Some coincidence theorems in wedges, cones, and convex sets

Existence of Nonnegative Solutions of a Semilinear Equation at Resonance with Linear Growth

“…Nonnegative solutions to some boundary value problems are obtained to illustrate the theory.The problem of existence of solutions in a convex set, or nonnegative solutions, for abstract semilinear equations at resonance has been recently considered by Nieto [20], Gaines and Santanilla [9], Mawhin and Rybakowski [19], and Santanilla [22]. They have considered the problem of existence of solutions to…”

“…The problem of existence of solutions in a convex set, or nonnegative solutions, for abstract semilinear equations at resonance has been recently considered by Nieto [20], Gaines and Santanilla [9], Mawhin and Rybakowski [19], and Santanilla [22]. They have considered the problem of existence of solutions to (1) Lu = Nu in a convex set, where L: dorn LcI->Z is a Fredholm operator of index zero, N:X -» Z is not necessarily linear and satisfies a compactness property relative to L, and X , Z are real Banach spaces.…”

“…In contrast to the results presented here, the arguments to obtain a priori bounds in those papers heavily depend upon the continuity of / on the first variable. For additional results on the subject, we mention the references in [9,12,17,22,23]. …”

“…According to our abstract result the nonnegative solutions to (3) and (6) Finally we shall mention that coincidence degree has been used by Gaines and Santanilla [9] and the author [22,23] to obtain nonnegative periodic solutions to systems of first and second order ordinary differential equations when the nonlinearity is continuous. In contrast to the results presented here, the arguments to obtain a priori bounds in those papers heavily depend upon the continuity of / on the first variable.…”

“…(ii) There exists R > 0 such that 4. The results in [9,19,22] do not assume (explicitly) any growth condition on the nonlinearity. However, they require the evaluation of a Brouwer degree.…”