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A fixed point theorem for weakly inward Aproper maps and application to a Picard boundary value problem
Fixed Point Theory and Applications volumeÂ 2016, ArticleÂ number:Â 71 (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.
1 Introduction
The purpose of this paper is to establish a fixed point theorem for weakly inward Aproper maps defined on cones in Banach spaces that generalizes a result in Deimling [1], p.254, for weakly inward maps defined on a cone in \(\mathbb{R}^{n}\). We use the fixed point index for weakly inward Aproper maps introduced by Lan and Webb [2] to obtain our new result. As an application, we obtain a positive solution to the Picard boundary value problem
under suitable conditions on f. This problem has been extensively studied; in particular, we refer to [3], where the concept of \(P_{\gamma} \)compact maps and quasinormal cones is used, [4], where the problem is formulated as a semilinear equation, [5], where f is allowed to take negative values, and [6], where positive solutions for threepoint boundary value problems are obtained. As mentioned in [5], in [3] and [4], examples were provided with conflicting hypotheses; our theorem will allow a different approach, which corrects the hypotheses of the analogous examples.
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 Î“ 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
Let K be a closed convex set, and \(f:K \to X\) be weakly inward on K, where \(I  f\) is Aproper. Suppose that
 (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 \(\lambda > 1\).
Then f has a fixed point in \(\{ x \in K:\rho < \Vert x \Vert < r\}\).
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, \(i_{K}(f,B_{r_{K}}\backslash B_{\rho_{K}}) = i_{K}(f,B_{r_{K}})  i_{K}(f,B_{\rho_{K}}) = \{ 0\}  \{ 1\} = \{  1\}\).
Since the index is not 0, the existence property implies that there exists a fixed point \(x \in K\) such that
â€ƒâ–¡
Remark 3.1
The conclusion of TheoremÂ 3.1 is valid if condition (a) holds for \(\Vert x \Vert = \rho\) and condition (b) holds for \(\Vert x \Vert = r\), that is,
 (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
We formulate the Picard boundary value problem
as a fixed point equation of the operator \(T:\overline{K}_{r} \to K\), \(K_{r} = \{ x \in K:\Vert x \Vert < r \}\),
where \(L:X \to Y\) is defined by \(Lx =  x''(t)\). Observe that (1) is equivalent to \(y = Ty\).
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
Under the above assumptions, suppose also that
 (aâ€²):

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 \(t \in [0, 1]\), \(p \in [0, r]\), \(q \in [  r, r]\), \(s_{1},s_{2} \in R^{ }\);
 (bâ€²):

\(f(t,p,q,s) < r\) for every \(t \in [0, 1]\), \(p \in [0, r]\), \(q \in [  r, r]\), \(s =  r\);
 (câ€²):

there are \(\rho \in (0, r)\), \(t_{0} \in [0, 1]\) such that \(f(t_{0},p,q,s) > \rho\) for \(p \in [0, \rho ]\), \(q \in [  \rho, \rho ]\), \(s =  \rho\).
Then there exists a positive solution \(x \in K\) to equation (1) with \(\rho < \Vert x \Vert _{X} < r\).
Proof
Since T maps K to K, T is weakly inward. Condition (aâ€²) 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 RemarkÂ 3.1, we first show that (bâ€²) implies (b). Let r be as in (bâ€²) 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 (câ€²) implies (a) of RemarkÂ 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 (câ€²) 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 \(\rho < \Vert x \Vert < r\).â€ƒâ–¡
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 TheoremÂ 4.1 is nonempty.
References
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Cremins, CT: Existence theorems for semilinear equations in cones. J. Math. Anal. Appl. 265, 447457 (2002)
Lan, K, Webb, JRL: AProperness of contracting and condensing maps. Nonlinear Anal. 49, 885895 (2002)
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)
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The author is grateful to the referees for their useful comments that have improved this paper.
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Cremins, C.T. A fixed point theorem for weakly inward Aproper maps and application to a Picard boundary value problem. Fixed Point Theory Appl 2016, 71 (2016). https://doi.org/10.1186/s136630160564x
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DOI: https://doi.org/10.1186/s136630160564x