A fixed point theorem for weakly inward Aproper maps and application to a Picard boundary value problem
 Casey T Cremins^{1}Email author
https://doi.org/10.1186/s136630160564x
© Cremins 2016
Received: 22 January 2016
Accepted: 17 June 2016
Published: 29 June 2016
Abstract
A fixed point theorem for weakly inward Aproper maps defined on cones in Banach spaces is established using a fixed point index for such maps. The result generalizes a theorem in Deimling (Nonlinear Functional Analysis, 1985) for weakly inward maps defined on a cone in \(\mathbb{R}^{n}\). We then apply the theorem to a Picard boundary value problem and obtain the existence of a positive solution.
Keywords
MSC
1 Introduction
2 Preliminaries
Let X be a Banach space, \(X_{n} \subset X\) a sequence of oriented finitedimensional subspaces, and \(P_{n}:X \to X_{n}\) a sequence of continuous linear projections such that \(P_{n} x \to x\) for each \(x \in X\).
Then X is called a Banach space with projection scheme \(\Gamma = \{ X_{n},P_{n} \}\).
A map \(f:\operatorname{dom} f \subset X \to X\) is said to be Aproper with respect to Γ if \(P_{n}f:X_{n} \to X_{n}\) is continuous for each n and for any bounded sequence \(\{ x_{n_{j}}x_{n_{j}} \in X_{n_{j}} \}\) such that \(f_{n_{j}}(x_{n_{j}}) \to y\), there exists a subsequence \(\{ x_{n_{j_{k}}} \}\) such that \(x_{n_{j_{k}}} \to x\) and \(f(x) = y\).
A closed convex set K in a Banach space X is called a cone if \(\lambda K \subset K\) for all \(\lambda \ge 0\) and \(K \cap \{  K \} = 0\).
Let \(K \subset X\) be a closed convex set. For each \(x \in K\), the set \(I_{K}(x) = \{ x + c(z  x):z \in K, c \ge 0 \}\) is called the inward set of x with respect to K. A map \(f:K \to X\) is called inward (respectively, weakly inward) if for all \(x \in K\), \(f(x) \in I_{K}(x)\) (\(f(x) \in \bar{I}_{K}(x)\)).
A map \(f:\overline{\Omega}_{K} \to X\) is said to be inward (respectively, weakly inward) on \(\overline{\Omega}_{K}\) relative to K if \(f(x) \in I_{K}(x)\) (respectively, \(f(x) \in \bar{I}_{K}(x)\)) for \(x \in \overline{\Omega}_{K}\), where \(\Omega \subset X\) is open and bounded with \(\Omega_{K} = \Omega \cap K \ne \emptyset\).
For the definition and properties of the LanWebb fixed point index, see [2].
3 An existence theorem for weakly inward Aproper maps
Theorem 3.1
 (a):

\(f(x)\not\le x\) on \(\Vert x \Vert = r\), and
 (b):

