARY VALUE PROBLEM YONGPING SUN, MIN ZHAO, AND SHUHONG

More information

Multiple positive solutions for a nonlinear three-point integral boundary-value problem

Multiple positive solutions for a nonlinear three-point integral boundary-value problem Int. J. Open Problems Compt. Math., Vol. 8, No. 1, March 215 ISSN 1998-6262; Copyright c ICSRS Publication, 215 www.i-csrs.org Multiple positive solutions for a nonlinear three-point integral boundary-value

More information

A continuous operator extending fuzzy ultrametrics

A continuous operator extending fuzzy ultrametrics A continuous operator extending fuzzy ultrametrics I. Stasyuk, E.D. Tymchatyn November 30, 200 Abstract We consider the problem of simultaneous extension of fuzzy ultrametrics defined on closed subsets

More information

Some Fixed Point Theorems for G-Nonexpansive Mappings on Ultrametric Spaces and Non-Archimedean Normed Spaces with a Graph

Some Fixed Point Theorems for G-Nonexpansive Mappings on Ultrametric Spaces and Non-Archimedean Normed Spaces with a Graph J o u r n a l of Mathematics and Applications JMA No 39, pp 81-90 (2016) Some Fixed Point Theorems for G-Nonexpansive Mappings on Ultrametric Spaces and Non-Archimedean Normed Spaces with a Graph Hamid

More information

Preprint Preprint Preprint Preprint

Preprint Preprint Preprint Preprint CADERNOS DE MATEMÁTICA 13, 69 81 May (2012) ARTIGO NÚMERO SMA# 362 (H, G)-Coincidence theorems for manifolds Denise de Mattos * Departamento de Matemática, Instituto de Ciências Matemáticas e de Computação,

More information

REMARKS ON THE SCHAUDER TYCHONOFF FIXED POINT THEOREM

REMARKS ON THE SCHAUDER TYCHONOFF FIXED POINT THEOREM Vietnam Journal of Mathematics 28 (2000) 127 132 REMARKS ON THE SCHAUDER TYCHONOFF FIXED POINT THEOREM Sehie Park 1 and Do Hong Tan 2 1. Seoul National University, Seoul, Korea 2.Institute of Mathematics,

More information

Introduction to Real Analysis Alternative Chapter 1

Introduction to Real Analysis Alternative Chapter 1 Christopher Heil Introduction to Real Analysis Alternative Chapter 1 A Primer on Norms and Banach Spaces Last Updated: March 10, 2018 c 2018 by Christopher Heil Chapter 1 A Primer on Norms and Banach Spaces

More information

MAT 570 REAL ANALYSIS LECTURE NOTES. Contents. 1. Sets Functions Countability Axiom of choice Equivalence relations 9

MAT 570 REAL ANALYSIS LECTURE NOTES. Contents. 1. Sets Functions Countability Axiom of choice Equivalence relations 9 MAT 570 REAL ANALYSIS LECTURE NOTES PROFESSOR: JOHN QUIGG SEMESTER: FALL 204 Contents. Sets 2 2. Functions 5 3. Countability 7 4. Axiom of choice 8 5. Equivalence relations 9 6. Real numbers 9 7. Extended

More information

Viscosity Iterative Approximating the Common Fixed Points of Non-expansive Semigroups in Banach Spaces

Viscosity Iterative Approximating the Common Fixed Points of Non-expansive Semigroups in Banach Spaces Viscosity Iterative Approximating the Common Fixed Points of Non-expansive Semigroups in Banach Spaces YUAN-HENG WANG Zhejiang Normal University Department of Mathematics Yingbing Road 688, 321004 Jinhua

More information

Extension of continuous functions in digital spaces with the Khalimsky topology

Extension of continuous functions in digital spaces with the Khalimsky topology Extension of continuous functions in digital spaces with the Khalimsky topology Erik Melin Uppsala University, Department of Mathematics Box 480, SE-751 06 Uppsala, Sweden melin@math.uu.se http://www.math.uu.se/~melin

More information

Journal of Inequalities in Pure and Applied Mathematics

Journal of Inequalities in Pure and Applied Mathematics Journal of Inequalities in Pure and Applied Mathematics http://jipam.vu.edu.au/ Volume 2, Issue 1, Article 12, 2001 ON KY FAN S MINIMAX INEQUALITIES, MIXED EQUILIBRIUM PROBLEMS AND HEMIVARIATIONAL INEQUALITIES

More information

PREVALENCE OF SOME KNOWN TYPICAL PROPERTIES. 1. Introduction

PREVALENCE OF SOME KNOWN TYPICAL PROPERTIES. 1. Introduction Acta Math. Univ. Comenianae Vol. LXX, 2(2001), pp. 185 192 185 PREVALENCE OF SOME KNOWN TYPICAL PROPERTIES H. SHI Abstract. In this paper, some known typical properties of function spaces are shown to