there exists \(\rho \in (0, r)\) such that \(\lambda x \not\le f (x)\) for \(\Vert x \Vert = \rho\) and \(\lambda > 1\).
Proof
Let \(B_{r} = \{ x \in X:\Vert x \Vert < r\}\), \(B_{r_{K}} = B_{r} \cap K\), \(B_{\rho} = \{ x \in X:\Vert x \Vert < \rho \}\), and \(B_{\rho_{K}} = B_{\rho} \cap K\). We show that \(i_{K}(f,B_{r_{K}}) = \{ 0\}\) and \(i_{K}(f,B_{\rho_{K}}) = \{ 1\}\), so that by the additivity property of the index \(i_{K}(f,B_{r_{K}}\backslash B_{\rho_{K}}) = i_{K}(f,B_{r_{K}})  i_{K}(f,B_{\rho_{K}}) = \{ 0\}  \{ 1\} = \{  1\} \ne \{ 0\}\), which implies the existence of a fixed point \(x \in K\) such that \(\rho < \Vert x \Vert < r\).
To show that \(i_{K}(f,B_{r_{K}}) = \{ 0\}\), suppose instead that \(i_{K}(f,B_{r_{K}}) \ne \{ 0\}\). Then we choose an a with \(\Vert f(x) \Vert \le a\) on \(\overline{B}_{r_{K}}\) and an \(e \in K\) with \(\Vert e \Vert > r + a\). Define the weakly inward Aproper homotopy \(H(x,t) = f(x) + te\). Now if \(H(x,t) = x\) for some \((x,t) \in \partial B_{r_{K}} \times [0, 1]\), then \(f(x) + te = x\), so that \(x \in K\) and \(x  f(x) = te \in K\) so \(f (x) \leq x\), which contradicts (a). Thus, H is an admissible homotopy, and \(i_{K}(H(x, 1),B_{r_{K}}) = i_{K}(f,B_{r_{K}}) \ne \{ 0\}\). Then there exists \(x \in B_{r_{K}}\) with \(f(x) + e = x\), so that \(\Vert e \Vert = \Vert x  f(x) \Vert \le \Vert x \Vert + \Vert f(x) \Vert \le r + a\), which contradicts \(\Vert e \Vert > r + a\). Hence, \(i_{K}(f,B_{r_{K}}) = \{ 0\}\).
Now we show that \(i_{K}(f,B_{\rho_{K}}) = \{ 1\}\). Define the weakly inward Aproper homotopy \(H(x,t) = tf(x)\).
If \(H(x,t) = x\) for some \((x,t) \in \partial B_{\rho_{K}} \times [0, 1]\), then \(t \ne 0\) (this would give \(0 = x\) on \(\partial B_{r_{K}}\)) and \(tf(x) = x\) and \(x \in K\), so that \(f(x) = \frac{1}{t} x \ge x\), which contradicts (b).
Thus \(H(x,t) \ne x\) on \(\partial B_{\rho_{K}} \times [0, 1]\).
By the homotopy property of the index, \(i_{K}(H(x,0),B_{\rho_{K}}) = i_{K}(H(x,1),B_{\rho_{K}}) = i_{K}(f,B_{\rho_{K}}) = \{ 1\}\).
Consequently, \(i_{K}(f,B_{r_{K}}\backslash B_{\rho_{K}}) = i_{K}(f,B_{r_{K}})  i_{K}(f,B_{\rho_{K}}) = \{ 0\}  \{ 1\} = \{  1\}\).
Remark 3.1
 (a):

\(f(x) \not\le x\) on \(\Vert x \Vert = \rho\), and
 (b):

\(\lambda x \not\le f (x)\) for \(\Vert x \Vert = r\) and \(\lambda > 1\).
We shall use these conditions in the following application.
4 Application
Let \(X = \{ x \in C^{2}[0, 1]:x(0) = x(1) = 0\}\), \(Y = C[0, 1]\), and \(K = \{ y \in C[1, 0]:y(t) \ge 0\}\) with norms \(\Vert x \Vert _{X} = \max \{ \Vert x \Vert _{Y},\Vert x' \Vert _{Y},\Vert x'' \Vert _{Y}\}\) and \(\Vert x \Vert _{Y} = \max_{t \in [0,1]}\{ \vert x(t) \vert \}\). Then L is a linear bounded isometric homeomorphism.
Theorem 4.1
 (a′):

there exist \(r > 0\) and \(k \in (0, 1)\) such that \(f:[0, 1] \times [0, r] \times [  r, r] \times R^{ } \to R^{ +}\) is continuous with \(\vert f(t,p,q,s_{1})  f(t,p,q,s_{2}) \vert \le k\vert s_{1}  s_{2} \vert \) for \(t \in [0, 1]\), \(p \in [0, r]\), \(q \in [  r, r]\), \(s_{1},s_{2} \in R^{ }\);
 (b′):

\(f(t,p,q,s) < r\) for every \(t \in [0, 1]\), \(p \in [0, r]\), \(q \in [  r, r]\), \(s =  r\);
 (c′):

there are \(\rho \in (0, r)\), \(t_{0} \in [0, 1]\) such that \(f(t_{0},p,q,s) > \rho\) for \(p \in [0, \rho ]\), \(q \in [  \rho, \rho ]\), \(s =  \rho\).
Proof
Since T maps K to K, T is weakly inward. Condition (a′) implies that T is \((\beta_{K})k\)ball contractive, where \(\beta_{K}\) is the ball measure of noncompactness associated with K, and thus \(\lambda I  T\) is Aproper with respect to the projection scheme \(\Gamma = \{ X_{n},P_{n}\}\) for every \(\lambda \ge \gamma\), \(\gamma \in (k, 1)\) (cf. [3]). To verify the remaining hypotheses of Remark 3.1, we first show that (b′) implies (b). Let r be as in (b′) and \(y \in K\) such that \(\Vert y \Vert _{Y} = r\). Then there exists \(x \in L^{  1}(K)\) such that \(Lx = y\) and \(\Vert x \Vert _{X} = \Vert y \Vert _{Y} = \Vert x'' \Vert _{Y}\), so that \(r = \Vert x'' \Vert _{Y} = \Vert x \Vert _{X}\) and there exists \(t_{0} \in [0, 1]\) such that \(y(t_{0}) = r\). Now since \(y = Lx\) for some \(x \in L^{  1}(K)\), we have that \(x(t) \in [0, r]\), \(x'(t) \in [  r, r]\) for all \(t \in [0, 1]\) and \(r =  x''(t_{0})\). Then if \(Ty = \lambda y\) for some \(\lambda > 1\) and \(y \in K\) with \(\Vert y \Vert _{Y} = r\), we would have \(f(t,x(t),x'(t),x''(t)) = \lambda y(t)\) for all \(t \in [0, 1]\), including \(t_{0}\), but then this implies \(\lambda r < r\), a contradiction. So (b) holds.
To show that (c′) implies (a) of Remark 3.1, let \(x \in K\) with \(\Vert x \Vert _{X} = \rho\). Then \(\Vert Lx \Vert _{Y} = \Vert  x'' \Vert _{Y} = \rho\), and there exists \(t_{1} \in [0, 1]\) such that \( x''(t_{1}) = \rho\) or \(x''(t_{1}) =  \rho\). So we have for \(t \in [0, 1]\) that \(x(t) \in [0, \rho ]\), \(x'(t_{1}) \in [  \rho, \rho ]\), and \(x''(t_{1}) =  \rho\). By (c′) we have \(Ty(t_{1}) = f(t_{1},x(t_{1}),x'(t_{1}),x''(t_{1})) > \rho\), and so (a) is satisfied.
Thus, there exists a solution to equation (1) with \(x \in K\) and \(\rho < \Vert x \Vert < r\). □
Example 4.1
The function \(f(t,x,x',x'') = 1 + \frac{3}{4}\sin x''\) with \(r = \frac{3\pi}{2}\) and \(\rho = \frac{\pi}{2}\) shows that the class of maps that satisfy the conditions of Theorem 4.1 is nonempty.
Declarations
Acknowledgements
The author is grateful to the referees for their useful comments that have improved this paper.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
 Deimling, K: Nonlinear Functional Analysis. Springer, Berlin (1985) View ArticleMATHGoogle Scholar
 Lan, K, Webb, JRL: A fixed point index for weakly inward Aproper maps. Nonlinear Anal. 28, 315325 (1997) MathSciNetView ArticleMATHGoogle Scholar
 Lafferriere, B, Petryshyn, WV: New positive fixed point and eigenvalue results for \(P_{\gamma} \)compact maps and applications. Nonlinear Anal. 13, 14271440 (1989) MathSciNetView ArticleGoogle Scholar
 Cremins, CT: Existence theorems for semilinear equations in cones. J. Math. Anal. Appl. 265, 447457 (2002) MathSciNetView ArticleMATHGoogle Scholar
 Lan, K, Webb, JRL: AProperness of contracting and condensing maps. Nonlinear Anal. 49, 885895 (2002) MathSciNetView ArticleMATHGoogle Scholar
 Infante, G: Positive solutions of some three point boundary value problems via fixed point index for weakly inward Aproper maps. Fixed Point Theory Appl. 2005, 177184 (2005) MathSciNetView ArticleMATHGoogle Scholar