More information

WEAK CONVERGENCE THEOREMS FOR EQUILIBRIUM PROBLEMS WITH NONLINEAR OPERATORS IN HILBERT SPACES

WEAK CONVERGENCE THEOREMS FOR EQUILIBRIUM PROBLEMS WITH NONLINEAR OPERATORS IN HILBERT SPACES Fixed Point Theory, 12(2011), No. 2, 309-320 http://www.math.ubbcluj.ro/ nodeacj/sfptcj.html WEAK CONVERGENCE THEOREMS FOR EQUILIBRIUM PROBLEMS WITH NONLINEAR OPERATORS IN HILBERT SPACES S. DHOMPONGSA,

More information

En busca de la linealidad en Matemáticas

En busca de la linealidad en Matemáticas En busca de la linealidad en Matemáticas M. Cueto, E. Gómez, J. Llorente, E. Martínez, D. Rodríguez, E. Sáez VIII Escuela taller de Análisis Funcional Bilbao, 9 Marzo 2018 1 Motivation 2 Everywhere differentiable

More information

Countably condensing multimaps and fixed points

Countably condensing multimaps and fixed points Electronic Journal of Qualitative Theory of Differential Equations 2012, No. 83, 1-9; http://www.math.u-szeged.hu/ejqtde/ Countably condensing multimaps and fixed points Tiziana Cardinali - Paola Rubbioni

More information

Convergence to Common Fixed Point for Two Asymptotically Quasi-nonexpansive Mappings in the Intermediate Sense in Banach Spaces

Convergence to Common Fixed Point for Two Asymptotically Quasi-nonexpansive Mappings in the Intermediate Sense in Banach Spaces Mathematica Moravica Vol. 19-1 2015, 33 48 Convergence to Common Fixed Point for Two Asymptotically Quasi-nonexpansive Mappings in the Intermediate Sense in Banach Spaces Gurucharan Singh Saluja Abstract.

More information

YET MORE ON THE DIFFERENTIABILITY OF CONVEX FUNCTIONS

YET MORE ON THE DIFFERENTIABILITY OF CONVEX FUNCTIONS PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 103, Number 3, July 1988 YET MORE ON THE DIFFERENTIABILITY OF CONVEX FUNCTIONS JOHN RAINWATER (Communicated by William J. Davis) ABSTRACT. Generic

More information

SOME REMARKS ON KRASNOSELSKII S FIXED POINT THEOREM

SOME REMARKS ON KRASNOSELSKII S FIXED POINT THEOREM Fixed Point Theory, Volume 4, No. 1, 2003, 3-13 http://www.math.ubbcluj.ro/ nodeacj/journal.htm SOME REMARKS ON KRASNOSELSKII S FIXED POINT THEOREM CEZAR AVRAMESCU AND CRISTIAN VLADIMIRESCU Department

More information

Strong Convergence Theorem by a Hybrid Extragradient-like Approximation Method for Variational Inequalities and Fixed Point Problems

Strong Convergence Theorem by a Hybrid Extragradient-like Approximation Method for Variational Inequalities and Fixed Point Problems Strong Convergence Theorem by a Hybrid Extragradient-like Approximation Method for Variational Inequalities and Fixed Point Problems Lu-Chuan Ceng 1, Nicolas Hadjisavvas 2 and Ngai-Ching Wong 3 Abstract.

More information

Continuous family of eigenvalues concentrating in a small neighborhood at the right of the origin for a class of discrete boundary value problems

Continuous family of eigenvalues concentrating in a small neighborhood at the right of the origin for a class of discrete boundary value problems Annals of the University of Craiova, Math. Comp. Sci. Ser. Volume 35, 008, Pages 78 86 ISSN: 13-6934 Continuous family of eigenvalues concentrating in a small neighborhood at the right of the origin for

More information

Layered Compression-Expansion Fixed Point Theorem

Layered Compression-Expansion Fixed Point Theorem Res. Fixed Point Theory Appl. Volume 28, Article ID 2825, pages eissn 258-67 Results in Fixed Point Theory and Applications RESEARCH ARTICLE Layered Compression-Expansion Fixed Point Theorem Richard I.

More information

Contents. Index... 15

Contents. Index... 15 Contents Filter Bases and Nets................................................................................ 5 Filter Bases and Ultrafilters: A Brief Overview.........................................................

More information

4 Countability axioms

4 Countability axioms 4 COUNTABILITY AXIOMS 4 Countability axioms Definition 4.1. Let X be a topological space X is said to be first countable if for any x X, there is a countable basis for the neighborhoods of x. X is said

More information

THE KNASTER KURATOWSKI MAZURKIEWICZ THEOREM AND ALMOST FIXED POINTS. Sehie Park. 1. Introduction

THE KNASTER KURATOWSKI MAZURKIEWICZ THEOREM AND ALMOST FIXED POINTS. Sehie Park. 1. Introduction Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 16, 2000, 195 200 THE KNASTER KURATOWSKI MAZURKIEWICZ THEOREM AND ALMOST FIXED POINTS Sehie Park Abstract. From the

More information

COMMON FIXED POINT THEOREMS FOR A PAIR OF COUNTABLY CONDENSING MAPPINGS IN ORDERED BANACH SPACES

COMMON FIXED POINT THEOREMS FOR A PAIR OF COUNTABLY CONDENSING MAPPINGS IN ORDERED BANACH SPACES Journal of Applied Mathematics and Stochastic Analysis, 16:3 (2003), 243-248. Printed in the USA c 2003 by North Atlantic Science Publishing Company COMMON FIXED POINT THEOREMS FOR A PAIR OF COUNTABLY

More information

AN EXTENSION OF THE NOTION OF ZERO-EPI MAPS TO THE CONTEXT OF TOPOLOGICAL SPACES

AN EXTENSION OF THE NOTION OF ZERO-EPI MAPS TO THE CONTEXT OF TOPOLOGICAL SPACES AN EXTENSION OF THE NOTION OF ZERO-EPI MAPS TO THE CONTEXT OF TOPOLOGICAL SPACES MASSIMO FURI AND ALFONSO VIGNOLI Abstract. We introduce the class of hyper-solvable equations whose concept may be regarded

More information

CHARACTERIZATION OF (QUASI)CONVEX SET-VALUED MAPS

CHARACTERIZATION OF (QUASI)CONVEX SET-VALUED MAPS CHARACTERIZATION OF (QUASI)CONVEX SET-VALUED MAPS Abstract. The aim of this paper is to characterize in terms of classical (quasi)convexity of extended real-valued functions the set-valued maps which are

More information

SOLUTION OF AN INITIAL-VALUE PROBLEM FOR PARABOLIC EQUATIONS VIA MONOTONE OPERATOR METHODS

SOLUTION OF AN INITIAL-VALUE PROBLEM FOR PARABOLIC EQUATIONS VIA MONOTONE OPERATOR METHODS Electronic Journal of Differential Equations, Vol. 214 (214), No. 225, pp. 1 1. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu SOLUTION OF AN INITIAL-VALUE

More information

ON THE CONVERGENCE OF MODIFIED NOOR ITERATION METHOD FOR NEARLY LIPSCHITZIAN MAPPINGS IN ARBITRARY REAL BANACH SPACES

ON THE CONVERGENCE OF MODIFIED NOOR ITERATION METHOD FOR NEARLY LIPSCHITZIAN MAPPINGS IN ARBITRARY REAL BANACH SPACES TJMM 6 (2014), No. 1, 45-51 ON THE CONVERGENCE OF MODIFIED NOOR ITERATION METHOD FOR NEARLY LIPSCHITZIAN MAPPINGS IN ARBITRARY REAL BANACH SPACES ADESANMI ALAO MOGBADEMU Abstract. In this present paper,

More information

A Direct Proof of Caristi s Fixed Point Theorem

A Direct Proof of Caristi s Fixed Point Theorem Applied Mathematical Sciences, Vol. 10, 2016, no. 46, 2289-2294 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2016.66190 A Direct Proof of Caristi s Fixed Point Theorem Wei-Shih Du Department

More information

PREPUBLICACIONES DEL DEPARTAMENTO DE MATEMÁTICA APLICADA UNIVERSIDAD COMPLUTENSE DE MADRID MA-UCM

PREPUBLICACIONES DEL DEPARTAMENTO DE MATEMÁTICA APLICADA UNIVERSIDAD COMPLUTENSE DE MADRID MA-UCM PREPUBLICACIONES DEL DEPARTAMENTO DE MATEMÁTICA APLICADA UNIVERSIDAD COMPLUTENSE DE MADRID MA-UCM 2009-13 Extremal equilibria for monotone semigroups in ordered spaces with application to evolutionary

More information

CHAPTER II THE HAHN-BANACH EXTENSION THEOREMS AND EXISTENCE OF LINEAR FUNCTIONALS

CHAPTER II THE HAHN-BANACH EXTENSION THEOREMS AND EXISTENCE OF LINEAR FUNCTIONALS CHAPTER II THE HAHN-BANACH EXTENSION THEOREMS AND EXISTENCE OF LINEAR FUNCTIONALS In this chapter we deal with the problem of extending a linear functional on a subspace Y to a linear functional on the

More information